Abstract

A previously developed technique for the calculation of modes of perfectly aligned unstable resonators in the limit of large Fresnel number is here extended to include effects of misalignment. It is shown how asymptotic techniques may be employed to simplify the calculation of near- and far-field patterns produced by such modes. The basic conclusion is that rectangular aperture unstable resonators are quite insensitive to misalignment, in the sense that the lowest-loss mode continues to be essentially diffraction limited as long as the feedback mirror remains well within the output beam.

© 1976 Optical Society of America

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References

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  1. P. Horwitz, J. Opt. Soc. Am. 63, 1528 (1973).
    [CrossRef]
  2. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 2241 (1969).
    [CrossRef] [PubMed]
  3. A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron QE-3, 156 (1967).
    [CrossRef]
  4. N. Bleistein, Commun. Pure Appl. Math. 19, 353 (1966).
    [CrossRef]
  5. M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

1973 (1)

1969 (1)

1967 (1)

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron QE-3, 156 (1967).
[CrossRef]

1966 (1)

N. Bleistein, Commun. Pure Appl. Math. 19, 353 (1966).
[CrossRef]

Arrathoon, R.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron QE-3, 156 (1967).
[CrossRef]

Bleistein, N.

N. Bleistein, Commun. Pure Appl. Math. 19, 353 (1966).
[CrossRef]

Horwitz, P.

Sanderson, R. L.

Siegman, A. E.

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron QE-3, 156 (1967).
[CrossRef]

Streifer, W.

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

N. Bleistein, Commun. Pure Appl. Math. 19, 353 (1966).
[CrossRef]

IEEE J. Quantum Electron (1)

A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron QE-3, 156 (1967).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (1)

M. Abramowitz, I. A. Stegun, Eds. Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964).

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Figures (12)

Fig. 1
Fig. 1

Geometry of a misaligned unstable resonator.

Fig. 2
Fig. 2

Lowest loss symmetric mode for Feff = 9.6, = 0.0. The dotted lines mark the position of the feedback mirror: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 3
Fig. 3

Lowest loss symmetric mode for Feff = 9.6, = 0.2: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 4
Fig. 4

Lowest loss symmetric mode for Feff = 9.6, = 0.5: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 5
Fig. 5

Diffraction limited beam for = 0: (a) far-field irradiance; (b) integrated far-field irradiance.

Fig. 6
Fig. 6

Diffraction limited beam for = 0.5: (a) far-field irradiance; (b) integrated far-field irradiance.

Fig. 7
Fig. 7

Lowest loss antisymmetric mode for Feff = 9.6, = 0.0: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 8
Fig. 8

Next-lowest loss symmetric mode for Feff = 9.6, = 0.0: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 9
Fig. 9

Lowest loss anti-symmetric mode for Feff = 9.6, = 0.5: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 10
Fig. 10

Next-lowest loss symmetric mode for Feff = 9.6, = 0.5: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 11
Fig. 11

One of two degenerate lowest loss modes for Feff = 9.78, = 0.1: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Fig. 12
Fig. 12

The other degenerate mode. Feff = 9.78, = 0.1: (a) near-field irradiance; (b) near-field phase; (c) far-field irradiance; (d) integrated far-field irradiance.

Equations (78)

