Abstract

A theory of the modes in a resonator bounded by the surface of a triaxial ellipsoid was given previously by Weinstein. We have applied this theory to characterize the modes observed in a large-aperture Brewster-window laser. Recognizing that Brewster windows impart astigmatism to the laser cavity, we can find an equivalent ellipsoidal cavity with which to associate the modes. The theory predicts various forms of mode structure, depending on the total astigmatism present. We have verified the theory experimentally and show that high-order modes having rectangular symmetry may be obtained even though a circular aperture is used in the cavity.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. H. Auston, IEEE QE-4, 420 (1968).
  2. R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
    [CrossRef]
  3. L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).
  4. D. Hanna, IEEE QE-5, 483 (1969).
  5. H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).
  6. D. Gloge and D. Marcuse, J. Opt. Soc. Am. 59, 1629 (1969).
    [CrossRef]
  7. H. Coddington, A System of Optics (Cambridge U. P., London, 1823), pp. 82–83.
  8. J. P. Goldsborough and E. B. Hodges, in Proceedings of the IEEE Electron Devices Conference, March, 1968, p. 88.
  9. H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
    [CrossRef]

1972 (1)

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

1970 (1)

H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
[CrossRef]

1969 (2)

1968 (1)

D. H. Auston, IEEE QE-4, 420 (1968).

1966 (1)

R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
[CrossRef]

Auston, D. H.

D. H. Auston, IEEE QE-4, 420 (1968).

Coddington, H.

H. Coddington, A System of Optics (Cambridge U. P., London, 1823), pp. 82–83.

Colombo, J.

H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
[CrossRef]

Dienes, A.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

Fisher, C. L.

H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
[CrossRef]

Gloge, D.

Goldsborough, J. P.

J. P. Goldsborough and E. B. Hodges, in Proceedings of the IEEE Electron Devices Conference, March, 1968, p. 88.

Hanna, D.

D. Hanna, IEEE QE-5, 483 (1969).

Hodges, E. B.

J. P. Goldsborough and E. B. Hodges, in Proceedings of the IEEE Electron Devices Conference, March, 1968, p. 88.

Ippen, E. P.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

Kogelnik, H. W.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

Ling, H.

H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
[CrossRef]

Marcuse, D.

Myers, R. A.

R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
[CrossRef]

Pole, R. V.

R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
[CrossRef]

Shank, C. V.

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

Weinstein, L. A.

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

Wieder, H.

R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
[CrossRef]

Appl. Phys. Lett. (1)

R. V. Pole, H. Wieder, and R. A. Myers, Appl. Phys. Lett. 8, 229 (1966).
[CrossRef]

IEEE (3)

D. H. Auston, IEEE QE-4, 420 (1968).

D. Hanna, IEEE QE-5, 483 (1969).

H. W. Kogelnik, E. P. Ippen, A. Dienes, and C. V. Shank, IEEE QE-8, 373 (1972).

J. Opt. Soc. Am. (1)

Rev. Sci. Instrum. (1)

H. Ling, J. Colombo, and C. L. Fisher, Rev. Sci. Instrum. 41, 1436 (1970).
[CrossRef]

Other (3)

L. A. Weinstein, Open Resonators and Open Waveguides (Golem, Boulder, Colo., 1969).

H. Coddington, A System of Optics (Cambridge U. P., London, 1823), pp. 82–83.

J. P. Goldsborough and E. B. Hodges, in Proceedings of the IEEE Electron Devices Conference, March, 1968, p. 88.

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.


Figures (15)

FIG. 1
FIG. 1

The A-type mode envelope in the first octant, showing the boundary surfaces η = θ1 and η = θ2.

FIG. 2
FIG. 2

The B-type mode envelope in the first octant, showing the boundary surfaces η = θ1 and ρ = θ2.

FIG. 3
FIG. 3

Geometry of a concave-mirror/Brewster-angle-window system in the (a) tangential and (b) sagittal planes.

FIG. 4
FIG. 4

Half-widths of the B-type mode envelope.

FIG. 5
FIG. 5

Sags of the B-type mode envelope.

