zn + an-1 zn-1 + ... + a0 = (z – zn) (z – zn-1) ... (z – z1)

and the Measurement of the Shortest

Rick Trebino

School of Physics

Georgia Institute of Technology

Atlanta, GA 30332

The Musical Score,

the Fundamental Theorem of Algebra,

Events Ever Created

In order to measure an event in time,you need a shorter one.

To study this event, you need a strobe light pulse that’s shorter.

But then, to measure the strobe light pulse, you need a detector whose response time is even shorter.

And so on…

So, now, how do you measure the shortest event?

Photograph taken by Harold Edgerton, MIT

The Dilemma

Ultrashort laser pulses are the shortest technological events ever created by humans.

It’s routine to generate pulses shorter than 10-13 seconds in duration,and researchers have generated pulses only a few fs (10-15 s) long.

Such a pulse is to one second as 5 cents is to the US national debt.

Such pulses have many applications in physics, chemistry, biology, and engineering. You can measure any event—as long as you’ve got a pulse that’s shorter.

So how do you measure the pulse itself?

You must use the pulse to measure itself. But that isn’t good enough. It’s only as short as the pulse. It’s not shorter.

Techniques based on using the pulse to measure itself have not sufficed.

A laser pulse has the time-domain electric field:

E I(t)1/ 2 exp [ it – i(t) ] }

Intensity Phase

(t) = Re {

Equivalently, vs. frequency:

exp[ -i (–0) ] }

Spectral Phase

(neglecting thenegative-frequencycomponent)

E() = Re {~

S()1/ 2

We must measure an ultrashort laser pulse’sintensity and phase vs. time or frequency.

Spectrum

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

tddtThe instantaneous frequency:

Example: “Linear chirp”

Pha

se,

(t)

time

time

Fre

quen

cy,

(t

)

time

We’d like to be able to measure,not only linearly chirped pulses,but also pulses with arbitrarily complex phases and frequencies vs. time.

The phase determines the pulse’s frequency (i.e., color) vs. time.

Even nice autocorrelations have ambiguities.These complex intensities have nearly Gaussian autocorrelations.Conclusions drawn from an autocorrelation are unreliable.

-80 -60 -40 -20 0 20 40 60 80

IntensityAmbiguous Intensity

Time

Intensity

-150 -100 -50 0 50 100 150

Autocorrelation

AutocorrelationAmbig AutocorGaussian

Delay

TimeIntensityPhase

Perhaps it’s time to ask how researchers in other fields deal with their waveforms…

Consider, for example, acoustic waveforms.

Autocorrelation and related techniques yield little information about the pulse.

It’s a plot of frequency vs. time, with information on top about the intensity.

The musical score lives in the “time-frequency domain.”

Most people think of acoustic waves in terms of a musical score.

A mathematically rigorous form of a muscial score is the “spectrogram.”

The spectrogram is a function of andτ.IτisτheseτospecτraoallτemporalslicesoEτ.

If E(t) is the waveform of interest, its spectrogram is:

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SpE (ω,τ) ≡ E(t) g(t−τ)exp(−iωt) dt−∞

∞

∫2

where g(t-τ) is a variable-delay gate function and τ is the delay.

Without g(t-τ), SpE(,τ) would simply be the spectrum.

The Spectrogram of a waveform E(t)We must compute the spectrum of the product: E(t) g(t-τ)( -g t τ)E(t)timeτ ( ) The spectrogram tells the color and intensity of E t at the timeτ.( -g t τ) contributes , only intensity not

( . ., ),phase i e color .to the signal pulse

( )E t contributes( . .,phase i e color),

.to the signal pulse( ) ( -E t g tτ)

10 20 30 40 50 60102030405060

SHG FROG trace--expanded

10 20 30 40 50 60102030405060

FROG trace--expanded

Fre

quen

cyF

requ

ency

Time

Delay

Spectrograms for Linearly Chirped Pulses

Properties of the Spectrogram

The spectrogram resolves the dilemma! It doesn’t need the shorter event! It temporally resolves the slow components and spectrally resolves the fast components.