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λ g ( x ) = ( i t / π ) 1 / 2 1 1 exp [ i t ( y x / M ) 2 ] g ( y ) d y ,
h = R 1 ( R 2 L ) L R 1 R 2 θ .
h a 1 = L θ a 1 g 2 g 1 g 2 1 = θ 2 F ( a 1 / λ ) ( g 1 g 2 1 ) .
F eff = ½ F [ M ( 1 / M ) ] ,
M = ( g 1 g 2 ) 1 / 2 + ( g 1 g 2 1 ) 1 / 2 ( g 1 g 2 ) 1 / 2 ( g 1 g 2 1 ) 1 / 2
= h a 1 = a 1 λ θ F eff m + 1 M 1 .
λ g ( x ) = ( i t / π ) 1 / 2 α β exp [ i t ( y x / M ) 2 ] g ( y ) d y ,
g ( y ) = h + n = 1 N [ f n α ( y ) E n α ( y ) + f n β ( y ) E n β ( y ) ] ,
( i t / π ) 1 / 2 α β exp [ i t ( y x / M ) 2 ] hdy = h { 1 + 1 2 ( i π t ) 1 / 2 [ E 1 β ( x ) β x / M E 1 α ( x ) α x / M ] } ,
( i t / π ) 1 / 2 α β exp [ i t ( y x / M ) 2 ] E n α ( y ) f n α ( y ) d y = a n 1 / 2 f n α [ r n α ( x ) ] E n + 1 α ( x ) + 1 2 ( i π t ) 1 / 2 a n 1 [ f n α ( β ) E 1 β ( x ) E n α ( β ) β r n α ( x ) f n α ( α ) E 1 α ( x ) E n α ( α ) α r n α ( x ) ] ,
( i t / π ) 1 / 2 α β exp [ i t ( y x / M ) 2 ] E n β ( y ) f n β ( y ) dy = a n 1 / 2 f n β [ r n β ( x ) ] E n + 1 β ( x ) + 1 2 ( i π t ) 1 / 2 a n 1 × [ f n β ( β ) E 1 β ( x ) E n β ( β ) β r n β ( x ) f n β ( α ) E 1 α ( x ) E n β ( α ) α r n β ( x ) ] ,
r n α ( x ) = M n 1 M n x M + α M n M n , r n β ( x ) = M n 1 M n x M + β M n M n .
( λ 1 ) h = a N 1 / 2 ( f n α E N + 1 α + f N β E N + 1 β ) ,
λ f n + 1 α ( x ) = a n 1 / 2 f n α [ r n α ( x ) ] ,
λ f n + 1 β ( x ) = a n 1 / 2 f n β [ r n β ( n ) ] .
λ f 1 α ( x ) = 1 2 ( i π t ) 1 / 2 { h α x / M + a n 1 n = 1 N [ f n α ( α ) E n α ( α ) α r n α ( x ) + f n β ( α ) E n β ( α ) α r n β ( x ) ] } ,
λ f 1 β ( x ) = 1 2 ( i π t ) 1 / 2 { h β x / M + a n 1 n = 1 N [ f n α ( β ) E n α ( β ) β r n α ( β ) + f n β ( β ) E n β ( β ) β r n β ( β ) ] } .
f n α ( x ) = M n 1 1 / 2 λ 1 n f 1 α [ S n 1 α ( x ) ] ,
f n β ( x ) = M n 1 1 / 2 λ 1 n f 1 β [ S n 1 β ( x ) ] ,
S n α ( x ) = M n 1 M n α M + x M n M n = r n x ( α ) ,
S n β ( x ) = M n 1 M n β M + x M n M n = r n x ( β ) ,
h = λ 1 N λ 1 ( 1 M 2 ) 1 / 2 [ f 1 α ( α / M ) E α + f 1 β ( β / M ) E β ]
E α = lim N E N α ( x ) = exp ( 2 π i F eff α 2 ) ,
2 ( i π t ) 1 / 2 f 1 α ( x ) = h / λ α x / M + n = 1 N λ n M n 1 1 / 2 M n × { E n α ( α ) f 1 α [ r n 1 α ( α ) ] α r n α ( x ) + E n β ( α ) f 1 β [ r n 1 α ( β ) ] α r n β ( x ) } ,
2 ( i π t ) 1 / 2 f 1 β ( x ) = h / λ β x / M + n = 1 N λ n M n 1 1 / 2 M n × { E n α ( β ) f 1 α [ r n 1 β ( α ) ] β r n α ( x ) + E n β ( β ) f 1 β [ r n 1 β ( β ) ] β r n β ( β ) } .
f 1 α ( x ) = α f 1 α ( 0 ) / ( α x / M ) ,
f 1 β ( x ) = β f 1 β ( 0 ) / ( β x / M ) .
