Abstract

By using a three-mirror arrangement, one of which is planar and two of which are spherical (paraboloidal), discrimination against transverse as well as longitudinal modes can be shown. Suitable design parameters are obtained.

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. W. Smith, Proc. IEEE 60, 422 (1965).
    [CrossRef]
  2. D. A. Kleinman and P. P. Kisliuk, Bell Syst. Tech. J. 41, 453 (1962).
    [CrossRef]
  3. H. Kogelnik and C. K. N. Patel, Proc. IRE 50, 2365 (1962).
    [CrossRef]
  4. P. W. Smith, IEEE J. Quantum Electron. 1, 343 (1965).
    [CrossRef]
  5. See, e.g., A. G. Fox and T. Li, IEEE J. Quantum Electron. 2, 774 (1966).
    [CrossRef]
  6. H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).
  7. W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

1966 (2)

See, e.g., A. G. Fox and T. Li, IEEE J. Quantum Electron. 2, 774 (1966).
[CrossRef]

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

1965 (2)

P. W. Smith, Proc. IEEE 60, 422 (1965).
[CrossRef]

P. W. Smith, IEEE J. Quantum Electron. 1, 343 (1965).
[CrossRef]

1962 (2)

D. A. Kleinman and P. P. Kisliuk, Bell Syst. Tech. J. 41, 453 (1962).
[CrossRef]

H. Kogelnik and C. K. N. Patel, Proc. IRE 50, 2365 (1962).
[CrossRef]

Fox, A. G.

See, e.g., A. G. Fox and T. Li, IEEE J. Quantum Electron. 2, 774 (1966).
[CrossRef]

Kisliuk, P. P.

D. A. Kleinman and P. P. Kisliuk, Bell Syst. Tech. J. 41, 453 (1962).
[CrossRef]

Kleinman, D. A.

D. A. Kleinman and P. P. Kisliuk, Bell Syst. Tech. J. 41, 453 (1962).
[CrossRef]

Kogelnik, H.

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

H. Kogelnik and C. K. N. Patel, Proc. IRE 50, 2365 (1962).
[CrossRef]

Li, T.

See, e.g., A. G. Fox and T. Li, IEEE J. Quantum Electron. 2, 774 (1966).
[CrossRef]

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

Louisell, W. H.

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

Patel, C. K. N.

H. Kogelnik and C. K. N. Patel, Proc. IRE 50, 2365 (1962).
[CrossRef]

Smith, P. W.

P. W. Smith, IEEE J. Quantum Electron. 1, 343 (1965).
[CrossRef]

P. W. Smith, Proc. IEEE 60, 422 (1965).
[CrossRef]

Other (7)

P. W. Smith, Proc. IEEE 60, 422 (1965).
[CrossRef]

D. A. Kleinman and P. P. Kisliuk, Bell Syst. Tech. J. 41, 453 (1962).
[CrossRef]

H. Kogelnik and C. K. N. Patel, Proc. IRE 50, 2365 (1962).
[CrossRef]

P. W. Smith, IEEE J. Quantum Electron. 1, 343 (1965).
[CrossRef]

See, e.g., A. G. Fox and T. Li, IEEE J. Quantum Electron. 2, 774 (1966).
[CrossRef]

H. Kogelnik and T. Li, Proc. IEEE 54, 1312 (1966).

W. H. Louisell, Quantum Statistical Properties of Radiation (Wiley, New York, 1973).

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 (5)

Fig. 1
Fig. 1

Mirror simulation using a thin dielectric plate. Wave incident from active medium.

Fig. 2
Fig. 2

Mirror simulation using a thin dielectric plate. Wave incident from free space.

Fig. 3
Fig. 3

Three-mirror arrangement for discriminating against transverse modes. Active region between spherical mirrors.

Fig. 4
Fig. 4

Three-mirror arrangement for discriminating against transverse modes. Active region between plane and nearer spherical mirror.

Fig. 5
Fig. 5

Symmetrical spherical-mirror arrangement for discriminating against transverse modes.

Tables (1)

Tables Icon

Table I Calculated spectrum showing the discrimination for each transverse mode.

