Abstract

A spatially partially coherent two-dimensional beamlike wave field with arbitrary distributions of the optical intensity and complex degree of spatial coherence is assumed incident on a diffraction grating with an arbitrary periodic permittivity modulation. Rigorous electromagnetic diffraction theory of gratings is used, together with the coherent-mode representation of partially coherent fields, to calculate near-field and far-zone distributions of the zeroth-order beam and the Bragg-diffracted beam. A volume grating with significant Bragg selectivity is shown to increase the spatial coherence of the diffracted beam, but the diffraction efficiency is simultaneously reduced.

© 1997 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. A 44, 165–169 (1956).
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  3. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  4. M. C. Gupta, S. T. Peng, “Diffraction characteristics of surface-relief gratings,” Appl. Opt. 32, 2911–2917 (1993).
    [CrossRef] [PubMed]
  5. E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
    [CrossRef]
  6. R. S. Chu, J. A. Kong, T. Tamir, “Diffraction of Gaussian beams by a periodically modulated layer,” J. Opt. Soc. Am. 67, 1555–1561 (1977).
    [CrossRef]
  7. M. D. McNeill, T.-C. Poon, “Gaussian-beam profile shaping by acousto-optic Bragg diffraction,” Appl. Opt. 33, 4508–4515 (1994), and references cited therein.
    [CrossRef] [PubMed]
  8. Y. Ohtsuka, “Modulation effects of a sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
    [CrossRef]
  9. Y. Ohtsuka, “Partial coherence controlled by a progressive ultrasonic wave,” J. Opt. Soc. Am. 69, 684–689 (1979).
    [CrossRef]
  10. Y. Imai, Y. Ohtsuka, “Optical coherence modulation by ultrasonic waves. 1: dependence of partial coherence on ultrasonic parameters,” Appl. Opt. 19, 542–547 (1980).
    [CrossRef] [PubMed]
  11. Y. Imai, Y. Ohtsuka, “Optical coherence modulation by ultrasonic waves. 2: application to speckle reduction,” Appl. Opt. 19, 3541–3544 (1980).
    [CrossRef] [PubMed]
  12. Y. Ohtsuka, Y. Arima, Y. Imai, “Acousto-optic 2-D profile shaping of a Gaussian laser beam,” Appl. Opt. 24, 2813–2819 (1995).
    [CrossRef]
  13. Y. Ohtsuka, “Modulation of optical coherence by ultrasonic waves,” J. Opt. Soc. Am. A 3, 1247–1257 (1986).
    [CrossRef]
  14. J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
    [CrossRef]
  15. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  16. R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).
  17. J. Perina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).
  18. E. Wolf, “New theory of partial coherence in the space-frequency domain. Part I: spectra and cross spectra of steady-state sources,” J. Opt. Soc. Am. 72, 343–351 (1982).
    [CrossRef]
  19. E. W. Marchand, E. Wolf, “Radiometry with sources of any state of coherence,” J. Opt. Soc. Am. 64, 1219–1226 (1974).
    [CrossRef]
  20. A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  21. A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
    [CrossRef]
  22. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  23. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
  24. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  25. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  26. D. Courjon, J. Bulabois, W. H. Carter, “Use of a holographic filter to modify the coherence of a light field,” J. Opt. Soc. Am. 71, 469–473 (1981).
    [CrossRef]
  27. A. Korpel, Acousto-optics (Marcel Dekker, New York, 1988), Sec. 3.2.3.

1995 (2)

Y. Ohtsuka, Y. Arima, Y. Imai, “Acousto-optic 2-D profile shaping of a Gaussian laser beam,” Appl. Opt. 24, 2813–2819 (1995).
[CrossRef]

A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

1994 (3)

1993 (1)

1990 (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

1986 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1982 (2)

1981 (1)

1980 (3)

1979 (1)

1978 (1)

1977 (1)

1976 (1)

Y. Ohtsuka, “Modulation effects of a sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
[CrossRef]

1974 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

1967 (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

1966 (1)

1956 (1)

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. A 44, 165–169 (1956).

Arima, Y.

Bulabois, J.

Burckhardt, C. B.

Carter, W. H.

Chu, R. S.

Courjon, D.

Friberg, A. T.

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Friberg, S. Y.

A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Gori, F.

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Gupta, M. C.

Huttunen, A. C.

A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Imai, Y.

Knop, K.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

Kong, J. A.

Korpel, A.

A. Korpel, Acousto-optics (Marcel Dekker, New York, 1988), Sec. 3.2.3.

Marchand, E. W.

McNeill, M. D.

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Noponen, E.

Ohtsuka, Y.

Peng, S. T.

Perina, J.

J. Perina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).

Phariseau, P.

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. A 44, 165–169 (1956).

Poon, T.-C.

Schell, A. C.

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

Starikov, A.

Tamir, T.

Tervonen, E.

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Turunen, J.

A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

E. Noponen, J. Turunen, “Binary high-frequency-carrier diffractive optical elements: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1097–1109 (1994).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

Wolf, E.

