Abstract

The self-consistent integral equation for the field distribution of the resonant modes in an inclined retroreflective grating resonator is solved in the limit of large Fresnel numbers. The transverse field distribution in the direction perpendicular to the grating grooves can be described in terms of Hermite-Gaussian functions provided that λ ≪ dw, where λ is the wavelength, d is the grating spacing, and w is the beam spot size.

© 1981 Optical Society of America

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References

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  1. H. H. Barrett, S. F. Jacobs, Opt. Lett. 4, 190 (1979).
    [CrossRef] [PubMed]
  2. V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
    [CrossRef]
  3. P. Mathieu, P.-A. Belanger, Appl. Opt. 19, 2262 (1980).
    [CrossRef] [PubMed]
  4. P. A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 602 (1980).
    [CrossRef] [PubMed]
  5. G. S. Zhou, Acta Phys. Sinica 27, 682 (1978).
  6. P. O. Clark, Proc. IEEE 53, 36 (1965).
    [CrossRef]

1980 (2)

1979 (1)

1978 (2)

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

G. S. Zhou, Acta Phys. Sinica 27, 682 (1978).

1965 (1)

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

Barrett, H. H.

Belanger, P. A.

Belanger, P.-A.

Clark, P. O.

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

Gerasimov, V. B.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Hardy, A.

Jacobs, S. F.

Kalinin, Yu. A.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Mathieu, P.

Orlov, V. K.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Sagalovich, A. Ya.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Siegman, A. E.

Virnik, Ya. Z.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Vorotilin, S. P.

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

Zhou, G. S.

G. S. Zhou, Acta Phys. Sinica 27, 682 (1978).

Acta Phys. Sinica (1)

G. S. Zhou, Acta Phys. Sinica 27, 682 (1978).

Appl. Opt. (2)

Opt. Lett. (1)

Proc. IEEE (1)

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

Sov. J. Quantum Electron. (1)

V. K. Orlov, Ya. Z. Virnik, S. P. Vorotilin, V. B. Gerasimov, Yu. A. Kalinin, A. Ya. Sagalovich, Sov. J. Quantum Electron. 8, 799 (1978).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Schematic representation of a light ray in a grating resonator.

Fig. 2
Fig. 2

Expanded view of a single grating groove.

Equations (44)

