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Measuring Ultrashort Laser Pulses II: FROG 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 FROG, XFROG, and TREEFROG Rick Trebino, Georgia Tech,

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Perhaps its time to ask how researchers in other fields deal with their waveforms… Consider, for example, acoustic waveforms. Autocorrelation and related techniques yield little information about the pulse.

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Its a plot of frequency vs. time, with information on top about the intensity. The musical score lives in the time-frequency domain. Most people think of acoustic waves in terms of a musical score.

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If E(t) is the waveform of interest, its spectrogram is: where g ( t - ) is a variable-delay gate function and is the delay. Without g ( t - ), Sp E (, ) would simply be the spectrum. Spectrogram

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Frequency Time Delay Spectrograms for Linearly Chirped Pulses

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Properties of the Spectrogram The spectrogram resolves the dilemma! It doesnt need the shorter event! It temporally resolves the slow components and spectrally resolves the fast components. 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 beand should not besignificantly shorter than E(t). Suppose we use a delta-function gate pulse: = The Intensity. No phase information!

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Polarization-Gate Geometry Spectro- meter Frequency-Resolved Optical Gating (FROG) Trebino, et al., Rev. Sci. Instr., 68, 3277 (1997). Kane and Trebino, Opt. Lett., 18, 823 (1993).

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FROG

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Frequency Time Delay FROG Traces for Linearly Chirped Pulses

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Frequency Intensity Time Delay FROG Traces for More Complex Pulses

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Unfortunately, spectrogram inversion algorithms require that we know the gate function. Substituting for E sig (t, ) in the expression for the FROG trace: yields: E sig (t, ) E(t) |E(t– )| 2 where: g(t– ) |E(t– )| 2 The FROG trace is a spectrogram of E(t).

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If E sig (t, ), is the 1D Fourier transform with respect to delay of some new signal field, E sig (t, ), then: So we must invert this integral equation and solve for E sig (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. and The input pulse, E(t), is easily obtained from E sig (t, ) : E(t) E sig (t, ) Instead, consider FROG as a two- dimensional phase-retrieval problem. Stark, Image Recovery, Academic Press, 1987.

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1D Phase Retrieval: Suppose we measure S ( ) and desire E(t), where: Given S(k x,k y ), there is essentially one solution for E(x,y) !!! It turns out that its possible to retrieve the 2D spectral phase!. Given S ( ), there are infinitely many solutions for E(t). We lack the spectral phase. 2D Phase Retrieval: Suppose we measure S(k x,k y ) and desire E(x,y) : These results are related to the Fundamental Theorem of Algebra. We assume that E(t) and E(x,y) are of finite extent. 1D vs. 2D Phase Retrieval Stark, Image Recovery, Academic Press, 1987.

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The Fundamental Theorem of Algebra states that all polynomials can be factored: f N-1 z N-1 + f N-2 z N-2 + … + f 1 z + f 0 = f N-1 (z–z 1 ) (z–z 2 ) … (z–z N–1 ) The Fundamental Theorem of Algebra fails for polynomials of two variables. Only a set of measure zero can be factored. f N-1,M-1 y N-1 z M-1 + f N-1,M-2 y N-1 z M-2 + … + f 0,0 = ? 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 2D phase retrieval is possible. Phase Retrieval and the Fundamental Theorem of Algebra

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1D Phase Retrieval and the Fundamental Theorem of Algebra The Fourier transform {F 0, …, F N-1 } of a discrete 1D data set, { f 0, …, f N-1 }, is: where z = e –ik The Fundamental Theorem of Algebra states that any polynomial, f N-1 z N-1 + … + f 0, can be factored to yield: f N-1 (z–z 1 ) (z–z 2 ) … (z–z N–1 ) So the magnitude of the Fourier transform of our data can be written: |F k | = | f N-1 (z–z 1 ) (z–z 2 ) … (z–z N–1 ) | where z = e –ik Complex conjugation of any factor(s) leaves the magnitude unchanged, but changes the phase, yielding an ambiguity! So 1D phase retrieval is impossible! Phase Retrieval & the Fund Thm of Algebra 2 polynomial!

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2D Phase Retrieval and the Fundamental Theorem of Algebra The Fourier transform {F 0,0, …, F N-1,N-1 } of a discrete 2D data set, { f 0.0, …, f N-1,N-1 }, is: where y = e –ik and z = e –iq But we cannot factor polynomials of two variables. So we can only complex conjugate the entire expression (yielding a trivial ambiguity). Only a set of polynomials of measure zero can be factored. So 2D phase retrieval is possible! And the ambiguities are very sparse. Phase Retrieval & the Fund Thm of Algebra 2 Polynomial of 2 variables!

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E sig (t, ) E(t) |E(t– )| 2 E sig (t, ) Generalized Projections

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E sig (t, ) E(t) Algorithm Find such that and is as close as possible to E sig (t, ) DeLong and Trebino, Opt. Lett., 19, 2152 (1994) Code is available commercially from Femtosoft Technologies.

