Abstract

The modes in an unstable resonator can be computed within the limit of a large Fresnel number using the asymptotic expansion of the diffraction integral, as shown by Horwitz, Butts, and Avizonis. The expansion is not valid for the points of interest around or beyond the shadow boundary of the output light. We use a better numerical representation, which extends the regions of use. The comparison of several cases with earlier work shows that the asymptotic theory can be successfully applied for all parameters without restrictions.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).
  2. A. E. Siegman, “Unstable Optical Resonators,” Appl. Opt. 13, 353–367 (1974).
    [CrossRef] [PubMed]
  3. P. Horwitz, “Asymptotic Theory of Unstable Resonator Modes,” J. Opt. Soc. Am. 63, 1528–1543 (1973).
    [CrossRef]
  4. P. Horwitz, “Modes in Misaligned Unstable Resonators,” Appl. Opt. 15, 167–178 (1976).
    [CrossRef] [PubMed]
  5. R. R. Butts, P. V. Avizonis, “Asymptotic Analysis of Unstable Laser Resonators with Circular Mirrors,” J. Opt. Soc. Am. 68, 1072–1078 (1978).
    [CrossRef]
  6. C. C. Sung, Y. Q. Li, M. E. Smithers, “Mode-Medium Instability in an Unstable Resonator,” Appl. Opt. 27, 58–65 (1988).
    [CrossRef] [PubMed]
  7. L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, New York, 1971).
  8. J. D. Murray, Asymptotic Analysis (Springer-Verlag, New York, 1984).
    [CrossRef]
  9. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser.55, p. 302.

1988 (1)

1978 (1)

1976 (1)

1974 (1)

1973 (1)

1961 (1)

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Avizonis, P. V.

Butts, R. R.

Fox, A. G.

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Horwitz, P.

Li, T.

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

Li, Y. Q.

Murray, J. D.

J. D. Murray, Asymptotic Analysis (Springer-Verlag, New York, 1984).
[CrossRef]

Siegman, A. E.

Sirovich, L.

L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, New York, 1971).

Smithers, M. E.

Sung, C. C.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant Modes in a Maser Interferometer,” Bell Syst. Tech. J. 40, 453 (1961).

J. Opt. Soc. Am. (2)

Other (3)

L. Sirovich, Techniques of Asymptotic Analysis (Springer-Verlag, New York, 1971).

J. D. Murray, Asymptotic Analysis (Springer-Verlag, New York, 1984).
[CrossRef]

Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Natl. Bur. Stand. Appl. Math. Ser.55, p. 302.

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 (6)

Fig. 1
Fig. 1

Geometry of the unstable resonator. R1 and R2 are the radii of mirrors 1 and 2, respectively. L is the distance between two mirrors. a1 is the diameter of the small mirror.

Fig. 2
Fig. 2

Intensity distribution inside the small mirror and outside the small mirror. The dashed line is Horwitz’s result, and the solid line is our modified result. The effective Fresnel number Feff = 16.4, M = 1.5. X is in the unit of a1.

Fig. 3
Fig. 3

Same as Fig. 2 except Feff = 1.5, M = 1.3.

Fig. 4
Fig. 4

Same as Fig. 2 except Feff = 0.5, M = 1.1.

Fig. 5
Fig. 5

Same as Fig. 2 except F = 6.25, M = 1.01, the dashed line is the Fox-Li accurate result, and the solid line is our result.

Fig. 6
Fig. 6

Same as Fig. 2 except F = 6.25, M = 1.01.

Equations (55)

