Dealing with error in FROG measurements

Random error (noise) and how to suppress it; error bars

Nonrandom error (systematic error), how to know when it’sthere, and how to correct for it

The FROG marginals

Extremely simple FROG beam geometry

Measuring Ultrashort Laser Pulses III: FROG tricks

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FROG trace--expandedRandomly remove about half of the datapoints from the trace.

Rick Trebino, Georgia Tech, rick.trebino@physics.gatech.edu

Random and Systematic Error in Pulse Measurement

Consider an autocorrelation measurement.

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FROG trace--expandedA FROG trace has N2 points.N pixelsN pixels

The FROG trace overdetermines the pulse. This has advantages.

1. Natural √N averaging occurs, reducing noise.2. Can perform filtering operations to reduce noise further.3. Can run algorithm with some points removed to determine error bars in the

intensity and phase—independent of the source of noise.4. Can identify the presence of systematic error—independent of the source.5. Can remove systematic error—independent of the source.6. Can understand distortions in the autocorrelation due to systematic error.

Advantages:

N pixelsThe intensity and phase (or frequency) have only 2N points.

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Inte

nsity

Noise and its Suppression in FROG

Noise can corrupt a FROG trace and yield an incorrect pulse measurement.

Fortunately, there are many techniques for suppressing the noise with minimal distortion to the retrieved pulse.

1. Background subtractionThe FROG trace should be an island in a sea of zeroes. Otherwise, data are missing. So we can subtract off any background.

2. Corner suppression

No data should be in the corners of the trace; what’s there can only be noise, so set it to ~zero by multiplying by exp(-r4/d4).

3. Low-pass filtering

Noise varies from pixel to pixel, that is, with a high frequency. The FROG trace has only slower variations.

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Time (pulse widths)

Frequency (1/pulse width)

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3.00

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Time (pulse widths)

Frequency (1/pulse width)

Without noise With noise

Fittinghoff, et al., JOSA B, 12, 1955 (1995).

FROG trace fora complex pulse:

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Time (pulse widths)

Frequency (1/pulse width)

Noise and its Suppression in FROG: Example

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Phase (radians)

Time (pulse widths)

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Normalized Intensity

Time (pulse widths)

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Intensity

Phase

This pulse has a narrow glitch inits intensity vs. time, and it hasa phase jump of ~2 radians, adifficult feature to reproduce.We’ll add noise to this trace.

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3.00

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Time (pulse widths)

Frequency (1/pulse width)

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3.00

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Time (pulse widths)

Frequency (1/pulse width)

Adding 10% additive noise turns this clear trace into this mess:

(Noise is Gaussian distributed with a mean of 10%.)

Note the resulting large background in the noisy trace.

Corrupting a FROG Trace with Noise

Time (pulse widths)

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Time (pulse widths)

Noise in the FROG trace can yield a noisy retrieved intensity and phase.

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Time (pulse widths)

Frequency (1/pulse width)

The retrieved pulse is very noisy!It looks nothing like the actual pulse.

Background at large delays yields wings in the intensity. Background at large frequency offsets yields noise in those wings.

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1.2ActualRetrieved

Subtracting off the background improves the retrieved intensity and phase.

Frequency (1/pulse width)

Time (pulse widths)

FROG trace with 10% addi-tive noise—after subtracting the mean of the noise

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Note the suppression of the wings and of the noise in the wings of the pulse.

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Time (pulse widths)

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Time (pulse widths)

Frequency (1/pulse width)

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Actual Retrieved

Time (pulse widths)

FROG trace with 10% addi-tive noise—after subtracting

off the background meanand suppressing the corners

Suppressing the corners of the trace also improves the retrieved intensity and phase.

Trace was multiplied by a “super-Gaussian”: exp(-r4/d4), where r =distance from trace center.

Note further improvement in the wings.

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Time (pulse widths)

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Time (pulse widths)

Low-pass filtering further improves the retrieved intensity and phase.