Equations (70)

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

E x ~ e α z + i θ ( z ) ,
θ ( z ) = ( ω z / c ) ( m + n + 1 ) tan 1 ( 2 z / k 0 w 0 2 ) .
E I = A e i θ ( z ) + α z + B e i θ ( z ) α z ,
E II = D exp i ( κ k 0 z ) + E exp i ( κ k 0 z ) ,
E III = C e i k 0 z .
t = C A = 2 k e i k 0 δ ( k + k 0 ) cos κ k 0 δ i [ ( k / κ ) + κ k 0 ] sin κ k 0 δ ,
k = ( k 0 i α ) ,
r = B A = ( k k 0 ) cos κ k 0 δ i [ ( k / κ ) k 0 κ ] sin κ k 0 δ ( k + k 0 ) cos κ k 0 δ i [ ( k / κ ) + k 0 κ ] sin κ k 0 δ .
μ = k 0 κ δ
t 2 [ 1 ( i α / k 0 ) ] 2 i [ μ + ( α / k 0 ) ]
r i [ μ ( α / k 0 ) ] 2 i [ μ + ( α / k 0 ) ] .
| r | 2 + | t | 2 1.
α / k 0 1 and α / k 0 μ ,
t 2 / ( 2 i μ ) = 2 | r | e i φ / μ = | t | e i φ ,
r i μ / ( 2 i μ ) = i | r | e i φ ,
tan φ = μ / 2 = | r | / | t | ,
| r | = μ ( 4 + μ 2 ) 1 2
| t | 2 + | r | 2 = 1.
| r | = sin φ ; | t | = cos φ .
r = i [ μ + ( α / k 0 ) ] / { 2 i [ μ + ( α / k 0 ) ] } ,
t = 2 / { 2 i [ μ + ( α / k 0 ) ] } .
r 0 = i | r 0 | exp ( i φ 0 ) = i μ 0 t 0 / 2 ,
| r 0 | = sin φ 0 = μ 0 ( 4 + μ 0 2 ) 1 2 ,
E I = A i θ ( z ) + B e i θ ( z ) ,
E II = C e α z + i θ ( z ) + D e α ( d ¯ z ) + i [ 2 θ ¯ θ ( z ) ] ,
θ ¯ θ ( d ¯ / 2 ) = ( k 0 d ¯ / 2 ) ( m + n + 1 ) tan 1 [ ( d ¯ / d ) ( ( 2 b / d ) 1 ) 1 2 ] .
D = r ¯ C ,
A = r 0 B .
E I = B [ r 0 e i θ ( z ) + e i θ ( z ) ] ,
E II = C [ e α z + i θ ( z ) + r ¯ e α ( d ¯ z ) + i [ 2 θ ¯ θ ( z ) ] ] .
θ θ ( d / 2 ) = ( k 0 d / 2 ) ( m + n + 1 ) tan 1 [ ( 2 b / d ) 1 ] 1 2 ,
C e [ ( α d / 2 ) + i θ ] = C r r ¯ e α [ d ¯ ( d / 2 ) ] + i [ 2 θ ¯ θ ] + B t r 0 e i θ ,
B e i θ = B r r 0 e i θ + C t r ¯ e α [ d ¯ ( d / 2 ) ] + i [ 2 θ ¯ θ ] .
e α ( d ¯ d ) = r ¯ { r + [ r 0 t 2 / ( e 2 i θ r r 0 ) ] } e 2 i Δ ,
Δ θ ¯ θ = [ k 0 ( d ¯ d ) / 2 ] + ( m + n + 1 ) { tan 1 [ ( 2 b / d ) 1 ] 1 2 tan 1 ( d ¯ / d ) [ ( 2 b / d ) 1 ] 1 2 } .
e α ( d ¯ d ) = | r ¯ | { | r | + [ | r 0 | | t | 2 / ( | r | | r 0 | + e i u ) ] } e i υ ,
u = 2 θ + φ + φ 0 = k 0 d 2 ( m + n + 1 ) tan 1 [ ( 2 b / d ) 1 ] 1 2 + sin 1 | r | + sin 1 | r 0 |
