Abstract

We introduce an algorithm to determine the relative modal weights of a laser beam with Hermite–Gaussian modes. The method is based on coherence theory of stable resonator modes in the space–frequency domain. Numerical simulations are presented that show that the algorithm is not sensitive to moderate levels of noise.

© 1989 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  2. Yu. Lyabavsky, V. Ovchinnikov, Solid-State Laser Technology (MIR, Moscow, 1975).
  3. P. W. Smith, Proc. IEEE 60, 422 (1972).
    [CrossRef]
  4. K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
    [CrossRef]
  5. J. P. Goldsborough, Appl. Opt. 3, 267 (1964).
    [CrossRef]
  6. T. H. Zachos, J. E. Ripper, IEEE J. Quantum Electron. QE-5, 29 (1969).
    [CrossRef]
  7. P. Spano, Opt. Commun. 33, 265 (1980).
    [CrossRef]
  8. E. G. Johnson, Appl. Opt. 25, 2967 (1986).
    [CrossRef] [PubMed]
  9. F. Gori, Opt. Commun. 34, 301 (1980).
    [CrossRef]
  10. A. Starikov, E. Wolf, J. Opt. Soc. Am. 72, 923 (1982).
    [CrossRef]
  11. H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
    [CrossRef]
  12. S. Lavi, R. Prochaska, E. Keren, Appl. Opt. 27, 3696 (1988).
    [CrossRef] [PubMed]
  13. E. Wolf, G. S. Agarwal, J. Opt. Soc. Am. A 1, 541 (1984).
    [CrossRef]
  14. L. Mandel, E. Wolf, J. Opt. Soc. Am. 66, 529 (1976).
    [CrossRef]
  15. The approximations made here are consistent with the physical model, in which the multimode wave field itself is expanded as U(x, z, ω) = Σnan(ω)ψn(x, z, ω). In this equation the coefficients an(ω) are random, uncorrelated (they represent slightly different frequency components), and satisfy 〈an*(ω)am(ω)〉 = λn(ω)δnm, where the angular brackets denote ensemble averaging. Definition of the cross-spectral density W(x1, x2, z; ω) = 〈U*(x1, z, ω)U(x2, z, ω)〉 then leads to relation (3) of the text.
  16. A. Messiah, Quantum Mechanics(Wiley, New York, 1958), Vol. I, Appendix B.
  17. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  18. A. T. Friberg, J. Turunen, J. Opt. Soc. Am. A 5, 713 (1988).
    [CrossRef]

1988

1986

1984

1982

1980

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
[CrossRef]

1978

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

1976

1972

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

1969

T. H. Zachos, J. E. Ripper, IEEE J. Quantum Electron. QE-5, 29 (1969).
[CrossRef]

1966

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1964

Agarwal, G. S.

Aiki, K.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Chinone, N.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Eichler, H. J.

H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
[CrossRef]

Enterlein, G.

H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
[CrossRef]

Friberg, A. T.

Goldsborough, J. P.

Gori, F.

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Ito, R.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Johnson, E. G.

Keren, E.

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Kuroda, T.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Langhans, D.

H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
[CrossRef]

Lavi, S.

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lyabavsky, Yu.

Yu. Lyabavsky, V. Ovchinnikov, Solid-State Laser Technology (MIR, Moscow, 1975).

Maeda, M.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Mandel, L.

Messiah, A.

A. Messiah, Quantum Mechanics(Wiley, New York, 1958), Vol. I, Appendix B.

Nakamura, M.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Ovchinnikov, V.

Yu. Lyabavsky, V. Ovchinnikov, Solid-State Laser Technology (MIR, Moscow, 1975).

Prochaska, R.

Ripper, J. E.

T. H. Zachos, J. E. Ripper, IEEE J. Quantum Electron. QE-5, 29 (1969).
[CrossRef]

Siegman, A. E.

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

Smith, P. W.

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

Spano, P.

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

Starikov, A.

Turunen, J.

Umeda, J.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

Wolf, E.

Zachos, T. H.

T. H. Zachos, J. E. Ripper, IEEE J. Quantum Electron. QE-5, 29 (1969).
[CrossRef]

Appl. Opt.

Appl. Phys.

