@prefix vivo: . @prefix edm: . @prefix ns0: . @prefix dcterms: . @prefix skos: . vivo:departmentOrSchool "Science, Faculty of"@en, "Physics and Astronomy, Department of"@en ; edm:dataProvider "DSpace"@en ; ns0:identifierCitation "Hickson, Paul; Burley, Gregory S. Single-image wavefront curvature sensing. Adaptive Optics in Astronomy, edited by Mark A. Ealey, Fritz Merkle. Proceedings of SPIE Volume 2201, 549, 1994."@en ; ns0:rightsCopyright "Hickson, Paul"@en ; dcterms:creator "Hickson, Paul"@en, "Burley, Gregory S."@en ; dcterms:issued "2011-09-20T18:01:43Z"@en, "1994"@en ; dcterms:description """A single defocused star image contains sufficient information to uniquely determine the spatial phase fluctuations of the incident wavefront. A sensor which responds to the intensity distribution in the image produces signals proportional to the wavefront curvature within the pupil and the radial slope at the pupil boundary. Unlike Roddier's differential curvature sensing technique, a single-image sensor does not cancel intensity fluctuations due to atmospheric scintillation. However, it has been shown that at typical astronomical sites the scintillation signal is negligibly small. A single-image curvature sensor can theoretically achieve a signal-to-noise ratio of order Q approximately equals r20/(lambda) z0 where r0 is Fried's correlation length, (lambda) is the wavelength, and z0 is the root-mean-square distance through the atmosphere, weighted by the refractive index structure constant C2n. This is more than adequate for AO systems whenever D/r0 4<52>1/2' () which is the ratio of variances of the signal produced by curvature and amplitude fluctuations. The angular brackets denote an ensemble average. For a Kolmogorov turbulence spectrum, it is shown4 that (5) /3Z0 where z0 is the C,-weighted RMS line of sight distance through the atmospheric turbulence, and C,is the refractive index structure constant. The curvature sensor is most efficient when the size of a subaperture is matched to the atmospheric seeing. This condition is !=/3a, (6) where 2a is the typical linear dimension of the region, in the entrance pupil, which is imaged onto the subaperture. This gives Q=!9. (7))'zo For astronomical applications, a will be at least as large as ro, which is typically O.3m(\\/O.5um)6/5 at the best astronomical sites7'8. An additional restriction comes from the C term. Observations from Mauna Kea7 indicate that ZO 4 km. Thus we have Q ;: 45(A/O.5um)715 , (8) which shows that the scintillation noise is typically less than about 2% of the curvature signal. It's effect will generally be negligible in comparison to photon noise. While the above analysis refers to measurements of wavefront curvature, is can also be shown4 that a similar result holds for the subapertures which measure the radial slope of the wavefront. The signal from these subapertures is typically larger than those from the curvature-sensing subapertures, while the scintillation noise is somewhat smaller. As a result, scintillation provides an even smaller contribution to the total noise. 4. APPLICATIONS Single image curvature sensors have potential applications in many areas. For active optics (aO) applications, high- spatial-frequency phase information can be obtained by projecting a defocussed star image onto a CCD detector. For integration times exceeding several minutes, atmospheric fluctuations effectively average to zero. Phase errors introduced by the optical system can then be estimated from the intensity distribution in the image. For example, a modal expansion can be made of the wavefront phase, and the expansion coefficients determined by least-squares analysis of the measured intensities9. A related application is for fast guiding of large telescopes. A small CCD can be attached to a guide probe (or illuminated by associated relay optics). This CCD can have a sufficient field of view to locate and acquire guide stars when positioned in the focal plane. By defocussing the image, either by moving the CCD or a relay lens, the CCD becomes a single image curvature sensor. The subaperture geometries are then determined by software. On-chip binning can be used to reduce read-noise. Such a sensor can provide information not only on wavefront slope (tracking error), but also defocus, astigmatism, and higher-order aberrations if desired. SPIE Vol. 2201 Adaptive Optics in Astronomy (1994) / 551 Downloaded from SPIE Digital Library on 20 Sep 2011 to 137.82.117.28. Terms of Use: http://spiedl.org/terms A final application that we will consider is in adaptive optics (AO). In AO systems, high quantum efficiency and low noise are of paramount importance in a wavefront sensor. With no reimaging, beamsplitting, or relay optics, single-image curvature sensors employing high-quantum-efficiency CCD detectors, offer the prospect of very high efficiency. As noted, the detector read noise associated with the measurement is smaller by a factor of than that of a differential system. In typical AO systems, the signal-to-noise ratio is limited by photon noise because bright guide stars are rarely sufficiently close to the object of interest. In this