Abstract

It is shown numerically that the diffractive transverse (Fox–Li) eigenmodes supported by an unstable cavity with tilted end mirrors can be computed by expanding these modes in terms of the fully aligned (aberration-free) eigenmodes of the same cavity. Circular mirror resonators are considered in which the aligned cavity eigenmodes can be decomposed into different azimuthal components. The biorthogonality property of the aligned cavity eigenmodes is used to obtain the coefficients in the modal expansion of the misaligned modes. Results are given for two different resonators: a conventional hard-edge unstable cavity with a small tilt of the output coupler and one that uses a graded reflectivity output mirror with a small tilt of the primary mirror. It is shown that the series expansion of the misaligned modes in terms of the aligned modes converges, and the converged eigenvalues are virtually identical to those computed by using the Prony method. Symmetry considerations and other new insights into the effects of a mirror tilt on the modes of a resonator are also discussed.

© 1992 Optical Society of America

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References

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    [CrossRef]
  5. M. E. Smithers, “Transverse-mode control in unstable optical resonators,” J. Opt. Soc. Am. 73, 1894 (1983).
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    [CrossRef]
  7. A. N. Chester, “Mode selectivity and mirror misalignment effects in unstable laser resonators,” Appl. Opt. 11, 2584–2590 (1972).
    [CrossRef] [PubMed]
  8. A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
    [CrossRef]
  9. A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 1729–2736 (1970).
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    [CrossRef] [PubMed]
  13. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 21.
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    [CrossRef]
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    [CrossRef]
  16. A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
    [CrossRef]
  17. B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
    [CrossRef]
  18. M. S. Bowers, S. E. Moody, “Numerical solution of the exact cavity equations of motion for an unstable optical resonator,” Appl. Opt. 29, 3905–3915 (1990).
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    [CrossRef] [PubMed]

1990 (1)

1989 (1)

1988 (1)

R. Hauck, N. Hodgson, H. Weber, “Misalignment sensitivity of unstable resonators with spherical mirrors,” J. Mod. Opt. 35, 165–176 (1988).
[CrossRef]

1986 (1)

1983 (2)

1981 (1)

1980 (1)

1979 (1)

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

1978 (1)

1976 (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

1975 (1)

1972 (1)

1970 (1)

A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 1729–2736 (1970).
[CrossRef]

1969 (1)

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

1963 (1)

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Bernabe, M. L.

Bowers, M. S.

Boyle, J. M.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Chester, A. N.

Dente, G. G.

Dongarra, J. J.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Garbow, B. S.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Hauck, R.

R. Hauck, N. Hodgson, H. Weber, “Misalignment sensitivity of unstable resonators with spherical mirrors,” J. Mod. Opt. 35, 165–176 (1988).
[CrossRef]

Hodgson, N.

R. Hauck, N. Hodgson, H. Weber, “Misalignment sensitivity of unstable resonators with spherical mirrors,” J. Mod. Opt. 35, 165–176 (1988).
[CrossRef]

Ikebe, Y.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Klema, V. C.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Krupke, W. F.

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Latham, W. P.

Li, T.

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

McCarthy, N.

Miller, H. Y.

A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 1729–2736 (1970).
[CrossRef]

Moler, C. B.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Moody, S. E.

Morin, M.

Murphy, W. D.

Oughstun, K. E.

Siegman, A. E.

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14, 1874–1889 (1975).
[CrossRef] [PubMed]

A. E. Siegman, H. Y. Miller, “Unstable optical resonator loss calculations using the Prony method,” Appl. Opt. 9, 1729–2736 (1970).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 21.

Smith, B. T.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

Smithers, M. E.

M. E. Smithers, “Transverse-mode control in unstable optical resonators,” J. Opt. Soc. Am. 73, 1894 (1983).

Sooy, W. R.

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

Sziklas, E. A.

Weber, H.

R. Hauck, N. Hodgson, H. Weber, “Misalignment sensitivity of unstable resonators with spherical mirrors,” J. Mod. Opt. 35, 165–176 (1988).
[CrossRef]

Appl. Opt. (7)

IEEE J. Quantum Electron. (2)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[CrossRef]

W. F. Krupke, W. R. Sooy, “Properties of an unstable confocal resonator CO2 laser system,” IEEE J. Quantum Electron. QE-5, 575–586 (1969).
[CrossRef]

J. Mod. Opt. (1)

R. Hauck, N. Hodgson, H. Weber, “Misalignment sensitivity of unstable resonators with spherical mirrors,” J. Mod. Opt. 35, 165–176 (1988).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

A. E. Siegman, “Orthogonality properties of optical resonator eigenmodes,” Opt. Commun. 31, 369–373 (1979).
[CrossRef]

Proc. IEEE (1)

A. G. Fox, T. Li, “Modes in a maser interferometer with curved and tilted mirrors,” Proc. IEEE 51, 80–89 (1963).
[CrossRef]

Other (3)

K. E. Oughstun, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 165–387.
[CrossRef]

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 21.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines—EISPACK Guide (Springer-Verlag, New York, 1976).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

a) Standing-wave aligned unstable cavity, b) model of the aligned cavity characterized by a single diffractive aperture (the output mirror) and by a single paraxial optical system.

Fig. 2
Fig. 2

a) Unstable resonator with the primary mirror tilted by an angle α from its aligned position, b) the equivalent lens guide system. The mirror tilt is modeled as a phase wedge.

