Measuring everything youve always wanted to know about
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 Georgia Tech School of Physics Atlanta, GA 30332 This work is funded by the NSF, Swamp Optics, and the Georgia Research Alliance.
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.
Frequency-Resolved Optical Gating (FROG) FROG is simply a spectrally resolved autocorrelation. Pulse to be measured This version uses SHG autocorrelation. Beam splitter E(t–t) Variable delay, t E(t) SHG crystal Camera Spectrometer Esig(t, t)= E(t)E(t-t) FROG uniquely determines the pulse intensity and phase vs. time for nearly all pulses. Its algorithm is fast (20 pps) and reliable.
SHG FROG traces for various pulses Cubic-spectralphase pulse Double pulse Intensity Frequency Self-phasemodulated pulse Frequency Time Delay SHG FROG traces are symmetrical, so it has an ambiguity in the direction of time, but it’s easily removed.
FROG easily measures very complex pulses SHG FROG trace with 1% additive noise Occasionally, a few initial guesses are necessary, but we’ve never found a pulse FROG couldn’t retrieve. Red = correct pulse; Blue = retrieved pulse
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.
GRating-Eliminated No-nonsense Observation of Ultrafast Incident Laser Light E-fields FROG (GRENOUILLE) 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. 2001.
The Fresnel biprism Crossing beams at a large angle maps delay onto transverse position. Input pulse Pulse #1 t = t(x) Here, pulse #1 arrives earlier than pulse #2 Here, the pulses arrive simultaneously Here, pulse #1 arrives later than pulse #2 Pulse #2 x Fresnel biprism Even better, this design is amazingly compact and easy to use, and it never misaligns!
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. Very Thin SHG crystal Thick crystal begins to separate colors. Thin SHG crystal Thick Very thick crystal acts like SHG crystal a spectrometer! Replace the crystal and spectrometer in FROG with a very thick crystal. Very thick crystal
Compare a GRENOUILLE measurement of a pulse with a tried-and-true FROG measurement of the same pulse: Measured FROG Retrieved Testing GRENOUILLE Retrieved pulse in the time and frequency domains
FROG Measured Compare a GRENOUILLE measurement of a complex pulse with a FROG measurement of the same pulse: GRENOUILLE Retrieved Testing GRENOUILLE Retrieved pulse in the time and frequency domains
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.
Angular dispersion is an example of a spatio-temporal distortion. In the presence of angular dispersion, the mean off-axis k-vector component kx 0 depends on frequency, w. x z Input pulse Prism Angularly dispersed output pulse
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, w. Tilted window Prism pair Input pulse Spatially chirped output pulse
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. Angularly dispersed pulse with spatial chirp and pulsefront tilt Input pulse Grating Prism Angularly dispersed pulse with spatial chirp and pulse-front tilt
Angular dispersion always causes pulsefront tilt! Angular dispersion: where g = dkx 0 /dw Inverse Fourier-transforming with respect to kx, ky, and kz yields: using the shift theorem Inverse Fourier-transforming with respect to w-w 0 yields: using the inverse shift theorem which is just pulse-front tilt!
The combination of spatial and temporal chirp also causes pulse-front tilt. Dispersive medium Spatially chirped pulse with pulse-front tilt, but no angular dispersion Spatially chirped v (red) > v (blue) g g input pulse 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
General theory of spatio-temporal distortions To understand the lowest-order spatio-temporal distortions, assume a complex Gaussian with a cross term, and Fouriertransform 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 Time vs. angle Angular dispersion
The imaginary parts of the pulse distortions: spatio-temporal phase distortions The imaginary part of Qxt yields: wave-front rotation. x z The electric field vs. x and z. Red = + Black = - Wavepropagation direction
The imaginary parts of the pulse distortions: spatio-temporal phase distortions The imaginary part of Rxw is wave-front-tilt dispersion. Plots of the electric field vs. x and z for different colors. w 1 x z w 2 w 3 There are eight lowest-order spatio-temporal distortions, but only two independent ones.
The prism pulse compressor is notorious for introducing spatio-temporal distortions. Wavelength tuning Prism Wavelength tuning Prism Fine GDD tuning Coarse GDD tuning (change distance between prisms) Wavelength tuning Even slight misalignment causes all eight spatio-temporal distortions!