case a single-image sensor should perform at least as well as a differential system. For high signal-to--noise ratio applications, a single-image sensor will ultimately be limited by scintillation noise. In that case, one can show4 that the structure function DAO of the AO corrected wavefront has the form D(r) Q22D(r) (9) where D is the structure function of the uncorrected wavefront. (The structure function described here is the sum of the mean square phase and intensity differences between two points on the wavefront separated by a distance r). For Kolmogorov turbulence, D has the form V(r) = 6.88(r/ro)513 . (10) From this we see that an AO system employing a single-image curvature sensor can theoretically increase the effective correlation length ?O by a factor Q6/. At good astronomical sites this corresponds to a scale which exceeds the diameters of the largest optical telescopes, indicating that scintillation noise will not be a limiting factor in AO applications of single-image curvature sensors. 5. LABORATORY TESTS With a simple optical system involving a short focal length lens and a pinhole illuminated by a light-emitting diode, we have recorded some single beyond-focus images on a small 64 x 64 CCD. The pinhole and CCD were centred on the optical axis of the lens. This arrangement allowed us to measure the primary spherical aberration of the system, which has the form (r) = —44r (11) where B is the Seidel coefficient of spherical aberration. Substituting this expression in Eq. (1), gives /I 4sB2y=r , (12) i.e. , the intensity fluctuation has a quadratic dependence on radius in the pupil. A series of images were produced from a 5 ms exposures followed by the usual CCD bias removal and flat fielding process. After computing the mean intensity Io over the aperture, the curvature map was extracted from 1(r) and on a pixel by pixel basis. Figure 1 shows a cross-section through the center of one of the processed curvature maps. The smooth curve is the best least-squares parabolic fit to the data. It has the form LI/Io —0.6(r/R)2. For the experimental setup used, f = 70 mm, S = 72.8 mm, R = 13.75 mm, and /3 = —0.040. Eq. (12) then gives B = 0.0156 mm which corresponds to 5.7 waves of spherical aberration 6. ACKNOWLEDGEMENTS We are pleased to acknowledge financial support from the Natural Sciences and Engineering Research Council of Canada. 552 ISPIE Vol. 2201 Adaptive Optics in Astronomy (1994) Downloaded from SPIE Digital Library on 20 Sep 2011 to 137.82.117.28. Terms of Use: http://spiedl.org/terms c-O.2(I) 4-' ctj 2: 0 Pixel number Figure 1. Sample cross-section of a 64x64 curvature map with a parabolic fit to the iI/Io curvature signal showing 5.7 waves of spherical aberration. 7. REFERENCES 1. F. Roddier, \"Curvature sensing: a diffraction theory,\" NOAO R&D Note 87-3 (1987). 2. F. Roddier, \"A new concept in adaptive optics: curvature sensing and compensation,\" Appl. Optics 27, 1223- 1225 (1988). 3. F. Roddier, C. Roddier and N. Roddier, \"Curvature sensing: a new wavefront sensing method,\" Proc. Soc. Photo-Opt. Instrum. Eng. 976, 203-209 (1988). 4. P. Hickson, \"Wavefront Curvature Sensing from a Single Defocussed Image,\" J. Opt. Soc. Am., in press (1994). 5. M. Borne and E. Wolf, Principles of Optics, 6th ed. (Pergamon Press Ltd., Oxford, 1980). SPIE Vol. 2201 Adaptive Optics in Astronomy (1994) / 553 —1 0 10 20 30 40 50 60 Downloaded from SPIE Digital Library on 20 Sep 2011 to 137.82.117.28. Terms of Use: http://spiedl.org/terms 6. D. L. Fried, \"Optical resolution through a randomly inhomogeneous medium for very long and very short exposures,\" J. Opi. Soc. Am. 56, 1372-1379 (1966). 7. F. Roddier, L. Cowie, J. E. Graves, A. Songaila, D. McKenna, J. Vernin, M. Azouit, J. L. Caccia, E. Limburg, C. Roddier, D. Salmon, S. Beland, D. Cowley, and S. Hill, \"Seeing at Mauna Kea: a joint UH-UN-NOAO- CFHT study,\" in Advanced Technology Opiical Telescopes IV, L. D. Barr, ed., Proc. Soc. Phoio-Opi. Insirtim. Eng. 1236, 485-491 (1990). 8. B. L. Ellerbroek, \"Adaptive optics performance predictions for large telescopes under good seeing conditions,\" in ESO Conf. on Progress in Telescope and Insfrumeniation Technologies, M.-H. Ulrich, ed., 411-413 (1992). 9. P. Hickson, \"Modal estimation of wavefront phase by means of curvature sensing,\" in preparation (1994). 554 ISPIE Vol. 2201 Adaptive Optics in Astronomy (1994) Downloaded from SPIE Digital Library on 20 Sep 2011 to 137.82.117.28. Terms of Use: http://spiedl.org/terms"@en ; edm:hasType "Conference Paper"@en ; edm:isShownAt "10.14288/1.0107621"@en ; dcterms:language "eng"@en ; ns0:peerReviewStatus "Reviewed"@en ; edm:provider "Vancouver : University of British Columbia Library"@en ; dcterms:publisher "Society of Photo-Optical Instrumentation Engineers (SPIE)"@en ; ns0:publisherDOI "10.1117/12.176090"@en ; dcterms:rights "Attribution-NonCommercial-NoDerivatives 4.0 International"@en ; ns0:rightsURI "http://creativecommons.org/licenses/by-nc-nd/4.0/"@en ; ns0:scholarLevel "Faculty"@en ; dcterms:title "Single-image wavefront curvature sensing."@en ; dcterms:type "Text"@en ; ns0:identifierURI "http://hdl.handle.net/2429/37492"@en .