Abstract

Within the formalism of the Wigner distribution function, a new parameter is proposed, which characterizes arbitrary tridimensional partially coherent beams and is invariant through ABCD optical systems. The relationship between such a parameter and the bidimensional concept of beam quality is analyzed. An absolute lower bound that the new parameter can reach is also shown.

© 1991 Optical Society of America

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References

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  1. W. L. Bohn, Th. Hall, “Effect of beam quality on the scaling of high-energy flow lasers,” in Gas Flow and Chemical Lasers, E. Rosenwaks, ed. (Springer, Berlin, 1987), pp. 90–95.
    [Crossref]
  2. Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
    [Crossref]
  3. S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
    [Crossref] [PubMed]
  4. A. Culoma, J. L. Paradis, D. Gerbert, “High power laser amplification,” in High Power Lasers: Sources, Laser-Material Interactions, E. W. Kreutz, A. Quenzer, D. Schücker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.801, 42–44 (1987).
    [Crossref]
  5. E. D. Rockower, “Laser beam-quality/aperture-shape scaling relation,” Appl. Opt. 25, 1394–1397 (1986).
    [Crossref] [PubMed]
  6. A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 696–697.
  7. K. E. Oughstum, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 167–387.
  8. J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.
  9. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,”J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [Crossref]
  10. R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
    [Crossref]
  11. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [Crossref]

1988 (3)

Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
[Crossref]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

S. Lavi, R. Prochaska, E. Keren, “Generalized beam parameters and transformation laws for partially coherent light,” Appl. Opt. 27, 3696–3703 (1988).
[Crossref] [PubMed]

1986 (1)

1985 (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

1979 (1)

Bastiaans, M. J.

Bohn, W. L.

W. L. Bohn, Th. Hall, “Effect of beam quality on the scaling of high-energy flow lasers,” in Gas Flow and Chemical Lasers, E. Rosenwaks, ed. (Springer, Berlin, 1987), pp. 90–95.
[Crossref]

Calvo, G.

J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.

Culoma, A.

A. Culoma, J. L. Paradis, D. Gerbert, “High power laser amplification,” in High Power Lasers: Sources, Laser-Material Interactions, E. W. Kreutz, A. Quenzer, D. Schücker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.801, 42–44 (1987).
[Crossref]

Delgado, J.

J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.

Gerbert, D.

A. Culoma, J. L. Paradis, D. Gerbert, “High power laser amplification,” in High Power Lasers: Sources, Laser-Material Interactions, E. W. Kreutz, A. Quenzer, D. Schücker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.801, 42–44 (1987).
[Crossref]

Hall, Th.

W. L. Bohn, Th. Hall, “Effect of beam quality on the scaling of high-energy flow lasers,” in Gas Flow and Chemical Lasers, E. Rosenwaks, ed. (Springer, Berlin, 1987), pp. 90–95.
[Crossref]

Hui, D.

Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
[Crossref]

Keren, E.

Kun, Y.

Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
[Crossref]

Lavi, S.

Li, Y. P.

Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
[Crossref]

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Neira, J. L. H.

J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.

Oughstum, K. E.

K. E. Oughstum, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 167–387.

Paradis, J. L.

A. Culoma, J. L. Paradis, D. Gerbert, “High power laser amplification,” in High Power Lasers: Sources, Laser-Material Interactions, E. W. Kreutz, A. Quenzer, D. Schücker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.801, 42–44 (1987).
[Crossref]

Prochaska, R.

Rockower, E. D.

Sanchez, M.

J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.

Siegman, A. E.

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 696–697.

Simon, R.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
[Crossref]

Opt. Commun. (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD-law,” Opt. Commun. 65, 322–328 (1988).
[Crossref]

Y. P. Li, D. Hui, Y. Kun, “Design of phase plates for changing the wavefront of lasers,” Opt. Commun. 66, 122–126 (1988).
[Crossref]

Other (5)

W. L. Bohn, Th. Hall, “Effect of beam quality on the scaling of high-energy flow lasers,” in Gas Flow and Chemical Lasers, E. Rosenwaks, ed. (Springer, Berlin, 1987), pp. 90–95.
[Crossref]

A. Culoma, J. L. Paradis, D. Gerbert, “High power laser amplification,” in High Power Lasers: Sources, Laser-Material Interactions, E. W. Kreutz, A. Quenzer, D. Schücker, eds., Proc. Soc. Photo-Opt. Instrum. Eng.801, 42–44 (1987).
[Crossref]

A. E. Siegman, Lasers (Oxford U. Press, Oxford, 1986), pp. 696–697.

K. E. Oughstum, “Unstable resonator modes,” in Progress in Optics XXIV, E. Wolf, ed. (North-Holland, Amsterdam, 1987), pp. 167–387.

