Abstract

A new computational method for unloaded optical resonators is developed based on the discrete Fourier analysis of information generated by repeated iterations of the optical field corresponding to transits between reflectors. The method is a straightforward extension of the propagating beam method developed earlier for optical fibers for extracting modal properties from numerical solutions to the paraxial scalar wave equation. The method requires computation of a field correlation function, whose Fourier transform reveals the eigenmodes as resonant peaks. Analysis of the location and breadth of these peaks determines the resonator eigenvalues. When the eigenvalues are known, additional discrete Fourier transforms of the field are used to generate the mode eigenfunctions. This new method makes possible the unambiguous identification and accurate characterization of the entire spectrum of transverse resonator modes.

© 1981 Optical Society of America

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References

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  1. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).
  2. H. K. V. Lotsch, Z. Naturforsch. Teil A: 20, 38 (1965).
  3. W. Streifer, J. Opt. Soc. Am. 55, 868 (1965).
    [CrossRef]
  4. P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
    [CrossRef]
  5. A. E. Siegman, R. Arrathoon, IEEE J. Quantum Electron. QE-3, 156 (1967).
    [CrossRef]
  6. R. L. Sanderson, W. Streifer, Appl. Opt. 8, 131 (1969).
    [CrossRef] [PubMed]
  7. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  8. E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
    [CrossRef] [PubMed]
  9. W. D. Murphy, M. L. Bernabe, Appl. Opt. 17, 2358 (1978).
    [CrossRef] [PubMed]
  10. A. E. Siegman, Ginzton Laboratory Report 3111 (1980).
  11. B. D. O'Neil, V. A. Hedin, J. L. Forgham, Air Force Weapons Laboratory Report AFWL-TR-78-15 (1978), p. 60.
  12. C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 458.
  13. M. D. Feit, J. A. Fleck, Appl. Opt. 17, 3990 (1978).
    [CrossRef] [PubMed]
  14. M. D. Feit, J. A. Fleck, Appl. Opt. 19, 1154 (1980).
    [CrossRef] [PubMed]
  15. M. D. Feit, J. A. Fleck, Appl. Opt. 19, 2240 (1980).
    [CrossRef] [PubMed]
  16. M. D. Feit, J. A. Fleck, Appl. Opt. 19, 3140 (1980).
    [CrossRef] [PubMed]
  17. M. D. Feit, J. A. Fleck, Appl. Opt. 20, 848 (1981).
    [CrossRef] [PubMed]
  18. The notation used in this paper is designed to maintain consistency with earlier Refs. 13–17.
  19. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  20. P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).
  21. Gibbs phenomena are known to result from truncating an infinite Fourier series that represents a discontinuous function. Fourier coefficients of the remaining untruncated terms, however, are not the discrete Fourier transform coefficients that would be computed from a finite number of values sampled from the same function. The latter would at least allow the correct function values to be recovered at the sampling points. For numerical comparisons of the effects of aperture apodization see Ref. 19.

1981

1980

1978

1976

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1975

1970

1969

1967

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

1966

P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
[CrossRef]

1965

H. K. V. Lotsch, Z. Naturforsch. Teil A: 20, 38 (1965).

W. Streifer, J. Opt. Soc. Am. 55, 868 (1965).
[CrossRef]

1961

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Arrathoon, R.

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

Bernabe, M. L.

Bevington, P. R.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Checcacci, P. F.

P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
[CrossRef]

Consortini, A.

P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
[CrossRef]

Feit, M. D.

Fleck, J. A.

Forgham, J. L.

B. D. O'Neil, V. A. Hedin, J. L. Forgham, Air Force Weapons Laboratory Report AFWL-TR-78-15 (1978), p. 60.

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Hedin, V. A.

B. D. O'Neil, V. A. Hedin, J. L. Forgham, Air Force Weapons Laboratory Report AFWL-TR-78-15 (1978), p. 60.

Li, T.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Lotsch, H. K. V.

H. K. V. Lotsch, Z. Naturforsch. Teil A: 20, 38 (1965).

Miller, H. Y.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Murphy, W. D.

