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Rick Trebino, Pablo Gabolde, Pam Bowlan, and Selcuk Akturk Measuring everything you’ve always wanted to know about an ultrashort laser pulse (but were afraid to ask) This work is funded by the NSF, Swamp Optics, and the Georgia Research Alliance. Georgia Tech School of Physics Atlanta, GA 30332

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We desire the ultrashort laser pulse’s intensity and phase vs. time or frequency. Light has the time-domain spatio-temporal electric field: IntensityPhase Equivalently, vs. frequency: Spectral Phase (neglecting the negative-frequency component) Spectrum Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the pulse.

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Frequency-Resolved Optical Gating (FROG) SHG crystal Pulse to be measured Variable delay, Camera Spec- trometer Beam splitter E(t)E(t) E(t– ) E sig (t, )= E(t)E(t- ) FROG uniquely determines the pulse intensity and phase vs. time for nearly all pulses. Its algorithm is fast (20 pps) and reliable. FROG is simply a spectrally resolved autocorrelation. This version uses SHG autocorrelation.

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SHG FROG traces for various pulses SHG FROG traces are symmetrical, so it has an ambiguity in the direction of time, but it’s easily removed. Frequency Intensity Time Delay Frequency Delay Self-phase- modulated pulse Cubic-spectral- phase pulse Double pulse

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The FROG algorithm even works well for very complex pulses TBP = 94.3 “Measured” traceRetrieved trace Pulse vs. timePulse vs. wavelength Reduced background in retrieved trace is due to noise reduction by algorithm. Grad student: Lina Xu

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FROG easily measures very complex pulses Occasionally, a few initial guesses are necessary, but we’ve never found a pulse FROG couldn’t retrieve. Red = correct pulse; Blue = retrieved pulse SHG FROG trace with 1% additive noise

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FROG Measurements of a 4.5-fs Pulse! Baltuska, Pshenichnikov, and Weirsma, J. Quant. Electron., 35, 459 (1999). FROG is now even used to measure attosecond pulses.

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GRating-Eliminated No-nonsense Observation of Ultrafast Incident Laser Light E-fields (GRENOUILLE) P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, Opt. Lett key innovations: A single optic that replaces the entire delay line, and a thick SHG crystal that replaces both the thin crystal and spectrometer. GRENOUILLE FROG

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Crossing beams at a large angle maps delay onto transverse position. Even better, this design is amazingly compact and easy to use, and it never misaligns! Here, pulse #1 arrives earlier than pulse #2 Here, the pulses arrive simultaneously Here, pulse #1 arrives later than pulse #2 Fresnel biprism = (x) x Input pulse Pulse #1 Pulse #2 The Fresnel biprism

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

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Testing GRENOUILLE Compare a GRENOUILLE measurement of a pulse with a tried-and-true FROG measurement of the same pulse: Retrieved pulse in the time and frequency domains GRENOUILLEFROG Measured Retrieved

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Testing GRENOUILLE GRENOUILLEFROG Measured Retrieved Retrieved pulse in the time and frequency domains Compare a GRENOUILLE measurement of a complex pulse with a FROG measurement of the same pulse:

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Spatio-temporal distortions Ordinarily, we assume that the electric-field separates into spatial and temporal factors (or their Fourier-domain equivalents): where the tilde and hat mean Fourier transforms with respect to t and x, y, z.

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Angular dispersion is an example of a spatio-temporal distortion. In the presence of angular dispersion, the mean off-axis k-vector component k x0 depends on frequency, Prism Input pulse Angularly dispersed output pulse x z

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Another spatio-temporal distortion is spatial chirp (spatial dispersion). Prism pairs and simple tilted windows cause spatial chirp. The mean beam position, x 0, depends on frequency, . Prism pair Input pulse Spatially chirped output pulse Input pulse Tilted window

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And yet another spatio-temporal distortion is pulse-front tilt. Gratings and prisms cause both spatial chirp and pulse-front tilt. The mean pulse time, t 0, depends on position, x. Prism Angularly dispersed pulse with spatial chirp and pulse-front tilt Input pulse Grating Angularly dispersed pulse with spatial chirp and pulse- front tilt Input pulse

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Angular dispersion always causes pulse- front tilt! Angular dispersion : where = dk x0 /d which is just pulse-front tilt! Inverse Fourier-transforming with respect to k x, k y, and k z yields: Inverse Fourier-transforming with respect to yields: using the shift theorem using the inverse shift theorem

