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Femtosecond laser pulse compression Chin, Roger S. 1991

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F E M T O S E C O N D L A S E R P U L S E C O M P R E S S I O N By Roger S. Chin Sc. (Honours Cooperative Applied Physics) University of Waterloo A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S P H Y S I C S We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A September 1991 (§) Roger S. Chin, 1991 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Physics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract Once the Spectra-Physics Femtosecond Laser System had arrived, it had to be character-ized. For further pulse compression, various techniques had to be considered. The best of these were chosen considering our needs and limitations. First, the Spectra-Physics Femtosecond Laser System is described and its 616 nm laser pulses are characterized. By using an autocorrelation technique based on the nonlinear optical characteristics of a potassium dihydrogen phosphate (KDP) crystal and assuming a particular intensity pulse shape (such as that described by a symmetric exponential decay), the pulse width (full width at half maximum) could be obtained. Assuming a pulse shape described by a symmetric exponential decay function, the "exponential" pulse width was measured to be 338 ± 6 fs. The nominal average power of the 82-MHz modelocked pulse train was 225 mW. The "exponential" pulse energy was 2.7 nJ with a peak pulse power of 2.8 kW. Theoretical calculations for fibre grating pulse compression are presented. Experi-mentally, I was able to produce 68 ± 1 fs (exponential) pulses at 616 nm. The average power was 55 mW. The "exponential" pulse energy was 0.67 nJ with a peak power of 3.4 kW. The pulse compressor consisted of a 30.8 ±0 .5 cm fibre and a grating compressor with the effective grating pair distance of 103.8 ± 1 cm. Various techniques were considered for further pulse compression. Fibre-grating pulse compression and hybrid mode locking appeared to be the most convenient and least expensive options while yielding moderate results. The theory of hybrid mode locking is presented. Experimentally, it was determined that with the current laser system tuned to 616 nm, DODCI is better than DQOCI based ii on pulse shape, power, stability and expense. The recommended DODCI concentration is 2-3 mmol/1. The shortest "exponential" pulse width was 250 fs. The average power was 185 mW. The exponential pulse,.energy was 2.3 nJ with a peak pulse power of 2.6 kW. A n attempt to increase the bandwidth of the laser pulse by replacing the one-plate birefringent plate with a pellicle severely limited the tunability of the dye laser and introduces copious noise. Attempts to reduce group velocity dispersion (responsible for pulse broadening) with a grating compressor was indeterminate, but did result in a slightly better pulse shape. Interferometric autocorrelation is recommended for such a study. A n increase or decrease from the nominal power output of the pulse compressor showed a decrease in pulse compression. m Table of Contents Abstract ii List of Tables viii List of Figures ix Acknowledgement xii 1 Introduction 1 2 The Femtosecond Laser System 3 2.1 The N d : Y A G Laser 3 2.1.1 Krypton Arc Lamp and Pump Chamber 4 2.1.2 Mode Locker 4 2.1.3 Breakdowns 5 2.2 The Pulse Compressor 6 2.3 The Dye Laser 6 2.4 System Performance 7 3 The Autocorrelator 8 3.1 Introduction 8 3.2 Theory 9 3.2.1 Autocorrelation Functions 9 3.2.2 Second-order Autocorrelation 10 iv 3.2.3 Selected Autocorrelation Functions . 13 3.2.4 Application 13 3.3 Construction 15 3.3.1 Components 15 3.3.2 Electronics 17 3.4 Calibration 19 3.4.1 Calibration Procedure 19 3.4.2 Subpicosecond Range 20 3.4.3 Femtosecond Range 20 3.5 Operation 21 3.5.1 Alignment . . • . 21 3.5.2 Electronic Adjustments 22 4 Pulse Compression Techniques 27 4.1 Theory 27 4.2 , Hybrid Mode Locking 29 4.2.1 Theory 29 4.2.2 DQOCI and D O D C I 30 5 Fibre Grating Pulse Compression 31 5.1 Theory 31 5.1.1 Self-focussing and Self-phase Modulation 31 5.1.2 G V D Compensation 34 5.2 Construction 36 5.2.1 Components . 37 5.2.2 Alignment 38 v 6 Measurements 40 6.1 Standard Femtosecond System Measurements . ." 40 6.1.1 Method 40 6.1.2 Results 42 6.1.3 Discussion 42 6.2 Subpicosecond to Femtosecond Pulse Compression 50 6.2.1 Results 50 6.2.2 Discussion 62 6.3 Hybrid Mode Locking 63 6.3.1 Method 63 6.3.2 Results 64 6.3.3 Discussion . ; 65 6.4 Replacement of One-plate Birefringent Filter with a Pellicle 76 6.4.1 Results 76 6.4.2 Discussion 79 6.5 Group Velocity Dispersion in Dye Laser 79 6.5.1 Results 79 6.5.2 Discussion 85 6.6 The Effect of Power on the Fibre-Grating Pulse Compressor 85 6.6.1 Results . 85 6.6.2 Discussion 91 7 Conclusions 92 Appendices ®4 A Start-up Procedure for the Femtosecond Laser System 94 vi A . l N d : Y A G Laser 94 A.1.1 Preliminary 94 A.1.2 N d : Y A G Power Supply 96 A.1.3 N d : Y A G Laser 97 A.2 Pulse Compressor 98 A.2.1 Alignment . . 98 A.2.2 Stabilization 99 A.3 Dye Laser . . . 100 B Shutdown Procedure for the Femtosecond Laser System 102 C Changing Dyes in the Dye Laser 104 D Theoretical Autocorrelation Functions: Proofs 108 D . l Hyperbolic Secant Squared 108 D.2 Gaussian 110 D.3 Lorentzian I l l D.4 Symmetric Two-sided Exponential 112 E The K D P Phase-matching Angle 114 F Curve Fitting 117 F . l Calibration Curve 118 F.2 Minuit Subroutine 121 Bibliography 128 vii List of Tables 3.1 Characteristics of various pulseshape models 14 6.2 Pulse width and autocorrelation standard deviation for various dye laser cavity lengths 48 6.3 Pulse width and autocorrelation standard deviation for selected compressed pulses 50 6.4 Pulse width and autocorrelation standard deviation for various cavity lengths of the dye laser with a pellicle 76 viii List of Figures 3.1 Autocorrelator 24 3.2 Autocorrelator Calibration Curve (subpicosecond range) 25 3.3 Autocorrelator Calibration Curve (femtosecond range) 26 5.4 Fibre-Grating Pulse Compressor 36 6.5 Pulse Width and Autocorrelation Standard Deviation vs. R C time constant 43 6.6 Autocorrelation Power vs. Time, for a very short dye laser cavity length 44 6.7 Autocorrelation Power vs. Time, for a short dye laser cavity length . . . 45 6.8 Autocorrelation Power vs. Time, for an optimal dye laser cavity length . 46 6.9 Autocorrelation Power vs. Time, for a long dye laser cavity length . . . . 47 6.10 Pulse Width vs. Fibre Length and Compression Distance (hyperbolic se-cant squared fit) 51 6.11 Pulse Width vs. Fibre Length and Compression Distance (gaussian fit). . 52 6.12 Pulse Width vs. Fibre Length and Compression Distance (lorentzian fit). 53 6.13 Pulse Width vs. Fibre Length and Compression Distance (exponential fit). 54 6.14 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (hyperbolic secant squared fit). 55 6.15 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (gaussian fit) 56 6.16 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (lorentzian fit) 57 ix 6.17 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (exponential fit) 58 6.18 Autocorrelation Power vs. Time, for the shortest compressed pulse . . . . 59 6.19 Autocorrelation Power vs. Time, for a compressed pulse (fibre too long) . 60 6.20 Autocorrelation Power vs. Time, for a compressed pulse (fibre too short) 61 6.21 Power, Pulse Width and Autocorrelation Standard Deviation vs. DQOCI Concentration (hyperbolic secant squared fit) 66 6.22 Power, Pulse Width and Autocorrelation Standard Deviation vs. DQOCI Concentration (gaussian fit) 67 6.23 Power, Pulse Width and Autocorrelation Standard Deviation vs. DQOCI Concentration (lorentzian fit) 68 6.24 Power, Pulse Width and Autocorrelation Standard Deviation vs. DQOCI Concentration (exponential fit) 69 6.25 Autocorrelation Power vs. Time, for a dye laser with the saturable dye, DQOCI ; 70 6.26 Power, Pulse Width and Autocorrelation Standard Deviation vs. DODCI Concentration (hyperbolic secant squared fit) 71 6.27 Power, Pulse Width and Autocorrelation Standard Deviation vs. DODCI Concentration (gaussian1 fit) 72 6.28 Power, Pulse Width and Autocorrelation Standard Deviation vs. DODCI Concentration (lorentzian fit) 73 6.29 Power, Pulse Width and Autocorrelation Standard Deviation vs. DODCI Concentration (exponential fit) 74 6.30 Autocorrelation Power vs. Time, for a dye laser with the saturable dye, DODCI 75 x 6.31 Autocorrelation Power vs. Time, for a dye laser with a pellicle and an optimal cavity length 77 6.32 Autocorrelation Power (experimental and theoretical) vs. Time, for a dye laser with a pellicle and a long cavity length 78 6.33 Pulse Width and Autocorrelation Standard Deviation vs. Grating Dis-tance (hyperbolic secant squared fit) . 80 6.34 Pulse Width and Autocorrelation Standard Deviation vs. Grating Dis-tance (gaussian fit) 81 6.35 Pulse Width and Autocorrelation Standard Deviation vs. Grating Dis-tance (lorentzian fit) . 82 6.36 Pulse Width and Autocorrelation Standard Deviation vs. Grating Dis-tance (exponential fit) 83 6.37 Autocorrelation Power vs. Time, for the laser system with grating com-pressor 84 6.38 Pirise Width and Autocorrelation Standard Deviation vs. 532 nm Power (hyperbolic secant squared fit) ". 86 6.39 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (gaussian fit) . 87 6.40 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (lorentzian fit) 88 6.41 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (exponential fit) 89 6.42 Autocorrelation Power vs. Time, for 1000 mW average power and a short fibre 90 xi Acknowledgement I would like to thank Dr. Jochen Meyer for his supervision and his guidance , A l Cheuck for his technical support, and Jim Booth and Alan Adams for the use of their equipment. Also I would like to thank Hubert Houtman, Abdul Elizzabi, Ross McKenna, Michel Laberge and Peter Zhu for their help in the lab. I would like to thank Eric Register for his strength and support during the completion of this thesis. Also, I would to thank Dr. Raymond Lam who continues to help me inves-tigate experimental methods for treating my disabling affliction with Hypernychthemeral Syndrome, hoping one day there will be a successful treatment. xn Chapter 1 Introduction The need for shorter laser pulses has been primarily in the area of spectroscopy. Shorter laser pulses result in increased time resolution of spectroscopic data of events such as those in atomic excitation or chemical reactions. Commercial femtosecond laser systems have been available for the last several years. The Spectra-Physics synchronous femtosecond laser system bought by the University of British Columbia (UBC) Plasma Group in Apri l 1988 was the first such commercial system in Canada. Other systems have since come on the market such as the Coherent® Satori™ femtosecond dye laser which compensates for group velocity dispersion (GVD) within the dye laser. Such a system exists at the University of British Columbia 1. Other femtosecond systems in Canada include colliding-pulse mode-locked (CPM) lasers at the National Research Council in Ottawa 2 and the University of Toronto3. A synchronous femtosecond laser system was chosen so that its nominally 400 fs4 pulses may be further compressed to typically 70 fs by a fibre grating pulse compressor and amplified by a regenerative dye amplifier. A n amplified femtosecond laser pulse was required for use in certain experiments within the Plasma Group. In one experiment, it is hoped that the femtosecond pulse can switch out a subpicosecond pulse from the pulse from a C 0 2 laser using the surface of a gallium arsenide (GaAs) crystal [1] as a reflective switch. 1contact Dr. Andrew Ng, U B C Physics, University of Brit ish Columbia, Vancouver, B C V 6 T 2A6 2contact Dr. Paul Corkum, National Research Council, Division of Physics, Ottawa, O N K 1 A OR6 3contact Dr. Geraldine Kenney-Wallace, U. Toronto, Lash Miller Laboratories, Toronto, O N M5S 1A1 4assuming an exponential pulse shape 1 Chapter 1. Introduction 2 Characterization of femtosecond laser pulses necessitated the construction of a di-agnostic instrument to determine its pulse shape and pulse width. A background-free second-order intensity autocorrelator which sufficed with its simplicity was chosen. The fibre grating compressor was constructed while a regenerative dye amplifier was received later. In addition, two other methods of pulse compression were tested: hybrid mode locking and replacement of the dye laser one-plate birefringent plate with a pellicle. Group velocity dispersion in the dye laser and the effect of power on the fibre grating pulse compressor were also investigated. In chapter 6, measurements of the standard femtosecond system with and without the above modifications (ie. pulse compressor, hybrid mode locking, replacement of the birefringent filter with a pellicle) are presented and discussed. Chapter 2 The Femtosecond Laser System The Spectra-Physics femtosecond laser system consists of the N d : Y A G laser, the Pulse Compressor and the Dye Laser. In this chapter, topics specific to the N d : Y A G laser are covered: list of its components, krypton arc lamp replacement, the mode locker and its nominal operating parameters, and its history of breakdowns. Also, an overview of the pulse compressor and the dye laser and notes on overall system performance are presented. 2.1 The N d : Y A G Laser The Spectra Physics Model 3800 C W N d : Y A G Laser System consists of the following: 1. S/P Model 3800 laser head and power supply (S/N 134) 2. S/P Model 3275-20195 acousto-optic stabilizer/Q-switch (S/N 117) • Newport Driver 64027-CL/OSB (S/N 0201) • Newport Remote N64027-CL/QSR (S/N 0188) 3. S/P Model 451-20057 mode locker power supply (S/N 733) 4. S/P Model 452A-46650 mode locker driver (S/N 454) 5. S/P Model 453-46332 mode locker stabilizer (S/N 366) Details on the N d : Y A G laser can be found in the manuals, [2,3]. During day-to-day operation several observations not mentioned in the manuals are noted below. 3 11 . . Chapter 2. The Femtosecond Laser System 4 2.1.1 Krypton Arc Lamp and Pump Chamber During replacement of the krypton arc lamp, several observations were made. There was significant deterioration of the gold coating of the elliptical pump cham-ber. Several (3 or 4) bare spots about one square millimeter in area had developed. On the other hand, deterioration of the krypton arc lamp was nominal and typical of characteristics noted in [4]. Separating the metal electrodes of the lamp from the electrical sockets was difficult. Gently twisting the contact and the metal electrode (not the quartz part of the lamp) solved this problem. Upon separation, grayish dirt was noted between the electrode and socket. Furthermore, replacement necessitated non-trivial re-alignment of the N d : Y A G optical cavity. . . . . . . During the second lamp replacement, the quartz flow tube around the flash lamp was observed to be broken. (Fortunately, the N d : Y A G rod is located upstream from the pump chamber, thereby, saving it from possible damage.) The fracture occurred on the bottom side where a spring had applied stress. The broken section was roughly triangular (1 cm longitudinally x 0.6 cm). The broken tube was replaced with one made by the U B C Physics glass shop. 2.1.2 Mode Locker Dust evaporated on the mode locker windows and/or incorrect Bragg angle can cause every other pulse in the pulse train to be of a different peak power. Therefore, it is necessary to purge dust from the dust tubes with nitrogen. Improper shielding of the laser's light monitor signal can be affected by ambient r.f. from the mode locker producing the above effect. Chapter 2. The Femtosecond Laser System 5 Operating Parameters The mode locker operating parameters are: • frequency: 41.1750 MHz • oven temperature: 7.5 (dial's scale is from 0 to 10) • rf power: full deflection 9.25 (dial's scale is from 0 to 10) • servo power: 75% to 80% • servo error: 0.0 Increasing the r.f. frequency, r.f. power or the oven temperature causes the servo error to swing more negative. This technique was used during initial installation when the steady-state servo error was too positive. Loss of mode locking overnight may have been caused by low ambient temperature. Increasing the r.f. frequency and oven temperature solved this problem. 