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The Wigner function in paraxial optics I. Matrix methods in Fourier optics

 

Roberto Ortega-Martínez^1*, Carlos J. Román-Moreno¹, Ana Leonor Rivera²

 

¹ Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Apartado Postal 70-186, 04510 México, D.F., México *roberto@aleph.cinstrum.unam.mx

² Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Cuernavaca, Morelos, México.

 

Recibido el 19 de noviembre de 2001.
Aceptado el 10 de junio de 2002.

 

Abstract

The paraxial regime of scalar wave optics has the same structure as non-relativistic quantum mechanics, with wavelength taking the place of the Planck constant. The Wigner function is a central tool to explore the phase space of a system; in optics, moreover, it can be produced by purely optical means. In this first part, we present a matrix-based formalism for the study of paraxial optical systems, and apply it to the description of a setup that, as will be seen in the second part, produces the Wigner function.

Keywords: Wave propagation; Fourier optics; space phase measurements.

 

Resumen

El régimen parxial de la óptica ondulatoria escalar tiene la misma estructura que la mecánica cuántica no relativista, donde la longitud de onda juega el papel de la constante de Planck. La función de Wigner es una herramienta central para explorar el espacio fase de un sistema; además, en óptica, se puede producir con medios exclusivamente ópticos. En esta primera parte, presentamos un formalismo matricial para el estudio de los sistemas ópticos parxiales y lo aplicamos en la descripción de un arreglo, que como veremos en la segunda parte, produce la función de Wigner.

Descriptores: Propagación de ondas; óptica de Fourier; mediciones de fase espacial.

 

PACS: 42.25.Bs; 42.30.Kq; 42.50.Dv

 

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Acknowledgements

The authors would like to thank Kurt B. Wolf for having unconditionally shared with us his vast knowledge and Neil Bruce (CCADET-UNAM) and Eugenio Ley Koo for the review of the text. C.J. Román had support from DGAPA-UNAM projects IN106595 and IN104597. Support of projects IN104198 Óptica Matemática and 1N108900 Detección Ultrarrápida y Ultrasensible by the Dirección General de Asuntos del Personal Académico, UNAM, are gratefully acknowledged.

 

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