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The combination of spatial and temporal chirp also causes pulse-front tilt. Dispersive medium Spatially chirped input pulse v g (red) > v g (blue) Spatially chirped pulse with pulse-front tilt, but no angular dispersion The theorem we just proved assumed no spatial chirp, however. So it neglects another contribution to the pulse-front tilt. The total pulse-front tilt is the sum of that due to dispersion and that due to this effect. Xun Gu, Selcuk Akturk, and Erik Zeek

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General theory of spatio-temporal distortions To understand the lowest-order spatio-temporal distortions, assume a complex Gaussian with a cross term, and Fourier- transform to the various domains, recalling that complex Gaussians transform to complex Gaussians: Time vs. angle Angular dispersion Pulse-front tilt Spatial chirp dropping the x subscript on the k Grad students: Xun Gu and Selcuk Akturk

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The imaginary parts of the pulse distortions: spatio-temporal phase distortions The imaginary part of Q xt yields: wave-front rotation. The electric field vs. x and z. Red = + Black = - Wave- propagation direction z x

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The imaginary parts of the pulse distortions: spatio-temporal phase distortions The imaginary part of R x is wave-front-tilt dispersion. Plots of the electric field vs. x and z for different colors. z x 11 22 33 There are eight lowest-order spatio-temporal distortions, but only two independent ones.

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The prism pulse compressor is notorious for introducing spatio-temporal distortions. Fine GDD tuning Prism Wavelength tuning Prism Coarse GDD tuning (change distance between prisms) Wavelength tuning Prism Even slight misalignment causes all eight spatio-temporal distortions!

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The two-prism pulse compressor is better, but still a big problem. Prism Wavelength tuning Periscope Wavelength tuning Prism Coarse GDD tuning Roof mirror Fine GDD tuning

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GRENOUILLE measures spatial chirp. -0-0 +0+0 SHG crystal Signal pulse frequency Delay Frequency +0+0 -0-0 Tilt in the otherwise symmetrical SHG FROG trace indicates spatial chirp! Fresnel biprism Spatially chirped pulse

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GRENOUILLE accurately measures spatial chirp. Measurements confirm GRENOUILLE’s ability to measure spatial chirp. Positive spatial chirp Negative spatial chirp Spatio-spectral plot slope (nm/mm)

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GRENOUILLE measures pulse-front tilt. Zero relative delay is off to side of the crystal Zero relative delay is in the crystal center SHG crystal An off-center trace indicates the pulse front tilt! Delay Frequency 0 Fresnel biprism Untilted pulse front Tilted pulse front

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GRENOUILLE accurately measures pulse-front tilt. Negative PFTZero PFTPositive PFT Varying the incidence angle of the 4 th prism in a pulse- compressor allows us to generate variable pulse-front tilt.

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Focusing an ultrashort pulse can cause complex spatio-temporal distortions. In the presence of just some chromatic aberration, simulations predict that a tightly focused ultrashort pulse looks like this: Ulrike Fuchs Increment between images: 20 fs (6 m). Focus Propagation direction Intensity vs. x & z (at various times) z x Measuring only I(t) at a focus is meaningless. We need I(x,y,z,t) !

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Time So we’ll need to be able measure, not only the intensity, but also the phase, that is, E(x,y,z,t), for even complex focused pulses. And we’ll also need great spectral resolution for such long pulses. And the device(s) should be simple and easy to use! Also, researchers now often use shaped pulses as long as ~20 ps with complex intensities and phases.

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We desire the ultrashort laser pulse’s intensity and phase vs. space and time or frequency. Light has the time-domain spatio-temporal electric field: IntensityPhase Equivalently, vs. frequency: Spectral Phase (neglecting the negative-frequency component) Spectrum Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the pulse.

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Strategy Then use it to help measure the more difficult one with a separate measurement device. Measure a spatially uniform (unfocused) pulse in time first. GRENOUILLE SEA TADPOLE STRIPED FISH

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Spectral Interferometry E unk E ref T Spectrometer Camera 1/T Frequency Beam splitter E ref E unk Measure the spectrum of the sum of a known and unknown pulse. Retrieve the unknown pulse E( ) from the cross term. With a known reference pulse, this technique is known as TADPOLE (Temporal Analysis by Dispersing a Pair Of Light E-fields). ~

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Retrieving the pulse in TADPOLE Interference fringes in the spectrum 00 Frequency The spectral phase difference is the phase of the result. IFFT 00 Frequency FFT 0 “Time” The “DC” term contains only spectra Filter & Shift 0 “Time” Filter out these two peaks The “AC” terms contain phase information Fittinghoff, et al., Opt. Lett. 21, 884 (1996). This retrieval algorithm is quick, direct, and reliable. It uniquely yields the pulse. Spectrum Keep this one.