2.1.3 Breakdowns m Power Supply Phase Detection Interlock The circuit which detects the correct phase of the 3-phase 220 V A C power supply failed Oct 25, 1988. This triggered a safety interlock preventing further operation of the power supply. This was repaired by a Spectra-Physics representative. This repair period left the laser system idle for 6 weeks. As a result, power output was initially low ( « 60% nominal) but returned to normal after 6 days of operation. It was speculated that the lack of circulation of the cooling water caused deposits inside the pump chamber. It is recommended on p.3-29 in the manual [2] that the cooling water Chapter 2. The Femtosecond Laser System 6 be circulated every week. Extended idle periods (greater than two months) necessitate complete draining of the cooling water system. Modelocker Stabilizer The mode locker stabilizer, which compares the mode locker driver's r.f. output with its reflection produced by the acousto-optic modulator, broke down March 28, 1989. The unit was sent to Spectra-Physics in Mountain View, C A , repaired and returned June 23, 1989. 2.2 The Pulse Compressor The Spectra-Physics Model 3695 Optical Pulse Compressor employed the fibre-grating pulse compression technique described in Chapter 5 to compress the N d : Y A G laser's 100 ps pulse to 5 ps. This pulse was shortened further to 3.5 ps by colHnear second harmonic generation (SHG). SHG was performed after pulse compression intentionly to improve the SHG efficiency which improves with peak pulse power. During intial installation, the fibre was shortened to eliminate stimulated Raman scattering. See p.3-18 of the manual [5]. It is suggested that the fibre not protrude too far from nor too close to the fibre chuck (about 3 mm) to prevent excessive thermal stresses on the chuck or the fibre during operation. 2.3 The Dye Laser The Spectra-Physics Model 3500 Ultrashort Pulse Dye Laser was pumped synchronously by the 800 mW output of the frequency doubler crystal in the pulse compressor. The gain dye used was Rhodamine 6G which has an tuning range of 575-635 nm in this system (refer to p.1-7 of the manual [6]). Chapter 2. The Femtosecond Laser System 7 2.4 System Performance The laser loses power (at 532 nm) as each session progresses causing cease of operations after 5 or 6 hours due to insufficient stable power. Since the N d : Y A G signal monitor does not show a decrease in power, it was assumed that this power decrease was due to the pulse compressor or the frequency doubler. Overheating in any of these elements could be the cause. The best average power output of the N d : Y A G was 14 W. The best fibre coupling attained in the pulse compressor was 65%. In general, alignment was easiest in the dye laser and most difficult in the pulse compressor. Iterative adjustments of various controls were necessary in all components making initial installation difficult. Chapter 3 The Autocorrelator In this chapter, a brief history of autocorrelation techniques, the theory of general au-tocorrelation functions and more specifically second-order autocorrelator functions are presented. The choice of a second-order intensity slow autocorrelation for an autocorre-lator was made based on the ability to provide the laser's pulse shape and pulse width despite its simplicity. Furthermore, the contruction, calibration and operation of an autocorrelator based on this autocorrelation are detailed. 3.1 Introduction Direct methods of measuring the pulse width using high-speed opto-electronic devices such as photodetectors and streak cameras have limited resolution in the subpicosecond and femtosecond regions. With the development of laser pulses in these regions, another method had to be developed. As a result, indirect methods based on correlation tech-niques (in particular, second-order autocorrelation using second-harmonic generation) were developed. Two-photon flourescence was popular initially [7,8], but was later replaced by second-harmonic generation (SHG) in nonlinear crystals such as potassium dihydrogen phosphate (KDP) [9]. A n autocorrelator based on this latter technique will be discussed in theory, after which its construction, calibration and operation will be described. 8 Chapter 3. The Autocorrelator 9 3.2 Theory Autocorrelation measurements provide a good technique for inferring the original signal when a direct measurement is not possible. There are different methods for autocorre-lation measurements. Selection of a specific autocorrelation technique is determined by the amount of information required on the original signal and the degree of difficulty tolerated in implementation. A general overview of autocorrelation is covered, featuring interferometric and intensity second-order autocorrelation. The practical application of this latter technique is then discussed. 3.2.1 Autocorrelation Functions As the name implies, autocorrelation is similar to cross-correlation but differs in that the functions or signals being correlated are all the same. There are several types of autocorrelation: background and background-free, fast and slow. The n-th order autocorrelation function of a signal I(t) is given by nntT T — N _ (/ (* )/ (*+ T 1 ) . - - J ( * + T n - 1 ) ) where the time average is defined by ([ ]> = r Km 7} ]dt (3.2) In general, higher orders than second-order are necessary to describe I(t) uniquely. However, the second-order and third-order autocorrelation functions (ie. g2 and g3) are sufficient to describe all higher orders. Thus, I(t) may be determined. Higher orders are more sensitive to pulse asymmetry. Chapter 3. The Autocorrelator , 10 Noise Noise is generated in a laser in several ways. Improper mode locking, vibration, dust or variable dye jet thickness all introduce noise. A laser operating in a large number of randomly phased modes can approximate gaussian or thermal noise. The effect of continuous thermal noise is to produce an autocorrelation trace with a constant background level and a correlation spike at Tj = 0. The contrast ratio between the noise spike and the noise background is (2n)!/(n!)2. The effect of a noise burst equal in intensity to the pulse intensity is to narrow the top half of the noise-free autocorrelation trace producing a substructure or shoulders in the lower half. The width of the spike is determined by the coherence time of the noise burst. 3.2.2 Second-order Autocorrelation For second-order autocorrelation; a Michelson interferometer type setup is used to split the laser pulse into two replicas and recombine them with a variable time delay. For interferometric autocorrelation, the standard Michelson interferometer setup is modified by replacing the output detector as follows. A lens focusses the two collinear output beams onto a nonlinear crystal (positioned at the correct phase-matching angle for SHG). A filter filters out the fundamental frequency while passing the second harmonic which is detected by an appropriate detector such as a photomuliplier. Thus, the superposition of the two electric fields are used to produce the SHG fre-quency. If the path difference is large enough, only one pulse will be present in the crystal at one time. However, in this configuration, this is sufficient to produce SHG since the phase matching condition is still satisfied for each individual pulse. Since the two collinear beams interfere with each other in the crystal, phase information is preserved in SHG. Chapter 3. The Autocorrelator 11 By contrast, for the intensity autocorrelation, phase information is lost. This is achieved by making the two collinear beams in the above setup non-collinear. This can be achieved by replacing a mirror in one of the arms with a right angle prism. The lens still focusses both beams onto the nonlinear crystal, but sum frequency generation does not occur unless both pulses are present in the crystal. This provides a background-free autocorrelation technique without interference (phase) effects. It is strictly an intensity autocorrelation. Interferometric Autocorrelation The interferometric autocorrelation is usually described as a background autocorrela-tion. Its function is denoted by a.subscript, B . Since the two replicas are superimposed, the autocorrelation is performed between the superposition and itself. Thus, Eqn. 3.1 becomes ( [{^( t ) + E2(t + r)P] [{E^t) + E2(t + T)P] ) (im)}2 + [Em?> (3.3) where Ej = £(t) cos[a>j + 4>j(t)] and g%(±oo) = 1. The second-order fast1 autocorrelation with background is {3.4) Intensity Autocorrelation The intensity autocorrelation is usually described as a background-free autocorrelation. Its function is usually denoted by a subscript, 0. explained in "Fast and Slow Autocorrelation", p. 12 Chapter 3. The Autocorrelator 12 9o{T) - -ji^wmwr [ } where Ej — £ ( t ) cos[u>j + (f>j(t)] and g$(0) = 1. The second-order fast autocorrelation without background is /OO E2(t)E2{t + r)dt 9oyr) = °° ,00 a (3.6) / E\t)dt J — OO Fast and Slow Autocorrelation In order to resolve the fast variations in an autocorrelation function, the control over the path difference must be to within 10% of the wavelength involved [10], such as with a piezoelectric crystal. In addition, wavefront distortion in the autocorrelator's optics need to be less than A/4. Thus, fast autocorrelation can measure gg or g$. When autocorrelation is slow, one measures G £ ( T i , T i , - - - , T n _ i ) = ( ^ ( r i . T i , - - - , ^ - ! ) ) ^ (3.7) Gn0(ruru • • • ,T n_ a) = {g^(ruru • • • , r n _ 1 ) ) r n (3.8) where the multidimensional "optical time average" is defined by ([])rtn = ——J ' / [}dT1dT1...dr1 (3.9) with uf < Tj < rpj t (3.10) Thus, the autocorrelation function sees only the pulse envelopes (ie. (J(TJ-I)) and not the phase part (ie. COS[UJ + (f>j(t)]). In an application, the time average is determined by the time constant of the detector employed. Chapter 3. The Autocorrelator 13 With a slowly varying pulse envelope, and phase deviation, <f>(t), the second-order fast autocorrelation function with background (gB) and without background (g^) become the slow autocorrelation functions, GB and GQ, respectively. /OO em2(t+r)dt uB = i + Z - ^ (3.11) / i\t)dt J—oo /OO I(t)I(t + r)dt = (3.12) / I2(t)dt J — OO where I(t) is the optical time-averaged pulse intensity. /OO t2m2{t+T)dt uB = -^20 (3.13) / i\t)dt J—oo (3.14) /OO I(t)I(t + r) <ft -OO  /OO i -OO l[G2B(r)-l] (3.15) 3.2.3 Selected Autocorrelation Functions In Table 3.1, four forms of I(t) are presented along with their bandwidth products ( AvAi), intensity pulse widths ( r p ) , autocorrelation functions ( Gl), and autocorrelation pulse widths, ( r G ) . 3.2.4 Application The intensity autocorrelation was chosen for ease of alignment. Alignment of an inter-ferometer is time-consuming and difficult. Slight changes in the incoming laser beam by Chapter 3. The Autocorrelator 14 Table 3.1: Characteristics of various pulseshape models: bandwidth products, inten-sity pulse widths, autocorrelation functions and autocorrelation pulse widths [11]. See Appendix for the proofs of the autocorrelation functions. I(t) AvAt IR T (y = i) rG T Hyperbolic Secant Squared I(t) — sech2a: 0.3148 1.7627 Gl(T) = ^j-[ycathy-l] sinn y 2.7196 0.6482 Gaussian I(t) = e~x2 0.4413 1.3863 Gl(r) = e-y2!* 2.3548 0.7071 Lorentzian K t ) - 1 0.2206 2 (12(T\ - 1 1 4 0.5000 Go(r) - x + ( f ) 2 Symmetric Exponential I(t) = e-W 0.1420 0.6931 Gg(T) = ( l + 2|y|) e- 2l»l 1.6783 0.4130 Chapter 3. The Autocorrelator 15 adjustments in the laser setup and pulse compressor would cause long delays between photographs. This was unacceptable due to the drift in operating parameters that the laser system undergoes during operation. 3.3 Construction The main purpose of an autocorrelator's construction is to split the incoming laser pulse allowing each half to travel different path lengths and recombining them to produce an autocorrelation signal dependent on the time difference produced. A simple Michelson interferometer provides a standard way of splitting a beam and then recombining it. One optical path in one arm is usually held constant (the "sta-tioriary" arm). The other optical path is changed through various means: shaker assem-blies [12,13], piezoelectric crystals [14,15], rotating prisms [16] or speakers [11,17,18,19]. In the case of a rotating beam splitter, both optical paths are changed simultaneously. The autocorrelator constructed employed a speaker from the above options for ease and economy. The layout of the autocorrelator is shown in Figure 3.1 and will be de-scribed in the Section 3.3.1. 3.3.1 Components The components of the autocorrelator, excluding the electronics, are: the K D P crys-tal, speaker, beamsplitter, right angle prism, a fine precision translation stage, another translation stage, retroreflector, plano-convex lens, visible cut filter, and photomultiplier housing. The crystal of potassium dihydrogen phosphate (KH 2P0 4) had a crystal aperture of 10 mm and a crystal thickness of 0.5 mm. (See Appendix E regarding thickness.) The crystal was mounted in a 2" gimbal with a line to indicate the vertical axis and Chapter 3. The Autocorrelator 16 was cut 58.6° from the optical axis. For K D P to produce the second harmonic of 616 nm, the phase matching angle from the optical axis must be 59.3° (See Appendix E). This angle must bisect the two converging input beams. The second-harmonic will exit the crystal along this bisector. Thus the crystal must be turned 0.7° counterclockwise. A n advantage for the difference between the phase matching angle and the angle at which the crystal was cut is to reduce retroreflections back to the laser. The speaker was laboratory surplus. It had an 8" diameter frame with a 7 |" diameter cone. The (d.c.) resistance of the coil was 3.7 f2. As a result, a 45 Cl resistor was placed in series with the speaker to match the 50 Q, output impedance of the sine wave generator. A n L-shaped mount was machined to mount the speaker vertically. A hole was cut through the vertical section of the mount to allow the laser beam to hit the retroreflector which was mounted on the centre of the cone. The beam splitter was originally anti-reflection coated for the wavelength of a ruby laser (694.3 nm). The right angle prism or rooftop prism was made from B K 7 . Its front face was 3.5 cm wide. ?' The fine precision translation stage was purchased from Newport Research Corpora-tion. Its translation screw was 80-pitch with a 2" travel. This translation stage moved' the right angle prism above from side to side. This enabled the two parallel beams to emerge a specific distance apart. When focussed by a lens, a specific convergence angle may be formed in this way. The retroreflector was purchased from Melles Griot . Its diameter was 15 mm and its depth was 11 mm. It was set in a thin hollow plexiglas cylinder with a set screw. The cylinder could be screwed into a lightweight mount which was glued to the speaker cone. This plano-convex lens achromat had a diameter of l | " and a focal length of 7". A n achromat was chosen in the event that the bandwidth of a femtosecond laser pulse was Chapter 3. The Autocorrelator 17 too great creating achromatic dispersion. It was plano-convex with the flat side facing the photomultiplier to prevent retroreflections. Since the fundamental frequency of the femtosecond laser was in the visible light region, a broadband visible cut filter (Corning GS7-54-1) was used to extract the second harmonic (ultraviolet light). The original 2" square was ground down at the corners to fit into the photomultiplier housing and was placed between the photomultiplier tube and the shutter. The photomultiplier housing was a light-tight cylinder on a vertical post. A shutter on the front could be closed when bright laboratory ceiling lights were necessary. The visible cut filter situated behind the shutter filtered most of the visible light except for some of the ultraviolet generated from the fluorescent lights. In addition, the photomultiplier was electrically insulated from the aluminum optical bench with a plexiglas base. 3.3.2 Electronics The electronic components of the autocorrelator are: sine wave generator, phase shifter, z-axis blanking