Algorithms exist to retrieve E(t) from its spectrogram.

The spectrogram essentially uniquely determines the waveform intensity, I(t), and phase, (t).

There are a few ambiguities, but they are “trivial.”

The gate need not be—and should not be—significantly shorter than E(t).

Suppose we use a delta-function gate pulse:

E(t)δ(t−τ) exp(−iωt)dt−∞

∞

∫2

= E(τ) exp(−iωτ)2

= E(τ)2

= The Intensity.

No phase information!

“Polarization-Gate” Geometry

Spectro-

meter

Frequency-Resolved Optical Gating (FROG)

Trebino, et al., Rev. Sci. Instr., 68, 3277 (1997). Kane and Trebino, Opt. Lett., 18, 823 (1993).

Frequency-Resolved Optical GatingEsig(t,τ)∝ ( )E t | ( -E t τ)|2 ( -E t τ)( )E ttimeτ Signal pulse2τ/3 , ( ),The signal pulse reflects the color of the gated pulse E t 2at the timeτ/3.

| ( -E t τ)|2 contributes , only intensity not

( . ., ),phase i e color .to the signal pulse

( )E t contributes( . ., ),phase i e color

.to the signal pulse

10 20 30 40 50 60102030405060

SHG FROG trace--expanded

10 20 30 40 50 60102030405060

FROG trace--expanded

Fre

quen

cyF

requ

ency

Time

Delay

FROG Traces for Linearly Chirped Pulses

Fre

quen

cyF

requ

ency

Inte

nsity

Time

Delay

FROG Traces for More Complex PulsesF

requ

ency

Delay

Unfortunately, spectrogram inversion algorithms require that we know the gate function.

Substituting for Esig(t,τ) in the expression for the FROG trace:

yields:

Esig(t,τ) E(t) |E(t–τ)|2

IFROG(ω,τ) ∝ Esig(t,τ) exp(−iωt)dt∫2

IFROG(ω,τ) ∝ E(t)g(t−τ) exp(−iωt) dt∫2

where: g(t–τ) |E(t–τ)|2

The FROG trace is a spectrogram of E(t).

If Esig(t,τ), is the 1D Fourier transform with respect to delay τ of some new signal field, Esig(t,), then:

IFROG(ω,τ) = ˆ E sig(t,Ω)exp(−iωt−iΩτ) dtdΩ∫∫2

So we must invert this integral equation and solve for Esig(t,).

This integral-inversion problem is the 2D phase-retrieval problem,for which the solution exists and is unique.And simple algorithms exist for finding it.

and

The input pulse, E(t), is easily obtained from Esig(t,): E(t) Esig(t,)

Instead, consider FROG as a two-dimensional phase-retrieval problem.

Stark, Image Recovery, Academic Press, 1987.

1D Phase Retrieval: Suppose we measure S() and desire E(t), where:

Given S(kx,ky), there is essentially one solution for E(x,y)!!!It turns out that it’s possible to retrieve the 2D spectral phase!

.

Given S(), there are infinitely many solutions for E(t). We lack the spectral phase.

2D Phase Retrieval: Suppose we measure S(kx,ky) and desire E(x,y):

These results are related to the Fundamental Theorem of Algebra.

We assume that E(t) and E(x,y) are of finite extent.

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S(ω) = E (t)exp(−iωt) dt−∞

∞

∫2

S(kx,ky) = E(x,y)exp(−ikxx−ikyy) dxdy−∞

∞

∫−∞

∞

∫2

1D vs. 2D Phase Retrieval

Stark, Image Recovery, Academic Press,

1987.

