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Measuring pulse-front tilt in ultrashort pulses using GRENOUILLE

Selcuk Akturk, Mark Kimmel, Patrick O’Shea, and Rick Trebino

Open Access Open Access

Abstract

We show that the spatio-temporal distortion, pulse-front tilt, is naturally, easily, and sensitively measured by the recently demonstrated, extremely simple variation of single-shot second-harmonic generation frequency-resolved optical gating (SHG FROG): GRENOUILLE. While GRENOUILLE traces are ordinarily centered on the zero of delay, a pulse with pulse-front tilt yields a trace whose center is shifted to a nonzero delay that is proportional to the pulse-front tilt. As a result, the trace-center shift reveals both the magnitude and sign of the pulse-front tilt—independent of the temporal pulse intensity and phase. The effects of pulse-front tilt can then easily be removed from the trace and the intensity and phase vs. time also retrieved, yielding a full description of the pulse in space and time.

©2003 Optical Society of America

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Figures (9)
Fig. 1.
Fig. 1. Elements that introduce angular dispersion also introduce pulse-front tilt.

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Fig. 2.
Fig. 2. Pulse-front tilt in single-shot SHG FROG: Pulses without pulse-front tilt (shown in green) yield traces centered on the crystal, where zero relative delay occurs. Pulses with pulsefront tilt (shown in red), also yield traces centered at the same point. Therefore, single-shot SHG FROG does not distinguish pulse-front tilt. (Magnitude of tilt exaggerated for clarity.) This is also true of other FROG beam geometries. Pulse-front tilt does, however, change the delay calibration, so this must be taken into account (using a reference double-pulse with known pulse separation as a calibration, as is commonly done).

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Fig. 3.
Fig. 3. Pulse-front tilt in GRENOUILLE: Pulses without pulse-front tilt (shown in green) yield traces centered at the crystal, where zero relative delay occurs. However pulses with pulsefront tilt (shown in red), cause the zero relative delay to be off to the side of the crystal, causing the trace to be shifted from the center of the crystal. The figure also shows calculation of pulsefront tilt from the amount of shift at the center of the trace. (Magnitude of tilt exaggerated for clarity.)

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Fig. 4.
Fig. 4. Theoretical dependence of the pulse-front tilt on angle of incidence for plane waves (blue curve) and Gaussian beams with various beam parameters (other colored curves). A 69° apex-angle fused-silica prism is used for calculation. The parameters are; d, the distance between beam waist and prism; s, the distance between prism and observation point; w, the beam width. The variation in pulse-front tilt is due to a complex combination of diffraction and beam expansion due to refraction at the prism faces.

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Fig. 5.
Fig. 5. Measured GRENOUILLE traces for pulses with very negative, slightly negative, zero, and slightly positive, and very positive pulse-front tilt. Notice that the trace displacement is proportional to the pulse-front tilt. The traces also possess some amount of shear, which indicates the existence of spatial chirp [1], but it has no effect on the center of the trace.

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Fig. 6.
Fig. 6. Theoretically predicted pulse-front tilt and the experimentally measured pulse-front tilt using GRENOUILLE. The pulse-front tilt was varied using the modified pulse compressor described in the text. Note that, GRENOUILLE easily measures even small amounts of pulsefront tilt, such as occurs when the prism angle is at angle of minimum deviation, and zero pulse-front tilt is obtained, which corresponds to the case of the pulse-compressor being considered to be “aligned.”

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Fig. 7.
Fig. 7. Experimentally measured and theoretically predicted pulse-front tilt for our experiment performed by measuring pulses that have passed through a single fused silica prism. In this case the pulse front tilt is always negative (Theory curve is the blue curve in Fig. 4).

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Fig. 8.
Fig. 8. Measurement of the intensity and phase of a pulse that does not have significant pulsefront tilt. The FWHM pulse width is 123.5 fs. The FROG error is 0.0039 (for a 128×128 array) for this measurement.

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Fig. 9.
Fig. 9. Measurement of the intensity and phase of a pulse with pulse-front tilt. The FWHM pulse width is 125.1 fs. Note that the pulse broadens temporally due to spectral narrowing induced by spectral lateral walk-off. The FROG error is 0.0038 (for a 128×128 array) for this measurement.

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Equations (15)

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I FROG SHG ω τ = ∣ ∫ − ∞ ∞ { E ( t ) exp [ i ω 0 t ] } { E ( t − τ ) exp [ i ω 0 ( t − τ ) ] } exp [ − i ω t ] dt ∣ 2
I FROG SHG pft ω τ =
∣ ∫ − ∞ ∞ E ( t + ζ x ) exp [ i ω 0 t ] E ( t + ζ x − τ ) exp [ i ω 0 ( t − τ ) ] exp [ − i ω t ] dt ∣ 2
∣ ∫ − ∞ ∞ E ( t ′ ) exp [ i ω 0 t ′ ] E ( t ′ − τ ) exp [ i ω 0 ( t ′ − ζ x − τ ) ] exp [ − i ω ( t ′ − ζ x ) ] d t ′ ∣ 2
= ∣ ∫ − ∞ ∞ E ( t ′ ) exp [ i ω 0 t ′ ] E ( t ′ − τ ) exp [ i ω 0 t ′ ] exp [ − i ω t ′ ] d t ′ ∣ 2
= I SHGFROG ω τ
∣ ∫ − ∞ ∞ E ( t + τ 0 + ζ x ) exp [ i ω 0 t ] E ( t − τ 0 + ζ x − τ ) exp [ i ω 0 ( t − τ ) ] exp [ − i ω t ] dt ∣ 2
∣ ∫ − ∞ ∞ E ( t ′ ) exp [ i ω 0 t ′ ] E ( t ′ − 2 τ 0 − τ ) exp [ i ω 0 ( t ′ − τ 0 − ζ x − τ ) ] exp [ − i ω ( t ′ − τ 0 − ζ x ) ] d t ′ ∣ 2
= ∣ ∫ − ∞ ∞ E ( t ′ ) exp [ i ω 0 t ′ ] E ( t ′ − 2 τ 0 − τ ) exp [ i ω 0 t ′ ] exp [ − i ω t ′ ] d t ′ ∣ 2
= I SHGFROG ( ω , τ + 2 τ 0 )
tan ⁡ ψ = c ∂ t ∂ x = c ς
tan ⁡ ψ = Δ x L
ζ = ∂ t ∂ x = Δ x Lc
I FROG SHG ω τ = I FROG SHG ptf ( ω , τ − 2 τ 0 )
E x t = I ( t + ς x ) exp [ i ( t + ς x ) ω 0 − i ϕ ( t + ς x ) ]
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