circuit, photomultiplier, low-pass R C filter, and an oscilloscope. The sine wave generator frequency generator used was the Function Generator F54 by Interstate Electronics Corp. The frequency and output voltage were variable. The output was at 50 $7 impedance. The speaker is basically an oscillator with a mass, spring constant and damping constant. The voltage applied to the speaker coils represents the driving force. The speaker motion follows the driving voltage except that there is a phase delay. If the driving voltage is to represent the displacement of the speaker, then it must be phase delayed to match the phase delay in the response of the speaker. Thus, the purpose of the phase shifter was to phase shift the driving voltage signal for the oscilloscope x-axis. The phase shift is set by a 10-turn potentiometer which adjusts the phase angle of Chapter 3. The Autocorrelator 18 the phase delay (independent of frequency of the input signal). This Z-axis blanking circuit was used to provide the blanking signal to the z-axis of the oscilloscope. In this way, the autocorrelation trace is swept in one direction only. The input trigger signal was taken from the sine wave generator. The photomultiplier was a RCA (Type C31034) 2" diameter tube. This 11-stage QUANTACON type had a GaAs photocathode and a UV transmitting window. The voltage specification was 2200 V (max.) and 1500 V (typ.). During the experiment, the operating voltage ranged from 1700 V to 2100 V. The spectral range is 200 to 930 nm. The high voltage power supply used was a Model 412B, John Fluke Mfg. Co., Inc. through an isolation transformer. The isolation transformer, which "isolates" the high voltage supply from ground, was used as a safety measure to prevent accidental electric shock. An low-pass RC filter was added to the output of the photomultiplier tube to average the autocorrelation scan generated by the 82-Mhz laser pulse train. A slow detector is usually used for autocorrelation techniques. The fast response time of the photomultiplier with the use of a high impedance (1 Mfi) oscilloscope input necessitated the addition of a low-pass RC filter. The RC time constant used was 44 fis (23 kHz). The oscilloscope used was a Tektronix oscilloscope capable of 100 MHz resolution. The following plug-in units were used: a dual trace amplifier (7A12), two time base (7B70 and 7B71), and an amplifier (7A29). Together, the dual trace amplifier and time base (7B70), produced the autocorrelation measurement scan. The dual trace amplifier's 1 MO input was set to -DC and its range usually used was in the 5 mV/div range. The time base (7B70) was switched from time base to EXTERNAL x-axis input which was adjusted to enable a full autocorrelation scan corresponding to the entire x-axis (ie. -5 to 5). The amplifier (7A29)'s 50 £1 input was used to monitor the Nd:YAG laser with the Chapter 3. The Autocorrelator 19 7B71 delaying time base. './ 3.4 Calibration 3.4.1 Calibration Procedure Calibration involves mapping the time difference between the arrival of the two replicas of the laser pulse into the nonlinear K D P crystal to the x-axis of the oscilloscope. The time delay between splitting of the input beam and the arrival times of each of the split beams at the K D P crystal is produced by the different spatial path lengths each takes through the autocorrelator (See Figure 3.1). The distance light had to travel in the "stationary" arm (ie. not the arm with the vibrating speaker) was changed by moving a fine translation stage. The increased light path difference induced is twice that of the translation of the stage. Finding how far the translation stage moved is calculated by taking the distance the translation stage moves per every turn of its screw (1 inch per 80 turns) and converting it to metres. Using the speed of light equation and the number of femtoseconds in a second, the light path difference is converted to a time difference in femtoseconds. This gives us time in femtoseconds as a function of the turns of screw in the translation stage. By moving the translation stage known steps (not only spatially, but also temporally), the position of the peak of a stable autocorrelation pulse could be plotted across the time axis of the oscilloscope. The resulting data are converted into a simple polynomial relating turns of the screw of the translation stage to time on the horizontal axis of the oscilloscope. As a result, the time across the oscilloscope screen may be determined by the above method. Note that an earlier attempt to directly measure the translation of the translation Chapter 3. The Autocorrelator 20 stage proved inaccurate and imprecise. Since the translation screw is of a very fine pitch, the turns of the screw was used to measure the translation indirectly. 3.4.2 Subpicosecond Range The settings for the subpicosecond range of the autocorrelator are: • Speaker voltage: 15.0 ± 0.1 V p p @ 29.9 ± 0.5 Hz • Phase shift setting: <f> = +86° ± 1° Figure 3.2 was used for the calibration curve. The fitted polynomial converting x-axis position on the oscilloscope to time in femtoseconds is given by: m- , ^  n l i n c h 2 .54xl0- 2 m 1015fs , n n n n n Time(x) = 2 x — x —— x — 4- (2.9979 x 108m/s) v ' 80 turn 1 inch sec v ' ' x {0.466,79 x} (3.16) Using 18 data points, the f-test rejection hypothesis yields only a 0.04% level of signifi-cance. The f-value for this polynomial of order 1 is 0.0498. 3.4.3 Femtosecond Range The settings for the femtosecond range of the autocorrelator are: • Speaker voltage: 2.00 ± 0.Q5 V p p @ 20.0 ± 0.4 Hz • Phase shift setting: </> = +78° ± 1° Figure 3.3 was used for the calibration curve. The fitted polynomial converting x-axis position on the oscilloscope to time in femtoseconds is given by: m- / x n l i n c h 2.54 x 10~2 m 1015fs / n n n r j n , r t 8 , N Time(x) = 2 x — x — — x (2.9979 x 108m/s) v ' . 80 turn 1 inch sec v ' ; x {0.196152x - (1.76209 x 10~3) x2} (3.17) Chapter 3. The Autocorrelator 21 Using 8 data points, the f-test rejection hypothesis yields only a 2.8% level of significance. The f-value for this polynomial of order 2 is 0.106. 3.5 Opera t ion 3.5.1 Al ignmen t Optical alignment of the beams through the autocorrelator was quite easy. It was similar to setting up a Michelson interferometer.' Alignment was achieved as follows. Measure both optical paths to ensure they are within a centimeter. Ensure that there is enough travel either way for the high precision translation stage. Adjust this stage if the two optical paths are not equal. Use a large sweep of the speaker first. In other words, use the subpicosecond range settings. Obviously, the initial autocorrelation signal will be weak. Thus, it is recom-mended that a slightly higher voltage setting be used on the photomultiplier or a more sensitive oscilloscope setting. The hardest part of alignment involves targeting the retroreflector so that the laser beam hits only one of the three reflective surfaces (to avoid its corners). Be sure the incident beam is in the horizontal plane before enterring the autocorrelator. Adjust the two beams so that they are focussed oh the same spot of the K D P crystal. This may be done by first moving the position of the converging lens to position the foci along a vertical line within the crystal. Then make the two foci coincide by adjusting the vertical angle of the prism. Using 616 nm wavelength, the angle of the converging beams should be approximately 5° with the K D P angle about 0.7° counterclockwise given the K D P crystal listed above. The convergence angle can be changed by translating the prism from side to side. As the frequency of the laser changes, the optimal angle of the crystal as does the angle of convergence of the two beams. Chapter 3. The Autocorrelator 22 Adjust the angle of the K D P crystal until an autocorrelation signal can be detected. The adjustment of the K D P angle does not require fine adjustments. However, the foci of the two incident beams may not overlap within the crystal as the crystal position changes. Once the K D P angle is correct, adjust the focus and the convergence of the two parallel beams for maximum signal. A few iterations of the above adjustments will be required to obtain the autocorrela-tion signal. Finally, change the fine precision translation stage to centre the autocorrelation signal on the oscilloscope and lower the photomultiplier high voltage supply as much as possible. This requires using the lowest voltage range on the oscilloscope (ie. 5 mV/div) . 3.5.2 Electronic Adjustments To provide the best linear response of the autocorrelator, several electronic adjustments were made. The driving voltage and frequency and the phase shifter were adjusted. If the driving voltage was too large or the driving frequency was too high or low, the nonlinear characteristics of the speaker bellows would be seen. In addition, the driving voltage determines the range of the time scan. (In other words, the larger the range of the speaker travel, the larger the range for time delay (r) for the autocorrelation scan.) The correct phase shift was determined as follows: 1. Disconnect the blanking circuit to the z-axis input of the oscilloscope. 2. Adjust the phase shifter until both autocorrelation signals overlap especially near the ends of the scan. Since there is a bit of hysteresis, both the blankable and the unblankable autocorrelation traces will not overlap in the centre of the scan. 3. Lock the adjustment knob of the phase shifter. Chapter 3. The Autocorrelator 23 4. Reconnect the blanking circuit to the z-axis input. Chapter 3. The Autocorrelator 24 Oscilloscope Z-axis Blanker Phase Sine Wave Shifter Generator Out In Out In Out Right Angle Prism Achromatic Lens Retrorefl PMTHH«^ :::"" | KDP Visible Cut Beamsplitter Filter Femtosecond Laser Beam Figure 3.1: Autocorrelator. Chapter 3. The Autocorrelator 25 Autocorrelator Calibration Curve (subpicosecond range) 12000 0 - | — . — i — i — | — . — i — i — | — i — | — i — | — i — | — i — | — i — i — i — -5 - 4 - 3 - 2 - 1 0 1 2 3 4 5 r Hor izonta l Axis (div) Figure 3.2: Autocorrelator Calibration Curve (subpicosecond range). Primary un-bankable trace: experimental (•) and fitted polynomial ( ). Blankable return trace: experimental (o) and fitted polynomial (• • •). See Equation 3.16 for the fitted polynomial used for calibration. Chapter 3. The Autocorrelator 26 Autocorrelator Calibration Curve (femtosecond range) 6000 1000 - | — i — | — . — | — H - , — • — | — , — | — , — | — , — | — • — | — i — | — . — -5 - 4 - 3 - 2 -1 0 1 2 3 4 5 Horizontal Axis (div) Figure 3.3: Autocorrelator Calibration Curve (femtosecond range). Primary unbank-able trace: experimental (•) and fitted polynomial ( ). See Equation 3.17 for the fitted polynomial used for calibration. Chapter 4 Pulse Compression Techniques In this chapter, the theory and methods of several pulse compression techniques are presented, with hybrid mode locking highlighted. 4.1 Theory The Fourier theory states that the "time-bandwidth" product AuAt>K (4.18) where K is a constant of the order of unity. The constant K may be determined by taking the Fourier transform of the desired pulse shape of pulse width, A t , and finding its bandwidth, Au (which is defined similarly to the pulse width except in the frequency domain). When the time-bandwidth products equals K, the result is a "transform-limited" or "bandwidth-limited" pulse. See Table 3.1 for values of transform-limited time-bandwidth products for selected pulse shapes. The pulse width, A t , of a transform-limited pulse may be decreased if the bandwidth, Au, is increased. One method is to use the nonlinear process of self-phase modulation (See Chapter 5). Another way is to increase the bandwidth of the gain of the laser. In the N d : Y A G laser, the mode locker extends the gain bandwidth by adding multiple acoustic frequencies to the longitudinal modes through Bragg diffraction [20]. 27 Chapter 4. Pulse Compression Techniques 28 Another way is to synchronously pump a laser. Dye lasers are preferred due to their shorter excited state lifetimes of the amplifying dye than in solid state lasers. Bandwidth is not a major problem since the dye used as a gain medium can be used over a continuous range of frequencies. A method for selecting the centre lasing frequency is a one-plate birefringent filter or pellicle but this limits the spectral bandwidth resulting in a longer laser pulse. (The birefringent filter has a higher finesse than the pellicle, thus, limiting the bandwith more.) Through synchronous pumping and a short excited state lifetime, the gain increases only when the centre of the resonating laser pulse is in the dye. Thus, the centre of the pulse is amplified more than the tails. This results in a shorter pulse. Pumping will no longer be synchronous with the resonating laser pulse if the cavity length deviates from that of the pumping laser. If the cavity is too long, a long pulse results because the pump pulse amplifies the leading edge of resonating laser pulse. If the cavity is too short, a shorter but weaker, unstable pulse results because the resonating laser pulse is just passing out of the jet when the pump pulse arrives. Another method is to use a saturable absorber in a dye laser [21]. This is often employed in synchronously pumped dye lasers. In addition to increasing the gain at the centre of the resonating pulse, the gain at tails of the pulse is reduced further by a saturable absorber. Refer to the following section (Section 4.2). In a colliding-pulse mode-locked (CPM) laser [22], a saturable absorber is placed at the opposite side to the gain dye cell in a ring cavity configuration. In a ring cavity, two pulses propagate in opposite directions. Mode locking is achieved when the two pulses collide in the saturable absorber. The peak intensity of the two pulses saturate the absorber when only one pulse would not be able to. As a result, the gain at the centre of the resonating pulse is greater than at the tails. Unfortunately, the C P M laser runs freely without any active synchronization. This is not preferred if one wants to amplify the Chapter 4. Pulse Compression Techniques 29 output with regenerative dye amplifiers which must be synchronously pumped [23,24], although novel amplifiers for C P M lasers have been devised [25] Frequency doubling in a nonlinear crystal also has a pulse shortening effect due to the second harmonic intensity being dependent on the square of the intensity of the fundamental intensity. A good overview of the above techniques and others is covered in reference [26]. 4.2 Hybrid Mode Locking 4.2.1 Theory The basic theory behind a saturable absorber is quite simple. When a certain concentra-tion of saturable absorber solution is placed within a laser cavity, the saturable absorber will absorb the lower intensity tails of the laser pulse in addition to any noise present in the system. However, only a limited amount of the main peak can be absorbed since the absorption of the dye has reached saturation. As a result, the gain for the central part of the pulse is higher than for the tails. This results in shortening the pulse. For a theoretical model, refer to [27]. The addition of saturable absorber dye in the laser cavity results in largely asymetric spectra with a broad wing on the long-wavelength side. As a result, exponential-like long tails occur in the pulse shape that are indicated in the SHG autocorrelation trace. This can be explained by self-phase modulation due to the optical Kerr effect in the mixed dye jet stream [28]. The frequency chirp and asymetric spectra can be compensated for by the introduction of high refractive index prisms in the cavity [29]. A commercial sj 'stem recently on the market, the Coherent® Satori™ femtosecond dye laser employs this compensation. Chapter 4. Pulse Compression Techniques 30 4.2.2 D Q O C I and D O D C I With an optically contacted absorber dye cell arrangement, DODCI with Rhodamine 6G has been able to produce 0.3-1.5 ps pulses in the tuning range, 592-617 nm. In fact, without a tuning element, DODCI tends to shift the lasing frequency to 615 nm. With DQOCI, pulses of 0.6-2 ps at 580-613 nm were produced [30]. DODCI's ground state peak extinction coefficient occurs at 590 nm. However, its optical isomer has an absorption coefficient at 615 nm [31] and is estimated to contribute ~10% to the saturable absorption. Using the same technique but starting with a shorter pulse, further pulse compression should be possible. Chapter 5 Fibre Grating Pulse Compression In this chapter, the theory of fibre grating pulse compression is covered. Nonlinear opti-cal mechanisms in the fibre such as fibre self-focussing and self-phase modulation occur in addition to group velocity dispersion which predominates. The grating compensates for this group velocity dispersion. The construction and alignment of such a fibre grat-ing pulse compressor is detailed. Results on optimizing the compressor is presented in chapter 6. 