Ultrashort Pulsed Laser Diagnostics Using a Second Harmonic Generation Frequency Resolved Optical Gating Apparatus (SHG FROG)

Ultrashort Pulsed Laser Diagnostics Using a Second Harmonic Generation Frequency Resolved Optical Gating Apparatus (SHG FROG) Eugene Torigoe Department of Physics and Astronomy, Binghamton University Linn Van Woerkom Research Group Ohio State University REU Summer 2 1

Introduction/Background: In the fields of physics, and chemistry there have been a growing number of experiments that require the implementation of ultrashort laser pulses. Ultrashort-pulsed laser systems are required because they allow experimenters to probe systems such as atoms with time scales proportional to electron orbits. In these types of experiments the width of the pulse is used to determine the amount of temporal resolution of the collected data, and therefore the length of the pulses is extremely important. But as the need for ever-shorter pulses has increased, so have the capabilities of the new laser systems. Today there are laser systems that can create pulses that are as short as a few femtoseconds (1^-15). As the technology progressed, however, there arose the difficulty of properly characterizing these pulses. When measuring something it is usually necessary that the probe being used be at least as small as the thing you are trying to study, so the question arose of how one could measure a pulse that could be as short as only a few femtoseconds. The problem was not only the ability to accurately quantify the duration of these pulses but also the frequency distributions along the pulses, given by the phase of the electric field of the pulse, which may have a major effect on the outcome of an experiment. While being compressed the pulse may gain a frequency distribution, known as a chirp. A pulse is said to have a positive chirp if along the pulse the frequency is greater at the leading edge than at the trailing edge, a negative chirp if the opposite is true. 2 The classical method of obtaining information about short pulses was a method known as autocorrelation. Autocorrelation solved the problem of finding a probe as 2

Aperture Neutral Density Filter Aperture Beamsplitter SHG Crystal Translation Stage Camera Connected to Computer Diagram #1: The schematic for an autocorrelator. The pulsed beam enters the apparatus polarized parallel to the surface of the table. The pulse is split into two identical pulses in the beam splitter, the two pulses travel equal optical distances and are recombined in the SHG crystal, which in this case is Potassium Dihydrogen Phosphate (KDP). Through nonlinear optical processes the two beams combine to output a frequency doubled signal beam. The signal beam is redirected into a CCD camera and is recorded. small as pulses being studied by using the pulses to probe themselves. Diagram #1 shows the schematic for an autocorrelation device. The pulse is split and then recombined in such a way that the two copies of the pulse recombine to both spatially and temporally overlap in a second harmonic generation (SHG) crystal. Second harmonic generation is a process under which the two pulses are crossed and vectorally combine to create a signal beam that is twice the frequency of the two initial pulses (Figure #2). Second harmonic generation is classified as a second order nonlinear optical effect because the intensity of the signal beam is dependent upon the intensities of the two input beams in a nonlinear manner. 1 This useful because it allows us to detect the moment when the pulses are exactly overlapped. If the interaction were not nonlinear then the intensity of the signal beam would be independent of whether the pulses were overlapped or slightly separated. 6 The autocorrelation method is also useful because the process in effect translates temporal information of the two initial pulses into spatial characteristics of the signal beam. The width of the pulse can be determined by taking two images of the signal pulse 3

Diagram #2: - When the two 8nm. wavelength pulses overlap inside of the SHG crystal they combine to form a signal beam with a wavelength of 4 nm. This effect in an Second Harmonic Generation (SHG) crystal was one of the discoveries that lead to the development of the nonlinear optics field of study. Nonlinear optics signifies that the phenomenon, in this case the intensity of the signal beam is dependent upon the intensity of the two input beams in a nonlinear manner. For SHG, Boyd writes that the signal beam tends to increase as the square of the intensity of the applied laser light. - The SHG Interaction is useful in the measurement of the properties of the ultrashort laser pulses because it allows us to transform temporal characteristics of the input pulses into spatial characteristics of the signal beam. at different delays, which can be achieved by moving the translation stage, and by using the following equation: t = K * Calibration * FWHM (Eq. 1) t is the width of the pulse, K is the pulse shape constant (For an assumed Gaussian pulse shape K =.771), and Calibration is the time delay to pixel calibration of the camera. The calibration is given by: d Calibration = (Eq. 2) pix * c d is the difference in the spatial delay between the two images taken, pix is the number of pixels difference between the peaks of the two images, and c is the speed of light. As the delay is varied between the two initial pulses, they are no longer exactly temporally overlapped in the crystal causing a side to side translation of the signal beam, the direction of translation depending on which beam is delayed. This phenomenon can be explained by using geometrical arguments of the pulse overlap region. 3,4 Appendix #1 shows an example of an autocorrelation measurement. 4