FIG. 6
FIG. 6

Limiting form of B-type mode envelope from which an equivalent envelope for A-type modes is derived.

FIG. 7
FIG. 7

Experimental Fizeau interferogram of an eroded Brewster-angle window.

FIG. 8
FIG. 8

Experimental examples of B-type modes for Pρ > 50.

FIG. 9
FIG. 9

Experimental examples of B-type modes for 1 < Pρ <50.

FIG. 10
FIG. 10

Experimental mode patterns near Pρ = 1 showing the transition from B-type to A-type modes.

FIG. 11
FIG. 11

Experimental examples of A-type modes for 0 < Pρ < 1.

FIG. 12
FIG. 12

Δρ/Wρ vs Pη from Eq. (28) (solid curve) and experimental points (dashed curve).

FIG. 13
FIG. 13

Δη/Wη vs Pρ from Eq. (27) (solid curve) and experimental points (dashed curve).

FIG. 14
FIG. 14

[ ( W η ) 2 + W η 2 ] / W 0 2 vs Nρ from Eq. (36) (solid curve) and experimental points (dashed curve).

FIG. 15
FIG. 15

[ ( W η ) 2 - W η 2 ] / W 0 2 vs ΔR from Eq. (34) (solid curve) and experimental points (dashed curve).

Equations (56)

