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Rick Trebino, Pablo Gabolde, Pam Bowlan, and Selcuk Akturk

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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.

Light has the time-domain spatio-temporal electric field:

Intensity

Phase

(neglecting the

negative-frequency

component)

Equivalently, vs. frequency:

Spectral

Spectrum

Phase

Knowledge of the intensity and phase or the spectrum and spectralphaseis sufficient to determine the pulse.

Frequency-Resolved Optical Gating (FROG)

FROG is simply a spectrally resolved autocorrelation.

This version uses SHG autocorrelation.

Pulse to be measured

Beam

splitter

Camera

E(t–t)

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.

Cubic-spectral-phase pulse

Self-phase-modulated pulse

Double pulse

Intensity

Frequency

Time

Frequency

Frequency

Delay

Delay

SHG FROG traces are symmetrical, so it has an ambiguity in the direction of time, but it’s easily removed.

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

Baltuska,

Pshenichnikov,

and Weirsma,

J. Quant. Electron.,

35, 459 (1999).

FROG is now even used to measure attosecond pulses.

FROG

2 key innovations: Asingle optic that replaces the entiredelay line,

and a thickSHG crystal that replaces both the thin crystal andspectrometer.

GRENOUILLE

P. O’Shea, M. Kimmel, X. Gu, and R. Trebino, Opt. Lett. 2001.

Pulse #1

Here, pulse #1 arrives

earlier than pulse #2

x

Here, the pulses

arrive simultaneously

Here, pulse #1 arrives

later than pulse #2

Pulse #2

Crossing beams at a large angle maps delay onto transverse position.

Input pulse

t = t(x)

Fresnel biprism

Even better, this design is amazingly compact and easy to use, and it never misaligns!

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.

Thick crystal begins to

separate colors.

Thin

SHG

crystal

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.

Very

Thin

SHG

crystal

GRENOUILLE

FROG

Measured

Retrieved

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

GRENOUILLE

FROG

Measured

Retrieved

Testing GRENOUILLE

Compare a GRENOUILLE measurement of a complex pulse with a FROG measurement of the same pulse:

Retrieved pulse in the time and frequency domains

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.

x

z

Angularlydispersed output pulse

Prism

Input pulse

In the presence of angular dispersion, the mean off-axis k-vector component kx0depends on frequency, w.

Tilted window

Prism pair

Input pulse

Input pulse

Spatially chirped output pulse

Spatially chirped output pulse

Prism pairs and simple tilted windows cause spatial chirp.

The mean beam position, x0, depends on frequency, w.

Angularly dispersed pulse with spatial chirp and pulse-front tilt

Angularly dispersed pulse with spatial chirp and pulse-front tilt

Input pulse

Input pulse

Prism

Grating

Gratings and prisms cause both spatial chirp and pulse-front tilt.

The mean pulse time, t0, depends on position, x.

Angular dispersion:

whereg = dkx0/dw

Inverse Fourier-transforming with respect tokx, ky,andkzyields:

using the shift theorem

Inverse Fourier-transforming with respect tow-w0yields:

using the inverse shift theorem

which is just 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

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

x

z

The imaginary part of Qxt yields: wave-front rotation.

Wave-propagation direction

The electric field vs. x and z.

Red = +Black = -

x

z

The imaginary part ofRxw is wave-front-tilt dispersion.

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.

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!

Coarse GDD tuning

Wavelength tuning

Roof mirror

Periscope

Prism

Prism

Fine GDD tuning

Wavelength tuning

Signal pulse frequency

2w+dw

Frequency

2wdw

Tilt in the otherwise symmetrical SHG FROG trace indicates spatial chirp!

Delay

-t0

+t0

Fresnel biprism

GRENOUILLE measures spatial chirp.

SHG

crystal

Spatially chirped pulse

-t0

+t0

Measurements confirm GRENOUILLE’s ability to measure spatial chirp.

Positive spatial chirp

Spatio-spectral plot slope (nm/mm)

Negative spatial chirp

Zero relative delay is off to side of the crystal

Zero relative delay is in the crystal center

Frequency

An off-center trace indicates the pulse front tilt!

0

Delay

Fresnel biprism

Tilted pulse front

SHG

crystal

Untilted pulse front

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

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

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)!

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!

