Abstract

A generalized correlation-based definition for moments of arbitrary order is introduced that can also accommodate mixed spatial and angular moments. Moreover, a transformation law for these moments for propagation through linear optical systems is derived. This law has the same form as the corresponding propagation law of the moments defined in terms of the Wigner distribution function. The correlation-based moments can be used to fully characterize beams of arbitrary states of coherence.

© 2003 Optical Society of America

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References

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  1. D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
    [Crossref]
  2. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749–759 (1932).
    [Crossref]
  3. M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
    [Crossref]
  4. M. J. Bastiaans, “The Wigner distribution function and Hamilton’s characteristics of a geometrical-optical system,” Opt. Commun. 30, 321–326 (1979).
    [Crossref]
  5. 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]
  6. S. A. Ponomarenko, E. Wolf, “Effective spatial and angular correlations in beams of any state of spatial coherence and an associated phase-space product,” Opt. Lett. 26, 122–124 (2001).
    [Crossref]
  7. R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
    [Crossref]
  8. H. M. Pedersen, J. J. Stamnes, “Radiometric theory of spatial coherence in free-space propagation,” J. Opt. Soc. Am. A 17, 1413–1420 (2000).
    [Crossref]
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    [Crossref]
  10. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [Crossref]
  11. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
    [Crossref]
  12. M. J. Bastiaans, T. Alieva, “Wigner distribution moments measured as fractional Fourier transform intensity moments,” Proc. SPIE 4829, 245–246 (2002).
  13. C. J. R. Sheppard, K. G. Larkin, “Focal shift, optical transfer function, and phase-space representations,” J. Opt. Soc. Am. A 17, 772–779 (2000).
    [Crossref]

2002 (1)

M. J. Bastiaans, T. Alieva, “Wigner distribution moments measured as fractional Fourier transform intensity moments,” Proc. SPIE 4829, 245–246 (2002).

2001 (1)

2000 (2)

1998 (1)

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
[Crossref]

1997 (1)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[Crossref]

1996 (1)

1994 (1)

1992 (1)

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

1986 (1)

1980 (1)

1979 (1)

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

1932 (1)

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

Alieva, T.

M. J. Bastiaans, T. Alieva, “Wigner distribution moments measured as fractional Fourier transform intensity moments,” Proc. SPIE 4829, 245–246 (2002).

Aye, T.

Bastiaans, M. J.

M. J. Bastiaans, T. Alieva, “Wigner distribution moments measured as fractional Fourier transform intensity moments,” Proc. SPIE 4829, 245–246 (2002).

M. J. Bastiaans, “Applications of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1237 (1986).
[Crossref]

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

Dragoman, D.

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[Crossref]

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]

Erwin, D. A.

Jannson, T.

Larkin, K. G.

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
[Crossref]

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

Mendlovic, D.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
[Crossref]

Neira, J. L. H.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

Pedersen, H. M.

Ponomarenko, S. A.

Sanchez, M.

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

Sheppard, C. J. R.

Stamnes, J. J.

Teague, M. R.

Tengara, I.

Wigner, E.

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

Wolf, E.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

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

Opt. Lett. (1)

Opt. Quantum Electron. (1)

R. Martinez-Herrero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, 1021–1026 (1992).
[Crossref]

Phys. Rev. (1)

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

Proc. SPIE (1)

M. J. Bastiaans, T. Alieva, “Wigner distribution moments measured as fractional Fourier transform intensity moments,” Proc. SPIE 4829, 245–246 (2002).

Prog. Opt. (2)

D. Dragoman, “The Wigner distribution function in optics and optoelectronics,” Prog. Opt. 37, 1–56 (1997).
[Crossref]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transformations in optics,” Prog. Opt. 38, 263–342 (1998).
[Crossref]

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Equations (19)

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Gp1, p2; ω=2π-2  Γr1, r2; ωexpir1p1-r2p2dr1dr2,
r2¯cor= r1-r22|Γr1, r2; ω|2dr1dr2 |Γr1, r2; ω|2dr1dr2,
p2¯cor=p1-p22|Gp1, p2; ω|2dp1dp2 |Gp1, p2; ω|2dp1dp2.
rmpn¯cor= r1-r2mp1-p2nΓr1, r2G*p1, p2expir1p1-r2p2dr1dr2dp1dp2 Γr1, r2G*p1, p2expir1p1-r2p2dr1dr2dp1dp2,
r2¯cor= r1-r22Γr1, r2G*p1, p2expir1p1-r2p2dr1dr2dp1dp2 Γr1, r2G*p1, p2expir1p1-r2p2dr1dr2dp1dp2,
 G*p1, p2expir1p1-r2p2dp1dp2=2π-2  Γ*ξ1, ξ2exp-iξ1p1-ξ2p2×expir1p1-r2p2dξ1dξ2dp1dp2=2π2Γ*r1, r2.
rmpn¯= r-r¯mp-p¯nWr, pdrdp Wr, pdrdp,
Wr, p= Γr+r2, r-r2expirpdr= Gp+p2, p-p2exp-irpdp.
rpo=ABCDrpi=Srpi,
Wor, p=WiDTr-BTp, -CTr+ATp.
Mj=rTpTrprTpT¯j times,
Mjo=SSj/2timesMjiSSTj-j/2times,
rmpn¯cor= rmpnWr, qWs, pexpirp-q+ipr-sdrdrdsdpdpdq2π4  W2r, pdrdp.
rmpn¯cor,o=Ar+BpmCr+Dpn¯cor,i,
 WDTr-BTq, -CTr+ATqWDTs-BTp, -CTs+ATpexpirp-q+ipr-sdrdsdpdq
 Wr, qWs, pexpip-qDTr-BTp+ir-s-CTr+ATpdrdsdpdq,
Aξ+BρmCξ+DρnWr, qWs, pexpip-qξ+ir-sρdrdsdpdqdξdρ.
Fαs=expiαi sin α  φrexp×i r2+s2cos α-2rs2 sin αdr,
rlpm¯α=rcos α-psin αlrsin α+pcos αm¯,

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