5.1 Theory The fibre grating pulse compressor consists of a fibre section to optically chirp 1 the laser pulse by self-phase modulation and a grating compressor section to compress the pulse broadened by positive group velocity dispersion (GVD) and the positive chirp. 5 . 1 . 1 Self-focussing and Self-phase Modulation Self-phase modulation occurs when the electric field is so intense that a nonlinear response occurs in the medium through which it propagates. In the case of a short pulse, even though the average power is modest, the peak power is extremely high. In the case of the Spectra-Physics femtosecond laser system, the peak power was 2.8 kW. This nonlinear response results in a positive chirp. 1 coined from radar technology, this means that lower frequencies are generated at the leading edge of the pulse and higher frequencies at the trailing edge. 31 Chapter 5. Fibre Grating Pulse Compression 32 Ignoring transient effects, the index of refraction can be expressed as n = n0 + n2 <E2> ... (5.19) where no is the linear index of refraction and n 2 is the nonlinear index of refraction which gives rise to a linear chirp (discussed later). < E2 > is the time average value of the square of the electric field. Letting the electric field be E = i f cos (kz + ut) (5.20) =» <E2> = \ \E\2 (5.21) If n 2 > 0 in the core of a single mode fibre, the increase in index of refraction as a function of radius is 8n(r) = n2222(r). This lens-like effect leads to self-focussing of the beam. If the index of refraction is substituted in k in Eqn. 5.20, an extra phase term occurs from the nonlinear term, leading to E = U cos(—z + ut) (5.22) c = ^ f c o s ( z -f u)t H z) (5.23) • c c Thus, the frequency modulation is - - ^ . ) : (5.24) Thus, for n 2 > 0, the frequency shift will be negative on the leading edge and positive on the trailing edge of the laser pulse propagating in the fibre. In other words, shorter Chapter 5. Fibre Grating Pulse Compression 3 3 wavelengths will lead longer wavelengths. Positive group velocity dispersion also causes this effect. A more accurate model of the evolution of an optical pulse in a single-mode fibre is described by the dimensionless nonlinear Schrodinger equation [32] , OV 7T d2V „ ' - f - 2 | V | 2 V ( 5 . 2 6 ) d(z/z0) 4 [d{t/TY where V is the complex pulse amplitude, t is the retarded time defined such that for any distance, z, along the fibre, the centre of the pulse is at t = 0. ZQ and T are normalization constants. As the pulse propagates along the fibre, the short pulse width and high peak power produces a frequency shift as in the first simpler model. As extra frequencies are gener-ated at the leading and trailing edges of the pulse and group velocity dispersion occurs, the peak becomes flatter and wider until a rectangular pulse is obtained. This continues until the intensity becomes too low to produce self-phase modulation. From a spectral point of view, the original pulse contains a narrow bandwidth of frequencies. As the pulse propagates, power from the centre frequency gets redistributed to side frequencies at the pulse tails. Like the intensity profile, the frequency profile becomes somewhat rectangu-lar, looking like a square pulse with a limited bandwidth (ie. there are overshoots at the edges of the square pulse). The result is a linear frequency chirp. If the input pulse envelope is assumed to be hyperbolic secant squared, I V{z = 0,t) = Asech( ;t/r) ( 5 . 2 7 ) and 2° = °-322$Wx (5-28) fp for 6 1 6 nm in fused silica fibre core. ( 5 . 2 9 ) 0.116 m _ 1 psec2 p Chapter 5. Fibre Grating Pulse Compression 34 „ ncXA„xt _ Pi = — — ^ x 10" 7 5.30 l D 7 r 2 r 0 ^ 2 A = JPlPi (5.31) where zo is the length required for G V D alone to approximately double the width of the input pulse, c is the speed of light (cm/s), r p is the input pulse width (psec), A is the centre wavelength of the input pulse (m), D(X) is the G V D (dimensionless), n 0 and n2 are the refractive index (dimensionless) and nonlinear coefficient (esu) of the fibre core, respectively, Aefi is the effective fibre core area (cm 2), P is the peak power of the input pulse (W), P i is the peak power required for self-phase modulation alone to approximately double the spectral width of a pulse in a fibre length of z0 (W), and A is the pulse amplitude (dimensionless). This pulse as it propagates through the fibre may be expressed in the frequency domain as V(z,u) = A(u>) e'*^ (5.32) where A(u>) and $(u;) are real function implicitly dependent on z as well. 5.1.2 G V D Compensat ion Anomalous dispersion may be used to compensate for a positive chirp. This can be described as adding a phase factor dependent on frequency in the phase portion of the equation for the chirped pulse (Eqn. 5.32). Vc(z,u) = A(u>) ei(>*M+**W (5.33) Thus, if $c(w) = — then all the frequency components will be in phase produc-ing the maximum pulse amplitude possible with the shortest possible pulse width. In reality, this is not achieveable. However, an approximate quadratic phase delay (in UJ) can be provided by a grating-pair compressor [33], or high-index prisms [34]. Chapter 5. Fibre Grating Pulse Compression 35 $c(a;) = $ o - c w 2 (5.34) where ao is the optimal compressor factor, a, a° ATrc2d2 cos2 7' (5.35) where b is the distance between gratings (m), A is the centre wavelength of the chirped pulse (m), c is the speed of light (m/s), d is the groove spacing (m), and 7 is the incident angle of the input beam to the normal of the first grating. In this experiment, a right angle prism was used so that a single grating could serve as both gratings of a grating-pair compressor (See Figure 5.4). d = 3.33 x 1 0 - 6 m, A = 0.616/im, and 7' = 2.8° ± 26'. Substituting in Eqn 5.35, For a hyperbolic secant squared pulse, the pulse width is r p = 1.7627 T from Table 3.1. Assuming TP = 563 fs from Table 6.2, and maximum power, P = 2.4 kW x 55% coupling efficiency, n 0 = 1.6, n 2 = 1.1 x 10~ 1 3 esu, and A e f j = 7r(2 /mi) 2 . z0 = 2.73 m (5.37) P i = 24.6 W (5.38) A = 7.32 (5.39) Interpolating Table 1 from [32] for A = 7.32, a0/T2 w 0.18896 (5.40) ZoPt/zo w 0.192 (5.41) T P /T = 4.8992 ' ' (5.42) Chapter 5. Fibre Grating Pulse Compression 36 From Laser" Fibre Chuck Fibre Holder Figure 5.4: Fibre-Grating Pulse Compressor. where zopt is the optimal fibre length when a quadratic compressor is used (ie. grating pair). Thus, b « 104 cm . (5.43) zopt « 52 cm i (5.44) T « 115 fs (5.45) 5.2 Construction The layout of the fibre-grating compressor is shown in Figure 5.4 amd will be described in Section 5.2.1. Chapter 5. Fibre Grating Pulse Compression 37 5.2.1 Components F ib re Section Newport Corporation (Model F-SV-1) single mode fibre was chosen over polarization preserving fibre due to a a major cost difference ($60 and $170 for 10 metres, respectively). Its optimal operating wavelength was 633 nm, its numerial aperture was 0.1, its core diameter was 4 fim, its cladding diameter was 125 //m, and its jacket diameter was 250 fj,m. It dispersion was <300 psec/km/nm and attenuation was 12 dB/km. The jacket was stripped by,soaking in methylene chloride for approximately four minutes. The jacket usually starts peeling away from the cladding at this time. Using a wad of lens tissue, the jacket was pulled away. Additional soaking may be required if the jacket does not pull away easily. Care must be taken not to damage the exposed fibre cladding. The fibre was cleaved using a F-CL1 Fibre Cleaver from Newport Corporation. The cleave was visually inspected with a microscope. A good cleave was confirmed by coupling a laser beam in the other end and checking for a good output beam profile when projected onto a screen. A Newport Corporation (Model F-1015) high precision steering lens fibre coupler employed a microscope objective (Newport M-20x). A steering lens, an AR-coated long-focal-length (254 mm) negative lens, was positioned near the back focal plane of the objective lens. A transverse translation of this lens causes a much smaller transverse translation of the focused beam across the fibre core. The translation ratio was approx-imately 32:1. Thus, the effective focal length of the system was 7.4 mm. A Newport Model F P H - J 4-jaw fibre chuck with a teflon liner was used to hold the fibre. A Newport Corporation precision fibre coupler (Model F-91) mounted a M- lOx mi-croscope objective which produced a wider beam than a M-20x microscope objective Chapter 5. Fibre Grating Pulse Compression 38 without clipping the edges of the beam profile. The wider beam reduces the beam in-tensity and thus reduces nonlinear effects in refractive media further down such as in the autocorrelator. In addition, the wider beam takes advantage of the resolving power available on the gratings. The position of the fibre chuck was changed to direct a parallel beam to the grating compressor. G r a t i n g Compressor The Milton Roy diffraction gratings (Catalog #35-99) had grating spacing of 300 grooves per millimeter, with a blaze wavelength of 7500 A and blaze angle of 6° 28'. It is 84% efficient at 7500 A. The blank is 58 mm x 58 mm x 10 mm of BSC2. These were mounted in mirror mounts which could be mounted on an optical track. The right angle prism whose front face measured 4.4 cm could also be mounted on an optical track. One diffraction grating could be used in conjection with a right angle prism on an optical track to eliminate the necessity for a second diffraction grating for a "grating pair" arrangement. The input diffraction grating also became the output diffraction grating. This grating was turned so that the blaze angle bisected angle between the incident and diffracted beams to maximize diffraction efficiency. The angle of the track (parallel to the diffracted beam) to the incoming chirped laser beam was 7.5° ± 1°. The theoretical angle was 10.6°. 5.2.2 Al ignmen t Coupling into the fibre presented the most difficult part of using the fibre-grating pulse compressor. Due to the small numerical aperture of the fibre, alignment involved not only the fibre holder controls but also the last routing mirror before the fibre holder Chapter 5. Fibre Grating Pulse Compression 39 optics. The former controlled the transverse translation of the focussed beam across the fibre core. The latter controlled the angle into the core. In addition, since the fibre was not polarization preserving, the re-positioning of the fibre was sometimes necessary to obtain a stronger autocorrelation signal. The fibre was kept as straight as possible to prevent losses and induced birefringence. The beam emerging from the fibre output coupler was directed at the diffraction grating. The first diffracted order was directed to the right angle prism where it was returned to the grating. The beam was diffracted a second time and directed into the autocorrelator. The beams between the diffraction grating and the right angle prism were made parallel to the optical track. Thus, changing the position of the right angle prism along the track did not change the position of the beam into the autocorrelator very much. A compressor with a grating pair would not be able to accomplish this. By adjusting the position of the right angle prism, the optical distance betweeen the successive diffraction points was changed. This distance is later referred to as the "grating distance". Thus, changing the grating distance changes the amount of compression of the pulse chirped by the fibre. 1 Chapter 6 Measurements In this chapter, measurements of the standard femtosecond system with and without a fibre grating pulse compressor are presented and discussed. Motivation for the following two experiments was discussed earlier in chapter 4 and their results are presented: hybrid mode locking, replacement of the one-plate birefringent filter with a pellicle. In addition, an investigation of the group velocity dispersion in the dye laser and the fibre-grating pulse compressor efficiency as a function of the femtosecond system's output power is made. 6.1 Standard Femtosecond System Measurements 6.1.1 Method The autocorrelation signal was filtered with a low-pass R C filter as discussed in Sec-tion 3.3.2. This signal was displayed on an oscilloscope. A Polaroid photograph was taken at 1/25 sec shutter speed to obtain one autocorrelation trace. The autocorrelation trace was digitized with the Talos C Y B E R G R A P H of the U B C Computing Centre. The software used was VDIGIT [35] which among other things provided skew correction. The trace was rescaled according to the autocorrelator calibration curve (Figures 3.2-3.3) before being fitted to the various theoretical autocorrelation functions. The theo-retical autocorrelation functions assumed either a hyperbolic secant squared, gaussian, lorentzian or symmetric one-sided exponential intensity pulse shape. Four parameters '.i 40 Chapter 6. Measurements 41 were adjusted for each fit of the autocorrelation data: vertical (height and background) and horizontal (pulse width of the intensity pulse and time shift). The errors for each parameter were calculated corresponding to a change in x2 equal to 1.0. See Appendix F for computer program details on fitting the various theoretical autocorrelation functions to the experimental data. The primary characteristic of the laser pulse is the pulse width. The error interval for the pulse width indicates how well a specific theoretical function fits the data compared with other theoretical functions. . Another important statistic variable is the autocorrelation standard deviation which was calculated during the fitting. When the minimization was done, the standard devia-tion of the autocorrelation data points from the theoretical autocorrelation function was calculated within a window of four times the pulse width centered on the pulse (arbi-trarily chosen to include only a major portion of the pulse). This gave one quantitative way of measuring the fit of the central peak irrespective of the background level, pulse shoulder (as with an incorrectly optimized pulse compressor) or satellite pulses (as with a too short dye laser cavity length). Reference will be made to an "optimal dye laser cavity length". This refers to the case when the dye laser cavity length was adjusted so that a minimum pulse width could be obtained without causing pulse instability or the formation of large satellite pulses. Instability was characterized by variability in the peak height and the pulse shape. The satellite pulse could not be greater than « 2 % of the peak height of the autocorrelation trace. The average power was measured by Laser Precision Corp. thermopile power meter (Model RT-20) which had 7 ranges from 10 mW to 20 W. The accuracy was ± 5 % . The wavelength was measured by an interference wedge ( E G & G , Electro-Optics Di-vision). The accuracy was ± 2 nm. u Chapter 6. Measurements 42 6.1.2 Results Effect of Low-pass R C Filter When the time constant of the R C filter was increased, no observable trend could be seen beyond the random error of the data (Figure 6.5). Due to the increased bandwidth of the laser pulse (as evidenced by the lack of fringes in a Fabry-Perot interferometer and a simple interference wedge observation), and possibly the increased bandwidth of the autocorrelation signal, the introduction of a low pass filter may be filter frequencies within the autocorrelation signal bandwidth and thus affect the measurements. Although this was a marginal concern, it was done anyway to ensure accuracy of the autocorrelation measurements. Pulse Measurements Figures 6.6-6.9 shows experimental autocorrelation data and the correponding the-oretical autocorrelation function fits for different dye laser cavity lengths. The vertical axis is scaled according to the exponential autocorrelation function (100% being the peak height). Table 6.2 shows the pulse width and autocorrelation standard deviation for various dye laser cavity lengths and assuming various pulse shapes. The average power peaks when the cavity is slightly longer than the optimal cavity length. As the dye laser cavity length is shortened, the average power decreases and fluctuates until lasing is stopped. 