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 TechSchool of Physics

Atlanta, GA 30332

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

Equivalently, vs. frequency:

Spectral Phase

(neglecting thenegative-frequency

component)

Spectrum

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

0( ) Re exp ( )( ) ( )E t iI tt t

( )( ) exp ( )SE i

Frequency-Resolved Optical Gating (FROG)

SHGcrystal

Pulse to be measured

Variable delay,

Camera

Spec-trometer

Beamsplitter

E(t)

E(t–)

Esig(t,)= E(t)E(t-)

2

( , ) ( , ) exp( )FROG sigI E t i t dt

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.

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.

10 20 30 40 50 60

10

20

30

40

50

60

SHG FROG trac e--ex panded

10 20 30 40 50 60

10

20

30

40

50

60

SHG FROG trac e--ex panded

10 20 30 40 50 60

10

20

30

40

50

60

SHG FROG trac e--ex panded

SHG FROG traces for more complex pulses

Self-phase-modulated pulse

Double pulseCubic-spectral-phase pulse

SHG FROG traces are symmetrized PG FROG traces.

Frequency

Frequency

Inte

nsi

ty

Time

Delay

Frequency

Delay

Self-phase-modulated

pulse

Cubic-spectral-phase pulse

Double pulse

FROG easily measures very complex pulsesOccasionally, a few initial guesses are necessary, but we’ve never found a pulse FROG couldn’t retrieve.

1.0

0.8

0.6

0.4

0.2

0.0

Inte

ns

ity

[a

.u.]

10005000-500-1000Time [fs]

-30

-20

-10

0

10

20

30

Ph

as

e [ra

d]

Temporal Intensity and Phase

1.0

0.8

0.6

0.4

0.2

0.0

Inte

ns

ity

[a

.u.]

900850800750Wavelength [nm]

-20

-10

0

10

20

Ph

as

e [ra

d]

Spectral Intensity and Phase

430

420

410

400

390

380

Wa

ve

len

gth

[n

m]

-1000 -500 0 500 1000Delay [fs]

Red = correct pulse; Blue = retrieved pulse

SHG FROG trace with 1% additive noise

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

(GRENOUILLE)

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

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

FROG

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 arrivesearlier than pulse #2

Here, the pulsesarrive simultaneously

Here, pulse #1 arriveslater than pulse #2

Fresnel biprism

= (x)

x

Input pulse

Pulse #1

Pulse #2

The Fresnel biprism

Very thin crystal creates broad SH spectrum in all directions. Standard autocorrelators and FROGs use such crystals.

VeryThinSHG

crystal

Thin crystal creates narrower SH spectrum in a given direction and so can’t be used for autocorrelators or FROGs.

ThinSHG

crystal

Thick crystal begins to separate colors.

ThickSHG crystalVery thick crystal acts like

a spectrometer! Replace the crystal andspectrometer 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

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

Measu

red

Retr

ieved

Testing GRENOUILLE

GRENOUILLE FROG

Measu

red

Retr

ieved

Retrieved pulse in the time and frequency domains

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

Spatio-temporal distortions

Ordinarily, we assume that the electric-field separates into spatial and temporal factors (or their Fourier-domain equivalents):

( , , , ) ( , , ) ( )xyz tE x y z t E x y z E t

where the tilde and hat mean Fourier transforms with respect to t and x, y, z.

ˆ ˆ( , , , ) ( , , ) ( )x y z xyz x y z tE k k k E k k k E

( , , , ) ( , , ) ( )xyz tE x y z E x y z E

ˆ ˆ( , , , ) ( , , ) ( )x y z xyz x y z tE k k k t E k k k E t

Angular dispersion is an example of a spatio-temporal distortion.

00

0ˆ ˆ( , , , ) [ , , ,) ](x

xx y z y z

dkk

dE k k k E k k

In the presence of angular dispersion, the mean off-axis k-vector component kx0 depends on frequency,

PrismInput pulse

Angularlydispersed output pulse

x

z

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, .

Prism pair

Input

pulseSpatially

chirped output pulse

Spatially chirped output

pulse

Input

pulse

Tilted window

00( , , , ) [ ,) , ,( ]

dx

dE x y z E x y z

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.

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

00( , , , ) [ , , , ]( )

dtt x x

dE x y z t E x y z

x

Angular dispersion always causes pulse-front tilt!

0 0ˆ ˆ( , , , ) [ ( ), , , ]x y z x y zE k k k E k k k

0( , , , ) ( , , , )E x y z t E x y z t x

Angular dispersion:

where = dkx0 /d

which is just pulse-front tilt!

Inverse Fourier-transforming with respect to kx, ky, and kz yields:

00

( )( , , , ) ( , , , )

i xE x y z E x y z e

Inverse Fourier-transforming with respect to yields:

using the shift theorem

using the inverse shift theorem

The combination of spatial and temporal chirp also causes pulse-front tilt.

