Supported by NSF and Georgia Institute of Technology

Numerical Simulations of an UltrasimpleUltrashort-Laser-Pulse Measurement

Device─GRENOUILLE

Xuan Liu and R. TrebinoSchool of Physics, Georgia Institute of

Technology, GA 30332 USAArlee V. Smith

Sandia National Laboratories, Albuquerque, NM 87185, USA

Outline

Measure ultrashortpulses using the FROG technique

NumericalSimulations of GRENOUILLE

Simplification of FROG set up: the

GRENOUILLE device

Frequency Resolved Optical Gating (FROG)

SHGcrystal

Pulse to be measured

Variable delay, τ

CameraSpec-

trometer

Beamsplitter

E(t)

E(t–τ)

Esig(t,τ)

FROG measures the full-intensity-and-phase of the pulse.

FROG involves gating the pulse with a variably delayed replica of itself in an instantaneous nonlinear-optical medium and then spectrally resolving the gated pulse vs. delay.

The FROG trace is a spectrogram of E(t). The mathematical expressionfor the FROG trace:

2( , ) ( ) ( ) exp( )FROGI E t g t i t dtω τ τ ω∝ − −∫

g(t–τ) = E(t–τ)

We must invert this integral equation and solve for the unknown pulse.Thisintegral-inversion problem is the 2D phase-retrieval problem, for which the solution exists and is (essentially) unique. And simple algorithms exist for finding it.

Run FROGretrieval program

The FROG Technique

Where is the gate function.

Outline

Measure ultrashortpulses using the FROG technique

NumericalSimulations of GRENOUILLE

Simplification of FROG set up: the

GRENOUILLE device

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

(GRENOUILLE)

FROG:Frequency-

ResolvedOpticalGating

GRENOUILLE:GRating-

EliminatedNo-nonsense

Observation ofUltrafastIncident

LaserLight

E-fields

A single optic (a Fresnel biprism) replaces the entire delay line, anda thick SHG crystal replaces both the thin crystal and spectrometer.

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

τ =τ(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 ina 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.

Crystal Thickness

Yields a complete single-shot FROG. Uses the standard FROG algorithm. Never misaligns. Is more sensitive. Measures spatio-temporal distortions*

Lens images position in crystal (i.e., delay, τ) to horizontal

position at camera

Topview

Sideview

Cylindricallens

FresnelBiprism

ThickSHG

Crystal

Imaging Lens

FT Lens

Camera

Lens maps angle (i.e.,wavelength) to vertical

position at camera

GRENOUILLE Beam Geometry

*Akturk, S., et al., Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE. Opt. Expr., 2003. 11(5): p. 491-501.Akturk, S., et al., Measuring spatial chirp in ultrashort pulses using single-shot Frequency-Resolved Optical Gating. Opt. Expr.,

2003. 11(1): p. 68-78

Testing GRENOUILLE

GRENOUILLE FROG

Mea

sure

dR

etrie

ved

Retrieved pulse in the time and frequency domains

Read more about

GRENOUILLE in the cover

story of OPN, June 2001

Compare a GRENOUILLE measurement of a pulse with a tried-and-true FROG measurement of the same pulse:

Outline

Measure ultrashortpulses using the FROG technique

NumericalSimulations of GRENOUILLE

Simplification of FROG set up: the

GRENOUILLE device

Nonlinear Process in GRENOUILLE

1 2 3ω ω ω+ =

All the plane wave pairs have to satisfy:

1 2 3y y yk k k+ =

3ω1ω

2ω

3k1k

2k

GRENOUILLE uses the second-order nonlinearities of a thick crystal withrelatively tightly focused and broadband pulses. We must allow not only SHGprocesses, but also all possible sum-frequency-generation processes, both collinear and non-collinear.

Sum Frequency Generation

No pump depletion:

3θ2θ

qkr1Pk

r

2Pkr Z

1θ3 3 3

3

1 1 2 21 2

( , ) cos

( ) ( )cos cos

eqz

o o

nkc

n nc c

ω ω θ θ

ω ω ω ωθ θ

Δ = −

−

Phase mismatch:

3 3 3 3

3

1 1 1 2 3 1 3 1 1 1

( , , )exp( )

( , , ) ( , , )

y effz

y y y y

E k z di i k z

z cnE k z E k k z dk d

ω ω

ω ω ω ω

∂= − Δ ×

∂− −

∫∫ %

Zero Delay Spectrum at Different Angles

The deviation angle vs. spectrum plot confirmed that the crystaldoes work like a spectrometer.

Wavelength (nm)

Deviation angle (Rad)

Simulated GRENOUILLE Trace

Tight Focus

Crystal Thickness

Simulated GRENOUILLE Trace and Its Retrieval

200 fs chirped pulse, The FROG (rms) error was 0.001148.

Conclusion

We numerically simulated the performance of GRENOUILLE, which involves the sum frequency generation of tightly focusedbroadband input beams.

Using the full Sellmeier equation, we can take into account all of the dispersion effects.

In order to improve GRENOUILLE’s spectral resolution when high accuracy is required, we numerically deconvolve the spectralresponse of the device with a efficiency curve that we derive fromour simulations. We show that accurate measurements are easily obtained for properly designed device.