Abstract

The higher-order correction terms of the electric field vector of a Gaussian beam are derived explicitly from the magnetic vector potential that is assumed to be Gaussian and linearly polarized at the z=0 plane. The correction terms are proved to satisfy exactly Lax’s recurrence equations [Phys. Rev. A 11, 1365 (1975)]. The electric field vector with correction terms of orders up to 3 is compared with the exact electric field vector of an integral form that is also derived from the magnetic vector potential.

© 1999 Optical Society of America

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  3. M. Couture, P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
    [CrossRef]
  4. G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
    [CrossRef]
  5. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  6. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  7. B. T. Landesman, H. H. Barrett, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A 5, 1610–1619 (1988).
    [CrossRef]
  8. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  9. H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
    [CrossRef]
  10. Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
    [CrossRef]
  11. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
    [CrossRef]
  12. A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
    [CrossRef]
  13. D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
    [CrossRef]
  14. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Cross polarization in laser beams,” Appl. Opt. 26, 1589–1593 (1987).
    [CrossRef] [PubMed]
  15. W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
    [CrossRef]
  16. P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
    [CrossRef] [PubMed]
  17. 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]
  18. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966), p. 140.
  19. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  20. S. Ramo, J. R. Whinnery, T. V. Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1994), p. 589.
  21. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 113–118.
  22. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 169–174.
  23. G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
    [CrossRef]
  24. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]

1998

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

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]

Q. Cao, X. Deng, “Corrections to the paraxial approximation of an arbitrary free-propagation beam,” J. Opt. Soc. Am. A 15, 1144–1148 (1998).
[CrossRef]

1996

1994

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

1992

1988

1987

1985

1983

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981

M. Couture, P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

1980

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

1979

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

1975

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973

1972

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Barrett, H. H.

Belanger, P.

M. Couture, P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Boyd, G. D.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Cao, Q.

Couture, M.

M. Couture, P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

Cullen, A. L.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Davis, L. W.

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

Deng, X.

Deschamps, G. A.

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

Duzer, T. V.

S. Ramo, J. R. Whinnery, T. V. Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1994), p. 589.

Erikson, W. L.

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Fukumitsu, O.

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 113–118.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Landesman, B. T.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Mukunda, N.

Pattanayak, D. N.

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
[CrossRef]

Ramo, S.

S. Ramo, J. R. Whinnery, T. V. Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1994), p. 589.

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 169–174.

Siegman, A. E.

Simon, R.

Singh, S.

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Sudarshan, E. C. G.

Takenaka, T.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 169–174.

Török, P.

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]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

Varga, P.

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]

P. Varga, P. Török, “Exact and approximate solutions of Maxwell’s equations for a confocal cavity,” Opt. Lett. 21, 1523–1525 (1996).
[CrossRef] [PubMed]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966), p. 140.

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. V. Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1994), p. 589.

Wünsche, A.

Yokota, M.

Yu, P. K.

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, “Confocal multimode resonator for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

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]

Opt. Lett.

Phys. Rev. A

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

M. Couture, P. Belanger, “From Gaussian beam to complex-source-point spherical wave,” Phys. Rev. A 24, 355–359 (1981).
[CrossRef]

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A 19, 1177–1179 (1979).
[CrossRef]

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Phys. Rev. E

W. L. Erikson, S. Singh, “Polarization properties of Maxwell–Gaussian laser beams,” Phys. Rev. E 49, 5778–5786 (1994).
[CrossRef]

Proc. IEEE

G. A. Deschamps, “Ray techniques in electromagnetics,” Proc. IEEE 60, 1022–1035 (1972).
[CrossRef]

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Proc. R. Soc. London, Ser. A

A. L. Cullen, P. K. Yu, “Complex source-point theory of the electromagnetic open resonator,” Proc. R. Soc. London, Ser. A 366, 155–171 (1979).
[CrossRef]

Other

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, London, 1966), p. 140.

S. Ramo, J. R. Whinnery, T. V. Duzer, Fields and Waves in Communication Electronics, 3rd ed. (Wiley, New York, 1994), p. 589.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 113–118.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 169–174.

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Figures (5)

Fig. 1
Fig. 1

Contour plot of the intensities of the exact electric-field components in the ξη plane; (a)–(c) are drawn at the ζ=0 plane and (d)–(f ) are drawn at the ζ=1.0 plane. f is assumed to be 1/2π in the calculation.

Fig. 2
Fig. 2

Electric field lines drawn at four different instants. The time interval between adjacent figures is a quarter of the temporal period. The arrows inside the curves show the directions of the electric field on the contours.

