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The Musical Score

The Musical Score. n. n-1. z. + a. z. + ... +. a. = (z – z. ) (z – z. ) ... (z – z. ). n-1. 0. n. n-1. 1. The Musical Score,. the Fundamental Theorem of Algebra,. and the Measurement of the Shortest. Events Ever Created. Rick Trebino. School of Physics.

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The Musical Score

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  1. The Musical Score n n-1 z + a z + ... + a = (z – z ) (z – z ) ... (z – z ) n-1 0 n n-1 1 The Musical Score, the Fundamental Theorem of Algebra, and the Measurement of the Shortest Events Ever Created Rick Trebino School of Physics Georgia Institute of Technology Atlanta, GA 30332

  2. The Dilemma In order to measure an event in time, you need a shorter one. To study this event, you need a strobe light pulse that’s shorter. Photograph taken by Harold Edgerton, MIT But then, to measure the strobe light pulse, you need a detector whose response time is even shorter. And so on… So, now, how do you measure the shortest event?

  3. Ultrashort laser pulses are the shortest technological events ever created by humans. It’s routine to generate pulses shorter than 10-13 seconds in duration, and researchers have generated pulses only a few fs (10-15 s) long. Such a pulse is to one second as 5 cents is to the US national debt.Such pulses have many applications in physics, chemistry, biology, and engineering. You can measure any event—as long as you’ve got a pulse that’s shorter. So how do you measure the pulse itself? You must use the pulse to measure itself. But that isn’t good enough. It’s only as short as the pulse. It’s not shorter. Techniques based on using the pulse to measure itself have not sufficed.

  4. We must measure an ultrashort laser pulse’sintensity and phase vs. time or frequency. A laser pulse has the time-domain electric field: w  i E (t) = Re { exp [ t – (t) ] } I(t)1/2 i 0 Intensity Phase (neglecting the negative-frequency component) Equivalently, vs. frequency: ~ S(ww0)1/2 w j w w E ( exp [ ( ) ] } ) = Re { -i – 0 Spectral Spectrum Phase Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the pulse.

  5. The phase determines the pulse’s frequency (i.e., color) vs. time. Phase,  (t) Frequency, w (t) time time time  The instantaneous frequency: Example: “Linear chirp” We’d like to be able to measure, not only linearly chirped pulses, but also pulses with arbitrarily complex phases and frequencies vs. time.

  6. Autocorr 2

  7. 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.

  8. 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.”

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

  10. Spectrogram

  11. Spectrograms for Linearly Chirped Pulses Frequency Time Frequency Delay

  12. Properties of the Spectrogram = The Intensity. No phase information! 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: The spectrogram resolves the dilemma! It doesn’t need the shorter event! It temporally resolves the slow components and spectrally resolves the fast components.

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

  14. FROG

  15. FROG Traces for Linearly Chirped Pulses Frequency Time Frequency Delay

  16. FROG Traces for More Complex Pulses Intensity Frequency Time Frequency Delay

  17. 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 yields: g(t–)|E(t–)|2 where: Unfortunately, spectrogram inversion algorithms require that we know the gate function.

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

  19. 1D vs. 2D Phase Retrieval We assume that E(t) and E(x,y) are of finite extent. 1D Phase Retrieval: Suppose we measure S(w) and desire E(t), where: Given S(w), there are infinitely many solutions for E(t). We lack the spectral phase. 2D Phase Retrieval: Suppose we measure S(kx,ky) and desire E(x,y): Stark, Image Recovery, Academic Press, 1987. Given S(kx,ky), there is essentially one solution for E(x,y)!!! It turns out that it’s possible to retrieve the 2D spectral phase! . These results are related to the Fundamental Theorem of Algebra.

  20. 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-1zM-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 2D phase retrieval is possible.

  21. Phase Retrieval & the Fund Thm of Algebra 2 1D Phase Retrieval and the Fundamental Theorem of Algebra The Fourier transform {F0 , … , FN-1} of a discrete 1D data set, { f0 , …, fN-1}, is: where z = e–ik polynomial! The Fundamental Theorem of Algebra states that any polynomial, fN-1zN-1 + … + f0 , can be factored to yield: fN-1 (z–z1 )(z–z2 ) … (z–zN–1) So the magnitude of the Fourier transform of our data can be written: |Fk| = | fN-1 (z–z1 )(z–z2 ) … (z–zN–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!

  22. Phase Retrieval & the Fund Thm of Algebra 2 2D Phase Retrieval and the Fundamental Theorem of Algebra The Fourier transform {F0,0 , … , FN-1,N-1} of a discrete 2D data set, { f0.0 , …, fN-1,N-1}, is: where y = e–ik and z = e–iq Polynomial of 2 variables! 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.

