Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Brandt, Christoph Tobias
Master of Applied Science - MASc
University of British Columbia
Multiphoton microscopy is a promising imaging technique capable of optical sectioning to acquire images from below the surface of tissue. A miniature objective with depth-scanning capability is needed for clinically adaptation. Imaging performance is dependent upon the ability to focus pulsed laser light. State-of-the-art endoscopes have a depth-scanning capability of ~200 μm and suffer from a significant decrease in resolution at the limit of their depth-scanning range. Zemax software is used to evaluate aspherical lenses for use as objectives, to evaluate the effect of scanning optics, and to design a custom water-dipping objective. Simulations show that the custom objective is capable of achieving diffraction-limited focusing for depths from 0 μm to 1400 μm in water by minimizing spherical aberration. A shape memory alloy actuator is used by the objective for depth-scanning across a 440 μm range, which is experimentally demonstrated by imaging fluorescent beads across the entire range and is more than double the range capability of MPM endoscopes in literature. Resolution measured ~0.7 μm laterally and ~10 μm axially and remained consistent across the entire depth-scanning range. This is a significant improvement to both our lab’s imaging ability and to the depth-scanning abilities of endoscopes presented in literature. Resolution is improved by the addition of corrective optics that reduces axial chromatic aberration. The effect of chromatic aberration on the focusing ability of the custom objective is investigated via simulation using Zemax. Simulations show that the 3 dB wavelengths of the laser’s 80 fs pulses focus at a different depth than the central wavelength and also that marginal rays and on-axis rays arrive at the focal point at different times. By adding CO to reduce chromatic aberration, both of these differences are significantly reduced, which improves the signal to noise ratio. An equation to calculate the difference in arrival time is derived, which shows that the difference is proportional to the square of the objective’s numerical aperture. The deep imaging capabilities of the water-dipping miniature MPM objective is demonstrated by imaging tissue and shows great promise for future clinical use.
MINIATURE OBJECTIVE LENS DESIGN FOR MULTIPHOTON MICROSCOPY by CHRISTOPH TOBIAS BRANDT BASc., The University of British Columbia, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Electrical and Computer Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) October 2022 © Christoph Tobias Brandt, 2022 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled: Miniature Objective Lens Design for Multiphoton Microcopy submitted by Christoph Tobias Brandt in partial fulfilment of the requirements for the degree of Master of Applied Science in Electrical and Computer Engineering Examining Committee: Shuo Tang, Professor, Electrical and Computer Engineering, UBC Supervisor Lukas Chrostowski, Professor, Electrical and Computer Engineering, UBC Supervisory Committee Member Sudip Shekhar, Associate Professor, Electrical and Computer Engineering, UBC Supervisory Committee Member iii Abstract Multiphoton microscopy is a promising imaging technique capable of optical sectioning to acquire images from below the surface of tissue. A miniature objective with depth-scanning capability is needed for clinically adaptation. Imaging performance is dependent upon the ability to focus pulsed laser light. State-of-the-art endoscopes have a depth-scanning capability of ~200 μm and suffer from a significant decrease in resolution at the limit of their depth-scanning range. Zemax software is used to evaluate aspherical lenses for use as objectives, to evaluate the effect of scanning optics, and to design a custom water-dipping objective. Simulations show that the custom objective is capable of achieving diffraction-limited focusing for depths from 0 μm to 1400 μm in water by minimizing spherical aberration. A shape memory alloy actuator is used by the objective for depth-scanning across a 440 μm range, which is experimentally demonstrated by imaging fluorescent beads across the entire range and is more than double the range capability of MPM endoscopes in literature. Resolution measured ~0.7 μm laterally and ~10 μm axially and remained consistent across the entire depth-scanning range. This is a significant improvement to both our lab’s imaging ability and to the depth-scanning abilities of endoscopes presented in literature. Resolution is improved by the addition of corrective optics that reduces axial chromatic aberration. The effect of chromatic aberration on the focusing ability of the custom objective is investigated via simulation using Zemax. Simulations show that the 3 dB wavelengths of the laser’s 80 fs pulses focus at a different depth than the central wavelength and also that marginal rays and on-axis rays iv arrive at the focal point at different times. By adding CO to reduce chromatic aberration, both of these differences are significantly reduced, which improves the signal to noise ratio. An equation to calculate the difference in arrival time is derived, which shows that the difference is proportional to the square of the objective’s numerical aperture. The deep imaging capabilities of the water-dipping miniature MPM objective is demonstrated by imaging tissue and shows great promise for future clinical use. v Lay Summary Multiphoton microscopy is a promising imaging technique that uses laser light to non-invasively image tissue below the tissue surface up to a depth of about 0.2 to 0.3 mm with submicron resolution. This capability to perform optical biopsies has the potential to improve patient care and would be especially useful in cancer detection. The ability to acquire high quality images across a wide range of depths is needed for wider clinical adaptation. This work presents the optical design of a novel miniature objective for use in multiphoton microscopy that enables the imaging of tissue below its surface over a large depth-scanning range. Imaging resolution is improved by the addition of corrective optics that minimize optical aberrations. The performance of the objective and corrective optics is characterized via simulation and experiment. The depth-scanning ability and imaging performance of the objective is demonstrated by imaging animal tissue. vi Preface The research presented in this thesis was done at the Biophotonics Laboratory at the University of British Columbia. I was responsible for the simulations, experiments and analysis presented herein. The research builds upon the work of current and former Biophotonic team members. Of particular importance are the contributions of Wentao Wu who made the optical assemblies used in experiment, and Qiaho Liu who integrated the use of a shape memory alloy actuator into the lab’s multiphoton microscope for depth scanning. Prof. Shuo Tang was involved throughout the research process, providing insight into proposed areas of research, reviewing simulation and experiment results, and editing the thesis manuscript. All the tissue imaging experiments were approved by the University of British Columbia’s committee on human ethics [certificate H16-02503] and biosafety [certificate B17-0066]. Permission to use Figure 4.7 was obtained from Wentao Wu, the author of the cited source. Contents of the research presented in this thesis has been presented at conferences and/or published in journal articles. • Christoph Brandt, Wentao Wu, Qihao Liu, and Shuo Tang, “Endoscopic MPM objective designed for depth scanning,” in Endoscopic Microscopy XVII, vol. 11937, pp. 13-19. SPIE, 2022. • Wentao Wu, Qihao Liu, Christoph Brandt, and Shuo Tang, “Dual-wavelength multimodal multiphoton microscope with SMA-based depth scanning,” Biomedical Optics Express 13, no. 5 (2022): 2754-2771. vii • Wentao Wu, Qihao Liu, Christoph Brandt, and Shuo Tang, “Multimodal multiphoton microscope with depth scanning,” in Multiphoton Microscopy in the Biomedical Sciences XXII, vol. 11965, pp. 124-129. SPIE, 2022. I was the primary investigator of the “Endoscopic MPM objective designed for depth scanning” conference paper; much of the content is included in the Abstract and in Chapter 2. I was responsible for the concept formation, simulation, experiment and manuscript composition. Wentao Wu built the depth-scanning objective that was used during experiments, and also provided his assistance during experiments. Qiaho Liu developed the lab’s ability to use a shape memory alloy actuator. Prof. Shuo Tang was the supervisory author and contributed toward concept formation and manuscript composition. The objective described in this thesis was featured in the journal article “Dual-wavelength multimodal multiphoton microscope with SMA-based depth scanning,” and also conference paper “Multimodal multiphoton microscope with depth scanning.” I contributed the simulation results featured in the journal article, some of which are included in Chapter 2. Contents from Chapters 1 to 5 are planned to be published in a future journal article. viii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary .................................................................................................................................v Preface ........................................................................................................................................... vi Table of Contents ....................................................................................................................... viii List of Tables ................................................................................................................................ xi List of Figures .............................................................................................................................. xii List of Abbreviations ................................................................................................................ xvii Acknowledgements .................................................................................................................... xix Dedication .....................................................................................................................................xx Chapter 1: Introduction ................................................................................................................1 1.1 Multiphoton Microscopy and Endoscopy ....................................................................... 1 1.2 Imaging Requirements for MPM endoscopy .................................................................. 3 1.3 Objectives used in MPM endoscopy............................................................................... 5 1.3.1 GRIN ........................................................................................................................... 5 1.3.2 Aspheric and Achromatic Doublet Objectives ......................................................... 10 1.3.3 Summary of MPM Endoscope Objectives ................................................................ 12 1.4 Research Motivation and Objectives ........................................................................... 14 1.5 Outline of Thesis ........................................................................................................... 15 Chapter 2: Objective Lens Design ..............................................................................................17 2.1 Background Information ............................................................................................... 17 2.1.1 Background on Point Spread Function and Wavefront Error ................................... 17 ix 2.1.2 Background on Optical Aberrations ......................................................................... 21 2.2 Analysis of Single Aspherical Lens as Dry Objective .................................................. 24 2.3 Design of a Water Dipping Objective with Depth Scanning Capability for MPM ...... 29 2.4 Impact of Scanning Optics ............................................................................................ 35 2.5 Conclusions ................................................................................................................... 39 Chapter 3: Chromatic Aberration Reduction ...........................................................................40 3.1 Background on Chromatic Aberration .......................................................................... 40 3.2 Reducing MPM Axial Chromatic Aberration ............................................................... 42 3.3 Effect on Lateral Color ................................................................................................. 47 3.4 Impact of Corrective Optics on Other Aberrations and Overall System Performance . 48 3.5 Consideration of Temporal Focusing............................................................................ 55 3.6 Conclusions ................................................................................................................... 60 Chapter 4: MPM Experimental Results ....................................................................................62 4.1 Experimental Setup of MPM ........................................................................................ 62 4.1.1 Beam Size ................................................................................................................. 64 4.1.2 Field of View ............................................................................................................ 67 4.1.3 Depth-Scanning Range ............................................................................................. 71 4.2 Resolution ..................................................................................................................... 72 4.3 Tolerance Analysis........................................................................................................ 77 4.4 Images of Biological Samples ...................................................................................... 84 4.5 Conclusions ................................................................................................................... 88 Chapter 5: Conclusion .................................................................................................................90 5.1 Summary ....................................................................................................................... 90 x 5.2 Future Work .................................................................................................................. 93 Bibliography .................................................................................................................................96 xi List of Tables Table 1.1 Endoscopic MPM Objective Performance Summary ................................................... 13 Table 2.1 Characteristics of Aspheric Lenses used as Objective Lenses ..................................... 27 Table 4.1 Characterization of Beam Diameter After PPLN ......................................................... 66 Table 4.2 Beam Angle after MEMS ............................................................................................ 70 Table 4.3 FOV .............................................................................................................................. 71 Table 4.4 MPM Resolution ........................................................................................................... 75 Table 4.5 Lens Manufacturing Tolerances ................................................................................... 78 Table 4.6 Assembly Tolerances .................................................................................................... 78 Table 4.7 Estimated Performance via Sensitivity Analysis .......................................................... 81 Table 4.8 Estimated Performance via Monte Carlo Analysis ....................................................... 82 xii List of Figures Figure 1.1 Jablonski energy diagrams illustrating the conversion of excitation photons to signal photons for fluorescent and harmonic generation processes in MPM [4]. ..................................... 2 Figure 1.2 Diffraction-limited focusing performance of GRIN lens GT-MO-080-018-810 versus field position. Diffraction-limited performance is assessed at 810 nm according to the Marechal criterion that RMS wavefront error is less than or equal to 0.07 waves [17]. ................................ 6 Figure 1.3 WD as a function of wavelength for lenses GT-MO-080-018-AC900-450 and GT-MO-080-018-810 [17]. ................................................................................................................... 7 Figure 2.1 Diffraction-limited PSF at the focal spot in water (n = 1.33) for a 0.58 NA objective lens. (a) PSF in the plane orthogonal to the optical axis. (b) Intensity distribution along the optical axis. ................................................................................................................................... 18 Figure 2.2 Wavefront error ‘W’ between an aberrated and reference wavefront [11]. ................ 19 Figure 2.3 Strehl ratio as a function of RMS wavefront error. ..................................................... 21 Figure 2.4 Illustration of under corrected SA in which beam’s outer marginal rays have a shorter focal length than the beam’s inner paraxial rays [12-15]. ........................................................... 21 Figure 2.5 . Illustration of the effect of field curvature and astigmatism on the curvature of focal surfaces: sagittal, medial, tangential and Petzval. Sagittal, medial, and tangential focal surfaces are defined by astigmatism and the Petzval surface by field curvature [11, 12]. ......................... 23 Figure 2.6 . Effect of pincushion (a) and barrel (b) type distortion when imaging a square [12-15]. ................................................................................................................................................ 23 Figure 2.7 (a) Optical layout used for individual aspherical lens simulation, which includes a lens (1), air gap (2), tissue (3). (b) RMS wavefront error of different aspherical lenses that are xiii manufactured by Lightpath. The horizontal dashed line indicates the diffraction limit of 0.072 waves as used by Zemax [29]. (c) Lens 355392 spot patterns at select field positions. (d) A plot of spot size versus image field position for lens 355392. ............................................................. 26 Figure 2.8 Sample induced SA whereby marginal rays and paraxial rays focus at different depths due to refraction by the sample. .................................................................................................... 29 Figure 2.9 (a) Optical layout of a single aspherical lens, 355392, acting as the objective. (b) The optical layout of the custom water-dipping objective focusing light into tissue. (1) Aspherical lens (352080, Lightpath), (2) PCX (32373, Edmund Optics), (3) Tissue. In both (a,b), the focal point is shifted by translating the aspherical lens. (c,d) SA wavefront aberration coefficients for (c) the single aspherical lens and (d) the custom objective. ......................................................... 31 Figure 2.10 Performance comparison of signal aspherical lens (355392) versus that of our custom water-dipping objective. (a) The on-axis RMS spot size as a function of the depth of focus in tissue. The Airy disk radii are plotted using dashed lines. (b) RMS wavefront error versus field plots. A heatmap of the RMS wavefront error versus field and the depth of focus in tissue is shown in (c) for lens 355352 and in (d) for the custom objective. The diffraction limit of 0.072 waves is indicated by the dashed black line in plots (b) to (d). .......................................... 