Abstract

A self-consistent perturbation method is developed for the determination of the normal modes and eigenvalues of optic cavities of small Fresnel numbers. The method permits direct determination of the field distribution and eigenvalue (i.e., the diffraction loss and resonant frequency) of a normal mode of any given order to within any desired accuracy without simultaneously solving for the other modes, and can be applied to cavities having end reflectors of arbitrary shape and curvature. The method is applied to solve the integral equation governing the relation between the normal modes and the geometry of the cavity for the particular case of infinite-strip parabolic cavities.

© 1965 Optical Society of America

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References

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  1. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
    [Crossref]
  2. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
    [Crossref]
  3. L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
    [Crossref]
  4. H. Schachter, “Resonant Modes of Optic Cavities,” Ph.D. dissertation, Polytechnic Institute of Brooklyn, June1964.
  5. G. D. Boyd and J. P. Gordon, Bell System Tech. J. 41, 1347 (1962).
    [Crossref]
  6. S. R. Barone, J. Appl. Phys. 34, 831 (1963).
    [Crossref]
  7. A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
    [Crossref]
  8. D. Gloge, Arch. Elect. Über. 18, 197 (1964).
  9. L. A. Vainshtein, Soviet Physics—JETP 17, 709 (1963).
  10. J. T. LaTourette, S. F. Jacobs, and P. Rabinowitz, Appl. Opt. 3, 981 (1964).
    [Crossref]
  11. We order the modes in order of increasing losses (or, equivalently, decreasing eigenvalues) and assign the index 1 to the dominant or lowest-order mode, the index 2 to the next higher-order mode, etc.
  12. We set mj=12∫-1+1ξjdξ=11+j.
  13. It was pointed out by one of the reviewers that the zeroth-order eigenfunctions and eigenvalues can also be obtained from Eq. (17) by expanding exp{−iπHτξ} in terms of the prolate-spheroidal wavefunctions14S0n(πH,ξ).
  14. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford California, 1957).

1964 (3)

1963 (3)

L. A. Vainshtein, Soviet Physics—JETP 17, 709 (1963).

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

1962 (1)

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

1961 (2)

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Barone, S. R.

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

Bergstein, L.

Boyd, G. D.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford California, 1957).

Fox, A. G.

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

Gloge, D.

D. Gloge, Arch. Elect. Über. 18, 197 (1964).

Gordon, J. P.

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

Jacobs, S. F.

LaTourette, J. T.

Li, T.

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

Rabinowitz, P.

Schachter, H.

L. Bergstein and H. Schachter, J. Opt. Soc. Am. 54, 887 (1964).
[Crossref]

H. Schachter, “Resonant Modes of Optic Cavities,” Ph.D. dissertation, Polytechnic Institute of Brooklyn, June1964.

Vainshtein, L. A.

L. A. Vainshtein, Soviet Physics—JETP 17, 709 (1963).

Appl. Opt. (1)

Arch. Elect. Über. (1)

D. Gloge, Arch. Elect. Über. 18, 197 (1964).

Bell System Tech. J. (3)

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 41, 1347 (1962).
[Crossref]

A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).
[Crossref]

G. D. Boyd and J. P. Gordon, Bell System Tech. J. 40, 489 (1961).
[Crossref]

J. Appl. Phys. (1)

S. R. Barone, J. Appl. Phys. 34, 831 (1963).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

A. G. Fox and T. Li, Proc. IEEE 51, 80 (1963).
[Crossref]

Soviet Physics—JETP (1)

L. A. Vainshtein, Soviet Physics—JETP 17, 709 (1963).

Other (5)

H. Schachter, “Resonant Modes of Optic Cavities,” Ph.D. dissertation, Polytechnic Institute of Brooklyn, June1964.

We order the modes in order of increasing losses (or, equivalently, decreasing eigenvalues) and assign the index 1 to the dominant or lowest-order mode, the index 2 to the next higher-order mode, etc.

We set mj=12∫-1+1ξjdξ=11+j.

It was pointed out by one of the reviewers that the zeroth-order eigenfunctions and eigenvalues can also be obtained from Eq. (17) by expanding exp{−iπHτξ} in terms of the prolate-spheroidal wavefunctions14S0n(πH,ξ).

C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford California, 1957).

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Figures (5)

Fig. 1
Fig. 1

Geometry of the infinite-strip parabolic cavity.