Fourier-transforming the trace, retaining only the center region, and transforming back.

The resulting intensity and phase now look very much like the actual curves!

Wavelength (1/pulse width)

Time (pulse widths)

FROG trace with 10% addi-tive noise after mean sub-traction, super-Gaussian filtering and lowpass filtering

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Time (pulse widths)

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Time (pulse widths)

Filtering summary: Always do it!

Dramatic improvements in the retrieval occur with little distortion. After filtering,10% additive noise yields ~1% error; even less with multiplicative noise.

With filtering

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Time (pulse widths)

Intensity:

Phase:

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Without filtering

Time

Repeat the above procedure several times, removing different points each time.

Calculate the mean and standard deviation of the intensity and phase (or frequency) for each time.

Munroe, et al., CLEO Proceedings, 1998.Press, et al., Numerical Recipes

We can place error bars on the retrieved intensity and phase using the “Bootstrap” method.

Fre

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Time

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(or

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FROG trace--expandedRandomly remove about half of the datapoints from the trace.

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(ar

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Time (arb. units)

Analytic intensity Retrieved intensity with noise

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Time (arb. units)

Analytic phase Retrieved phase with noise

Intensity Phase

Errors in the intensity are similar everywhere (slightly larger at the peak). Because the noise was ad-ditive, noise exists in the wings also.

Errors in the phase are muchlarger in the wings, where theintensity is near-zero and thephase is necessarily undefined.

Introducing 1% additive noise to the FROG trace:

Error Bars in the Intensity and Phase Using the Bootstrap Method—Theory

Error Bars in the Intensity and Phase Using the Bootstrap Method—Exp’t

Intensity Phase

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(ar

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Time (fs)

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hase

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Errors in the intensity are muchlarger at the peak. Because the noise was multiplicative, there isalmost no noise in the wings.

The phase error is low, exceptin the wings, where, as before,the intensity is near-zero and thephase is necessarily undefined.

In practice, SHG FROG traces have mostly multiplicative noise:

Variation in spectral response of optics

Variation in spectral response of camera

Dispersion of nonlinearity

Group-velocity mismatch/phase-matching bandwidth

Variable alignment of beam overlap

Unknown

Check? Correct?

√

√

√

√

√

√

Source:

√

√

√

√

Possibly!

Sources of Systematic Error in FROG

It is possible, not only to check for systematic error, but also to correct it in most pulse measurements using FROG, even when its origin in unknown.

Geometrical time-smearing could yield systematic error.

Avoiding Geometrical Time-Smearing

Wavelength-dependent SHG phase-matching efficiency yields systematic error.

Even very thin SHG crystals may lack sufficient bandwidth for a 10-fs pulse.

Group-velocity mismatch yields a wavelength-dependent SHG efficiency.Usually, it’s a sinc2 curve, but even when two such curves fortuitously overlap, there’s wavelength-dependent SHG efficiency:

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350 nm400 nm

Second Harmonic Wavelength (nm)

Phase-matched wave-length

60-µm thick KDP crystal

Phase-matching efficiency vs. wavelength

It’s impossible to achieve the desired flat curve.

Taft, et al., J. Selected Topics in Quant. Electron., 3,

575 (1996)

The delay marginal is the integral of the FROG trace over all frequencies. It is a function of delay only.

The frequency marginal is the integral of the FROG trace over all delays: It is a function of frequency only.

Mω(ω) ≡ I FROG(ω,τ)d∫ τ

The FROG marginals can be related to easily meas-ured quantities:

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SHG FROG trace--expanded

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DelayMτ(τ) ≡ I FROG(ω,τ)d∫ ω

Mω(ω) = The Autoconvolutionof the Spectrum

Mτ(τ) = The Autocorrelation

The marginals are essential in checking for systematic error.

The FROG Marginals

DeLong, et al., JQE, 32, 1253 (1996).