Dispersive medium

Spatially chirped input

pulse

vg(red) > vg(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

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:

2 2( , ) exp 2xx xt ttE x t Q x Q xt Q t

2 2( , ) exp 2xx xE x R x R x R

2 2ˆ ( , ) exp 2kk kE k S S k Sk

2 2ˆ ( , ) exp 2kk kt ttE k t P k P kt P t

Time vs. angle

Angular dispersion

Pulse-front tilt Spatial chirp

dropping the x subscript on the k

Grad students: Xun Gu and Selcuk Akturk

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

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

The electric field vs. x and z.

Red = +Black = -

Wave-propagation direction

z

x

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

The imaginary part of Rx is wave-front-tilt dispersion.

Plots of the electric field vs. x and z for different colors.

z

x1

2

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.

Fine GDD tuning

Prism

Wavelength tuning

Wavelength tuning

Prism

Coarse GDD tuning (change distance between

prisms)

Wavelength tuning

Wavelength tuning

Prism

Prism

Even slight misalignment causes all eight spatio-temporal distortions!

The two-prism pulse compressor is

better, but still a bigproblem.

Prism

Wavelength tuning

Periscope

Wavelength tuning

Prism

Coarse GDD tuning

Roof mirror

Fine GDD tuning

GRENOUILLE measures spatial chirp.

-0

+0

SHGcrystal

Signal pulse frequency

DelayF

requ

ency

+0-0

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

Fresnel

biprism

Spatially

chirped pulse

GRENOUILLE accurately measures spatial chirp.

Measurements confirm GRENOUILLE’s ability to measure spatial chirp.

Positive spatial chirp

Negative spatial chirp

Sp

ati

o-s

pect

ral plo

t sl

ope

(nm

/mm

)

GRENOUILLE measures pulse-front tilt.

Zero relative delay is off to side of the crystal

Zero relative delay is in the crystal center

SHGcrystal

An off-center trace indicates the pulse front

tilt!Delay

Frequency

0

Fresnel biprism

Untilted pulse front

Tilted pulse front

GRENOUILLE accurately measures pulse-front tilt.

Negative PFT Zero PFT Positive PFT

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

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 FuchsIncrement between images: 20 fs (6 m).

Focus

Propagation direction

Intensity vs. x & z (at various times)

zx

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!

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

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

Equivalently, vs. frequency:

Spectral Phase

(neglecting thenegative-frequency

component)

Spectrum

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

0( , , , ) Re exp ( )( , , , , , ), ()I x y z t xE x y tz i t yt z

( , , , ) ( ( , ,, , )e p, ) x,S x y z xE x y z i y z

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 TADPOLESTRIPED FISH

Spectral Interferometry

EunkEref

T

Spectrometer Camera

1/T

Frequency

Beam splitter

ErefEunk

Measure the spectrum of the sum of a known and unknown pulse.

Retrieve the unknown pulse E() from the cross term.

( ) ( ) ( ) 2 ( ) ( ) cos[ ( ) ( ) ]SI ref unk ref unk unk refS S S S S T

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

0 Frequency

The spectral phase difference is the phase of the result.

IFFT

0 Frequency

FFT

0 “Time”

The “DC” termcontains only spectra

Filter&

Shift

0 “Time”

Filterout thesetwo peaks

The “AC” termscontain 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.

SI is very sensitive!

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

FROG’s sensitivity:

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

TADPOLE ‘s sensitivity: 1 zeptojoule = 10-21 J

Spectral Interferometry does not have the problems that plague SPIDER.

cos[ ( ) ( ) ]unk ref T

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:cos[ ( ) ( ) ]unk unk T

This is very different from standard SI’s cross-term cosine:

*J.R. Birge, R. Ell, and F.X. Kärtner, Opt. Lett., 2006. 31(13): p. 2063-5.

Linear chirp (dunk/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!

cos[ ( / ) ]unkd d T

> 10p< p~ p /100 ~ 100p

p = pulse bandwidth; p = pulse length

Desired quantity

% Difference between 10-fs and 20-fs traces

-10

-5

0

5

10

650 700 750 800 850 900 950

Wavelength (nm)

Inte

ns

ity

(%

)

Examples of ideal SPIDER traces

SPIDER Traces (10-fs flat-phase and a 20-fs chirped pulse)

0

0.5

1

1.5

2

2.5

3

3.5

4

650 700 750 800 850 900 950

Wavelength (nm)

Inte

ns

ity

(a

rb u

nit

s) 10 fs

20 fs

Even if the separation, T, were known precisely, SPIDER cannot measure pulses accurately.

Inte

nsi

ty (

%)

These two pulses are very different

but have very similar SPIDER

traces.