υ = 2 Δ + φ + φ ¯ = [ ( d ¯ / d ) 1 ] + 2 ( m + n + 1 ) { ( d ¯ / d ) tan 1 [ ( 2 b / d ) 1 ] 1 2 + tan 1 ( d ¯ / d ) [ ( 2 b / d ) 1 ] 1 2 } .
tan υ = sin u / [ T cos u + S ] ,
T = [ 1 + | r | 2 ] / [ 1 | r | 2 ] ; S = ( | r | / | r 0 | ) [ ( 1 + | r 0 | 2 ) / ( 1 | r | 2 ) ] ,
| r | | r ¯ | e α ( d ¯ d ) = cos υ sin υ ( cos u + | r | | r 0 | ) / sin u ,
S = T = [ 1 + | r 0 | 2 ] / [ 1 | r 0 | 2 ] ,
T tan υ = tan ( u / 2 ) .
sin u = 2 T tan υ [ T 2 tan 2 υ + 1 ] ,
cos u = [ T 2 tan 2 υ 1 ] / [ T 2 tan 2 υ + 1 ] .
α ( d ¯ d ) = ln [ ( 1 + | r 0 | 2 ) / 2 | r | | r 0 | ] ln [ cos υ ] .
1 cos υ < 0.
d ¯ / d = 1 + ( p / q ) ,
P u = { q π if q is even 2 q π if q is odd ,
P υ = { p π if q is even 2 p π if q is odd .
υ = ( p u / q ) + 2 ( m + n + 1 ) × { [ 1 + ( p / q ) ] tan 1 [ ( 2 b / d ) 1 ] 1 2 tan 1 [ 1 + ( p / q ) ] [ ( 2 b / d ) 1 ] 1 2 } [ ( 2 p / q ) 1 ] sin 1 | r 0 | + sin 1 | r ¯ | ,
u = k 0 d 2 ( m + n + 1 ) tan 1 [ ( 2 b / d ) 1 ] 1 2 + 2 sin 1 | r 0 | .
cos υ = 1
υ = ( 2 a + 1 ) π ,
u = 2 π s ,
2 [ 1 + ( p / q ) ] tan 1 [ ( 2 b / d ) 1 ] 1 2 tan 1 [ 1 + ( p / q ) ] [ ( 2 b / d ) 1 ] 1 2 [ ( 2 p / q ) 1 ] sin 1 | r 0 | + sin 1 | r | = [ 2 a + 1 ( 2 s p / q ) ] π
α min ( d ¯ d ) = ln [ ( 1 + | r 0 | 2 ) / 2 | r | | r 0 | ] .
D = α min / α .
ν = ( c / 2 π d ) { u + 2 ( m + n + 1 ) tan 1 [ ( 2 b / d ) 1 ] 1 2 2 sin 1 | r 0 | + N P u } .
Δ ν = ( c / 2 π d ) { Δ u + P u Δ N + 2 ( m + n ) tan 1 [ ( 2 b / d ) 1 ] 1 2 } ,
Δ ν = ( c / 2 π d ) P u
tan u = sin υ / [ T cos υ + S ] ,
T = ( 1 + | r | 2 ) / ( 1 | r | 2 ) ; S = ( | r ¯ | / | r | ) [ ( 1 + | r ¯ | 2 ) / ( 1 | r ¯ | 2 ) ]
| r | | r 0 | e α d = cos u + sin u ( cos υ + | r | | r ¯ | ) / sin υ .
T tan u = tan ( υ / 2 ) ,
α d = ln [ ( 1 + | r | 2 ) 2 | r | | r 0 | ] ln ( cos u )
( 2 b / d ) = 1.4 ; p / q = ( d ¯ / d ) 1 = 10 ; r = r 0 = 0.5450 ; r = 0.9500.
α min ( d ¯ d ) = 0.225139 cm .
P u = 2 π , P υ = 20 π .
( d ¯ / d ) 1 p / q = 10 ( 2 b / d ) 4 ,