Abstract

Gaussian beams have been widely used for propagating electromagnetic waves in free space and in certain other optical systems. It has been suggested that recurring forms of such beams might also be useful for propagation in planar or rectangular metal waveguides. Experimental verification of the recurrence of the Gaussian field distribution in metal waveguides is reported here.

© 2001 Optical Society of America

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References

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  1. 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]
  2. L. W. Casperson, “Gaussian light beams in inhomogeneous media,” Appl. Opt. 12, 2434–2441 (1973).
    [CrossRef] [PubMed]
  3. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995), and references therein.
  4. See, for example, L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997), and references therein.
  5. D. E. Weston, “A moiré fringe analog of sound propagation in shallow water,” J. Acoust. Soc. Am. 32, 647–654 (1960).
    [CrossRef]
  6. D. E. Weston, “Sound focusing and beaming in the interference field due to several shallow-water modes,” J. Acoust. Soc. Am. 44, 1706–1712 (1968).
    [CrossRef]
  7. L. A. Rivlin, V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318–321 (1968).
    [CrossRef]
  8. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  9. R. Ulrich, “Light-propagation and imaging in planar optical waveguides,” Nouv. Rev. Opt. 6, 253–262 (1975).
    [CrossRef]
  10. R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
    [CrossRef]
  11. R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
    [CrossRef]
  12. A. Simon, R. Ulrich, “Fiber-optical interferometer,” Appl. Phys. Lett. 31, 77–79 (1977).
    [CrossRef]
  13. O. Bryngdahl, “On light distribution in optical waveguides,” J. Opt. Soc. Am. 68, 310–315 (1978).
    [CrossRef]
  14. R. Ulrich, T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
    [CrossRef]
  15. J. C. Campbell, T. Li, “Electro-optic multimode waveguide switch,” Appl. Phys. Lett. 33, 710–712 (1978).
    [CrossRef]
  16. E. E. Grigor’eva, A. T. Semenov, “Waveguide image transmission in coherent light (review),” Sov. J. Quantum Electron. 8, 1063–1073 (1978).
    [CrossRef]
  17. D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
    [CrossRef]
  18. E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).
  19. E. F. Kuester, D. C. Chang, “Imaging and propagation of beams in metallic or dielectric waveguides,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Boston, Mass., 1984), pp. 185–194.
    [CrossRef]
  20. A. R. Mahnad, E. F. Kuester, “Image formation in circular waveguides and optical fibers,” IEEE MTT-S International Microwave Symposium Digest, 122–124 (1983).
  21. L. W. Casperson, “Gaussian beams in hollow metal waveguides,” J. Opt. Soc. Am. A 17, 1115–1123 (2000).
    [CrossRef]

2000

1997

1995

1982

E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).

1981

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

1978

O. Bryngdahl, “On light distribution in optical waveguides,” J. Opt. Soc. Am. 68, 310–315 (1978).
[CrossRef]

R. Ulrich, T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
[CrossRef]

J. C. Campbell, T. Li, “Electro-optic multimode waveguide switch,” Appl. Phys. Lett. 33, 710–712 (1978).
[CrossRef]

E. E. Grigor’eva, A. T. Semenov, “Waveguide image transmission in coherent light (review),” Sov. J. Quantum Electron. 8, 1063–1073 (1978).
[CrossRef]

1977

A. Simon, R. Ulrich, “Fiber-optical interferometer,” Appl. Phys. Lett. 31, 77–79 (1977).
[CrossRef]

1975

R. Ulrich, “Light-propagation and imaging in planar optical waveguides,” Nouv. Rev. Opt. 6, 253–262 (1975).
[CrossRef]

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

1973

1968

D. E. Weston, “Sound focusing and beaming in the interference field due to several shallow-water modes,” J. Acoust. Soc. Am. 44, 1706–1712 (1968).
[CrossRef]

L. A. Rivlin, V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318–321 (1968).
[CrossRef]

1965

1960

D. E. Weston, “A moiré fringe analog of sound propagation in shallow water,” J. Acoust. Soc. Am. 32, 647–654 (1960).
[CrossRef]

Ankele, G.

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

Bryngdahl, O.

Campbell, J. C.

J. C. Campbell, T. Li, “Electro-optic multimode waveguide switch,” Appl. Phys. Lett. 33, 710–712 (1978).
[CrossRef]

Casperson, L. W.

Chang, D. C.

E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

E. F. Kuester, D. C. Chang, “Imaging and propagation of beams in metallic or dielectric waveguides,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Boston, Mass., 1984), pp. 185–194.
[CrossRef]

Dow, G. S.

E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).

Grigor’eva, E. E.

E. E. Grigor’eva, A. T. Semenov, “Waveguide image transmission in coherent light (review),” Sov. J. Quantum Electron. 8, 1063–1073 (1978).
[CrossRef]

Hall, D. G.

Kamiya, T.

Kogelnik, H.

Kuester, E. F.

E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

E. F. Kuester, D. C. Chang, “Imaging and propagation of beams in metallic or dielectric waveguides,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Boston, Mass., 1984), pp. 185–194.
[CrossRef]

A. R. Mahnad, E. F. Kuester, “Image formation in circular waveguides and optical fibers,” IEEE MTT-S International Microwave Symposium Digest, 122–124 (1983).

Li, T.

J. C. Campbell, T. Li, “Electro-optic multimode waveguide switch,” Appl. Phys. Lett. 33, 710–712 (1978).
[CrossRef]

Mahnad, A. R.