Beam Splitter SHG Crystal Imaging Spectrometer CCD Camera Translation Stage Diagram #3: This is a schmatic diagram for an SHG FROG. Like the autocorrelation device the two beams are combined in the SHG crystal and a frequency doubled signal beam is created. This signal beam is then sent through an imaging spectrometer which outputs the beams frequency as a function of distance. The output of the spectrometer is what is known as the FROG trace, and is captured by a CCD camea. The width of the pulse and the frequency distribution of the pulse is then determined using an algorithm in a computer connected to the camera. The amount of characterization of the pulses that one gains from the autocorrelation method is by no means exhaustive. One of the major problems with autocorrelation is the fact that one must guess upon the actual shape of the pulse to find the width of the pulse. For example, if the pulse were actually a Sech 2 pulse the correct value for K would be.6452, resulting in a 1% error in the measured length of the pulse. Another problem of autocorrelation is that it does not shed any light upon the frequency fluctuations of the pulse. 4 In 1993 a group at Sandia National Laboratory, led by Rick Trebino solved these difficulties by developing a method known as Frequency Resolved Optical Gating (FROG). 2 Just as the name of the method indicates, not only is this method capable of finding the intensity and the phase of the electric filed of the pulse to a very high degree of accuracy. From information about the intensity and the phase the width of the pulse and the frequency distribution along the pulse can then be found directly. Although there are many different types of FROG s the type of geometry I will focus on is what is 5

known as the Second Harmonic Generation (SHG) FROG (Diagram #3). Under this FROG geometry, an SHG crystal is used just like in the autocorrelation device to resolve the time axes, but additionally the signal beam is sent through an imaging spectrometer, which uses diffraction gratings to separate the light of the signal beam, in effect spatially representing the frequency of the signal beam. The spectrometer outputs to a camera the images of the signal beam after it has be separated into its component wavelengths, known as FROG traces. Although at this point a trace is on the time, frequency axes the characteristics of the pulses must still be retrieved because the trace does not directly tell us the width of the pulse or the phase. To retrieve this information the trace is sent through a FROG algorithm, which uses constraints to iterate to a unique solution of both the phase and intensity of the pulse as a function of time. The intensity as a function of time will give us the width of the pulse. The most common algorithm used is a type known as generalized projections (GP). GP is an iterative method that uses projections between two different constraint sets to arrive at a unique solution that satisfies both constraints (Diagram #4). For SHG these two constraints are as follows: 3 Constraint #1: E SIG ( t, τ) = E( t)* E( t τ) (Eq. 3) Constraint #2: I FROG ( ϖ, τ) = dt * E ( t, )* exp( i t) SIG τ ϖ 2 2 = ( ϖ, τ) (Eq. 4) E SIG 6

Constraint 1 Constraint 2 Diagram #4: Schematic diagram of generalized projections. The algorithm projects onto values from one constraint to the other until the unique solution for the electric field is determined. It should be noted that GP only guarantees convergence when the two sets are convex. In the diagram the sets are convex if a line drawn from any two points in the sets never leaves the boundaries of the set. E SIG is the electric field of the signal beam, I FROG is the intensity of the signal beam, t is time, τ is the delay, ω is the frequency. The GP algorithm uses Fourier transforms to project from the frequency domain, to the time domain, and vice versa. The solution is found once a value of E SIG (t,τ) is found, from which E(t) can be directly calculated. My Project: This summer I worked in the Ohio State University Physics Department, in the Linn Van Woerkom research group, in the REU program funded by the NSF. My project for the summer was to build a second harmonic generation frequency resolved optical gating apparatus (SHG FROG) to be used to run diagnostics on the laser systems. Diagram #5 is a schematic of the SHG FROG I built. The Van Woerkom Lab has two identical laser systems, the two systems create 8nm wavelength, roughly 12 fs pulses at a repetition rate of a kilohertz. The laser system in room Smith 58 is known as the Hans laser, and the laser system in room Smith 46 is known as the Franz laser. In the Van Woerkom lab the lasers are used for a variety of different types of research from the 7