6.1.3 Discussion As expected, the pulse width increased with increasing dye laser cavity length as discussed in Chapter 4. The theoretical functions fitting the autocorrelation data are from best to worst fit: Chapter 6. Measurements 4 3 Pulse Width and Autocorrelation Standard Deviation vs. R C constant 400 CD 3 100 H 0 0 Time Constant (/Lis) 10 20 30 40 50 5- 5 I s 5 5 5 • n o V V o V 8 0 e • © 8 • 8 B o © o l 0 1 10 1 1 20 1 1 1 30 40 i 50 Time Constant (ju,s) o - 6 c o - A O - 2 0 > a) a -I-" co Figure 6.5: Pulse Width and Autocorrelation Standard Deviation (hyperbolic secant squared (•), gaussian (o), lorentzian (o) and exponential (v) vs. R C time constant. Some of the pulse width error bars may be too short to be shown. Chapter 6. Measurements 44 Autocorrelation Power vs. Time (very short dye laser cavity length) Figure 6.6: Autocorrelation Power (experimental (o) and theoretical: hyperbolic secant squared ( ), gaussian ( ), lorentzian ( ) and exponential (• • •) ) vs. Time, for a very short dye laser cavity length. Exponential pulse energy of 2.3 nJ and peak power of 4.1 kW. Chapter 6. Measurements 45 Autocorrelation Power vs. Time (short dye laser cavity length) -4000 -2000 0 2000 4000 Time (fs) Figure 6.7: Autocorrelation Power (experimental (o) and theoretical: hyperbolic secant squared ( ), gaussian ( ), lorentzian ( ) and exponential (• • •) ) vs. Time, for a short dye laser cavity length. Exponential pulse energy of 2.3 nJ and peak power of 3.0 kW. Chapter 6. Measurements 4 6 Autocorrelation Power vs. Time (optimal dye laser cavity length) o o 1 1 I 1 ~I 1 I 1 I - 4 0 0 0 - 2 0 0 0 0 2000 4000 Time (fs) Figure 6.8: Autocorrelation Power (experimental (o) and theoretical: hyperbolic secant squared ( ), gaussian ( ), lorentzian ( ) and exponential (•••) ) vs. Time, for an optimal dye laser cavity length. Exponential pulse energy of 2.3 nJ and peak power of 2.4 kW. Chapter 6. Measurements 47 Autocorrelation Power vs. Time (long dye laser cavity length) o 1 r 1 1 : — i 1 1 -4000 -2000 0 2000 4000 Time (fs) Figure 6.9: Autocorrelation Power (experimental (o) and theoretical: hyperbolic secant squared ( ), gaussian (— — ) , lorentzian ( ) and exponential (•••) ) vs. Time, for a long dye laser cavity length. Exponential pulse energy of 2.6 nJ and peak power of 1.3 kW. Chapter 6. Measurements 48 Table 6.2: Pulse width (with error interval in brackets) and autocorrelation standard deviation (percentage of peak power) for various dye laser cavity lengths. Cavity Pulse Width (fs) and Length Autocorrelation Standard Deviation sech^ Gaussian Lorentzian Exponential Very 329 (11,-H) 368 (15,-14) 229 (4,-4) 198 (3,-3) Short or (3.4%,~3.2%) or (4.0%,-3.8%) or (1.6,-1.6) or (1.7%,-1.6%) 1.22% 1.56% 0.69% 0.76% Short 445 (13,-13) 496 (17,-17) 317 (4,-4) 269 (4,-4) or (3.0%,~2.9%) or (3.5%,~3.3%) or (1.3%,-1.3%) or (1.6%,-1.5%) 1.31% 1.78% 0.24% 0.49% Optimal 563 (17,-17) 630 (22,-22) 399 (8,-8) 338 (6,-6) or (3.1%,"3.0%) or (3.5%,-3.4%) or (1.9%,-1.9%) or (1.9%,-1.8%) 1.21% 1.66% 0.35% 0.44% Long 1180 (114,-108) 1343 (130,-125) 779 (75,-68) 690 (59,-56) or (9.7%,~9.2%) or (9.7%,~9.3%) or (9.6%,-8.8%) or (8.6%,~8.1%) 2.93% 3.23% 1.79% 2.36% Chapter 6. Measurements 49 exponential, lorentzian, hyperbolic secant squared and gaussian. This was indicated by the error intervals for the pulse width (Table 6.2). Looking at the "Exponential" column of the table, one can see that the smallest autocorrelation standard deviation occurs at the optimal cavity length. The next fitted function that is narrower than the exponential is the hyperbolic secant squared. If one looks at the "sech2" column, again it shows that the best fit occurs at the optimal cavity length although it does not degrade as much as the exponential fit as the cavity length is shortened. This could indicate that if one could shorten the cavity length without producing the characteristic satellite pulses, one could yield a pulse better fitted to a hyperbolic secant squared. The use of a saturable absorber such as DODCI or DQOCI to eUminate these satellite pulses will be discussed later in this chapter. Group velocity dispersion in the dye laser would cause a deviation from the transform-limited hyperbolic secant squared pulse shape. It seems that this caused the pulse shape to resemble a symmetric one-sided exponential pulse shape which differs only slightly as compared to the either the lorentzian or gaussian pulse shapes. A n exponential pulse shape fit resulted in a pulse width of 338 ± 6 fs, whereas a hyperbolic secant squared yielded a higher pulse width of 563 ± 17 fs. This showed that the assumed pulse shape can change the pulse width dramatically. As a result, when a pulse width is quoted the assumed pulse shape must be specified too. A n exponential pulse shape can occur with self-phase modulation (SPM) in the dye laser (See Section 4.2). For all four cavity lengths, the autocorrelation data peaks beyond all theoretical autocorrelation functions (Figure's 6.6-6.9 show autocorrelation data and four theoretical autocorrelation functions normalized for an exponential pulse). This may be due to the coherence spike generated by random noise discussed in Section 3.2.1. This might also explain why the base was slightly wider than for an exponential pulse shape. Chapter 6. Measurements 50 Table 6.3: Pulse width (with error interval in brackets) and autocorrelation standard de-viation (percentage of peak power) for selected compressed pulses. See Figures 6.18-6.20. Fig. Pulse Width (fs) and Autocorrelation Std. Deviation sech2 Gaussian Lorentzian Exponential 114.2 (3.1,-3.0) 127.7 (4.1,-3.9) 79.7 (1.4,-1.4) 67.7 (1.0,-1.0) 6.18 or (2.7%,"2.6%) or (3.2%,"3.1%) or (1.8%,-1.8%) or (1.5%,-1.5%) 1.58% 1.95% 0.61% 1.10% 163.4 (6.9,-6.5) 181.7 (7.8,-7.3) 116.0 (7.2,-6.8) 96.8 (5.4,-5.1) 6.19 or (4.2%,-4.0%) or (4.3%,-4.0%) or (6.2%,-5.8%) or (5.6%,-5.2%) 2.37% 2.45% 1.95% 2.66% 140.0 (5.1,-4.9) 155.1 (6.6,-6.2) 102.0 (2.2,-2.2) 85.2 (1.8,-1.8) 6.20 or (3.7%,-3.5%) or (3.7%,"3.5%) or (2.2%,-2.1%) or (2.1%,-2.1%) 2.00% . 2.36% 0.95% 1.88% The lower half of the pulse broadened when the dye laser cavity was lengthened (Fig-ure 6.9) or shortened (Figure 6.6-6.7). This indicates that more noise was present at these cavity lengths, especially the long cavity length. As the dye cavity was shortened, the autocorrelation trace became asymmetric. It was not due to an intrinsic asymmetry in the autocorrelator as can be disproved by the symmetry of the autocorrelation trace for the optimal cavity length. Since average power fluctuates at these cavity lengths, a probable cause for the asymmetry is the instability that results at these extreme cavity lengths. For example, when the satellite pulses form, they compete with the main central pulse for power. 6.2 Subpicosecond to Femtosecond Pulse Compression 6.2.1 Results Figures 6.10-6.13 show the pulse width as a function of the compressor's fibre length Chapter 6. Measurements > Pulse Width vs. Fibre Length and Compression Distance (hyperbolic secant squared fit) C~5 i s o n s o n 4-0 T 3 0 n 2.0 1 -\ o n 6 0 r^- , „ _ , , G r a t i n g D i s t a n o e ( o <-r^ ") 6- Measurements Fibre r . ^ , P u I s e W ^ t h Length "l u m vs. a n d C o m p r e s s i o n (gaussian fit) *' ance i S e W * v , F i b r e L e n g t h a n d C o i « P r e S S i o n Dirt. a n c e (gaussian fit). Chapter 6. Measurements 53 Pulse Width vs. Fibre Length and Compression Distance (lorentzian fit) Figure 6.12: Pulse Width vs. Fibre Length and Compression Distance (lorentzian fit). Chapter 6. Measurements 54 Pulse Width vs. Fibre Length and Compression Distance (exponential fit) Figure 6.13: Pulse Width vs. Fibre Length and Compression Distance (exponential fit). Chapter 6. Measurements 55 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (hyperbolic secant squared fit) Figure 6.14: Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (hyperbolic secant squared fit). Chapter 6. Measurements 56 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (gaussian fit) Figure 6.15: Autocorrelation Standard Deviation vs. Distance (gaussian fit). Fibre Length and Compression Chapter 6. Measurements 57 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (lorentzian fit) Figure 6.16: Autocorrelation Standard Deviation vs. Distance (lorentzian fit). Fibre Length and Compression Chapter 6. Measurements 58 Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (exponential fit) Figure 6.17: Autocorrelation Standard Deviation vs. Fibre Length and Compression Distance (exponential fit). Chapter 6. Measurements 59 Autocorrelation Power vs. Time for the shortest compressed pulse I I I . i | I I I I | I I I I j I I I I -1000 -500 0 500 1000 Time (fs) Figure 6.18: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— — ) , lorentzian ( ) and exponential (• • •) ) vs. T i m e , for the shortest compressed pulse. Exponential pulse energy of 0.67 nJ and peak p o w e r of 3.4 kW. Fibre length: 30.8 ± 0.5 cm. Grating distance: 103.8 ± 1 cm. Chapter 6. Measurements 60 Autocorrelation Power vs. Time for a compressed pulse (fibre too long) CD o Q_ c o 1_ l_ o o o •*-> < 100% -80% -60% -40% -20% -0% 1000 -500 0 500 1000 Time (fs) Figure 6.19: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— — ) , lorentzian ( ) and exponential (•••) ) vs. Time, for a compressed pulse. Exponential pulse energy of 0.67 nJ and peak power of 2.4 kW. Fibre length: 31.9 ± 0.05 cm. Grating distance: 108.8 ± 1 cm. Chapter 6. Measurements 61 Autocorrelation Power vs. Time for a compressed pulse (fibre too short) -1000 - 5 0 0 0 500 1000 Time (fs) Figure 6.20: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (• • •) ) vs. Time, for a compressed pulse. Exponential pulse energy of 0.67 nJ and peak power: 2.7 kW. Fibre length: 29.85 ± 0.05 cm. Grating distance: 108.8 ± 1 cm. Chapter 6. Measurements 62 and grating pair distance, thus showing the optimal fibre length and grating pair distance for minimum pulse width. Figures 6.14-6.17 show the autocorrelation standard deviation, thus showing the optimal fibre length and grating pair distance for best pulse shape. Table 6.3 shows the characteristics of the pulses shown in Figures 6.18-6.20. These pulses are among the shortest produced by the fibre-grating pulse compressor. The maximum fibre transmission that could be achieved was 55%. The grating com-pressor transmission was 45%. Total transmission through the complete fibre-grating pulse compressor was 25%. 6.2.2 Discussion Since the transmission of the compressor section was 45%, high index prisms may be used to improve the transmission and provide continuous G V D compensation through zero. Two valleys at near right angles occur in the contour plots. Each roughly parallels an axis. The valley nearly parallelling the grating distance axis indicates that a slightly longer grating distance is required when longer fibres are used. This is due to greater G V D in a longer fibre. The deeper valley in the pulse width contour plot occurs along the grating distance axis, whereas in the autocorrelation standard deviation contour plot it occurs along the fibre length axis. For an exponential pulse shape, the 90 fs pulse width contour is bounded by the following parameters: fibre length, 30.5 to 31.4 cm, and grating distance, 88 to 110 cm. Correspondingly, the recommended pulse compressor parameters are: fibre length, 31.0 ± 0.4 cm, and grating distance, 99 ± 11 cm. For a hyperbolic secant squared pulse shape, the 140 fs pulse width contour is bounded by the following parameters: fibre length, 30.6 to 31.0 cm, and grating distance, 99 to 106 cm. Chapter 6. Measurements 63 Correspondingly, the recommended pulse compressor parameters are: fibre length, 30.8 ± 0.2 cm, and grating distance, 103 d b 4 cm. Note that in the autocorrelation trace of compressed pulses contain more noise than for the uncompressed pulses. Due to the very short pulses involved, the effect of vibrations and other sources of noise becomes greater. In particular, the pulse compressor needs to be more isolated from vibrations such as that from the dye circulator system and the Nd:YAG cooling water system. Conventional optical mounts are not sufficient for femtosecond study. From Section 5.1.2, the following estimates were made: • Grating distance: 104 cm. • Fibre length: 52 cm. ' • Compressed pulse width: 115 fs (hyperbolic secant squared). These figures compare well with the experimental results, except for the fibre length which was overestimated. However, this was expected since the theory assumes a rectangular shape and a linear frequency chirp. Thus, a theoretical pulse could utilize a longer fibre experiencing more self-phase modulation than experimentally. 6.3 Hybrid Mode Locking 6.3.1 Method DQOCI (l,3'-diethyl-4,2'-quinolyoxacarbocyanine iodide) and DODCI (3,3'-diethyl-oxa-dicarbocyanine iodide) were supplied by Exciton, Inc. DQOCI has a molecular weight of 470.35 while DODCI has a molecular weight of 486.35. (Rhodamine 6G or Rhodamine 590 Chloride has a molecular weight of 479.02.) The DODCI crystal powder is blue while the DQOCI is green. (Rhodamine 6G is a reddish powder.) Chapter 6. Measurements 64 Solutions were prepared by weighing the appropriate amounts of powder and dissolv-ing it in some Rhodamine 6G dye solution assisted by an ultrasonic bath. Both dyes resulted in a principally blue solution. Initial stock solutions of 0.05 M were made. How-ever, the DQOCI solution gelled when it was left overnight. Decreasing the concentration to 0.033 M solved this problem. Later, to ensure that all the dye was dissolved, the so-lutions were diluted further to produce 0.001 M . The DQOCI solution seemed to have a thin film of particles floating on it. Further use of the ultrasonic bath failed to dissolve it. It was assumed that these particles were impurities. The use of hyperdermic needle syringes avoided the drawing of these impurities. The solutions were injected into the dye circulator reservoir during the dye circulator's operation. 10 ml and 5 ml hypodermic needle syringes were used for the DQOCI and DODCI solutions, respectively. Greater precision by using the 5 ml syringe resulted in more data points in the DODCI experiment than in the DQOCI experiment which was conducted first. Five minutes were allowed to elapse to allow thorough mixing throughout the dye circulatory system. Afterwards, the dye laser cavity length was adjusted. As the cavity length changed, the output beam changed direction slightly necessitating the realignment of the autocorrelator. Occassionally, as time progressed, adjustments would be necessary to the N d : Y A G and pulse compressor. Polaroid photographs were taken at 1/2 sec shutter speeds to obtain an average of 15 autocorrelation traces. 6.3.2 Results After the injection of the saturable dye into the dye reservoir, pulse stability initally decreased. As the concentration of the saturable dye became more uniform throughout the dye system, the stability returned to normal (approximately three minutes). Chapter 6. Measurements 65 D Q O C I Figures 6.21-6.24 show the pulse widths and the autocorrelation standard deviation as the DQOCI concentration was increased. Figure 6.25 shows a typical pulse with DQOCI hybrid mode locking. D O D C I Figures 6.26-6.29 show the pulse widths and the autocorrelation standard deviation as the DODCI concentration was increased. Figure 6.30 shows a typical pulse with DODCI hybrid mode locking. 