Abstract

We deal with the propagation of broadband Schell-model sources in nonlinear media with finite relaxation time. The approach is based on a study of the Wigner distribution function and on a separation of scales technique between the microscopic random fluctuations of the field and the macroscopic intensity profile. The regime in which the nonlinearity is strong and slow is considered. Precise results are obtained for the small- and large-scale characteristics of the pulse: optical intensity profile, speckle radius, and typical intensity profile of the speckle spots.

© 2003 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge UniversityCambridge, UK, 1995).
  2. A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [Crossref]
  3. F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
    [Crossref]
  4. P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
    [Crossref]
  5. R. Simon and N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [Crossref]
  6. F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
    [Crossref]
  7. F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
    [Crossref]
  8. J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
    [Crossref]
  9. W. Wang and E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
    [Crossref]
  10. B. Gross and J. T. Manassah, “Modification of a quasi-monochromatic beam spatial coherence function through propagation in a two-dimensional Kerr medium,” Opt. Lett. 17, 166–168 (1992).
    [Crossref] [PubMed]
  11. J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Propagation and amplification of incoherent pulses in dispersive and nonlinear media,” J. Opt. Soc. Am. B 15, 2773–2781 (1998).
    [Crossref]
  12. V. P. Nayyar, “Propagation of partially coherent Gaussian Schell-model sources in nonlinear media,” J. Opt. Soc. Am. B 14, 2248–2253 (1997).
    [Crossref]
  13. W. T. Coffey, “On the direct calculation of the Kerr effect of an assembly of dipolar molecules,” Chem. Phys. 143, 171–183 (1990).
    [Crossref]
  14. A. C. Newell and J. W. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).
  15. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied physics (Springer-Verlag, Berlin, 1984), pp. 9–75.
  16. G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).
  17. V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
    [Crossref]
  18. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).
  19. R. Adler, The Geometry of Random Fields (Wiley, New York, 1981).
  20. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [Crossref]
  21. R. J. Leveque, Numerical Methods for Conservation Laws (Birkhauser, Basel, 1992).

2000 (1)

J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
[Crossref]

1998 (2)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Propagation and amplification of incoherent pulses in dispersive and nonlinear media,” J. Opt. Soc. Am. B 15, 2773–2781 (1998).
[Crossref]

1997 (1)

1993 (1)

1992 (2)

1990 (1)

W. T. Coffey, “On the direct calculation of the Kerr effect of an assembly of dipolar molecules,” Chem. Phys. 143, 171–183 (1990).
[Crossref]

1987 (1)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

1986 (1)

1983 (1)

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

1982 (1)

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

1981 (1)

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
[Crossref]

1979 (1)

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

1974 (1)

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Adler, R.

R. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

Aleshkevich, V. A.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
[Crossref]

Bastiaans, M. J.

Borghi, R.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

Coffey, W. T.

W. T. Coffey, “On the direct calculation of the Kerr effect of an assembly of dipolar molecules,” Chem. Phys. 143, 171–183 (1990).
[Crossref]

De Santis, P.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Friberg, A. T.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Garnier, J.

J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
[Crossref]

J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Propagation and amplification of incoherent pulses in dispersive and nonlinear media,” J. Opt. Soc. Am. B 15, 2773–2781 (1998).
[Crossref]

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied physics (Springer-Verlag, Berlin, 1984), pp. 9–75.

Gori, F.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Gouédard, C.

J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
[Crossref]

J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Propagation and amplification of incoherent pulses in dispersive and nonlinear media,” J. Opt. Soc. Am. B 15, 2773–2781 (1998).
[Crossref]

Gross, B.

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Lebedev, S. S.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
[Crossref]

Leveque, R. J.

R. J. Leveque, Numerical Methods for Conservation Laws (Birkhauser, Basel, 1992).

Manassah, J. T.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge UniversityCambridge, UK, 1995).

Matveev, A. N.

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
[Crossref]

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

Migus, A.

Moloney, J. W.

A. C. Newell and J. W. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

Mukunda, N.

Nayyar, V. P.

Newell, A. C.

A. C. Newell and J. W. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

Palma, C.

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

Pasmanik, G. A.

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Santarsiero, M.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

Simon, R.

Sudol, R. J.

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

Vicalvi, S.

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

Videau, L.

J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
[Crossref]

J. Garnier, L. Videau, C. Gouédard, and A. Migus, “Propagation and amplification of incoherent pulses in dispersive and nonlinear media,” J. Opt. Soc. Am. B 15, 2773–2781 (1998).
[Crossref]

Wang, W.

W. Wang and E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[Crossref]

Wolf, E.