The Fundamental Theorem of Algebra states that all polynomials can be factored:

fN-1 zN-1 + fN-2 zN-2 + … + f1 z + f0 = fN-1 (z–z1 ) (z–z2 ) … (z–zN–1)

The Fundamental Theorem of Algebra fails for polynomials of two variables.Only a set of measure zero can be factored.

fN-1,M-1 yN-1 zM-1 + fN-1,M-2 yN-1 zM-2 + … + f0,0 = ?

Why does this matter?

The existence of the 1D Fundamental Theorem of Algebra implies that 1D phase retrieval is impossible.

The non-existence of the 2D Fundamental Theorem of Algebra implies that 2D phase retrieval is possible.

Phase Retrieval and the Fundamental Theorem of Algebra

1D Phase Retrieval and the Fundamental Theorem of Algebra

The Fourier transform {F0 , … , FN-1} of a discrete 1D data set, { f0 , …, fN-1}, is:

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Fk ≡ fme−imk

m=0

N−1

∑ = fmzm

m=0

N−1

∑ where z = e–ik

The Fundamental Theorem of Algebra states that any polynomial,fN-1zN-1 + … + f0 , can be factored to yield: fN-1 (z–z1 ) (z–z2 ) … (z–zN–1)

So the magnitude of the Fourier transform of our data can be written:

|Fk| = | fN-1 (z–z1 ) (z–z2 ) … (z–zN–1) | where z = e–ik

Complex conjugation of any factor(s) leaves the magnitude unchanged,but changes the phase, yielding an ambiguity! So 1D phase retrieval isimpossible!

polynomial!

2D Phase Retrieval and the Fundamental Theorem of Algebra

The Fourier transform {F0,0 , … , FN-1,N-1} of a discrete 2D data set, { f0.0 , …, fN-1,N-1}, is:

Fk,q ≡ fm,p e−imk

p =0

N−1

∑ e−ipq

m=0

N−1

∑ = fm,p ymzp

p =0

N−1

∑m=0

N−1

∑where y = e–ik and z = e–iq

But we cannot factor polynomials of two variables. So we can only complex conjugate the entire expression (yielding a trivial ambiguity).

Only a set of polynomials of measure zero can be factored.So 2D phase retrieval is possible! And the ambiguities are very sparse.

Polynomial of 2 variables!

Generalized ProjectionsThe Solution!

Initial guess for Esig(t,τ)Aproecτionmapsτhecurrenτguessorτheaveormτoτheclosesτpoinτinτheconsτrainτseτ.Convergenceisguaranτeedorconvexseτs,buτgenerallyoccurseveniτhnonconvexseτs.

Set of waveforms that satisfy- :nonlinear optical constraint

Set of waveforms that :satisfy data constraint

Esig(t,τ) E(t) |E(t–τ)|2

IFROG(ω,τ) ∝ ∫Esig(t,τ) exp(−iωt) dt2

Esig(t,τ)

QuickTime™ and aQuickDraw decompressor

are needed to see this picture.

Single-shot FROG gives real-time feedbackon laser performance.

A grating pulse compressor requires the precise grating spacing,or the pulse will be chirped (positively or negatively). This can be a difficult alignment problem.

Note that the trace is rotated by 90˚.

Data taken by Toth and coworkers

Sho

rtes

t pu

lse

leng

th

Year

Shortest pulse vs. year

Plot prepared in 1994, reflecting the state of affairs at that time.

The measured pulse spectrum had two humps,and the measured autocorrelation had wings.

Data courtesy of Kapteyn and Murnane, WSU

Despite different predictions for the pulse shape, both theories were consistent with the data.

Two different theories emerged, and both agreed with the data.

From Harvey et. al, Opt. Lett.,v. 19, p. 972 (1994)

From Christov et. al, Opt. Lett., v. 19, p. 1465 (1994)

FROG distinguishes between the theories.

Taft, et al., J. Special Topics in Quant. Electron., 3, 575 (1996).

SHG FROG Measurements of a 4.5-fs Pulse!