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SI is very sensitive! 1 microjoule = J 1 nanojoule = J 1 picojoule = J 1 femtojoule = J 1 attojoule = J FROG’s sensitivity: TADPOLE has measured a pulse train with only 42 zeptojoules (42 x J) per pulse. TADPOLE ‘s sensitivity: 1 zeptojoule = J

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Spectral Interferometry does not have the problems that plague SPIDER. Recently* it was shown that a variation on SI, called SPIDER, cannot accurately measure the chirp (or the pulse length). SPIDER’s cross-term cosine: This is very different from standard SI’s cross-term cosine: *J.R. Birge, R. Ell, and F.X. Kärtner, Opt. Lett., (13): p Linear chirp ( d unk /d ) and T are both linear in and so look the same. Worse, T dominates, so T must be calibrated—and maintained— to six digits! The linear term of unk is just the delay, T, anyway! > 10 p < p ~ p /100 ~ 100 p p = pulse bandwidth; p = pulse length Desired quantity

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Examples of ideal SPIDER traces Even if the separation, T, were known precisely, SPIDER cannot measure pulses accurately. Intensity (%) These two pulses are very different but have very similar SPIDER traces.

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More ideal SPIDER traces Unless the pulses are vastly different, their SPIDER traces are about the same. Intensity (%) Practical issues, like noise, make the traces even more indisting- uishable.

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Unknown Mode-matching is important—or the fringes wash out. Beams must be perfectly collinear — or the fringes wash out. Phase stability is crucial — or the fringes wash out. The interferometer is difficult to work with. To resolve the spectral fringes, SI requires at least five times the spectrometer resolution. Spectrometer Spectral Interferometry: Experimental Issues

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SEA TADPOLE SEA TADPOLE has all the advantages of TADPOLE—and none of the problems. And it has some unexpected nice surprises! Fibers Camera x Cylindrical lens Reference pulse Unknown pulse Spherical lens Grating Spatially Encoded Arrangement (SEA) SEA TADPOLE uses spatial, instead of spectral, fringes. Grad student: Pam Bowlan

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Why is SEA TADPOLE a better design? Single mode fibers assure mode- matching. Fibers maintain alignment. Our retrieval algorithm is single shot, so phase stability isn’t essential. And any and all distortions due to the fibers cancel out! Collinearity is not only unnecessary; it’s not allowed. And the crossing angle is irrelevant; it’s okay if it varies.

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We retrieve the pulse using spatial fringes, not spectral fringes, with near-zero delay. 1D Fourier Transform from x to k The delay is ~ zero, so this uses the full available spectral resolution! The beams cross, so the relative delay, T, varies with position, x.

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SEA TADPOLE theoretical traces (mm)

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More SEA TADPOLE theoretical traces (mm)

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SEA TADPOLE measurements SEA TADPOLE has enough spectral resolution to measure a 14-ps double pulse

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An even more complex pulse… An etalon inside a Michelson interferometer yields a double train of pulses, and SEA TADPOLE can measure it, too.

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SEA TADPOLE achieves spectral super- resolution! The SEA TADPOLE cross term is essentially the unknown-pulse complex electric field. This goes negative and so may not broaden under convolution with the spectrometer point-spread function. Blocking the reference beam yields an independent measurement of the spectrum using the same spectrometer.

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SEA TADPOLE spectral super-resolution When the unknown pulse is much more complicated than the reference pulse, the interference term becomes: Sine waves are eigenfunctions of the convolution operator.

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Scanning SEA TADPOLE: E(x,y,z,t) By scanning the input end of the unknown-pulse fiber, we can measure E( ) at different positions yielding E(x,y,z,ω). So we can measure focusing pulses!

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E(x,y,z,t) for a theoretically perfectly focused pulse. E(x,z,t) Pulse Fronts Color is the instantaneous frequency vs. x and t. Uniform color indicates a lack of phase distortions. Simulation

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Measuring E(x,y,z,t) for a focused pulse. Simulation Measurement 810 nm 790 nm Aspheric PMMA lens with chromatic (but no spherical) aberration and GDD. f = 50 mm NA = 0.03

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Spherical and chromatic aberration Simulation Measurement Singlet BK-7 plano-convex lens with spherical and chromatic aberration and GDD. f = 50 mm NA = nm 790 nm

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A ZnSe lens with chromatic aberration Simulation Measurement Singlet ZnSe lens with massive chromatic aberration (GDD was canceled). 804 nm 796 nm

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SEA TADPOLE measurements of a pulse focusing A ZnSe lens with lots of chromatic aberration. Lens GDD was canceled out in this measurement, to better show the effect of chromatic aberration. 796 nm 804 nm

