Abstract

The self-consistent integral equation for the field distribution of the resonant modes in a resonator with a tilted retroreflecting roof mirror is solved. The field distribution in the direction of the roof can be described in terms of Hermite-Gaussian functions. The beam matrix for a retroreflecting roof is found, and a new type of resonator which does not need precise alignment is proposed.

© 1981 Optical Society of America

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References

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  1. G. Toraldo di Francia, Appl. Opt. 4, 1267 (1965).
    [CrossRef]
  2. W. K. Kahn, Appl. Opt. 6, 865 (1967).
    [CrossRef] [PubMed]
  3. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  4. P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
    [CrossRef]
  5. P. F. Checcaci, A. Consortini, A. Scheggi, Appl. Opt. 65, 1567 (1966).
    [CrossRef]
  6. L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).
  7. See Ref. 3, Eq. (22). The term x1′ tanθ in our Eq. (3) is due to rotation of the left-hand reference plane. Here we have assumed that (tanθ)2 ≪ 1.
  8. P. O. Clark, Proc. IEEE 53, 36 (1965).
    [CrossRef]
  9. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  10. J. -P. Taché, Appl. Opt. 19, 4214 (1980), Eq. (23).
    [CrossRef] [PubMed]
  11. L. W. Casperson, Appl. Opt. 20, 2243 (1981), Eq. (10).
    [CrossRef] [PubMed]

1981 (1)

1980 (1)

1967 (1)

1966 (3)

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

P. F. Checcaci, A. Consortini, A. Scheggi, Appl. Opt. 65, 1567 (1966).
[CrossRef]

1965 (2)

1962 (1)

L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).

1961 (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Bergstein, L.

L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).

Casperson, L. W.

Checcaci, P. F.

P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

P. F. Checcaci, A. Consortini, A. Scheggi, Appl. Opt. 65, 1567 (1966).
[CrossRef]

Clark, P. O.

P. O. Clark, Proc. IEEE 53, 36 (1965).
[CrossRef]

Consortini, A.

P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

P. F. Checcaci, A. Consortini, A. Scheggi, Appl. Opt. 65, 1567 (1966).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Kahn, W.

L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).

Kahn, W. K.

Kogelnik, H.

Li, T.

H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
[CrossRef] [PubMed]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Scheggi, A.

P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

P. F. Checcaci, A. Consortini, A. Scheggi, Appl. Opt. 65, 1567 (1966).
[CrossRef]

Shulman, C.

L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).

Taché, J. -P.

Toraldo di Francia, G.

Appl. Opt. (6)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Proc. IEEE (2)

P. F. Checcaci, A. Consortini, A. Scheggi, Proc. IEEE, 54, 1329 (1966).
[CrossRef]

P. O. Clark, Proc. IEEE 53, 36 (1965).
[CrossRef]

Proc. IRE Eng. (1)

L. Bergstein, W. Kahn, C. Shulman, Proc. IRE Eng. 50, 1833 (1962).

Other (1)

See Ref. 3, Eq. (22). The term x1′ tanθ in our Eq. (3) is due to rotation of the left-hand reference plane. Here we have assumed that (tanθ)2 ≪ 1.

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Figures (3)

Fig. 1
Fig. 1

Schematic representation of a light ray in a resonator with a retroreflecting roof.

Fig. 2
Fig. 2

Low loss resonator which does not need fine angular alignment.

Fig. 3
Fig. 3

Schematic representation for calculating path length differences.

Equations (43)

