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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) Rick Trebino, Pablo Gabolde, Pam Bowlan, and Selcuk Akturk For the past decade and a half, my group has been developing techniques for measuring ultrashort laser pulses. And like nearly everyone else who has such pulses, we’ve concentrated on their behavior in time. But a technique that we recently developed has also proved to be able to measure a couple of spatio-temporal distortions. And because such capability hasn’t generally been available, most pulses we’ve measured are contaminated with them. This has gotten us interested in these distortions and has taken us on a journey that has led to the development of not one, but two, different techniques for measuring the complete spatio-temporal field of an ultrashort laser pulse. Georgia Tech School of Physics Atlanta, GA 30332 This work is funded by the NSF, Swamp Optics, and the Georgia Research Alliance.

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**We desire the ultrashort laser pulse’s intensity and phase vs**

We desire the ultrashort laser pulse’s intensity and phase vs. time or frequency. Light has the time-domain spatio-temporal electric field: Intensity Phase (neglecting the negative-frequency component) Equivalently, vs. frequency: Spectrum Spectral Phase 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)**

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

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**SHG FROG traces for various pulses**

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

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**The FROG algorithm even works well for very complex pulses**

“Measured” trace Retrieved trace Reduced background in retrieved trace is due to noise reduction by algorithm. Pulse vs. wavelength Pulse vs. time TBP = 94.3 Grad student: Lina Xu

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**FROG easily measures very complex pulses**

SHG FROG trace with 1% additive noise 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

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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)**

FROG 2 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 P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, Opt. Lett

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

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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 Very thin crystal creates broad SH spectrum in all directions. Standard autocorrelators and FROGs use such crystals. Thin crystal creates narrower SH spectrum in a given direction and so can’t be used for autocorrelators or FROGs. Thin SHG crystal Very 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

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Testing GRENOUILLE GRENOUILLE FROG Measured Retrieved 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

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

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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 kx0 depends on frequency, w. 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, x0, depends on frequency, w. Prism pair Input pulse Spatially chirped output 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, t0, depends on position, x. Grating Angularly dispersed pulse with spatial chirp and pulse-front tilt Input pulse Prism Angularly dispersed pulse with spatial chirp and pulse-front tilt Input pulse

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**Angular dispersion always causes pulse-front tilt!**

where g = dkx0 /dw Inverse Fourier-transforming with respect to kx, ky, and kz yields: using the shift theorem Inverse Fourier-transforming with respect to w-w0 yields: using the inverse shift theorem which is just pulse-front tilt!

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**Spatially chirped input pulse**

The combination of spatial and temporal chirp also causes pulse-front tilt. Spatially chirped pulse with pulse-front tilt, but no angular dispersion Dispersive medium Spatially chirped input pulse vg(red) > vg(blue) 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: Grad students: Xun Gu and Selcuk Akturk Pulse-front tilt Spatial chirp dropping the x subscript on the k Angular dispersion Time vs. angle

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**The imaginary parts of the pulse distortions: spatio-temporal phase distortions**

The imaginary part of Qxt yields: wave-front rotation. z x Wave-propagation direction The electric field vs. x and z. Red = + Black = -

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**The imaginary parts of the pulse distortions: spatio-temporal phase distortions**

The imaginary part of Rxw is wave-front-tilt dispersion. z x w1 Plots of the electric field vs. x and z for different colors. w2 w3 There are eight lowest-order spatio-temporal distortions, but only two independent ones.

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**Coarse GDD tuning (change distance between prisms)**

The prism pulse compressor is notorious for introducing spatio-temporal distortions. Wavelength tuning Wavelength tuning Prism Prism Prism Prism Fine GDD tuning Wavelength tuning Wavelength tuning Coarse GDD tuning (change distance between prisms) 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.**

Coarse GDD tuning Wavelength tuning Roof mirror Periscope Prism Prism Fine GDD tuning Wavelength tuning

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**GRENOUILLE measures spatial chirp.**

Fresnel biprism Signal pulse frequency SHG crystal Spatially chirped pulse -t0 +t0 Delay Frequency 2w+dw 2wdw +t0 -t0 Tilt in the otherwise symmetrical SHG FROG trace indicates spatial chirp!

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**GRENOUILLE accurately measures spatial chirp.**

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

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**GRENOUILLE measures pulse-front tilt.**

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

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**GRENOUILLE accurately measures pulse-front tilt.**

Varying the incidence angle of the 4th prism in a pulse-compressor allows us to generate variable pulse-front tilt. Negative PFT Zero PFT Positive PFT

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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: Intensity vs. x & z (at various times) Focus We’re now very good at measuring spatially smooth ultrashort pulses at the outputs of lasers. But nearly everyone focuses his ultrashort pulses, and it’s at the focus that we need to know its properties. Unfortunately, complex spatio-temporal distortions occur to focused ultrashort pulses. x z Propagation direction Ulrike Fuchs Increment between images: 20 fs (6 mm). Measuring only I(t) at a focus is meaningless. We need I(x,y,z,t)!

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**Also, researchers now often use shaped pulses as long as ~20 ps with complex intensities and phases.**

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!

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**We desire the ultrashort laser pulse’s intensity and phase vs**

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: Intensity Phase (neglecting the negative-frequency component) Equivalently, vs. frequency: Spectrum Spectral Phase Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the pulse.