study of multiphoton ionization of noble gases, to the study of light emitting polymers. It is important that the lab have a system that is capable of finding both the frequency distribution and the width of the pulses they use in their experimental work. 5 From Franz Flipper mirror Neutral Density Aperture Filter Aperture Inverter SHG Crystal Translation Stage CCD Camera Blue Nuetral Density Filter Imaging Spectrometer From Hans Diagram #5: Picture of the SHG FROG I bulit in the Van Woerkom lab in Smith 68. A flipper mirror is used to switch between the beam from the Hans laser and the beam from the Franz laser located in Smith 46. The size of the apparatus is limited to the size of the FROG bench which is 2 feet by 1.5 feet. Building of the SHG FROG: The majority of the summer was spent on the building of the apparatus, which included the alignment and the cleaning of optics. A lot of time was spent making sure that the beams were perfectly clean when they entered the SHG crystal because any aberrations that were present in the input pulses were then also present in the SHG signal and were major causes of bad looking traces. Additionally the FROG needed to be 8

versatile enough to work around other experiments being performed in the lab, so an elaborate network of flipper mirrors was constructed to navigate around other peoples work. Even before the beams reached the first aperture the beams had been reflected by a number of mirrors. From the Hans laser to the SHG apparatus the beam was reflected through 5 mirrors, and from the Franz laser to the apparatus the beam needed to be reflected through 8 mirrors. The setup was quite prone to misalignment, and the task of tracking down the sources of aberrations in the beams was quite difficult. The first part of the construction was to build an autocorrelation device. Appendix #1 shows some of the autocorrelation data taken. By successfully building an autocorrelation device it was ensured that the optical path lengths between the beam splitter and the SHG crystal (which in this apparatus was Potassium Dihydrogen Phosphate) were exactly the same the separated pulses. In order to achieve the creation of the frequency doubled signal pulse the two copies of the input pulse needed to be perfectly overlapped both spatially and temporally. Once the signal beam was achieved, many hours were spent changing, and cleaning mirrors and other optics, paying close attention to the image of the autocorrelation trace on a monitor. To then build the FROG apparatus the signal beam was redirected into the imaging spectrometer. The most difficult part of doing that was setting up the inverter assembly to properly redirect the signal beam into the imaging spectrometer. The inverter was necessary because the output of the spectrometer set the axis for the frequency as the horizontal direction, so the signal beam was rotated 9 degrees so that it would translate up and down for delay. The resulting FROG trace was set on a delay vs. wavelength axes (Diagram #6). 9

2 15 Diagram #6: Picture of a FROG trace captured on August 8, 2 1 5 5 1 15 2 25 3 The FROG traces did not look so good so to improve them the SHG crystal was moved farther away from the mirrors to decrease the angle between the two input beams. This technique greatly increased the resolution of the signal beam. The wavelength and delay axes needed to be calibrated, to find the wavelength/pixel, and the delay/pixel values of out image. To find the wavelength calibration an Ar lamp source was placed at the input of the imaging spectrometer and images of the output spectra were recorded. Using published values for the Ar spectra the published values for the wavelength separation between the peaks were compared to the measured separation of corresponding peaks in pixels. To find the calibration for the time delay axis images of the Frog trace at two different time delays were recorded. The amount of time delay could be calculated by taking the difference of the translation stage reading multiplying by 2 and then dividing by the speed of light. The number of pixels the trace translated between the two images was calculated and divided the time difference by that measurement to find the delay to pixel calibration. The UnFrog Algorithm: 1

The next step was to then retrieve the electric field of the pulses using an algorithm. The algorithm that we used was an algorithm program named UnFROG, written by the Bern Kohler Group in the Chemistry Department at Ohio State. 8 A former undergraduate student named Ryan Barnett was primarily responsible for compiling and setting the program up on the computers in the Van Woerkom lab. The UnFrog program is based upon the generalized projections iterative algorithm. The UnFrog program accepts input in the form of a tab delimited text file. The files need to be converted to a specific type of text format in order for them to work in the program. I wrote a function script named ConvertFrog() in Igor Pro to automatically convert image files to the correct text file format. The program prompts the user to select a Tiff image file, the type of file that is taken by the CCD camera, and automatically converts the image file into the correct text format (Appendix #2). The text format that the UnFrog program accepts has a header which lists the values for the length and height of the trace matrix on the first line, then a line of 2 separated zeros, a space then a list of the delay values for each of the rows of the matrix, another space then a list of the wavelength values for each of the columns of the matrix, another space and then the square matrix with each component of the matrix representing the intensity measurement for each pixel of the image. The program is set to recognize the center of the wavelength axis to be at 4 nm, and the center of the delay axis to be at delay. Changing the center variable in the function script on Igor Pro can change the center wavelength to match the center of the imaging spectrometer if it is not at 4 nm. As long as the imaging spectrometer is set at center of 4 nm the labels should matchup perfectly with the actual values for the image (assuming the calibration is correct). The 11