W. Wang and E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge UniversityCambridge, UK, 1995).

Chem. Phys. (1)

W. T. Coffey, “On the direct calculation of the Kerr effect of an assembly of dipolar molecules,” Chem. Phys. 143, 171–183 (1990).
[Crossref]

J. Mod. Opt. (2)

F. Gori, M. Santarsiero, R. Borghi, and S. Vicalvi, “Partially coherent sources with helicoidal modes,” J. Mod. Opt. 35, 539–554 (1998).
[Crossref]

W. Wang and E. Wolf, “Propagation of Gaussian Schell-model beams in dispersive and absorbing media,” J. Mod. Opt. 39, 2007–2021 (1992).
[Crossref]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

Opt. Commun. (5)

F. Gori, G. Guattari, and C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[Crossref]

A. T. Friberg and R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
[Crossref]

F. Gori, “Mode propagation of the field generated by Collett–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[Crossref]

P. De Santis, F. Gori, G. Guattari, and C. Palma, “An example of a Collett–Wolf source,” Opt. Commun. 29, 256–260 (1979).
[Crossref]

J. Garnier, C. Gouédard, and L. Videau, “Propagation of a partially coherent beam under the interaction of small and large scales,” Opt. Commun. 176, 281–297 (2000).
[Crossref]

Opt. Lett. (1)

Sov. J. Quantum Electron. (1)

V. A. Aleshkevich, S. S. Lebedev, and A. N. Matveev, “Self-interaction of a noncoherent light beam,” Sov. J. Quantum Electron. 11, 647–649 (1981).
[Crossref]

Sov. Phys. JETP (1)

G. A. Pasmanik, “Self-interaction of incoherent light beams,” Sov. Phys. JETP 39, 234–238 (1974).

Other (6)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge UniversityCambridge, UK, 1995).

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960).

R. Adler, The Geometry of Random Fields (Wiley, New York, 1981).

A. C. Newell and J. W. Moloney, Nonlinear Optics (Addison-Wesley, Redwood City, Calif., 1992).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed., Vol. 9 of Topics in Applied physics (Springer-Verlag, Berlin, 1984), pp. 9–75.

R. J. Leveque, Numerical Methods for Conservation Laws (Birkhauser, Basel, 1992).

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Figures (2)

Fig. 1
Fig. 1

Intensity profiles of the partially coherent beams in dimensionless variables. The initial profile has a Gaussian shape: (a) β¯=0 (linear regime), (b)–(f) β¯=0.70, 0.98, 1.25, 1.41, 2.11, respectively.

Fig. 2
Fig. 2

Same as Fig. 1 but the initial profile has a super-Gaussian shape with order 2n=6.

Equations (77)

