Abstract

Generalized vectorial Rayleigh–Sommerfeld diffraction integrals are developed for the cross-spectral-density matrices of spatially partially coherent beams. Using the Gaussian Schell-model (GSM) beam as an example, we derive the expressions for the propagation of cross-spectral-density matrices and intensity of partially coherent vectorial nonparaxial beams, and the corresponding far-field asymptotic forms, beyond the paraxial approximation. The propagation of the vectorial nonparaxial GSM beams are evaluated and analyzed. It is shown that a 3×3 cross-spectral-density matrix or a vector theory is required for the exact description of nonparaxial GSM beams.

© 2004 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  2. G. P. Agrawal, D. N. Pattanayak, “Gaussian beam propagation beyond the paraxial approximation,” J. Opt. Soc. Am. 69, 575–578 (1979).
    [CrossRef]
  3. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [CrossRef]
  4. H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
    [CrossRef]
  5. A. Wünsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  6. S. R. S. Seshadri, “Virtual source for a Hermite–Gauss beam,” Opt. Lett. 28, 595–597 (2003).
    [CrossRef] [PubMed]
  7. A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
    [CrossRef]
  8. J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
    [CrossRef]
  9. W. H. Carter, “Electromagnetic field of a Gaussian beam with an elliptical cross section,” J. Opt. Soc. Am. 62, 1195–1201 (1972).
    [CrossRef]
  10. K. Duan, B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29, 800–802 (2004).
    [CrossRef] [PubMed]
  11. X. Zeng, C. Liang, Y. An, “Far-field propagation of an off-axis Gaussian wave,” Appl. Opt. 38, 6253–6256 (1999).
    [CrossRef]
  12. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
    [CrossRef]
  13. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  15. K. Duan, B. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. (to be published).
  16. A. T. Friberg, R. J. Sudol, “Propagation parameters of Gaussian Schell-model beams,” Opt. Commun. 41, 383–387 (1982).
    [CrossRef]
  17. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [CrossRef] [PubMed]

2004 (1)

2003 (1)

2002 (3)

A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
[CrossRef]

1999 (1)

1998 (1)

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

1992 (1)

1990 (1)

1985 (1)

1982 (1)

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

1979 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1972 (1)

Agrawal, G. P.

An, Y.

Carter, W. H.

Chen, C. G.

Ciattoni, A.

A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Duan, K.

K. Duan, B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29, 800–802 (2004).
[CrossRef] [PubMed]

K. Duan, B. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. (to be published).

Ferrera, J.

Friberg, A. T.

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

Fukumitsu, O.

Heilmann, R. K.

Konkola, P. T.

Laabs, H.

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Liang, C.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lü, B.

K. Duan, B. Lü, “Partially coherent nonparaxial beams,” Opt. Lett. 29, 800–802 (2004).
[CrossRef] [PubMed]

K. Duan, B. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. (to be published).

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

Mandel, L.

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

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Nemoto, S.

Pattanayak, D. N.

Porto, P. D.

A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

Schattenburg, M. L.

Seshadri, S. R. S.

Sudol, R. J.

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

Takenaka, T.

Tervo, J.

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

Turunen, J.

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

Wolf, E.

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

Wünsche, A.

Yokota, M.

Zeng, X.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Commun. (4)

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

H. Laabs, “Propagation of Hermite–Gaussian-beams beyond the paraxial approximation,” Opt. Commun. 147, 1–4 (1998).
[CrossRef]

A. Ciattoni, B. Crosignani, P. D. Porto, “Vectorial analytical description of propagation of a highly nonparaxial beam,” Opt. Commun. 202, 17–20 (2002).
[CrossRef]

J. Tervo, J. Turunen, “Angular spectrum representation of partially coherent electromagnetic fields,” Opt. Commun. 209, 7–16 (2002).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Other (3)

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1966).

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

K. Duan, B. Lü, “Propagation properties of vectorial elliptical Gaussian beams beyond the paraxial approximation,” Opt. Laser Technol. (to be published).

