Abstract

We present analytical solutions valid for large Fresnel number of the Fresnel–Kirchhoff integral equation for marginally stable resonators, for the specific case of flat circular mirrors. The asymptotic approaches used for curved mirrors have been extended to the waveguide region given by m<1+1/N. The resonator modes are expressed in terms of a slowly varying core term similar in form to the electromagnetic fields of a closed resonator and a small, rapidly oscillating term arising from diffraction around the mirror edge.

© 1979 Optical Society of America

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References

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  1. R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1071–1076 (1978).
    [CrossRef]
  2. P. Horowitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1542 (1973).
    [CrossRef]
  3. L. Weinstein, Open Resonators and Open Waveguides (Golem Press, Boulder, Colorado, 1969).
  4. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  5. A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

1978 (1)

R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1071–1076 (1978).
[CrossRef]

1973 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961 (1)

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Avizonis, P. V.

R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1071–1076 (1978).
[CrossRef]

Butts, R. R.

R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1071–1076 (1978).
[CrossRef]

Fox, A.

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Horowitz, P.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

Weinstein, L.

L. Weinstein, Open Resonators and Open Waveguides (Golem Press, Boulder, Colorado, 1969).

Bell Syst. Tech. J. (1)

A. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).

J. Opt. Soc. Am. (2)

R. R. Butts, P. V. Avizonis, “Asymptotic analysis of unstable resonators with circular mirrors,” J. Opt. Soc. Am. 68, 1071–1076 (1978).
[CrossRef]

P. Horowitz, “Asymptotic theory of unstable resonator modes,” J. Opt. Soc. Am. 63, 1528–1542 (1973).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (1)

L. Weinstein, Open Resonators and Open Waveguides (Golem Press, Boulder, Colorado, 1969).

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Figures (1)

Fig. 1
Fig. 1

Comparison of calculated and numerical amplitude distributions for the four lowest-order modes. The dotted lines are calculated amplitudes, and solid lines are numerical results. Numerical results from Refs. 4 and 5.

Tables (1)

Tables Icon

Table 1 Magnitude and Phases of Eigenvalues

Equations (14)

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λ f ( x ) = 2 i l + 1 π N 0 1 y d y J l ( 2 π N x y ) × exp [ - i π N ( x 2 + y 2 ) f ( y ) ,
f ( x ) = J l ( a x ) ,             λ = exp ( i a 2 / 4 π N ) .
λ f = K f - f ( 1 ) F 1 ( x ) ,
F k ( x ) = 2 i l + 1 π N k 1 y d y J l ( 2 π N k x y ) × exp [ - i π N k ( x 2 + y 2 ) ] ,
K F k = F k + 1 .
f ( x ) = f ( 1 ) k = 1 F k ( x ) / λ k ,
m > 1 + 1 / N
k > k F k ( x ) / λ k = r = 0 c r J l ( b r l x ) .
r = 0 c r ( λ - λ r ) J l ( b r l x ) = F k + 1 ( x ) / λ k .
F k ( 1 ) ~ - 1 2 + o ( 1 / N ) ,
0 1 x J l ( b x ) F k ( x ) d x ~ - k J l ( b ) / 2 π i N .
r = 0 c r J l ( b r l ) ~ r = 0 2 i ( k + 1 ) N J l ( b r l ) J l ( b r l x ) ( a 2 - b r l 2 ) N r + o ( 1 / N ) ,
f ( x ) = - f ( 1 ) { 2 i N ( k + 1 ) J l ( a x ) / [ a J l ( a ) ] + k = 1 k F k ( x ) / λ k } ,
J l ( a ) + ( 1 + k / 2 ) a J l ( a ) / [ 2 i N ( k + 1 ) ] = 0.

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