The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams

Selcuk Akturk; Xun Gu; Pablo Gabolde; Rick Trebino

doi:10.1364/OPEX.13.008642

Optics Express
Vol. 13,
Issue 21,
pp. 8642-8661
(2005)
•https://doi.org/10.1364/OPEX.13.008642

The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams

Selcuk Akturk, Xun Gu, Pablo Gabolde, and Rick Trebino

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Selcuk Akturk, Xun Gu, Pablo Gabolde, and Rick Trebino, "The general theory of first-order spatio-temporal distortions of Gaussian pulses and beams," Opt. Express 13, 8642-8661 (2005)

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Abstract

We present a rigorous, but mathematically relatively simple and elegant, theory of first-order spatio-temporal distortions, that is, couplings between spatial (or spatial-frequency) and temporal (or frequency) coordinates, of Gaussian pulses and beams. These distortions include pulse-front tilt, spatial dispersion, angular dispersion, and a less well-known distortion that has been called “time vs. angle.” We write pulses in four possible domains, xt, xω, kω, and kt; and we identify the first-order couplings (distortions) in each domain. In addition to the above four “amplitude” couplings, we identify four new spatio-temporal “phase” couplings: “wave-front rotation,” “wave-front-tilt dispersion,” “angular temporal chirp,” and “angular frequency chirp.” While there are eight such couplings in all, only two independent couplings exist and are fundamental in each domain, and we derive simple expressions for each distortion in terms of the others. In addition, because the dimensions and magnitudes of these distortions are unintuitive, we provide normalized, dimensionless definitions for them, which range from -1 to 1. Finally, we discuss the definitions of such quantities as pulse length, bandwidth, angular divergence, and spot size in the presence of spatio-temporal distortions. We show that two separate definitions are required in each case, specifically, “local” and “global” quantities, which can differ significantly in the presence of spatio-temporal distortions.

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Fig. 1. The effect of the WFR. The movie shows the wave fronts in (x,t) domain as a function of time and position (file size 2.26 MB).

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Fig. 2. The effect of the WFD. The movie shows the wave fronts in (x,ω) as a function of frequency and position (file size 2.06 MB).

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Fig. 3. Intensity profile of a pulse expressed in the four different domains. This pulse simultaneously has all four spatio-temporal amplitude couplings: PFT, SPC, AGD and TVA, as can be seen from the tilted images.

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Fig. 4. Intensity profile of a pulse expressed in the four different domains. This pulse has significant PFT and SPC but very small AGD and TVA, as can be seen from the tilt of the traces.

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Tables (3)

Table 1. Summary of the relations of spatio-temporal distortions in all four domains.

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Table 2 The correlation coefficients of the pulse field shown in Figure 3.

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Table 3 The correlation coefficients of the pulse field shown in Fig. 4.

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Equations (86)

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E (x, t) = \exp {- i \frac{π}{λ_{0}} {(\begin{matrix} x \\ - t \end{matrix})}^{T} Q^{- 1} (\begin{matrix} x \\ t \end{matrix})} =

\exp [- i \frac{π}{λ_{0}} ({(Q^{- 1})}_{11} x^{2} + {(Q^{- 1})}_{12} xt - {(Q^{- 1})}_{21} xt - {(Q^{- 1})}_{22} t^{2})]

K = [\begin{matrix} \frac{\partial x_{out}}{\partial x_{in}} & \frac{\partial x_{out}}{\partial θ_{in}} & 0 & \frac{\partial x_{out}}{\partial v_{in}} \\ \frac{\partial x_{out}}{\partial x_{in}} & \frac{\partial θ_{out}}{\partial θ_{in}} & 0 & \frac{\partial θ_{out}}{\partial v_{in}} \\ \frac{\partial t_{out}}{\partial x_{in}} & \frac{\partial t_{out}}{\partial θ_{in}} & 1 & \frac{\partial t_{out}}{\partial v_{in}} \\ 0 & 0 & 0 & 1 \end{matrix}] = [\begin{matrix} A & B & 0 & E \\ C & D & 0 & F \\ G & H & 1 & I \\ 0 & 0 & 0 & 1 \end{matrix}]

Q_{out} = {[\begin{matrix} A & 0 \\ G & 1 \end{matrix}] Q_{in} + [\begin{matrix} B & \frac{E}{λ_{0}} \\ H & \frac{I}{λ_{0}} \end{matrix}]} \cdot {[\begin{matrix} C & 0 \\ 0 & 0 \end{matrix}] Q_{in} + [\begin{matrix} D & \frac{F}{λ_{0}} \\ 0 & 1 \end{matrix}]}^{- 1}