6.3.3 Discussion As the dye concentration increased, the pulse width decreased slightly while the standard deviation increased. This can be explained by a more intense peak caused by the saturable absorber dye (Figure 6.25 and 6.30). With DODCI, the pulse width reduced from « 3 7 0 to « 3 0 0 fs (exponential) using 1 to 50 mmol/1 DODCI. The shortest pulse width recorded at 245 fs (exponential) occurred at 3 mmol/1 DODCI. Since the power decreases monotically, the lower end of the range of DODCI concentration, namely 2 or 3 mmol/1 DODCI is recommended. With DQOCI, the pulse width reduced from « 330 to « 250 fs (exponential) using « 1 0 mmol/1 DODCI. The shortest pulse width recorded at 240 fs (exponential) occurred when the experiment was stopped at 11 mmol/1 DQOCI. At this concentration, the laser power had deteroiated to 150 mW. DODCI started with a slightly longer pulse (due to variations from day to day oper-ation) than DQOCI but both ended at roughly the same short pulse. However, DODCI shortened the pulse before laser power declined too much whereas DQOCI shortened the i Chapter 6. Measurements 66 Power, Pulse Width and Autocorrelation Standard Deviation vs. D Q O C I Concentration (hyperbolic secant squared fit) • DQOCI (mmol/1) 0 .0.1 1 10 0.1 DQOCI (mmol/l) 1 r 10 Figure 6.21: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DQOCI Concentration (hyperbolic secant squared fit). Chapter 6. Measurements 67 Power, Pulse Width and Autocorrelation Standard Deviation vs. D Q O C I Concentration (gaussian fit) CD o Q_ E c <N ro m -t-> JO 3 Q-250 200 150-1 100 700 600 500 H 400 300 -200 -100 -0 0 DQOCI (mmol / l ) 0.1 1 10 _ _ l L _ I 1 L_ • D • 0.1 1 DQOCI (mmol / l ) 10 30 D 0) Q . 20 .2 - 10 > Q - M CO 0 Figure 6.22: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DQOCI Concentration (gaussian fit). Chapter 6. Measurements 68 Power, Pulse Width and Autocorrelation Standard Deviation vs. D Q O C I Concentration (lorentzian fit) 0 0 DQOCI ( m m o l / 1 ) 0.1 1 j i 0.1 1 DQOCI ( m m o l / l ) 10 L 10 Figure 6.23: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DQOCI Concentration (lorentzian fit). Chapter 6. Measurements 69 Power, Pulse Width and Autocorrelation Standard Deviation vs. D Q O C I Concentration (exponential fit) DQOCI (mmol/1) 0.1 1 0 0.1 1 DQOCI (mmol / l ) 10 Figure 6.24: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DQOCI Concentration (exponential fit). Chapter 6. Measurements 70 Autocorrelation Power vs. Time for a dye laser with the saturable dye, DQOCI o - 4 0 0 0 - 2 0 0 0 0 2000 4000 Time (fs) Figure 6.25: Autocorrelation Power (experimental (o) and theoretical: hyperbolic secant squared ( ), gaussian (— — ) , lorentzian ( ) and exponential (•••) ) vs. Time, for a dye laser with the saturable dye, DQOCI. DQOCI concentration: 12.2 mmol/l. Exponential pulse width of 239 (22,~ 20) fs, energy of 2.0 nJ, and peak power of 2.8 kW. Chapter 6. Measurements 71 Power, Pulse Width and Autocorrelation Standard Deviation vs. D O D C I Concentration (hyperbolic secant squared fit) 0 250 <D 200 H o E c D Q_ 150 S 100 700 -600 -500 -400 300 -200 -100 0 o DODCI (mmol / l ) 0.1 1 10 100 _i L • • • • D [ J - 30 - 20 o o C L - 10 0 > Q CO 0.1 1 10 DODCI (mmol / l ) 100 Figure 6.26: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DODCI Concentration (hyperbolic secant squared fit). Chapter 6. Measurements 72 Power, Pulse Width and Autocorrelation Standard Deviation vs. D O D C I Concentration (gaussian fit) CD o Q_ E c CN ro m co sz •+-• cu _co 3 Q_ 0 250 - T — V 200 -150 -100 700 -600 -500 -400 -300 -200 -100 -0 0 DODCI (mmol / l ) 0.1 1 10 I i I . L _ 100 • 8 •• • • • • Q [ V - 30 - 20 o o cu Q. - 10 0 > a xi •+-> CO 0.1 1 10 DODCI (mmol / l ) 100 Figure 6.27: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DODCI Concentration (gaussian fit). Chapter 6. Measurements 73 Power, Pulse Width and Autocorrelation Standard Deviation vs. D O D C I Concentration (lorentzian fit) cu o Q_ E c CM to m 250 2 0 0 -1 5 0 -1 0 0 4 0 0 A 3 0 0 H 2 0 0 A 1 0 0 0 0 -±yy--y y--*y y-o DODCI ( m m o l / l ) 0.1 1 10 _i L X 1 0 0 • • • • D c . ' i i 3 0 S a. h 2 0 .2 -r-> D '> a) a V 10 0 to 0.1 1 10 DODCI ( m m o l / l ) 1 0 0 Figure 6.28: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DODCI Concentration (lorentzian fit). Chapter 6. Measurements 74 Power, Pulse Width and Autocorrelation Standard Deviation vs. D O D C I Concentration (exponential fit) 0 2 5 0 a> 2 0 0 -\ o 0_ E c sz X J 1 5 0 -ro 1 0 0 L O 4 0 0 -3 0 0 a) 2 0 0 — Q_ 1 0 0 A o v . 0 . DODCI ( m m o l / l ) 0.1 1 10 1 0 0 i • • • D Q [ T * n 5 •1 im1. i V V V V -I—I 1—I 1—I 1—I— 0.1 1 10 1 0 0 DODCI ( m m o l / l ) h 3 0 g Q. h 2 0 o *> cu Q - 10 "D CO 0 Figure 6.29: Power (•), Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. DODCI Concentration (exponential fit). Chapter 6. Measurements 75 Autocorrelation Power vs. Time i. for a dye laser with the saturable dye, DODCI o o 1 , 1 1 n 1 I 1 I -4000 -2000 0 2000 4000 Time (fs) Figure 6.30: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (•• •) ) vs. Time, for a dye laser with the saturable dye, DODCI. DODCI concentration: 2.7 mmol/1. Exponential pulse width of 234(20," 19) fs,. energy of 2.4 nJ, and peak power of 3.5 kW. Chapter 6. Measurements 76 Table 6.4: Pulse width (with error interval in brackets) and autocorrelation standard deviation (percentage of peak power) for various cavity lengths of the dye laser with a pellicle. Cavity Pulse Width (fs) and Length Autocorrelation Standard Deviation sech^  Gaussian Lorentzian Exponential Optimal 453 (44,-41) 508 (51,-47) 302 (31,-29) 269 (26,-24) or (9.7%,"9.1%) or (10.0%,"9.3%) or (10.3%,"9.6%) or (9.7%,"8.9%) 4.11% 4.63% 2.81% 3.45% Long 683 (111,-105) 777 (127,-120) 417 (65,-57) 399 (58,-54) or (16%,-15%) or (16%,-15%) or (16%,-14%) or (15%,-14%) 3.98% '. 4.15% 2.66% 3.65% pulse while simultaneously reducing laser power. . Earlier in Chapter 4, it was stated that DODCI should be better than DQOCI for hybrid mode locking at 615 nm. This was confirmed. With this femtosecond laser system at the wavelength of 615 nm, DODCI seems to produce better hybrid mode locking characteristics (pulse width and power) than DQOCI. DODCI is easier to prepare as a solution and is less expensive than DQOCI. 6.4 Replacement of One-plate Birefringent Filter with a Pellicle 6.4.1 Results For a brief experiment, the one-plate birefringent filter was replaced with a pellicle of unknown thickness. The pellicle angle used was 45° ± 10° with a diameter of 52 mm. Changes in the pellicle angle changed the centre lasing frequency slightly but peak power and peak stability was achieved at 598 ± 2 nm. This was the frequency that was used. Table 6.4 shows the pulse widths (assuming different pulse shapes) and autocorrelation standard deviation for two cavity lengths. Figures 6.31-6.31 show the pulse shapes that Chapter 6. Measurements 77 Autocorrelation Power vs. Time for^  a dye laser with a pellicle and an optimal cavity length CD -j 1 1 . 1 r 1 1 1 -4000 -2000 0 2000 4000 T i m e ( f s ) Figure 6.31: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (•••) ) vs. Time, for a dye laser with a pellicle and an optimal cavity length. Exponential pulse energy of 1.7 nJ and peak power of 2.2 kW. Chapter 6. Measurements 78 Autocorrelation Power vs. Time for a dye laser with a pellicle and a long cavity length CD o 0_ c o %-> _D CD i _ i _ o o o •+-> D < 100% -80% -60% -40% -20% -0% -•4000 - 2 0 0 0 0 2000 T i m e ( f s ) 4000 Figure 6.32: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (• • •) ) vs. Time, for a dye laser with a pellicle and a long cavity length. Exponential pulse energy of 1.7 nJ and peak power of 1.5 kW. Chapter 6. Measurements i. 79 result. As with the one-plate birefringent filter, the pulse was closest to an exponential pulse shape. When the cavity length was shorter than what was optimal, the pulse power and shape became unstable very quickly. However, no satellite pulses were observed as was the case with the one-plate birefringent filter. 6.4.2 Discussion If the instability at shorter cavity lengths could be reduced, the pulse width could have been shorter. The instability may be due to movement of the pellicle and the necessity for a precise cavity length at this pulse width. A pellicle with a smaller diameter or a thicker membrane to reduce vibration would be more appropriate. An indication of the instability is seen in the autocorrelation trace. The top half of the peak is narrower than the bottom half. This shows that noise was present in the laser pulse. j 6.5 Group Velocity Dispersion in Dye Laser 6.5.1 Results For another brief experiment, the compressor portion of the full fibre-grating pulse compressor was tried to shorten the dye laser pulse. Components of the dye laser pro-duce positive GVD which may have been corrected by the compressor which introduces negative GVD. , t•,, Unfortunately, depending on ^ the position of the right angle prism in the compressor, the laser beam was obstructed by the prism's mount. Thus, only a very limited range of short grating distances were allowed and tried (Figures 6.33-6.36). Figure 6.37 shows Chapter 6. Measurements 80 Pulse Width and Autocorrelation Standard Deviation vs. Grating Distance (hyperbolic secant squared fit) -5 Grating Distance (cm) 0 5 10 15 20 25 30 800 « 700 -JZ = 600 -CD Q_ 500 -400 • i i i i i t i i i i i  I — J . ' * 1 L_l I 1 I I I L. D CD h 1.5 °-h 1.0 J *-> o > CD I- 0.5 Q -t-> in — 1 — I — I — I — | — I — I — I — I — | — I — I — I — I — | — I — I — I — I — | — I — I — I — r - ] — I — I — I — I — | — I — I — I 5 0 5 10 15 ; 20 25 30 0.0 Grating Distance (cm) Figure 6.33: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. Grating Distance (hyperbolic secant squared fit). i" Chapter 6. Measurements 81 Pulse Width and Autocorrelation Standard Deviation vs. Grating Distance (gaussian fit) - 5 0 Grat ing D is tance ( cm) 5 10 15 20 25 30 800 w 700 -~ 600 H CD =• 500 H 400 • • i i i i i i i i i i ' i i i i i i i t i i i i -i—i—i—i—|—i—i i I—|— i—i— i—i—|—I—I— i—i—|— i—i—i—r- |— i— i—I—i—|— i—i— i—r -5 0 5.. 10 15 20 25 30 D CD 1.5 °-1.0 J -+-" .2 *> CD 0.5 a - M CO 0.0 Grat ing D is tance ( cm) Figure 6.34: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. Grating Distance (gaussian fit). Chapter 6. Measurements 82 Pulse Width and Autocorrelation Standard Deviation vs. Grating Distance (lorentzian fit) Grating Distance (cm) - 5 5 0 0 w 4 0 0 -•5 3 0 0 H => 2 0 0 H 1 0 0 0 5 10 1 5 2 0 2 5 3 0 l l l l I l l l l I l \ I I | | l l l I I I I | I l l i I I I l l l T—i—i—i—|—i—i—i—i—|—n—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—r T o 1.5 °-1.0 J > Qi 0 . 5 Q -t-> CO 0 . 0 •5 0 5 10 15 2 0 2 5 3 0 Grating Distance (cm) Figure 6.35: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. Grating Distance (lorentzian fit). Chapter 6. Measurements 8 3 Pulse Width and Autocorrelation Standard Deviation vs. Grating Distance (exponential fit) Grating Distance (cm) - 5 0 5 10 15 20 25 30 500 « 400 5 300 H cu 3 200 -| 100 J I I I I I I L-J I I I I I I I I I I I I L i i I i i i t I i i i i - 1.5 - 1.0 0.5 i i i i i i i i i i i i i i I i i i i i i i i i I i i i i i i i i 5 0 5 10 15 20 25 30 Grating Distance (cm) 0.0 D CD CL C o > cu Q xi -t-> CO Figure 6.36: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. Grating Distance (exponential fit). Chapter 6. Measurements 8 4 Autocorrelation Power vs. Time for laser system with grating compressor - 4 0 0 0 - 2 0 0 0 0 2000 4000 Time (fs) Figure 6.37: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (• • •) ) vs. Time, for laser system with grating compressor (grating distance = 6.3 cm) Chapter 6. Measurements 85 experimental autocorrelation data for a grating distance of 6.3 cm. 6.5.2 Discussion The pulse width varied as much as fifty femtoseconds. As a result, no conclusive evidence was found in improving the pulse width. If there was an improvement, it was inpercepti-ble. However, the autocorrelation standard deviation improved for the hyperbolic secant squared and gaussian fits, worsened for the lorentzian fit and remained the same for the exponential. Recall that the order for size of the base is (from narrow to wide): gaus-sian, hyperbolic secant squared, exponential and lorentzian. This would indicate that pulse shape changed slightly from an exponential shape to more of a hyperbolic secant squared. In Figure 6.37, one can see that the peak decreased to the theoretical values without increasing the base of the pulse. A grating compressor at the dye laser output reduced power by 55% with little or no improvement in pulse width. As a result, the benefit of a simple grating compressor at the dye laser output is small. If a further decrease power can be tolerated, the full fibre-grating pulse compressor is recommended. 6.6 The Effect of Power on the Fibre-Grating Pulse Compressor 6.6.1 Results Pulse power was increased to test its effects on a, laser pulse after being pulse com-pressed by a 28.05 cm fibre and.a grating compressor (99 cm grating distance). Recall the optimal fibre length is 30.8 cm and the corresponding grating distance is 104 cm. Figures 6.38-6.41 show pulse width and autocorrelation standard deviation assuming var-ious pulse shapes as the pulse power was increased. There was a considerable shoulder under the peak (Figure 6.42). (The curve fitting neglected the shoulder and just fitted Chapter 6. Measurements 86 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (hyperbolic secant squared fit) 500 400 -CO •5 300 CD _co D_ 200 100 600 700 800 900 1000 1100 532 nm Power (mW) 1200 o CD CL c o o *> CD Q T7 •*-> CO Figure 6.38: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. 532 nm Power (hyperbolic secant squared fit). Chapter 6. Measurements 87 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (gaussian fit) 500 400 -•5 300 -CD D_ 200 -100 Q. c o D *> Q CO 600 700 800 900 1000 1100 532 nm Power (mW) 1200 Figure 6.39: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. 532 nm Power (gaussian fit). Chapter 6. Measurements 88 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (lorentzian fit) 400 300 -CO -C cu _co 3 0_ 200 -100 600 700 .800 900 1000 1100 'i 532 nm Power (mW) 1200 Figure 6.40: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. 532 nm Power (lorentzian fit). Chapter 6. Measurements 89 Pulse Width and Autocorrelation Standard Deviation vs. 532 nm Power (exponential fit) 400 300 -0) — D_ 200 100 D CI. c o > Q CO 600 700 800 900 1000 1100 532 nm Power (mW) 1200 Figure 6.41: Pulse Width (o) and Autocorrelation Standard Deviation (v) vs. 532 nm Power (exponential fit). Chapter 6. Measurements 90 Autocorrelation Power vs. Time for 1000 mW average power and a short fibre CCD —2000 -1000 0 1000 2000 Time (fs) Figure 6.42: Autocorrelation Power (experimental (o) and theoretical: hyperbolic se-cant squared ( ), gaussian (— —), lorentzian ( ) and exponential (• • •) ) vs. Time, for 1000 mW average power (200 mW greater than the nominal 800 mW) and a 28.05 cm fibre (3 cm shorter than the optimal fibre length for 800 mW). Chapter 6. Measurements 91 the peak). 6.6.2 Discussion According to the theory discussed in Chapter 5, if a higher pulse peak power is used, a shorter fibre length is necessary.. Conversely, if a shorter fibre is used (as in this test), greater pulse power would be necessary. The above results confirmed this. They also show the extent of this effect. A decrease in fibre length from 31.0 cm to 28.05 cm resulted in a shift of the optimal average power from 800 mW to 1000 mW. Changing the pulse power could be used to fine-tune the fibre length without actually changing the fibre length. However, due to the limited range of stable power delivered by the laser system, this would be impractical. Cleaving the fibre to the correct length is simpler than adjusting the pulse power. Chapter 7 Conclusions Operation and characterization of the Spectra-Physics femtosecond laser system was per-formed through the diagnostics of a background-free second-order intensity autocorrelator which was constructed after considering its theory of operation. 