values for the wavelength and the delay in the program are based on calibration measurements I took for the spectrometer and camera, and may therefore be off by some amount. The only problem with the ConvertFrog() program is that it outputs a 1 megabyte text file. Because of this a second program named SConvertFrog() was created. SConvertFrog will work if the trace is well centered on the screen, the resulting output file is a lot smaller because the matrix used is a lot smaller. Once you load the file onto the program you can then run the algorithm. There were a few problems that arose in the use of the algorithm, the first is that the program expects that the higher intensity pixels will have higher numerical values, but the CCD camera assigns higher numerical values to the lower intensity readings. To overcome this difficulty all one needs to do is invert the colors of the image by pressing the invert button in the Frog Image window of the UnFrog program. The second difficulty is that one often forgets to change the setting of the geometry type to SHG before they run the algorithm. If the setting is incorrect the algorithm will apply the wrong constraints to the traces and the retrieved information will be incorrect. Once the algorithm is finished running it makes 4 graphs, and saves them as _outlip, _out-tip, out-wip files. The four graphs are of intensity as a function of delay (stored in the tip file), phase as a function of delay (stored in the tip file), intensity as a function of wavelength (stored in the lip file), and phase as a function of wavelength (stored in the lip file). By analyzing these graphs one may then interpret the different characteristics of the pulses. 7 12

Conclusion/What I was Finally Able to Accomplish: Due to the limit of time in the REU program I was only able to spend a limited amount of time actually working with the UnFrog algorithm. For me to really understand the program I would need a significant amount of extra time to gain experience in retrieving the pulse information, and interpreting the graphical outputs. I was, however able to characterize and categorize a few of the traces I took images for. Appendix #3 shows my attempts at trying to interpret these traces. Thanks: I would first like to thank all of the people in the LVW group. Prof. Van Woerkom (sorry about my inability to call you Linn) for all of the on the spot explanations and encouragement, Moon for helping me find reference materials and for helping me analyze all of the strange phenomena that seemed to pop up, Glen for answering questions and for attempting to optimize the Franz laser, Patch for Igor Pro assistance, Susan keeping me up to date on all of the REU events and physics department gossip, Matt for helping me find optics on the shelves when I couldn t find them myself, and Richard for not getting in my way. Of course I would like to thank Prof. Palmer, and Prof. Van Woerkom (twice over) for organizing the program. In all honesty the Ohio State REU has to be the most organized in the country, just look at all those activities on the schedule. And of course none of this would be possible without the deep pockets of the NSF, which is gracious enough to fund programs like the REU s to encourage and inspire students to pursue the fields of science. 13

References: 1) Boyd, Robert W. Nonlinear Optics San Diego: Academic Press Inc. (1992) 2) R. Trebino, K.W. DeLong, D.N. Fittinghoff, J.N. Sweetster, M.A. Krmbugel, B.A. Richman, D.J. Kane, Rev. Sci Instrum. 68 (9), September 1997 3) K.W. DeLong, R. Trebino, J. Hunter, and W.E. White, J. Opt. Soc. Am. B, Vol. 11, No. 11, November 1994 4) A. Brun, P. Georges, G. LeSaux, and F. Salin, J. Phys. D: Appl. Phys. 24 (1991), pp.1125-1233 5) LVW Group Website, http://www.physics.ohio-state.edu/~lvw/lvwgroup.html 6) Conversations with Prof. Linn Van Woerkom, June 28 August 18, 2 Department of Physics, Ohio State University 7) Conversations with Mark Moon Walker, June 28 August 18, 2 Department of Phsyics, Ohio State University 8) Conversation with Prof. Bern Kohler Department of Chemistry, Ohio State University 14