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. C. Schell, “A technique for the determination of the radiation pattern of a partially coherent aperture,” IEEE Trans. Antennas Propag. 15, 187–188 (1967).
[CrossRef]

J. Appl. Phys. (1)

J. Turunen, E. Tervonen, A. T. Friberg, “Acousto-optic control and modulation of spatial coherence by synthetic acousto-optics holograms,” J. Appl. Phys. 67, 49–59 (1990).
[CrossRef]

J. Opt. Soc. Am. (8)

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

Opt. Commun. (2)

Y. Ohtsuka, “Modulation effects of a sound wave on the mutual coherence function of light,” Opt. Commun. 17, 234–237 (1976).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Phys. Rev. E (1)

A. C. Huttunen, S. Y. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Proc. Indian Acad. Sci. A (1)

P. Phariseau, “On the diffraction of light by progressive supersonic waves,” Proc. Indian Acad. Sci. A 44, 165–169 (1956).

Other (3)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer, Berlin, 1980).

J. Perina, Coherence of Light (Reidel, Dordrecht, The Netherlands, 1985).

A. Korpel, Acousto-optics (Marcel Dekker, New York, 1988), Sec. 3.2.3.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Rigorously evaluated diffraction efficiencies η-1 (solid curve) and η0 (dotted curve) as functions of the angle of incidence θ: fully coherent plane wave and grating 1 (a binary surface-relief grating).

Fig. 2
Fig. 2

Same as Fig. 1, but for grating 2 (a holographic volume grating).

Fig. 3
Fig. 3

Far-zone diffraction of a Gaussian Schell-model field with w=σ=10λ by grating 2. Solid curve: A0,0ϕ, Δϕ, h) when Δr=0. Dotted curve: A-1, -1(Δϕ, Δϕ, h). Dashed curve: A0,0ϕ, Δϕ, h). Here Δϕ=ϕ-θB if m=0 and Δϕ=ϕ+θB if m=-1.

Fig. 4
Fig. 4

Same as Fig. 1, but for (a) grating 3(a) and (b) grating 3(b) (photorefractive volume grating).

Fig. 5
Fig. 5

Intensity distributions 100W-1, -1(x ,x, h) (solid curve) and W0,0(x, x, h) (dotted curve) for a Gaussian Schell-model field with w=σ=10λ.

Equations (35)

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

E(x, h)=Aim=- Tm exp(i2πux)exp(i2πmx/d),
Ei(x, 0)=- Ai(u, 0)exp(i2πux)du.
E(x, h)=m=- Em(x, h),
Em(x, h)=exp(i2πmx/d)- Ai(u, 0)×Tm(u)exp(i2πux)du
A(u, h)=-E(x, h)exp(-i2πux)dx=m=-Am(u, h),
Am(u, h)=Ai(u-m/d, 0)Tm(u-m/d)
Wi(x1, x2, 0)=Ei*(x1, 0)Ei(x2, 0),
Ai(u1, u2, 0)=Ai*(u1, 0)A(u2, 0)
W(x1, x2, h)=E*(x1, h)E(x2, h).
W(x1, x2, h)=m=-n=- Wmn(x1, x2, h),
Wmn(x1, x2, h)
=exp[-i2π(mx1-nx2)/d]×- Ai(u1, u2, 0)Tm*(u1)Tn(u2)×exp[-i2π(u1x1-u2x2)]du1du2.
A(u1, u2, h)=- W(x1, x2, h)×exp[i2π(u1x1-u2x2)]dx1dx2,=m=-n=- Amn(u1, u2, h),
Amn(u1, u2, h)=Ai(u1-m/d, u2-n/d, 0)×Tm*(u1-m/d)Tn(u2-n/d).
W(x1, x2, z)=[W(x1, x1, z)W(x2, x2, z)]1/2×μ(x1, x2, z).
A(u1, u2, z)=[A(u1, u1, z)×A(u2, u2, z)]1/2ν(u1, u2, z).
Amm(u, u, h)=|Tm(u-m/d)|2Ai(u-m/d,u-m/d, 0),
νmm(u1, u2, h)=Tm*(u1-m/d)|Tm(u1-m/d)|Tm(u2-m/d)|Tm(u2-m/d)|×νi(u1-m/d, u2-m/d, 0).
Wi(x1, x2, 0)=q=0λqψq*(x1, 0)ψq(x2, 0).
- Wi(x1, x2, 0)ψq(x1)dx1=λqψq(x2).
W(x1, x2, h)=q=0 λqψq*(x1, h)ψq(x2, h),
ψq(x, h)=m=- ψqm(x, h)=m=- exp(i2πmx/d)- aq(u, 0)Tm(u)×exp(i2πux)du,
aq(u, 0)=- ψq(x, 0)exp(-i2πux)dx
Wmn(x1, x2, h)=q=0 λqψqm*(x1, 0)ψqn(x2, 0).
Jqm(x)=- aq(u, 0)Tm(u)exp(i2πux)du,
Wmm(x1, x2, h)=q=0 λqJqm*(x1)Jqm(x2)×exp[-i2πm(x1-x2)/d].
Wi(x1, x2, 0)=exp[-(x12+x22)/w2]×exp[-(x1-x2)2/2σ2]×exp[-iπ(x1-x2)/d],
Ai(u1, u2, 0)
=πw2β exp-πwβ2u1-1/2d2+u2-1/2d2×exp-12πw21-β2u1-u22,
ψqx, 0=12qq!2πw2β1/4Hqx22βexp-x2w2β×expiπx/d,
λq=2πwβ1+β1-β1+β2,
aqu, 0=-iq2π2qq!2πw2β1/4Hqwβ2u-12d×exp-14w2βu-12d2;
r(x, z)=n12n32when0x<d/2whend/2x<d
r(x, z)=¯r+Δr sin(2πx/d),
η-1=n3 cos ϕ-1n1 cos θ|T-1|2

Metrics