J. L. H. Neira, J. Delgado, G. Calvo, M. Sanchez, “Beam quality in high-power laser amplification,” in High Power Lasers and Laser-Machining Technology, M. L. Guillard, A. Quenzer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1132, 1–8 (1989), and references therein.

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Equations (48)

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Γ ( x , s , z ) = V ( x + s / 2 , z ) V * ( x - s / 2 , z ) ,
x = ( x 1 + x 2 ) / 2 ,             s = x 1 - x 2 .
h ( x , u , z ) - + exp ( - i k u s ) Γ ( x , s , z ) d s ,
- + h ( x , u , z ) d x d u = 1 ,
x - + x h ( x , u , z ) d x d u ,
x 2 - + x 2 h ( x , u , z ) d x d u ,
u 2 - + u 2 h ( x , u , z ) d x d u ,
Q = x 2 u 2 - x u 2 ,
Q 1 / 4 k 2 ,             k = 2 π / λ ,
P = [ W 2 Ψ Ψ t Φ 2 ] = [ x 2 x y x u x v x y y 2 y u y v x u y u u 2 u v x v y v u v v 2 ] ,
W 2 = [ x 2 x y x y y 2 ] ,             Ψ = [ x u x v y u y v ] , Φ 2 = [ u 2 u v u v v 2 ] ,
P = MPM t .
Q 3 D = x 2 u 2 - xu 2 = ( x 2 + y 2 ) ( u 2 + v 2 ) - ( x u + y v ) 2 .
G β = [ cos β sin β 0 0 - sin β cos β 0 0 0 0 cos β sin β 0 0 - sin β cos β ] .
cot 2 β = 2 u v v 2 - u 2 ,
u 2 β = v 2 β ,
M z = [ 1 0 z 0 0 1 0 z 0 0 1 0 0 0 0 1 ] .
x u β z = u 2 β z + x u β ,
z = - x u u 2 β ,
Q 3 D = Q 3 D β z = x 2 β z u 2 β z + x 2 β z v 2 β z + ( y 2 β z v 2 β z - y v β z 2 ) + y 2 β z u 2 β z ,
Q x = x 2 u 2 - x u 2 1 / 4 k 2 ,
Q y = y 2 v 2 - y v 2 1 / 4 k 2 ,
Q 3 D = Q x + Q x + Q y + Q y + y v β z 2 1 / k 2 .
Q 3 D = 4 Q x = 4 Q y = 1 / k 2 .
x y = 0.
u v = 0.
cot 2 θ = x 2 i - y 2 i 2 x y i ,
cot 2 θ = u 2 i - v 2 i 2 u v i .
z ( x 2 + y 2 ) = 0
x u + y u = 0 ,
z c = x u 0 + u v 0 u 2 0 + v 2 0 .
J = t r ( W 2 Φ 2 - Ψ Ψ ) ,
J = x 2 u 2 - x u 2 + y 2 v 2 - y v 2 + 2 x y u v - 2 x v y u ,
J = Q x + Q y - 2 S ,             S = x v y u - x y u v ,
M - 1 = LM t L L = M t LM ,
L = i [ 0 - 1 1 0 ] = L - 1 .
J = t r ( W 2 Φ 2 - Ψ Ψ ) = ( 1 / 2 ) t r ( P L P L ) ,
J = ( 1 / 2 ) t r ( MPM t LMPM t L ) = ( 1 / 2 ) t r ( P M t LMPM t LM ) .
J = ( 1 / 2 ) t r ( P L P L ) = ( 1 / 2 ) t r ( PLPL ) = J             ( Q . E . D . ) .
M = [ 1 B 0 1 ] ,             B = [ d a a d ] = B t ,
P = [ W 2 Ψ Ψ Φ 2 ] ,             Φ 2 = [ u 2 0 0 v 2 ] .
P = [ W 2 + B Ψ t + Ψ B + B Φ 2 B Ψ + B Φ 2 Ψ t + Φ 2 B Φ 2 ] ,
Ψ + B Φ 2 = [ x u + d u 2 x v + a v 2 y u + a u 2 y v + d v 2 ] .
P = [ x 2 x y 0 0 x y y 2 y u 0 0 y u u 2 0 0 0 0 ( v 2 ] .
J = x 2 u 2 + y 2 v 2 = Q x + Q y 1 / 2 k 2 .
J = x 2 u 2 + y 2 v 2 + 2 x u y v - 2 x v y u = Q c + A c ,
Q c = x 2 u 2 + y 2 v 2 ,
A c = 2 x u y v - 2 x v y u .

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