  23. Generalized Projections Esig(t,)  E(t) |E(t–)|2 Esig(t,)

  24. Kohler traces

  25. Single-shot FROG gives real-time feedbackon 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 coworkers

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

  27. The measured pulse spectrum had two humps, and the measured autocorrelation had wings. 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. 1465 (1994) 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.

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

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

  30. Ultracomplex pulse: Ultrabroadband Continuum Ultrabroadband continuum was created by propagating 1-nJ, 800-nm, 30-fs pulses through 16 cm of Lucent microstructure fiber. Retrieved intensity and phase Measured trace Kimmel, Lin, Trebino, Ranka, Windeler, and Stentz, CLEO 2000. This pulse has a time-bandwidth product of > 1000, and is the most complex ultrashort pulse ever measured.

  31. Sensitivity of FROG

  32. Measuring Ultraweak Ultrashort Light Pulses 1/t t E unk E ref frequency Camera Spectrometer Because ultraweak ultrashort pulses are almost always created by much stronger pulses, a stronger reference pulse is always available. Use Spectral Interferometry This involves no nonlinearity! ... and only one delay! FROG + SI = TADPOLE (Temporal Analysis by Dispersing a Pair Of Light E-fields) Froehly, et al., J. Opt. (Paris) 4, 183 (1973) Lepetit, et al., JOSA B, 12, 2467 (1995) C. Dorrer, JOSA B, 16, 1160 (1999) Fittinghoff, et al., Opt. Lett., 21, 884 (1996).

  33. Sensitivity of Spectral Interferometry (TADPOLE) – 6 1 microjoule = 10 J – 9 1 nanojoule = 10 J A pulse train containing only 42 zepto- joules (42 x 10-21 J) per pulse has been measured. That’s one photon every five pulses! Fittinghoff, et al., Opt. Lett. 21, 884 (1996). – 12 1 picojoule = 10 J – 15 1 femtojoule = 10 J – 18 1 attojoule = 10 J TADPOLE can measure pulses – 21 with as little energy as: 1 zeptojoule = 10 J

  34. Unpolarized light doesn’t exist… POLLIWOG (POLarization-Labeled Interference vs. Wavelength for Only a Glint*) * Glint = “a very weak, very short pulse of light”

  35. Application of POLLIWOG Measurement of the variation of the polarization state of the emission from a GaAs-AlGaAs multiple quantum well when heavy-hole and light-hole excitons are excited elucidates the physics of these devices. Excitation-laser spectrum and hh and lh exciton spectra Evolution of the polarization of the emission: time (fs) A. L. Smirl, et al., Optics Letters, Vol. 23, No. 14 (1998)

  36. Pulse to be measured SHG crystal SHG crystal Camera Can we simplify FROG? FROG has 3 sensitive alignment degrees of freedom (q, f of a mirror and also delay). The thin crystal is also a pain. Pulse to be measured Camera Spec- trom- eter Variable delay 2 alignment q parameters q (q, f) q Crystal must be very thin, which hurts sensitivity. 1 alignment q parameter q (delay) q Remarkably, we can design a FROG without these components!

  37. The angular width of second harmonic varies inversely with the crystal thickness. Thin crystal creates narrower SH spectrum in a given direction and so can’t be used for autocorrelators or FROGs. Suppose white light with a large divergence angle impinges on an SHG crystal. The SH generated depends on the angle. And the angular width of the SH beam created varies inversely with the crystal thickness. Very thin crystal creates broad SH spectrum in all directions. Standard autocorrelators and FROGs use such crystals. Very Thin SHG crystal Thick crystal begins to separate colors. Thin SHG crystal Thick SHG crystal Very thick crystal acts like a spectrometer! Why not replace the spectrometer in FROG with a very thick crystal? Very thick crystal

  38. GRating-Eliminated No-nonsense Observationof Ultrafast Incident Laser Light E-fields(GRENOUILLE) Patrick O’Shea, Mark Kimmel, Xun Gu and Rick Trebino, Optics Letters, 2001; Trebino, et al., OPN, June 2001.

  39. GRENOUILLE Beam Geometry

  40. GRENOUILLE FROG Measured: Retrieved: Really Testing GRENOUILLE Even for highly structured pulses, GRENOUILLE allows for accurate reconstruction of the intensity and phase. Retrieved pulse in the time and frequency domains

  41. FROG has a few advantages!

  42. To learn more, see the FROG web site! www.physics.gatech.edu/frog Or read the cover story in the June 2001 issue of OPN Or read the book!

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