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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Franco Gori
  • Vol. 30, Iss. 7 — Jul. 1, 2013
  • pp: 1381–1386

Propagation equation of Hermite–Gauss beams through a complex optical system with apertures and its application to focal shift

Sun Peng, Guo Jin, and Wang Tingfeng  »View Author Affiliations

JOSA A, Vol. 30, Issue 7, pp. 1381-1386 (2013)

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Based on the generalized Huygens–Fresnel diffraction integral (Collins’ formula), the propagation equation of Hermite–Gauss beams through a complex optical system with a limiting aperture is derived. The elements of the optical system may be all those characterized by an ABCD ray-transfer matrix, as well as any kind of apertures represented by complex transmittance functions. To obtain the analytical expression, we expand the aperture transmittance function into a finite sum of complex Gaussian functions. Thus the limiting aperture is expressed as a superposition of a series of Gaussian-shaped limiting apertures. The advantage of this treatment is that we can treat almost all kinds of apertures in theory. As application, we define the width of the beam and the focal plane using an encircled-energy criterion and calculate the intensity distribution of Hermite–Gauss beams at the actual focus of an aperture lens.

© 2013 Optical Society of America

OCIS Codes
(050.0050) Diffraction and gratings : Diffraction and gratings
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

ToC Category:
Fourier Optics and Signal Processing

Original Manuscript: May 2, 2013
Revised Manuscript: May 29, 2013
Manuscript Accepted: May 31, 2013
Published: June 20, 2013

Sun Peng, Guo Jin, and Wang Tingfeng, "Propagation equation of Hermite–Gauss beams through a complex optical system with apertures and its application to focal shift," J. Opt. Soc. Am. A 30, 1381-1386 (2013)

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  1. L. Vicari and F. Bloisi, “Matrix representation of axisymmetric optical systems including spatial filters,” Appl. Opt. 28, 4682–4686 (1989). [CrossRef]
  2. X. Y. Tao, N. R. Zhou, and B. D. Lü, “Recurrence propagation equation of Hermite-Gaussian beams through a paraxial optical ABCD system with hard-edge aperture,” Optik 114, 113–117 (2003). [CrossRef]
  3. H. Kogelnik and T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]
  4. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970). [CrossRef]
  5. E. Feldheim, “Équations intégrales pour les polynomes d'Hermite à une et plusieurs variables, pour les polynomes de Laguerre, et pour les fonctions hypergéométriques les plus générales,” Proc. Kon. Ned. Akad. v. Wetensch. 43, 224–239 (1940).
  6. J. J. Wen and A. Breazeale, “A diffraction beam field expressed as the superpositon of Gaussion beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988). [CrossRef]
  7. P. L. Greene and D. G. Hall, “Focal shift in vector beams,” Opt. Express 4, 411–419 (1999). [CrossRef]
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, 1964).

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