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Figures (5)

Fig. 1
Fig. 1

(a) Intensity distributions I(x, y, 20zR), (b) corresponding contour lines of I(x, y, 20zR), and (c) intensity distributions of Iz(x, y, 20zR) of a vectorial nonparaxial GSM beam at the plane z=20zR. The calculation parameters are f=0.01, fσ=0.01.

Fig. 2
Fig. 2

(a) Intensity distributions of I(x, y, 20zR), (b) corresponding contour lines of I(x, y, 20zR), and (c) intensity distributions of Iz(x, y, 20zR) of a vectorial nonparaxial GSM beam at the plane z=20zR. The calculation parameters are f=0.8, fσ=0.01.

Fig. 3
Fig. 3

(a) Intensity distributions of I(x, y, 20zR), (b) corresponding contour lines of I(x, y, 20zR), and (c) intensity distributions of Iz(x, y, 20zR) of a vectorial nonparaxial GSM beam at the plane z=20zR. The calculation parameters are f=0.01, fσ=0.5.

Fig. 4
Fig. 4

Iz,max(x, 0, 60zR)/I(0, 0, 60zR) at the plane z=60zR of a vectorial nonparaxial GSM beam.

Fig. 5
Fig. 5

(a) Far-field divergence angles θx, θy, and θp of a vectorial nonparaxial GSM beam versus f-1 for different values of fσ=0.01, 0.2, and 0.6. (b) Far-field divergence angles θx, θy, and θp of a vectorial nonparaxial GSM beam versus fσ-1 for different values of f=0.01, 0.3, and 0.6.

Equations (75)