E (x, t) \propto \exp {{\tilde{Q}}_{xx} x^{2} + 2 {\tilde{Q}}_{xt} xt - {\tilde{Q}}_{tt} t^{2}}

Q_{in} = {[\begin{matrix} Q_{11} & Q_{12} \\ - Q_{12} & Q_{22} \end{matrix}] = i \frac{λ_{0}}{π} [\begin{matrix} {\tilde{Q}}_{xx} & {\tilde{Q}}_{xt} \\ - {\tilde{Q}}_{xt} & {\tilde{Q}}_{tt} \end{matrix}]}^{- 1}

{\tilde{Q}}_{xx} = - i \frac{π}{λ_{0} R (z)} - \frac{1}{w^{2} (z)}

{\tilde{Q}}_{tt} = - iβ + \frac{1}{τ^{2}}

E (x, t) \propto \exp {{\tilde{Q}}_{xx} x^{2} + 2 {\tilde{Q}}_{xt} xt - {\tilde{Q}}_{tt} t^{2}}

Re {{\tilde{Q}}_{xx}} \leftrightarrow beam spot size (BSS)

Im {{\tilde{Q}}_{xx}} \leftrightarrow wave - front curvature (WFC)

Re {{\tilde{Q}}_{tt}} \leftrightarrow temporal pulse width (TPW)

Im {{\tilde{Q}}_{tt}} \leftrightarrow temporal chirp (TCH)

Re {{\tilde{Q}}_{xt}} \leftrightarrow pulse - front tilt (PFT)

Im {{\tilde{Q}}_{xt}} \leftrightarrow wave - front rotation (WER)

E (x, ω) = \frac{1}{2 π} \int E (x, t) e^{- iωt} dt

\int_{- \infty}^{\infty} \exp (- p^{2} x^{2} \pm qx) dx = \exp (\frac{q^{2}}{4 p}) \frac{\sqrt{π}}{∣ p ∣}

E (x, ω) \propto \exp {R_{xx} x^{2} + 2 R_{xω} xω - R_{ωω} ω^{2}}

R_{xx} = {\tilde{Q}}_{xx} + \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{tt}}

R_{xω} = \frac{- i}{2} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{tt}}

R_{ωω} = \frac{1}{{4 \tilde{Q}}_{tt}}

Re {R_{xx}} \leftrightarrow beam spot size (BSS)

Im {R_{xx}} \leftrightarrow wave - front curvature (WFC)

Re {R_{ωω}} \leftrightarrow band width (BDW)

Im {R_{ωω}} \leftrightarrow frequency chirp (FCH)

Re {R_{xω}} \leftrightarrow spatial chirp (SPC)

Im {R_{xω}} \leftrightarrow wave - front - tilt dispersion (WFD)

E (k, ω) \propto \exp {S_{kk} k^{2} + 2 S_{kω} kω - S_{ωω} ω^{2}}

S_{kk} = \frac{1}{4 R_{xx}}

S_{kω} = \frac{i}{2} \frac{R_{xω}}{R_{xx}}

S_{ωω} = R_{ωω} + \frac{R_{xω}^{2}}{R_{xx}}

[\tilde{Q}] = [\begin{matrix} {\tilde{Q}}_{xx} & {\tilde{Q}}_{xt} \\ - {\tilde{Q}}_{xt} & {\tilde{Q}}_{tt} \end{matrix}]

[S] = [\begin{matrix} S_{kk} & S_{kω} \\ - S_{kω} & S_{ωω} \end{matrix}]

[S] = \frac{1}{4} {[{\tilde{Q}}^{T}]}^{- 1}

Re {S_{kk}} \leftrightarrow angular div ergence (ADV)

Im {S_{kk}} \leftrightarrow angular phase - front curvature (APC)

Re {S_{ωω}} \leftrightarrow band width (BDW)

Im {S_{ωω}} \leftrightarrow frequency chirp (FCH)

Re {S_{kω}} \leftrightarrow angular dispersion (AGD)

Im {S_{kω}} \leftrightarrow angular spectral chirp (ASC)