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Distortions are more pronounced for a tighter focus. 814 nm 787 nm Simulation Experiment Singlet BK-7 plano-convex lens with a shorter focal length. f = 25 mm NA = 0.06

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SEA TADPOLE measurements of a pulse focusing A BK-7 lens with some chromatic and spherical aberration and GDD. f = 25 mm. 787 nm 814 nm

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Focusing a pulse with spatial chirp and pulse- front tilt. 790 nm 812 nm Aspheric PMMA lens. f = 50 mm NA = 0.03 Simulation Experiment

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Single-shot measurement of E(x,y,z,t). Multiple holograms on a single camera frame: STRIPED FISH Array of spectrally-resolved holograms Spatially and Temporally Resolved Intensity and Phase Evaluation Device: Full Information from a Single Hologram Grad student: Pablo Gabolde

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Holography Measure the integrated intensity I(x,y) of the sum of known and unknown monochromatic beams. Extract the unknown monochromatic field E(x,y) from the cross term. Camera Spatially uniform, monochromatic reference beam Unknown beam Object

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Frequency-Synthesis Holography for complete spatio-temporal pulse measurement Performing holography using a well-characterized ultrashort pulse and measuring a series of holograms, one for each frequency component, yields the full pulse in the space-frequency domain. Performing holography with a monochromatic beam yields the full spatial intensity and phase at the beam’s frequency ( 0 ): E(x,y,t) then acts as the initial condition in Maxwell’s equations, yielding the full spatio-temporal pulse field: E(x,y,z,t). This approach is called “Fourier-Synthesis Holography.”

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STRIPED FISH

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The 2D diffraction grating creates many replicas of the input beams. Unknown Reference 50 μm Glass substrate Chrome pattern Using the 2D grating in reflection at Brewster’s angle removes the strong zero-order reflected spot.

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The band-pass filter spectrally resolves the digital holograms

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STRIPED FISH Retrieval algorithm Complex image Intensity and phase vs. ( x, y, λ )

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Typical STRIPED FISH measured trace

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Measurements of the spectral phase Group delay Group-delay dispersion

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Reconstruction procedure λ = 782 nmλ = 806 nmλ = 830 nm CCD camera Spatially-chirped input pulse 2-D Fourier transform of H(x,y) Reconstructed phase φ(x,y) at λ = 830 nm 2-D digital hologram H(x,y) Reconstructed intensity I(x,y) at λ = 830 nm Intensity of the entire pulse (spatial reference) Takeda et al, JOSA B 72, (1982). IFFT FFT

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Results for a pulse with spatial chirp Reconstructed phase at the same wavelengths λ = 782 nm λ = 806 nm λ = 830 nm Reconstructed intensity for a few wavelengths (wrapped phase plots) λ = 782 nm λ = 806 nm λ = 830 nm x y Contours indicate beam profile

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The spatial fringes depend on the spectral phase! (a) (b) Zero delay With GD

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A pulse with temporal chirp, spatial chirp, and pulse-front tilt. x [mm] t [fs] 803 nm 777 nm = 4.5 mrad 797 nm 775 nm x [mm] t [fs] = 11.3 mrad Suppressing the y-dependence, we can plot such a pulse: where the pulse-front tilt angle is:

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Complete electric field reconstruction Pulse with horizontal spatial chirp

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Complete 3D profile of a pulse with temporal chirp, spatial chirp, and pulse-front tilt 797 nm 775 nm Dotted white lines: contour plot of the intensity at a given time.

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The Space-Time-Bandwidth Product x y t STRIPED FISH can measure pulses with STBP ~ 10 6 ~ the number of camera pixels. SEA TADPOLE can do even better (depends on the details)! After numerical reconstruction, we obtain data “cubes” E(x,y,t) that are ~ [200 by 100 pixels] by 50 holograms. Space-Bandwidth Product (SBP) Time-Bandwidth Product (TBP) Space-Time-Bandwidth Product (STBP) = How complex a pulse can STRIPED FISH measure?

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Corner cube Prism Wavelength tuning GDD tuning Roof mirror Periscope Single-prism pulse compressor is spatio-temporal-distortion-free!

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Beam magnification is always one. d out d in

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The total dispersion is always zero. The dispersion depends on the direction through the prism.

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A zoo of techniques! GRENOUILLE easily measures E(t) (and spatial chirp and pulse-front tilt). SEA TADPOLE measures E(x,y,z,t) of focused and complex pulses (multi-shot). STRIPED FISH measures E(x,y,z,t) of a complex (unfo- cused) pulse on a single shot.

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To learn more, visit our web sites… And if you read only one ultrashort-pulse-measurement book this year, make it this one! You can have a copy of this talk if you like. Just let me know!

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