Fig. 2
Fig. 2

Relative power loss, Pn(H,g)=1−| γ ˜ n|2, of the four low-order modes, (1), (2), (3), (4), of infinite-strip cavities having parabolic end reflectors of focal distances 1 2 p 1 4 l [ + 1.0 g = 1 - ( 1 / p ) - 1.0 ] as a function of the Fresnel number H=2a2l for 2H≤1.0.

Fig. 3
Fig. 3

Departure, δ ν n ( H , g ) = ν n - ν = ( c / 2 ( r ) 1 2 l ) [ arg ( γ ˜ n * ) / π ], of the resonant frequencies νn of the four low-order modes, (1), (2), (3), (4), of infinite-strip plane-parallel (g=+1.0) and parabolic-confocal (g=0) cavities from the resonant frequency ν = N c / 2 ( r ) 1 2 l of a plane-parallel Fabry–Perot interferometer of length l and infinite lateral extent as a function of the Fresnel number H for 2H⩽1.0.

Fig. 4
Fig. 4

Relative intensity distribution |fn(ξ; H,g)|2 (over the reflecting surfaces) of the four low-order modes, (1), (2), (3), (4), of infinite-strip cavities having parabolic end-reflectors of focal distances 1 2 p 1 4 l ( g 1.0 ) for Fresnel numbers H→0 and H=0.5. The distributions are normalized such that

Fig. 5
Fig. 5

Phase, arg{fn(ξ; H,g)}, of the relative field distribution fn(ξ; H,g) (over the reflecting surfaces) of the four low-order modes, (1), (2), (3), (4), of infinite-strip parabolic cavities. The solid graphs show the phase arg{fn(ξ; 0,g)}=arg{fn(ξ; H,0)} for parabolic cavities of Fresnel numbers H→0 and arbitrary curvature g, and for confocal cavities (g=0) of arbitrary Fresnel number H; the dashed graphs show for comparison the phase ±arg{fn(ξ; 0.5, ±1.0)} of plane-parallel (g=+1.0) and concentric (g=−1.0) cavities of Fresnel number H=0.5.

Equations (75)