34 Figure 2.11 Comparison between simulation results using a single achromatic doublet lens (a,c,e) and using a pair of achromatic doublet lenses (b,d,f) as the scan lens. (a, b) Optical layout of the scanning optics. (c, d) Seidel coefficients of the lenses and their sum for a field position of 0.092 mm (e, f) Spot diagrams at selected field locations. The Airy disk is shown as black circles. (g) Spot size as a function of field position. (h) RMS wavefront error as function of field position comparing the use of a single (dashed) or pair (solid) of achromatic doublets as the scan lens at selected depths of focus: 0 mm, 0.2 mm and 0.4 mm. .................................................................. 38 xiv Figure 3.1 Illustration of axial chromatic aberration in which shorter wavelength light has a shorter focal length than longer wavelength light. [11, 12]. ........................................................ 41 Figure 3.2 Illustration of lateral color in which the chief ray of the shorter wavelength intersects the image plane at a lower height than chief ray of the longer wavelength [11, 12]. ................... 41 Figure 3.3 Optical layout of the MPM scanning optics, including the CO in the red rectangle. .. 43 Figure 3.4 Plot of the focal distances at various wavelength: central wavelength (790 nm), -3dB wavelengths (783 nm and 797 nm), and -10 dB wavelengths (775 nm and 805 nm). Plot (a) is with CO and (b) is without CO. The objective’s aspherical lens and PCX lens are separated by a distance of 1.12 mm so as to achieve a nominal on-axis depth of focus of 0.2 mm below the tissue surface. ................................................................................................................................ 46 Figure 3.5 Analysis of lateral color. (a) Lateral color dependance on field position for -3dB wavelengths of 783 nm and 797 nm. (b,c) Spot diagrams without (b) and with (c) the use of CO at selected positions: 0 mm, 0.092 mm, and 0.25 mm. The Airy disk is shown in spot diagrams as a black circle and has a radius of 0.82 µm. (d, e) Chief ray (red) and marginal ray (blue) without (d) and with (e) the use of CO. The back focal plane of the objective is indicated by the arrow. 48 Figure 3.6 Comparison of monochromatic and chromatic aberrations by the optical system. (a) and (b) Seidel aberration and chromatic aberration coefficients of the optics without and with CO in place, respectively. Calculations are for a field position of 0.092 mm and at a depth of focus of 0.2 mm in tissue. (c) SA coefficient as function of depth of focus in tissue without and with CO. (d) Aberration coefficients of the elements in the CO. The SA bars for Achromat 1 and PCV 2 are truncated and have respective values of 21 µm and -18 µm. .................................................. 50 Figure 3.7 Absolute value of select Zernike standard coefficients are plotted as a function of field position. (a) is without CO. (b) is with CO .................................................................................. 52 xv Figure 3.8 Comparison between simulation results of MPM scanning optics without (a,c,e) and with (b,d,f) CO in place. (a, b) RMS wavefront error at a depth of focus of 0.2 mm. (c, d) RMS wavefront error as a heatmap. (e, f) Strehl ratio as a heatmap. The diffraction limit is indicated by the dashed black line at an RMS wavefront error of 0.072 wave in plots (a) to (d) and at a Strehl ratio of 0.8 in plot I and (f). .......................................................................................................... 54 Figure 3.9 RMS spot size radius versus field position for the MPM imaging optics with and without the CO (at a depth of focus in tissue of 0.2 mm). The Airy disk radius is 0.82 µm. ....... 55 Figure 3.10 An illustration of a fully illuminated singlet lens focusing rays in air. The delay between pulse front and phase front is shown [49, 50]. ................................................................ 56 Figure 3.11 PDT between off-axis rays at position ‘r’ and the on-axis ray with and without the use of CO (a) Analytically calculated as per Equation 3.10 (b) Determined via ray tracing. Radial position, r, is measured in front of the objective. .......................................................................... 59 Figure 4.1 Schematic of the MPM set-up. SMF: single mode fiber; CL: collimating lens; QWP: quarter-wave plate; HWP: half-wave plate; HWP: half-wave plate; FL: focusing lens; PPLN: periodically poled Lithium Niobate crystal; F: filter; MEMS mirror: micro-electro-mechanical scanning mirror; SL: scan lens; TL: tube lens; DM: dichroic mirror; CO: corrective optics OBJ: Objective; L: lens; MMF: multimode fiber; PMT: photomultiplier tube. .................................... 63 Figure 4.2 MPM scanning optics layout (a) without CO and (b) with CO. (i) Ray tracing for on-axis (blue beam) and off-axis (green beam) focusing. (ii) Paths of the marginal ray (blue ray) and the chief ray (red ray). ................................................................................................................... 67 Figure 4.3 The chief ray (fletched ray in red) reflecting off the MEMS mirror and passing through the scanning optics. The spacing of optical elements is set according to the focal length of the scan, tube and objective lenses. The rotated MEMS mirror is shown in black; its neutral xvi position is shown in grey; ∅ is the mechanical angle of rotation of the mirror. The horizontal black line indicates the optical axis of the scanning optics. ......................................................... 68 Figure 4.4 MEMS angle of rotation versus applied voltage [55] ................................................. 69 Figure 4.5 Lateral (a,c) and axial (b,d) FWHM values of the intensity profiles of individual 100 nm beads plotted against their radial field position relative to the image center (a,b) and their depth (c,d). .................................................................................................................................... 74 Figure 4.6 (a) On-axis coma observed at a bead cluster located inside of yellow circle. (b) On-axis astigmatism observed that is observed when alignment is particularly poor. The FOV is 51 μm by 78 μm. ................................................................................................................................ 76 Figure 4.7 (a) The objective assembly is made by attaching the aspherical lens and PCX lens to the SMA using 3D printed parts. Printed parts and lenses are glued in place. [40]. (b) The CO assembly is made by securing lenses within a 3D printed sleeve; their spacing is controlled using 3D printed spacers......................................................................................................................... 79 Figure 4.8 Induced RMS wavefront error from the worst offending parameter perturbations. .... 81 Figure 4.9 Bimodal images obtained ex-vivo of mouse without the CO in place (a) inner ear, (b) kidney. Image depth is indicated on each image. 2PEF is shown in red and SHG in green. The FOV is 64 μm by 100 μm; the scalebar is 16 μm [61]. ................................................................. 85 Figure 4.10 Bimodal images obtained ex-vivo of mouse tissue with the CO in place (a) kidney, (b) small intestine and (c) colon. Image depth is indicated on each image. 2PEF is shown in red and SHG in green. The FOV is 102 µm by 156 µm; the scalebar is 25 µm. ................................ 87 Figure 4.11 2PEF images obtained from a yew leaf (a) at the leaf center (b) near the leaf margin, (c) leaf midrib. Image depth below the surface is indicated on each image. The FOV is 102 µm by 156 µm; the scalebar is 25 µm. ................................................................................................ 88 xvii List of Abbreviations 1PEF One-Photon Excitation Fluorescence 2PEF Two-Photon Excitation Fluorescence 3PEF Three-Photon Excitation Fluorescence AO Adaptive Optics CA Clear Aperture CARS Coherent Anti-Stokes Raman Scattering CO Corrective Optics DM Dichroic Mirror FOV Field of View FWHM Full Width at Half Maximum GRIN Gradient-Index HWP Half Wave Plate MEMS Microelectromechanical systems MMF Multimode Fiber MPM Multiphoton Microscope NA Numerical Aperture PCV Plano Concave PCX Plano Convex PMT Photomultiplier Tube PPLN Periodically Poled Lithium Niobate PSF Point Spread Function xviii PTD Propagation Time Difference QWP Quarter Wave Plate RMS Root Mean Square SA Spherical Aberration SHG Second-Harmonic Generation SMA Shape Memory Alloy SNR Signal to Noise Ratio THG Third-Harmonic Generation WD Working Distance xix Acknowledgements I would like to extend my gratitude to Prof. Shuo Tang for the opportunity for me to pursue my Master’s degree, for providing continual guidance to navigate my graduate student career, and for providing funding to support my research. I would also like to thank current and former Biophotonics lab members Qiaho Liu, Mukhlasur Rahman Tanvir, Wentao Wu, Sina Maloufi and Xin Zhou for teaching me the basics on conducting optical experiments, helping me conduct experiments, and providing feedback on my manuscripts. I would like to thank my father and mother Harry and Marianne Brandt, father and mother-in-law Jim and Cori Terepocki, family friend Arno Penner, and wife Tiffany Brandt. Your support throughout the years has been invaluable and has helped me to achieve my academic goals. xx Dedication To my wife Tiffany, thank you for your unwavering support throughout my graduate studies and to my children Matthias, Asher and Norah who bring much joy to my life. In memory of Ryan Ridder, Ernst Neudorf and Lisa Terepocki; may this research contribute to improved early cancer detection. 1 Chapter 1: Introduction 1.1 Multiphoton Microscopy and Endoscopy Multiphoton microscopy (MPM) is an imaging modality that enables high-resolution three-dimensional imaging of thick tissues [1]. MPM focuses pulsed near-infrared (NIR) light below tissue surface to induce nonlinear signals. By raster-scanning the laser across the tissue an image with submicron resolution can be generated. NIR light can penetrate deeper into tissue than ultraviolet and visible light due to the absorption coefficient in tissue being lower and since longer wavelengths are less susceptible to scattering [2, 3]. At low intensities, NIR light is non-ionizing, which is an important safety consideration. Signals are generated through a number of nonlinear photon interactions: two-photon excitation fluorescence (2PEF), second-harmonic generation (SHG), three-photon excitation fluorescence (3PEF), and third-harmonic generation (THG) [4]. These processes require a high photon density to generate a signal, as such signals are only generated from within the focal volume. This allows for optical sectioning without the use of pinholes as is required in confocal microscopy. The energy relationship that governs fluorescence processes for 2PEF, 3PEF and also for SHG and THG processes are illustrated in Figure 1.1. Note that an emitted fluorescence photon has a lower energy than the energy input photons needed to induce fluorescence, whereas the energy of SHG and THG are equal to the energy of the input photons. This allows photons generated via fluorescence, SHG or THG to be distinguished from each other, allowing fluorescence, SHG and THG imaging to be done simultaneously. The intensity of the signal generated is sensitive to the excitation intensity as it is proportional to the square of the excitation intensity in the case of 2PEF 2 and SHG or to its cube in the case of 3PEF and THG, so relatively small changes to the excitation intensity can have a large effect on the signal intensity [5]. Figure 1.1 Jablonski energy diagrams illustrating the conversion of excitation photons to signal photons for fluorescent and harmonic generation processes in MPM [4]. The peak intensity of the excitation photons can be increased to induce a stronger signal and increase the signal to noise ratio (SNR) by focusing to a smaller spot, by increasing laser power, and by decreasing the pulse duration. Increasing signal intensity by changing parameters comes with limitations and tradeoffs. Laser power is constrained by the thermal limit at which tissue is damaged, thus putting a limit on the increase in signal intensity that is possible by increasing laser power. Decreasing the pulse width increases laser bandwidth, thus making the pulse more susceptible to dispersion and chromatic aberration which negatively affects laser imaging. A smaller spot size, can be achieved by using an objective with a higher NA, this has the added advantage of increasing image resolution. Achieving a higher NA is a challenge due to aberrations and size limits. 3 Benchtop MPM technology is well developed and microscope objectives control optical aberrations to achieve a small focal volume within tissue and generate high quality images whose resolution reaches the theoretical limit. Benchtop systems however are bulky and have made limited inroads to clinical use; the DermaInspect and MPTflex systems by JenLab are notable exceptions [6, 7]. An early version of the DermaInspect features an endoscopic gradient-index (GRIN) objective that is mounted in a fixed position, against which the patient must be positioned. Improvements to the system have been made since its original production including the addition of an optical mirror arm for easier patient positioning. The MPTflex has a reduced equipment footprint relative to the DermaInspect and offers an articulating arm to position the objective. An optional micro endoscopic tip with an NA of 0.8 can be added to the objective of either system. However, the imaging head that holds the objective remains bulky. To achieve wider adaptation in clinical use, the size of the imaging head and objective needs to be miniaturized, ideally small enough to allow for endoscopy and/or be mounted in a handheld probe, which could be connected to a benchtop laser via a flexible optical fiber. 1.2 Imaging Requirements for MPM endoscopy An MPM endoscope system not only needs to have a small objective but must also have high imaging quality for it to be useful as a clinical tool. General imaging requirements applicable to MPM include a high resolution, a high SNR, and a wide field of view (FOV). MPM also requires deep penetration of light into tissue and a depth-scanning capability to maximize functionality. An objective with a high NA and a low amount of aberration is needed to satisfy these requirements and achieve high imaging quality. 4 MPM resolution is determined by the focal spot size, which is dependent upon the NA of the objective and the excitation wavelength. Increasing the NA of an objective enables light to focus to a smaller spot size, thus improving resolution. The diffraction-limited lateral and axial resolution for 1PEF are respectively given by Equations 1.1 and 1.2 [5-9]. 𝑅𝑎𝑦𝑙𝑒𝑖𝑔ℎ 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑥𝑦 = 0.61𝜆𝑁𝐴 (1.1) 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑧 = 2𝜆𝑛𝑁𝐴2 (1.2) Resolution of two-photon processes (2PEF and SHG) and three-photon processes (3PEF and THG) improve upon one-photon excitation fluorescence (1PEF) by a factor of √2 and √3 respectively due to the nonlinear relationship between excitation and signal intensities [5, 10],. In practice however, resolution is lower than the diffraction limit due to optical aberrations. Spherical aberration (SA), coma, astigmatism, axial chromatic aberration and lateral color affect the focal spot size and thus need to be kept to a minimum to maximize resolution. SA and axial chromatic aberration affect imaging across the entire FOV whereas coma, astigmatism and lateral color only affect off-axis imaging [11, 12]. SA needs to be kept to a minimum for a large range of focal depths in order to allow for effective depth scanning. When focusing light below the tissue surface, the amount of SA induced is dependent upon the depth below the tissue surface at which light is focused and increases when moving away from the depth of optimal focus. This restricts the depth range over which a high-quality image can be acquired and limits the functionality of the endoscope. Additionally, an 5 endoscope should have a wide FOV. This requires that aberrations be kept low across the entire FOV. The SNR of an MPM endoscope can be increased by increasing the amount of signal photons generated, and also by increasing the proportion of signal photon detected [16]. Reducing aberrations allows light to come to a smaller focus, thus generating more excitation photons. Since the conversion of excitation photons to signal photons is dependent on the square of the excitation intensity for two-photon processes (2PEF and SHG) and the cube for three photon processes (3PEF and THG), a large increase in signal can be obtained from a relatively small improvement in focus. Increasing the NA of an objective serves to both increase the number of signal photons generated and the number of photons detected since the objective in an MPM endoscope is normally used to focus excitation photons and collect signal photons simultaneously. 1.3 Objectives used in MPM endoscopy The last 10 years have seen a development of compact MPM probes and endoscopes. Miniature objective lenses typically use either GRIN lenses or aspherical lenses due to their high NA and small size. Combining multiple achromatic doublet lenses as an objective has also been reported. 