Abstract

On the basis of the Rayleigh–Sommerfeld diffraction integral, a closed-form propagation expression for the Wigner distribution function of partially coherent nonparaxial beams in free space is derived for what is to our knowledge the first time. The propagation of spatially fully coherent nonparaxial beams is treated as a special case of our general result. Application of the result is illustrated with the nonparaxial propagation of partially coherent anisotropic Gaussian–Schell-model beams and TEM11-mode Hermite–Gaussian beams.

© 2004 Optical Society of America

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References

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  1. M. J. Bastiaans, J. Opt. Soc. Am. A 3, 1227 (1986).
    [CrossRef]
  2. M. J. Bastiaans, Optik 82, 173 (1989).
  3. K. B. Wolf, M. A. Alonso, and G. W. Forbes, J. Opt. Soc. Am. A 16, 2476 (1999).
    [CrossRef]
  4. C. J. R. Sheppard and K. G. Larkin, J. Opt. Soc. Am. A 18, 2486 (2001).
    [CrossRef]
  5. C. J. R. Sheppard and K. G. Larkin, Opt. Lett. 26, 968 (2001).
    [CrossRef]
  6. K. Duan and B. Lü, Opt. Lett. 29, 800 (2004).
    [CrossRef] [PubMed]
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. Y. Li and E. Wolf, Opt. Lett. 6, 256 (1982).
    [CrossRef]

2004 (1)

2001 (2)

1999 (1)

1989 (1)

M. J. Bastiaans, Optik 82, 173 (1989).

1986 (1)

1982 (1)

Y. Li and E. Wolf, Opt. Lett. 6, 256 (1982).
[CrossRef]

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Equations (27)

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Woρo1,ρo2,z=hρo1,ρi1h*ρo2,ρi2×Wiρi1,ρi2,0dρi1dρi2,
Fρ,q,z=Wρ+ρ2,ρ-ρ2,zexp-iq·ρdρ,
Foρo,qo,z=Fiρi,qi,0×Kρo,qo,ρi,qidρidqi,
Kρo,qo,ρi,qi=14π2hρo+12ρo,ρi+12ρi×h*ρo-12ρo,ρi-12ρiexp-iqo·ρo-qi·ρi×dρodρi,
hρo,ρi=1iλexpikrrcos θ,
Kρo,qo,ρi,qi=14π2z2λ2expikr1-r2r12r22×exp-iqo·ρo-qi·ρidρodρi,
r1=xo+xo2-xi-xi22+yo+yo2-yi-yi22+z21/2,
r2=xo-xo2-xi+xi22+yo-yo2-yi+yi22+z21/2.
r1-r2xo-xixo-xi+yo-yiyo-yir0,
Kρo,qo,ρi,qi=4π2z2λ2r04δkρo-ρir0-qo×δkρo-ρir0-qi.
Foρo,qo,z=z2r02Fiρo-r0kqo,qo,0.
Kpρo,qo,ρi,qi=δρo-ρi-zkqoδqo-qi,
Fopρo,qo,z=Fipρo-zkqo,qo,0,
Wiρi1,ρi2,0=exp-xi12+xi22w0x2-yi12+yi22w0y2×exp-xi12-xi222σ0x2-(yi12-yi222σ0y2,
Fiρi,qi,0=2πk2fx2+fσx2fy2+fσy21/2×exp-ui22fx2+fσx2k2-vi22fy2+fσy2k2-2fx2k2xi2-2fy2k2yi2,
Foρo,qo,z=2z2πr02k2fx2+fσx2fy2+fσy21/2×exp-uo22fx2+fσx2k2-2fx2k2xo-r0kuo2×exp-vo22fy2+fσy2k2-2fy2k2yo-r0kvo2,
Woρo1,ρo2,z=14π2Foρo1+ρo22,qo,z×expiρo1-ρo2·qodqo=z2r02s1s2exp-k2fx2xo1+xo222s12-k2fy2yo1+yo222s22×exp-fx2fσx2k2xo1-xo222s12-fy2fσy2k2yo1-yo222s22×expi4fx2fx2+fσx2k3r0xo12-xo22s12+4fy2fy2+fσy2k3r0yo12-yo22s22,
Ioρo,z=14π2Foρo,qo,zdqo=z2r02s1s2exp-2fxo2k2xo2s12-2fyo2k2yo2s22,
Wρ+ρ2,ρ-ρ2,z=Eρ+ρ2,zE*ρ-ρ2,z,
Fρ,q,z=Eρ+ρ2,zE*ρ-ρ2,z×exp-iρ·qdρ,
Eiρi,0=H12xw0H12yw0exp-ρi2w02,
Fiρi,qi,0=8πf6k6exp-qi2+4k4f4ρi22f2k2×-f2k2+ui2+4f4k4xi2-f2k2+vi2+4f4k4yi2,
Foρo,qo,z=8πz2r02f6k6×exp-qo2+4k4f4ρo-r0kqo22f2k2×-f2k2+uo2+4f4k4xo-r0kuo2×-f2k2+vo2+4f4k4yo-r0kvo2.
Ioρo,z=14π2Foρo,qo,zdqo=4z2f4k4r021+4f4k2r0234xo24yo2 exp-2f2k2ρo21+4f4k2r02.
Foxo,uo,z=z2r02Fixo-r0kuo,uo,0,
Ioxo,z=z2r02s1exp-2fx2k2xo2s12,
Ioxo,z=2z2f2k2r021+4f4k2r023/24xo2×exp-2f2k2xo21+4f4k2r02,

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