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γ ˜ f ( x ) = 1 ( i l λ ) 1 2 - a + a f ( x 1 ) × exp [ i π λ l ( g x 2 + g x 1 2 - 2 x x 1 ) ] d x 1 ,
g = 1 - ( l / p )
P n = 1 - γ ˜ n 2
ν n = ν + [ arg ( γ ˜ n * ) / π ] ( ν / N ) ,
ξ = x / a ,
H = 2 a 2 / λ l ,
γ ˜ f ( ξ ) = ( H i 2 ) 1 2 - 1 + 1 f ( τ ) × exp [ i π 2 H ( g ξ 2 + g τ 2 - 2 ξ τ ) ] d τ .
γ ˆ = ( i / 2 H ) 1 2 γ ˜
h = i π H ,
γ ˆ f ( ξ ) = 1 2 - 1 + 1 f ( τ ) exp [ 1 2 h ( g ξ 2 + g τ 2 - 2 ξ τ ) ] d τ .
K ( ξ , τ ) = exp { 1 2 h ( g ξ 2 + g τ 2 - 2 ξ τ ) }
γ ˆ ( h , g ) f ( ξ ; h , g ) = 1 2 k = 0 h k - 1 + 1 1 k ! [ 1 2 ( g ξ 2 + g τ 2 - 2 ξ τ ) ] k f ( τ ; h , g ) d τ .
f ( ξ ; h , g ) = k = 0 h k f ( k ) ( ξ ; g )
γ ˆ ( h , g ) = k = 0 h k C ( k ) ( g ) .
k = 0 { h k j = 0 k [ C ( j ) f ( k - j ) ( ξ ) ] } = k = 0 { h k j = 0 k [ 1 2 k - j ( k - j ) ! u = 0 k - j ( - 2 ) u ( k - j u ) g k - j - u × s = 0 k - j - u ( k - j - u s ) M 2 k - 2 j - u - 2 s ( j ) ξ u + 2 s ] } ,
M j ( k ) = 1 2 - 1 + 1 ξ j f ( k ) ( ξ ) d ξ
j = 0 k [ C ( j ) f ( k - j ) ( ξ ) ] = j = 0 k { 1 2 k - j ( k - j ) ! u = 0 k - j [ ( - 2 ) u ( k - j u ) g k - j - u s = 0 k - j - u ( k - j - u s ) M 2 k - 2 j - u - 2 s ( j ) ξ u + 2 s ] } ,
G ( ξ ) = f ( ξ ) e i 1 2 π H g ξ 2 = f ( ξ ) e 1 2 h g ξ 2 ,
γ ˆ ( h , g ) G ( ξ ; h , g ) e - h g ξ 2 = 1 2 - 1 + 1 G ( τ ; h , g ) e - h ξ τ d τ .
G ( ξ ; h , g ) = k = 0 h k G ( k ) ( ξ ; g )
k = 0 { h k j = 0 k [ ( - 1 ) j j ! ( g ξ 2 ) j u = 0 k - j C ( u ) G ( k - j - u ) ( ξ ) ] } = k = 0 { h k j = 0 k [ ( - 1 ) j j ! M ˜ j ( k - j ) ( ξ ) j ] } ,
M ˜ j ( k ) = 1 2 - 1 + 1 ξ j G ( k ) ( ξ ) d ξ
j = 0 k { ( - 1 ) j j ! [ u = 0 k - j C ( u ) G ( k - j - u ) ( ξ ) ] ( g ξ 2 ) j } = j = 0 k [ ( - 1 ) j j ! M ˜ j ( k - j ) ξ j ]
C ( 0 ) G ( 0 ) ( ξ ) = M ˜ 0 ( 0 ) .
f n ( 0 ) ( ± 1.0 ) = G n ( 0 ) ( ± 1.0 ) = ( ± 1 ) n - 1 .
f 1 ( 0 ) ( ξ ) = G 1 ( 0 ) ( ξ ) = 1.0
C 1 ( 0 ) = 1.0.
C ( 1 ) G ( 0 ) ( ξ ) = - M 1 ( 0 ) ξ .
f 2 ( 0 ) ( ξ ) = G 2 ( 0 ) ( ξ ) = ξ ,
C 2 ( 1 ) = - 1 3 .
C ( 2 ) G ( 0 ) ( ξ ) = ( 1 / 2 ! ) M ˜ 2 ( 0 ) ( ξ 2 - m 2 ) .
f 3 ( 0 ) ( ξ ) = G 3 ( 0 ) ( ξ ) = [ ( 1 · 3 ) / 2 ! ] ( ξ 2 - 1 3 )
C 3 ( 2 ) = 2 ! / [ 5 ( 1 · 3 ) 2 ] = 2 / 45.
C ( 3 ) G ( 0 ) ( ξ ) = - ( 1 / 3 ! ) M ˜ 3 ( 0 ) [ ξ 3 - ( m 4 / m 2 ) ξ ] ,
f 4 ( 0 ) ( ξ ) = G 4 ( 0 ) ( ξ ) = [ ( 1 · 3 · 5 ) / 3 ! ] [ ξ 3 - ( 3 / 5 ) ξ ]