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i Ez+12k0 ΔE+k0nnln0 E=0,
τRnnlt+nnl=n2|E|2,
nnl(z, r, t)=n2τR-t|E|2(z, r, s)exp[(t-s)/τR]ds.
C0,r,t(ρ, τ)=E0(r1, t1)E0*(r2, t2),
C0,r,t(ρ, t)=I0(r, t)exp-|ρ|2ρ02-τ2τ02.
I0(r, t)=I0exp-|r|pr0p-tqt0q,
Cz,r,t(ζ, ρ, τ)  E(z1, r1, t1)E*(z2, r2, t1),
nnl(z, r, t)=n2τR-t|E|2(z, r, s)exp[(t-s)/τR]ds.
nnl(z, r, t)=n2|E|2(z, r, t),
i Ez+12k0 ΔE+k0n2n0 |E|2E=0,
E˜N,01Nj=1NEj,0,
E˜N1Nj=1NEj.
i E˜Nz+12k0 ΔE˜N+k0n2n01Nj=1N|Ej|2Ej=0.
|Ej|2=|E1|2,j=1,, N.
|E˜N|2=1Nj,l=1NEjEl*.
|E˜N|2=1NjlEjEl*+1Nj=1N|Ej|2.
i E˜Nz+12k0 ΔE˜N+k0n2n0 |E˜N|2E˜N=0.
γ3-D(z, r; ζ, ρ)
Cz,r,t(ζ, ρ, 0)=Ez+ζ2, r+ρ2, tE*z-ζ2, r-ρ2, t
|E(Z+ζ, R+ρ)||E(Z, R)| |γ3-D(Z, R; ζ, ρ)||γ3-D(Z, R; 0, 0)|.
C0,r,t(ρ, τ)=ft(τ)γ0(r, ρ).
Cz,r,t(ζ, ρ, τ)=ft(τ)γ3-D(z, r; ζ, ρ),
W0(r, k)=1(2π)2exp(ik  ρ)γ0(r, ρ)dρ,
γ0(r, ρ)=E0r-ρ2E0*r+ρ2.
W(z, r, k)=1(2π)2exp(ik  ρ)γ(z, r, ρ)dρ,
γ(z, r, ρ)=Ez, r-ρ2E*z, r+ρ2.
|E(z, r)|2=W(z, r, k)dk.
Wz+kk0rW+k0n2n0LW=0.
LZ(r, k)=-iexp(-ipr)Vˆ(p)×Zr, k+p2-Zr, k-p2dp,
V(z, r)  |E|2(z, r)=W(z, r, k)dk.
γ0(r, ρ)=I0γ¯0rr0, ρρ0,
W(z, r, k)=I0ρ02W¯zz0 , rr0, ρ0k,
krW(z, r, k)
=I0ρ02ρ0r0 [k¯r¯W¯(z¯, r¯, k¯)]z¯=z/z0,r¯=r/r0,k¯=kρ0,
Wz (z, r, k)=I0ρ02z0W¯z¯ (z¯, r¯, k¯)z¯=z/z0,r¯=r/r0,k¯=kρ0,
V(z, r)=I0[V¯(z¯, r¯)]z¯=z/z0,r¯=r/r0,
V¯(z¯, r)=W¯(z¯, r¯, k¯)dk¯,
Vˆ(z, p)=I0r02[V¯^(z¯, p¯)]z¯=z/z0p¯=pr0,
LW(z, r, k)=I02ρ02[L¯W¯(z¯, r¯, k¯)]z¯=z/z0,r¯=r/r0,k¯=kρ0,
L¯Z(r¯, k¯)=-iexp(-ip¯r¯)V¯^(p¯)Zr¯, k¯+ρ0r0p¯2-Zr¯, k¯-ρ0r0p¯2dp¯,
z0=k0r0ρ02,
β¯=n2I0k02ρ022n0,
W¯z¯+12k¯r¯W¯+β¯r0ρ0L¯W¯=0.
β¯=P0/Pc.
L¯Z(r¯, k¯)=-i ρ0r0exp(-ip¯r¯)V¯^(p¯)[k¯Z(r¯, k¯)p¯]dp¯
L¯Z(r¯, k¯)=ρ0r0 r¯V¯k¯Z(r¯, k¯).
W¯z¯+12k¯r¯W¯+β¯r¯V¯k¯W¯=0,
γ(z, r, ρ)=W(z, r, k)exp(-ik  ρ)dk.
i γ3-Dζ+12k0 Δργ3-D+k0n2n0 V(z, r)γ3-D=0,
W3-D(z, r; κ, k)1(2π)3exp(ik  ρ+iκζ)γ3-D(z, r; ζ, ρ)dρdζ.
 κ-12k0 |k|2+k0n2n0 V(z, r)W3-D=0,
W3-D(z, r; κ, k)dκ=W(z, r, k),
W3-D(z, r; κ, k)=W(z, r, k)δ [κ-K(z, r, k)],
K(z, r, k)=12k0 |k|2-k0n2n0 V(z, r),
γ3-D(z, r; ζ, ρ)=dkW(z, r, k)×exp[-ik  ρ-iK(z, r, k)ζ].
W¯z¯+12 k¯cos(θ) W¯r¯-1r¯sin(θ) W¯θ
+β¯V¯r¯cos(θ) W¯k¯-1k¯sin(θ) W¯θ=0.
W¯z¯+r¯uk¯2 W¯+k¯β¯u V¯r¯ W¯
+u(1-u2)k¯2r¯+β¯k¯V¯r¯W¯=0.
γ0(r, ρ)=I0exp-|r|2r02-|ρ|2ρ02,
W¯0(r¯, k¯)=14πexp-|r¯|2-|k¯|24.
γ0(r, ρ)=I0exp-|r|6r06-|ρ|2ρ02.
Wi,j,lδz+ulkj2δr (Wi+1/2, j,l-Wi-1/2, j,l)
+βulδkVir (Wi,j+1/2,l-Wi,j-1/2,l)+1-ul2δu
×kj2ri+βkjVir(Wi,j,l+1/2-Wi,j,l-1/2)=0.
Wi+1/2, j,l=Wi,j,l+δr2 σriiful>g0,
Wi+1/2, j,l=Wi+1, j,l-δr2 σri+1iful<0,
σri=1δr (Wi+1,j,l-Wi,j,l)ϕ(θr),
θr=Wi,j,l-Wi-1, j,lWi+1, j,l-Wi,j,l,
ϕ(θ)=θ+|θ|1+|θ|.
Wi,j+1/2,l=Wi,j,l+δk2 σkj ifβulVir>0,
Wi,j+1/2,l=Wi,j+1,l-δk2 σkj+1 ifβulVir<0,
Wi,j,l+1/2=Wi,j,l+δu2 σul 
if kj2ri+βkjVir>0,
Wi,j,l+1/2=Wi,j,l+1-δu2 σul+1
if kj2ri+βkjVir<0,
Vir=δkδurj,lkj(Wi+1/2, j,l-Wi-1/2, j,l).

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