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ρ ( x 1 , x 2 ; y 1 , y 2 ) = ρ ( x 1 , x 2 ) + ρ ( y 1 , y 2 ) ,
ρ ( x 1 , x 2 ) = L / 2 + d ( x s + ξ ) tan θ + 2 x s tan θ ( d / L ) tan θ ( x 2 cos θ + ξ x s ) + ( 1 / 2 L ) × [ ( ξ + x s ) 2 + g 2 x 2 2 2 x 2 ( ξ + x s ) ] ,
ρ ( y 1 , y 2 ) = L / 2 + 1 2 L ( y 1 2 + g 2 y 2 2 2 y 1 y 2 ) ,
ρ ( x 1 , x 2 ; y 1 , y 2 ) = ρ ( x 1 , x 2 ) + ρ ( y 1 , y 2 ) ,
ρ ( x 1 , x 2 ) = L / 2 + ( 1 / 2 L ) [ x 1 2 + g 2 x 2 2 2 x 1 x 2 ] + x 1 tan θ .
γ x E x ( 2 ) ( x 2 ) = ( i / λ L ) E x ( 2 ) ( x 2 ) d x 2 d x 1 1 ( 1 + x 1 L tan θ ) 2 × exp { i k [ ρ ( x 1 , x 2 ) + ρ ( x 1 , x 2 ) ] } ,
γ y E y ( 2 ) ( y 2 ) = ( i / λ L ) E y ( 2 ) ( y 2 ) d 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 ,
E y ( 2 ) ( y 2 ) = ϕ m ( B 2 y 2 ) ,
ϕ m ( B 2 y 2 ) = N m ( 2 ) H m ( 2 B 2 y 2 ) exp ( B 2 2 y 2 2 ) ,
N m ( 2 ) = ( 2 B 2 ) 1 / 2 ( 2 m m ! π ) 1 / 2 ,
B 2 = ( π / λ L ) 1 / 2 [ g 2 ( 1 g 2 ) ] 1 / 4 ,
γ y m = σ m ,
σ m = exp { i k L + i ( m + 1 2 ) [ π 2 + tan 1 1 2 g 2 1 ( 1 2 g 2 ) 2 ] } .
B 1 = ( π λ L ) 1 / 2 [ 1 g 2 ( 1 g 2 ) ] 1 / 4 .
d sin θ = p λ ,
γ x E x ( 2 ) ( x 2 ) = i λ L exp [ i k ( L + d ) ] d x 2 E x ( 2 ) ( x 2 ) exp [ i k g 2 2 L ( x 2 2 + x 2 2 ) ] × s = ( 1 2 x s L tan θ ) exp { i k L [ x s 2 ( x 2 + x 2 d tan θ ) x s d x 2 tan θ ] } × d / 2 cos θ d / 2 cos θ d sin θ ( 1 2 ξ L tan θ ) exp { i k L [ ξ 2 ( x 2 x 2 + d tan θ ) ξ ] } d ξ ,
d cos θ ( 1 tan θ ) exp [ i k 2 L ( x 2 x 2 + d tan θ ) d sin θ ] .
s = [ 1 2 tan θ L ( s 1 2 ) d cos θ ] d cos θ ( 1 tan θ ) × exp i k L [ ( s 1 2 ) 2 d 2 cos 2 θ ( x 2 + x 2 d tan θ ) × ( s 1 2 ) d cos θ d x 2 tan θ ] .
exp ( i k L d x 2 tan θ ) ( 1 tan θ ) ( 1 2 tan θ L x s ) × exp { i k L [ x s 2 ( x 2 + x 2 d tan θ ) x s ] } d x s .
γ x E x ( 2 ) ( x 2 ) = exp ( π 4 i ) 2 λ L exp [ i k ( L + d ) ] ( 1 tan θ ) × d x 2 E x ( 2 ) ( x 2 ) [ 1 tan θ L ( x 2 + x 2 ) ] × exp { i k 4 L [ ( 2 g 2 1 ) ( x 2 2 + x 2 2 ) 2 x 2 x 2 ] } .
exp ( π 4 i i k L ) 2 λ L exp { i k 4 L [ ( 2 g 2 1 ) ( x 2 2 + x 2 2 ) 2 x 2 x 2 ] } = m σ m ϕ m ( B 2 x 2 ) ϕ m ( B 2 x 2 ) .
γ x E x ( 2 ) ( x 2 ) = exp ( i k d ) ( 1 tan θ ) ( 1 tan θ L x 2 ) m σ m C m ϕ m ( B 2 x 2 ) ,
C m = ϕ m ( B 2 x 2 ) ( 1 tan θ L x 2 ) E x ( 2 ) ( x 2 ) d x 2 .
ϕ n ( B 2 x 2 ) ( 1 tan θ L x 2 )
γ x C n = exp ( i k d ) ( 1 tan θ ) m σ m C m R m n ,
R m n = ( 1 tan θ L x 2 ) 2 ϕ m ( B 2 x 2 ) ϕ n ( B 2 x 2 ) d x 2 = { 1 n = m tan θ L B 2 m + 1 n = m + 1 tan θ L B 2 m n = m 1 for the rest .
E x m ( 2 ) ( x 2 ) = exp ( i k d ) ( 1 tan θ ) ( 1 x L tan θ ) [ ϕ m ( B 2 x 2 ) + C m 1 ϕ m 1 ( B 2 x 2 ) + C m + 1 ϕ m + 1 ( B 2 x 2 ) ] ,
C m 1 = tan θ 2 L B 2 ( m 1 g 2 ) 1 / 2 exp ( i χ ) ,
C m + 1 = tan θ 2 L B 2 ( m + 1 1 g 2 ) 1 / 2 exp ( i χ ) ,
χ = tan 1 ( g 2 1 g 2 ) 1 / 2 ,
γ x m = ( 1 tan θ ) σ m exp ( i k d ) .
| E x m ( 2 ) ( x 2 ) | = ( 1 tan θ ) | ϕ m ( B 2 x 2 ) | = ( 1 tan θ ) N m ( 2 ) | H m ( 2 B 2 x 2 ) | exp ( B 2 x 2 2 ) ,
E x m ( 1 ) ( x 1 ) = exp ( i k d i k L 2 ) exp ( i k x 1 tan θ ) ( 1 tan θ ) × ( 1 x 1 L tan θ ) [ χ m ϕ m ( B 1 x 1 ) + A m 1 χ m 1 × ϕ m 1 ( B 1 x 1 ) + A m + 1 χ m + 1 ϕ m + 1 ( B 1 x 1 ) ] ,
A m 1 = i C m 1 g 2 1 / 2 exp ( i χ )
A m + 1 = i C m + 1 g 2 1 / 2 exp ( i χ )
χ m = exp [ i ( m + 1 2 ) ( π 2 tan 1 g 2 1 g 2 ) ] .
( π L λ ) 1 / 2 [ 1 g 2 ( 1 g 2 ) ] 1 / 4 m tan θ ,
E m n ( 2 ) ( x 2 , y 2 ) = exp ( i k d ) ( 1 tan θ ) ϕ m ( B 2 x 2 ) ϕ n ( B 2 y 2 ) ,
E m n ( 1 ) ( x 1 , y 1 ) = exp ( i k d i k L / 2 ) ( 1 tan θ ) × exp ( i k x 1 tan θ ) ϕ m ( B 1 x 1 ) ϕ n ( B 1 y 1 ) .
ν = c 2 L + d [ q + 1 π ( m + n + 1 ) cos 1 g 2 ] ,
ν = p c / ( 2 d sin θ ) ,
δ 2 tan θ .

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