O'Neil, B. D.

B. D. O'Neil, V. A. Hedin, J. L. Forgham, Air Force Weapons Laboratory Report AFWL-TR-78-15 (1978), p. 60.

Sanderson, R. L.

Scheggi, O.

P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
[CrossRef]

Siegman, A. E.

E. A. Sziklas, A. E. Siegman, Appl. Opt. 14, 1874 (1975).
[CrossRef] [PubMed]

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

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

A. E. Siegman, Ginzton Laboratory Report 3111 (1980).

Streifer, W.

Sziklas, E. A.

Wylie, C. R.

C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 458.

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Bell Syst. Tech. J.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

IEEE J. Quantum Electron.

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

J. Opt. Soc. Am.

Proc. IEEE

P. F. Checcacci, A. Consortini, O. Scheggi, Proc. IEEE 54, 1329 (1966).
[CrossRef]

Z. Naturforsch. Teil A

H. K. V. Lotsch, Z. Naturforsch. Teil A: 20, 38 (1965).

Other

A. E. Siegman, Ginzton Laboratory Report 3111 (1980).

B. D. O'Neil, V. A. Hedin, J. L. Forgham, Air Force Weapons Laboratory Report AFWL-TR-78-15 (1978), p. 60.

C. R. Wylie, Advanced Engineering Mathematics (McGraw-Hill, New York, 1966), p. 458.

P. R. Bevington, Data Reduction and Error Analysis for the Physical Sciences (McGraw-Hill, New York, 1969).

Gibbs phenomena are known to result from truncating an infinite Fourier series that represents a discontinuous function. Fourier coefficients of the remaining untruncated terms, however, are not the discrete Fourier transform coefficients that would be computed from a finite number of values sampled from the same function. The latter would at least allow the correct function values to be recovered at the sampling points. For numerical comparisons of the effects of aperture apodization see Ref. 19.

The notation used in this paper is designed to maintain consistency with earlier Refs. 13–17.

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Figures (9)

Fig. 1
Fig. 1

Both an optical fiber waveguide and an optical resonator can be viewed as periodic arrays of lenses. The field in an optical fiber is observed in planes midway between the equivalent lenses, whereas the field of an optical resonator is customarily observed in a plane passing through the equivalent lenses.

Fig. 2
Fig. 2

Intensity profile for random input field.

Fig. 3
Fig. 3

Mode spectrum for 1-D strip resonator excited by random input field showing resonances corresponding to fifteen modes.

Fig. 4
Fig. 4

Comparison between eigenfunctions generated by Fox-Li and spectral method for (a) lowest order symmetric mode and (b) lowest order antisymmetric mode.

Fig. 5
Fig. 5

Additional mode eigenfunctions for strip resonator generated by spectral method.

Fig. 6
Fig. 6

Mode spectra generated by exciting resonator with individual spectrally generated eigenfunctions, showing contributions due to small amounts of other modes present.

Fig. 7
Fig. 7

Mode spectra for resonator with flat circular reflectors, generated for different angular symmetries with input field proportional to cos .

Fig. 8
Fig. 8

Plot of complex propagation constants for modes of resonator with flat circular reflectors. Numerals designate the corresponding angular mode index v.

Fig. 9
Fig. 9

Sample radial mode eigenfunctions f μ ν ( r ) generated by spectral method for resonator with flat circular reflectors.

Tables (4)

Tables Icon

Table I Complex Propagation Constants, Phase Shift, and Attenuation per Transit for 1-D Strip Resonator, Comouted from Soectrum

Tables Icon

Table II Comparison of Propagation Constants for 1-D Strip Resonator Computed by Fox-Li and Spectral Method

Tables Icon

Table III Magnitudes of the Orthogonality Integrals 〈un(x)un′ (x)〉 for the First Eight Spectrally Generated Eiaenfunctions of a Planar Resonator

Tables Icon

Table IV Complex Propagation Constants for Resonator with Flat Circular Reflectors

Equations (44)