C 4 ( 3 ) = - 3 ! / [ 7 ( 1 · 3 · 5 ) 2 ] = - 2 / 525.
M 0 ( 0 ) = M 1 ( 0 ) = M 2 ( 0 ) = = M n - 1 ( 0 ) = 0 ,
C ( 0 ) = C ( 1 ) = C ( 2 ) = = C ( n - 1 ) = 0 ,
C n + 1 ( n ) G n + 1 ( 0 ) ( ξ ) = [ ( - 1 ) n / n ! ] M n + 1 , n ( 0 ) × ( ξ n + a 1 ξ n - 1 + + a n - 1 ξ + a n ) ,
f n + 1 ( 0 ) ( ξ ) = G n + 1 ( 0 ) ( ξ ) = P n ( ξ )
C n + 1 ( n ) = ( - 1 ) n n ! ( 2 n + 1 ) M n 2 = ( - 1 ) n 2 2 n ( n ! ) 3 ( 2 n + 1 ) [ ( 2 n ) ! ] 2 ,
P n ( ξ ) = 1 2 n k = 0 [ n / 2 ] ( - 1 ) k ( n k ) ( 2 n - 2 k n ) ξ n - 2 k = 1 ( 2 n + 1 ) M n × k = 0 [ n / 2 ] ( - 1 ) k ( n ! ) 2 ( 2 n - 2 k ) ! ( 2 n ) ! k ! ( n - k ) ! ( n - 2 k ) ! ξ n - 2 k
P n ( ± 1.0 ) = ( ± 1.0 ) n ,
M n = 1 2 - 1 + 1 ξ n P n ( ξ ) d ξ = 2 n ( n ! ) 2 ( 2 n + 1 ) ! = n ! 1 · 3 · 5 · 7 ( 2 n + 1 ) .
f n ( 0 ) ( ξ ) = Lim H 0 { f n ( ξ ; H ) } = P n - 1 ( ξ ) ,
γ ˜ n ( 0 ) = Lim H 0 { γ ˜ n ( H ) } = e - i ( n - 1 2 ) ( π / 2 ) × [ ( n - 1 ) ! ( 2 n - 1 ) [ 1 · 3 · 5 ( 2 n - 3 ) ] 2 ( π 2 ) n - 1 ] ( 2 H ) n - 1 2 ,
- 1 + 1 P n ( ξ ) P m ( ξ ) d ξ = 2 2 n + 1 δ n , m ,
C 1 ( 0 ) = 1.0
f 1 ( 0 ) ( ξ ) = P 0 ( ξ ) = 1.0
M 1 , j ( 0 ) = 1 2 - 1 + 1 ξ j f 1 ( 0 ) ( ξ ) d ξ { m j = 1 / ( j + 1 ) , when j is even , 0 , otherwise .
M 1 , j ( k ) = 1 2 - 1 + 1 ξ j f 1 ( k ) ( ξ ) d ξ = 0 ,             when j is odd ,
C 1 ( 0 ) f 1 ( 1 ) ( ξ ) + C 1 ( 1 ) f 1 ( 0 ) ( ξ ) = 1 2 g ( ξ 2 + m 2 ) + M 1 , 0 ( 1 ) .
M 1 , 0 = 1 2 - 1 + 1 f 1 ( ξ ) d ξ = M 1 , 0 ( 0 ) = 1.0 ,
M 1 , 0 ( k ) = M 1 , 0 δ k , 0 .
C ( 0 ) f ( 1 ) ( ξ ) + C ( 1 ) f ( 0 ) ( ξ ) = 1 2 g ( ξ 2 + m 2 ) .
C 1 ( 1 ) = 1 3 g
f 1 ( 1 ) ( ξ ) = 1 2 g ( ξ 2 - 1 3 ) .
C 1 ( 0 ) f 1 ( 2 ) ( ξ ) + C 1 ( 1 ) f 1 ( 1 ) ( ξ ) + C ( 2 ) f ( 0 ) ( ξ ) = ( 1 / 2 ! 2 2 ) [ g 2 ξ 4 + 2 ( 2 + g 2 ) m 2 ξ 2 + g ( 3 m 4 - 2 m 2 2 ) ] .
C 1 ( 2 ) = ( 1 / 18 ) ( 1 + 9 / 5 g 2 )
f 1 ( 2 ) ( ξ ) = [ 1 / ( 2 ! 2 2 ) ] [ g 2 ξ 4 + 2 3 ( 2 - g 2 ) ξ 2 - ( 1 / 45 ) ( 20 - g 2 ) ] .
C 1 ( 3 ) = ( 13 / 270 ) g [ 1 + ( 45 / 91 ) g 2 ] , f 1 ( 3 ) ( ξ ) = [ 1 / ( 3 ! 2 3 ) ] g [ g 2 ξ 6 + ( 4 - g 2 ) ξ 4 - ( 1 / 15 ) ( 8 - g 2 ) ξ 2 - ( 1 / 315 ) ( 196 - 11 g 2 ) ] ,
C 2 ( 0 ) = 0 , C 2 ( 1 ) = - m 2 = - 1 3 ,
f 2 ( 0 ) ( ξ ) = P 1 ( ξ ) = ξ ,
M 2 , j ( 0 ) = { m j + 1 = ( j + 2 ) - 1 , when j is odd , 0 , otherwise ,