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Eα(ρ, z)=-12π(z=0)Eα(ρ0, 0) zexp(ikR)Rd2ρ0,(α=x,y),
Ez(ρ, z)=12π(z=0)Ex(ρ0, z) xexp(ikR)R+Ey(ρ0, 0) yexp(ikR)Rd2ρ0,
R=[(x-x0)2+(y-y0)2+z2]1/2.
W^0(ρ10, ρ20, 0)
=Wxx0(ρ10, ρ20, 0)Wxy0(ρ10, ρ20, 0)0Wyx0(ρ10, ρ20, 0)Wyy0(ρ10, ρ20, 0)0000,
Wˆ(ρ1, ρ2, z)
=Wxx(ρ1, ρ2, z)Wxy(ρ1, ρ2, z)Wxz(ρ1, ρ2, z)Wyx(ρ1, ρ2, z)Wyy(ρ1, ρ2, z)Wyz(ρ1, ρ2, z)Wzx(ρ1, ρ2, z)Wzy(ρ1, ρ2, z)Wzz(ρ1, ρ2, z),
Wαβ*(ρ1, ρ2, z)=Wβα(ρ2, ρ1, z).
Wxx(ρ1, ρ2, z)
=Ex*(ρ1, z)Ex*(ρ2, z)=12π2(z=0)Ex(ρ10, z) zexp(ikR1)R1d2ρ10*×(z=0)Ex(ρ10, z) zexp(ikR2)R2d2ρ20=12π2(z=0)Wxx0(ρ10, ρ20, 0)× zexp(-ikR1)R1zexp(ikR2)R2d2ρ10d2ρ20.
Wαβ(ρ1, ρ2, z)
=12π2(z=0)Wαβ0(ρ10, ρ20, 0)× zexp(-ikR1)R1zexp(ikR2)R2d2ρ10d2ρ20
(α, β=x, y),
Wzz(ρ1, ρ2, z)
=12π2(z=0)Wxx0(ρ10, ρ20, 0)× x1exp(-ikR1)R1x2exp(ikR2)R2+2Wxy0(ρ10, ρ20, 0) x1exp(-ikR1)R1×y2exp(ikR2)R2+Wyy0(ρ10, ρ20, 0)× y1exp(-ikR1)R1y2exp(ikR2)R2d2ρ10d2ρ20,
Wxz(ρ1, ρ2, z)
=-12π2(z=0)Wxx0(ρ10, ρ20, 0)× zexp(-ikR1)R1x2exp(ikR2)R2+Wxy0(ρ10, ρ20, 0) zexp(-ikR1)R1×y2exp(ikR2)R2d2ρ10d2ρ20,
Wyz(ρ1, ρ2, z)
=-12π2(z=0)Wyy0(ρ10, ρ20, 0)× zexp(-ikR1)R1y2exp(ikR2)R2+Wyx0(ρ10, ρ20, 0) zexp(-ikR1)R1×x2exp(ikR2)R2d2ρ10d2ρ20,
xexp(ikR)R=ik-1R(x-x0)R2exp(ikR),
xexp(ikR)Rik (x-x0)R2exp(ikR).
Wαβ(ρ1, ρ2, z)
=zλ2(z=0)Wαβ0(ρ10, ρ20, 0) ×exp[ik(R2-R1)]R12R22d2ρ10d2ρ20(α, β=x, y),
Wzz(ρ1, ρ2, z)
=1λ2(z=0)[Wxx0(ρ10, ρ20, 0)×(x1-x10)(x2-x20)+2Wxy0×(ρ10, ρ20, 0)(x1-x10)(y2-y20)+Wyy0(ρ10, ρ20, 0)(y1-y10)(y2-y20)]×exp[ik(R2-R1)]R12R22d2ρ10d2ρ20,
Wxz(ρ1, ρ2, z)
=-1λ2(z=0)[Wxx0(ρ10, ρ20, 0)z×(x2-x20)+Wxy0(ρ10, ρ20, 0)z×(y2-y20)] exp[ik(R2-R1)]R12R22d2ρ10d2ρ20,
Wyz(ρ1, ρ2, z)
=-1λ2(z=0)[Wyy0(ρ10, ρ20, 0)z×(y2-y20)+Wyx0(ρ10, ρ20, 0)z×(x2-x20)] exp[ik(R2-R1)]R12R22d2ρ10d2ρ20.
Wαβ0(ρ10, ρ20,z=0)
=exp-|ρ10-ρ20|22σ02exp-ρ102+ρ2022w02,α=β=x0,otherwise,
Riri+xi02+yi02-2xixi0-2yiyi02ri(i=1,2),
ri=(xi2+yi2+zi2)1/2,
--(ax2+bx+c)exp[d(x2+y2)+f(x+y)]dxdy=-π4d3 (4cd2+af2-2ad-2bdf)exp-f22d,
Wxx(ρ1, ρ2, z)=-k2z24s1s2+k4fσ4exp[ik(r2-r1)]r12r22×exp-k24s1x12+y12r12×expk2s14s1s2+k4fσ4x2r2-k2fσ22s1x1r12+y2r2-k2fσ22s1y1r12,
Wzz(ρ1, ρ2, z)
=2k4fσ2s1(4s1s2+k4fσ4)32s2+k4fσ42s1+k2x2r2-k2fσ22s1x1r12+ik2s2+k4fσ42s1×x2r2-k2fσ22s1x1r1 x2+1+f2fσ2x1-1+f2fσ2 2s2+k4fσ42s12x1x2×exp[ik(r2-r1)]r12r22exp-k24s1x12+y12r12×expk2s14s1s2+k4fσ4x2r2-k2fσ22s1x1r12+y2r2-k2fσ22s1y1r12,