1.3.1 GRIN GRIN lens GT-MO-080-018-810, GRINTech, stands out as having the highest NA of GRIN lenses used in MPM endoscopes in current academic literature. Endoscopes that use this lens in some cases underfill the lens, which reduces their effective NA but also increases the potential for a wider diffraction-limited FOV. Figure 1.2 plots the maximum NA that is diffraction limited for a 6 given field position of GT-MO-080-018-810 [17]. The lens has a maximum diffraction-limited NA of 0.8, though is restricted to a fairly narrow half FOV of 15 µm; at a half FOV of 50 μm, the diffraction-limited NA drops to 0.48. Furthermore, the lens is optimized for operation at a fixed working distance (WD) of 200 μm, thus limiting its usefulness in depth-scanning designs. Adaptive optics (AO) has been used to extend the diffraction-limited FOV and depth-scanning range of GRIN lenses [18, 19]. However, the addition of AO adds cost, complexity, and size to the design. Figure 1.2 Diffraction-limited focusing performance of GRIN lens GT-MO-080-018-810 versus field position. Diffraction-limited performance is assessed at 810 nm according to the Marechal criterion that RMS wavefront error is less than or equal to 0.07 waves [17]. GRIN lens (GT-MO-080-018-AC900-450, GRINTech) with an NA of 0.8 is presented as an achromatic version of GRIN lens GT-MO-080-018-810 [17]. It is designed to be chromatically compensated at two design wavelengths: 900 nm and 450 nm, so that they have equal focal lengths. This is advantageous for two-photon endomicroscopy where the excitation wavelength is at 900 nm and the emission wavelength of SHG is at 450 nm. However, chromatic aberration still exists 7 at other wavelengths such as around the broad bandwidth of the excitation light. A comparison of the focal depth versus wavelength of these two GRIN lenses is shown in Figure 1.3. Axial chromatic aberration at the design excitation wavelength of 900 nm is still present (the slope is nonzero), and is approximately equal in magnitude but with opposite sign to that of lens GT-MO-080-018-810. This has negative ramification on spatial and temporal focusing for MPM imaging where the excitation femtosecond pulse has a relatively broad bandwidth. Several papers using GRIN lens as objective are reviewed and summarized below. Figure 1.3 WD as a function of wavelength for lenses GT-MO-080-018-AC900-450 and GT-MO-080-018-810 [17]. In their 2011 paper, D. Rivera, et al, presented a compact endoscope that used a 0.8 NA GRIN lens (GT-MO-080-018-810, GRINTech) [21]. The laser beam is transmitted through an optical fiber to the endoscopic probe, thus separating the laser source from the probe head and providing flexibility to the user when positioning the probe next to the sample. However, dispersion caused the pulse width to nearly triple in duration from 110 fs at the laser output to 290 fs at the endoscope. Lateral resolution is measured to be 0.8 µm and axial resolution to be 10 µm. A 110 µm x 110 µm FOV 8 is attained using two-layer piezoelectric actuator which raster scans the fiber tip located just in front of the GRIN lens. Depth scanning is performed by translating samples using an axial stage; an approach not suitable for clinical application. A. Li, et al. published a paper in 2018 on the use of a GRIN objective lens with a 0.8 NA (GT-MO-080-018-810, GRINTech) in a handheld probe with depth-scanning capability and whose functionality was demonstrated by in-vivo imaging [22]. An achromatic doublet lens, which couples light into a GRIN relay lens in front of the GRIN objective lens, is translated by a piezoelectric micro translation stage to perform depth scanning and a microelectromechanical systems (MEMS) scanner is used for lateral scanning. Simulations show that diffraction limited imaging can be achieved for depths of ~100 µm to ~270 µm. This range is still less than the potential light penetration depth for many tissues [23]. The MPM systems operates at an NA of 0.66 to 0.59 depending on the imaging depth. The FOV (120 µm) and depth-scanning range (~200 µm) of the probe are limited by the drop off in signal intensity due to optical aberrations. Resolution is measured to be 0.833 µm laterally, and 6.11 µm axially. Z. Qin et al. successfully demonstrated the use of an endoscopic MPM GRIN objective (GT-MO-080-018-810, GRINTech) with a 0.8 NA as part of a benchtop MPM in conjunction with AO to image a mouse brain in-vivo and achieving diffraction-limited focusing in a paper published in 2020 [19]. The AO implementation uses a Shack-Hartmann wavefront sensor to directly measure the wavefront while a deformable mirror modifies the wavefront. A lateral resolution of 0.46 µm and an axial resolution of 2.3 µm is achieved. In addition to improving resolution, the AO allow for a wider FOV than what is otherwise achieved with the GRIN lens. A circular FOV with a 9 diameter of 300 µm is attained with the help of image post-processing by stitching together images from 50 µm by 50 µm subregions. A depth-scanning range of 300 µm is attained. The extra size from incorporating AO make it impractical for use in a handheld probe, though it demonstrates the imaging performance that can be achieved with diffraction-limited focusing. A. Dilipkumar et al. made a portable MPM system featuring a probe tip that has two configurations: a front-view configuration that uses a 0.5 NA GRIN lens (NEM-100-25-10-860-DL-ST, GRINTech) as the objective, and a side-view objective that is a combination of two GRIN lenses (GT-LFRL-100-125-20-NC and GT-LFRL-100-013-50-NC, GRINTech) and has a combined NA of 0.38 [24]. Lateral scanning is done using a pair of galvanometric scanning mirrors, for a FOV of 290 µm by 290 µm, which is significantly larger than the FOV of probes using the 0.8 NA GRIN lens. The relative increase in FOV over the 0.8 NA GRIN lens is attained at the expense of a decrease in resolution. The MPM achieves a lateral resolution of 0.9 µm with the 0.5 NA GRIN lens and 2.2 µm with the 0.38 NA GRIN lens, which remain fairly constant across the depth-scanning range. Axial resolution however decreases with focal depth. It decreased by half from ~15 µm to ~30 µm for the 0.5 NA GRIN lens and ~30 µm to ~60 µm for the 0.38 NA GRIN lens when imaging the surface versus at a depth of 180 µm. State-of-the-art miniature MPM systems use a GRIN objective due to their small size and their high NA. However, a high NA is only achievable for a very small FOV, thus a wider FOV at a lower NA is commonly used. This is the large drawback when using GRIN lenses. 10 1.3.2 Aspheric and Achromatic Doublet Objectives Objectives made of aspherical lenses and achromatic doublets have been used for MPM endoscopic imaging. Their FOV is generally larger than what is achieved using GRIN lenses, though their NA is somewhat lower. G. Ducourthial et al. presented an endoscopic probe in their 2015 paper [26]. The objective is comprised of three achromatic doublets (NT65-569, f = 6 mm; NT65-568, f = 4 mm; NT65-567, f = 3 mm, Edmund Optics). The objective has a combined NA of 0.45. Resolution is measured to be 0.83 µm laterally and 12 µm axially. A piezoelectric tube is used to induce resonant movement of a fiber tip for lateral spiral scanning and achieve a circular FOV of 450 µm during slow scanning and 250 µm during fast scanning. Depth scanning from the surface of the sample to a depth of 300 µm is achieved by moving the sample. Y. Zhao, et al. presented simulation and experimental imaging results of a handheld MPM probe, which used an aspherical lens as a dry objective in a 2016 paper [27]. The performance of three different aspherical lenses (352140, 352150, and 355392, Lightpath) with respective NA values of 0.55, 0.50 and 0.64 respectively, are simulated. Simulated point spread functions (PSFs) have full width at half maximum (FWHM) values of 0.88 µm, 0.96 µm and 0.75 µm respectively, whereas the experimental resolution of lenses 352140 and 352150 are considerably higher and measured 1.22 µm and 1.3 µm respectively. A FOV of ~150 µm by ~230 µm is achieved using a MEMS scanning mirror. Depth scanning is achieved by moving the sample. 11 An MPM endoscopic probe is presented in a 2018 paper by F Akhoundi et al. The MPM operates at a wavelength of 1700 nm and is thus suited for imaging THG rather than 2PEF and SHG [28]. The objective is made of two aspherical lenses. The first lens (354430-C, Lightpath) has an NA of 0.2 and collimates light emitted from a fiber, the second lens (354140-C, Lightpath) has an NA of 0.58 NA and focuses light for imaging [35]. The paper reports an NA of 0.5 for the endoscope. Using the THG signal, it achieves a resolution of 2.2 µm laterally and 12.7 µm axially. Signal intensity remains within 90% of the peak for a FOV of 400 µm. At a FOV of 800 µm the signal intensity at the edge of the FOV is reduced to about 50%. A resonant piezo electric tube performs lateral spiral scanning. The endoscope has a large WD of 1 mm in water. An achromatic miniature objective with an NA of 0.51 for use in CARS imaging using wavelengths of 817 nm (pump wavelength) and 1064 nm (Stokes wavelength) is presented by R. Mittal, et. al.[20]. The first element is a double convex lens (67-595, Edmund Optics) that collimates a diverging beam. The second element is a custom negative achromatic doublet that is fabricated by JenOptik, whose role is to expand the beam. The third and fourth elements are a pair of achromatic doublet lenses (49-923, Edmund Optics). The fifth element is an aspherical lens (355390, Lightpath), which is followed by a coverslip as the sixth element. The objective corrects for chromatic aberrations over the range of 810 nm to 1064 nm. A lateral resolution of 0.648 µm and axial resolution of 4.9 µm are achieved. Lateral scanning is performed using two galvanometric mirrors to obtain a FOV of 190 µm x 190 µm. 12 1.3.3 Summary of MPM Endoscope Objectives A summary of representative MPM endoscopes is presented in Table 1.1. Sub-micron lateral resolution is achieved by many of the endoscopes. Axial resolution values are substantially higher and are often greater than 10 µm. The highest resolution achieved by an endoscopic MPM is attained using the 0.8 NA GRIN lens (GT-MO-080-018-810, GRINTech). Even in this case, there is a large discrepancy between theoretical resolution values (0.40 µm lateral, 1.7 µm axial) and experimental values (0.833 µm lateral, 6.1 µm axial) indicating that a sufficient level of aberrations is present to prevent diffraction-limited performance. Only when AO is are theoretical values attained [19]. Moreover, resolution degrades as the FOV is increased and as the WD is moved away from the optimal working depth, which ultimately limits the depth-scanning range and FOV of many of the designs. AO has not yet been implemented for use in an MPM handheld endoscope, presumably due to their size and complexity. Attaining near theoretical resolution performance across a wide FOV and large depth-scanning range for a high NA objective is an on-going area of research in MPM endoscope design. 13 Table 1.1 Endoscopic MPM Objective Performance Summary Reference Laser Objective Lens NA Lateral Resolution (µm)** Axial Resolution (µm)** FOV (µm) Depth-Scanning Range (µm) Setup D. Septier, et al. (2021) [25] Ytterbium 40 to 80 fs 640-940 nm 1147-2676 nm GRIN w/ Fiber 0.7 2PEF:1.1 (0.57)* 3PEF: (0.65)* 2PEF: 14.9 (3.5)* 3PEF: 5.3 (4.1*) 200 (circular) Compact Probe Z. Qin, et al. (2020) [19] Ti:Sapphire 920 nm GRIN w/ coupling Microscope Obj. 0.8 0.46 (0.50)* 2.3 (2.7)* 300 (circular) 300 Benchtop MPM +AO A. Dilipkumar, et al. (2019) [24] Fiber laser 780 nm 80 fs 500 mW GRIN w/ coupling Microscope Obj. 0.38 2.2 (0.89)* 30 to 60 (10)* 290 x 290 (square) 200 Cage System 0.5 0.9 (0.67)* 15 to 30 (5.9)* 200 F. Akhoundi, et al. (2018) [28] Fiber laser 1700 nm 400 fs Two Element Compound Aspherical 0.5 THG: 2.2 (1.2)* THG: 12.7 (10)* 400 to 800 (circular) Compact Probe A. Li, et al. (2018) [22] Ti:Sapphire 750 nm 120 fs GRIN w/ coupling Achromatic Doublet 0.8 0.833 – 1.193 (0.40)* 6.11 (2.2)* 120 (circular) 200 Handheld Probe Y. Zhao, et al. (2016) [27] Fiber laser 1580 nm 250 fs 180 mW Aspherical 0.5 1.3 (0.68)* 16.2 (5.9)* 150 x 230 Handheld Probe 0.55 1.22 (0.62)* 13 (4.9)* G. Ducourthial, et al. (2015) [26] Ti:Sapph 810 nm 40 fs Three Element Compound Achromatic Doublet 0.45 0.83 (0.78)* 12 (7.5)* 250 (fast) 450 (slow) (circular) Compact Probe C. Wang, N Ji (2013) [18] Ti:Sapphire 900 nm GRIN w/ coupling Microscope Obj. 0.8 (0.49)* 3.8 – 4.1 (2.6)* 205 x 205 (square) 200 Benchtop MPM + AO 0.45 (0.86)* 12 – 25 (8.4)* 500 R. Mittal, et. al. (2013) [20] Ti:sapphire 817 nm (pump) Nd:vanadate 1064 nm (Stokes) Five Element Compound Aspherical / Achromatic Doublet 0.51 CARS: 0.65 CARS: 4.9 190 x 190 (square) Miniature Objective D. Rivera, et al. (2011) [21] Ti:Sapphire 800 nm GRIN w/ Fiber 0.8 0.8 (0.43)* 10 (2.4)* 110 x 110 (square) Compact Probe *Theoretical resolution based on Equations 1.1 and 1.2, scaled by factors of √2 and √3 for two- and three-photon processes respectively **Resolution is for a two-photon process unless otherwise indicated. 14 1.4 Research Motivation and Objectives Current miniature MPM systems are beginning to incorporate depth-scanning capabilities into their design to exploit the three-dimensional imaging capability of MPM, which is needed to perform optical sectioning in a clinical setting. However, their depth-scanning capabilities are restricted to a relatively small depth-scanning range and require the precise axial positioning of tissue so that the tissue lies within the depth-scanning range of the objective. Additionally, the axial resolution of endoscopes is fairly low, thus reducing the performance of optical sectioning. A possible reason for the low axial resolution is chromatic aberration which has not been considered in current MPM endoscope designs. Reducing chromatic aberration also has the potential to improve the temporal focusing of pulses, thus improving SNR. The objectives of this thesis are to advance MPM endoscopy to enable depth scanning over a wide range and to reduce the effects of chromatic aberration on the focusing of femtosecond pulses. A key challenge is to maintain a wide diffraction-limited FOV while fulfilling these objectives since adding optical elements to correct for SA or reducing chromatic aberration will most likely induce other types of aberrations that may degrade performance and limit the FOV. Fulfilling these objectives will bring MPM endoscopy closer to clinical use. The main contributions I have made include: (1) the optical design of a novel miniature objective that enables depth scanning over a wide range by reducing the sample introduced SA across the entire depth-scanning range; (2) the optical design of an optical assembly that corrects for axial chromatic aberration as well as coma and astigmatism to improve both on-axis and off-axis focusing. These contributions are made in conjunction with Wentao Wu who made the optical 15 assemblies and also Qihao Liu, who developed the use of a shape memory alloy (SMA) actuator for axial scanning. 1.5 Outline of Thesis This dissertation is organized as follows: In Chapter 2, background theory on the focusing of light and optical aberrations is given. The design of a miniature water-dipping objective that is specially designed for depth scanning by minimizing SA is presented. The focusing performance of the custom objective on its own and the impact that scanning optics have on aberrations and their effect on focusing performance is simulated using Zemax software. In Chapter 3, the focusing performance of the scanning optics and objective lens are further investigated to include the bandwidth of the pulsed laser light in the analysis. The design of CO that correct for axial chromatic aberration is presented along with a description of the design process. The effect that the CO have on aberrations and the overall focusing performance is investigated via simulation using Zemax software. The effect that axial chromatic aberration has on temporal focusing is investigated. In Chapter 4 the objective is incorporated with the lab’s MPM system. Analysis is done on the system as a whole and its functionality experimentally demonstrated. Theoretical beam size, the potential for vignetting, FOV and depth-scanning capabilities of the MPM are reviewed and experimentally measured where appropriate. Tolerance analysis, which takes into account lens 16 manufacturing and assembly tolerances, is conducted to estimate the performance of the experimental MPM set-up. Resolution is experimentally measured and the imaging capability of the MPM is demonstrated by imaging mouse tissues ex-vivo. In Chapter 5 key results are summarized and suggestions for future work are presented. 17 Chapter 2: Objective Lens Design In Chapter 2, we investigate the effect that SA has on the focusing capability of an aspherical lens when focusing below the surface of tissue and present a custom objective that is designed to minimizes SA across a large depth scanning range. The effect that scan and tube lenses have on on-axis and off-axis focusing is investigated in terms of RMS wavefront error, RMS spot size and Seidel aberrations. 2.1 Background Information 2.1.1 Background on Point Spread Function and Wavefront Error The objective lens in an MPM system focuses light within the sample being imaged, so as to generate a nonlinear signal. The diffraction-limited focal spot in the plane orthogonal to the optical axis is described by a Bessel function shaped PSF according to Equation 2.1 [5, 11]. 𝐼 = 𝐼0 [𝐽1 (2𝜋𝑁𝐴𝑟𝜆) ×𝜆𝜋𝑁𝐴𝑟]2 (2.1) where J1 is the Bessel function, r is the radial distance from the center, λ is the excitation wavelength, and NA is the numerical aperture (NA) of the objective. The intensity distribution of the PSF along the optical axis for diffraction-limited focusing is given by Equation 2.2 [5]. 𝐼 = 𝐼0 [𝑠𝑖𝑛 (𝜋𝑁𝐴2𝑧2𝜆𝑛) ×2𝜆𝑛𝜋𝑁𝐴2𝑧]2 (2.2) 18 where z is the distance along the optical axis from the focal spot, n is the index of refraction of the medium in which the light comes to a focus. The diffraction-limited PSF is shown in Figure 2.1(a) for the lateral direction and in Figure 2.1(b) for the axial direction. The figure is for focusing in water (n = 1.33) at wavelength of 790 nm for an objective lens with an NA of 0.58. The distance from peak intensity to the first zero crossing in the lateral direction is known as the Airy disk radius. The Airy disk radius corresponds to the Rayleigh resolution of a microscope and can be calculated using Equation 1.1. The first zero crossing in the axial direction corresponds to the 1PEF axial resolution and can be calculated using Equation 1.2. Figure 2.1 Diffraction-limited PSF at the focal spot in water (n = 1.33) for a 0.58 NA objective lens. (a) PSF in the plane orthogonal to the optical axis. (b) Intensity distribution along the optical axis. (a) (b) 19 Aberrations from the lenses and optical system make the focusing to be worse than the diffraction-limited spot. The imperfect focusing of light can be quantified in terms of wavefront error, which is the distance between the aberrated wavefront of light that has passed through an optical system and a reference wavefront that is centered on a reference image point on the imaging plane as illustrated in Figure 2.2. In MPM imaging simulations, the reference imaging plane is set to the curved plane that minimizes wavefront error, and not the paraxial image plane. In a three-dimensional sample, the MPM imaging plane is normally curved. Figure 2.2 Wavefront error ‘W’ between an aberrated and reference wavefront [11]. The RMS of the wavefront error is used to quantify the wavefront error in a single number. It is calculated using the optical path length differences between rays that form the wavefront of the optical system being analyzed and the corresponding rays that form the ideal wavefront that is centered on the intersection of the chief ray with the image plane. The RMS wavefront error is calculated according to Equation 2.3, 20 𝑊𝑅𝑀𝑆 = [∑(𝑊𝑖2 ∙ 𝑤𝑖)∑ 𝑤𝑖− [∑(𝑊𝑖 ∙ 𝑤𝑖)∑ 𝑤𝑖]2]12⁄ (2.3) where 𝑊𝑖 is the optical path length difference for ray ‘i’ and 𝑤𝑖 is the corresponding weight of the ray [29-31]. Summation is done for all rays included in the analysis. ∑(𝑊𝑖2∙𝑤𝑖)∑ 𝑤𝑖 is the weighted mean of the square of the optical path difference values and [∑(𝑊𝑖∙𝑤𝑖)∑ 𝑤𝑖]2 is the square of the weighted mean optical path difference [29, 31]. Subtracting out the mean causes the RMS value to be referenced to the mean rather than to the chief ray and accounts for the shift of the center of the focal point away from the chief ray [29]. In statistical terms, the RMS wavefront error is the standard deviation of the wavefront error. Aberrations distort the PSF, which causes the peak intensity of the PSF to be reduced and the distribution of energy in the PSF to be shift from inside the Airy disk to the surrounding rings [12]. The ratio between the peak intensity of the distorted PSF and that of the undistorted PSF is called the Strehl ratio. It decreases as the RMS wavefront error increases according to Equation 2.4, 𝑆𝑡𝑟𝑒ℎ𝑙 𝑅𝑎𝑡𝑖𝑜 = 𝑒−(2𝜋𝜔)2 (2.4) where 𝜔 is the RMS wavefront error in terms of waves [12]. A plot of Equation 2.4, which relates RMS wavefront error to Strehl ratio, is shown in Figure 2.3. Optical systems with a Strehl ratio above 0.8 or an RMS wavefront error below 0.072 waves are called diffraction limited [12, 29]. 21 Figure 2.3 Strehl ratio as a function of RMS wavefront error. 2.1.2 Background on Optical Aberrations The primary aberrations, known as Seidel aberrations include SA, coma, astigmatism, field curvature, and distortion [12-15]. They are typically calculated in terms of their effect on the chief ray and marginal ray. The chief ray is the central ray of the laser beam when focusing at the edge of the FOV; the marginal ray is the edge-most ray of a beam when focusing on-axis [13]. SA is the variation of focal length with aperture. Figure 2.4 shows an example of under corrected SA in which the beam’s marginal rays come to a focus closer to the lens than paraxial rays. Figure 2.4 Illustration of under corrected SA in which beam’s outer marginal rays have a shorter focal length than the beam’s inner paraxial rays [12-15]. 22 Coma is the variation of magnification with aperture, meaning that the height at which rays come to a focus depends in part on the location that rays pass through a lens [12-15]. This results in comet-like shape on the image plane when focusing off-axis. Astigmatism is the variation in focal length between tangential and sagittal rays, which results in multiple focal points after the lens when focusing off-axis. In the case of under-corrected astigmatism, tangential rays have a shorter focal length than sagittal rays and form a ‘focal line’ rather than a circular spot. Similarly, the sagittal rays also form a ‘focal line’ that is behind and orthogonal to the tangential ‘focal line’. Between these two focal lines, a circular focal spot called the medial focus is formed [12-15]. Astigmatism also causes the image plane to be curved, the effect of which is better understood in conjunction with field curvature. Field curvature is the characterization of an optical system to have a curved focal plane and in the absence of other aberrations will focus light perfectly on a curved surface known as the Petzval surface. The addition of astigmatism results in the formation of a curved tangential and sagittal focal surfaces that are offset from the Petzval surface, with the medial surface half-way in between [11]. Figure 2.5 illustrates the case of under-corrected field curvature, an inward bending Petzval surface, and a case of under-corrected astigmatism. The distance between the sagittal, medial, tangential and Petzval surface are equal. 