The two-prism pulse compressor is better, but still a big problem. Coarse GDD tuning Wavelength Roof tuning mirror Periscope Prism Wavelength tuning Prism Fine GDD tuning
GRENOUILLE measures spatial chirp. Spatially chirped pulse Fresnel biprism SHG crystal -t 0 Signal pulse frequency Tilt in the otherwise symmetrical SHG FROG trace indicates spatial chirp! Frequency +t 0 2 w+dw 2 w dw -t 0 Delay +t 0
GRENOUILLE accurately measures spatial chirp. Positive spatial chirp Negative spatial chirp Spatio-spectral plot slope (nm/mm) Measurements confirm GRENOUILLE’s ability to measure spatial chirp.
GRENOUILLE measures pulse-front tilt. Fresnel biprism Tilted pulse front Zero relative delay is off to side of the crystal SHG crystal Untilted pulse front An off-center trace indicates the pulse front tilt! Frequency Zero relative delay is in the crystal center 0 Delay
GRENOUILLE accurately measures pulse -front tilt. Varying the incidence angle of the 4 th prism in a pulsecompressor allows us to generate variable pulse-front tilt. Negative PFT Zero PFT Positive PFT
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) x z Focus Propagation direction Increment between images: 20 fs (6 mm). Ulrike Fuchs Measuring only I(t) at a focus is meaningless. We need I(x, y, z, t)!
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!
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 Equivalently, vs. frequency: Spectrum Phase (neglecting the negative-frequency component) Spectral Phase Knowledge of the intensity and phase or the spectrum and spectral phase is sufficient to determine the pulse.
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. SEA TADPOLE STRIPED FISH
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 Eref Eunk Eref Beam splitter Eunk Frequency Spectrometer Camera With a known reference pulse, this technique is known as TADPOLE (Temporal Analysis by Dispersing a Pair Of Light E-fields).
Retrieving the pulse in TADPOLE Interference fringes in the spectrum FFT w 0 Frequency Spectrum The spectral phase difference is the phase of the result. IFFT The “DC” term contains only spectra Filter out these two peaks The “AC” terms contain phase information 0 “Time” Filter & Shift Keep this one. w 0 Frequency 0 “Time” This retrieval algorithm is quick, direct, and reliable. It uniquely yields the pulse. Fittinghoff, et al. , Opt. Lett. 21, 884 (1996).
SI is very sensitive! 1 microjoule = 10 -6 J 1 nanojoule = 10 -9 J FROG’s sensitivity: 1 picojoule = 10 -12 J 1 femtojoule = 10 -15 J 1 attojoule = 10 -18 J TADPOLE ‘s sensitivity: 1 zeptojoule = 10 -21 J TADPOLE has measured a pulse train with only 42 zeptojoules (42 x 10 -21 J) per pulse.
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: Desired quantity ~ wp /100 < tp > 10 wp ~ 100 tp Linear chirp (djunk/dw w) and w. T are both linear in w and so look the same. Worse, w. T dominates, so T must be calibrated—and maintained— to six digits! 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. , 2006. 31(13): p. 2063 -5.
Examples of ideal SPIDER traces Intensity (%) 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.
More ideal SPIDER traces Intensity (%) Unless the pulses are vastly different, their SPIDER traces are about the same. Practical issues, like noise, make the traces even more indistinguishable.
Spectral Interferometry: Experimental Issues The interferometer is difficult to work with. Phase stability is crucial—or the fringes wash out. Mode-matching is important—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.
SEA TADPOLE Camera x Cylindrical lens l SEA TADPOLE uses spatial, instead of spectral, fringes. Reference pulse Fibers Unknown pulse Spatially Encoded Arrangement (SEA) Grating Spherical lens Grad student: Pam Bowlan SEA TADPOLE has all the advantages of TADPOLE—and none of the problems. And it has some unexpected nice surprises!
Why is SEA TADPOLE a better design? Fibers maintain alignment. Single mode fibers assure modematching. 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! Our retrieval algorithm is single shot, so phase stability isn’t essential.
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. 1 D Fourier Transform from x to k The delay is ~ zero, so this uses the full available spectral resolution!
(mm) SEA TADPOLE theoretical traces
(mm) More SEA TADPOLE theoretical traces
SEA TADPOLE measurements SEA TADPOLE has enough spectral resolution to measure a 14 -ps double pulse. -8 -6 6 8
An even more complex pulse… An etalon inside a Michelson interferometer yields a double train of pulses, and SEA TADPOLE can measure it, too.