Baltuska, Pshenichnikov, and Weirsma,J. Quant. Electron., 35, 459 (1999).

Measurement of Ultrabroadband Continuum

Ultrabroadband continuum was created by propagating 1-nJ, 800-nm, 30-fs pulses through 16 cm of Lucent microstructure fiber. The 800-nm pulse was measured with FROG,so it made an ideal known gate pulse.

This pulse has a time-bandwidth product of > 1000, and is the most complex ultrashort pulse ever measured.

XFROG trace

Retrieved intensity and phase

Kimmel, Lin, Trebino, Ranka, Windeler, and Stentz, CLEO 2000.

Sensitivity of FROG

Because ultraweak ultrashort pulses are almost always created by much stronger pulses, a stronger reference pulse is always available.

Use Spectral Interferometry

This involves no nonlinearity! ... and only one delay!

EunkEref

τ

Spectrometer Camera

frequency

FROG + SI = TADPOLE (Temporal Analysis by Dispersing a Pair Of Light E-fields)

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SSI(ω)=Sref(ω)+Sunk(ω)+2 Sref(ω) Sunk(ω)cos[ϕunk(ω)−ϕref(ω) +ωτ]

Froehly, et al., J. Opt. (Paris) 4, 183 (1973)Lepetit, et al., JOSA B, 12, 2467 (1995)C. Dorrer, JOSA B, 16, 1160 (1999)Fittinghoff, et al., Opt. Lett., 21, 884 (1996).

Measuring Ultraweak Ultrashort Light Pulses

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

with as little energy as: 10TADPOLE can measure pulses

1 zeptojoule = –21 J

A pulse train containing only 42 zepto-joules (42 x 10-21 J) per pulse has beenmeasured.

That’s one photon every five pulses!

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

Sensitivity of Spectral Interferometry (TADPOLE)

...there is, however, light whose polarization state changes too rapidly to be measured with the available apparatus!

POLLIWOG (POLarization-Labeled Interference vs. Wavelength for Only a Glint*)

* Glint = “a very weak, very short pulse of light”

Unpolarized light doesn’t exist…

Measurement of the variation of the polarization state of the emission from a GaAs-AlGaAs multiple quantum well when heavy-hole and light-hole excitons are excited elucidates the physics of these devices.

Evolution of the polarization of the emission:

A. L. Smirl, et al., Optics Letters, Vol. 23, No. 14 (1998)

Application of POLLIWOG

Excitation-laser spectrum and hh and lh exciton spectra

time (fs)

2 alignment parameters (,)

Can we simplify FROG?

SHGcrystal

Pulse to be measured

Variable delay

CameraSpec-trom-eter

FROG has 3 sensitive alignment degrees of freedom (, of a mirror and also delay).

The thin crystal is also a pain.

1 alignment parameter (delay)

Crystal must be very thin, which hurts sensitivity.

SHGcrystal

Pulse to be measured

Remarkably, we can design a FROG without these components!

Camera

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! Why not replace the 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 generated depends on the angle. And the angular width of the SH beam created varies inversely with the crystal thickness.

The angular width of second harmonic varies inversely with the crystal thickness.

GRating-Eliminated No-nonsense Observationof Ultrafast Incident Laser Light E-fields

(GRENOUILLE)

Patrick O’Shea, Mark Kimmel, Xun Gu and Rick Trebino, Optics Letters, 2001;Trebino, et al., OPN, June 2001.

GRENOUILLE Beam Geometry

Really Testing GRENOUILLE

Even for highly structured pulses, GRENOUILLE allows for accurate reconstruction of the intensity and phase.

GRENOUILLE FROG

Measured:

Retrieved:

Retrieved pulse in the time and frequency domains

FROG has a few advantages!

www.physics.gatech.edu/frog

To learn more, see the FROG web site!

Or read the cover story in the June 2001 issue of OPN Or read the book!