Abstract

The propagation of a Gaussian beam in a homogeneous, isotropic, local, linear, and nonmagnetic dielectric medium is studied using the angular spectrum representation for the electric field. The electric field associated with the Gaussian beam inside the dielectric medium consists of the paraxial result and higher-order non-Gaussian correction terms. It is shown that the second-order correction term satisfies an equation consistent with the recent work of Lax, Louisell, and McKnight. Numerical results showing the corrections to the paraxial approximation are presented.

© 1979 Optical Society of America

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  1. H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); H. Kogelnik, "On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss and Gain Variation," Appl. Opt. 4, 1562–1569 (1965).
  2. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975) Chap. 6 and references cited therein.
  3. See A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971), p. 317.
  4. See, for example, T. W. Cole, Progress in Optics, Vol. XV, edited by E. Wolf (North Holland, Amsterdam, 1977), p. 193.
  5. M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).
  6. For a discussion of the angular spectrum representation, see, for example, P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields, (Pergamon, Oxford, 1966); J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968) Sec. 3.7; J. R. Shewell and E. Wolf, "Inverse Diffraction and a Reciprocity Theroem," J. Opt. Soc. Am. 58, 1596–1603 (1968).
  7. W. H. Carter, "Electromagnetic Field of A Gaussian Beam with an Elliptical Cross section," J. Opt. Soc. Am. 62, 1195–1201 (1972).
  8. G. Goubau and F. Schwering, "On the Guided Propagation of Electromagnetic Wave Beams", IRE Trans. AP9, 248–256 (1961).
  9. G. N. Watson, A treatize on the Theory of Bessel Functions, (Cambridge University, Cambridge, 1966).
  10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Academic, New York, 1965), p. 716, p. 1037.
  11. The restriction z < l probably arises from the interchange of the order of summation in Eq. (21).
  12. A. G. van Nie, "Rigorous Calculation of the Electromagnetic Field of Wave Beam," Phillips Research Reports, 19, 378–394 (1964). We thank the anonymous reviewer for bringing this paper to our attention. Although corrections to the paraxial result were formulated in this paper, the correction factor is calculated explicitly for the Gaussian mode at the beam center only. The correction factor [Eq. (27) of Ref. 12] is in disagreement with the one obtained in our analysis [Eq. (24)]. We refer the reader to the Appendix where we briefly comment on the paper by van Nie and point out the possible reason for the above disagreement.
  13. The higher-order fields in the expansion [Eq. (22)] will satisfy the differential equations which may be obtained following the procedure of Ref. 5.

1975

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).

1972

1964

A. G. van Nie, "Rigorous Calculation of the Electromagnetic Field of Wave Beam," Phillips Research Reports, 19, 378–394 (1964). We thank the anonymous reviewer for bringing this paper to our attention. Although corrections to the paraxial result were formulated in this paper, the correction factor is calculated explicitly for the Gaussian mode at the beam center only. The correction factor [Eq. (27) of Ref. 12] is in disagreement with the one obtained in our analysis [Eq. (24)]. We refer the reader to the Appendix where we briefly comment on the paper by van Nie and point out the possible reason for the above disagreement.

1961

G. Goubau and F. Schwering, "On the Guided Propagation of Electromagnetic Wave Beams", IRE Trans. AP9, 248–256 (1961).

Carter, W. H.

Clemmow, P. C.

For a discussion of the angular spectrum representation, see, for example, P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields, (Pergamon, Oxford, 1966); J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968) Sec. 3.7; J. R. Shewell and E. Wolf, "Inverse Diffraction and a Reciprocity Theroem," J. Opt. Soc. Am. 58, 1596–1603 (1968).

Cole, T. W.

See, for example, T. W. Cole, Progress in Optics, Vol. XV, edited by E. Wolf (North Holland, Amsterdam, 1977), p. 193.

Goubau, G.

G. Goubau and F. Schwering, "On the Guided Propagation of Electromagnetic Wave Beams", IRE Trans. AP9, 248–256 (1961).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Academic, New York, 1965), p. 716, p. 1037.

Kogelnik, H.

H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); H. Kogelnik, "On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss and Gain Variation," Appl. Opt. 4, 1562–1569 (1965).

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).

Li, T.

H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); H. Kogelnik, "On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss and Gain Variation," Appl. Opt. 4, 1562–1569 (1965).

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Academic, New York, 1965), p. 716, p. 1037.

Schwering, F.

G. Goubau and F. Schwering, "On the Guided Propagation of Electromagnetic Wave Beams", IRE Trans. AP9, 248–256 (1961).

Siegman, A. E.

See A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971), p. 317.

van Nie, A.G.

A. G. van Nie, "Rigorous Calculation of the Electromagnetic Field of Wave Beam," Phillips Research Reports, 19, 378–394 (1964). We thank the anonymous reviewer for bringing this paper to our attention. Although corrections to the paraxial result were formulated in this paper, the correction factor is calculated explicitly for the Gaussian mode at the beam center only. The correction factor [Eq. (27) of Ref. 12] is in disagreement with the one obtained in our analysis [Eq. (24)]. We refer the reader to the Appendix where we briefly comment on the paper by van Nie and point out the possible reason for the above disagreement.

Watson, G. N.

G. N. Watson, A treatize on the Theory of Bessel Functions, (Cambridge University, Cambridge, 1966).

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975) Chap. 6 and references cited therein.

IRE Trans.

G. Goubau and F. Schwering, "On the Guided Propagation of Electromagnetic Wave Beams", IRE Trans. AP9, 248–256 (1961).

J. Opt. Soc. Am.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, "From Maxwell to Paraxial Optics," Phys. Rev. A 11, 1365–1370 (1975).

Other

For a discussion of the angular spectrum representation, see, for example, P. C. Clemmow, The Plane Wave Spectrum Representations of Electromagnetic Fields, (Pergamon, Oxford, 1966); J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, New York, 1968) Sec. 3.7; J. R. Shewell and E. Wolf, "Inverse Diffraction and a Reciprocity Theroem," J. Opt. Soc. Am. 58, 1596–1603 (1968).

G. N. Watson, A treatize on the Theory of Bessel Functions, (Cambridge University, Cambridge, 1966).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, (Academic, New York, 1965), p. 716, p. 1037.

The restriction z < l probably arises from the interchange of the order of summation in Eq. (21).

A. G. van Nie, "Rigorous Calculation of the Electromagnetic Field of Wave Beam," Phillips Research Reports, 19, 378–394 (1964). We thank the anonymous reviewer for bringing this paper to our attention. Although corrections to the paraxial result were formulated in this paper, the correction factor is calculated explicitly for the Gaussian mode at the beam center only. The correction factor [Eq. (27) of Ref. 12] is in disagreement with the one obtained in our analysis [Eq. (24)]. We refer the reader to the Appendix where we briefly comment on the paper by van Nie and point out the possible reason for the above disagreement.

The higher-order fields in the expansion [Eq. (22)] will satisfy the differential equations which may be obtained following the procedure of Ref. 5.

H. Kogelnik and T. Li, "Laser Beams and Resonators," Proc. IEEE 54, 1312–1329, (1966); H. Kogelnik, "On the Propagation of Gaussian Beams of Light Through Lenslike Media Including those with a Loss and Gain Variation," Appl. Opt. 4, 1562–1569 (1965).

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1975) Chap. 6 and references cited therein.

See A. E. Siegman, An Introduction to Lasers and Masers, (McGraw-Hill, New York, 1971), p. 317.

See, for example, T. W. Cole, Progress in Optics, Vol. XV, edited by E. Wolf (North Holland, Amsterdam, 1977), p. 193.

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