h = λ 1 N λ 1 [ A f 1 α ( 0 ) + B f 1 β ( 0 ) ] ,
2 ( i π t ) 1 / 2 f 1 α ( 0 ) = h α λ + n = 1 N λ n [ A n α f 1 α ( 0 ) + A n β f 1 β ( 0 ) ] ,
2 ( i π t ) 1 / 2 f 1 β ( 0 ) = h β λ + n = 1 N λ n [ B n α f 1 α ( 0 ) + B n β f 1 β ( 0 ) ] ,
A = E α ( 1 M 2 ) 1 / 2 , B = E β ( 1 M 2 ) 1 / 2 , and A n α = M n 1 1 / 2 M n α E n α ( α ) [ α r n α ( 0 ) ] [ α r n 1 α ( α ) / M ] , A n β = M n 1 1 / 2 M n β E n β ( α ) [ α r n β ( 0 ) ] [ β r n 1 α ( β ) / M ] , B n α = M n 1 1 / 2 M n α E n α ( β ) [ β r n α ( 0 ) ] [ α r n 1 β ( α ) / M ] , B n β = M n 1 1 / 2 M n β E n β ( β ) [ β r n β ( 0 ) ] [ β r n 1 β ( β ) / M ] .
[ ( λ 1 ) λ N + P A α ] f 1 α ( 0 ) = P A β f 1 β ( 0 ) ,
[ ( λ 1 ) λ N P B β ] f 1 β ( 0 ) = P B α f 1 α ( 0 ) ,
P A α = A / α + ( λ 1 ) n = 1 N λ n n A n α , P A β = B / α + ( λ 1 ) n = 1 N λ n n A n β , P B α = A / β + ( λ 1 ) n = 1 N λ n n B n α ,
P B β = B / β + ( λ 1 ) Σ λ n n B n β .
P 1 f 1 α ( 0 ) = P 2 f 1 β ( 0 ) ,
P 3 f 1 β ( 0 ) = P 4 f 1 α ( 0 )
P 1 P 3 = P 2 P 4 .
f 1 α ( 0 ) / f 1 β ( 0 ) = ( P 2 P 3 / P 1 P 4 ) 1 / 2 .
I = α β exp [ i t ( a y 2 2 b y + c ) ] g ( y ) d y
g ( y ) = γ 0 + γ 1 ( y y 0 ) + γ 2 ( y y 0 ) ( y α ) + ( y y 0 ) ( y α ) ( y β ) G ( y ) .
g ( y ) = γ 0 + [ γ 1 + γ 2 ( y 0 α ) ] ( y y 0 ) + γ 2 ( y y 0 ) 2 ,
γ 0 = g ( y 0 ) , γ 1 = [ g ( α ) g ( y 0 ) ] / ( α y 0 ) , γ 2 = [ g ( β ) / ( β y 0 ) g ( α ) / ( α y 0 ) + g ( y 0 ) / ( α y 0 ) g ( y 0 ) / ( β y 0 ) ] / ( β α ) .
I n = α β exp [ i t ( a y 2 2 b y + c ) ] ( y y 0 ) n d y ,
I 0 = exp [ i t ( c b 2 / a ) ] [ E ( δ β / ) E ( δ α / ) ] , I 1 = i 2 a t { exp [ i t ( a β 2 2 b β + c ) ] exp [ i t ( a α 2 2 b α + c ) ] } , I 2 = i 2 a t { δ β exp [ i t ( a β 2 2 b β + c ) ] δ α exp [ i t ( a α 2 2 b α + c ) ] } + 0 ( 3 ) ,
E ( x ) = 0 x exp ( i π 2 x 2 ) d x = C ( x ) i S ( x ) ,
I = g ( y 0 ) exp [ i t ( c b 2 / a ) ] [ E ( δ β / ) E ( δ α / ) ] + i 2 a t { [ g ( β ) g ( y 0 ) ] exp [ i t ( a β 2 2 b β + c ) ] δ β [ g ( α ) g ( y 0 ) ] exp [ i t ( a α 2 2 b α + c ) ] δ α } .
E ( x ) = ( 2 i ) 1 / 2 sgn ( x ) + ϕ ( x ) exp ( i π 2 x 2 ) ,
α β exp [ i t ( y x / M ) 2 ] E n β ( y ) f n β ( y ) d y = n f n β [ r n β ( x ) ] × { ( 2 i ) 1 / 2 [ sgn ( δ β ) sgn ( δ α ) ] E n + 1 β ( x ) + ϕ ( δ β / n ) E n β ( β ) ϕ ( δ α / n ) E n β ( α ) } + i 2 a n t ( { [ f n β ( β ) f n β [ r n β ( x ) ] ] } × E n β ( β ) β r n β ( x ) { f n β ( α ) f n β [ r n β ( x ) ] } E n β ( α ) α r n β ( x ) ) ,
f n β ( x ) = M n 1 1 / 2 λ 1 n f 1 β ( x / M n ) .
f n β ( α ) f n β [ r n β ( x ) ] α r n β ( x ) = f n β [ r n β ( x ) ] M n ( β α / M n ) .