23 Figure 2.5 . Illustration of the effect of field curvature and astigmatism on the curvature of focal surfaces: sagittal, medial, tangential and Petzval. Sagittal, medial, and tangential focal surfaces are defined by astigmatism and the Petzval surface by field curvature [11, 12]. Distortion is the variation of magnification with image height. Figure 2.6 illustrates its effect when imaging a square. The resulting pincushion or barrel shaped image is due to the change in radial distance that points along the edge of square have with respect to the center of the square. Since a point at a corner of the square is further away from the center of the square than the midpoint of an edge, the magnification due to distortion is greater. Figure 2.6 . Effect of pincushion (a) and barrel (b) type distortion when imaging a square [12-15]. (a) (b) 24 In terms of the effect on the PSF, SA is critical to control since it distorts the shape of the PSF and is the only Seidel aberration to affect both on-axis and off-axis focusing; the other four Seidel aberrations only affect off-axis focusing [11, 12]. Coma and astigmatism also distort the shape of the PSF and as such need to be kept low to achieve a wide diffraction-limited FOV. Low field curvature and distortion are needed to ensure that the image produced during MPM accurately represents the sample being imaged since they affect the position of the PSF; however, they do not change the shape of the PSF. Therefore, the presence of field curvature or distortion does not affect the SNR of an MPM system. 2.2 Analysis of Single Aspherical Lens as Dry Objective Aspherical lenses offer the advantages of small diameter, high NA, and low SA. Thus, they are good candidates for objective lenses in MPM systems. The characteristics of a selection of high NA aspherical lenses are simulated using Zemax and the results are shown in Figure 2.7 and Table 2.1. For all the simulations in this chapter, the design wavelength is set at 790 nm which is the center wavelength of our laser used for MPM imaging. Figure 2.7(a) shows the optical layout used in simulations of single aspherical lenses, where (1), (2) and (3) indicate the lens, air, and tissue (modeled by water with refractive index of n=1.335 [17, 32-34]), respectively. The clear aperture (CA) of the lens is fully illuminated by a collimated beam of uniform intensity. Simulations are performed at the depth at which the lowest RMS wavefront error is obtained. Figure 2.7(b) shows the RMS wavefront error versus imaging field position. The RMS wavefront error is the lowest for the on-axis position which corresponds to the 25 center of the imaging field. As the imaging point moves to an off-axis position, the RMS wavefront error increases dramatically. In general, when the NA of the lens is higher, the diffraction-limited FOV is smaller. As shown however, there is a range in the breadth of the diffraction-limited FOV of lenses with an NA of 0.55 and so the focusing ability of a lens should be checked prior to selection for use as an objective to ensure appropriate performance. Other lens parameters, such as the shape, focal length, design wavelength, or glass can also affect the lens performance. Lens 355392 has a relative high NA of 0.64 and short focal length of 2.74 mm. It has been used as an objective for MPM in a previous paper by Yuan Zhao et al. [27]. Thus, Lens 355392 is selected as a representative candidate of single aspherical lens as objective for MPM and its performance is further analyzed. Figure 2.7(c) shows the spot diagrams for lens 355392 at on-axis position, and off-axis positions of 0.02, 0.04, and 0.06 mm from the optical axis. The spot size increases significantly as the off-axis position moves further away from the optical axis. Coma pattern is seen at the off-axis field positions. A plot of the RMS spot size as a function of field position is shown in Figure 2.7(d). An increase in spot size with increasing field position is observed. The spot size remains below the Airy disk radius of 0.84 µm for a field position up to 0.022 mm, comparable to its diffraction-limited half FOV of 0.025 mm in terms of RMS wavefront error shown in Figure 2.7(b). Table 2.1 summaries the performance of a list of high NA aspherical lenses. Several aspherical lenses (e.g. 352080) are designed to work with a laser window positioned in front of the focal spot. Such lenses have been designed to compensate for the SA of a certain thickness of glass window. The optimal depth of focus in tissue is determined using Zemax as the tissue depth at which the 26 on-axis RMS wavefront error is lowest; additionally, the air gap between the lens and tissue is adjusted during simulation to lower the RMS wavefront error. The depth of focus in tissue when focusing off-axis is somewhat shallower than when focusing on-axis, resulting in a curved focal plane, whose curvature is recorded in Table 2.1. Figure 2.7 (a) Optical layout used for individual aspherical lens simulation, which includes a lens (1), air gap (2), tissue (3). (b) RMS wavefront error of different aspherical lenses that are manufactured by Lightpath. The horizontal dashed line indicates the diffraction limit of 0.072 waves as used by Zemax [29]. (c) Lens 355392 spot patterns at select field positions. (d) A plot of spot size versus image field position for lens 355392. (a) (b) (c) (d) 27 Table 2.1 Characteristics of Aspheric Lenses used as Objective Lenses Part Number NA1 CA (mm) 1 Design Wavelength (nm) 1 Laser Window Thickness (mm) Airy Disk Radius (µm) 2,3 Effective Focal Length at 790 nm (mm) 2 Optimal depth of focus in tissue (mm) 2 Focal plane radius of curvature (mm) 2 Diffraction- limited half FOV (mm) 2 352080 0.55 4.29 780 1.2 0.88 3.89 1.326 -1.91 0.078 355230 0.55 5.07 780 0.25 0.88 4.51 0.268 -1.60 0.035 355390 0.55 3.6 830 none 0.87 2.75 0.055 -4.91 0.004 355660 0.60 3.6 1550 none 0.79 2.91 0.092 -1.10 0.046 355392 0.64 3.6 830 none 0.75 2.74 0.011 -1.03 0.025 354330 0.68 5.00 830 none 0.71 3.10 0.027 -3.37 0.003 1 Specification obtained from the AMS Technologies Precision Molded Aspheric Lenses Catalog [35] 2 Zemax simulation result. 3 Values confirmed via calculation according to Equation 1.1. 28 An inward curvature of the FOV is present in all of the aspherical lenses that are analyzed and is taken into account during RMS wavefront error analysis. The curvature is due off-axis aberrations, which add a field dependance to the focal distance. This curvature does not affect resolution and only affects the flatness of the image, and is thus not an issue for thick tissue imaging. Curvature may be noticed when imaging a flat surface, such as a silicon chip. The Airy disk radius is simulated for the case when the CA is fully illuminated. The trend of Airy disk radius decreasing with an increase in lens NA is observed. The optimal depth of focus in tissue for lens 352080 is at 1.328 mm due to not having been used in conjunction with a glass window. Lens 355660 has a deeper optimal depth of focus below the tissue surface than the other aspherical lenses designed to be used without a glass window due to having been designed for use at a wavelength of 1550 nm rather than the 790 nm that is simulated. Lens 352080 offers a reasonably high NA of 0.55 and diffraction-limited half FOV of 0.078 mm. Its diffraction-limited full FOV is 0.156 mm, which is greater than the FOV of 0.110 - 0.120 mm attained by endoscopes that use GRIN lens (GT-MO-080-018-810, GRINTech) listed in Table 1.1. These properties make it a good choice for use as an objective. Most importantly however, is that it has been designed to work with a glass window of 1.2 mm which is a necessary property when we design a custom water dipping objective that is described in Section 2.3. 29 2.3 Design of a Water Dipping Objective with Depth Scanning Capability for MPM MPM is a deep-tissue imaging system where depth scanning over a wide range is needed. In this section, we design a custom objective that enables depth scanning, while also maintaining high focusing performance over the entire depth-scanning range. Minimizing SA is a key consideration in objective lens design [36, 37]. The challenge of depth scanning is that the SA changes as the beam is focused to different depths in tissue, which is called sample induced SA [38]. Figure 2.8 illustrates sample induced SA when focusing light deep into tissue using an aspherical lens that is optimized to focus near the tissue surface. Focusing to a deeper depth is achieved by reducing the distance between the aspherical lens and the tissue. Due to the large difference in refractive index between air (n=1) and tissue (n=1.33 to 1.5 or more [39]), refraction occurs at this interface. The marginal and paraxial rays are refracted by different amounts depending on their incident angle on the interface according to Snell’s law, and thus they focus to different on-axis positions and induce SA. Figure 2.8 Sample induced SA whereby marginal rays and paraxial rays focus at different depths due to refraction by the sample. 30 Figure 2.9 shows depth scanning performance with a single aspherical lens versus our custom water-dipping objective comprised of an aspherical lens plus a plano-convex lens (PCX) (which acts as a glass window). Figure 2.9(a) shows the optical layout for depth-scanning analysis with a single aspherical lens 355392. The change in the depth of focus is achieved by changing the distance between the lens and tissue. Figure 2.9(c) plots the SA, at individual surfaces and their summation, as a function of depth of focus. The SA from the front and back surfaces of the aspherical lens remain constant and they cancel out, so that when focusing at the tissue surface there is effectively no total SA. However, when focusing at deeper depth below the tissue surface, the air/tissue interface induces a SA that is proportional to the depth. Thus, the total SA increases as depth is increased, which results in deterioration of the focal spot size. To address the challenge of sample induced SA, a water dipping objective is designed using off-the-shelf components. By having water as the immersion medium, the refraction at the tissue surface is minimized due to the small difference in index of refraction thus reducing SA [38]. Figure 2.9(b) shows the optical layout of the custom objective which is comprised of an aspherical lens (352080, Lightpath) mounted in an SMA actuator in front of a PCX lens (32-473, f = 18 mm, Edmund Optics) [40, 41]. Depth-scanning is achieved by translating the aspherical lens along the z axis using an SMA actuator while the PCX remains stationary. Tissue is modeled by the water layer. In the simulation, the index mismatch between the immersion water (n=1.335) and cells/tissue (n = 1.36–1.38 for cells / n = 1.42 for soft tissue) is neglected [38, 39]. Figure 2.9(d) plots the SA, of each surface and their summation, as a function of depth of focus in tissue for the custom water-dipping objective. The front and back surfaces of the aspherical 31 lens induce a constant amount of SA, regardless of the focusing depth; whereas the amount of SA induced by the front and back of the PCX lens varies with focal distance. The amount of SA induced by the back surface of the PCX increases along with scanning depth; however, this increase is offset by a nearly equal amount of increase in negative SA from the front of the PCX resulting in a low overall SA. A low SA is achieved throughout the entire potential scanning range from a depth of 0-1.44 mm. Figure 2.9 (a) Optical layout of a single aspherical lens, 355392, acting as the objective. (b) The optical layout of the custom water-dipping objective focusing light into tissue. (1) Aspherical lens (352080, Lightpath), (2) PCX (32373, Edmund Optics), (3) Tissue. In both (a,b), the focal point is shifted by translating the aspherical lens. (c,d) SA wavefront aberration coefficients for (c) the single aspherical lens and (d) the custom objective. (c) (d) (b) (a) 32 The aspherical lens 352080 is originally designed to be used in air in conjunction with a 1.2 mm thick glass window. In our design, the PCX replaces the 1.2 mm glass window. By adding curvature to the front surface of the glass window and changing its thickness, in other words replacing the window with a PCX lens, the optimal depth of focus at which SA is minimum can not only be brought to be near the surface, but also it is possible to achieve relatively low overall SA over a large range in depth of focus below the tissue surface. The SA across this range is sufficiently low to attain diffraction-limited focusing over a large depth-scanning range. Aspherical lens 352080 is chosen for use in the custom objective as it has the largest diffraction-limited FOV and has a relatively high NA in comparison to other off-the shelf aspherical lenses that are designed to be used with a 1.2 mm glass window. Possible PCX lens solutions are determined for different combinations of depth of focus in tissue between 0 mm and 1.4 mm. The PCX lens 32-473, from Edmund Optics has a radius of 9.3 mm, a center thickness of 1.42 mm and is made of N-BK7 glass. It is chosen for use in the compound objective since the radius and thickness values are a close match to one of the possible PCX lens solutions and because the objective’s SA remains low across a large depth of focus range. The RMS spot size and RMS wavefront error are further compared in Figure 2.10 for the single aspherical lens (355392) and our custom water dipping objective (352080+32473). Figure 2.10(a) compares the on-axis RMS spot size as a function of depth of focus in tissue. In the case of lens 355392, the spot size is shown to increase dramatically with depth. The RMS spot size is less than the Airy disk radius (0.75 µm) only up to a depth of 70 µm, considerably less than which is experimentally achievable when imaging tissue [23]. In comparison, the on-axis spot size of the custom objective remains below its Airy disk radius (0.83 µm to 0.77 µm) for the depth of focus 33 from 0 to 1.44 mm and has minima at depths of 0.20 mm and 1.21 mm where SA is lowest. The Airy disk radius decreases with the depth since the NA of the objective increases when the aspherical lens is moved closer to the PCX during depth scanning due to greater refraction by the PCX; the NA increases from 0.58 to 0.63 when increasing the depth of focus from 0 mm to 1.44 mm. Maintaining a small spot size throughout the potential scanning range of the custom objective extends the depth from which the MPM can acquire images. In practice, the imaging depth will typically be limited to about 0.2 - 0.3 mm for 2PEF and SHG imaging due to light absorption and scattering from tissue, though MPM has been used to acquire images from brain tissue at a depth of 1 mm [23, 42]. Figure 2.10(b) shows the comparison of RMS wavefront error as a function of field position at selected depths of focus in tissue. When imaging the tissue surface (depth = 0 mm), lens 355392 has a diffraction-limited half FOV of 0.025 mm; however, diffraction-limited focusing is limited to only near the tissue surface. In comparison, the diffraction-limited half FOV of the custom objective is of 0.116 mm at a depth 0 mm, and increases to 0.138 mm at a depth of 0.2 mm. The RMS wavefront error is further plotted as a function of field position and depth of focus below the tissue surface as a 2D heatmap for the single aspherical lens (355392) and the custom objective (352080+PCX) in Figure 2.10(c) and Figure 2.10(d), respectively. Lens 355392 achieves on-axis diffraction limited-focusing up to a depth of focus in tissue of 0.08 mm and has a maximum diffraction-limited half FOV of 0.025 mm at a depth of 0.01 mm. In comparison, the custom objective achieves diffraction-limited focusing over a wide range of depth of focus in tissue from 0 mm to 1.44 mm, which is achieved by scanning the gap between the aspherical lens and PCX 34 from 1.275 mm to 0 mm. The custom objective has a maximum diffraction-limited half FOV of 0.139 mm at a depth of focus of 0.25 mm, and remains fairly wide across the depth scanning range. Figure 2.10 Performance comparison of signal aspherical lens (355392) versus that of our custom water-dipping objective. (a) The on-axis RMS spot size as a function of the depth of focus in tissue. The Airy disk radii are plotted using dashed lines. (b) RMS wavefront error versus field plots. A heatmap of the RMS wavefront error versus field and the depth of focus in tissue is shown in (c) for lens 355352 and in (d) for the custom objective. The diffraction limit of 0.072 waves is indicated by the dashed black line in plots (b) to (d). (a) (b) (c) (d) 35 2.4 Impact of Scanning Optics In MPM, laser beam scanning is performed in X and Y directions to obtain a 2D image. This is carried out by a 2D MEMS mirror that can tilt in two dimensions. Between the MEMS scanner and objective, a scan lens and a tube lens form a 4f relay system which also function as a beam expander [43]. The scan and tube lens also affect focusing performance and are a source of optical aberrations. Whereas analyses in Sections 2.2 and 2.3 consider the performance of the objective on its own, this section includes the scan and tube lens in the analysis. Furthermore, analysis in Sections 2.2 and 2.3 assume the general case of the objective being illuminated by a beam of uniform intensity. In practice however the beam has a Gaussian profile whose 1/e2 diameter fills the back aperture of the objective. Zemax analysis in Sections 2.4 and onward are analyzed with a Gaussian beam profile. Zemax adjusts the weights of rays used for RMS wavefront error and RMS spot size calculations to model the beam profile, which generally results in slightly lower error values. Endoscopic MPM systems often use achromatic doublet lenses as the scan and tube lens, which impart optical aberrations that distort the wavefront of the laser beam. The resulting RMS wavefront error can be reduced by replacing the achromatic doublet lens used as the scan lens with a pair of achromatic doublet lenses [44]. Figure 2.11 shows the system characteristics when the effect of scan and tube lens are included. The left column shows the results when using a single achromatic doublet lens (45-792, f = 20.0 mm, Edmund Optics) as the scan lens, and the right column shows the results when using a pair of achromatic doublet lenses (45-796, f = 40.0 mm, Edmund Optics) as the scan lens. The 36 respective optical layouts are shown in Figure 2.11(a) and 2.13(b). The pair of 45-796 lenses has an effective focal length of 20.7 mm when placed in a barrel 54-513 from Edmund Optics, which separates the lenses by 3.2 mm. The tube lens (#45-798, f = 60.0 mm, Edmund Optics) is a single achromatic doublet lens. The Seidel coefficient contributions of the scan, tube and objective lens as well as the combined sum from all three lenses are shown in Figure 2.11(c) for the case when using a single achromatic doublet lens as the scan lens. The Seidel coefficients S1 to S5 corresponds to SA, coma, astigmatism, field curvature and distortion [11, 29]. The scan lens is observed to induce greater coma and astigmatism than the tube lens, and so offers a greater opportunity to improve off-axis focusing capability. Figure 2.11(d) shows the Seidel coefficient values when using a pair of achromatic doublet lenses as the scan lens. By using a pair of lenses rather than a single lens, lens surface curvature is reduced, reducing the angle of incidence at surfaces. This lowers the amount of coma, astigmatism and distortion, which improves off-axis focusing [13]. The effect on aberrations by changing the scan lens is summarized as follows: • SA is increased by a minor amount. The SA induced by the objective is sufficiently low to allow for diffraction-limited focusing, so this minor increase from the scan lens is insignificant. • The magnitude of both coma and astigmatism induced by the scan lens are reduced and their signs become opposite to their respective sums thus lowering the net coma and astigmatism present. This reduction lowers off-axis RMS wavefront error. 