## Matrix method for beam propagation using Gaussian Hermite polynomials

Applied Optics, Vol. 29, Issue 6, pp. 802-808 (1990)

http://dx.doi.org/10.1364/AO.29.000802

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### Abstract

Using the properties of Hermite polynomials, a simple first-order matrix differential equation is developed which describes the propagation of an arbitrary field through an inhomogeneous medium and which can be solved exactly. This method handles both spatially varying refractive index and linear absorption and diffraction. As examples, it is applied to an etalon and a graded index optical fiber.

© 1990 Optical Society of America

**History**

Original Manuscript: July 5, 1989

Published: February 20, 1990

**Citation**

R. McDuff, "Matrix method for beam propagation using Gaussian Hermite polynomials," Appl. Opt. **29**, 802-808 (1990)

http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-29-6-802

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### References

- H. Kogelnik, “On the Propagation of Gaussian Beams of Light Through Lenslike Media Including Those with a Loss or Gain Variation,” Appl. Opt. 4, 1562–1569 (1965). [CrossRef]
- M. D. Feit, J. A. Fleck, “Light Propagation in Graded-Index Optical Fibers,” Appl. Opt. 17, 3990–3998 (1978). [CrossRef] [PubMed]
- A. Sharma, S. Banerjee, “Method for Propagation of Total Fields for Beams through Optical Waveguides,” Opt. Lett. 14, 96–98 (1989). [CrossRef] [PubMed]
- W. J. Firth, E. M. Wright, “Theory of Gaussian Beam Optical Bistability,” Opt. Commun. 40, 233–238 (1982). [CrossRef]
- A. E. Siegman, Lasers (Oxford U.P., London, 1986).
- I. W. Busbridge, “Some Integrals Involving Hermite Polynomials,” J. London Math. Soc. 23, 133–141 (1948). [CrossRef]
- J. H. Marburger, F. S. Felber, “Theory of a Lossless Nonlinear Fabry-Perot Interferometer,” Phys. Rev. A, 177, 335–342 (1978). [CrossRef]
- R. P. Riesz, R. Simon, “Reflection of a Gaussian Beam from a Dielectric Slab,” J. Opt. Soc. Am. A 2, 1809–1817 (1985). [CrossRef]
- The far-field irradiance is derived from the Fourier transform of the electric field at the surface. Since the F.T. of a GH polynomial is also a GH polynomial of the same order it follows that the F.T. of E(y)=∑i=0∞fiψi(y) exp(iβy) isEFT(k)=πr0∑i=0∞Cifi exp[-r02(β-k)2/4]Hi(r0(β-k)2)ii,where k = [(2π)/λ] sin θ.
- W. Nasalski, T. Tamir, L. Lin, “Displacement of the Intensity Peak in Narrow Beams Reflected at a Dielectric Interface,” J. Opt. Soc. Am. A 5, 132–140 (1988). [CrossRef]
- There seems to be an inconsistency in the work of Riesz and Simon,6 since there are two graphs of reflection amplitude vs sin θ for the case nd = 2.7 with each having the peak of the spatial frequency distribution on different sides of the etalon reflectivity minima [Figs.2(b) and 3(b)]. This by Eq. (14) gives different signs to the angular shift, either towards or away from the normal. Since the etalon reflectivity minima occurs at sinθ ~0.57 and the spectral peak occurs at sinθ = 0.5 (since the angle of incidence is 30°) Δα the angular shift should be −2.8°, i.e., toward the normal instead of away as their work suggests.
- R. McDuff, unpublished.
- W. N. Bailey, “Some Integrals Involving Hermite Polynomial,” J. London Math. Soc. 23, 291–297 (1948). [CrossRef]

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