Appendix 1: Autocorrelation Data Analysis Diagram 1 Diagram 2 2 2 15 15 1 1 5 5 1 2 3 1 2 3 Diagrams 1 and 2 - Images captured of the signal beam from a Second Harmonic Generation (SHG) crystal. Diagram 1 was taken at a reading of 42.79 mm on the translation stage, and Diagram 2 at a reading of 42.84 mm. As the delay between the pulses is increased in the crystal a translation to the right is produced in the signal beam. 1 Diagram 3 12 14 16 18 2 22 24 5 1 15 2 25 3 Diagram 3 - The two plots on the graph represent the intensity of Diagrams 1 and 2 integrated over their y axes. Using the Autocorrelation method one may then measure the width of the pulses being examined. Pulse Width = K * (Delta delay between pulses in meters/ Number of pixels between peaks) * FWHM / c K = Pulse Shape constant. For an assumed Gaussian pulse, K =.771 (Parentheses) = Delay to pixel calibration FWHM = Full W idth at Half Maximum (in pixels). Should be the same for both pulses. c = speed of light. For data given above the pulse width was measured to be about 15 fs.

Appendix #2 #pragma rtglobals=1 // Use modern global access method. Function ConvertFrog() Variable outputfile Open/M = "Save output file as:"/c = "TEXT"/T = "TEXT" outputfile as "Output File In Unfrog Format" if (outputfile!= ) //if user cancels the naming of the output file fprintf outputfile,"%d\t%d\r", 32, 32 fprintf outputfile, "%d\t%d\r",, fprintf outputfile, "\r" variable delcal = 7.3 variable delay = //Size of the matrix //calibration for fs per pixel delay = ((delcal/2)+delcal*159)*-1 do if (delay >= ) fprintf outputfile, "%.2f\r" delay else fprintf outputfile, "%+.2f\r" delay endif delay = delay + delcal while (delay <= ((delcal/2)+delcal*159)) //List of Delays fprintf outputfile, "\r" variable wavecal =.93 variable center = 4 wavelength at the center of the image variable wavelength = //calibration for wavelength per pixel //wavelength setting of the spectrometer of the wavelength = ((wavecal/2)+wavecal*159)*-1+center do //List of wavelengths fprintf outputfile,"%.4f\r" wavelength wavelength = wavelength + wavecal while (wavelength < ((wavecal/2)+wavecal*159+center)+wavecal) fprintf outputfile, "\r\r" Execute "ReadTiff/N = wackyfile/i" //Execute used because ReadTiff is an XOP. //Makes line following it as if it were written in //command line. Duplicate wackyfile, imagefile Edit imagefile.id matrixtranspose imagefile //correctly orient the matrix InsertPoints,4, imagefile //make the matrix square by adding rows to the //top and bottom InsertPoints 28,4, imagefile imagefile [,39] []= 255 //filling the rows with zero intensity '255' imagefile [28,319] []= 255 Redimension/S imagefile //make them single floating point values imagefile = imagefile/255 //divide all cells by 255 Variable row = Variable column =

do do //matrix fprintf outputfile, "%.7f\t" imagefile [row] [column] column = column +1 while (column <= 319) //single float 7 places after //the decimal point fprintf outputfile, "\r" row = row +1 column = while (row <= 319) RemoveFromTable imagefile.id KillWaves /A/Z KillStrings/A/Z KillVariables/A/Z Close outputfile endif End ///////////////////////////////////////////////////////////////////////////////////////// /////////////////// Function SConvertFrog() size // If the trace is reasonably centered it will create a smaller matrix, that will be //faster to convert and smaller in file Variable outputfile Open/M = "Save output file as:"/c = "TEXT"/T = "TEXT" outputfile as "Output File In Unfrog Format" if (outputfile!= ) //if user cancels the naming of the output file fprintf outputfile,"%d\t%d\r", 24, 24 fprintf outputfile, "%d\t%d\r",, fprintf outputfile, "\r" variable delcal = 7.3 variable delay = //calibration for fs per pixel delay = ((delcal/2)+delcal*119)*-1 do if (delay >= ) fprintf outputfile, "%.2f\r" delay else fprintf outputfile, "%+.2f\r" delay endif delay = delay + delcal while (delay <= ((delcal/2)+delcal*119)) fprintf outputfile, "\r" variable wavecal =.93 variable center = 4 variable wavelength = //calibration for wavelength per pixel //spectrometer setting for the wavelength at the //center of the image wavelength = ((wavecal/2)+wavecal*119)*-1+center do