Light has the time-domain spatio-temporal electric field:

Intensity

Phase

(neglecting the

negative-frequency

component)

Equivalently, vs. frequency:

Spectral

Spectrum

Phase

Knowledge of the intensity and phase or the spectrum and spectralphaseis sufficient to determine the pulse.

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

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

Frequency

Eref

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).

w0

w0

Frequency

Frequency

The “DC” term

contains only

spectra

The “AC” terms

contain phase information

Interference fringes

in the spectrum

FFT

“Time”

0

Filter

out these

two peaks

Filter

&

Shift

Spectrum

The spectral phase difference is the phase of the result.

IFFT

Keep this one.

“Time”

0

This retrieval algorithm is quick, direct, and reliable.

It uniquely yields the pulse.

Fittinghoff, et al., Opt. Lett. 21, 884 (1996).

FROG’s sensitivity:

TADPOLE ‘s sensitivity:

1 zeptojoule = 10-21 J

1 microjoule = 10-6 J

1 nanojoule = 10-9 J

1 picojoule = 10-12 J

1 femtojoule = 10-15 J

1 attojoule = 10-18 J

TADPOLE has measured a pulse train with only 42 zeptojoules (42 x 10-21 J) per pulse.

~ 100tp

~ wp /100

< tp

> 10wp

wp = pulse bandwidth; tp = pulse length

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/dww) 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

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.

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 (%)

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 (%)

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.

Camera

x

Cylindrical lens

Spatially Encoded Arrangement (SEA)

SEA TADPOLE usesspatial, instead of spectral, fringes.

l

Reference pulse

Fibers

Grating

Grad student: Pam Bowlan

Unknown pulse

Spherical lens

SEA TADPOLE has all the advantages of TADPOLE—and none of the problems. And it has some unexpected nice surprises!

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!

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!

(mm)

(mm)

(mm)

(mm)

-8 -6

6 8

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

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

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.

When the unknown pulse is much more complicated than the reference pulse, the interference term becomes:

Sine waves are eigenfunctions of the convolution operator.

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,z,t)

Simulation

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.

f = 50 mmNA = 0.03

Measurement

810 nm

Simulation

790 nm

Singlet BK-7 plano-convex lens with spherical and chromatic aberration and GDD.

f = 50 mmNA = 0.03

Measurement

Simulation

810 nm

790 nm

Singlet ZnSe lens with massive chromatic aberration (GDD was canceled).

Measurement

804 nm

Simulation

796 nm

804 nm

796 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.

Experiment

Singlet BK-7 plano-convex lens with a shorter focal length.

f = 25 mmNA = 0.06

814 nm

Simulation

787 nm

814 nm

787 nm

A BK-7 lens with some chromatic and spherical aberration and GDD.

f = 25 mm.

Experiment

Aspheric PMMA lens.

f = 50 mmNA = 0.03

812 nm

Simulation

790 nm

Array of spectrally-resolved holograms

Spatially and

Temporally

Resolved

Intensity and

Phase

Evaluation

Device:

Full

Information from a

Single

Hologram

Grad student: Pablo Gabolde

Spatially uniform, monochromatic reference beam

Unknown beam

Camera

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.

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.”

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.

Intensity

and phase

vs. (x,y,λ)

Complex image

Group delay

Group-delay dispersion

IFFT

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)

2-D digital hologramH(x,y)

Reconstructed intensity I(x,y) at λ = 830 nm

CCD camera

Reconstructed phaseφ(x,y) at λ = 830 nm

λ = 782 nm

λ = 806 nm

λ = 830 nm

Spatially-chirped input pulse

x

y

Contours indicate beam profile

Reconstructed intensity for a few wavelengths

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)

Zero delay

With GD

(b)

(a)

797 nm

y= 4.5 mrad

y = 11.3 mrad

775 nm

803 nm

x [mm]

t [fs]

x [mm]

t [fs]

777 nm

Suppressing the y-dependence, we can plot such a pulse:

where the pulse-front tilt angle is:

Pulse with horizontal spatial chirp

797 nm

775 nm

Dotted white lines: contour plot of the intensity at a given time.

y

t

x

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.

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)!

Corner cube

Periscope

Prism

GDD tuning

Roof mirror

Wavelength tuning

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

din

dout

The dispersion depends on the direction through the prism.

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.

www.physics.gatech.edu/frog

www.swampoptics.com

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!