E (k, t) \propto \exp {P_{kk} k^{2} + 2 P_{kt} kt - P_{tt} t^{2}}

P_{kk} = S_{kk} + \frac{S_{kω}^{2}}{S_{ωω}}

P_{kt} = \frac{i}{2} \frac{S_{kω}}{S_{ωω}}

P_{tt} = \frac{1}{{4 S}_{ωω}}

[P] = \frac{1}{4} {[R^{T}]}^{- 1}

[R] = [\begin{matrix} R_{xx} & R_{xω} \\ - R_{xω} & R_{ωω} \end{matrix}] [P] = [\begin{matrix} P_{kk} & P_{kt} \\ - P_{kt} & P_{tt} \end{matrix}]

Re {P_{kk}} \leftrightarrow angular div ergence (ADV)

Im {P_{kk}} \leftrightarrow angular phase - front curvature (APC)

Re {P_{tt}} \leftrightarrow temporal pulse width (TPW)

Im {P_{tt}} \leftrightarrow temporal chirp (TCH)

Re {P_{kt}} \leftrightarrow time vs . angle (TVA)

Im {P_{kt}} \leftrightarrow angular temporal chirp (ATC)

Δ x = {[\frac{\int_{- \infty}^{\infty} x^{2} I (x, t) dx - {(\int_{- \infty}^{\infty} xI (x, t) dx)}^{2}}{\int_{- \infty}^{\infty} I (x, t) dx}]}^{\frac{1}{2}}

Δ x_{L} (t) = {[\frac{\int_{- \infty}^{\infty} x^{2} I (x, t) dx - {(\int_{- \infty}^{\infty} xI (x, t) dx)}^{2}}{\int_{- \infty}^{\infty} I (x, t) dx}]}^{\frac{1}{2}}

Δ x_{G} = {[\frac{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} x^{2} I (x, t) dxdt}{\int_{- \infty}^{\infty} \int_{- \infty}^{\infty} I (x, t) dxdt}]}^{\frac{1}{2}}

Δ x_{L} = \frac{1}{2} {[- \frac{1}{{\tilde{Q}}_{xx}^{R}}]}^{\frac{1}{2}}

Δ x_{G} = \frac{1}{2} {[- \frac{{\tilde{Q}}_{tt}^{R}}{{\tilde{Q}}_{xx}^{R} {\tilde{Q}}_{tt}^{R} + {\tilde{Q}}_{xt}^{R 2}}]}^{\frac{1}{2}}

Δ t_{L} = \frac{1}{2} {[- \frac{1}{{\tilde{Q}}_{tt}^{R}}]}^{\frac{1}{2}}

Δ t_{G} = \frac{1}{2} {[- \frac{{\tilde{Q}}_{xx}^{R}}{{\tilde{Q}}_{xx}^{R} {\tilde{Q}}_{tt}^{R} + {\tilde{Q}}_{xt}^{R 2}}]}^{\frac{1}{2}}

E (x, ω) \propto \exp [- {(\frac{x - \frac{\partial x_{0}}{\partial ω} ω}{4 Δ x_{L}})}^{2} - \frac{ω^{2}}{4 Δ ω_{G}^{2}}]

E (x, ω) \propto \exp [R_{xx}^{R} x^{2} + 2 R_{xω}^{R} xω - R_{ωω}^{R} ω^{2}]

= \exp [R_{xx}^{R} {(x + \frac{R_{xω}^{R}}{R_{xx}^{R}} ω)}^{2} - (\frac{R_{xω}^{R 2}}{R_{xx}^{R}} + R_{ωω}^{R}) ω^{2}]

Δ t_{L} = \frac{1}{2} [- \frac{1}{R_{xx}^{R}}] = \frac{w}{2}

\frac{\partial x_{0}}{\partial ω} = - \frac{R_{xω}^{R}}{R_{xx}^{R}}

Δ ω_{G} = \frac{1}{2} {(\frac{R_{xx}^{R}}{R_{xx}^{R 2} + R_{ωω}^{R} R_{xx}^{R}})}^{\frac{1}{2}}

E (x, ω) = \exp [- \frac{x^{2}}{4 Δ x_{G}^{2}} - {(\frac{ω - \frac{\partial ω_{0}}{\partial x} x}{4 Δ ω_{L}})}^{2}]

= \exp [(\frac{R_{xω}^{R 2}}{R_{ωω}^{R}} + R_{xx}^{R}) x^{2} - R_{ωω}^{R} {(ω - \frac{R_{xω}^{R}}{R_{ωω}^{R}} x)}^{2}]

Δ x_{G} = \frac{1}{2} {(- \frac{R_{ωω}^{R}}{R_{xω}^{R 2} + R_{ωω}^{R} R_{xx}^{R}})}^{\frac{1}{2}}