37 • There is no change to field curvature. • Distortion is significantly reduced. This does not change the wavefront error, though will help straight features in the tissue appear straight in the image. Spot diagrams at selected field positions when using a single achromatic doublet lens as the scan lens and as a pair of lenses are shown in Figure 2.11(e) and in Figure 2.11(f), respectively. The central portion of the spot diagrams is denser when using the achromatic doublet pair in place of the single achromatic doublet for the same field position. A quantitative comparison between the spot sizes using the different scan lenses is shown as a plot of the RMS spot size as a function of field position in Figure 2.11(g). The RMS spot size when using the achromatic doublet pair is smaller than when using a single achromatic doublet throughout the FOV. The diffraction-limited half FOV according to RMS spot size is 0.16 mm when using the single lens as the scan lens and 0.22 mm when using a pair of lenses. The RMS wavefront error is compared in Figure 2.11(h). The diffraction-limited half FOV according to wavefront error analysis is increased from ~0.1 mm to ~ 0.15 mm when a pair of achromatic doublets are used as the scan lens. The lower RMS wavefront error which is attainable when using the pair of achromatic doublet lenses results in a greater concentration of energy within the focal point [12]. It also allows for less precise alignment when constructing the MPM system while still achieving an acceptable level of RMS wavefront error across the FOV. Currently, the FOV attained by our MPM system (102 μm x 156 μm) is limited by the scanning angle of the MEMS mirror. In the future, a better MEMS mirror with a wider scanning angle can be used to increase the imaging FOV to match with that provided by our improved optical design. 38 Figure 2.11 Comparison between simulation results using a single achromatic doublet lens (a,c,e) and using a pair of achromatic doublet lenses (b,d,f) as the scan lens. (a, b) Optical layout of the scanning optics. (c, d) Seidel coefficients of the lenses and their sum for a field position of 0.092 mm (e, f) Spot diagrams at selected field locations. The Airy disk is shown as black circles. (g) Spot size as a function of field position. (h) RMS wavefront error as function of field position comparing the use of a single (dashed) or pair (solid) of achromatic doublets as the scan lens at selected depths of focus: 0 mm, 0.2 mm and 0.4 mm. (a) (b) (c) (d) (e) (f) (g) (h) 39 2.5 Conclusions The use of aspherical lenses as objectives for use in MPM is explored via simulation. They are shown to achieve diffraction-limited performance across a relatively small FOV in comparison to the objectives from literature listed in Table 1.1. They also suffer from sample induced SA and only achieve a shallow diffraction-limited depth-scanning range. Thus, they cannot take full advantage of the three-dimensional imaging capabilities of MPM. A custom water-dipping objective is designed by combining an aspherical lens (352080, Lightpath) with a PCX lens (32-473, Edmund Optics). The unique combination of lenses nearly cancels out sample induced SA across a large depth scanning range from 0 mm to 1.4 mm below the tissue surface and allows for diffraction-limited focusing across this entire range. The aspherical lens is mounted in an SMA to enable depth scanning across a 0.44 mm range. This is greater than the depth scanning range of state-of-the-art endoscopes listed in literature (Table 1.1). Since light scattering and light absorption typically limits MPM imaging depth to ~0.2-0.3 mm below the tissue surface, the depth scanning range offered by the SMA is adequate for MPM endoscopy. The scanning optics also affect the diffraction-limited FOV that can be achieved. By using an achromatic doublet pair as the scan lens in place of a single achromatic doublet lens as is common in endoscopic MPM systems, the wide diffraction-limited FOV offered by an objective can be maintained. The reduction of the angle of incidence from changing the scan lens reduces coma and astigmatism and improves off-axis focusing. The custom objective together with the scanning optics achieve a diffraction-limited half FOV of up to 0.146 mm in simulations. 40 Chapter 3: Chromatic Aberration Reduction In Chapter 2, the monochromatic focusing performance of our custom objective and scanning optics is analyzed at a single design wavelength of 790 nm. In Chapter 3 we consider the broad bandwidth of the pulsed laser used by our MPM and present CO that are designed to minimize the effect of chromatic aberration. The effect that chromatic aberration has on spatial and temporal focusing is investigated. 3.1 Background on Chromatic Aberration The performance of a lens-based optical system is wavelength dependent since the index of refraction of glass with which lenses are made varies with wavelength. The resulting aberration due to the bandwidth of the light passing through an optical system can be described in terms of axial chromatic aberration as well as lateral color [12]. The effect of axial chromatic aberration from a lens is illustrated in Figure 3.1. It is a measure of an optical system’s focal length dependence on wavelength [12]. For most materials, shorter wavelengths have a higher index of refraction than longer wavelengths and undergo a greater amount of refraction [13]. Thus, singlet lenses have shorter focal lengths at shorter wavelengths. 41 Figure 3.1 Illustration of axial chromatic aberration in which shorter wavelength light has a shorter focal length than longer wavelength light [11, 12]. Lateral color is illustrated in Figure 3.2. It is a measure of the difference in lateral position at the focal plane between chief rays of different wavelengths, and it is induced when a chief ray passes through a lens away from the center of the lens [12]. In other words, it occurs when a light beam is off center to the axis of the lens through which it is passing. Figure 3.2 Illustration of lateral color in which the chief ray of the shorter wavelength intersects the image plane at a lower height than chief ray of the longer wavelength [11, 12]. An achromatic doublet lens is a simple and effective type of lens that minimizes chromatic aberration by combining two lenses: a converging lens and a diverging lens. The power of the doublet lens is the sum of the power of the two individual lenses as per Equation 3.1; 42 ∅ = ∅1 + ∅2 (3.1) where ∅ is the power of a lens, and the subscripts refer to first and second lenses of the doublet [11, 13]. Whether the doublet is converging or diverging depends on whichever of the two lenses has a greater power. The power of a lens is equal to the reciprocal of its focal length. The higher power lens is made of crown glass, which has relatively low chromatic dispersion, whereas the lower power lens is made of flint glass, which has relatively high chromatic dispersion. A doublet lens is made to be achromatic by satisfying Equation 3.2, ∅1𝑣1=∅2𝑣2 (3.2) where 𝑣 is the Abbe number of the lens glass [11, 13]. Abbe number is inversely related to dispersion, thus a glass with a lower Abbe number has higher dispersion. 3.2 Reducing MPM Axial Chromatic Aberration MPM requires a femtosecond pulsed laser for exciting the nonlinear signals. We use a frequency doubled Er-doped fiber laser which has a center wavelength of 790 nm and pulse width of 80 fs. Femtosecond pulses have a relatively broad bandwidth. The -3dB bandwidth of the laser is ~14 nm [47]. In the case of a transform-limited Gaussian shaped pulse, the time-bandwidth product equals 0.44. For the 14 nm bandwidth, the transform limited pulse width is 66 fs. Thus, our pulse width is close to transform-limited. The relatively broad bandwidth can result in chromatic 43 aberration of the pulse as it passes through the MPM system and increase the size of the focal spot in the tissue. The high NA aspherical lens used in our custom objective is made of a single type of glass (ECO-550). The aspherical lens is not achromatic which can deteriorate the focusing capability. For correcting chromatic aberration, our approach is to add corrective optics (CO) between the tube lens and objective [27]. This design is inspired by the work of R. Mittal, et. al, which uses a pair of off-the-shelf achromatic doublet lenses as well as a custom diverging achromatic doublet in conjunction with an aspherical lens [20]. The optical design we use is composed of two plano-concave (PCV) lenses and two converging achromatic doublet lenses, all of which are off-the-shelf, as shown in an optical layout with the scanning optics in Figure 3.3. The PCV lenses are used to compensate against axial chromatic aberration whereas the achromatic doublet lenses are used to control beam size and beam collimation. Figure 3.3 Optical layout of the MPM scanning optics, including the CO in the red rectangle. An optimization routine of Zemax is used to optimize the design, including lens selection and lens spacing. A merit function quantifies the deviation between target values and simulated values of selected parameters in a single number. The optimization routine tries to reduce the merit function. The merit function is calculated according to Equation 3.3 44 𝑀𝐹 = √∑ 𝑤𝑖(𝑉𝑖 − 𝑇𝑖)2∑ 𝑤𝑖 (3.3) where Vi is the simulated value of an operand, Ti is the target value, and wi is the weighting factor [29]. Summations are done for all i operands of the merit function. The optimization routine changes selected lens system parameters, including lens spacing, lens thickness, lens curvature, and lens materials to reduce the value of the merit function. The merit function has the goals listed below, which can be evaluated using the given four-letter Zemax operands. Weightings are set on a trial-and-error basis; if a given parameter accounted for almost all of the merit function value after optimization, its weight is reduced; if a target is not being met, its value is increased. Additionally, a requirement can be relaxed by reducing its weighting. Goals of the merit function: 1. The minimization of axial chromatic aberration. AXCL is used to measure the axial chromatic aberration and has a target value of zero. 2. The minimization of lateral chromatic aberration. LACL is used to measure the lateral chromatic aberration and has a target value of zero. 3. The minimization of RMS wavefront error across a FOV up to 0.15 mm. OPDX is used to measure wavefront error and has a target value of zero. 45 4. That the output beam after the CO is collimated. EFLY can be used to measure the effective focal length of the CO and a target of infinity can be achieved by setting a target value of zero for its inverse. The compensating optics’ actual focal length is 200 mm; this requirement is relaxed in order to reduce RMS wavefront error. 5. That the beam diameter be kept to the same size before and after CO. PARY can be used to measure the marginal ray height prior to the aspherical lens in the objective and has a target value of 2.145 mm, corresponding to half the CA of the aspherical lens. 6. That a distance of 8 mm between the compensating optics and the objective’s aspherical lens is obtained. This is the closest distance that these components can be practically placed due to size constraints imposed by the SMA in which the aspherical lens is mounted. TTHI is used to measure the distance between the last element in the CO and the aspherical lens in the objective and has a target value of 8 mm. 7. That two lenses not occupy the same location. MNCA is used to constrain the distance between lens elements to be greater than zero. At the start of optimization, an initial layout of the proposed CO is entered into Zemax, which consists of two achromatic doublet lenses and two PCV lenses. The spacing of the lenses and the radius of the PCV lenses are set as variables to be changed during optimization. The distance of the space after the CO is set using a chief ray height solve so that the chief ray has a height of zero at the back focal plane of the objective; this is necessary in order to maintain a 4f relay between the MEMS scanner and the objective throughout the optimization process. Optimization trials using different pairs of achromatic doublet lenses are performed and the performance of the 46 resulting designs is compared. Finally, achromatic doublet lenses (#49-951, f = 22.5 mm, Edmund Optics) and PCV lenses (#45-914, f = -18mm, Edmund Optics) are selected for use in the CO. Figure 3.4 shows the effect of axial chromatic aberration by plotting the focal depth versus the field position at various wavelengths, center (790 nm), -3 dB bandwidth (783 - 797 nm), and -10 dB bandwidth (775 - 805 nm) of the laser. The 783 nm / 797 nm wavelengths focus 1.1 µm shallower / deeper than the 790 nm wavelength without the addition of the CO. This difference is reduced to 0.3 µm with the addition of the CO. The corresponding -10 dB wavelengths focus 2.3 µm from the central wavelength without CO in place, which is reduced to 0.6 µm with CO. These decreases demonstrate the reduction in the axial chromatic aberration, which has the potential to increase SNR and axial resolution. Figure 3.4 Plot of the focal distances at various wavelength: central wavelength (790 nm), -3dB wavelengths (783 nm and 797 nm), and -10 dB wavelengths (775 nm and 805 nm). Plot (a) is with CO and (b) is without CO. The objective’s aspherical lens and PCX lens are separated by a distance of 1.12 mm so as to achieve a nominal on-axis depth of focus of 0.2 mm below the tissue surface. (a) (b) 47 3.3 Effect on Lateral Color Figure 3.5 shows the impact of lateral color by the optical system with and without the CO in place. Figure 3.5(a) plots lateral color as a function of field position. Lateral color increases with field position and at a field position of 0.092 mm (the maximum half FOV of the MPM), lateral color is increased from 0.07 µm to 0.20 µm by the addition of the CO. These values are both relatively small in comparison with the Airy disk radius value of 0.82 µm. There is no lateral color when focusing on-axis. Figures 3.5(b) and (c) show spot diagrams without and with the CO at select positions. On-axis (IMA=0.000 mm), no lateral color is present and the spot pattern of each wavelength form concentric circles. As the field position increases sufficiently far off-axis (e.g. IMA= 0.250 mm), the effect of lateral color becomes apparent and the spot pattern formed by the different wavelengths are shifted laterally; a larger shift is observed when using CO due to the greater amount of lateral color. The lateral color induced by a lens is proportional to the height of the chief ray at the lens and also the power of the lens [12]. Thus, even though the aspherical has a high power, it induces a relatively low amount of lateral color since the chief ray is only off-axis by tens of microns at the aspherical lens. Since the chief ray crosses the optical axis at the back focal plane of the objective, located between the CO and objective (Figure 3.5(e)), the PCV lenses in the CO that reduce axial chromatic aberration, actually increase lateral color of the lens system. The net effect of lateral color on MPM imaging however is minimal regardless of whether or not CO are in place since it 48 is much less than either the Airy disk radius or theoretical lateral resolution (~0.5 µm) across the half FOV of the MPM. Without Corrective Optics: 0.2 mm WD With Corrective Optics: 0.2 mm WD Figure 3.5 Analysis of lateral color. (a) Lateral color dependance on field position for -3dB wavelengths of 783 nm and 797 nm. (b,c) Spot diagrams without (b) and with (c) the use of CO at selected positions: 0 mm, 0.092 mm, and 0.25 mm. The Airy disk is shown in spot diagrams as a black circle and has a radius of 0.82 µm. (d, e) Chief ray (red) and marginal ray (blue) without (d) and with (e) the use of CO. The back focal plane of the objective is indicated by the arrow. 3.4 Impact of Corrective Optics on Other Aberrations and Overall System Performance When adding CO to the MPM system to reduce chromatic aberration, it also impacts monochromatic aberrations that are described by the Seidel coefficients. The monochromatic (a) (b) (c) (d) (e) 49 aberrations are calculated at a single wavelength and do not consider wavelength dependance. Seidel and chromatic aberration coefficients are shown in Figure 3.6(a) for the case without CO and in Figure 3.6(b) for the case with CO. The Seidel coefficients S1 to S5 corresponds to SA, coma, astigmatism, field curvature and distortion [11, 29]. The effect on aberrations from the CO can be summarized as follows: • SA is somewhat increased, shifting the SA versus depth of focus curve upward as shown in Figure 3.6(c) and moving the point of zero SA to a deeper depth. This change increases the RMS wavefront error at shallow depths and decreases it at deeper depths. • The magnitude of coma and astigmatism are reduced, thus reducing off-axis RMS wavefront error. • Field curvature is not affected. • Distortion becomes more negative. This does not affect wavefront error. • Axial color is reduced, but lateral color is increased. Aberrations from the scan, tube and objective lenses are nearly the same whether or not CO are used because the path of the marginal ray, which is used to compute the aberration coefficients, is nearly the same for either case. The change in the sum of the aberration values is predominantly introduced by the CO. A breakdown of the aberration contribution from each of the CO’s elements is shown in Figure 3.6(d). The PCV lenses induce negative Seidel coefficient values, whereas the achromatic lenses induce positive Seidel coefficient values such that the sum is fairly low relative to the coefficient values of the individual lenses. Without this balance of aberration coefficients, the performance of the MPM would be much reduced. The PCV lenses in the CO are shown to be responsible for the reduction in the magnitude of axial color and the increase of lateral color. 50 Figure 3.6 Comparison of monochromatic and chromatic aberrations by the optical system. (a) and (b) Seidel aberration and chromatic aberration coefficients of the optics without and with CO in place, respectively. Calculations are for a field position of 0.092 mm and at a depth of focus of 0.2 mm in tissue. (c) SA coefficient as function of depth of focus in tissue without and with CO. (d) Aberration coefficients of the elements in the CO. The SA bars for Achromat 1 and PCV 2 are truncated and have respective values of 21 µm and -18 µm. Seidel coefficient analysis provides a practical means of assessing the primary aberrations present in an optical system and identifying the optical elements responsible for aberrations. Seidel coefficients are based on paraxial imaging and are especially suited to determine the effect that (a) (b) (c) (d) 51 aberrations have on imaging at the paraxial focus. Alternatively, aberrations can be quantified using Zernike polynomials that describe a two-dimensional wavefront using orthogonal polynomial terms. Zernike standard coefficients quantify the RMS wavefront error of each particular Zernike term and can thus be used to determine the relative contribution of each aberration toward the overall wavefront error. Higher order aberrations beyond the primary aberrations described by Seidel coefficients are typically described using Zernike polynomials. Figure 3.7 shows Zernike standard coefficients of selected Zernike terms as a function of field position with and without CO. Primary SA, primary coma, primary astigmatism and also secondary coma (a higher-order aberration), are included in the plots. Secondary coma is included since it is sufficiently large to affect the overall RMS wavefront error. The total RMS wavefront error is computed by calculating the RMS of the Zernike coefficients 3 to 37, which includes primary and higher order aberrations. The total RMS wavefront error computed in this manner is somewhat higher the value obtained using Equation 2.3, since the computed RMS wavefront error is referenced to where the chief ray intercepts the image plane, rather than to the center of the focal point [29]. The CO reduces coma and astigmatism thus improving off-axis focusing. At larger field positions, astigmatism is the dominant aberration and is responsible for almost all of the overall wavefront error and limits the diffraction-limited FOV. At smaller field positions, SA is the dominant aberration when the CO is in place. The effect of secondary coma is greater than that of primary coma for much of the FOV; though relative to astigmatism both types of coma have a relatively minor effect on the total wavefront error. Comparing with CO versus without CO, the total RMS wavefront error is reduced, which is attributed to the reduction of astigmatism. This analysis is done at a depth of 0.2 mm below the tissue surface in both cases for consistency. 