Abstract

The Wigner-distribution-function description of light beams is extended to aberrated optical systems. The simulations performed show that the different types of aberrations can be identified separately by the use of experimental devices that display projections of the Wigner distribution function of a two-dimensional beam.

© 1996 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 5.
  2. H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).
  3. H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, U.K.1970).
  4. C. H. F. Velzel, J. L. F. de Meijere, “Characteristic functions and the aberrations of symmetric optical systems. I. Transverse aberrations when the eikonal is given,” J. Opt. Soc. Am. A 5, 246–250 (1988);“Characteristic functions and the aberrations of symmetrical optical systems. II. Addition of aberrations,” J. Opt. Soc. Am. A 5, 251–256 (1988);“Characteristic functions and the aberrations of symmetrical optical systems. III. Calculation of eikonal coefficients,” J. Opt. Soc. Am. A 5, 1237–1243 (1988).
    [CrossRef]
  5. A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 105–158.
    [CrossRef]
  6. V. I. Mańko, K. B. Wolf, “The influence of spherical aberration on Gaussian beam propagation,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 207–226.
    [CrossRef]
  7. J. Ojeda-Castaneda, “Focus-error operator and related special functions,” J. Opt. Soc. Am. 73, 1042–1047 (1983).
    [CrossRef]
  8. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [CrossRef]
  9. M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  10. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  11. D. Onciul, “Invariant properties of general astigmatic beams through first-order optical systems,” J. Opt. Soc. Am. A 10, 295–298 (1993).
    [CrossRef]
  12. D. Dragoman, “Higher-order moments of the Wigner distribution function in first-order optical systems,” J. Opt. Soc. Am. A 11, 2643–2646 (1994).
    [CrossRef]
  13. A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).
  14. M. J. Bastiaans, “The Wigner distribution function and Hamilton's characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
    [CrossRef]
  15. J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, U.K.1962).
  16. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).
  17. R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).
  18. E. C. G. Sudarshan, N. Mukunda, R. Simon, “Realization of first-order optical systems using thin lenses,” Opt. Acta 32, 855–872 (1985).
    [CrossRef]
  19. H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
    [CrossRef]
  20. R. Bamler, H. Glunder, “The Wigner distribution function of two-dimensional signals: coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
    [CrossRef]
  21. H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
    [CrossRef]
  22. N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
    [CrossRef]
  23. J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 4, p. 236.
  24. J. D. Lawson, The Physics of Charged-Particle Beams (Clarendon, Oxford, 1988), Chap. 4.

1994 (1)

1993 (1)

1992 (2)

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

1988 (1)

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]

1983 (3)

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

J. Ojeda-Castaneda, “Focus-error operator and related special functions,” J. Opt. Soc. Am. 73, 1042–1047 (1983).
[CrossRef]

R. Bamler, H. Glunder, “The Wigner distribution function of two-dimensional signals: coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

1980 (1)

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
[CrossRef]

1979 (2)

M. J. Bastiaans, “The Wigner distribution function and Hamilton's characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first-order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

1932 (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 4, p. 236.

Bamler, R.

R. Bamler, H. Glunder, “The Wigner distribution function of two-dimensional signals: coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Bartelt, H. O.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
[CrossRef]

Bastiaans, M. J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 5.

Brenner, K. H.

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
[CrossRef]

Buchdahl, H. A.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, U.K.1970).

de Meijere, J. L. F.

Dragoman, D.

Dragt, A. J.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 105–158.
[CrossRef]

Forest, E.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 105–158.
[CrossRef]

Glunder, H.

R. Bamler, H. Glunder, “The Wigner distribution function of two-dimensional signals: coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Haase, T.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

Hodgson, N.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

Kostka, R.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

Lawson, J. D.

J. D. Lawson, The Physics of Charged-Particle Beams (Clarendon, Oxford, 1988), Chap. 4.

Lohmann, A. W.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).

Manko, V. I.

V. I. Mańko, K. B. Wolf, “The influence of spherical aberration on Gaussian beam propagation,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 207–226.
[CrossRef]

Mukunda, N.

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

Ojeda-Castaneda, J.

J. Ojeda-Castaneda, “Focus-error operator and related special functions,” J. Opt. Soc. Am. 73, 1042–1047 (1983).
[CrossRef]

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Onciul, D.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

Simon, R.

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

Streibl, N.

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Sudarshan, E. C. G.

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

Synge, J. L.

J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, U.K.1962).

Velzel, C. H. F.

Weber, H.

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

Wigner, E.

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 5.

Wolf, K. B.