616 nm laser pulses from the Spectra-Physics Femtosecond Laser System were char-acterized. The exponential pulse width was 338 ± 6 fs. The nominal average power was 225 mW average power (with a 82-Mhz pulse train). The exponential pulse energy was 2.7 nJ with a peak pulse power of 2.8 kW. Pulse compression was achieved with the construction of a fibre grating pulse com-pressor after discussion of its theory. Pulse compression was able to produce a 68 ± 1 fs (exponential) pulse at 616 nm. The average power was 55 mW. The exponential pulse energy was 0.67 nJ with a peak power of 3.4 kW. The pulse compressor consisted of a 30.8 ± 0.5 cm fibre and a grating compressor with the effective grating pair distance of 103.8 ± 1 cm. In addition, two other methods of pulse compression, hybrid mode locking and re-placement of the one-plate birefringent plate with a pellicle were discussed and tested. Group velocity dispersion in the 'dye laser and the effect of power on the efficiency of the pulse compressor were also investigated. It was determined that with the current laser system tuned at 616 nm, DODCI is better than DQOCI for hybrid mode locking. The recommended DODCI concentration is 2-3 mmol/l. The shortest exponential pulse width was 250 fs. The average power was • 92 Chapter 7. Conclusions 93 185 mW. The exponential pulse'energy was 2.3 nJ with a peak pulse power of 2.6 kW. An attempt to increase the bandwidth of the laser pulse by replacing the one-plate birefringent plate with a pellicle severely limited the tunability of the dye laser and introduces copious noise. Attempts to reduce group velocity dispersion with a grating compressor was indeter-minate, but did result in a slightly better pulse shape. Interferometric autocorrelation is recommended for such a study. Finally, deviation from nominal input power on pulse compression showed a degrada-tion as predicted earlier in chapter 5. Appendix A Start-up Procedure for the Femtosecond Laser System The daily procedure for starting the femtosecond laser system is listed below. Please read each step completely before executing. The steps have been ordered to speed startup time. A . l N d : Y A G Laser A.1.1 Preliminary 1. Turn on power to optical bench (power bars). Mode Locker 2. Turn on Model 451 Mode Locker Chassis (mode locker's power supply). 3. Use the technique described in the "Model 3800, CW Nd:YAG Laser System" manual (pp. 3-18 to 3-23)[2] to set mode locker to the following parameters. (Setup time is K.30 minutes if mode locker has been turned has been turned off.) • Mode locking frequency: 41.1750 MHz • Oven: 7.5 (full scale = 9.25) • RF Power: 9.25 (full scale) • RF Servo Error: 0.0 • RF Servo Power: «70% 94 Appendix A. Start-up Procedure for the Femtosecond Laser System 95 4. Turn on Model 3225 Controller (frequency doubler crystal temperature stabilizer). This can be left on between startups. 5. Turn on circuit breaker #27,29,31 (Panel BLDD, South Wall, Room A024, Chem-istry/Physics Building Block A, U.B.C.) City Water 6. Turn CITY WATER on (southeast sink). 7. Ensure water flows out of drain hose properly. 8. Check for water leaks from CITY WATER cooling hoses. Nitrogen Purge 9. Turn regulator off (counterclockwise). 10. Open gas cylinder valve (counterclockwise). 11. Turn regulator on just enough to allow a positive pressure in the dust tubes. Listen to the regulator for just a hint of gas flow. Model 3800 N d : Y A G 12. Turn on main circuit breaker, CB1, of Nd:YAG power supply (upper left corner on back of power supply). 13. Turn on deionized water pump by inserting key into power supply front panel. 14. If desired, (a) Remove cover of Nd:YAG laser. Appendix A. Start-up Procedure for the Femtosecond Laser System 96 (b) Insert interlock defeat key. (c) Skip next step. 15. Remove access panel near output aperture (to access intracavity shutter switch (red lighted push button) and extracavity shutter). 16. Close intracavity shutter by pressing red button repeatedly until it is in the up position. 17. Close extracavity shutter by moving its lever outward. CAUTION: Fail-ure to do this WILL result in damage to pulse compressor! A. 1.2 NdrYAG Power Supply 18. Confirm OFF button is lit (red). If not, check interlocks. 1 19. Record lamp hours meter reading. 20. Press ON button (green). Green light will light. If not, wait 30 seconds or so. 21. Set to CURRENT MODE. 22. Set SHUTTER to CLOSED. 23. Press LAMP START (white). • Lamp current meter will indicate lamp start. • If lamp does not start, wait 30 seconds. • Initial lamp current will be «25 A. • Final lamp current should be ^28.0 A. Appendix A. Start-up Procedure for the Femtosecond Laser System 97 A.1.3 N d : Y A G Laser 24. Check mode locker for (a) servo lock (if not locked, RF servo power will be 50%). (b) no oscillations in SERVO ERROR. 25. Set SHUTTER to OPEN. Oscilloscope 26. Turn power on. 27. Connect SIG OUT from laser head to oscilloscope's input with BNC cable. 28. Connect SYNC OUT from mode locker to oscilloscope's EXTERNAL trigger. 29. Set dials for nominal pulse: 60 mV @ 82 MHz (new battery in the laser's light monitor circuit). 30. Check if extracavity shutter is closed. 31. Open intracavity shutter by pressing red button in laser head. • Lasing should occur as indicated by the oscilloscope. • SIG OUT signal will increase from 40 mV to 60 mV after 10 minutes. Mode Locker 32. Set mode locker FREQUENCY - FINE ADJ knob to centre position. Appendix A. Start-up Procedure for the Femtosecond Laser System 98 33. Adjust Nd:YAG cavity length to eliminate self-Q-switching («0.875 (0.75 + 0.125) on micrometer). 34. Adjust FINE AD J knob for best mode locked pulses. Turning micrometer counter-clockwise will move final position of FINE ADJ knob counterclockwise. A.2 Pulse Compressor A.2.1 Alignment 35. Remove cover of Model 3695 Pulse Compressor. CAUTION: Do not place hand in 12 Watt infrared beam! 36. Turn half-wave plate (attenuator) to "low" (black line). Preliminary Fibre Alignment 37. Set light meter scale to 10 W. 38. Zero light meter. 39. Insert light detector head between fibre holder and attentuator. 40. Open extracavity shutter. 41. Adjust attenuator for 2 W (fibre input power). 42. Close extracavity shutter. 43. Insert light detector head between fibre's output end and the diffraction grating. 44. Open extracavity shutter. Appendix A. Start-up Procedure for the Femtosecond Laser System 99 45. Ensure light transmitted through fiber (non-zero fibre output power). If not, adjust input beam steering optics (right set of knobs on fibre holder). Main Fibre Alignment 46. Repeat steps in Preliminary Fibre Alignment, but for 4 W input power and at least 60% fibre transmission (2.4 W output power). • Improve fibre transmission by adjusting the right set of knobs on the fibre holder and the last (second) routing mirror (just before pulse compressor). When "walking" the input beam (ie. adjusting both the fibre holder and the routing mirror simultaneously), adjust the two upper corresponding knobs together independent of the two lower corresponding knobs. ' i 1 i A.2.2 Stabilization 47. Check for no self-Q-switching. CAUTION: Self-Q-switching CAN damage doubler crystal! 48. Place light detector head between Model 3275 Remote Controller's light detector and dye laser. 49. On Remote Controller, (a) Switch MODE to Q-SWITCH. (b) Turn Q-SWITCH - RF LEVEL to zero (fully counterclockwise). (c) Switch MONITOR to LIGHT LEVEL. 50. Set light meter to 3 W and zero meter. 51. Open extracavity shutter. Appendix A. Start-up Procedure for the Femtosecond Laser System 100 52. Turn attenuator to "high" (red line). 53. Maximize 532 nm output by (a) Adjusting FREQUENCY-FINE ADJ knob. (b) Adjusting KDP doubler angle. 54. If necessary, adjust attenuator in Remote Controller's light detector for full deflec-tion on Remote Controller's light meter. 55. On Remote Controller, (a) Switch MODE to STABILIZED. (b) Switch SERVO LOOP to ON. 56. Adjust LIGHT LEVEL knob for 800 mW of 532 nm power. 57. Green SERVO LOCK light should be on steady. If not, adjust FREQUENCY-FINE ADJ knob very slightly (about 1/8 of a turn). 58. Close extracavity shutter. 59. Remove light detector head from beam path. A.3 Dye Laser 1 60. Push dye drain tube in. 61. Open Nd:YAG's extracavity shutter. 62. Turn dye circulator on. !. 63. Increase dye.pressure to 120 psi. Appendix A. Start-up Procedure for the Femtosecond Laser System 101 64. Place light detector head at dye laser output. 65. Set light meter to 300 mW and zero meter. 66. Pull dye drain tube out. 67. Adjust pump beam steering mirrors to maximize output (large round knobs near dye drain tube). 68. Adjust dye cavity length to optimize pulse. CAUTION: Do not put hand in 532 nm beam. 69. Adjust FREQUENCY-FINE ADJ knob very slightly (1/20 of a turn) to maximize output. (Note that this has a finer resolution in narrowing pulse width than the coarser control of the dye cavity length adjustment.) 70. Nominal femtosecond pulse characteristics should be as follows. • 225 mW average power If low power, it's usually due to pump beam alignment and focussing. • 340 fs (exponential) or 560 fs (sech2) If too long, it's usually due to a long dye cavity length. If unstable, it's usually due to a short dye cavity length. Appendix B Shutdown Procedure for the Femtosecond Laser System The procedure for shutting down the femtosecond laser system is listed below. 1. Close Nd:YAG's extracavity shutter. 2. Close Nd:YAG's intracavity shutter (press red button in laser head). 3. Press Nd:YAG power supply's OFF button (red). 4. Push dye drain tube in. 5. Decrease dye circulator pressure to 80 psi. 6. Turn off dye circulator. 7. Turn off nitrogen gas (first the regulator then the main valve). 8. Put on all covers. 1 • • 9. Turn off mode locker (if system will be idle for more than several days). 10. Turn off oscilloscope. 11. Remove SIG OUT connection to oscilloscope (to prevent any battery drain in laser's light monitor circuit). 102 Appendix B. Shutdown Procedure for the Femtosecond Laser System 103 12. After 10 minutes of lamp current off, switch key on Nd:YAG power supply to OFF. 13. Turn off circuit breaker, CB1, on Nd:YAG power supply (optional). 14. Turn off CITY WATER supply at sink. 15. Turn off circuit breaker #27,29,31, Panel BLDD (optional). Appendix C Changing Dyes in the Dye Laser The following outlines an alternative procedure to be used while changing dyes in the dye circulator. This procedure is recommended when a critical dye change is required (eg. when rinsing saturable absorber from the system). 1. Obtain the following: • chemical waste disposal containers with appropriate identification tags. (En-sure there is enough to hold the old dye and subsequent rinses.) • latex gloves. • two 5/8" wrenches. • one 1" wrench. • one adjustable wrench (to 1/2"). • deionized water/methanol rinse mixture (1:1) — 1.4 litres is required for each rinse. — Note: the circulator pump cannot be primed with a pure methanol rinse. • 1.5 litres of dye mixture. 2. Ensure adequate ventilation (for methanol vapours). 3. Place paper towels on floor. (This will soak up any dye that gets through the plastic sheet.) 104 Appendix C. Changing Dyes in,;tie Dye Laser 105 4. Place the plastic sheet over the paper towels. 5. Put the dye circulator on the plastic sheet. Put paper towels under the circulator to prevent tearing of the plastic sheet. 6. Ensure the drain tube is pushed over the dye jet nozzle. 7. Empty the dye from the circulator system into a waste container. (a) Remove the check valve from the dye supply line. • Catch any spillage with a container below. (b) Put the dye supply tube into a waste container. (c) Turn the circulator on (to pump the dye into the waste container). (d) When no more dye can be pumped out, drain the reservoir. (e) Rinse the reservoir. (f) Reassemble the check valve into the dye supply line. 8. Drain the filter housing. (a) Remove the top tube from the filter housing by releasing its clip. (b) Slightly loosen the 1" bolt at the bottom of the filter housing (enough to faciliate removal later but not enough to allow leaking). (c) Remove the filter by loosening two small bolts below the filter. (d) Remove the right tube from filter housing. • Catch any spillage with a container below. Appendix C. Changing Dyes in the Dye Laser 106 (e) Drain the dye in the filter housing. i. Drain the dye through the right hole by tilting the filter housing. ii. Loosen the 1" bolt to increase the drain flow. (f) Remove the filter by removing the 1" bolt. (g) Rinse the filter and the filter housing. (h) Reassemble the filter housing without a filter. • If the old filter is going to be reused, it should be soaked in methanol while rinsing is being done and replaced after the last rinse. 9. Rinse the circulator system; (a) Fill the reservoir with the rinse mixture. (b) Prime the circulator pump. (c) Fill the reservoir with more rinse mixture. (d) Let the rinse mixture circulate for 5-10 minutes. 10. Empty the rinse from the circulator system into a waste container as above. 11. Drain the filter housing as above. 12. Repeat steps 9 to 11 until the rinse is clear. • Two repeats are only necessary when the type of dye is not being changed. At least five rinses are necessary when a critical dye change is being made. • Clean the dye jet nozzle if necessary before the last rinse. (Usually just the rinses will suffice.) • Install a new filter (or rinsed old filter) after the last draining. Appendix C. Changing Dyes in the Dye Laser 107 • Check dye jet nozzle area for spillage of rinse. • The drain tube from the laser may be rinsed by bending it so that the jet stream hits different areas of the tube. 13. Remove and rinse the check valve ensuring no debris (usually filter material) is in valve. 14. Fill the reservoir with the new dye. 15. Prime the circulator pump. 16. Fill the reservoir with the remaining amount of dye. 17. Check for leaks. 18. After an overnight wait, check that the check valve is working. If it is not, air will enter the dye supply line via the dye jet nozzle. Appendix D Theoretical Autocorrelation Functions: Proofs The background-free second-order autocorrelation function from Eqn. 3.15 is /OO I{t)I(t + T)dt = (D.46) J—oo where I(t) is the intensity of the laser pulse. Four cases are presented each with a different intensity profile: hyperbolic secant squared, gaussian, lorentzian, and symmetric two-sided exponential. D . l Hyperbolic Secant Squared The hyperbolic secant squared pulse shape is described by I(t) = sech2x where x = — (D-47) Also letting y = ^ (ie. adjust time scale), the numerator in Equation D.46 becomes /oo poo I(x)I(x + y)dx = / sech2x sech2(x + y) dx (D.48) -oo J—oo = J°° sech2x (l - tanh2(x + y)) dx (D.49) r0? , •> / „ ( tanhx + tanht/ \ 2 \ , = /«•• sech2x 1 - r- r~ d x (D-50) J-oo \ \1 + tanhx tanhj/y / /oo sech2x dx — -oo f sech2x ( t a n h x + t a n h y ) 2 2 t J x (D.51) -'-oo (1 + tanhx tanhy) 108 Appendix D. Theoretical Autocorrelation Functions: Proofs 109 tanh x = 2 - A - A f°° , i (tanh x + tanh y)2 . where A = sech x — : — : . . o dx J — OO With a change of variables, let u = tanh x let 6 = tanhy Substituting, (u + b)2 (1 + tanhx tanh?/) du . o — = sech a; dx •1 -L dx (1 + bu)2 u 2 + 2bu + b2 (1 + bu)2 du u (1 + bu)2 1 + bu _ 26 26u (1 + bu)2 du + J: du + 6^ITH " i l n ( 1 + 6 u )L + 62 (1 + bu)2 du 1 6 2 ( l + 6zi) 62 -1 1 1 + - l n ( l + 6u) + J - l 62 b(l + bu) 1 + - 4 262 1-6 , + ( l - 6 2 ) . (D.52) (D.53) (D.54) (D.55) (D.56) (D.57) (D.58) (D.59) (D.60) (D.61) Appendix D. Theoretical Autocorrelation Functions: Proofs 1 1 0 = | ( 2 - 6 = ) + 2 1 .6 V b3. In 1 + 1 - 6 , Substituting A (Equation D.62) into GQ (Equation D . l ) , Gl(r) = 2 6 2 ( 2 - 6 ' ) = - £ ( i - * ) -Substituting u and 6, 2 6 1 b2 - 1 6 2 6 3 J In In ' l + feN 1 - 6 , > 1 + 6 N 1 - 6 , G2(r) = " 4sech2y tanh2y 4 sinh2y 4 n sech2y , / l + tanhy' 2 — r r In — 1 r1-tanh y \ 1 - tanhy/ sech2y tanh3y sech2y tanh3y sinh y 4 — 2 ~ [ycothy - 1 ] In (e-2y) (-2y) sinh y It can be shown using l'Hospital's rule that lim [y cothy-(D.62) ( D . 6 3 ) (D.64) (D.65) (D.66) (D.67) (D.68) (D.69) y-+o sinh2y Normalizing, the autocorrelation function assuming a hyperbolic secant squared intensity pulse shape is Gl(r) = sinh2y [y cothy - 1 ] (D.70) D.2 Gaussian The gaussian intensity pulse shape is described by I(t) = e"*2 where x = ^ ( D . 