## Propagation of Gaussian-apodized paraxial beams through first-order optical systems via complex coordinate transforms and ray transfer matrices |

JOSA A, Vol. 29, Issue 9, pp. 1860-1869 (2012)

http://dx.doi.org/10.1364/JOSAA.29.001860

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### Abstract

We investigate the linear propagation of Gaussian-apodized solutions to the paraxial wave equation in free-space and first-order optical systems. In particular, we present complex coordinate transformations that yield a very general and efficient method to apply a Gaussian apodization (possibly with initial phase curvature) to a solution of the paraxial wave equation. Moreover, we show how this method can be extended from free space to describe propagation behavior through nonimaging first-order optical systems by combining our coordinate transform approach with ray transfer matrix methods. Our framework includes several classes of interesting beams that are important in applications as special cases. Among these are, for example, the Bessel–Gauss and the Airy–Gauss beams, which are of strong interest to researchers and practitioners in various fields.

© 2012 Optical Society of America

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(080.2730) Geometric optics : Matrix methods in paraxial optics

(140.0140) Lasers and laser optics : Lasers and laser optics

(260.1960) Physical optics : Diffraction theory

(260.2030) Physical optics : Dispersion

(080.4295) Geometric optics : Nonimaging optical systems

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: March 21, 2012

Revised Manuscript: July 17, 2012

Manuscript Accepted: July 17, 2012

Published: August 15, 2012

**Citation**

T. Graf, D. N. Christodoulides, M. S. Mills, J. V. Moloney, S. C. Venkataramani, and E. M. Wright, "Propagation of Gaussian-apodized paraxial beams through first-order optical systems via complexcoordinate transforms and ray transfer matrices," J. Opt. Soc. Am. A **29**, 1860-1869 (2012)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-9-1860

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### References

- R. Grella, “Fresnel propagation and diffraction and paraxial wave equation,” J. Opt. 13, 367–374 (1982). [CrossRef]
- T. M. Pritchett and A. D. Trubatch, “A differential formulation of diffraction theory for the undergraduate optics course,” Am. J. Phys. 72, 1026–1034 (2004). [CrossRef]
- M. V. Berry and N. L. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979). [CrossRef]
- M. Levy, Parabolic Equation Methods for Electromagnetic Wave Propagation, IEE Electromagnetic Waves Series (Institution of Electrical Engineers, 2000).
- W. L. Siegmann and D. Lee, “Aspects of three-dimensional parabolic equation computations,” Comput. Math. Appl. 11, 853–862 (1985), Special Issue on Computational Ocean Acoustics. [CrossRef]
- P. W. Milonni and J. H. Eberly, Laser Physics (Wiley, 2010).
- A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973). [CrossRef]
- A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989). [CrossRef]
- E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986). [CrossRef]
- J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987). [CrossRef]
- J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]
- D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005). [CrossRef]
- J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz–Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005). [CrossRef]
- F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987). [CrossRef]
- M. A. Bandres and J. C. Gutiérrez-Vega, “Airy–Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007). [CrossRef]
- J. McLeod, “The axicon: a new type of optical element,” J. Opt. Soc. Am. 44, 592–597 (1954). [CrossRef]
- M. S. Mills, G. A. Siviloglou, N. Efremidis, T. Graf, E. M. Wright, J. V. Moloney, and D. N. Christodoulides are preparing a manuscript to be called “Optical bullets with hydrogen-like symmetries.”
- A. E. Siegman. Lasers (University Science, 1986).
- A. Belafhal and L. Dalil-Essakali, “Collins formula and propagation of Bessel-modulated Gaussian light beams through an ABCD optical system,” Opt. Commun. 177, 181–188 (2000). [CrossRef]
- Z. Mei, D. Zhao, X. Wei, F. Jing, and Q. Zhu, “Propagation of Bessel-modulated Gaussian beams through a paraxial ABCD optical system with an annular aperture,” Optik 116, 521–526 (2005). [CrossRef]
- N. Zhou and G. Zeng, “Propagation properties of Hermite–cosine–Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Opt. Commun. 232, 49–59 (2004). [CrossRef]
- S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
- J. A. Arnaud, Beam and Fiber Optics (Academic, 1976), Chap. 2.
- M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996). [CrossRef]
- X. Liu and K.-H. Brenner, “Minimal optical decomposition of ray transfer matrices,” Appl. Opt. 47E88–E98 (2008). [CrossRef]
- S. Y. Shin and L. B. Felsen, “Gaussian beam modes by multipoles with complex source points,” J. Opt. Soc. Am. 67, 699–700 (1977). [CrossRef]
- G. P. Agrawal and D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979). [CrossRef]
- T. Takenaka, M. Yokota, and O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985). [CrossRef]
- L. W. Casperson, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 20, 2243–2249 (1981). [CrossRef]
- G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, 1989).
- J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill Physical and Quantum Electronics Series (Roberts, 2005).
- A. Yariv, Optical Electronics, The Holt, Rinehart, and Winston Series in Electrical Engineering (Saunders College, 1991).
- M. A. Bandres and M. Guizar-Sicairos, “Paraxial group,” Opt. Lett. 34, 13–15 (2009). [CrossRef]
- National Institute of Standards and Technology, Digital Library of Mathematical Functions (29August2011), http://dlmf.nist.gov/ .
- E. Jones, T. Oliphant, and P. Peterson, “SciPy: open source scientific tools for Python” (2001), http://www.scipy.org .
- J. D. Hunter, “Matplotlib: a 2D graphics environment,” Comput. Sci. Eng. 9, 90–95 (2007). [CrossRef]

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