52 Figure 3.7 Absolute value of select Zernike standard coefficients are plotted as a function of field position. (a) is without CO. (b) is with CO The overall lens system performance is assessed by RMS wavefront error analysis for the central wavelength of 790 nm as well the -3dB bandwidth wavelengths of 783 nm and 797 nm. Polychromatic RMS wavefront error is calculated according to Equation 2.3 using one common reference wavefront from the 790 nm wavelength and thus is a separate calculation and not an average of the individual wavelength RMS wavefront errors. The 790 nm wavelength is given a weight of 1, and the 783 nm and 797 nm wavelengths are given weights of 0.5 for the RMS wavefront error computation (Equation 2.3). The curved plane across which the polychromatic RMS wavefront error is lowest is used as the image reference plane for the RMS wavefront error computation. Figure 3.8(a,b) shows the RMS wavefront error of the lens system without and with the CO in place. Without the CO, a significant difference between the RMS wavefront of the central (790 (a) (b) 53 nm) and -3dB (783 nm and 797 nm) is present. The difference in RMS wavefront error between wavelengths is reduced when CO are used due to reduced axial chromatic aberration. A larger diffraction-limited half FOV is obtained (0.150 mm) when using the CO in comparison when CO are not used (0.132 mm). The larger diffraction-limited half FOV is due to a reduction in astigmatism and axial chromatic aberration. Polychromatic RMS wavefront error of the MPM scanning optics without and with CO in place is shown as 2D heatmaps as functions of focal depth and field position in Figures 3.8(c) and (d). The polychromatic diffraction-limited half FOV throughout the objective’s 440 µm z-scanning range is increased by the addition of the CO. With the CO, the RMS wavefront error is reduced most notably in depths greater than 0.1 mm. The SA from the CO causes the tissue depth at which the lowest on-axis RMS wavefront error occurs to shift from a depth of 0.214 mm to 0.271 mm. Performance of the MPM scanning optics is also shown in terms of 2D heatmaps of Strehl ratio in Figures 3.8(c) and (d). The impact on SNR is better understood in terms of Strehl ratio, since it is proportional to the peak of the PSF and is also the ratio of irradiance within the Airy disk of the PSF to the total irradiance [47]. The significance of a high Strehl ratio is especially important for MPM, whose signal is proportional to the square of the excitation intensity. The addition of CO, increases the Strehl ratio for the FOV and depth-scanning range of the MPM. Much of the FOV has a Strehl ratio above 0.95 with the CO; whereas without the CO it is mostly in the range of 0.9 to 0.95. 54 Figure 3.8 Comparison between simulation results of MPM scanning optics without (a,c,e) and with (b,d,f) CO in place. (a, b) RMS wavefront error at a depth of focus of 0.2 mm. (c, d) RMS wavefront error as a heatmap. (e, f) Strehl ratio as a heatmap. The diffraction limit is indicated by the dashed black line at an RMS wavefront error of 0.072 wave in plots (a) to (d) and at a Strehl ratio of 0.8 in plot I and (f). (a) (b) (c) (d) (e) (f) 55 The performance of the MPM scanning optics in terms of RMS spot size is shown in Figure 3.9. The net effect of the CO is to reduce spot size. Without CO, the diffraction-limited half FOV according to RMS spot size is 0.22 mm without CO; with CO, it is extended to 0.25 mm. The on-axis reduction in spot size is due to the reduced axial chromatic aberration. The reduced spot size at wider field positions is a result of reduced axial chromatic aberration, coma and astigmatism. Figure 3.9 RMS spot size radius versus field position for the MPM imaging optics with and without the CO (at a depth of focus in tissue of 0.2 mm). The Airy disk radius is 0.82 µm. 3.5 Consideration of Temporal Focusing Since MPM uses femtosecond pulses to obtain high intensities at the focal point needed to excite nonlinear signals, the temporal focusing of pulses is critical. In addition to the spatial shift in focus, chromatic aberration also affects the temporal focusing, causing the pulse front along the marginal ray to come to a focus sooner, than the pulse front along on-axis ray [5, 9, 50]. Due to the difference between phase and group velocities, the pulse front (propagating at group velocity) may be delayed 56 with respect to the phase front. This effect is referred to as propagation time difference (PTD). A large PTD can deteriorate the SNR of MPM. PTD between rays can be determined by considering a fully illuminated singlet focusing a collimated beam as shown in Figure 3.10. Variables r, r0, L0, L1, L2, L3, L4, f are distances defined in Figure 3.10. ΔT(r) is the time difference between phase and pulse fronts. Figure 3.10 An illustration of a fully illuminated singlet lens focusing rays in air. The delay between pulse front and phase front is shown [49, 50]. The optical path length is the same for all rays as per Fermat’s principle. Therefore, the optical path length of an off-axis ray and the on-axis ray are equal, as given in Equation 3.4 [49]. 𝐿1 + 𝑛𝐿2 + 𝐿3 + 𝐿4 = 𝐿0𝑛 + 𝑓 (3.4) The group velocity in a medium is calculated according to Equation 3.5, 57 𝑣𝑔 =𝑐𝑛 − 𝜆𝑑𝑛𝑑𝜆 (3.5) where λ is the vacuum wavelength, c the speed of light, and n is the index of refraction of the glass [49, 50]. In air, the group velocity is reduced to phase velocity c. The propagation time needed for a pulse front to pass through the lens and arrive at the focal point is calculated as 𝑇(𝑟) =𝐿1 + 𝐿3 + 𝐿4𝑐+ 𝐿2𝑐(𝑛 − 𝜆𝑑𝑛𝑑𝜆) (3.6) In the case of the marginal ray (𝑟 = 𝑟0), the path length L2 in the lens approaches zero. As the pulse propagates completely in air, the group velocity is reduced to phase velocity. Thus, the pulse front and phase front overlaps for the marginal ray. Thus, the propagation time of the phase front can be obtained as 𝑇(𝑟0) =𝐿0𝑛𝑐+𝑓𝑐 (3.7) In the case of a ray emanating from position ‘r’, the temporal delay between the phase front and pulse front is calculated by subtracting Equation 3.7 from 3.6. In applying paraxial approximations, as done by Z. Bor, the difference reduces to Equation 3.8 [49]. 58 ∆𝑇(𝑟) = 𝑟02 − 𝑟22𝑐𝑓(𝑛 − 1)(−𝜆𝑑𝑛𝑑𝜆) (3.8) This time delay can be further written in terms of longitudinal chromatic aberration as given in Equation 3.9 [49, 50]. The equation is found to have general validity, making it suitable to analyze a lens system and not just a single lens [49]. The equation shows that the time delay is proportional to the axial chromatic aberration of an optical system and justifies the claim that temporal focusing is improved by reducing axial chromatic aberration. ∆𝑇(𝑟) = 𝑟02 − 𝑟22𝑐𝑓2𝜆𝑑𝑓𝑑𝜆 (3.9) The amount of time by which the ray from position ‘r’ arrives at the focal point before the on-axis ray is obtained by computing ∆𝑇(0) − ∆𝑇(𝑟) . The result is given in Equation 3.10, from which we can see that the time delay between the on-axis ray and a ray from position ‘r’ is independent of lens diameter, r0 [50]. ∆𝑇𝑟𝑎𝑦𝑠(𝑟) = 𝑟22𝑐𝑓2𝜆𝑑𝑓𝑑𝜆 (3.10) The equation has a quadratic dependance on 𝑟 𝑓⁄ , which can be expressed in terms of the lens NA (for r = r0) to generate Equation 3.11. Due to the quadratic dependence of NA, it is of critical importance to minimize chromatic aberration in high NA objectives. 59 ∆𝑇𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙 𝑟𝑎𝑦 = 𝑁𝐴22𝑐𝜆𝑑𝑓𝑑𝜆 (3.11) A plot of Equation 3.10 for the custom objective is shown in Figure 3.11(a). The 𝑑𝑓 𝑑𝜆⁄ value is obtained from Figure 3.4(b,c) as 158 (unitless) without CO and 42 without CO. Ray position r = 2.145 mm, corresponds to position of the marginal ray. The PTD between the marginal ray and on-axis ray is 72 fs without CO and 19 fs with CO. Figure 3.11 PDT between off-axis rays at position ‘r’ and the on-axis ray with and without the use of CO (a) Analytically calculated as per Equation 3.10 (b) Determined via ray tracing. Radial position, r, is measured in front of the objective. Other than using the analytical expression in Equation 3.10, we can directly track the propagation of ray paths via Zemax simulation and then calculate the respective propagation time. The PTD between off-axis and on-axis rays calculated by Zemax simulation is shown in Figure 3.11(b). This result is very similar to the one obtained by the analytical expression in Figure 3.11(a). The radial (a) (b) 60 position of rays, ‘r’, is measured prior to the objective. Analysis is done for the case of on-axis focusing at a depth of focus of 0.2 mm below the tissue surface. Without the use of CO, the marginal ray is found to arrive at the focal point 58 fs prior to the arrival of the on-axis ray. This arrival time difference is comparable with the laser’s 80 fs pulse width, which can potentially broaden the femtosecond pulse and thus reduce SNR. With the CO in place, the time difference between the marginal ray and on-axis ray is reduced to 15 fs, which is expected to have minimal impact on the focusing of the femtosecond pulse. Based on our analysis, the propagation time difference is comparable with the pulse width for the focusing system without the CO. This can reduce the SNR in MPM imaging. With the CO, the propagation time different is significantly reduced and thus has the potential to improve SNR. 3.6 Conclusions The femtosecond laser pulses used for MPM imaging are subject to chromatic aberration that degrades focusing performance. The axial chromatic aberration of the objective and scanning optics is reduced by the addition of CO. The addition of CO reduces the difference in focal depth between the laser’s -10dB wavelengths (783 and 797 nm) and central wavelength (790 nm) from 2.3 µm to 0.6 µm. The CO also minimize coma and astigmatism, thus increasing the diffraction-limited FOV. Simulations show that the addition of CO lowers the RMS wavefront error and Strehl ratio across the FOV of the MPM. A diffraction-limited half FOV of up to 0.155 mm is achieved. The temporal focusing of marginal versus on-axis rays is also affected by chromatic aberration. The difference in arrival time at the focal spot between the marginal ray and the on-axis ray is 61 proportional to axial chromatic aberration and to the square of the NA. Therefore, this is of particular concern in MPM as high NA lenses are used. The time difference between the rays for our scanning optics and objective is 58 fs without the CO and 15 fs with the CO. In comparison, the laser pulse width is 80 fs, thus the addition of the CO significantly reduces the propagation time delay and thus has the potential to increase SNR. 62 Chapter 4: MPM Experimental Results In Chapters 2 and 3, the focusing capability of the objective design and scanning optics is analyzed via simulation. Results therein represent the best-case performance attainable by the design and not what is necessarily achievable in the lab as the simulations do not include real-world errors associated with implementing the design. In Chapter 4, the FOV, resolution, depth penetrating and imaging abilities of MPM are experimentally assessed. The effect of lens fabrication tolerances and assembly tolerances on MPM performance is investigated to provide insight on experimental resolution results and to suggest future means of improving performance. 4.1 Experimental Setup of MPM The objective and scanning optics are incorporated into an MPM system as shown in Figure 4.1. An Erbium-doped fiber laser generates 80 fs pulses, centered at a wavelength of 1580 nm. A fiber pigtail (LPC-02-1300/1550-9/125-S-1.4, Oz Optics) is used to collimate the laser beam as light exits the fiber. The combination of the quarter wave plate (QWP) and half wave plate (HWP) are used to produce vertically oriented linear polarization, which is needed for frequency doubling by a Magnesium-doped periodically poled Lithium Niobate (PPLN) crystal (MSHG1550-0.5-0.3, Covesion) [51]. The focusing lens FL (A375TM-C, f = 7.5 mm, Thorlabs) focuses the pulses on the PPLN for frequency doubling to obtain 790 nm pulses, while the collimating lens CL2 (A397TM-B, f = 11 mm, Thorlabs) collimates the beam. A filter (RG9, Schott) is used to remove the 1580 nm fundamental wavelength from the 790 nm excitation wavelength used for imaging. The scan lens is comprised of a pair of achromatic doublet lenses (45-796, f = 40.0 mm, Edmund Optics) which have an effective focal length of 20.7 mm at 790 nm. The tube lens is a single achromatic doublet lens (#45-798, f = 60 mm, Edmund Optics). Scanning in the lateral xy 63 directions is performed by a MEMS scanning mirror (SN: S2008, Mirrorcle); axial scanning in the z direction is performed by an SMA actuator (ASA10080-101, Actuator Solutions) that translates the aspherical lens in the objective. Figure 4.1 Schematic of the MPM set-up. SMF: single mode fiber; CL: collimating lens; QWP: quarter-wave plate; HWP: half-wave plate; HWP: half-wave plate; FL: focusing lens; PPLN: periodically poled Lithium Niobate crystal; F: filter; MEMS mirror: micro-electro-mechanical scanning mirror; SL: scan lens; TL: tube lens; DM: dichroic mirror; CO: corrective optics OBJ: Objective; L: lens; MMF: multimode fiber; PMT: photomultiplier tube. SHG and 2PEF signals are generated from within the tissue and propagate backward through the objective and CO. A dichroic mirror (DM) (FF665-Di02, Semrock) separates the signal from the excitation light and is coupled into a MMF to allow the signal to be detected at a distance from the 64 sample. A second DM (FF414-Di01, Semrock) separates the 2PEF and SHG signals. The 2PEF and the SHG signals are further selected by two bandpass filters (FF01-550/88, Semrock) and (FF01-390/40, Semrock) respectively, before reaching two photomultiplier tubes (PMTs). Optics are selected to achieve several goals: maximize power transmission, fully fill the back aperture of the objective, achieve high resolution while maintaining a large FOV, avoid vignetting, and minimize aberrations. 4.1.1 Beam Size In order to fully utilize the NA of an objective, the back aperture of the objective lens should be fully illuminated [52]. In practical terms, an objective is considered fully filled if the 1/e2 beam diameter of a laser beam with a Gaussian distribution illuminates the back aperture of the objective [53]. The 1/e2 beam diameter contains ~86% of the power; thus, filling the objective with a beam of this diameter offers a good balance between uniformly illuminating the back aperture and power transmission [53]. For the MPM layout shown in Figure 4.1, the 1/e2 diameter of the beam nearly fills the objective. This however results in a somewhat larger PSF at the focal spot than the case with a uniformly illuminated beam. Simulations show the resulting FWHM of the PSF at the focal point to measure 0.77 µm and that for the hypothetical case of a laser with a uniform profile, the FWHM would be 0.70 µm. The beam diameter can be increased by increasing the focal length ratio between the collimating lens CL2 and the focusing lens FL on either side of the PPLN or by increasing the focal length ratio of the tube lens to scan lens. Either of these approaches has its own trade-offs. Increasing the beam diameter prior to the MEMS mirror, requires that the mirror is of sufficient size to avoid 65 vignetting; however, a larger mirror reduces the scanning speed of the MEMS. Increasing the focal length ratio of the tube lens to scan lens decreases the FOV. Computation of the beam diameter at the objective is as follows. According to the specification of the collimating lens CL1 located at the output of the laser fiber into free space, the output at 1580 nm has a 1/e2 beam diameter of 1.4 mm. The beam expansion ratio between the collimating lens (CL2) and focusing lens (FL) across the PPLN is 11:7.5. The 790 nm beam produced due to frequency doubling in the PPLN is smaller than the 1580 nm excitation beam by a factor of √2. The beam expansion ratio between the tube lens and scan lens is 60:20.7. The 790 nm beam diameter after the tube lens is 4.21 mm (1.4 𝑚𝑚√2×117.5×6020.7). Thus the 1/e2 diameter of the 790 nm beam just slightly underfills the 4.29 mm CA. The diameter of the 790 nm and 1580 nm beam after the PPLN is experimentally measured using the knife edge method in the horizontal and vertical directions for the case with 1:1 beam expansion across the PPLN for the following lenses CL2 (A375TM-B, f = 7.5 mm, Thorlabs) and F (A375TM-C, f = 7.5 mm, Thorlabs) [54]. Results are shown in Table 4.1. Results are within 7 % of the calculated value, indicating that the beam is nearly circular and confirm the √2 scaling of the frequency doubled beam diameter. 66 Table 4.1 Characterization of Beam Diameter After PPLN Wavelength Measurement Direction 1/e2 Beam Diameter 1580 Horizontal 1.46 mm (1.4 mm)* Vertical 1.42 mm (1.4 mm)* 790 Vertical 0.96 mm (1 mm)* * Calculated values after the PPLN for 1:1 beam expansion Optics are also chosen to avoid vignetting, which for example occurs when all or part of an off-axis ray bundle is clipped at a lens by not passing through the CA [11]. The beam path through the scanning optics for on-axis and off-axis focusing (𝜃 = 4.15°) and that of the chief ray and marginal ray are shown in Figure 4.2. The position of the ray that is most off-axis at any particular lens is the sum of the absolute height of the marginal ray and absolute height of the chief relative to the optical axis; the sum must be less than or equal to half of the CA of the lens to avoid vignetting [11]. By determining the position of the ray that is furthest off-axis (the worst case), it’s possible to confirm that vignetting does not occur. In our case, the ray that is furthest off-axis is located at the tube lens and is 3.65 mm off-axis. This is less than half the CA of the tube lens (4.5 mm), scan lens (4.5 mm) and CO (4.05 mm), thus vignetting does not occur at any of these lenses. 67 Figure 4.2 MPM scanning optics layout (a) without CO and (b) with CO. (i) Ray tracing for on-axis (blue beam) and off-axis (green beam) focusing. (ii) Paths of the marginal ray (blue ray) and the chief ray (red ray). Vignetting at the MEMS mirror is checked by comparing the size of the MEMS mirror to the height of the marginal ray at the MEMS mirror. The MEMS mirror is circular and has a radius of 1.2 mm; however, since it is at a 45-degree angle to the incoming laser beam, it appears as an oval, with a semi-minor axis of 0.85 mm. In comparison, the marginal ray height at the MEMS mirror is 0.74 mm, thus vignetting does not occur at the MEMS mirror. 4.1.2 Field of View The FOV of the MPM can be determined by tracing out the path of the chief ray through the scanning optics as illustrated in Figure 4.3. The angle of the chief ray relative to the optical axis is marked with θ. The calculation of the FOV given in Equation 4.1, relies on the scanning optics forming a 4f relay system between the MEMS mirror and the back focal plane of the objective lens. (a) (i) (ii) (b) (i) (ii) 68 𝐻𝑎𝑙𝑓 𝐹𝑂𝑉 = 𝑓𝑜𝑏𝑗𝑒𝑐𝑡𝑖𝑣𝑒𝑓𝑠𝑐𝑎𝑛 ∙ 𝑡𝑎𝑛 𝜃𝑓𝑡𝑢𝑏𝑒 (4.1) Figure 4.3 The chief ray (fletched ray in red) reflecting off the MEMS mirror and passing through the scanning optics. The spacing of optical elements is set according to the focal length of the scan, tube and objective lenses. The rotated MEMS mirror is shown in black; its neutral position is shown in grey; ∅ is the mechanical angle of rotation of the mirror. The horizontal black line indicates the optical axis of the scanning optics. The amount by which the MEMS mirror rotation affects the beam angle after the MEMS depends on the axis of rotation. Since the angle formed by the incident and reflected rays are equal when measured with respect to the normal of mirror, the reflected beam angle in the x direction (𝜃𝑥) is simply double the rotation of MEMS mirror about the y axis (∅𝑥). This is the situation illustrated in Figure 4.3 and described by Equation 4.2. 