α β exp [ i t ( y x / M ) 2 ] E n β ( y ) f n β ( y ) d y = f n β [ r n β ( x ) ] × { 1 2 ( i a n t / π ) 1 / 2 [ sgn ( δ β ) sgn ( δ α ) ] E n + 1 β ( x ) + ( 2 a n t / π ) 1 / 2 × [ ϕ ( δ β / n ) E n β ( β ) ϕ ( δ α / n ) E n β ( α ) ] + i 2 a n t M n [ E n β ( β ) β β / M n E n β ( α ) β α / M n ] } .
G ( p , q ) = [ d x d y α β d x γ δ d y ] g x ( x ) g y ( y ) × exp ( ipx ) exp ( iqy ) ,
G ( p , q ) = g x ( x ) exp ( ipx ) d x g y ( y ) exp ( iqy ) d y α β g x ( x ) exp ( ipx ) d x γ δ g y ( y ) exp ( iqy ) d y .
g x ( x ) = λ x 1 ( i t / π ) 1 / 2 α β exp [ i t ( x x / M ) 2 ] g x ( x ) d x ,
g x ( x ) exp ( ipx ) = λ x 1 ( i t / π ) 1 / 2 d x α β d x × exp ( ipx ) exp [ i t ( x x / M ) 2 ] g x ( x ) .
g x ( x ) exp ( ipx ) d x = λ x 1 M ex p ( i p 2 M 2 4 t ) α β exp ( iM p x ) g x ( x ) d x .
g x ( p ) = α β exp ( i p x ) g x ( x ) d x , g y ( q ) = γ δ exp ( i q y ) g y ( y ) d y ,
G ( p , q ) = λ x 1 λ y 1 M 2 exp [ i M 2 4 t ( p 2 + q 2 ) ] g x ( M p ) g y ( M q ) g x ( p ) g y ( q ) .
G ( p , 0 ) = λ x 1 λ y 1 M 2 exp ( i M 2 p 2 4 t ) g x ( M p ) g x ( p ) .
I n = α β exp ( i p x ) E n β ( x ) f n β ( x ) d x = M n 1 1 / 2 λ 1 n β f 1 β ( 0 ) × α β exp [ i t ( β x / M n ) 2 / M n 1 + i p x ] β x / M n d x = M n M n 1 1 / 2 λ 1 n β f 1 β ( 0 ) exp [ itn ( ξ 2 + 2 β ξ ) ] × [ ξ / ( β β / M n ) ξ / ( β α / M n ) ] ,
x ( y ) = y + x ( exp i π 2 z 2 ) z x d z .
x x I ( x ) d x , where I ( 0 ) is the central maximum ,
r n α ( x ) = M n 1 M n x M + α M n M n , S n α ( x ) = r 1 α ( r 2 α { . . . [ r n α ( x ) ] . . . } ) , M n = 1 + M 2 + . . . + M 2 n , n = 1 , 2 , 3 . . . , M 0 = 1 .
M r 1 + M 2 r = M r
M n 1 = M 2 M n 1 .
M n M r 1 = M 2 r M n r .
M r 1 M n r 1 M 2 + M n = M n r M r .
M r 1 ( M n r 1 ) + M n = M r 1 M n r + ( M n M r 1 ) .
M r 1 M n r + M 2 r M n r = M n r M r ,
r n 1 α [ r n α ( x ) ] = M n 2 M n 1 ( M n 1 M n x M 2 + α M n + 1 M n ) + α M n 1 M n 1 = M n 2 M n x M 2 + α M n M n 1 M n 1 ( M 2 M n 2 + M n ) .
r n 1 α [ r n α ( x ) ] = M n 2 M n x M 2 + α M n 1 M 1 M n .
r n 2 α { r n 1 α [ r n α ( x ) ] } = M n 3 M n 2 ( M n 2 M n x M 3 + α M n M 1 M n ) + α M n 2 M n 2 = M n 3 M n x M 3 + α M n M n 2 M n 2 ( M 1 M n 3 M 2 + M n ) .
r n 2 α { r n 1 α [ r n α ( x ) ] } = M n 3 M n x M 3 + α M n 2 M 2 M n .
r n m α ( r n m + 1 α { . . . [ r n α ( x ) ] . . . } ) = M n m 1 M n x M m + 1 + α M n m M m M n .
S n α ( x ) = M n 1 M n α M + x M n M n .
S n α ( x ) = r n x ( α ) .
f n α ( x ) f 1 α ( x / M n )

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