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 105–158.
[CrossRef]

V. I. Mańko, K. B. Wolf, “The influence of spherical aberration on Gaussian beam propagation,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 207–226.
[CrossRef]

J. Mod. Opt. (1)

H. Weber, “Wave optical analysis of the phase space analyser,” J. Mod. Opt. 39, 543–559 (1992).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Acta (2)

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

R. Bamler, H. Glunder, “The Wigner distribution function of two-dimensional signals: coherent-optical generation and display,” Opt. Acta 30, 1789–1803 (1983).
[CrossRef]

Opt. Appl. (1)

A. W. Lohmann, J. Ojeda-Castaneda, N. Streibl, “The influence of wave aberrations on the Wigner distribution,” Opt. Appl. 13, 465–471 (1983).

Opt. Commun. (2)

M. J. Bastiaans, “The Wigner distribution function and Hamilton's characteristics of a geometric-optical system,” Opt. Commun. 30, 321–326 (1979).
[CrossRef]

H. O. Bartelt, K. H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 3232–38 (1980).
[CrossRef]

Opt. Quantum Electron. (1)

N. Hodgson, T. Haase, R. Kostka, H. Weber, “Determination of laser beam parameters with the phase space beam analyser,” Opt. Quantum Electron. 24927–949 (1992).
[CrossRef]

Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
[CrossRef]

Other (10)

A. J. Dragt, E. Forest, K. B. Wolf, “Foundations of a Lie algebraic theory of geometrical optics,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 105–158.
[CrossRef]

V. I. Mańko, K. B. Wolf, “The influence of spherical aberration on Gaussian beam propagation,” in Lie Methods in Optics, J. Sanchez-Mondragon, K. B. Wolf, eds. (Springer-Verlag, Berlin, 1986), pp. 207–226.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, Oxford, 1970), Chap. 5.

H. A. Buchdahl, Optical Aberration Coefficients (Dover, New York, 1968).

H. A. Buchdahl, An Introduction to Hamiltonian Optics (Cambridge U. Press, Cambridge, U.K.1970).

J. L. Synge, Geometrical Optics: An Introduction to Hamilton's Method (Cambridge U. Press, Cambridge, U.K.1962).

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

R. K. Luneburg, Mathematical Theory of Optics (University of California, Berkeley, 1966).

J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), Chap. 4, p. 236.

J. D. Lawson, The Physics of Charged-Particle Beams (Clarendon, Oxford, 1988), Chap. 4.

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Figures (8)

Fig. 1
Fig. 1

Projection of (a) a normalized WDF [WDF/WDF(0, 0, 0, 0)], and (b) its intersection with planes parallel to the (x, y, px, py ) plane on the (y = py = 0) plane for an unaberrated Gaussian beam.

Fig. 2
Fig. 2

Same as Figs. 1, but spherical aberration is present: Lkω4/4 = 0.1.

Fig. 3
Fig. 3

Same as Figs. 1, but coma is present: Pkω4/4 = 0.05.

Fig. 4
Fig. 4

Same as Figs. 1, but astigmatism or field curvature is present: Nkω4/4 = Mkω4/2 = 0.2.

Fig. 5
Fig. 5

Same as Figs. 1, but distortion (pin-cushion aberration) is present: Okω4/4 = 0.2.

Fig. 6
Fig. 6

Same as Figs. 1, but distortion (barrel-type aberration) is present: Okω4/4 = −0.2.

Fig. 7
Fig. 7

Projection of (a) a normalized WDF and (b) its intersection with planes parallel to the (x, y, px, py ) plane on the (y = 0.5, py = 0) plane for a Gaussian beam in the presence of astigmatism: Mkω4/4 = 0.01. The maximum height of the normalized WDF is 0.78.

Fig. 8
Fig. 8

Same as Figs. 7, but field curvature is present: Nkω4/4 = 0.01. The maximum height of the normalized WDF is 0.739.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