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We must find such that and is as close as possible to E sig (t, ). Applying the Signal Field Constraint The way to do this is to find the field, E(t), that minimizes: Once we find the E(t) that minimizes Z, we write the new signal field as: This is the new signal field in the iteration. DeLong and Trebino, Opt. Lett., 19, 2152 (1994)

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Kohler traces

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Second-harmonic generation (SHG) is the strongest NLO effect. Second-Harmonic-Generation FROG Kane and Trebino, JQE, 29, 571 (1993). DeLong and Trebino, JOSA B, 11, 2206 (1994).

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Frequency Time Delay SHG FROG traces are symmetrical with respect to delay.

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Frequency Intensity Time Delay SHG FROG traces for complex pulses

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SHG FROG Measurements of a Free-Electron Laser SHG FROG works very well, even in the mid-IR and for difficult sources. Richman, et al., Opt. Lett., 22, 721 (1997).

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Shortest pulse length Year Shortest pulse vs. year Plot prepared in 1994, reflecting the state of affairs at that time.

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The measured pulse spectrum had two humps, and the measured autocorrelation had wings. 10-fs spectra and autocorrs Data courtesy of Kapteyn and Murnane, WSU Despite different predictions for the pulse shape, both theories were consistent with the data. Two different theories emerged, and both agreed with the data. From Harvey et. al, Opt. Lett., v. 19, p. 972 (1994) From Christov et. al, Opt. Lett., v. 19, p (1994)

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FROG distinguishes between the theories. Taft, et al., J. Special Topics in Quant. Electron., 3, 575 (1996).

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SHG FROG Measurements of a 4.5-fs Pulse! Baltuska, Pshenichnikov, and Weirsma, J. Quant. Electron., 35, 459 (1999).

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Pulse retrieval remains equivalent to the 2D phase-retrieval problem. Many interactions have been used, e.g., polarization rotation in a fiber. Generalizing FROG to arbitrary nonlinear-optical interactions

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FROG traces of the same pulse for different geometries

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FROG geometries: Pros and Cons Second- harmonic generation Third- harmonic generation Transient- grating Polarization- gate Self- diffraction most sensitive; most accurate tightly focused beams useful for UV & transient-grating experiments simple, intuitive, best scheme for amplified pulses useful for UV

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Generalized Projections with an Arbitrary Medium Response DeLong, et al., Opt. Lett. 20, 486 (1995).

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Single-shot FROG

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Single-Shot Polarization-Gate FROG Kane and Trebino, Opt. Lett., 18, 823 (1993).

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Single-shot FROG gives real-time feedback on laser performance. A grating pulse compressor requires the precise grating spacing, or the pulse will be chirped (positively or negatively). This can be a difficult alignment problem. Note that the trace is rotated by 90˚. Data taken by Toth and cowrkers

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This eliminates the need for a bulky expensive spectrometer as well as the need to align the beam through a tiny entrance slit (which would involve three sensitive alignment parameters)! We can use the focus of the beam in the nonlinear medium as the entrance slit for a home-made imaging spectrometer (in multi-shot and single-shot FROG measurements). FROG allows a very simple imaging spectrometer. Pulses from first half of FROG Collimating Lens Camera or Linear Detector Array Grating Imaging Lens Nonlinear Medium Signal pulse

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When a known reference pulse is available: Cross-correlation FROG (XFROG) The XFROG trace (a spectrogram): Unknown pulse Camera E(t) E g (t– ) SFG crystal Lens Known pulse If a known pulse is available (it need not be shorter), then it can be used to fully measure the unknown pulse. In this case, we perform sum-frequency generation, and measure the spectrum vs. delay. Spectro- meter XFROG completely determines the intensity and phase of the unknown pulse, provided that the gate pulse is not too long or too short. If a reasonable known pulse exists, use XFROG, not FROG. Linden, et al., Opt. Lett., 24, 569 (1999).

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XFROG example: Ultrabroadband Continuum Ultrabroadband continuum was created by propagating 1-nJ, 800- nm, 30-fs pulses through 16 cm of Lucent microstructure fiber. The 800-nm pulse was measured with FROG, so it made an ideal known gate pulse. This pulse has a time-bandwidth product of ~ 4000, and is the most complex ultrashort pulse ever measured. XFROG trace Retrieved intensity and phase Kimmel, Lin, Trebino, Ranka, Windeler, and Stentz, CLEO 2000.

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Measuring two pulses simultaneously

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1D vs. 2D Blind Deconvolution

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TREEFROG is still under active study, and many variations exist. It will be useful in excite-probe spectroscopic measurements, which involve crossing two pulses with variable relative delay at a sample. TREEFROG Example DeLong, et al., JOSA B, 12, 2463 (1995).

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