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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 17, Iss. 9 — Sep. 1, 2000
  • pp: 1556–1564

High-aperture diffraction of a scalar, off-axis Gaussian beam

Paul John Cronin, Peter Török, Peter Varga, and Carol Cogswell  »View Author Affiliations

JOSA A, Vol. 17, Issue 9, pp. 1556-1564 (2000)

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A scalar treatment for Gaussian beams offset from the optic axis and then focused by a high-numerical-aperture lens is presented. Such a theory is required for describing certain types of Doppler microscopes, i.e., when the measurement is simultaneously performed by more than a single beam axially offset and then focused by a lens. Analytic expressions for the intensity in the focal region of the high-aperture lens are derived. From these expressions we calculate the intensity in the focal region with parameters of beam size, beam offset, and the numerical aperture of the lens. The relative location and variation of the intensity around the focal region are discussed in detail. We show that for small-diameter Gaussian beams the Strehl ratio increases above unity as the beam is offset from the optic axis. This is explained by the increase in the effective numerical aperture of the offset beam compared with the one collinear with the optic axis. From examining the focal distribution, we conclude that it rotates for small beam size and that increasing beam diameter causes the focused distribution to rotate and shear, i.e., to distort. We also show that the distortion of the distribution increases with increasing numerical aperture.

© 2000 Optical Society of America

OCIS Codes
(050.1220) Diffraction and gratings : Apertures
(050.1960) Diffraction and gratings : Diffraction theory
(080.1010) Geometric optics : Aberrations (global)
(260.1960) Physical optics : Diffraction theory

Original Manuscript: October 20, 1999
Revised Manuscript: April 28, 2000
Manuscript Accepted: April 28, 2000
Published: September 1, 2000

Paul John Cronin, Peter Török, Peter Varga, and Carol Cogswell, "High-aperture diffraction of a scalar, off-axis Gaussian beam," J. Opt. Soc. Am. A 17, 1556-1564 (2000)

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  1. V. S. Ignatovski, “Diffraction by a lens of arbitrary aperture,” Trans. Opt. Inst. (Petrograd) 1, 1–36 (1919) (in Russian).
  2. E. Wolf, “Electromagnetic diffraction in optical systems. I. An integral representation of the image field,” Proc. R. Soc. London, Ser. A 253, 349–357 (1959). [CrossRef]
  3. B. Richards, E. Wolf, “Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. London, Ser. A 253, 358–379 (1959). [CrossRef]
  4. A. Yoshida, T. Asakura, “Electromagnetic field near the focus of Gaussian beams,” Optik (Stuttgart) 41, 281–292 (1974).
  5. P. Varga, P. Török, “The Gaussian wave solution of Maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998). [CrossRef]
  6. H. H. Hopkins, “The Airy disc formula for systems of high relative aperture,” Proc. Phys. Soc. London 55, 116–128 (1943). [CrossRef]
  7. A. Yoshida, T. Asakura, “Effect of aberrations on off-axis Gaussian beams,” Opt. Commun. 14, 211–214 (1975). [CrossRef]
  8. A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the presence of third-order spherical aberration in the optical system,” Opt. Commun. 19, 387–392 (1976). [CrossRef]
  9. A. Yoshida, T. Asakura, “Diffraction patterns of off-axis Gaussian beams in the optical system with astigmatism and coma,” Opt. Commun. 25, 133–136 (1978). [CrossRef]
  10. A. Yoshida, “Spherical aberration in beam optical systems,” Appl. Opt. 21, 1812–1816 (1982). [CrossRef] [PubMed]
  11. C. J. R. Sheppard, P. Török, “Approximate forms for diffraction integrals in high numerical aperture focusing,” Optik (Stuttgart) 105, 77–82 (1997).
  12. Y. Li, F. T. S. Yu, “Intensity distribution near the focus of an apertured focused Gaussian beam,” Opt. Commun. 70, 1–7 (1989). [CrossRef]
  13. B. Lu, W. Huang, B. Zhang, B. Cai, “Focal shift in apertured Gaussian beams and relation with the lens focus,” Optik (Stuttgart) 99, 8–12 (1995).
  14. Y. Yeh, H. Z. Cummins, “Localised fluid flow measurement with a He–Ne laser spectrometer,” Appl. Phys. Lett. 4, 176–178 (1964). [CrossRef]
  15. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. (Academic, London, UK, 1994).
  16. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK1999).
  17. P. Török, Z. Laczik, C. J. R. Sheppard, “Effect of half-stop lateral misalignment on imaging of dark-field and stereoscopic confocal microscopes,” Appl. Opt. 35, 6732–6739 (1996). [CrossRef] [PubMed]

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