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**Strategy Measure a spatially uniform (unfocused) pulse in time first.**

GRENOUILLE Then use it to help measure the more difficult one with a separate measurement device. STRIPED FISH SEA TADPOLE

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**Spectral Interferometry**

Measure the spectrum of the sum of a known and unknown pulse. Retrieve the unknown pulse E(w) from the cross term. ~ 1/T T Eunk Eref Eref Frequency Eunk Spectrometer Camera Beam splitter 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**

The “DC” term contains only spectra The “AC” terms contain phase information Interference fringes in the spectrum FFT w0 Frequency “Time” Filter out these two peaks Spectrum Filter & Shift The spectral phase difference is the phase of the result. IFFT Keep this one. w0 Frequency “Time” This retrieval algorithm is quick, direct, and reliable. It uniquely yields the pulse. Fittinghoff, et al., Opt. Lett. 21, 884 (1996).

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**SI is very sensitive! 1 microjoule = 10-6 J 1 nanojoule = 10-9 J**

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

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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: Linear chirp (djunk/dw w) and wT are both linear in w and so look the same. Worse, wT dominates, so T must be calibrated—and maintained—to six digits! Desired quantity > 10wp < tp ~ wp /100 ~ 100tp wp = pulse bandwidth; tp = pulse length This is very different from standard SI’s cross-term cosine: The linear term of junk is just the delay, T, anyway! *J.R. Birge, R. Ell, and F.X. Kärtner, Opt. Lett., (13): p

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**Examples of ideal SPIDER traces**

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

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**More ideal SPIDER traces**

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

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**Spectral Interferometry: Experimental Issues**

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

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**SEA TADPOLE x Spatially Encoded Arrangement (SEA)**

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

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**Why is SEA TADPOLE a better design?**

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

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**We retrieve the pulse using spatial fringes, not spectral fringes, with near-zero delay.**

The beams cross, so the relative delay, T, varies with position, x. 1D Fourier Transform from x to k The delay is ~ zero, so this uses the full available spectral resolution!

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**SEA TADPOLE theoretical traces**

(mm) (mm)

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**More SEA TADPOLE theoretical traces**

(mm) (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!**

Blocking the reference beam yields an independent measurement of the spectrum using the same spectrometer. For comparison we obtained independent spectra of the unknown pulse using the same device by blocking the reference arm beam. The fast features in the unknown spectrum that we retrieved from the SEA TADPOLE trace are not washed out as they are in the independent spectrum. The improvement in spectral resolution will not be this great for all pulses. Because double Pulses have zero crossings in their fields, which is what is being resolved here, the contrast or depth of the features in the spectrum is not harmed by the smearing due to the spectral response function. Because the field will always be a wider (or fatter) function than the spectrum, there is still a smaller improvement in resolution when there are no zero crossings in the field. 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.

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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(w) 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) Simulation In all of our experiments and simulations, the beams had spot sizes of 3 mm and bandwidths of 25 nm and lenses with diameters of 24.5 mm. Pulse Fronts Color is the instantaneous frequency vs. x and t. Uniform color indicates a lack of phase distortions.

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**Measuring E(x,y,z,t) for a focused pulse.**

Aspheric PMMA lens with chromatic (but no spherical) aberration and GDD. f = 50 mm NA = 0.03 Measurement 810 nm Simulation 790 nm

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**Spherical and chromatic aberration**

Singlet BK-7 plano-convex lens with spherical and chromatic aberration and GDD. f = 50 mm NA = 0.03 Measurement Simulation 810 nm 790 nm

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**A ZnSe lens with chromatic aberration**

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

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**SEA TADPOLE measurements of a pulse focusing**

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

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**Distortions are more pronounced for a tighter focus.**

Experiment Singlet BK-7 plano-convex lens with a shorter focal length. f = 25 mm NA = 0.06 814 nm Simulation 787 nm

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**SEA TADPOLE measurements of a pulse focusing**

787 nm 814 nm A BK-7 lens with some chromatic and spherical aberration and GDD. f = 25 mm.

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**Focusing a pulse with spatial chirp and pulse- front tilt.**

Experiment Aspheric PMMA lens. f = 50 mm NA = 0.03 812 nm I used the -1 order of a reflective grating (400g/mm) to create the angular dispersion, so I think that the results make sense but please let me know if you do not agree. The focusing lens was the ashpere that we used in the Optics Express paper( NA = 0.03 in this case). Simulation 790 nm

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**Array of spectrally-resolved holograms**

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 Spatially uniform, monochromatic reference beam**

Camera Spatially uniform, monochromatic reference beam Unknown beam Object 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.

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**Frequency-Synthesis Holography for complete spatio-temporal pulse measurement**

Performing holography with a monochromatic beam yields the full spatial intensity and phase at the beam’s frequency (w0): 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. 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.**

Glass substrate Chrome pattern Unknown Reference 50 μm 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**

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

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**Results for a pulse with spatial chirp**

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

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**The spatial fringes depend on the spectral phase!**

Zero delay With GD (b) (a)

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**A pulse with temporal chirp, spatial chirp, and pulse-front tilt.**

Suppressing the y-dependence, we can plot such a pulse: x [mm] t [fs] 803 nm 777 nm y = 4.5 mrad 797 nm 775 nm y = 11.3 mrad where the pulse-front tilt angle is: Pablo Gabolde made this nice picture.

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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**

How complex a pulse can STRIPED FISH measure? x y t 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) = STRIPED FISH can measure pulses with STBP ~ 106 ~ the number of camera pixels. SEA TADPOLE can do even better (depends on the details)!

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**Single-prism pulse compressor is spatio-temporal-distortion-free!**

Corner cube Prism Wavelength tuning GDD tuning Roof mirror Periscope

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**Beam magnification is always one.**

din dout

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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…**

You can have a copy of this talk if you like. Just let me know! And if you read only one ultrashort-pulse-measurement book this year, make it this one!

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Appendix A : Fourier transform

Appendix A : Fourier transform

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