Abstract

The scalar Gaussian wave, which is the usual solution of the eigenfunction problem for an open laser resonator, does not satisfy Maxwell’s equations. We obtain the exact solution by means of the Hertz vector instead of the more usual electric field. The exact solution (that satisfies Maxwell’s equations) is paraxially approximated, and the approximation fails to satisfy Maxwell’s equations. It is shown that for the Gaussian wave the electric and the magnetic vectors are not interchangeable in coordinates, which is a consequence of the inhomogeneous boundary condition.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1960).
  2. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  3. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1994), Chap. 3.1, p. 81.
  4. D. G. Hall, Opt. Lett. 21, 9 (1996).
    [CrossRef] [PubMed]
  5. D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
    [CrossRef]
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Chap. 8.3, p. 374.
  7. R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. California Press, Berkeley, Calif., 1966), p. 59.
  8. A. L. Cullen, P. K. Yu, Proc. R. Soc. London Ser. A 366, 155 (1979).
    [CrossRef]
  9. L. W. Davis, Phys. Rev. A 19, 1177 (1979).
    [CrossRef]
  10. G. Joos, Lehrbuch der theoretischen Physik, 7th ed. (Akademische Verlagsgesellschaft, Leipzig, 1950), Chap. 7, p. 296.
  11. M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

1996

1979

A. L. Cullen, P. K. Yu, Proc. R. Soc. London Ser. A 366, 155 (1979).
[CrossRef]

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[CrossRef]

1975

M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

1972

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

1966

1960

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1960).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Chap. 8.3, p. 374.

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1960).

Cullen, A. L.

A. L. Cullen, P. K. Yu, Proc. R. Soc. London Ser. A 366, 155 (1979).
[CrossRef]

Davis, L. W.

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1960).

Hall, D. G.

Joos, G.

G. Joos, Lehrbuch der theoretischen Physik, 7th ed. (Akademische Verlagsgesellschaft, Leipzig, 1950), Chap. 7, p. 296.

Kogelnik, H.

Lax, M.

M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

Li, T.

Luisell, W. H.

M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. California Press, Berkeley, Calif., 1966), p. 59.

McKnight, W. B.

M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

Pohl, D.

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1994), Chap. 3.1, p. 81.

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1994), Chap. 3.1, p. 81.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Chap. 8.3, p. 374.

Yu, P. K.

A. L. Cullen, P. K. Yu, Proc. R. Soc. London Ser. A 366, 155 (1979).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Pohl, Appl. Phys. Lett. 20, 266 (1972).
[CrossRef]

Bell Syst. Tech. J.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1960).

Opt. Lett.

Phys. Rev.

M. Lax, W. H. Luisell, W. B. McKnight, Phys. Rev. 11, 1365 (1975).

Phys. Rev. A

L. W. Davis, Phys. Rev. A 19, 1177 (1979).
[CrossRef]

Proc. R. Soc. London Ser. A

A. L. Cullen, P. K. Yu, Proc. R. Soc. London Ser. A 366, 155 (1979).
[CrossRef]

Other

G. Joos, Lehrbuch der theoretischen Physik, 7th ed. (Akademische Verlagsgesellschaft, Leipzig, 1950), Chap. 7, p. 296.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1959), Chap. 8.3, p. 374.

R. K. Luneburg, Mathematical Theory of Optics, 2nd ed. (U. California Press, Berkeley, Calif., 1966), p. 59.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1994), Chap. 3.1, p. 81.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Equations (28)

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

Δ P μ c 2 2 P t 2 = 0 .
Φ = P ,
A = μ c P t ,
E = Φ 1 c A t ,
B = × A .
E = μ c 2 2 P t 2 + ( P ) ,
B = μ c × P t .
Δ Q v + k 2 Q v = 0 ,
u ( x , y , z ) = 1 k 2 Ψ ( x , y , z ) exp ( i k z ) ,
Ψ ( x , y , z ) = exp { i [ p ( z ) + k x 2 + y 2 2 q ( z ) ] } .
E = [ I Ψ ( x , y , z ) exp ( i k z ) + u x ] exp ( i ω t ) ,
B = μ i k × I u ( x ) exp ( i ω t ) ,
E = E 0 + e ,
B = B 0 + b ,
E 0 = I Ψ ( x , y , z ) exp [ i ( ω t k z ) ] ,
B 0 = J μ Ψ ( x , y , z ) exp [ i ( ω t k z ) ] ,
e = 1 k 2 [ I 2 Ψ x 2 + J 2 Ψ y x + K ( i k Ψ x + 2 Ψ z x ) ] × exp [ i ( ω t k z ) ] ,
b = i k μ ( J Ψ z K Ψ y ) exp [ i ( ω t k z ) ] ,
e x = ( i k q x 2 q 2 ) E 0 , e y = x y q 2 E 0 , e z = x q ( x 2 + y 2 2 q 2 1 ) E 0 , b x = 0 , b y = ( i k q + x 2 + y 2 2 q 2 ) E 0 , b z = y q E 0 .
P x = 1 k 2 V ( x , y , z , t ) , P y = P z = 0 ,
F ( k x , k y ) = 1 2 π v ( x , y ) exp [ i ( k x x + k y y ) ] d x d y ,
V ( x , y , z , t ) = exp ( i ω t ) 2 π F ( k x , k y ) × exp [ i ( k z x + k y y + k z z ) ] d k x d k y ,
E = exp ( i ω t ) 2 π k 2 g ( k x , k y ) F ( k x , k y ) × exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
g ( k x , k y ) = ( k 2 k x 2 ) I k x k y J k x k z K .
B = exp ( i ω t ) 2 π k μ m ( k x , k y ) F ( k x , k y ) × exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
m ( k x , k y ) = k z J k y K .
v ( x , y ) = Ψ ( x , y , 0 ) = exp ( x 2 + y 2 w 0 2 ) ,
E = w 0 2 exp ( i ω t ) 4 k 2 g ( k x , k y ) × exp [ w 0 2 ( k x 2 + k y 2 ) 4 ] × exp [ i ( k x x + k y y + k z z ) ] d k x d k y .

Metrics