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

ψ ( x , y ) = u ( x ) v ( y ) .
γ u ( x ) = ( i F ) 1 / 2 - 1 1 exp ( - i π F [ g 0 × ( x 2 + y 2 ) - 2 x y ] ) u ( y ) d y ,
F eff = F / 2 ( M - 1 / M ) ,
M = ( g 0 + 1 ) 1 / 2 + ( g 0 - 1 ) 1 / 2 ( g 0 + 1 ) 1 / 2 - ( g 0 - 1 ) 1 / 2 .
g ( x ) = exp ( i π F eff x 2 ) u ( x ) ,
λ g ( x ) = ( i t / π ) 1 / 2 - 1 1 exp [ - i t ( y - x / M ) 2 ] g ( y ) d y ,
E n ( x ) = exp { - i t [ ( 1 + x / M n ) 2 M n - 1 ] } ,
M n = m = 0 n M - 2 m ,             n = 0 , 1 ,
g ( x ) = h ( x ) + n = 1 N [ E n ( x ) f n ( x ) + E n ( - x ) f n ( - x ) ] ,
I n ( x ) = ( i t / π ) 1 / 2 - 1 1 exp ( - i t ( y - x / M ) 2 ) E n ( y ) f n ( y ) d y ,
I n ( x ) = a n - 1 / 2 E n + 1 ( x ) f n [ r n ( x ) ] - [ E n ( - 1 ) f n ( - 1 ) E 1 ( x ) d n - + x / M + E n ( 1 ) f n ( 1 ) E 1 ( x ) d n + + x / M ] ,
I n , 0 = a n - 1 / 2 E n + 1 ( x ) f n [ r n ( x ) ] ,
I n , ± 1 = - E n ( ± 1 ) f n ( ± 1 ) E 1 ( x ) d n ± + x / M ,
λ n f n + 1 ( x ) = M n - 1 / 2 f 1 [ s n ( x ) ] ,
s n ( x ) = r 1 ( r 2 { [ r n ( x ) ] } ) ,
f 1 ( x ) = f 1 ( 0 ) / ( 1 + x / M ) ,
f 1 ( 0 ) = - λ - 1 h ( - 1 ) [ 1 + n = 1 N A n λ - n ] - 1 ,
A n = M n - 1 - 1 / 2 [ E n ( - 1 ) 1 + s n - 1 ( - 1 ) / M + E n ( 1 ) 1 + s n - 1 ( 1 ) / M ] .
λ N + 1 + ( A 1 - 1 ) λ N + ( A 2 - A 1 ) λ N - 1 + + ( A N + 1 - A N ) = 0 ,
N = { log ( 100 F eff ) / log ( M ) ( M is not close to 1 ) 100 F ( M 1 ) .
I ( x ) = ( i t / π ) 1 / 2 - 1 1 exp [ - i t ( y - x / M ) 2 ] g ( y ) d y ,
I 0 = ( i t / π ) 1 / 2 y 0 - δ y 0 + δ exp [ - i t ( y - y 0 ) ] 2 g ( y ) d y ,
I 0 = k = 0 g ( 2 k ) ( y 0 ) ( 2 k ) ! Γ ( k + 1 2 ) · π - 1 / 2 ( i t ) - k = g ( y 0 ) + o ( t - 1 ) .
I n , 1 = ( i t / π ) 1 / 2 1 - δ 1 exp [ - i t ( y - y 0 ) 2 ] g ( y ) d y ,
p ( y ) = p ( 1 ) + p ( 1 ) ( y - 1 ) + p ( 1 ) / 2 ( y - 1 ) 2 .
y = y 0 + [ ( 1 - y 0 ) 2 - s 2 ] 1 / 2 .
I n , 1 = k = 0 exp [ - i t ( 1 - y 0 ) 2 ] g ( k ) ( 1 ) I ( k ) ( t ) / k !
I ( k ) ( t ) = ( i t / π ) 1 / 2 0 δ 1 × exp ( i t s 2 ) { - ( 1 - y 0 ) + [ ( 1 - y 0 ) 2 - s 2 ] 1 / 2 } k [ ( 1 - y 0 ) 2 - s 2 ] 1 / 2 d s 2 ,
I ( 0 ) ( t ) = - ½ Q ( 1 - y 0 t 1 / 2 ) ,
I ( 1 ) ( t ) = - ½ ( i π t ) - 1 / 2 R ( 1 - y 0 t 1 / 2 ) ,
Q ( x ) = ( 2 i ) 1 / 2 [ f r ( x ) - i f i ( x ) ] ,
R ( x ) = 1 - ( 2 π ) 1 / 2 x [ f i ( x ) - i f r ( x ) ] ,
{ f i ( x ) = 1 + β 0 x 2 + β 1 x + β 2 x 2 , f r ( x ) = 1 2 + γ 1 x + γ 2 x 2 + γ 3 x 3 .
β k = ( 2 / π ) k / 2 β k 0 , γ k = ( 2 / π ) k / 2 γ k 0 ,
{ Q ( x ) = 0.55 / ( i 1 / 2 x ) , x , Q ( x ) = 1 , x 0 ,
{ R ( x ) = 1 / x , x , R ( x ) = 1 x 0.
I ( k ) ( t ) O ( t - k / 2 ) ,