Wxz(ρ1, ρ2, z)
=8π2zs1λ2(4s1s2+k4fσ4)2exp[ik(r2-r1)]r12r22×exp-k24s1x12+y12r12 ikk2fσ22s1x1r1-x2r2+4s1s2+k4fσ22s1 x2expk2s14s1s2+k4fσ2 ×k2fσ22s1x1r1-x2r22+k2fσ22s1y1r1-y2r22,
si=ik2r1+(-1)i-1k22 (f2+fσ2)(i=1,2).
Ix(x, y, z)=Wxx(ρ, ρ, z)=1Az2r2exp-k2f2A (x2+y2),
Iy(x, y, z)=Wyy(ρ, ρ, z)=0,
Iz(x, y, z)=Wzz(ρ, ρ, z)=1A3Afσ2+x2k2fσ2(fσ2-f2)-Bk2x21+k2(f2+fσ2)2r2×exp-k2f2A (x2+y2),
(r)=k2f2(f2+2fσ2)r2,
A=1+(r),
B=Afσ2(f2+2fσ2)-A2(f2+fσ2)2+r2k2fσ6(f2+fσ2).
I(x, y, z)=TrWˆ(ρ, ρ, z)=Ix(x, y, z)+Iy(x, y, z)+Iz(x, y, z)=1Az2r2+1A3Afσ2+x2k2fσ2(fσ2-f2)-Bk2x21+k2(f2+fσ2)2r2×exp-k2f2A (x2+y2),
Izc(x, y, z)=k2f4x2(1+k2f4r2)2×exp-k2f21+k2f4r2 (x2+y2),
Ic(x, y, z)=1(1+k2f4r2)z2r2+k2f4x21+k2f4r2×exp-k2f21+k2f4r2 (x2+y2).
Iz(0, 0, z)=fσ2[1+k2f2(f2+2fσ2)z2]2.
Wp(ρ1, ρ2, z)=w02w2(z)exp-ρ12+ρ222w2(z)×exp-|ρ1-ρ2|22σ2(z)exp-ik ρ12-ρ222R(z),
(z)=k2f2(f2+2fσ2)z2,
w(z)=1kf [1+(z)]1/2,
σ(z)=1kfσ [1+(z)]1/2,
R(z)=z1+1(z).
Ip(x, y, z)=w02w2(z)exp-x2+y2w2(z).
Riri-x1xi0+yiyi0ri(i=1,2).
Wfxx(ρ1, ρ2, z)
=-k2z2k4fσ4-4s2exp[ik(r2-r1)]r12r22exp-k24sx12+y12r12×expk2sk4fσ4-4s2x2r2-k2fσ22sx1r12+y2r2-k2fσ22sy1r12,
Wfzz(ρ1, ρ2, z)
=2k4fσ2s(k4fσ4-4s2)3k4fσ4-4s22s+k2x2r2-k2fσ22sx1r12+k4fσ4-4s22s2x1x2k2fσ2ikr1-2s-ik k4fσ4-4s22sk2fσ22sx1r1-x2r2×x2-x1k2fσ2ikr1-2sexp[ik(r2-r1)]r12r22×exp-k24sx12+y12r12expk2sk4fσ4-4s2×x2r2-k2fσ22sx1r12+y2r2-k2fσ22sy1r12,
Wfxz(ρ1, ρ2, z)
=8π2zsλ2(k4fσ4-4s2)2exp[ik(r2-r1)]r12r22×exp-k24sx12+y12r12ikk2fσ22sx1r1-x2r2+k4fσ4-4s22s x2expk2sk4fσ4-4s2×k2fσ22sx1r1-x2r22+k2fσ22sy1r1-y2r22,
s=k22 (f2+fσ2).
Ifx(x, y, z)=Wfxx(ρ, ρ, z)=1(r)z2r2exp-1f2+2fσ2x2+y2r2,
Ify(x, y, z)=Wfyy(ρ, ρ, z)=0,
Ifz(x, y, z)=Wfzz(ρ, ρ, z)=1k4f4(f2+2fσ2)3×(f2+2fσ2)fσ2+f2x2r2+k2f2(f2+2fσ2)2x21r4×exp-1f2+2fσ2x2+y2r2.
If(x, y, z)=1k4f4(f2+2fσ2)3(f2+2fσ2)fσ2+f2x2r2+k2f2(f2+2fσ2)2(x2+z2)1r4×exp-1f2+2fσ2x2+y2r2.
Ifzc(x, y, z)=1k4f81r2+k2f4x2r4exp-1f2x2+y2r2,
Ifc(x, y, z)=1k2f4r4x2+z2+1k2f4x2r2×exp-1f2x2+y2r2.
Wfp(ρ1, ρ2, z)=1(z)exp-ρ12+ρ222k-2(f2+fσ2)-1(z)×expρ1ρ2k-2fσ-2(z)exp-ik ρ12-ρ222z.
Ifp(x, y, z)=1(z)exp-k2f2x2+y2(z).
θα=limztan-1wa(z)z(α=x,y),
I(0, 0, z)=2I(wx(z), 0, z),
I(0, 0, z)=2I(0, wy(z), z),

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