SEA TADPOLE achieves spectral superresolution! Blocking the reference beam yields an independent measurement of the spectrum using the same spectrometer. 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.
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.
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!
E(x, y, z, t) for a theoretically perfectly focused pulse. Simulation E(x, z, t) Pulse Fronts Color is the instantaneous frequency vs. x and t. Uniform color indicates a lack of phase distortions.
Aspheric PMMA lens with chromatic (but no spherical) aberration and GDD. Measurement Measuring E(x, y, z, t) for a focused pulse. 810 nm 790 nm Simulation f = 50 mm NA = 0. 03
Singlet BK-7 plano-convex lens with spherical and chromatic aberration and GDD. Measurement Spherical and chromatic aberration 810 nm 790 nm Simulation f = 50 mm NA = 0. 03
Singlet Zn. Se lens with massive chromatic aberration (GDD was canceled). Measurement A Zn. Se lens with chromatic aberration Simulation 804 nm 796 nm
SEA TADPOLE measurements of a pulse focusing 796 nm 804 nm A Zn. Se lens with lots of chromatic aberration. Lens GDD was canceled out in this measurement, to better show the effect of chromatic aberration.
Singlet BK-7 plano-convex lens with a shorter focal length. Experiment Distortions are more pronounced for a tighter focus. 814 nm 787 nm Simulation f = 25 mm NA = 0. 06
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.
Aspheric PMMA lens. Experiment Focusing a pulse with spatial chirp and pulsefront tilt. f = 50 mm NA = 0. 03 Simulation 812 nm 790 nm
Single-shot measurement of E(x, y, z, t). Multiple holograms on a single camera frame: STRIPED FISH Spatially and Temporally Resolved Intensity and Phase Evaluation Device: Full Information from a Single Hologram Array of spectrally-resolved holograms Grad student: Pablo Gabolde
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. Spatially uniform, monochromatic reference beam Unknown beam Camera Object
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 ( w 0): 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. ”
STRIPED FISH
The 2 D diffraction grating creates many replicas of the input beams. Glass substrate Chrome pattern Unknown Reference 50 μm Using the 2 D grating in reflection at Brewster’s angle removes the strong zero-order reflected spot.
The band-pass filter spectrally resolves the digital holograms
STRIPED FISH Retrieval algorithm Intensity and phase vs. (x, y, λ) Complex image
Typical STRIPED FISH measured trace
Measurements of the spectral phase Group delay Group-delay dispersion
Reconstruction procedure Takeda et al, JOSA B 72, 156 -160 (1982). FFT 2 -D Fourier transform of H(x, y) Intensity of the entire pulse (spatial reference) IFFT 2 -D digital hologram H(x, y) CCD camera Reconstructed intensity I(x, y) at λ = 830 nm Reconstructed phase φ(x, y) at λ = 830 nm λ = 830 782 nm 806 Spatially-chirped input pulse
Results for a pulse with spatial chirp Reconstructed intensity for a few wavelengths Contours indicate beam profile 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)
The spatial fringes depend on the spectral phase! Zero delay With GD (b) (a)
A pulse with temporal chirp, spatial chirp, and pulse-front tilt. Suppressing the y-dependence, we can plot such a pulse: y = 11. 3 4. 5 mrad 797 803 nm nm where the pulse -front tilt angle is: xx [[m mm m] ] ] tt [[ffss] 775 nm 777 nm
Complete electric field reconstruction Pulse with horizontal spatial chirp
Complete 3 D 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.
The Space-Time-Bandwidth Product How complex a pulse can STRIPED FISH measure? After numerical reconstruction, we obtain data “cubes” E(x, y, t) that are ~ [200 by 100 pixels] by 50 holograms. y t x Space-Bandwidth Product (SBP) Time-Bandwidth Space-Time-Bandwidth = Product (TBP) 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)!
Single-prism pulse compressor is spatio-temporal-distortion-free! Corner cube Periscope Prism GDD tuning Roof mirror Wavelength tuning
Beam magnification is always one. din dout
The total dispersion is always zero. The dispersion depends on the direction through the prism.
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 (unfocused) pulse on a single shot.
To learn more, visit our web sites… www. physics. gatech. edu/frog You can have a copy of this talk if you like. Just let me know! www. swampoptics. com And if you read only one ultrashort-pulse-measurement book this year, make it this one!
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