fprintf outputfile, "%.4f\r" wavelength wavelength = wavelength + wavecal while (wavelength < ((wavecal/2)+wavecal*119+center)+wavecal) fprintf outputfile, "\r\r" Execute "ReadTiff/N = wackyfile/i" //Execute used because ReadTiff is an XOP. //Makes line following it as if it were written in //command line. Duplicate wackyfile, imagefile Edit imagefile.id matrixtranspose imagefile //correctly orient the matrix DeletePoints/M=1, 4, imagefile //remove columns from table to make the matrix //square DeletePoints/M=1 239, 4, imagefile Redimension/S imagefile //make them single floating point values imagefile = imagefile/255 //divide all cells by 255 Variable row = Variable column = do do fprintf outputfile, "%.7f\t" imagefile [row] [column] column = column +1 while (column <= 239) //single float 7 places after //the decimal point fprintf outputfile, "\r" row = row +1 column = while (row <= 239) RemoveFromTable imagefile.id KillWaves /A/Z KillStrings/A/Z KillVariables/A/Z Close outputfile endif End

Appendix 3: Hans Pulse Compressor Analysis This is the retrieved intensity and phase information for pulses from the Hans laser system on August 16, 2. The three images were taken at different compressor settings of the laser system. The pulse compressor is an apparatus consisting of a a grating, which separates the frequency components of the pulse, and a translation stage that is used to advance or delay different frequencies witin the pulse. When the frequencies of the pulse are set by the compressor to be in phase the pulse width is minimized. But if the compressor is in the wrong position not only will the pulse be larger but the it will also gain a chirp (freqency distribution) Retrieved Intensity and Phase vs. Delay Intensity 1..8.6.4.2 'Intensity' 'Phase' 15 1 5-5 -1 Phase Image 1: The translation stage is set too far in one direction. The pulse is wider and there should be an associated chirp. -15-2 -1 1 2 Delay Intensity 1..8.6.4.2 'Intensity' 'Phase' 15 1 5-5 -1 Phase Image 2: This is the pulse when it has been minimized by eye. You can see from the intensity profile that the pulse is indeed more compressed.. -15-2 -1 1 2 Delay Intensity 1..8.6.4.2 'Intensity' 'Phase' 15 1 5-5 -1 Phase Image 3: This is the pulse after the compressor has been set too far in the other direction. The chirp associated with this measured pulse should be chirped in the opposite direction as the pulse in image 1.. -15-2 -1 1 2 Delay

Retrieved Intensity and Phase vs. Wavelength These graphs were retrieved from the UnFrog algorithm. They are most basically the spectrographs of the pulses, with phase information. I'm not quite sure how to interpret this data. From my basic understand of how the compressor works, all three of these graphs should look the same in their intensity profiles, but as you can see they do not. Intensity 1..8.6.4.2. 'Intensity' 'Phase' 15 1 5-5 -1-15 Phase Image 1: This graph seems to indicate that the pulse encompasses a large range of wavelengths. It is also expected that there should be a single main peak around 4 (in thiscase 44 for an input 88nm beam).but for some reason we see two. 398 4 42 44 46 Wavelength 48 41 1..8 'Intensity' 'Phase' 15 1 Intensity.6.4.2 5-5 -1 Phase Image 2: This is the minimized beam. It is a nice gaussian shape like we would expect, with a center at 44nm.. -15 398 4 42 44 46 Wavelength 48 41 Intensity 1..8.6.4.2. 398 4 42 44 46 Wavelength 48 'Intensity' 'Phase' 41 15 1 5-5 -1-15 Phase Image 3: This is an especially confusing graph. Not only does the center of the beam translate from 44 to 42 but the wavelength range seems to be shorter than the other pulses. You can see a resemblance of this peak with the smaller peak from image one. This could possibly be a systematic problem with the compressor.

Frequency Distribution wrt Delay From the phase as a function of delay one can find the frequency of the pulse as a function of delay by taking the derivative of the phase with respect to time and multiplying that value by -1. The data should then tell you whether your data is chirped or not. 1. Frequency.5. -.5 Image 1: As we expected there was a chirp in this pulse. Since SHG is a second order nonlinearity we cannot know whether it is positively or negatively chriped but only that it has a chirp. -1. -1-5 5 1 Delay 1. Frequency.5. -.5-1. Image 2: Since this is the supposed minimized beam there should be no chirp. But we observe one nonetheless. But it is at least not as steep as the chirp from image 1. -1-5 5 1 Delay 1. Frequency.5. -.5 Image 3: For some reason the chirp on this pulse is almost identical to that of image 2. We would expect a much larger slope. -1. -1-5 5 1 Delay Note: It is likely that most of this data is misleading. I have not spent much time working with the algorithm program so there is alot of room for me to improve my abilities in decoding these Frog traces.