𝜃𝑥 = 2∅𝑥 (4.2) 69 In order for the beam to be reflected in the y direction however, the MEMS mirror rotates about the x = z axis, which is at 45 degrees to the beam. Using vector algebra, it can be found that the angle of the reflected beam (𝜃𝑦) is √2 times the angle of rotation of the MEMS mirror (∅𝑦) as described by Equation 4.3. This is equation is only valid for small angles. 𝜃𝑦 = √2∅𝑦 (4.3) The rotation of the MEMS mirror as a function of input voltage is given in Figure 4.4 as per the product specification sheet. The MPM delivers a sinusoidal voltage waveform for scanning in the x direction and a sawtooth waveform for scanning in the y direction [10]. The peak voltage of both waveforms is ±75 V relative to a 70 V bias. The resulting angle, ∅, depends on the axis of rotation and the direction of rotation in addition to the applied voltage. Figure 4.4 MEMS angle of rotation versus applied voltage [55] 70 The angle of reflection of the MEMS mirror is experimentally confirmed by shining the reflected laser beam onto grid paper located about one and a half meters from the MEMS mirror. The position of the laser is marked on the paper, from which the beam angle can be calculated. A comparison of computed and experimentally determined values of ±θx and ±θy is presented in Table 4.2. The computed and experimental results agree within error bounds. Table 4.2 Beam Angle after MEMS Scanning Direction Applied Voltage (V) Mechanical angle ∅ from specification sheet (deg.) Computed optical angle θ after MEMS (deg.) Experiment optical angle θ after MEMS (deg.) +x +75 1.76 ± 0.05 3.49 ± 0.10 3.59 ± 0.04 -x -75 -1.70 ± 0.05 3.40 ± 0.10 3.47 ± 0.04 +y +75 1.69 ± 0.05 2.39 ± 0.07 2.29 ± 0.04 -y -75 -1.69 ± 0.05 2.39 ± 0.07 2.33 ± 0.04 The FOV is experimentally confirmed by imaging a silicon chip that is secured to a translation stage and translating a feature from one edge of the image to the opposite edge. The translation stage is equipped with Vernier markings on its thumbscrews that allow the position to be read to 1 µm. Experimental, calculated, and Zemax simulated FOV results are compared in Table 4.3. Calculated values are computed using Equation 4.1 using values of fscan = 20.73 mm, ftube = 60.0 mm and fobj = 3.658 mm. Focal length values are determined for the scan lens and objective lens using Zemax as they are compound lenses. Zemax is also used to confirm that the nominal 60 mm focal length of the tube lens is correct for operation at 790 nm. Simulated FOV results are determined using Zemax. Results agreed with each other within error. The FOV ratio of 156 µm to 102 µm is consistent with the 512-pixel to 328-pixel aspect ratio of the images produced by the 71 MPM and is comparable to the FOV of endoscopes in literature that use the 0.8 NA GRIN objective (GT-MO-080-018-810, GRINTech). The FOV is limited by the scanning angle of the MEMS mirror and also by its diameter, which is much smaller than the back aperture of the objective and thus the scan and tube lens need to expand the beam at the expense of the FOV. Table 4.3 FOV Direction of tilt Full FOV (µm) Computation by Equation 4.1 Simulation by Zemax Experimental ± x without CO (θx = +3.59° / -3.47°) 156 ± 1 156 ± 1 156 ± 2 ± x with CO (θx = +3.59° / -3.47°) N/A 155 ± 1 156 ± 2 ± y without CO (θy = +2.29° / -2.33°) 102 ± 1 102 ± 1 102 ± 2 ± y with CO (θy = +2.29° / -2.33°) N/A 101 ± 1 102 ± 2 4.1.3 Depth-Scanning Range Depth-scanning is achieved by translating the aspherical lens in the objective [40, 41, 56, 57]. The SMA can translate the aspherical lens by 370 µm, which corresponds to a 440 µm focal shift in tissue as a result of refraction at the front and back surfaces of the PCX which respectively have an air/glass and a glass/tissue interface. In other words, the focus moves 1.2 µm for every 1 µm translation of the aspherical lens. This ratio is determined via Zemax simulation and is also confirmed experimentally [40]. The depth scanning range of 440 µm is larger than the scanning range of any of the MPM systems listed in Table 1.1 and is beyond the typical maximum penetration depth in tissue of 200 µm to 300 µm achieved for 2PEF and SHG imaging of tissue 72 [23]. The SMA/aspherical lens is spaced relative to the PCX lens to achieve possible depths of focus from the surface of the PCX to a depth of 440 µm in tissue. 4.2 Resolution Imaging resolution of MPM is determined by the PSF. In the case of 1PEF, the distance from the peak intensity to the first zero crossing of the diffraction pattern (Figure 2.1) is one model that is used to determine the theoretical resolution and can be calculated using Equations 1.1 and 1.2 for lateral and axial directions respectively. In the case of two-photon imaging, resolution is improved by a factor of √2 since the signal intensity is proportional to the square of the excitation intensity [5]. However, resolution is typically determined experimentally by measuring the FWHM, which is different than the distance from the peak intensity to the first zero crossing, and is a potential reason why the experimental and calculated values may not match for a diffraction-limited MPM. Equations derived specifically to determine two-photon lateral and axial resolution according to the FWHM have been determined by fitting the square of a simulated PSF to a Gaussian function [52]. The lateral FWHM of the fitted profile is calculated according to Equation 4.4 for objectives with an NA <= 0.7 and according to Equation 4.5 for NA > 0.7 [52]. 𝐹𝑊𝐻𝑀𝑥𝑦 = √2𝑙𝑛20.320𝜆𝑁𝐴 for NA<=0.7 (4.4) 𝐹𝑊𝐻𝑀𝑥𝑦 = √2𝑙𝑛20.325𝜆𝑁𝐴0.91 for NA> 0.7 (4.5) The FWHM of the fitted profile in the axial direction is calculated according to Equation 4.6 [52], 73 𝐹𝑊𝐻𝑀𝑧 = √2𝑙𝑛2 [0.532𝜆𝑛 − √𝑛2 − 𝑁𝐴2] (4.6) where n is the index of the refraction of the tissue being imaged. The resolution of the MPM is experimentally measured by imaging 100 nm fluorescent beads (19774, Polysciences) that are suspended within a gelatin matrix and then calculating the FWHM of a Gaussian curve fitted to the cross-sectional intensity of a bead. Beads at various depths across the entire depth-scanning range of 440 µm and for field positions of up to 40 µm are measured. Imaging is done at a FOV smaller than maximum possible in order to increase the image’s sampling rate to more accurately measure the FWHM. The measured FWHM of beads as function of depth and field position are presented in Figure 4.5 for both with and without the CO in place. Measured values are fairly consistent regardless of depth and field position, indicating that the aberration level is fairly constant across the imaging range. This suggests that SA, which has plagued previous depth-scanning attempts, is being adequately controlled. This is also in contrast to the performance of endoscopes in literature, which report a decrease in resolution as the imaging depth is moved away from an optimal depth, though quite often only a single resolution value is reported [18, 22, 24]. For example, A. Li et. al. reports a single axial resolution value of 6.11 µm for their endoscope, which is taken at a depth of 200 µm; lateral resolution is reported as 0.83 µm at a depth of 200 µm and increases to almost 1.2 µm at a depth of 300 µm. It’s likely that the axial resolution has a similar decrease in resolution. In 74 comparison, our custom object has a fairly consistent resolution across its 440 µm depth-scanning range of ~0.65 µm laterally and ~10 µm axially. Figure 4.5 Lateral (a,c) and axial (b,d) FWHM values of the intensity profiles of individual 100 nm beads plotted against their radial field position relative to the image center (a,b) and their depth (c,d). The mean of the measured FWHM values is presented in Table 4.4; measurement errors correspond to respective standard deviations of the measured values. The CO are observed to slightly increase both lateral and axial resolution. In comparison to the measured resolution values of endoscopes listed in Table 1.1, the measured axial resolution is higher than all 2PEF values of (a) (b) (c) (d) 75 endoscopes with an NA less than or equal to 0.7. The measured lateral resolution is higher than all 2PEF values including those utilizing the 0.8 NA GRIN lens (GT-MO-080-018-810, GRINTech), except for where Z. Qin, et al used AO[19]. The higher resolution is achieved despite our objective having a slightly lower NA of 0.58. Table 4.4 MPM Resolution MPM Configuration Lateral Resolution (µm) Axial Resolution (µm) Experiment w/o CO 0.69 ± 0.06 11.1 ± 1.3 Experiment w/ CO 0.65 ± 0.05 9.9 ± 2.0 1PEF Resolution from Equations 1.1 and 1.2 divided by √2 0.59 (NA = 0.58) 4.4 (NA = 0.58) 2PEF Resolution from Equations 4.4 and 4.6 0.51 (NA = 0.58) 3.7 (NA = 0.58) Theoretical resolution values as computed using either of the models in Equations 1.1 and 1.2, which are based on the distance between the peak and zero-crossing of a PSF or on Equations 4.4 and 4.6, which are based on the FWHM are not attained. This indicates that the MPM is not operating at the diffraction limit. The fairly large difference in axial resolution suggests that significant improvement to resolution and the quality of images shown in Section 4.4 is possible. In our case, the presence of on-axis coma is identified by comparing the diffraction pattern of a large bead cluster shown in Figure 4.6(a) to known diffraction pattern of aberrations [58, 59]. On-axis coma is a result of the misalignment of optical lenses [13]. Therefore, the misalignment of lenses is believed to be the reason why diffraction-limited performance is not attained. Figure 4.6(a) is post-processed by increasing its brightness and contrast to reveal the diffraction pattern that is otherwise hidden. The diffraction pattern is only observed at the large bead cluster due to the large signal it generates; the signal from the 100 nm beads is too small for the diffraction pattern 76 to be observed in the image. The image is taken with the CO in place; however, a similar pattern at a bead cluster is also observed without the CO. Possible means for increasing resolution and reducing aberrations are given in Section 5.2. On a separate occasion, on-axis astigmatism is observed when imaging 500 nm beads as shown in Figure 4.6(b) [60]. Prior to imaging the beads on this occasion, the collimated beam after the tube lens is observed to be distorted, indicating particularly poor alignment. Consequently, this image of on-axis astigmatism is not indicative of the capability of the MPM, though speaks to the degradation in performance and possible aberration present when alignment is poor. The vertical and horizontal lines are separated by about 24 µm in depth and correspond to the tangential and sagittal focal lines present in astigmatism. The axial resolution is measured to be ~15 µm in this case; significantly worse than the resolution obtained with better alignment (Table 4.4). Figure 4.6 (a) On-axis coma observed at a bead cluster located inside of yellow circle. (b) On-axis astigmatism observed that is observed when alignment is particularly poor. The FOV is 51 μm by 78 μm. (a) (b) 77 Changes to the system’s SNR when adding the CO are evaluated by comparing the average peak pixel intensity of beads with and without CO in place. The comparison, revealed that for the same excitation laser power, no difference in peak pixel intensity is observed. 4.3 Tolerance Analysis In practice, lenses contain small manufacturing deformations and are misaligned to some degree within the optical setup which can affect the ability to focus light [45]. Perturbated optical setups are simulated during tolerancing to provide insight into the anticipated performance and to identify tolerancing parameters that have the biggest impact on performance [13, 46]. The manufacturing tolerances of the lenses are obtained from their respective specification sheets wherever possible. A summary of relevant specifications is shown in Table 4.5 for lenses obtained from Edmund Optics and Lightpath. Lens radius and surface irregularity is checked by counting the fringes (Newton’s rings) that result when the lens is placed next to a test plate and illuminated by a mercury lamp [13]. Lens wedge refers to the angle between the mechanical and optical axis of a lens [13]. The lenses are all of high quality. The spherical lenses from Edmund Optics are manufactured with precision quality. The aspherical lens from Lightpath is originally made for optical data storage applications thus requiring a high level of performance and is advertised as being diffraction limited. 78 Table 4.5 Lens Manufacturing Tolerances Parameter Edmund Optics (Spherical Lenses) Lightpath (Aspherical Lens) Radius 3 fringes 3 fringes 1 Thickness ± 0.05 mm ± 0.05 mm Lens Wedge ± 2 arcmin 2 ± 4 arcmin 1 Surface Irregularity 0.5 fringes 1 fringe 1, 3 1 Obtained from https://www.lightpath.com/capabilities/fabrication/ 2 Zemax preset value for Edmund Optics precision lenses 3 Cannot be modeled for aspherical surfaces in Zemax The tolerance with which lenses are positioned in assemblies and the tolerance of the positioning of the assemblies within the MPM setup depends on the manufacturing tolerance of the mounts and the alignment techniques used. A summary of estimated assembly tolerances of lenses based on empirical experience is given in Table 4.6. Table 4.6 Assembly Tolerances Components Parameter Tolerance Airspace Lens Decenter Lens Tilt Scan Lens Assembly, Tube Lens CO Assembly, Objective Assembly ± 100 µm ± 50 µm ± 0.5° degrees Lenses within Scan Lens Assembly ± 50 µm ± 50 µm ± 0.24 degrees Lenses within CO Assembly ± 50 µm ± 25 µm ± 0.32 degrees / ± 0.16 degrees* Aspherical Lens within Objective Assembly N/A ± 100 µm ± 0.67 degrees PCX Lens within Objective Assembly N/A ± 50 µm ± 0.95 degrees * ± 0.16 degrees for PCV lens closest to the objective, ± 0.32 degrees for other lenses Lens assemblies are mounted to an optical cage system to achieve a common axis. The optical components are positioned axially using the MPM’s laser in conjunction with an alignment disk 79 for alignment according to internal laboratory techniques; it is estimated that the axial precision is 0.1 mm. The decentration and tilt of the assemblies depends on the manufacturing tolerances of the optical cage system parts, and are used to infer alignment tolerance values. A 3D printer (Form3, Formlabs) is used to print the mounts in the objective that hold the aspherical lens, SMA, and PCX lens (Figure 4.7(a)), as well as the mount for the CO, which is made of a sleeve and spacers (Figure 4.7(b)). Tolerances are estimates based on the 3D printer’s resolution of 25 µm. For lenses whose position depends on multiple printed parts, the combined tolerance of the printed parts is used. In general, the tilt tolerance of lenses is estimated to be the arctangent of the ratio of the axial tolerance to the lens diameter. The tilt of the aspherical and PCX lenses is checked by eye; resulting in these lenses having a higher tilt tolerance. Figure 4.7 (a) The objective assembly is made by attaching the aspherical lens and PCX lens to the SMA using 3D printed parts. Printed parts and lenses are glued in place. [40]. (b) The CO assembly is made by securing lenses within a 3D printed sleeve; their spacing is controlled using 3D printed spacers. In the case of the scan lens, which is composed of an achromatic doublet pair, lenses are inserted from both ends of the mounting tube. This means that during fabrication, the tube is removed from (a) (b) 80 its chuck, flipped around and realigned prior to cutting the second hole. Therefore, the tolerance is estimated to be twice the resolution of precision machining. Sensitivity analysis is performed whereby the system’s optical performance is assessed by perturbing (changing) the value of each parameter one at a time according to the tolerance value and computing the increase in RMS wavefront error. The induced RMS wavefront error of the worst offending perturbations is shown in Figure 4.8. Sensitivity results with and without the CO are almost the same. The analysis reveals that the MPM performance is particularly affected by the alignment of the lenses within the objective and their manufacturing tolerance; as such improving the tilt and centration of the lenses within the objective is critical for improving performance. The top five sources of error are due to alignment of lenses; the sixth largest source of error is the manufacturing tolerance of the aspherical lens’ wedge that induces an RMS wavefront error of 0.05 waves, which is almost sufficient to increase the total RMS wavefront error above the 0.072 wave diffraction limit on its own. The tilt on the first achromatic doublet lens in the CO is the main source of tolerancing error from the CO and induces an RMS wavefront error of 0.05 waves, which negates the RMS wavefront error reduction offered by the CO. 81 Figure 4.8 Induced RMS wavefront error from the worst offending parameter perturbations. The estimated total RMS wavefront error according to sensitivity analysis over a half FOV of 0.092 mm is given in Table 4.7. The total RMS wavefront error is estimated by taking the root sum square of the individual errors and the nominal wavefront error [29]. The analysis predicts that total RMS wavefront error is well above the diffraction limit of 0.072 waves and that the total error is equal to the RMS wavefront error induced from tolerancing. It also predicts slightly lower performance with the CO than without, which indicates that the error induced from tolerancing the components of the CO is slightly more than the reduction in error that the CO are designed provide. Table 4.7 Estimated Performance via Sensitivity Analysis Optical system without CO Optical system with CO RMS Wavefront Error without Tolerancing Error (waves) 0.043 0.028 Estimated Induced RMS Wavefront Error due to Tolerancing (waves) 0.34 0.35 Estimated Total RMS Wavefront Error after Tolerancing (waves) 0.34 0.36 82 Overall performance is statistically estimated via Monte Carlo analysis for the case with CO and the case without. In each trial, the values of individual parameters are perturbed within their respective tolerance range according to a normal distribution. Analysis is done to determine the system’s RMS wavefront error and RMS spot size. The mean and standard deviation of both metrics is presented in Table 4.8. It shows that the sensitivity analysis, which combines errors according to the root sum square, overestimated the overall effect of perturbations. Two-photon lateral resolution is estimated by dividing the RMS spot size by √2. Measured resolution is higher than the estimated mean, though lies within the estimated error. The error is rather large at about 35% of the mean, which indicates that other assemblies may perform significantly better or worse than the current assembly. Monte Carlo analysis also predicts the performance to be worse with the CO than without, which is consistent with the sensitivity analysis results but contrary to measurement results. Presumably the degradation in wavefront error related to tolerancing the CO is greater than the improvement that they are designed to provide. Monte Carlo and sensitivity analyses however, do not take into account improvements in temporal focusing, and may be why the measured resolution is better with the CO in place than without. Table 4.8 Estimated Performance via Monte Carlo Analysis Optical system w/o CO Optical system w/ CO RMS Wavefront Error (waves) 0.15 ± 0.05 0.16 ± 0.06 RMS Spot Size (μm) 1.1 ± 0.4 1.2 ± 0.5 Estimated Two-Photon Resolution (μm) 0.8 ± 0.3 0.8 ± 0.3 Tolerancing analysis shows why diffraction-limited focusing is not achieved experimentally as both the sensitivity analysis and Monte Carlo analysis are above the diffraction limit of 0.072 waves. As shown in Figure 4.8, any of the top three sources tolerancing error: aspherical lens tilt, 83 aspherical lens decenter, and PCX tilt each contribute more RMS wavefront error on their own than the 0.072-wave diffraction limit. Thus, any of these perturbations is sufficient to prevent diffraction-limited focusing on their own. As such, improving their alignment in future assemblies is critical toward achieving better resolution. Active alignment techniques, such as passing a laser beam through the optics during assembly to evaluate lens alignment before they are secured in place could significantly improve their alignment and improve overall system performance. For example, reducing the PCX lens tilt, the aspherical lens tilt and the aspherical lens decenter by half reduces their individually induced RMS wavefront error by about half and would reduce the estimated overall RMS wavefront error to 0.23 waves with CO and to 0.21 waves without CO as per sensitivity analysis. Monte Carlo analysis, which perturbs parameters according to a normal distribution across the tolerance range rather than by the tolerance value limit as is the case in sensitivity analysis, predicts RMS wavefront error at 0.075 ± 0.028 waves with CO and 0.097 ± 0.027 waves without CO, showing better performance with the CO that is almost diffraction limited. In the hypothetical case of perfect alignment for the entire optical set-up, some error from lens manufacturing tolerance values will remain. This presents a best-case scenario of what is achievable by improving alignment. When only including lens manufacturing-based tolerances, sensitivity analysis estimates a total RMS wavefront error of 0.085 waves for both cases of with or without CO. In comparison, Monte Carlo analysis suggests that diffraction-limited resolution would be achieved as the predicted RMS wavefront error is 0.043 ± 0.009 waves with CO and 0.053 ± 0.007 waves without CO and showing improved performance by using CO. The 84 aspherical lens’ manufacturing wedge tolerance value of ±4 arcmin induces a relatively large amount of RMS wavefront error of 0.05 waves on its own, and thus the actual lens wedge value would largely determine performance in the case of perfect alignment. Adaptive optics could be used to improve performance and achieve diffraction-limited focusing while allowing somewhat looser alignment tolerances. Tolerance analysis should be conducted prior to the selection of adaptive optics hardware to ensure that they are able to provide sufficient wavefront error correction. 