See graphically the effects of systematic error on autocorrelations

Check SHG FROG trace by comparing the spectrum autoconvolution and the frequency marginal

Correct SHG FROG trace by multiplying trace by ratio of the spectrum autoconvolution and the frequency marginal

Applications of the SHG FROG Marginals

See graphically the effects of systematic error on autocorrelations

Check SHG FROG trace by comparing the spectrum autoconvolution and the frequency marginal

Correct SHG FROG trace by multiplying trace by ratio of the spectrum autoconvolution and the frequency marginal

Taft, et al., J. Selected Topics in Quant. Electron., 3,

575 (1996)

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350370390410430450470490

Delay (fs)-40 -20 0 20 40

350370390410430450470490

Delay (fs)

measured retrieved

Delay (fs) Delay (fs)

Independently measured spectrum

Correcting for Systematic Error: Example

Attempts to measure a ~10-fs pulse produced this trace and pulse:

Usually, systematic error yields poor convergence. Here, however, despite good convergence, the retrieved spectrum disagrees with the independently measured spectrum. (This is due to insufficient phase-matching bandwidth in a 60-µm KDP crystal.)

Retrieved pulseFROG trace FROG trace

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FROG MarginalAutoconvolution

Wavelength (nm)

Comparing the FROG frequency marginal with the spectrum autoconvolution

Although they should agree, they don’t! This is because the SHG crystal did not phase-match the longer wavelengths of the pulse.

Forcing the frequency marginal to agree with the spectrum autoconvolution yields an improved trace.

Multiplying the measured FROG trace by the ratio of the spectrum autoconvolution and frequency marginal:

Independently measured spectrum

Retrieved pulse

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350370390410430450470490

Delay (fs)-40 -20 0 20 40

350370390410430450470490

Delay (fs)

measured retrieved

Delay (fs) Delay (fs)

FROG trace FROG trace

corrected

The retrieved spectrum now agrees with the measured spectrum.

The spectral phase has also changed.

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6807658509351020Wavelength (nm)

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Predicted pulse

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6807658509351020Wavelength (nm)Measured pulse

The corrected pulse can now be used in comparisons with theory.

This pulse measurement verifies that material dispersion is the pulse-length-limiting effect in this laser.

The measured and predicted pulse vs. time:

Wavelength Wavelength

The FROG marginals can be used to understand the effects of systematic error on autocorrelation measurements.

Why pulse autocorrelations can appear narrower when using a thick crystal

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α=∞α=0.60α=0.27Time

α = 0.60 α = 0.27α = ∞

SHG FROG trace

Gaussian spectrum w/cubic spectral phase

Delay Delay Delay

Thick crystal: Thicker crystal:

Thick crystal

suppresses wings!

= phase-matching bandwidth / pulse bandwidthα

Thin crystal:

CroppedSHG FROG trace

Very croppedSHG FROG trace

Delay

Autocorrelation

Correct (thin-crystal) autocorrelationα =

Incorrect (thick-crystal)Autocorrelations α 6 α 27 It’s difficult to know

if the crystal is thin enough!

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

This is the opposite of the usual condition!

In GRENOUILLE, the GVM must be large!

In GRENOUILLE, the GVD must still be small.

Putting it all together

GVM is usually much greater than GVD.

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:

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

Advantages of GRENOUILLE

Disadvantages of GRENOUILLE

It currently only works for pulses between ~ 40 fs and ~ 300 fs.

Like other single-shot techniques, it requires good spatial beam quality.

Improvements on the horizon:

Inclusion of GVD and GVM in FROG code to extend the range of operation to shorter and longer pulses.

Folded beam geometry for even more compact arrangement.

Disadvantages of FROG and its relatives

FROG requires taking a lot of data. While this can be done easily with a readily available camera, and it allows error checking and correcting, multi-shot FROG measurements can take minutes.

The algorithm can be slow, also taking minutes for complex pulses. (There is, however, a new algorithm, based on singular-value decomposition, which is much faster: < 1 sec.)

SHG FROG has an ambiguity in the direction of time.

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!