I 1 = - g ( 1 ) / 2 exp [ - i t ( 1 - y 0 ) 2 ] Q ( 1 - y 0 t 1 / 2 ) - g ( 1 ) / 2 exp [ - i t ( 1 - y 0 ) 2 ] R ( 1 - y 0 t 1 / 2 ) · ( i π t ) - 1 / 2 + o ( t - 2 ) .
I ( x ) = ( 1 + μ ) / 2 g ( y 0 ) - μ + / 2 [ g ( 1 ) Q ( 1 - y 0 t 1 / 2 ) + g ( 1 ) · ( i π t ) - 1 / 2 R ( 1 - y 0 t 1 / 2 ) ] - μ - / 2 [ g ( - 1 ) Q ( 1 - y 0 t 1 / 2 ) + g ( - 1 ) · ( i π t ) - 1 / 2 R ( 1 - y 0 t 1 / 2 ) ] + O ( t - 1 ) ,
Q ( x ) 0.55 / ( i 1 / 2 x ) .
1 - y 0 F - 1 / 2 ,
I ( x ) = ½ g ( 1 ) - ½ g ( 1 ) · ( i π t ) - 1 / 2 - ½ g ( - 1 ) exp ( - 4 i t ) Q ( 2 t 1 / 2 ) + O ( t - 1 ) .
I ( x ) = ½ { g ( 1 ) exp [ - i t ( 1 - y 0 ) 2 ] Q ( 1 - y 0 t 1 / 2 ) + g ( - 1 ) × exp [ - i t ( 1 + y 0 ) 2 ] Q ( 1 + y 0 t 1 / 2 ) } + ½ { g ( 1 ) × exp [ - i t ( 1 - y 0 ) 2 ] R ( 1 - y 0 t 1 / 2 ) } + g ( - 1 ) × exp [ - i t ( 1 + y 0 ) 2 ] R ( 1 - y 0 t / 1 / 2 ) } . ( i π t ) - 1 / 2 + O ( t - 1 ) .
I n ( x ) = ½ ( 1 + η ) a n - ½ E n + 1 ( x ) f n [ r n ( x ) ] - ½ a n - 1 / 2 { η + E n ( 1 ) f n ( 1 ) E 1 ( - x ) Q [ a n - 1 / 2 ( d + - x / M ) t 1 / 2 ] + η - E n ( 1 ) f n ( 1 ) E 1 ( - x ) Q [ a n - 1 / 2 ( d - + x / M ) t 1 / 2 ] } - a n - 1 / 2 × [ E n ( 1 ) f n ( 1 ) E 1 ( - x ) · R ( d + - x / M t 1 / 2 ) + E n ( 1 ) f n ( 1 ) × E 1 ( - x ) · R ( d - + x / M t 1 / 2 ) ] · ( i π t ) - 1 / 2 ,
u 0 ( x ) = ( 1 + x / M ) t 1 / 2 , u n ± ( x ) = a n - 1 / 2 ( d n ± + x / M ) t 1 / 2 ,
f n ( x ) = λ - n + 1 M n - 1 - 1 / 2 f 1 [ s n - 1 ( x ) ] ,
f 1 ( x ) = - ½ ( λ - 1 Q [ u 0 ( x ) ] + n = 1 N M n - 1 - 1 / 2 a n - 1 / 2 λ - n · { E n ( - 1 ) f 1 [ s n - 1 ( - 1 ) ] · Q * [ u n - ( x ) ] + E n ( 1 ) f 1 [ s n - 1 ( 1 ) ] · Q * [ u n + ( x ) ] } ) ,
Q * ( x ) = Q ( x ) + ( ixt ) - 1 / 2 M n - 1 M n - 1 · f 1 [ s n - 1 ( - 1 ) ] f 1 [ s n - 1 ( - 1 ) ] R ( x ) .
f 1 ( x ) = f 1 ( 0 ) B ( x ) Q [ ( 1 + x / M ) t 1 / 2 ] Q ( t 1 / 2 ) ,
f 1 ( 0 ) = λ - 1 / 2 ( t 1 / 2 ) [ 1 + 1 2 n = 1 N A n * λ - n ] - 1 ,
A n * = ( E n ( - 1 ) B [ s n - 1 ( - 1 ) ] Q * [ u n - ( 0 ) ] Q { u 0 [ s n - 1 ( - 1 ) ] } / Q ( t 1 / 2 ) + E n ( 1 ) B [ s n - 1 ( 1 ) ] Q * [ u n + ( 0 ) ] · Q { u 0 [ s n - 1 ( 1 ) ] } / Q ( t 1 / 2 ) ) × ( M n - 1 a n ) - 1 / 2 ,
B ( x ) = 1 + N λ - n c n ( x ) 1 + n = 1 λ - n c n ( 0 ) ,
c n ( x ) = ( M n - 1 a n ) - 1 / 2 { E n ( - 1 ) f 1 [ s n - 1 ( - 1 ) ] · Q * [ u n - ( x ) ] / Q [ u 0 ( x ) ] + E n ( 1 ) f 1 [ s n - 1 ( 1 ) ] · Q * [ u n + ( x ) ] / Q [ u 0 ( x ) ] } .
λ N + 1 + ( A 1 * / 2 - 1 ) λ N + ( A 2 * - A 1 * ) / 2 λ N - 1 + + ( A N + 1 * - A N * ) / 2 = 0.
g out ( x ) = ( i t / π ) 1 / 2 - 1 1 exp [ - i t ( y - x ) 2 ] g ( y ) d y ,

Metrics