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

x 2 a 2 - θ + y 2 b 2 - θ + z 2 c 2 - θ = 1 ,
a = R x L / 2 , b = R y L / 2 , c = L / 2.
θ = ξ ,             - ξ c 2 ,
θ = η ,             c 2 η b 2 ,
θ = ρ ,             b 2 ρ a 2 .
D ( θ ) = ( a 2 - θ ) ( b 2 - θ ) ( c 2 - θ )
ξ ¯ 1 2 0 ξ d θ D ( θ ) , η ¯ 1 2 c 2 η d θ - D ( θ ) , ρ ¯ 1 2 b 2 ρ d θ D ( θ ) ,
( η - ρ ) 2 Φ ξ ¯ 2 - ( ρ - ξ ) 2 Φ η ¯ 2 + ( ξ - η ) 2 Φ ρ ¯ 2 + k 2 ( ξ - η ) ( ξ - ρ ) ( η - ρ ) Φ = 0
d 2 X d ξ ¯ 2 + k 2 P ( θ ) X = 0 ,
d 2 Y d η ¯ 2 - k 2 P ( θ ) Y = 0 ,
d 2 Z d ρ ¯ 2 + k 2 P ( θ ) Z = 0 ,
P ( θ ) = ( θ - θ 1 ) ( θ - θ 2 ) .
A type c 2 θ 1 θ 2 b 2 , B type c 2 < θ 1 b 2 < θ 2 , C type θ 1 < c 2 θ 2 < b 2 , D type θ 1 < c 2 < b 2 θ 2 .
X ( ξ ) = [ P ( ξ ) ] - 1 / 4 sin cos { k 2 0 ξ P ( θ ) D ( θ ) d θ } .
k 0 ξ P ( θ ) D ( θ ) d θ = N ξ π .
k θ 1 θ 2 P ( θ ) - D ( θ ) d θ = 2 π ( N η A + 1 2 ) ,
k b 2 a 2 P ( θ ) D ( θ ) d θ = N ρ A π
k θ 1 b 2 P ( θ ) - D ( θ ) d θ = π ( N η B + 1 2 ) ,
k θ 2 a 2 P ( θ ) D ( θ ) d θ = π ( N ρ B + 1 2 ) .
t 2 = cos 2 θ i { n cos 2 θ r t 1 n cos 2 θ r / cos 2 θ i - e / cos θ r + cos θ i - n cos θ r r t } - 1 ,
s 2 = { n s 1 n - e / cos θ r + cos θ i - n cos θ r r s } - 1
t 1 = s 1 = - R + z 0 - R .
R t = t 2 + z 0 + e / cos θ t 2 ,
R s = s 2 + Z 0 + e / cos θ s 2 .
Δ R R t - R s .
1 r s ( 1 + n 2 ) - 1 r t = e R 2 n 4 .
N ξ π = k { 2 c - b 2 - θ 1 b 2 - c 2 α 1 - a 2 - θ 2 a 2 - c 2 α 2 } ,
( N η B + 1 2 ) π = k 2 b 2 - θ 1 b 2 - c 2 ,
( N ρ B + 1 2 ) π = k 2 a 2 - θ 2 a 2 - c 2 ,
x 2 a 2 - θ 1 + y 2 b 2 - θ 1 - z 2 θ 1 - c 2 = 1 ,
x 2 a 2 - θ 2 - y 2 θ 2 - b 2 - z 2 θ 2 - c 2 = 1.
B x 2 a 2 - c 2 ,
B y 2 b 2 - c 2 ,
1 2 L Δ R = 1 2 L ( R x - R y ) = a 2 - b 2 .
x 2 1 2 L Δ R + ( B y λ / 2 π ) ( N η B + 1 2 ) + y 2 ( B y λ / 2 π ) ( N η B + 1 2 ) - z 2 ( B y / 2 ) 2 - ( B y λ / 2 π ) ( N η B + 1 2 ) = 1 ,
x 2 ( B x λ / 2 π ) ( N ρ B + 1 2 ) - y 2 1 2 L Δ R - ( B x λ / 2 π ) ( N ρ B + 1 2 ) - z 2 ( B x / 2 ) 2 - ( B x λ / 2 π ) ( N ρ B + 1 2 ) = 1
W η 2 ( 0 , y , z ; N η B ) = B y λ 2 π ( N η B + 1 2 ) × { 1 + z 2 ( B y / 2 ) 2 - ( B y λ / 2 π ) ( N η B + 1 2 ) } ,
W ρ 2 ( x , 0 , z ; N ρ B ) = B x 2 π ( N ρ B + 1 2 ) × { 1 + z 2 ( B x / 2 ) 2 - ( B x λ / 2 π ) ( N ρ B + 1 2 ) } .
Δ η W η = 1 - ( ( W η 2 - W ρ 2 ) W ρ 2 W ρ 2 W η 2 + W ρ 2 W η 2 ) 1 / 2 ,
Δ ρ W ρ = ( ( W η 2 + W ρ 2 ) W η 2 W ρ 2 W η 2 + W ρ 2 W η 2 ) 1 / 2 - 1 ,
W η 2 = { L 2 Δ R + B y λ 2 π ( N η B + 1 2 ) } × { 1 + z 2 ( B y / 2 ) 2 - ( B y λ / 2 π ) ( N η B + 1 2 ) } , W ρ 2 = { L 2 Δ R - B x λ 2 π ( N ρ B + 1 2 ) } × { 1 + z 2 ( B x / 2 ) 2 - ( B x λ / 2 π ) ( N ρ B + 1 2 ) }
Δ η / W η = 1 - 1 - 1 / P ρ B
Δ ρ / W ρ = 1 + 1 / P η B - 1 ,
P η B = Δ R · L ( B λ / π ) ( N η B + 1 2 ) , P ρ B = Δ R · L ( B λ / π ) ( N ρ B + 1 2 ) ,
W η 2 = W 0 2 ( N η B + 1 2 )
W ρ 2 = W η 2 - W η 2 ,
Δ ρ = W η - W ρ .
P η A = W ρ 2 / W η 2 ,
P η A = Δ R · L ( B λ / π ) ( N η B + 1 2 ) .
P η A = Δ R · L ( B λ / π ) ( W η 2 / W 0 2 ) .
W η 2 - W η 2 W 0 2 = Δ R · L ( B λ / π ) .
2 π W 0 ( N ρ A + 1 2 ) .
2 π 1 2 [ ( W η ) 2 + W η 2 ] = 2 π W 0 N ρ A + 1 2
W η 2 + W η 2 W 0 2 = 2 ( N ρ A + 1 2 ) .
W η W 0 = ( N ρ A + 1 2 ) + π 2 ( Δ R · L B λ ) ,
W η W 0 = ( N ρ A + 1 2 ) - π 2 ( Δ R · L B λ ) .