H. J. Eichler, G. Enterlein, D. Langhans, Appl. Phys. 23, 299 (1980).
[CrossRef]

IEEE J. Quantum Electron.

K. Aiki, M. Nakamura, T. Kuroda, J. Umeda, R. Ito, N. Chinone, M. Maeda, IEEE J. Quantum Electron. QE-14, 89 (1978).
[CrossRef]

T. H. Zachos, J. E. Ripper, IEEE J. Quantum Electron. QE-5, 29 (1969).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

P. Spano, Opt. Commun. 33, 265 (1980).
[CrossRef]

F. Gori, Opt. Commun. 34, 301 (1980).
[CrossRef]

Proc. IEEE

P. W. Smith, Proc. IEEE 60, 422 (1972).
[CrossRef]

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Other

The approximations made here are consistent with the physical model, in which the multimode wave field itself is expanded as U(x, z, ω) = Σnan(ω)ψn(x, z, ω). In this equation the coefficients an(ω) are random, uncorrelated (they represent slightly different frequency components), and satisfy 〈an*(ω)am(ω)〉 = λn(ω)δnm, where the angular brackets denote ensemble averaging. Definition of the cross-spectral density W(x1, x2, z; ω) = 〈U*(x1, z, ω)U(x2, z, ω)〉 then leads to relation (3) of the text.

A. Messiah, Quantum Mechanics(Wiley, New York, 1958), Vol. I, Appendix B.

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

Yu. Lyabavsky, V. Ovchinnikov, Solid-State Laser Technology (MIR, Moscow, 1975).

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Figures (1)

Fig. 1
Fig. 1

Numerical simulations demonstrating the accuracy of the modal strengths calculated by the proposed algorithm in the presence of noise.

Equations (14)

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W n ( x 1 , x 2 ; ω ) = λ n ( ω ) ψ n * ( x 1 , ω ) ψ n ( x 2 , ω ) ,
A L ( x 1 , x 2 ; ω ) ψ n ( x 1 ; ω ) d x 1 = α n ( ω ) ψ n ( x 2 , ω ) .
W ( x 1 , x 2 , z ; ω ) n λ n ( ω ) ψ n * ( x 1 , z , ω ) ψ n ( x 2 , z , ω ) .
W ( x 1 , x 2 , z ; ω ) ψ n ( x 1 , z , ω ) d x 1 = λ n ( ω ) ψ n ( x 1 , z , ω ) .
ϕ n ( x , z ) = [ 2 π w 2 ( z ) ] 1 / 4 1 2 n n ! H n [ 2 x w ( z ) ] exp [ x 2 w 2 ( z ) ] ,
α n ( x , z ) = k z ( n + 1 / 2 ) arctan ( λ z / π w 0 2 ) k x 2 / 2 R ( z ) ,
W ( x 1 , x 2 ) = W R ( x 1 , x 2 ) exp [ i k ( x 1 2 , x 2 2 ) / 2 R ] ,
λ n = [ ϕ n ( x 2 ) ] 1 W R ( x 1 , x 2 ) ϕ n ( x 1 ) d x 1 .
[ ϕ n ( x 01 , w ) ] 1 W R ( x , x 01 ) ϕ n ( x , w ) d x = [ ϕ n ( x 02 , w ) ] 1 W R ( x , x 02 ) ϕ n ( x , w ) d x ,
W R ( x , x 0 ) = exp [ ( x 2 + x 0 2 ) / 2 σ I 2 ] exp [ ( x x 0 ) 2 / 2 σ g 2 ] ,
W R ( x , x 0 ) = exp [ ( x 2 + x 0 2 ) / 2 σ I 2 ] ,
V P ( ω ) = 2 [ I 1 ( ω ) I 2 ( ω ) ] 1 / 2 I 1 ( ω ) + I 2 ( ω ) | μ ( x 1 , x 2 , ω ) | .
V ( ω ) = 2 t 1 t 2 t 1 I ( x 1 , ω ) + t 2 I ( x 2 , ω ) | W ( x 1 , x 2 , ω ) | .
W R ( x 1 , x 2 , ω ) = ½ sgn ( W R ) [ t 1 / t 2 ] 1 / 2 I ( x 1 , ω ) + ( t 2 / t 1 ) 1 / 2 ( x 2 , ω ) V ( ω ) .

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