V p = L r 1 4 M r 4 N r 0 2 r 1 2 + O r 0 2 r 2 + P r 1 2 r 2 .
Δ x = V p x 1 , Δ y = V p y 1 .
W ( x , y , p x , p y ) = φ ( x + x 2 , y + y 2 ) φ ( x x 2 , y y 2 ) × exp ( ikx p x + iky p y ) d x d y ,
W 1 ( x 1 , y 1 , p x 1 , p y 1 ) = 1 4 π 2 [ K ( x 0 , y 0 , p x 0 , p y 0 , x 1 , y 1 , p x 1 , p y 1 ) ] × [ W 0 ( x 0 , y 0 , p x 0 , p y 0 ) ] d x 0 d y 0 d p x 0 d p y 0 ,
K ( x 0 , y 0 , p x 0 , p y 0 , x 1 , y 1 , p x 1 , p y 1 ) = 16 π 4 | g ( x 0 , y 0 , x 1 , y 1 ) | 2 δ ( p x 0 + V x 0 ) × δ ( p y 0 + V y 0 ) δ ( p x 1 V x 1 ) δ ( p y 1 V y 1 ) ,
= 4 π 2 | h ( p x 0 , p y 0 , x 1 , y 1 ) | 2 δ ( x 0 M p x 0 ) × δ ( y 0 M p y 0 ) δ ( p x 1 M p x 1 ) δ ( p y 1 M p y 1 ) ,
= 4 π 2 | h ¯ ( x 0 , y 0 , p x 1 , p y 1 ) | 2 δ ( p x 0 + M ¯ x 0 ) × δ ( p y 0 + M ¯ y 0 ) δ ( x 1 + M ¯ p x 1 ) δ ( y 1 + M ¯ p y 1 ) ,
= | f ( p x 0 , p y 0 , p x 1 , p y 1 ) | 2 δ ( x 0 T p x 0 ) × δ ( y 0 T p y 0 ) δ ( x 1 + T p x 1 ) δ ( y 1 + T p y 1 ) .
V M = M ¯ T = x 0 p x 0 + y 0 p y 0 , T M = M ¯ V = x 1 p x 1 + y 1 p y 1 .
p x 0 = V x 0 , p y 0 = V y 0 , p x 1 = V x 1 , p y 1 = V y 1 ,
x 0 = M p x 0 , y 0 = M p y 0 , p x 1 = M x 1 , p y 1 = M y 1 ,
p x 0 = M ¯ x 0 , p y 0 = M ¯ y 0 , x 1 = M ¯ p x 1 , y 1 = M ¯ p y 1 ,
x 0 = T p x 0 , y 0 = T p y 0 , x 1 = T p x 1 , y 1 = T p y 1 ,
V ( x 0 , y 0 , x 1 , y 1 ) = A ( x 0 2 + y 0 2 ) + D ( x 1 2 + y 1 2 ) 2 ( x 0 x 1 + y 0 y 1 ) 2 B + V p + L opt ,
p x 0 = x 1 A x 0 B + 2 M x 1 ( x 0 x 1 + y 0 y 1 ) + N x 0 ( x 1 2 + y 1 2 ) O [ 2 x 0 ( x 0 x 1 + y 0 y 1 ) + x 1 ( x 0 2 + y 0 2 ) ] P x 1 ( x 1 2 + y 1 2 ) ,
p y 0 = y 1 A y 0 B + 2 M y 1 ( x 0 x 1 + y 0 y 1 ) + N y 0 ( x 1 2 + y 1 2 ) O [ 2 y 0 ( x 0 x 1 + y 0 y 1 ) + y 1 ( x 0 2 + y 0 2 ) ] P y 1 ( x 1 2 + y 1 2 ) ,
p x 1 = D x 1 x 0 B L x 1 ( x 1 2 + y 1 2 ) 2 M x 0 ( x 0 x 1 + y 0 y 1 ) N x 1 ( x 0 2 + y 0 2 ) + O x 0 ( x 0 2 + y 0 2 ) + P [ 2 x 1 ( x 0 x 1 + y 0 y 1 ) + x 0 ( x 1 2 + y 1 2 ) ] ,
p y 1 = D y 1 y 0 B L y 1 ( x 1 2 + y 1 2 ) 2 M y 0 ( x 0 x 1 + y 0 y 1 ) N y 1 ( x 0 2 + y 0 2 ) + O y 0 ( x 0 2 + y 0 2 ) + P [ 2 y 1 ( x 0 x 1 + y 0 y 1 ) + y 0 ( x 1 2 + y 1 2 ) ] .
φ 0 ( x 0 , y 0 ) = exp ( x 0 2 + y 0 2 ω 2 ) .
W 0 ( x 0 , y 0 , p x 0 , p y 0 ) = 2 π ω 2 exp [ 2 ( x 0 2 + y 0 2 ) ω k 2 ω 2 2 ( p x 0 2 + p y 0 2 ) ] .
W 0 ( x , y , p x , p y ) = 2 π ω 2 exp [ ( x 2 + y 2 ) p x 2 + p y 2 4 ] .
W 1 unab ( x , y , p x , p y ) = 2 π ω 2 exp [ ( x p x ) 2 ( y p y ) 2 p x 2 + p y 2 4 ] .
( x p x ) 2 + ( y p y ) 2 + p x 2 + p y 2 4 = ln ( 1 m ) .

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