Abstract

A method based on moiré deflectometry for direct determination of the number of transverse modes of radiation of a light beam is presented. An expression for the number of transverse modes of a light beam, as a function of the beam divergence, is derived. We demonstrate the method for a Cu-vapor laser-beam analysis.

© 1983 Optical Society of America

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References

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  1. L. D. Landau, E. M. Lifshitz, Statistical Physics (Pergamon, London, 1969), p. 145.
  2. S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, 1969), p. 329.
  3. This number of radiation modes should not be confused with the so-called cavity longitudinal modes, which represent the number of cavity eigenfrequencies.
  4. D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
    [CrossRef]
  5. C. R. Prasad et al., Rev. Sci. Instrum. 50, 1161 (1979).
    [CrossRef] [PubMed]
  6. S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
    [CrossRef]
  7. S. Lavi et al., Appl. Opt. 20, 1145 (1981).
    [CrossRef] [PubMed]
  8. O. Kafri, Opt. Lett. 5, 555 (1980); Phys Bull. 33, 197 (1982).
    [CrossRef] [PubMed]
  9. J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1981), p. 19.
  10. A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).
  11. Z. Karny, O. Kafri, Appl. Opt. 21, 3326 (1982).
    [CrossRef] [PubMed]

1982 (1)

1981 (1)

1980 (2)

O. Kafri, Opt. Lett. 5, 555 (1980); Phys Bull. 33, 197 (1982).
[CrossRef] [PubMed]

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

1979 (1)

C. R. Prasad et al., Rev. Sci. Instrum. 50, 1161 (1979).
[CrossRef] [PubMed]

1978 (1)

D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
[CrossRef]

Cowley, J. M.

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1981), p. 19.

Durelli, A. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Kafri, O.

Karny, Z.

Klein, M. P.

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Korobkim, V. V.

D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
[CrossRef]

Landau, L. D.

L. D. Landau, E. M. Lifshitz, Statistical Physics (Pergamon, London, 1969), p. 145.

Lavi, S.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, Statistical Physics (Pergamon, London, 1969), p. 145.

Lipson, H.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, 1969), p. 329.

Lipson, S. G.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, 1969), p. 329.

Parks, V. J.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

Prasad, C. R.

C. R. Prasad et al., Rev. Sci. Instrum. 50, 1161 (1979).
[CrossRef] [PubMed]

Serov, P. V.

D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
[CrossRef]

Sorscher, S. M.

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Vlasov, D. V.

D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
[CrossRef]

Appl. Opt. (2)

Opt. Lett. (1)

Rev. Sci. Instrum. (2)

C. R. Prasad et al., Rev. Sci. Instrum. 50, 1161 (1979).
[CrossRef] [PubMed]

S. M. Sorscher, M. P. Klein, Rev. Sci. Instrum. 51, 98 (1980).
[CrossRef]

Sov. J. Quantum Electron. (1)

D. V. Vlasov, V. V. Korobkim, P. V. Serov, Sov. J. Quantum Electron. 8, 1380 (1978).
[CrossRef]

Other (5)

L. D. Landau, E. M. Lifshitz, Statistical Physics (Pergamon, London, 1969), p. 145.

S. G. Lipson, H. Lipson, Optical Physics (Cambridge U. Press, Cambridge, 1969), p. 329.

This number of radiation modes should not be confused with the so-called cavity longitudinal modes, which represent the number of cavity eigenfrequencies.

J. M. Cowley, Diffraction Physics (North-Holland, Amsterdam, 1981), p. 19.

A. J. Durelli, V. J. Parks, Moiré Analysis of Strain (Prentice-Hall, Englewood Cliffs, N.J., 1970).

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Figures (5)

Fig. 1
Fig. 1

Schematic description of the setup, a light beam of diameter α traversing two Ronchi rulings G1 and G2. The photodetector PD is wired to a recorder for measuring the fringe contrast.

Fig. 2
Fig. 2

Ideal moiré-fringe intensity profile (triangular) smeared by the broadening that is due to a beam divergence δα.

Fig. 3
Fig. 3

(a) Densitogram of the moiré pattern produced by a beam of a Cu-vapor laser having confocal unstable resonator. λ = 0.5106 μm, Δ = 28 mm, p = 1 12 mm, a = 24 mm. (b) Same as (a), but the laser cavity has flat mirrors.

Fig. 4
Fig. 4

(a) Relative intensity and n across the beam described in Fig. 3(a). (b) Same as (a) but for the flat mirrors.

Fig. 5
Fig. 5

(a) Deflectogram of the Cu-laser beam when Δ = 84 mm. (b) Reference deflectogram taken at Δ = 0.

Equations (12)

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N = ν 2 c 3 δ ν δ Ω V ,
m = δ ν δ t ;
n = δ Ω / ( λ 2 / a 2 ) ;
N = 2 n m .
p = p 2 sin ( θ / 2 ) p θ .
h α Δ / θ .
δ α δ Ω 2 n λ a .
δ h = 2 n λ Δ / a θ .
c = ( I i ) / i ,
i = I Δ δ α 2 p = I n λ Δ p a .
n = ( p a λ Δ 1 1 + c ) 2 .
n = n i I j / I j .

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