Fig. 3
Fig. 3

Peak magnitudes of the electric-field components, which contain the correction terms of orders up to 3, calculated and compared with those of the exact electric field. (a)–(c) Percent errors of the peak magnitudes of the three components, which are calculated for three different values of f; (d),(e) locations of the peaks. In (d) and (e) the predictions of the exact electric field are drawn in the upper half-plane, and those of the corrected electric field are drawn in the lower half-plane.

Fig. 4
Fig. 4

Contours of the beam power density, or the z component of the Ponyting vector, drawn at different ξη planes. The vertical axis is the ξ axis, and the horizontal axis is the η axis. Solid curves, scalar Gaussian beam; dashed curves, the exact electric field; dotted curves, the electric field with correction terms of orders up to 3. f is assumed to be 1/2π in (a)–(c) and 1/3 in (d)–(f ).

Fig. 5
Fig. 5

Total beam power crossing the transverse plane calculated as a function of f. Solid line, scalar Gaussian beam; dashed curve, the exact electric field; dotted curve, the prediction by the electric field with correction terms of orders up to 3. Note that the total beam power is conserved along the ζ axis.

Equations (19)

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

A(x, y, z)=xˆ -i4πkc0f 2 c=-b=-×exp-14f 2 (b2+c2)+i bf ξ+i cf η+iζ 1f 2 (1-b2-c2)1/2dbdc,
B=14πc0f 2 c=-b=-[(1-b2-c2)1/2yˆ-czˆ]×exp-14f 2 (b2+c2)+i bf ξ+i cf η+iζ 1f 2 (1-b2-c2)1/2dbdc,
E=14πf 2c=-b=-×[(1-b2)xˆ-bcyˆ-b(1-b2-c2)1/2zˆ]×exp-14f 2 (b2+c2)+i bf ξ+i cf η+iζ 1f 2 (1-b2-c2)1/2dbdc.
expi ζf 2 (1-b2-c2)1/2
=expi ζf 2 [1-½(b2+c2)]×1+m=1t=1mr=0m+t (m+t)!r!(m+t-r)!×Cm, tf 2mζt-ibf2(m+t-r)-icf2r,
Cm,t=i22m+t (2m)!m(m+t)!(t-1)!(m-t)!,t1.
A=xˆ-ikc0×ψ0,0+m=1t=1mr=0m+tf 2mCm,tζt(m+t)!r!(m+t-r)!×ψ2(m+t-r),2r.
ψm,n=1(1+i2ζ)1/2m+n+2Hmξ(1+i2ζ)1/2
×Hnη(1+i2ζ)1/2exp-ξ2+η21+i2ζ+i ζf 2.
Ex=ψ0,0+f 2ψ2,0-iζ8(ψ4,0+2ψ2,2+ψ0,4)+m=2f 2mKm,
Ey=f 2ψ1,1+m=2f 2mMm,
Ez=-ifψ1,0-f 3i2 (ψ3,0+ψ1,2)+18 ζ(ψ5,0+2ψ3,2+ψ1,4)-im=2f 2m+1Nm,
By=1c0 ψ0,0+f 212 (ψ2,0+ψ0,2)-iζ8 (ψ4,0+2ψ2,2+ψ0,4)+m=2f 2mOm,
Bz=1c0 -ifψ0,1-f 3 18 ζ(ψ4,1+2ψ2,3+ψ0,5)-im=2f 2m+1Pm,
Km=t=1mr=0m+t (m+t)!r!(m+t-r)! Cm,tζtψ2(m+t-r),2r+t=1m-1r=0m+t-1 (m+t-1)!r!(m+t-r-1)!×Cm-1,tζtψ2(m+t-r),2r,
Mm=t=1m-1r=0m+t-1 (m+t-1)!r!(m+t-r-1)!×Cm-1,tζtψ2(m+t-r)-1,2r+1,
Nm=t=1mr=0m+t (m+t)!r!(m+t-r)! Cm,tζtψ2(m+t-r)+1,2r+12 t=1m-1r=0m+t (m+t)!r!(m+t-r)!×Cm-1,tζtψ2(m+t-r)+1,2r-it=1m-1r=0m+t-1 (m+t-1)!r!(m+t-r-1)!×Cm-1,ttζt-1ψ2(m+t-r)-1,2r,
Om=t=1mr=0m+t (m+t)!r!(m+t-r)! Cm,tζtψ2(m+t-r),2r+12 t=1m-1r=0m+t (m+t)!r!(m+t-r)! Cm-1,tζtψ2(m+t-r),2r-it=1m-1r=0m+t-1 (m+t-1)!r!(m+t-r-1)!×Cm-1,ttζt-1ψ2(m+t-r-1),2r,
Pm=t=1mr=0m+t (m+t)!r!(m+t-r)! Cm,tζtψ2(m+t-r),2r+1.

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