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Δ = d sec θ - d tan θ sec θ ( x 1 + x 2 ) / L ,
ρ ( x 1 y 1 ; x 2 , y 2 ) = ρ ( x 1 , x 2 ) + ρ ( y 1 , y 2 ) ,
ρ ( x 1 , x 2 ) L / 2 + x 1 tan θ + ( x 1 2 + g 2 x 2 2 - 2 x 1 x 2 ) / 2 L ,
ρ ( y 1 , y 2 ) L / 2 + ( y 1 2 + g 2 y 2 2 - 2 y 1 y 2 ) / 2 L ,
g 2 = 1 - L / R 2 .
ρ ( x 1 , y 1 ; x 2 , y 2 ) = ρ ( x 1 , x 2 ) + ρ ( y 1 , y 2 ) ,
ρ ( x 1 , x 2 ) = L / 2 + x 1 tan θ + ( x 1 2 + g 2 2 x 2 2 - 2 x 1 x 2 ) / 2 L .
γ E ( 2 ) ( x 2 , y 2 ) = d x 2 d y 2 K ( x 2 , y 2 ; x 2 , y 2 ) E ( 2 ) ( x 2 , y 2 ) ,
K ( x 2 , y 2 ; x 2 , y 2 ) = ( i / λ ) 2 d x 1 d y 1 exp { [ - i k ρ ( - x 1 , y 1 ; x 2 , y 2 ) - i k ρ ( x 1 , y 1 ; x 2 , y 2 ) ] } [ ρ ( - x 1 , y 1 ; x 2 , y 2 ) + ρ ( x 1 , y 1 ; x 2 , y 2 ) ] - 1 ,
γ x E x ( 2 ) ( x 2 ) = ( i / λ L ) d x 2 E x ( 2 ) ( x 2 ) d x 1 ( 1 - x 1 tan θ / L ) - 1 · ( 1 + x 1 tan θ / L ) - 1 exp { - i k [ ρ ( - x 1 , x 2 ) + ρ ( x 1 , x 2 ) ] } ,
γ y E y ( 2 ) ( y 2 ) = ( i / λ L ) d y 2 E y ( 2 ) ( y 2 ) · d y 1 exp { - i k [ ρ ( y 1 , y 2 ) + ρ ( y 1 , y 2 ) ] } ,
E ( 2 ) ( x 2 , y 2 ) = E x ( 2 ) ( x 2 ) E y ( 2 ) ( y 2 ) ,
γ = γ x γ y ,
γ x E x ( 2 ) ( x 2 ) = exp ( i π / 4 - i k L ) ( 2 λ L ) - 1 / 2 · - d x 2 E x ( 2 ) ( x 2 ) exp { - i k 4 L [ ( 2 g 2 - 1 ) ( x 2 2 + x 2 2 ) + 2 x 2 x 2 ] } .
E y ( 2 ) ( y 2 ) = ϕ n ( y 2 / w 2 ) ,
ϕ n ( y 2 / w 2 ) = N n ( 2 ) H n ( 2 y 2 / w 2 ) exp ( - y 2 2 / w 2 2 ) ,
N n ( 2 ) = ( 2 / w 2 ) 1 / 2 ( 2 n n ! π ) - 1 / 2 ,
w 2 = ( π / λ L ) - 1 / 2 [ g 2 ( 1 - g 2 ) ] - 1 / 4 .
γ y n = σ n ,
σ n = exp [ - i k L + i ( n + ½ ) ( π / 2 + tan - 1 { ( 1 - 2 g 2 ) [ 1 - ( 1 - 2 g 2 ) 2 ] - 1 / 2 } ) ] .
( 2 λ L ) - 1 exp { - i π / 4 - i k L - i k [ ( 2 g 2 - 1 ) ( x 2 2 + x 2 2 ) - 2 x 2 x 2 ] / 4 L } = m σ m ϕ m ( x 2 / w 2 ) ϕ m ( x 2 / w 2 ) .
γ x E x ( 2 ) ( x 2 ) = σ m C m ϕ m ( x 2 / w 2 ) ,
C m = - d x 2 E ( 2 ) ( - x 2 ) ϕ m ( x 2 / w 2 ) = - d x 2 E ( 2 ) ( x 2 ) ϕ m ( - x 2 / w 2 ) .
E x m ( 2 ) ( x 2 ) = ϕ m ( x 2 / w 2 ) ,
γ x = ( - 1 ) m σ m ,
- ϕ m ( x 2 / w 2 ) ϕ n ( - x 2 / w 2 ) d x 2 = ( - 1 ) m δ m n .
ν m n q = ( c / 2 L ) [ q - m / 2 + ( m + n + 1 ) ( π ) - 1 cos - 1 g 2 1 / 2 ] .
E x m ( 1 ) = χ m ( 1 - x 1 tan θ / L ) exp ( - i k x 1 tan θ ) ϕ m ( 1 ) ( x 1 / w 1 ) ,
χ m = exp ( - i k L / 2 + i ( m + / 2 1 ) { π / 2 - tan - 1 [ g 2 / ( 1 - g 2 ) ] 1 / 2 } ) ,
Φ m ( 1 ) ( x 1 / w 2 ) = N m ( 1 ) H m ( 2 x 1 / w 1 ) exp ( - x 1 2 / w 1 2 ) ,
N m ( 1 ) = ( 2 / w 1 ) 1 / 2 ( 2 m m ! π 1 / 2 ) - 1 / 2 ,
w 1 = ( π / λ L ) - 1 / 2 [ g 2 - 1 ( 1 - g 2 ) ] - 1 / 4 .
F m ( 1 ) ( x 1 ) = ( i / λ z ) 1 / 2 ( 1 - x 1 tan ϕ / z ) exp ( - i k z / 2 + i k x 1 tan θ ) · d x 2 F m ( x 2 ) exp [ - i k ( x 1 2 + x 2 2 + 2 x 1 x 2 ) / ( 2 z ) ] ,
F m ( 3 ) ( x 2 ) = ( i / λ z ) 1 / 2 d x 1 F m ( 1 ) ( x 1 ) ( 1 + x 1 tan θ / z ) - 1 · exp [ - i k z / 2 - i k x 1 tan θ - i k ( x 1 2 + x 2 2 - 2 x 1 x 2 ) / 2 z ] .
F m ( 3 ) ( x 2 ) = ( - 1 ) m ( 2 λ z ) - 1 / 2 exp ( - i k z ) · - d x 2 F m ( x 2 ) exp [ - i k ( x 2 2 + x 2 2 - 2 x 2 x 2 ) / ( 4 z ) ] .
F m ( x 3 ) = ( 2 λ z ) - 1 / 2 exp ( - i k z ) - d x 2 F m ( x 2 ) · exp [ - i k ( x 2 2 + x 3 2 - 2 x 2 x 3 ) / ( 4 z ) ] .
( A B C D ) = ( - 1 0 0 - 1 ) .
( A B C D ) = ( 1 L 0 1 ) ( 1 0 - cos θ f 1 ) ( - 1 0 0 - 1 ) ( 1 0 - cos θ f 1 ) × ( 1 L 0 1 ) ( - 1 0 0 - 1 ) , = ( 1 L 0 1 ) ( 1 0 - 2 cos θ f 1 ) ( 1 L 0 1 ) ,
R = f / cos θ .
Δ = P 1 S = d / cos ( θ - β ) ,
Δ d sec θ - d ( x 2 - x 1 ) tan θ sec θ / L .
x 1 = - x 1 .
ρ ( x 1 , x 2 ) = L / 2 + x 1 tan θ + ( x 1 2 + g 2 x 2 2 - 2 x 1 x 2 ) / 2 L , = L / 2 - x 1 tan θ + ( x 1 2 + g 2 x 2 2 + 2 x 1 x 2 ) / 2 L .

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