A. R. Mahnad, E. F. Kuester, “Image formation in circular waveguides and optical fibers,” IEEE MTT-S International Microwave Symposium Digest, 122–124 (1983).

Rivlin, L. A.

L. A. Rivlin, V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318–321 (1968).
[CrossRef]

Semenov, A. T.

E. E. Grigor’eva, A. T. Semenov, “Waveguide image transmission in coherent light (review),” Sov. J. Quantum Electron. 8, 1063–1073 (1978).
[CrossRef]

Shul’dyaev, V. S.

L. A. Rivlin, V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318–321 (1968).
[CrossRef]

Simon, A.

A. Simon, R. Ulrich, “Fiber-optical interferometer,” Appl. Phys. Lett. 31, 77–79 (1977).
[CrossRef]

Tovar, A. A.

Ulrich, R.

R. Ulrich, T. Kamiya, “Resolution of self-images in planar optical waveguides,” J. Opt. Soc. Am. 68, 583–592 (1978).
[CrossRef]

A. Simon, R. Ulrich, “Fiber-optical interferometer,” Appl. Phys. Lett. 31, 77–79 (1977).
[CrossRef]

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

R. Ulrich, “Light-propagation and imaging in planar optical waveguides,” Nouv. Rev. Opt. 6, 253–262 (1975).
[CrossRef]

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Weston, D. E.

D. E. Weston, “Sound focusing and beaming in the interference field due to several shallow-water modes,” J. Acoust. Soc. Am. 44, 1706–1712 (1968).
[CrossRef]

D. E. Weston, “A moiré fringe analog of sound propagation in shallow water,” J. Acoust. Soc. Am. 32, 647–654 (1960).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

A. Simon, R. Ulrich, “Fiber-optical interferometer,” Appl. Phys. Lett. 31, 77–79 (1977).
[CrossRef]

J. C. Campbell, T. Li, “Electro-optic multimode waveguide switch,” Appl. Phys. Lett. 33, 710–712 (1978).
[CrossRef]

Arch. Elektr. Uebertrag.

E. F. Kuester, G. S. Dow, D. C. Chang, “Coupling and imaging of Gaussian beams in parallel dielectric slab waveguides,” Arch. Elektr. Uebertrag. 36, 417–435 (1982).

IEEE Trans. Microwave Theory Tech.

D. C. Chang, E. F. Kuester, “A hybrid method for paraxial beam propagation in multimode optical waveguides,” IEEE Trans. Microwave Theory Tech. MTT-29, 923–933 (1981).
[CrossRef]

J. Acoust. Soc. Am.

D. E. Weston, “A moiré fringe analog of sound propagation in shallow water,” J. Acoust. Soc. Am. 32, 647–654 (1960).
[CrossRef]

D. E. Weston, “Sound focusing and beaming in the interference field due to several shallow-water modes,” J. Acoust. Soc. Am. 44, 1706–1712 (1968).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nouv. Rev. Opt.

R. Ulrich, “Light-propagation and imaging in planar optical waveguides,” Nouv. Rev. Opt. 6, 253–262 (1975).
[CrossRef]

Opt. Commun.

R. Ulrich, “Image formation by phase coincidences in optical waveguides,” Opt. Commun. 13, 259–264 (1975).
[CrossRef]

Radiophys. Quantum Electron.

L. A. Rivlin, V. S. Shul’dyaev, “Multimode waveguides for coherent light,” Radiophys. Quantum Electron. 11, 318–321 (1968).
[CrossRef]

Sov. J. Quantum Electron.

E. E. Grigor’eva, A. T. Semenov, “Waveguide image transmission in coherent light (review),” Sov. J. Quantum Electron. 8, 1063–1073 (1978).
[CrossRef]

Other

E. F. Kuester, D. C. Chang, “Imaging and propagation of beams in metallic or dielectric waveguides,” in Hybrid Formulation of Wave Propagation and Scattering, L. B. Felsen, ed. (Nijhoff, Boston, Mass., 1984), pp. 185–194.
[CrossRef]

A. R. Mahnad, E. F. Kuester, “Image formation in circular waveguides and optical fibers,” IEEE MTT-S International Microwave Symposium Digest, 122–124 (1983).

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Figures (4)

Fig. 1
Fig. 1

Series of transverse intensity profiles of a normalized on-axis Gaussian beam interacting with flat waveguide surfaces located at position x = ±0.5d, where d is the distance between the waveguide surfaces. The beam is polarized parallel to the surfaces, the input waist spot size is w 0 = 0.2d, and the propagation distance between successive profiles is z = 0.1z 0 (ten plots per Rayleigh length). The original near-Gaussian profile recurs after approximately eight Rayleigh lengths.

Fig. 2
Fig. 2

Schematic representation of the experimental setup for demonstrating the recurrence of a Gaussian beam in a planar metal waveguide. The beam from a helium–neon laser is polarized by polarizer P, focused by lens L, transmitted through waveguide W, and propagated to detector array D.

Fig. 3
Fig. 3

Intensity profile of the elliptical Gaussian beam incident upon a detector array after transmission through a planar waveguide. The elliptical shape results from the greater diffraction of the initially symmetric Gaussian beam in the direction parallel to the waveguide surfaces.

Fig. 4
Fig. 4

Intensity profile of the detected beam in the direction perpendicular to the waveguide surfaces.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

z0=πw02/λ,
z=z0πw0/d2,
z=d2/λ.
z=f1+f/z12,
w0=fλπw11+f/z121/2.
z=f,
w0=fλπw1.

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