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Mu = γ u ,
2 i k z = 2 x 2 + 2 y 2 ,
f ( t ) = i = 1 N K i exp ( p i t ) ,
F n = v n v 2 N , n = 1 , 2 , , 2 N ,
P 1 ( z ) = ( x , y , 0 ) ( x , y , z ) ,
( x , y , Re β n ) = 0 Z ( x , y , z ) w ( z ) exp ( i Re β n z ) d z ,
E ( ω , x , y , z ) = ( ω , x , y , z ) exp ( i k z ) ,
2 i k z = 2 + k 2 { [ n ( x , y ) n 0 ] 2 1 } ,
( x , y , z + Δ z ) = exp ( i Δ z 4 k 2 ) exp ( i χ ) × exp ( i Δ z 4 k 2 ) ( x , y , z ) ,
χ ( x , y ) = k 2 { [ n ( x , y ) n 0 ] 2 1 } Δ z ,
2 = 2 x 2 + 2 y 2 .
exp ( i Δ z 4 k 2 )
( x , y , z ) = m = N / 2 + 1 N / 2 n = N / 2 + 1 N / 2 m n ( z ) exp [ 2 π i L 0 ( m x + n y ) ] ,
m n ( z + Δ z 2 ) = m n ( z ) exp [ i Δ z 4 k ( 2 π L 0 ) 2 ( m 2 + n 2 ) ] ,
( x , y , z + Δ z 2 ) = exp ( i Δ z 2 k 2 ) exp ( i χ ) ( x , y , z Δ z 2 ) ,
( x , y , z ) = n , j A n j u n j ( x , y ) exp ( i β n z ) ,
γ n = exp ( i β n L ) .
Δ β max = π / Δ z .
Δ β min = π / Z ,
P 1 ( z ) = ( x , y , 0 ) ( x , y , z ) = ( x , y , 0 ) ( x , y , z ) d x d y ,
P 1 ( z ) = n , j A n j 2 exp ( i β n z ) .
w ( z ) = { 1 cos 2 π z Z , 0 z Z , 0 , z > Z ,
P 1 ( β ) = n , j A n j 2 1 ( β β n ) = n W n 1 ( β β n ) ,
W n = j A n j 2 ,
1 ( β β n ) = exp [ i ( β β n ) Z ] 1 i ( β β n ) Z ½ ( exp { i [ ( β β n ) Z + 2 π ] } 1 i [ ( β β n ) Z + 2 π ] + exp { i [ ( β β n ) Z 2 π ] } 1 i [ ( β β n ) Z 2 π ] ) .
P 1 ( β ) = W n 1 ( β β n ) .
( x , y , β ) = 1 Z 0 Z ( x , y , z ) w ( z ) exp ( i β z ) d z = n , j A n j u n j ( x , y ) 1 ( β β n ) .
( x , y , β r n ) = j A n j u n j ( x , y ) 1 ( β r n β n ) + n , j A n j u n j ( x , y ) 1 ( β r n β n ) ,
( x , y , β n ) = j A n j u n j ( x , y ) 1 ( β r n β n ) .
u n ( x , y ) = const × 0 Z ( x , y , z ) w ( z ) exp ( i β n z ) d z = const × ( x , y , β n ) .
( x , y ) = u ( x , y ) ,
u ( x , y ) = 1 ( x , y ) .
( x , y , 0 ) = F ( r ) [ r ν { cos ν θ sin ν θ } + r ν + 1 { cos ( ν + 1 ) θ sin ( ν + 1 ) θ } ] ,
u μ ν ( r , θ ) = { cos ν θ sin ν θ } f μ ν ( r ) ,
f μ ν ( r ) = const × 0 Z ( x , 0 , z ) w ( z ) exp ( i β μ ν z ) d z = const × ( x , 0 , β μ ν ) ,
u n ( x ) = const × 0 Z ( x , y ) w ( z ) exp ( i β n z ) d z .
β r ( cm 1 )
β i ( cm 1 )
β r ( cm 1 )
β i ( cm 1 )
β r ( cm 1 )
β i ( cm 1 )
β r ( cm 1 )
β i ( cm 1 )

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