Wigner matrix formalism for phase-modulated signals

H. Coïc; C. Rouyer; N. Bonod

doi:10.1364/JOSAA.408363

Journal of the Optical Society of America A
Vol. 38,
Issue 1,
pp. 124-139
(2021)
•https://doi.org/10.1364/JOSAA.408363

Wigner matrix formalism for phase-modulated signals

H. Coïc, C. Rouyer, and N. Bonod

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H. Coïc, C. Rouyer, and N. Bonod, "Wigner matrix formalism for phase-modulated signals," J. Opt. Soc. Am. A 38, 124-139 (2021)

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Abstract

Laser beams can carry multi-scale properties in space and time that ultimately impact their quality. The study of their evolution along complex optical sequences is of crucial interest, especially in high-intensity laser chains. For such analysis, results obtained with standard numerical methods strongly depend on the sampling. In this paper, we develop an analytic model for a sinusoidal phase modulation inside a sequence of first-order optics elements based on the Wigner matrix formalism. A Bessel decomposition of the Wigner function gives pseudo-Wigner functions that obey the general ABCD matrix law transformation without approximations and sampling considerations. Applied to a Gaussian beam, explicit expressions are obtained for the projections of the Wigner function in the sub-spaces and give a powerful tool for analyzing the laser beam properties. The formalism is established in the spatial and temporal domains and can be used to evaluate the impact of the phase noise on the beam properties and is not limited to small modulation depths. For the sake of illustration, the model is applied to the Talbot effect with the analysis of the propagation in the spatial and phase-space domains. A comparison with full numerical calculations evidences the high accuracy of the analytic model that retrieves all the features of the diffracted beam.

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( $n, l$ )	$J_{n} J_{l}$	Resulting Wigner Function
(0, 0)	$1 - \frac{m^{2}}{2}$	$\begin{array}{l} W_{n + l = 0} (X, K_{x}) \\ = \exp [- 2 {(X - Z K_{x})}^{2}] \exp [- \frac{1}{2} K_{x}^{2}] \\ \times {1 - \frac{m^{2}}{2} (1 + \cos [2 u (X - Z K_{x})])} \end{array}$
(−1, 1)	$- \frac{m^{2}}{4}$
(1, −1)	$- \frac{m^{2}}{4}$
(0, 1)	$\frac{m}{2}$	$\begin{array}{l} W_{n + l = 1} (X, K_{x}) = m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(1, 0)	$\frac{m}{2}$
(−1, 0)	$- \frac{m}{2}$	$\begin{array}{l} W_{n + l = - 1} (X, K_{x}) = - m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(0, −1)	$- \frac{m}{2}$
(1, 1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + u)}^{2}] \\ \times {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, 2)	$\frac{m^{2}}{8}$
(2, 0)	$\frac{m^{2}}{8}$
(−1, −1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = - 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - u)}^{2}] \\ {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, −2)	$\frac{m^{2}}{8}$
(−2, 0)	$\frac{m^{2}}{8}$

$u [i] (m)^{- 1}$	3.1	21.6	32.9	33.5	34.0	43.5	46.0	53.48	57.7	59.6
$m (u [i])$ $(\times 100)$	9.93	7.46	5.08	4.95	4.48	3.05	2.65	1.67	1.24	1.08
$Φ [i]$ (rad)	5.69	2.22	3.76	0.455	2.06	0.27	1.61	4.79	1.24	5.26

( $n, l$ )	$J_{n} J_{l}$	Resulting Wigner Function
(0, 0)	$1 - \frac{m^{2}}{2}$	$\begin{array}{l} W_{n + l = 0} (X, K_{x}) \\ = \exp [- 2 {(X - Z K_{x})}^{2}] \exp [- \frac{1}{2} K_{x}^{2}] \\ \times {1 - \frac{m^{2}}{2} (1 + \cos [2 u (X - Z K_{x})])} \end{array}$
(−1, 1)	$- \frac{m^{2}}{4}$
(1, −1)	$- \frac{m^{2}}{4}$
(0, 1)	$\frac{m}{2}$	$\begin{array}{l} W_{n + l = 1} (X, K_{x}) = m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(1, 0)	$\frac{m}{2}$
(−1, 0)	$- \frac{m}{2}$	$\begin{array}{l} W_{n + l = - 1} (X, K_{x}) = - m \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - \frac{u}{2})}^{2}] \cos [u (X - Z K_{x})] \end{array}$
(0, −1)	$- \frac{m}{2}$
(1, 1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} + u)}^{2}] \\ \times {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, 2)	$\frac{m^{2}}{8}$
(2, 0)	$\frac{m^{2}}{8}$
(−1, −1)	$\frac{m^{2}}{4}$	$\begin{array}{l} W_{n + l = - 2} (X, K_{x}) = \frac{m^{2}}{4} \exp [- 2 {(X - Z K_{x})}^{2}] \\ \times \exp [- \frac{1}{2} {(K_{x} - u)}^{2}] \\ {1 + \cos [2 u (X - Z K_{x})]} \end{array}$
(0, −2)	$\frac{m^{2}}{8}$
(−2, 0)	$\frac{m^{2}}{8}$

$u [i] (m)^{- 1}$	3.1	21.6	32.9	33.5	34.0	43.5	46.0	53.48	57.7	59.6
$m (u [i])$ $(\times 100)$	9.93	7.46	5.08	4.95	4.48	3.05	2.65	1.67	1.24	1.08
$Φ [i]$ (rad)	5.69	2.22	3.76	0.455	2.06	0.27	1.61	4.79	1.24	5.26

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Equations (101)

Journal of the Optical Society of America A