\frac{\partial ω_{0}}{\partial x} = \frac{R_{xω}^{R}}{R_{ωω}^{R}}

Δ ω_{L} = \frac{1}{2} {[\frac{1}{R_{ωω}^{R}}]}^{\frac{1}{2}}

\frac{\partial ω_{0}}{\partial x} = \frac{\frac{\partial x_{0}}{\partial ω}}{{(\frac{\partial x_{0}}{\partial ω})}^{2} + \frac{Δ x_{L}^{2}}{Δ ω_{G}^{2}}}

ρ_{xω} \equiv \frac{\int\int xωI (x, ω) dxdω}{\int\int I (x, ω) dxdω} \frac{1}{Δ x_{G} Δ ω_{G}}

ρ_{xω} = \frac{R_{xω}^{R}}{\sqrt{- R_{xx}^{R}} R_{ωω}^{R}}

Δ x_{L} (ω) = Δ x_{G} \sqrt{1 - ρ_{xω}^{2}}

Δ ω_{L} (x) = Δ ω_{G} \sqrt{1 - ρ_{xω}^{2}}

{\tilde{Q}}_{xt}^{R} = \frac{1}{2 {∣ R_{ωω} ∣}^{2}} [- R_{ωω}^{R} R_{xω}^{I} + R_{ωω}^{I} R_{xω}^{R}]

PET = - 2 WFD + 2 FCH \times FRG

PET = AGD + 2 FCH \times FRG

SPD = 2 ASC - 2 AGD \times APC

AGD = - 2 ATC + 2 TVA \times TCH

TVA = - 2 WFR + 2 PFT \times WFC

{\tilde{Q}}_{xt} = \frac{i}{2} FRG + {\tilde{Q}}_{tt} PFT

WFR = \frac{FRG}{2} + TCH \times PET

WFD = - \frac{AGD}{2} + SPD \times WFC

ASC = - \frac{TVA}{2} + AGD \times FCH

ATC = \frac{PFT}{2} + TVA \times APC

(x,t)	(x,ω)	(k,ω)	(k,t)
Q͂_xt	$\frac{i}{2} \frac{R_{xω}}{R_{ωω}}$	$\frac{1}{4} \frac{S_{kω}}{S_{kk} S_{ωω} + S_{kω}^{2}}$	$- \frac{i}{2} \frac{P_{kt}}{P_{kk}}$
$- \frac{i}{2} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{tt}}$	R_xω	$- \frac{i}{2} \frac{S_{kω}}{S_{kk}}$	$\frac{1}{4} \frac{P_{kt}}{P_{kk} P_{tt} + P_{kt}^{2}}$
$\frac{1}{4} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{xx} {\tilde{Q}}_{tt} + {\tilde{Q}}_{xt}^{2}}$	$\frac{i}{2} \frac{R_{xω}}{R_{xx}}$	S_kω	$- \frac{i}{2} \frac{P_{kt}}{P_{tt}}$
$\frac{i}{2} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{xx}}$	$\frac{1}{4} \frac{R_{xω}}{R_{kk} R_{ωω} + R_{xω}^{2}}$	$\frac{i}{2} \frac{S_{kω}}{R_{ωω}}$	P_kt

(x,t)	(x,ω)	(k,ω)	(k,t)
Q͂_xt	$\frac{i}{2} \frac{R_{xω}}{R_{ωω}}$	$\frac{1}{4} \frac{S_{kω}}{S_{kk} S_{ωω} + S_{kω}^{2}}$	$- \frac{i}{2} \frac{P_{kt}}{P_{kk}}$
$- \frac{i}{2} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{tt}}$	R_xω	$- \frac{i}{2} \frac{S_{kω}}{S_{kk}}$	$\frac{1}{4} \frac{P_{kt}}{P_{kk} P_{tt} + P_{kt}^{2}}$
$\frac{1}{4} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{xx} {\tilde{Q}}_{tt} + {\tilde{Q}}_{xt}^{2}}$	$\frac{i}{2} \frac{R_{xω}}{R_{xx}}$	S_kω	$- \frac{i}{2} \frac{P_{kt}}{P_{tt}}$
$\frac{i}{2} \frac{{\tilde{Q}}_{xt}}{{\tilde{Q}}_{xx}}$	$\frac{1}{4} \frac{R_{xω}}{R_{kk} R_{ωω} + R_{xω}^{2}}$	$\frac{i}{2} \frac{S_{kω}}{R_{ωω}}$	P_kt

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