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Autocorrelation and related techniques yield,little information about the pulse. Perhaps it s time to ask how researchers in other fields deal with. their waveforms,Consider for example acoustic waveforms. Most people think of acoustic waves,in terms of a musical score. It s a plot of frequency vs time with information on top about the intensity. The musical score lives in the time frequency domain 3. A mathematically rigorous form of a,muscial score is the spectrogram. If E t is the waveform of interest its spectrogram is. E t g t exp i t dt, where g t is a variable delay gate function and is the delay.
Without g t SpE would simply be the spectrum,The spectrogram is a function of and. It is the set of spectra of all temporal slices of E t 4. The Spectrogram of a waveform E t, We must compute the spectrum of the product E t g t. g t contributes,g t only intensity not,E t contributes. phase i e color phase i e color,to the signal pulse to the signal pulse. The spectrogram tells the color and intensity of E t at the time. Spectrograms for Linearly Chirped Pulses, Negatively chirped Unchirped pulse Positively chirped.
pulse pulse,SHG FROG trace expanded,FROG trace expanded. 10 20 30 40 50 60,10 20 30 40 50 60, Like a musical score the spectrogram visually displays the. frequency vs time 6,Properties of the Spectrogram, Algorithms exist to retrieve E t from its spectrogram. The spectrogram essentially uniquely determines the waveform intensity. I t and phase t,There are a few ambiguities but they are trivial. The gate need not be and should not be significantly shorter than E t. Suppose we use a delta function gate pulse,E t t exp i t dt.
E The Intensity,No phase information, The spectrogram resolves the dilemma It doesn t need the shorter. event It temporally resolves the slow components and spectrally. resolves the fast components,Frequency Resolved Optical Gating FROG. FROG involves gating the pulse with a variably delayed replica. of the pulse in an instantaneous nonlinear optical medium and. then spectrally resolving the gated pulse,Pulse to be. measured Polarization Gate Geometry,Wave plate Esig t E t E t. splitter 45 rotation Camera,of polarization,Instantaneous nonlinear.
optical medium IFROG Esig t e i t dt, Use any fast nonlinear optical interaction SHG self diffraction etc. Trebino et al Rev Sci Instr 68 3277 1997 8,Kane and Trebino Opt Lett 18 823 1993. Frequency Resolved Optical Gating,Esig t E t E t,Signal pulse. E t contributes E t 2 contributes,phase i e color,only intensity not. to the signal pulse,phase i e color,to the signal pulse.
The signal pulse reflects the color of the gated pulse E t. at the time 2 3 9,FROG Traces for Linearly Chirped Pulses. Negatively chirped Unchirped pulse Positively chirped. pulse pulse,FROG trace expanded,SHG FROG trace expanded. 10 20 30 40 50 60,10 20 30 40 50 60, The FROG trace visually displays the frequency vs time. FROG Traces for More Complex Pulses,Self phase Cubic spectral Double pulse. modulated pulse phase pulse,FROG trace expanded FROG trace expanded.
FROG trace expanded,10 20 30 40 50 60,10 20 30 40 50 60. 10 20 30 40 50 60,The FROG trace is a spectrogram of E t. Substituting for Esig t in the expression for the FROG trace. Esig t E t E t 2,IFROG Esig t exp i t dt,IFROG E t g t exp i t dt. where g t E t 2,Unfortunately spectrogram inversion algorithms. require that we know the gate function 12,Instead consider FROG as a two.
dimensional phase retrieval problem, If Esig t is the 1D Fourier transform with respect to. delay of some new signal field Esig t then, The input pulse E t is easily obtained from Esig t E t Esig t. IFROG E sig t exp i t i dt d, So we must invert this integral equation and solve for Esig t. This integral inversion problem is the 2D phase retrieval problem. for which the solution exists and is unique, And simple algorithms exist for finding it Stark Image Recovery 13. Academic Press 1987,1D vs 2D Phase Retrieval, 1D Phase Retrieval Suppose we measure S and desire E t where.
S E t exp i t dt, Given S there are infinitely many solutions We assume that. for E t We lack the spectral phase E t and E x y are. of finite extent, 2D Phase Retrieval Suppose we measure S kx ky and desire E x y. E x y exp ikx x iky y dx dy,Image Recovery, Given S kx ky there is essentially one solution for E x y Academic Press. It turns out that it s possible to retrieve the 2D spectral phase. These results are related to the Fundamental Theorem of Algebra. Phase Retrieval and the Fundamental Theorem of Algebra. The Fundamental Theorem of Algebra states that all polynomials can. be factored,fN 1 zN 1 fN 2 zN 2 f1 z f0 fN 1 z z1 z z2 z zN 1. The Fundamental Theorem of Algebra fails for polynomials of two variables. Only a set of measure zero can be factored,fN 1 M 1 yN 1 zM 1 fN 1 M 2 yN 1 zM 2 f0 0.
Why does this matter, The existence of the 1D Fundamental Theorem of Algebra implies that. 1D phase retrieval is impossible, The non existence of the 2D Fundamental Theorem of Algebra implies that. Laser Pulses II FROG 10 20 30 40 50 60 10 20 30 40 50 60 SHG FROG trace expanded 10 20 30 40 50 60 10 20 30 40 50 60 FROG trace expanded The Musical Score and the Spectrogram Frequency Resolved Optical Gating FROG 1D vs 2D Phase Retrieval FROG as a 2D Phase retrieval Problem Second harmonic generation SHG FROG and other geometries Measuring the shortest event ever created Single shot