Fig. 3
Fig. 3

Eigenvalue magnitudes as a function of the primary mirror tilt angle for the graded reflectivity unstable cavity computed by using the present mode expansion method. Also shown are the symmetry of the expansion coefficients and the results from the Prony method. The effect of the mirror tilt is to destroy the azimuthal symmetry of the cavity and to split the l = ±1 modes as shown.

Fig. 4
Fig. 4

Intracavity intensity distribution incident upon the output mirror for the first three dominant modes with a 40-μrad tilt of the primary mirror for the resonator described in Table I.

Fig. 5
Fig. 5

Intensity distribution of the first two dominant modes for primary mirror tilts of 0, 10, 20, and 30 μrad for the resonator described in Table I.

Fig. 6
Fig. 6

Eigenvalue magnitudes as a function of the output mirror tilt angle for the hard-edged unstable cavity listed in Table IV computed by using the present mode-expansion method. Also shown are the symmetry of the expansion coefficients and the results from the Prony method.

Fig. 7
Fig. 7

Absolute values of the expansion coefficients for the dominant mode with the output mirror tilted 80 μrad for the hard-edged unstable resonator listed in Table IV.

Fig. 8
Fig. 8

Same as Fig. 7 for the second mode.

Fig. 9
Fig. 9

Absolute values of the expansion coefficients for the dominant mode with the output mirror tilted 100 μrad for the hard-edged unstable resonator listed in Table IV.

Fig. 10
Fig. 10

Same as Fig. 9 for the second mode.

Fig. 11
Fig. 11

Intracavity intensity distribution incident upon the output mirror for the first five dominant modes with a 100-μrad tilt of the output mirror for the hard-edged unstable resonator described in Table IV. The expansion coefficients for the first three modes and the fifth mode obey the symmetry relation Alp = (−1)lAlp and the fourth mode obeys A0p = 0, Alp = (−1)l+1Alp.

Tables (6)

Tables Icon

Table I Parameters of the Optical Resonator Used for Computing the Modes for a Tilt of the Primary Mirror

Tables Icon

Table II First Five Eigenvalues Computed from the Present Elgenmode Expansion Method for a Primary Mirror Tilt of 40 μrad as a Function of Basis Set Size for the Optical Resonator Described in Table I

Tables Icon

Table III Comparison of the Eigenvalues Versus Various Primary Mirror Tilt Angles Computed from the Present Eigenmode Expansion Method to Those Computed from the Prony Method for the Resonator Described in Table I

Tables Icon

Table IV Parameters for the Optical Resonator Used for Computing the Modes for a Tilt of the Output Mirror

Tables Icon

Table V First Five Eigenvalues Computed from the Present Eigenmode Expansion Method for an Output Mirror Tilt of 100 μrad as a Function of Basis Set Size for the Optical Resonator Described in Table IV

Tables Icon

Table VI Comparison of the Eigenvalues Versus Various Output Mirror Tilt Angles Computed from the Present Eigenmode Expansion Method to Those Computed by Using the Prony Method for the Resonator Described in Table IV

Equations (15)

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λ n ϕ n ( r ) = d r K ( r , r ) ϕ n ( r ) ,
γ l p u l p ( r ) = k i l + 1 B 0 a r d r ρ ( r ) u l p ( r ) J l ( k r r B ) × exp [ i k 2 B ( A r 2 + D r 2 ) ] ,
υ l p ( r ) = β l p ρ ( r ) u l p ( r ) exp [ i k 2 B ( A D ) r 2 ] ,
0 2 π d θ exp [ i ( l l ) θ ] 0 a r d r υ l p ( r ) u l p ( r ) = 2 πδ l l 0 a r d r ρ ( r ) u l p ( r ) u l p ( r ) exp [ i k 2 B ( A D ) r 2 ] = δ l l ' δ p p ' ,
ϕ n ( r ) = l = p = 0 A l p n u l p ( r ) exp ( i l θ ) .
λ n A l ' p ' n = l = p = 0 K l p , l p A l p n ,
K l p , l p = d r d r υ l p ( r ) exp ( i l θ ) K ( r , r ) u l p ( r ) exp ( i l θ ) .
K ( r , r ) = i k 2 π B ρ ( r ) exp ( 2 i k α r sin θ ) × exp { i k 2 B [ A r 2 + D r 2 2 r r cos ( θ θ ) ] } .
exp ( i x sin θ ) = m = exp ( i m θ ) J m ( x ) , exp ( i x cos θ ) = m = i m exp ( i m θ ) J m ( x ) .
K l p , l p = 2 πγ l p 0 a r dr ρ ( r ) u l p ( r ) u l p ( r ) × exp [ i k 2 B ( A D ) r 2 ] J l l ( 2 k α r ) .
K ( r , r ) = i k 2 π B ρ ( r ) exp { i k 2 B [ A r 2 + D r 2 2 r r cos ( θ θ ) + 4 α 2 L 2 4 α L ( r sin θ + r sin θ ) ] .
K l p , l p = 2 π i k B exp ( 2 i k α 2 L 2 B ) 0 a r d r ρ ( r ) u l p ( r ) × exp ( i k 2 B A r 2 ) 0 a r d r ρ ( r ) u l p ( r ) exp [ i k 2 B A r 2 ] × m = i m J l m ( 2 k α L r B ) J m l ( 2 k α L r B ) J m ( k r r B ) .
K l p , l p = ( 1 ) l + l K l p , l p , K l p , l p = ( 1 ) l + l K l p , l p .
A l p n = ( 1 ) l A l p n
A 0 p n = 0 , A l p n = ( 1 ) l + 1 A l p n .

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