4.4 Images of Biological Samples The imaging capability of the MPM are demonstrated by the ex-vivo imaging of biological samples. Images from the sample surface and from below the surface are obtained. In samples where both 2PEF and SHG signals are detected, images from each modality are given a false color and then combined to produce a bimodal composite image. The imaging of biological samples with CO in place is done for the setup shown in Figure 4.1. imaging of biological samples without CO was done earlier in my research and used a different PPLN collimation lens (CL2) and scan lens (SL) [61]. A pair of achromatic doublets (55-271, f = 25 mm, Edmund Optics) with an effective focal length of 13.3 mm at 790 nm are used as the scan lens. Additionally, an aspherical lens (A375TM-B, f = 7.5 mm, Thorlabs) is used as collimating lens CL2 after the PPLN. The net result is that size of the beam illuminating the objective is not changed; however, the FOV is smaller and measures 64 μm by 100 μm instead of 102 μm by 156 μm [61]. 85 Mouse kidney and ear tissues are imaged ex-vivo using the MPM setup without the CO in place to acquire image stacks, with images taken every 1.2 μm. Representative bimodal, label-free images from the three-dimensional stacks are shown in Figure 4.9. The 2PEF signal reveals tissue structure below the tissue surface and collagen is observed by the SHG signal near the surface. Images at depths up to 150 μm/110 μm are acquired for the kidney/ear tissues respectively. Renal tubules and renal cortical interstitium can be seen by the 2PEF signal in the kidney. A honeycomb structure is revealed by the 2PEF signal in the mouse ear. Attenuation induced by light scattering and absorption in tissue reduces excitation light strength at deeper depths, thus decreasing image quality due to a lower SNR. This limits the usable depth range over which images from tissue can be acquired to well below the scanning range of the SMA. Figure 4.9 Bimodal images obtained ex-vivo of mouse without the CO in place (a) inner ear, (b) kidney. Image depth is indicated on each image. 2PEF is shown in red and SHG in green. The FOV is 64 μm by 100 μm; the scalebar is 16 μm [61]. (b) 0 μm 30 μm 60 μm 90 μm 120 μm 150 μm (a) 0 μm 24 μm 43 μm 67 μm 86 μm 110 μm Kidney Inner Ear 86 Images of mouse kidney, colon and small intestine obtained with the CO in place are shown in Figure 4.10 and demonstrate the depth penetration ability of the MPM. Images are obtained using ~30 mW of power as measured in front of the objective, which is also the case for images obtained without the CO. A maximum penetration depth of 180 μm is achieved when imaging the small intestine. A lesser penetration depth of 90 µm is achieved when imaging mouse tissue with the CO in place, which is less than the 150 µm depth achieved without the CO. Imaging is done using different kidneys, which may be why a different depth is achieved. Collagen is imaged near the tissue surface by the SHG signal; the fibrous nature of the collagen is seen in Figure 4.10(a,c) of the kidney and small intestine. A honeycomb structure is imaged by the 2PEF signal in the colon (Figure 4.10(b)). Sub-surface images of the renal cortical interstitium and renal tubules are obtained by the 2PEF signal in the kidney [22]. Images obtained both with and without CO are fairly grainy. The addition of the CO did not result in an observed increase in image quality, which is to be expected when considering that both setups have comparable experimental resolution. 87 Figure 4.10 Bimodal images obtained ex-vivo of mouse tissue with the CO in place (a) kidney, (b) small intestine and (c) colon. Image depth is indicated on each image. 2PEF is shown in red and SHG in green. The FOV is 102 µm by 156 µm; the scalebar is 25 µm. Selected regions of a yew leaf are imaged to reveal surface and sub-surface cells using the 2PEF signal. The MPM shown in Figure 4.11 with CO is used to obtain images. Images from the leaf center reveal stomata and papillae on the leaf surface and closed guard cells that are recessed from the surface and are at the center of the stomata are shown in Figure 4.11(a). Images from near the leaf margin and midrib are shown in Figure 4.11(b,c) and reveal a second layer of cells below the surface cells. A ‘shadow’ of the cell wall from surface cells is seen on the second layer of cells due to the absorption and scattering of photons. Image intensity is strong throughout the FOV even (a) 0 μm 18 μm 36 μm 54 μm 72 μm 90 μm (b) 0 μm 18 μm 36 μm 60 μm 84 μm 108 μm (c) 0 μm 36 μm 72 μm 108 μm 144 μm 180 μm Small Intestine Colon Kidney z 88 at image corners that are ~90 um from the image center, this is in contrast to images from the endoscope of A. Li, et al. which uses the 0.8 NA GRIN lens (GT-MO-080-018-810, GRINTech), which show a strong decrease in intensity at ~50 µm from the image center [22]. Figure 4.11 2PEF images obtained from a yew leaf (a) at the leaf center (b) near the leaf margin, (c) leaf midrib. Image depth below the surface is indicated on each image. The FOV is 102 µm by 156 µm; the scalebar is 25 µm. 4.5 Conclusions The objective and CO are successfully integrated into our lab’s MPM. Experiments are carried out to characterize the performance of the designed objectives. The resolution is found to be fairly (a) 0 μm 6 μm 12 μm 18 μm 24 μm 30 μm Leaf (b) 0 μm 12 μm 24 μm 48 μm 72 μm 96 μm (c) 0 μm 18 μm 36 μm 54 μm 72 μm 90 μm Leaf Midrib Leaf Margin \ Leaf Center 89 consistent regardless of the imaging depth, indicating that SA is being adequately minimized by the objective. Our MPM achieved a high resolution across the 440 µm depth scanning range, without any degradation in resolution, thus achieving the research objective of enabling depth scanning over a large depth-scanning range and improving upon the abilities of state-of-the-art endoscopes in literature, which have a depth-scanning range of ~200 µm and suffer from a significant decrease in resolution at the limit of their depth-scanning range. However, the measured resolution is lower than the theoretical limit, indicating that diffraction-limited performance is not being achieved and is most likely due to the presence of on-axis coma from misaligned lenses. Tolerance analysis revealed that focusing performance is particularly sensitive to the alignment of the lenses within the objective and the objective as a whole. Their misalignment accounts for much of the wavefront error present in the MPM according to the analysis. The addition of the CO provides a minor improvement in resolution, which is possibly due to improved temporal focusing. The capabilities of the MPM and objective are also demonstrated by imaging biological tissues to a depth of ~180 µm below the surface of the tissue. Scattering and absorption of photons limit the penetration depth, which largely depends on the tissue being imaged. No drop-off in image intensity at the image corners is observed at ~90 µm from the image center, this is in contrast to GRIN based endoscopes, which may have a marked decrease in image intensity near the edge of their FOV. 90 Chapter 5: Conclusion 5.1 Summary MPM is a promising imaging technique capable of acquiring images from below the surface of tissue. Clinical adaptation of the technique requires that the objective is capable of depth-scanning and is also small in size. The MPM imaging quality is dependent on being able to focus pulsed laser light to as small of a spot as possible; this ability is compromised by the presence of optical aberrations. SA and chromatic aberration are of particular concern as they affect the whole FOV. SA may be induced when changing the focusing depth during depth scanning. A potentially large source of SA is sample induced SA, which is a caused by the difference in index of refraction between tissue and the surrounding medium, thus limiting the depth scanning range. Chromatic aberration is caused by the wavelength dependance of index of refraction and is of particular concern for high NA lenses that are used in MPM. Chromatic aberration also causes marginal rays and on-axis rays to arrive at the focal point at different times, which reduces both SNR and resolution. Neither SA during depth scanning nor chromatic aberration is being thoroughly addressed in published MPM endoscope designs, which formed the basis for the motivation of my research. The optical design of a custom water-dipping objective that has a high NA and is capable of depth-scanning is presented in Chapter 2. The objective is made of an aspherical lens mounted in an SMA and located in front of a PCX lens. The SMA performs depth-scanning by translating the aspherical lens, while the PCX is in contact with the tissue being imaged. A thin water layer 91 between the PCX lens and tissue minimizes sample induced SA. The radius and thickness of the PCX lens are chosen so that SA remains low during depth-scanning as the depth of the focal point below the tissue surface is changed. The optical design allows for diffraction-limited focusing throughout a 1400 µm depth-scanning range; however, the range is limited to 440 µm by the SMA. Simulations showed that the size of the diffraction-limited FOV is primarily affected by the choice in aspherical lens and also by the choice of scan lens. By using an achromatic doublet pair in place of a single achromatic lens as the scan lens, off-axis aberrations are reduced and a wider diffraction-limited FOV is possible. The effect that chromatic aberration has on the focusing ability of the custom objective is investigated and the design of CO to reduce chromatic aberration is presented in Chapter 3. The 3dB wavelengths of the 80 fs laser pulses are found to focus ~1.1 µm from the central wavelength; this difference is minimized to 0.3 µm with the addition of CO. Due to the distance between the objective and the CO, lateral color, which affects off-axis focusing, is increased. The increase in lateral color however only becomes apparent at a very wide FOV. The CO design also considers its effect on monochromatic aberrations; both coma and astigmatism, which affect off-axis focusing, are reduced by the addition of CO. A diffraction-limited FOV of up to 300 µm is attained by the objective with the CO in place. The reduction in axial chromatic aberration by the CO also reduces the propagation time difference between marginal rays and the on-axis ray from 58 fs without the CO to 15 fs with the CO. The propagation time difference is comparable with the pulse width for the focusing system without the CO, which can reduce the SNR in MPM imaging. With the CO, the propagation time different is significantly reduced and thus has the potential to improve 92 SNR. The time difference is dependent on the square of an objective’s NA and so the higher the NA, the greater the effect on image quality. The custom objective and CO are integrated into an MPM system whose laser emits 80 fs pulses at 1580 nm that are frequency doubled to 790 nm via a PPLN as described in Chapter 4. Optics are chosen to minimize aberration, avoid vignetting and also to maximize the FOV, power throughput and beam expansion. Some of these goals are conflicting, for example increasing beam expansion reduces power throughput and may cause the FOV to be decreased, thus a compromise between goals is needed. Equations relating the MEMS scan angle, beam angle and FOV are presented and are shown to agree with experimental results. The FOV is 102 µm x 156 µm, which is comparable to endoscopes in literature utilizing the 0.8 NA GRIN objective (GT-MO-080-018-810, GRINTech). The FOV of the MPM is limited by the scanning angle of the MEMS mirror and its diameter. The experimental performance of the MPM is analyzed in Chapter 4 by the imaging of 100 nm fluorescent beads. The objective is able to image beads across the entire 440 µm depth-scanning range, which is limited by the range of the SMA. Resolution measures 0.69 µm ± 0.06 µm laterally and 11.1 µm ± 1.3 µm axially without CO, and 0.65 µm ± 0.05 µm and 9.9 µm ± 2.0 µm with CO. Lateral and axial resolution values remain constant with respect to imaging depth and FOV, which is contrast to state-of-the-art endoscopes that have a much smaller depth-scanning capability of ~200 μm. They also suffer from a significant decrease in resolution at the limit of their depth-scanning range such that their axial resolution is halved or that sub-micron lateral resolution is no longer achieved. 93 On-axis coma is observed to be present and is suspected of being the primary reason for not achieving theoretical resolution values of 0.59 µm (lateral resolution) and 4.4 µm (axial resolution). Tolerance analysis reveals that the alignment of the objective and the lenses within are responsible for most of the tolerance related performance degradation. Tolerance analysis shows that diffraction-limited performance will not be achieved by the MPM and estimates lateral resolution to be 0.8 ± 0.4 µm both with and without CO. However, even in the hypothetical case of much improved alignment, diffraction-limited performance may not be possible due to the large amount of RMS wavefront error induced by the aspherical lens’ wedge manufacturing tolerance. The depth-scanning ability of the objective is also demonstrated by imaging mouse tissue ex-vivo. Maximum penetration depth in the range of 90 µm to 180 µm are obtained, depending on the tissue being imaged and is limited by the scattering and absorption of light. Image intensity remained strong throughout the entire FOV, which is in contrast to images from GRIN lenses, which show a degradation in intensity at the edge of their FOV. 5.2 Future Work The objective’s imaging performance and suitability for use in endoscopy can be improved. Imaging performance can either be improved by reducing aberrations, increasing the NA, increasing beam expansion, or a combination thereof. Experimental resolution measurements indicate that the objective is not operating at the diffraction limit. By examining images of fluorescent beads, the presence of on-axis coma is observed. 94 According to tolerance analysis, the misalignment of the objective and lenses within are the major cause of performance degradation. Using a microscope or active alignment techniques during the assembly of the objective to check / evaluate alignment and improve the tilt and decenter of its constituent lenses could lead to improved performance. The implementation of AO could reduce wavefront error to a sufficient extent to achieve diffraction-limited focusing. A sensorless approach could be used to minimize the increase in size of the MPM. The only additional hardware required for a sensorless approach would be a deformable mirror or spatial light modulator to modulate the wavefront and lenses to expand the beam to the size needed by the wavefront modulator [62]. A possible sensor-less approach involves imaging a bead and changing the wavefront modulation via a deformable mirror or spatial light modulator to maximize its peak intensity [62]. Once the optimal modulation at multiple imaging depth is determined; a look-up table of modulation parameters could be established to avoid having to determine the modulation parameters during actual imaging of tissue, thus allowing fast scanning [19]. Increasing the NA of the objective is another approach that could be pursued to improve resolution. Perhaps the largest increase could be achieved by using a higher NA aspherical lens in the objective; a custom lens would most likely have to be fabricated for this purpose. Alternatively, the PCX lens in the objective could be replaced with a custom lens that has a shorter focal length (resulting in a higher objective NA). A custom doublet lens may be best suited for this purpose as this would increase the flexibility of the design to also meet the other design requirement of minimizing SA during depth scanning. Lastly, another alternative is to redesign the CO to be 95 somewhat focusing, which would increase the angle of the marginal ray, and hence increase the overall NA. Further investigation on the potential benefits and ramifications of these approaches is needed. Illuminating the objective with an even greater expanded beam to fully utilize the NA of the objective is another avenue for increased resolution. In the current experimental setup, the beam is expanded so that the 1/e2 diameter fills the back aperture of the objective, by expanding the beam by a factor of √2 , the 1/e diameter would fill the back aperture. According to simulations the FWHM of the lateral PSF of focal spot would be reduced from 0.77 µm to 0.73 µm, a 5% decrease. Lateral resolution is anticipated to improve accordingly. Simulations show that a uniform beam, produces a 0.70 µm PSF, a 9% reduction and represents the maximum possible improvement by increasing the size of the beam. The beam expansion would ideally be performed by a redesigned set of CO. This avoids reducing the FOV, which occurs when expanding the beam using the scan and tube lenses; or expanding the beam at the PPLN and having to use a larger MEMS mirror with a slower scan rate. Regardless of where the beam is expanded, it results in power loss, so a higher input power from the laser is needed to compensate. Further investigation on the effect that the difference in propagation time between marginal rays and on-axis rays has on MPM resolution and SNR is needed to determine the benefit of a further reduction in chromatic aberration. Such an investigation would have application beyond the objective presented in this thesis. 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Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Tang, Shuo
Miniature objective lens design for multiphoton microscopy
Text