More ideal SPIDER tracesSPIDER Traces (100-fs flat phase and a 200-fs

linearly chirped pulse)

0

0.5

1

1.5

2

2.5

3

3.5

4

780 785 790 795 800 805 810 815 820

Wavelength (nm)

Inte

ns

ity

(a

rb

un

its

)

100 fs

200 fs

% Difference between 100-fs and 200-fs pulse traces

-10

-5

0

5

10

780 785 790 795 800 805 810 815 820

Wavelength (nm)

Inte

nsit

y (

%)

Unless the pulses are vastly different, their SPIDER traces are about the same.

Inte

nsi

ty (

%)

Practical issues, like noise, make the traces even more indisting-

uishable.

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

SEA TADPOLE

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

Fibers

Camera

xCylindrical

lens

Reference pulse

Unknown pulse

Spherical lens

Grating

Spatially Encoded

Arrangement (SEA)

SEA TADPOLE

uses spatial,

instead of spectral, fringes.

Grad student:

Pam Bowlan

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.

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!

( ) ( ) exp ( ) ( )ref unk ref unkS S i

( , ) ( ) ( ) 2 ( ) ( ) cos ( ) s( ) 2 inref unk ref unk unk refS S S S Sx kx

The beams cross, so the relative delay, T, varies with position, x.

SEA TADPOLE theoretical traces(m

m) (m

m)

More SEA TADPOLE theoretical traces

(mm

)

(mm

)

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 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.

SEA TADPOLE spectral super-resolution

( ) ( ) exp[ ( ( ) ( ))] ( )

( ) exp[ ( )] ( )

ref unk ref unk

unk unk

S S i H

S i H

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() 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.

E(x,z,t)

Pulse Fronts

Color is the instantaneous frequency vs. x and t.

Uniform color indicates a lack of phase distortions.

Sim

ula

tion

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

Sim

ula

tion

M

easu

rem

en

t

810 nm

790 nm

Aspheric PMMA lens with chromatic (but no spherical) aberration and GDD.

f = 50 mmNA = 0.03

Spherical and chromatic aberration

Sim

ula

tion

M

easu

rem

en

t

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

f = 50 mmNA = 0.03

810 nm

790 nm

A ZnSe lens with chromatic aberration

Sim

ula

tion

M

easu

rem

en

t

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

804 nm

796 nm

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

Distortions are more pronounced for a tighter focus.

814 nm

787 nm

Sim

ula

tion

Exp

eri

men

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

f = 25 mmNA = 0.06

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

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

790 nm

812 nm

Aspheric PMMA lens.

f = 50 mmNA = 0.03

Sim

ula

tion

Exp

eri

men

t

Single-shot measurement of E(x,y,z,t). Multiple holograms on a single camera frame: STRIPED FISH

Array of spectrally-resolved hologramsSpatially and

TemporallyResolvedIntensity andPhaseEvaluationDevice:

FullInformation from aSingleHologram

Grad student: Pablo Gabolde

Holography

( , ) ( , ) ( , ) 2 ( , ) ( , ) cos[ ( , ) ( , )]holo ref unk ref unk unk refI x y I x y I x y I x y I x y x y x y

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

Frequency-Synthesis Holography for complete spatio-temporal pulse measurement

0( , , )E x y

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.

1

1 1( , , ) ( , , ) ( , , )

2 2k

ni ti t

k

k

E x y t d e E x y e E x y

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.”

STRIPED FISH

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.

The band-pass filter spectrally resolves the digital holograms

STRIPED FISH Retrieval algorithm

Complex image

Intensityand phasevs. (x,y,λ)

Typical STRIPED FISH measured trace

Measurements of the spectral phase

Group delay

Group-delay dispersion

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, 156-160 (1982).

IFFT

FFT

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

The spatial fringes depend on the spectral phase!

(a)

(b)

Zero delay

With GD

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

0tcx

x [mm] t [fs]

803 nm

777 nm

= 4.5 mrad797 nm

775 nmx [mm] t [fs

]

= 11.3 mrad

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

where the pulse-front tilt angle is:

Complete electric field reconstruction

Pulse with horizontal spatial chirp

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.

The Space-Time-Bandwidth Product

x

y

t

STRIPED FISH can measure pulses with STBP ~ 106 ~ 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-BandwidthProduct (SBP)

Time-BandwidthProduct (TBP)

Space-Time-BandwidthProduct (STBP)

=

How complex a pulse can STRIPED FISH measure?

Corner cube

Prism

Wavelength tuning

GDD tuning

Roof mirror

Periscope

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

Beam magnification is always one.

1 32 4

1 1= = =M M

M M

1 2 3 4 = 1totM = M M M M

1M

2Mdout

din

/out inM d d

The total dispersion is always zero.

1 2 2 3 4 4D M D D M D

31 24

2 3 4 3 4 4

0tot

DD DD D

M M M M M M

1D

2D

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 (unfo-

cused) pulse on a single shot.

To learn more, visit our web sites…

www.swampoptics.com

www.physics.gatech.edu/frog

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!