7 1 ) Appendix D. Theoretical Autocorrelation Functions: Proofs 111 Also letting y = £ (ie. adjust time scale), the numerator in Equation D.46 becomes /oo I(x) I(x + y) dx -oo Letting f*g be the convolution of f(x) and g(x), /oo I(x)I(x + y)dx = Fl(-y) -oo Letting y' = — y and using the convolution theorem for Fourier Transforms, /oo I(x)I(x + y)dx = ri(y') -oo = ^ / _ ~ ^ [ / ( x ) ] ^ [ 7 ( x ) ] e « ^ ' c i a 2^7-00 1^ /2 J I \/2 J 1 _ z = 2 6 2 (D.72) (D.73) (D.74) (D.75) (D.76) (D.77) (D.78) Normalizing (ie. taking expression D.78 and dividing by itself when y = 0), the auto-correlation function assuming a gaussian intensity pulse shape is Gl(r) = e-V (D.79) D.3 Lorentzian The lorentzian intensity pulse shape is described by i(t) = . w h e r e x = f (D-8°) Appendix D. Theoretical Autocorrelation Functions: Proofs 112 Also letting y = j, (ie. adjust time scale), the numerator in Equation D.46 becomes /oo I(x)I(x + y)dx -oo Letting f*g be the convolution of f(x) and g(x), /oo I(x)I(x + y)dx = ri(-y) -oo Letting y' = —y and using the convolution theorem for Fourier Transforms, /oo I(x)I(x + y)dx = ri(y') -oo = T~x [n2e~2a] 1 (D.81) (D.82) (D.83) (D.84) (D.85) (D.86) U + ( 0 J Normalizing (ie. taking expression D.87 and dividing by itself when y = 0), the auto-correlation function assuming a lorentzian intensity pulse shape is (D.88) 7T — < 2 D.4 Symmetric Two-sided Exponential The symmetric two-sided exponential intensity pulse shape is described by I(t) = e~2W where x = ^ (D.89) Also letting y = ^ (ie. adjust time scale), and letting y > 0 without loss of generality, the numerator in Equation D.46 becomes /oo too I(x)I(x + y)dx = / e~2W e~2|*+y| dx (D.90) -oo J—oo Appendix D. Theoretical Autocorrelation Functions: Proofs 113 J—oo J—y + I e - 2 x e ~ 2 ^ d x Jo e + 4 x d x + e~2y / dx -co J—y /•oo / e"4a Jo = e + e -2tf = e 2 y = e-2y ,+4x -y dx ,—4x -4 Normalizing (ie. taking expression D.94 and dividing by itself when y = 0), G2(r) = e-2y(l + 2y) (DM) (D.92) (D.93) (D.94) (D.95) Since the autocorrelation function is symmetric, the autocorrelation function assuming a symmetric two-sided exponential intensity pulse shape is G20(r) = e2^ (1 + 2|2/|) (D.96) Appendix E The K D P Phase-matching Angle KDP has belongs to the symmetry group 42m (non-inversion symmetry). The indices of refraction for the ordinary and extraordinary ray at 616 nm and 308 nm (interpolation from Table 16.3 in [36]). n£ = 1.467840 n2e" = 1.496720 n£ = 1.508628 n 2" = 1.543883 Since n" < n", KDP is a negative uniaxial crystal. Two types of phase matching are possible: Type I, in which both incident photons have the same polarization, and Type II, in which they are orthogonal. In this experi-ment, Type I second-harmonic generation was used. In this case, the input fundamental wavevector(s) and the second-harmonic wavevector must add up to zero. This condition, phase matching, leads to the following equations [36]. n 2 e w ( 0 m ) = < if n e < n a (E.97) 1 cos2flm sin2flm or, solving for 0m, sin2 0 - ( r a")~ 2 ~ ( no")~ 2 CE 99) Substituting for the values for KDP above, 0m •= 59.3° (E.100) 114 Appendix E. The K D P Phase-matching Angle 115 Since the femtosecond pulse has a large bandwidth, the phase matching angle must be forgiving enough to allow for the extra bandwidth. Otherwise, the efficiency of SHG will decrease. The phase matching bandwidth [37] is SXi = ±(1.39 /im/cm)-^-Z7r/ dXx 2~?>XT (E.101) where / is in centimeters and A is in microns. Using Table 16.3 [36], for Ai = 0.616 and A2 = 0.308 /xm, SXX » 2 fs, which explains the necessity for a photomultiplier tube at the upper operating high voltage range. Pulse broadening of the incident fundamental beam an occur if the crystal is too thick. This is due to the group velocity mismatch between the high and low frequency components of the pulse. The difference in propagation time through the crystal [38] is dk du (E.102) where u± are the highest and lowest frequency components of the pulse. For a crystal thickness, / = 0.5 mm, and a 60 fs pulse, T<J < 6 fs. Pulse broadening in the second harmonic can also occur if the crystal is too thick. The peak amplitude of the second harmonic saturates and clipping occurs. This is due to the group velocity mismatch between the fundamental and the second harmonic pulses. The pulse width should be larger than the group delay [37], <"»> Using Table 16.3 [36], for Ai ± 0.616 pm. and A2 = 0.308 pm, TD < 20 fs, which is less than the shortest pulse obtained in this experiment (67 fs). However, for noncollinear autocorrelation, it does not matter that the second harmonic is broadened in time. With collinear autocorrelation, this would decrease the contrast ratio. Appendix E. The K D P Phase-matching Angle 116 i Changes in crystal thickness, /, cause opposite effects in the phase matching band-width, 8X1, and group delay, TD. With collinear autocorrelation, a thinner crystal is recommended to increase the phase matching bandwidth, increase the SHG efficiency, and decrease the group velocity dis-persion of the fundamental beam. Appendix F Curve Fitting The following are the programs used to analyse data in this thesis. 1. Polynomial curve fitting program (used for the calibration curves of autocorrelator). 2. Minuit1 subroutine for curve fitting the theoretical autocorrelation functions to the autocorrelator data. 1 minimization program from C E R N [39] 117 Appendix F. Curve Fitting F . l Calibration Curve C Program to plot the c a l i b r a t i o n curve f o r the autocorrelator. C This w i l l smoothly f i t experimental data C F i l e 3 = input data f i l e C F i n a l l i n e i n header s t a r t s with — C F i l e 6 = output data f i l e PROGRAM CALIB C Character s t r i n g to copy header from input f i l e to output f i l e CHARACTER*100 HDR CHARACTER*1 HDRRAY(IOO) EQUIVALENCE (HDR.HDRRAY) C The number of points to plo t INTEGER*4 NP/1/ C Maximum degree of polynomial which w i l l be f i t t e d . I f LK=.FALSE. C then K ( f i n a l ) w i l l be the degree of polynomial chosen INTEGER*4 K/30/ C Array of independent variable values (including equal error bars) REAL*8 XRAY(300), XERAY(300) C Array of dependent variable values (including equal error bars) REAL*8 YRAY(300), YERAY(300) C Array of f i t t e d values of corresponding to XRAY REAL*8 YFRAY(300), YF1RAY(300) C Array of residuals (YRAY(i) - YFRAY(I)) REAL*8 YDRAY(300) C Array of weights asscoiated with the XRAY REAL*8 WTRAY(300)/300*1.0/ C NWT = 0 i f array of weights, WTRAY, are to default to 1.0 C NWT <> 0 i f array of weights, WTRAY, i s to be used INTEGER*4 NWT/0/ C On e x i t , c o e f f i c i e n t s of the generated polynomials REAL*8 S(31) C On e x i t , the contents of t h i s array determine the best f i t REAL*8 SIGMA(31) C On e x i t , (alpha)l,...,(alpha)KI REAL*8 A(30) C On e x i t , (beta)1,...,(beta)KI Appendix F. Curve Fitting REAL*8 B(30) C On e x i t , weighted sum of squares of the residuals REAL*8 SS.SSO C f l a g set to .FALSE, i f a polynomial of degree less than or C or equal to K ( i n i t i a l ) i s to be f i t t e d to the data C f l a g set to .TRUE, i f a polynomial of degree equal to C K ( i n i t i a l ) i s to be f i t t e d to the data L0GICAL*4 LK/.FALSE./ C On e x i t , the c o e f f i c i e n t s of the polynomial of degree K ( f i n a l ) REAL*8 P(31) C I f K reaches NP-2 then t h i s flag' goes true L0GICAL*4 KOVER/.FALSE./ C f-value REAL*8 F C Copy header 5 READ(3,500) HDR WRITE(6,500) HDR 500 FORMAT(A80) IF ((HDRRAY(l).NE.'-').OR.(HDRRAY(2).NE.'-')) GOTO 5 C Read i n data DO 10 NP=1,300 10 READ(3,*,END=20) XRAY(NP), XERAY(NP), YRAY(NP), YERAY(NP) 20 NP = NP - 1 WRITE(6,1000) NP 1000 FORMAT('*** PROGRAM CALIB ***'/'NUMBER OF DATA POINTS: ',13) LK = .FALSE. K=NP-2 C C a l l subroutine DOLSF to f i t data to polynomial CALL DOLSF(K,NP,XRAY,YRAY,YFRAY,YDRAY,WTRAY,NWT, + S,SIGMA,A,B,SS,LK,P) C P r i n t out the c o e f f i c i e n t s of each of the terms i n the polynomial DO 30 1=0,K WRITE(6,2000) I,P(I+1) 2000 FORMAT('COEFFICIENT OF V**',I2, ' TERM = >.G20.10) 30 CONTINUE Appendix F. Curve Fitting C Calculate f-value SSO=SS LK=.TRUE. IF (K.EQ.(NP-2)) THEN KOVER = .TRUE. K = K - 1 ELSE KOVER = .FALSE. K = K + 1 ' ENDIF C C a l l subroutine DOLSF to f i t data to polynomial CALL DOLSF(K,NP,XRAY,YRAY,YF1RAY,YDRAY,WTRAY,NWT, + S,SIGMA,A,B,SS,LK,P) IF (KOVER) THEN K = K + 1 F = SSO SSO = SS SS = F ENDIF F=(SSO-SS)/SSO*(NP-K) C P r i n t f-value IF (KOVER) THEN WRITE(6,2999) K 2999 FORMAT('Maximum degree of polynomial reached',12) ENDIF WRITE(6,3000) K-l.F 3000 FORMAT('The f-value f o r a polynomial order ',12, + ' i s ' ,G20.10/'~') C Copy o r i g i n a l data and f i t t e d data to .OUT f i l e WRITE(6,4000) 4000 FORMAT( 'EXPT X DATA',2X,3X,'+/- error',3X, + 2X,'EXPT Y DATA',2X,3X,'+/- error',3X, + 2X,'FITTED Y DATA') DO 60 1=1,NP WRITE(6,5000) XRAY(I),XERAY(I), YRAY(I),YERAY(I), YFRAY(I) 5000 F0RMAT(5(G13.5,2X)) 60 CONTINUE STOP END Appendix F. Curve Fitting F.2 Minuit Subroutine C This subroutine i s to be used by MINUIT to f i t data to a C hyperbolic secant squared pulse (light intensity). C Arguments: NPAR = number of variable parameters (max.15) C GRAY = vector into which the derivatives are to be put C CHISQR = values of the function to be minimized C PRAY = vector containing the external parameter values C IFLAG = 1 for i n i t i a l i z i n g entry, C read i n data C =2 for normal entry with gradient, C calculate the derivatives i n G and C value of CHISQR C =3 for terminating entry, C print f i n a l parameter values, C value of CHISQR, C plot (or output) original and f i t t e d data C =4 for normal entry without gradient, C calculate the value of CHISqR only C =5 not used (reserved) C SUBROUTINE FCN(NPAR,G,CHISQR,P,IFLAG) IMPLICIT REAL*8 (A-H,0-Z) C0MM0N/MINERR/ERP(30),ERN(30) REAL+8 G(20) REAL*8 CHISQR REAL*8 P(20) C Character string to copy header from input f i l e to output f i l e CHARACTER*80 HDR C Character string for f i l e name CHARACTER*80 FN C The number of data points to plot INTEGER*4 NP C Array of data points REAL*8 XARRAY(500), YARRAY(500) C Min and Max of data points REAL*4 XMIN, XMAX, YMIN, YMAX C Time variable scaled by 1/T and time shifted as i n literature REAL*8 Yl C Variable for calibration curves REAL*8 CALIB(2,2) Appendix F. Curve Fitting 122 REAL*8 TURNS2FS REAL*8 T2TAU(4) C Variable for calibration curve to be used C 1 = wide range (subpicosecond) C 2 = narrow range (femtosecond) INTEGER*4 AC C Variable for choice of theoretical function to f i t C 1 = hyperbolic secant squared C 2 = lorentzian C 3 = gaussian C 4 = exponential INTEGER*4 FIT C Variable for square of standard deviation of data points around C the theoretical curve REAL*8 SIGMA2 INTEGER*4 NSIGMA C Variables for plotting REAL*4 DX(500), DYE(500), DYF(500) C Array containing the theoretical data points REAL*8 YFRAY(500) C Array for printing the date and time CHARACTER*4 DAT(7) GOTO (100,200,400,400,500,600) IFLAG RETURN 100 CONTINUE SIGMA2=1.0 T2TAU(1) T2TAU(2) T2TAU(3) T2TAUC4) 1.762747174D0 2D0 1.665109222D0 0.69314718D0 TURNS2FS = 1D0/80D0*2.54D-2*2D0*1D15/2.997925D8 CALIB(1,1)=0.4667938691D0 *TURNS2FS CALIB(1,2)=0.0D0 *TURNS2FS CALIB(2,1)=0.1961521936D0 *TURNS2FS CALIB(2,2)=-0.1762093423D-02 *TURNS2FS AC=0 Appendix F. Curve Fitting 123 FIT=0 C — Read i n / Write out header — WRITE(7,800) '==' 800 FORMAT(A2) READ(3,900) FN WRITE(7,900) FN 105 READ(3,900) HDR IF (HDR.EQ.'AUTOCORRELATOR #1') IF (HDR.EQ.'AUTOCORRELATOR #2') IF (HDR.Eq.'FIT = SECH"2') IF (HDR.EQ.'FIT = LORENTZIAN') IF (HDR.EQ.'FIT = GAUSSIAN') IF (HDR.Eq.'FIT = EXPONENTIAL') WRITE(7,900) HDR 900 FORMAT(A80) IF (HDR(1:2) .NE.' —') GOTO 105 C — Check to see i f autocorrelator and f i t s p e c i f i e d — IF (AC.Eq.O) THEN WRITE(7,*) 'AUTOCORRELATOR NOT SPECIFIED' STOP ENDIF IF (FIT.EQ.O) THEN WRITE(7,*) 'TYPE OF FIT NOT SPECIFIED' STOP ENDIF jl i C — Read i n data — DO 110 NP=1,500 READ(3,*,END=120) XARRAY(NP), YARRAY(NP) 110 CONTINUE 120 NP = NP - 1 AC=1 AC=2 FIT=1 FIT=2 FIT=3 FIT=4 C — Convert X-axis p o s i t i o n to time using c a l i b r a t i o n equation — DO 130 1=1,NP XARRAY(I) = ( CALIB(AC,1) + CALIB(AC,2)*XARRAY(I) ) * XARRAY(I) 130 CONTINUE C — Insert new header — CALL CDATE(DAT) Appendix F. Curve Fitting 124 WRITE(7,1000) DAT 1000 FORMAT('*** PROGRAM PULSE ***'/7A4) GOTO (131,132,133,134) FIT 131 WRITE(7,1001) 1001 FORMAT( 'Kt) = SECH(X)**2 where X=t/T' / + 'G(tau) = 3*[Y*C0TH(Y)-1] / SINH(Y)**2 where Y=tau/T' ) GOTO 139 132 WRITE(7,1002) 1002 FORMAT('I(t) = 1/(1+X**2) where X=t/T' / + 'G(tau) = l/(l+(Y/2)**2) where Y=tau/T' ) GOTO 139 133 WRITE(7,1003) 1003 FORMAT('I(t) = EXP(-X**2) where X=t/T' / + 'G(tau) = EXP(-(Y**2)/2 where Y=tau/T' ) GOTO 139 134 WRITE(7,1004) 1004 FORMAT('I(t) = EXP(-2*ABS(X)) where X=t/T' / + 'G(tau) = (1+2*ABS(Y)) * EXP(-2*ABS(Y)) where Y=tau/T' ) GOTO 139 139 WRITE(7,1100) NP 1100 FORMAT('NUMBER OF DATA POINTS: ',13) C — Determine min and max of data — XMAX = XARRAY(l) XMIN = XMAX YMAX = YARRAY(l) YMIN = YMAX , DO 140 1=1,NP IF (XARRAY(NP).GT.XMAX) XMAX = XARRAY(NP) IF (XARRAY(NP).LT.XMIN) XMIN = XARRAY(NP) IF (YARRAY(NP).GT.YMAX) YMAX = YARRAY(NP) IF (YARRAY(NP).LT.YMIN) YMIN = YARRAY(NP) 140 CONTINUE 200 CONTINUE Appendix F. Curve Fitting C — Calculate gradient function here — 400 CONTINUE C — Calculate theoretical function corresponding to data points C — P(l) = TIME SCALE FACTOR (is 1/T i n autocorrrcamoplp) C — P(2) = TIME SHIFT (to be used to center peak at 0 fs) C — P(3) = INTENSITY AMPLITUDE (MAX HEIGHT) C — p(4) = INTENSITY BACKGROUND (ZERO HEIGHT) DO 410 1=1,NP Yl = P(1)*(XARRAY(I)-P(2)) GOTO (411,412,413,414) FIT C *** Hyperbolic secant squared *** 411 CONTINUE IF ((Y1.GT.30).OR.(Y1.LT.-30)) THEN C — Assume extreme t a i l s of pulse as equal to 0.0 — YFRAY(I) = P(4) ELSE C -- Calculate main pulse — IF ((Yl.LT.lE-2).AND.(Yl.GT.-lE-2)) THEN C — Near peak so1 equate to peak — YFRAY(I) = P(3) + P(4) ELSE C — Away from peak — YFRAY(I) = P(3) * (3* (Y1/DTANH(Y1)-1)/DSINH(Y1)**2) + P ENDIF ENDIF GOTO 410 C *** Lorentzian *** 412 CONTINUE YFRAY(I) = P(3)/(l+(Yl/2)**2) + P(4) GOTO 410 C *** Gaussian *** 413 CONTINUE YFRAY(I) = P(3)*DEXP(-(Yl**2)/2) + P(4) GOTO 410 C *** Exponential *** 414 CONTINUE Appendix F. Curve Fitting 126 Yl = ABS(Yl) YFRAY(I) = P(3)*(1+2*Y1)*DEXP(-2*Y1) + P(4) GOTO 410 410 CONTINUE C — Calculate Chi Squared — CHISQR =0.0 DO 420 1=1,NP CHISQR = CHISQR + (YFRAY(I) - YARRAY(I))**2/SIGMA2 420 CONTINUE IF (IFLAG.NE.3) RETURN C — Calculate errors ensuring they're correct — C DO 310 I=1,NPAR C IF (ERP(I).LT.O) ERP(I)=0 C IF (ERN(I).GT.O) ERN(I)=0 C 310 CONTINUE C — Calculate pulse width with errors — P(NPAR+1) = T2TAU(FIT)/P(1) ERP(NPAR+1) = T2TAU(FIT)/(P(1)+ERN(1)) - P(NPAR+1) ERN(NPAR+1) = T2TAU(FIT)/(P(1)+ERP(1)) - P(NPAR+1) — Output results for f i n a l parameter values --WRITE(7,1201) P(l),ERP(1),ERN(l) 1201 FORMAT(' TIME SCALE FACTOR (1/T): ',G13. 5, ' +/-: ',2(G13.5,2X)) WRITE(7, 1202) P(2),ERP(2),ERN(2) 1202 FORMAT(; TIME SHIFT : ' ,G13 5, ' +/-: ',2(G13.5,2X)) WRITE(7, 1203) P(3),ERP(3),ERN(3) 1203 FORMAT(' INTENSITY AMPLITUDE : ' ,G13 5, ' +/-: ',2(G13.5,2X)) WRITE(7 1204) P(4),ERP(4),ERN(4) 1204 FORMAT(1 INTENSITY BACKGROUND : ' ,G13 5, ' +/-: ',2(G13.5,2X)) WRITE(7 1205) P(5),ERP(5),ERN(5) 1205 FORMAT( PULSE WIDTH (tau-p) : ' ,G13 5, ' +/-: ',2(G13.5,2X)) WRITE(7 ,1300) CHISQR ;: 1300 FORMAT( 'CHI SQUARED = \G13.5) WRITE(7 ,1310) CHISQR * SIGMA2 / NSIGMA 1310 FORMAT( 'S.D.(POP) = \G13.5) Appendix F. Curve Fitting 127 C — Output data (experimental and f i t t e d ) — WRITE(7,1400) 1400 FORMAT(' — V'EXPT X DATA',4X, 'EXPT Y DATA' ,4X, 'FITTED Y DATA') DO 320 1=1,NP WRITE(7,1500) XARRAY(I)-P(2), + (YARRAY(I)-P(4))/P(3), (YFRAY(I)-P(4))/P(3) 1500 F0RMAT(3(G13.5,2X)) 320 CONTINUE C — Plot data — DO 330 1=1,NP DX(I) = XARRAY(I)-P(2) DYE(I) = (YARRAY(I)-P(4))/P(3) DYF(I) = (YFRAY(I)-P(4))/P(3) 330 CONTINUE NP = -NP CALL ALAXIS(FN,20,' ',1) CALL ALSCAL(0.0,0.0,-0.15,1.35) CALL ALGRAF (DX, DYE, NP-1) CALL ALGRAF(DX,DYF,NP,0) CALL ALDONE RETURN 500 CONTINUE RETURN 600 CONTINUE C — Calculate Standard Deviation of Data i n window C — of four times the pulse width centered on pulse. — SIGMA2 = 0D0 NSIGMA = 0 DO 415 1=1,NP Y l = P(1)*(XARRAY(I)-P(2)) IF ( ABS(Yl).LT.2*T2TAU(FIT) ) THEN SIGMA2 = SIGMA2 + (YARRAY(l)-YFRAY(l))**2 NSIGMA = NSIGMA +1 ENDIF 415 CONTINUE SIGMA2 = SIGMA2/NSIGMA RETURN END Bibliography [1] J. Meyer and A. Y. Elezzabi. 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