M 2 , j ( k ) = 0 , when j is even ,
M n , n - 1 = 1 2 - 1 + 1 ξ n - 1 f n ( ξ ) d ξ = M n , n - 1 ( 0 )
M n , n - 1 ( k ) = M n , n - 1 δ k , 0 ,
- 1 + 1 f n ( ξ ; g , H ) f m ( ξ ; g , H ) d ξ = k = 0 [ δ n , m C n , k ( g ) H k ] ,
γ ˜ 1 ( H , g ) = e - i ( π / 4 ) ( 2 H ) 1 2 [ 1 + i 1 3 g ( π H ) - 1 18 ( 1 + 9 5 g 2 ) ( π H ) 2 - i 13 270 g ( 1 + 45 91 g 2 ) ( π H ) 3 + 67 16 200 ( 1 + 2518 469 g 2 + 75 67 g 4 ) ( π H ) 4 + ] ,
γ ˜ 2 ( H , g ) = e - i ( 3 π / 4 ) ( 1 3 ) ( π 2 ) ( 2 H ) 3 2 [ 1 + i 3 5 g ( π H ) - 3 50 ( 1 + 25 7 g 2 ) ( π H ) 2 - i 87 1750 g ( 1 + 875 783 g 2 ) ( π H ) 3 + 543 245 000 ( 1 + 15 626 1629 g 2 + 30 625 5973 g 4 ) ( π H ) 4 + ] ,
γ ˜ 3 ( H , g ) = e - i ( 5 π / 4 ) ( 2 45 ) ( π 2 ) 2 ( 2 H ) 5 2 [ 1 + i 11 21 g ( π H ) - 5 882 ( 1 + 553 25 g 2 ) ( π H ) 2 + i 107 6174 g ( 1 - 15 043 17 655 g 2 ) ( π H ) 3 - 3725 1 555 848 ( 1 + 1 405 122 204 875 g 2 + 103 439 9 685 000 g 4 ) ( π H ) 4 + ] ,
γ ˜ 4 ( H , g ) = e - i ( 7 π / 4 ) ( 2 525 ) ( π 2 ) 3 ( 2 H ) 7 2 [ 1 + i 23 45 g ( π H ) - 7 4050 ( 1 + 40 041 539 g 2 ) ( π H ) 2 + i 11 437 2 004 750 × g ( 1 - 4 024 809 1 040 767 g 2 ) ( π H ) 3 - 1 232 609 3 969 405 000 ( 1 + 1 133 608 958 112 167 419 g 2 - 8 176 298 031 785 171 933 g 4 ) ( π H ) 4 + ] ;
f 1 ( ξ ; H , g ) = 1 + i 1 3 g ( π H ) { 1 2 [ - 1 + 3 ξ 2 ] } - 1 9 ( π H ) 2 { 1 2 [ - ( 1 - 1 20 g 2 ) + 3 ( 1 - 1 2 g 2 ) ξ 2 + 9 4 g 2 ξ 4 ] } - i 8 135 g ( π H ) 3 { 1 32 [ - 7 ( 1 - 11 196 g 2 ) - 6 ( 1 - 1 8 g 2 ) ξ 2 + 45 ( 1 - 1 4 g 2 ) ξ 4 + 45 4 g 2 ξ 6 ] } + 11 2025 ( π H ) 4 { 1 88 [ - 17 ( 1 + 43 238 g 2 - 29 1904 g 4 ) - 30 ( 1 + 139 27 g 2 - 11 56 g 4 ) ξ 2 + 135 ( 1 + 1 6 g 2 + 1 24 g 4 ) ξ 4 + 225 4 g 2 ( 6 - g 2 ) ξ 6 + 675 16 g 4 ξ 8 ] } + ,
f 2 ( ξ ; H , g ) = ξ [ 1 + i 1 5 g ( π H ) { 1 2 [ - 3 + 5 ξ 2 ] } - 1 25 ( π H ) 2 { 1 2 [ - 3 ( 1 - 17 28 g 2 ) + 5 ( 1 - 3 2 g 2 ) ξ 2 + 25 4 g 2 ξ 4 ] } - i 8 875 g ( π H ) 3 { 1 32 [ 27 ( 1 - 227 972 g 2 ) - 170 ( 1 - 3 8 g 2 ) ξ 2 + 175 ( 1 - 3 4 g 2 ) ξ 4 + 875 1 H g 2 ξ 6 ] } + 3 30 625 ( π H ) 4 { 1 24 [ 875 ( 1 + 1307 2034 g 2 - 4909 178 992 g 4 ) - 1190 ( 1 - 449 612 g 2 + 227 1224 g 4 ) ξ 2 + 339 ( 1 - 47 10 g 2 + 51 40 g 4 ) ξ 4 + 6125 4 g 2 ( 2 - g 2 ) ξ 6 + 30 625 48 g 4 ξ 8 ] } + ] ,
- 1 + 1 f n ( ξ ; H , g ) 2 d ξ = - 1 + 1 f n ( ξ ; 0 , g ) 2 d ξ = 2 2 n - 1 .