Abstract

The full polarization property of volume holographic grating diffraction is investigated theoretically. With a simple volume grating model, the diffracted fields and Mueller matrices are first derived from Maxwell’s equations by using the Green’s function algorithms. The formalism is derived for the general case that the diffraction beam and the grating wave vector are not in the plane of incidence, where s waves and p waves are not decoupled. The derived photon-momentum relations determine the Bragg angle selectivity. The parameters of diffraction strength related to the hologram-writing process and material are defined and are not necessarily small in general. The diffracted-beam profiles are analytically calculated by using the known grating shape function. This theory has provided a fundamental understanding of the polarization phenomena of a real holographic diffraction grating device. The derived algorithm would provide a simulation-analysis tool for the engineering design of real holographic beam combiner/splitter devices.

© 2004 Optical Society of America

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References

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  1. M. S. Shahriar, J. Riccobono, W. Weathers, “Holographic beam combiner,” in Proceedings of the IEEE International Conference on Microwaves and Optics (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 10–14.
  2. M. S. Shahriar, J. Riccobono, W. Weathers, “Highly Bragg selective holographic laser beam combiner,” presented at the Solid State and Diode Laser Technology Review, Albuquerque, N.M., June 5–8, 2000.
  3. M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
    [CrossRef]
  4. Digital Optical Technologies, Inc., “Holographic beam combiner for ladars, printers, fiber amplifiers and cancer treatment,” MDA SBIR Phases I & II program 1999–2003.
  5. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1969).
    [CrossRef]
  6. C. B. Burckhardt, “Diffraction of a plane wave at a sinusoidally stratified dielectric grating,” J. Opt. Soc. Am. 56, 1502–1509 (1966).
    [CrossRef]
  7. F. G. Kaspar, “Diffraction by thick, periodically stratified gratings with complex dielectric constant,” J. Opt. Soc. Am. 63, 37–45 (1973).
    [CrossRef]
  8. S. Kessler, R. Kowarschik, “Diffraction efficiency of volume holograms,” Opt. Quantum Electron. 7, 1–14 (1975).
    [CrossRef]
  9. R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
    [CrossRef]
  10. M. G. Moharam, T. K. Gaylord, J. R. Leger, “Diffractive optics modeling,” J. Opt. Soc. Am. A 12, 1026–1027 (1995).
    [CrossRef]
  11. J. R. Leger, M. G. Moharam, T. K. Gaylord, “Diffractive optics modeling: introduction [to the feature issue],” Appl. Opt. 34, 2399–2400 (1995).
    [CrossRef] [PubMed]
  12. G. Bao, D. C. Dobson, J. A. Cox, “Mathematical studies in rigorous grating theory,” J. Opt. Soc. Am. A 12, 1029–1042 (1995).
    [CrossRef]
  13. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  14. T. W. Nee, “Surface-irregularity-enhanced subband resonance of seminconductos. I. General theory,” Phys. Rev. B 29, 3225–3238 (1984).
    [CrossRef]
  15. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).
  16. T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, S. Shahriar, “Polarization of holographic grating diffraction. II. Experiment,” J. Opt. Soc. Am. A 21, 532–539 (2004).
    [CrossRef]
  17. R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw Hill, New York, New York, 1995), Vol. II, pp. 22.1–22.37.
  18. S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), pp. 60.1–60.24.
  19. M. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 44.
  20. R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 149.
  21. S. F. Nee, “Polarization of specular reflection and near-specular scattering by a rough surface,” Appl. Opt. 35, 3570–3582 (1996).
    [CrossRef]

2004 (1)

2003 (1)

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

1996 (1)

1995 (4)

1984 (1)

T. W. Nee, “Surface-irregularity-enhanced subband resonance of seminconductos. I. General theory,” Phys. Rev. B 29, 3225–3238 (1984).
[CrossRef]

1975 (2)

S. Kessler, R. Kowarschik, “Diffraction efficiency of volume holograms,” Opt. Quantum Electron. 7, 1–14 (1975).
[CrossRef]

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

1973 (1)

1969 (1)

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

1966 (1)

Alferness, R.

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 149.

Bao, G.

Bashara, N. M.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 149.

Burckhardt, C. B.

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw Hill, New York, New York, 1995), Vol. II, pp. 22.1–22.37.

Cox, J. A.

Dobson, D. C.

Gaylord, T. K.

Grann, E. B.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

Kaspar, F. G.

Kessler, S.

S. Kessler, R. Kowarschik, “Diffraction efficiency of volume holograms,” Opt. Quantum Electron. 7, 1–14 (1975).
[CrossRef]

Kleinschmit, M.

T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, S. Shahriar, “Polarization of holographic grating diffraction. II. Experiment,” J. Opt. Soc. Am. A 21, 532–539 (2004).
[CrossRef]

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

Kogelnik, H.

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

Kowarschik, R.

S. Kessler, R. Kowarschik, “Diffraction efficiency of volume holograms,” Opt. Quantum Electron. 7, 1–14 (1975).
[CrossRef]

Leger, J. R.

Moharam, M. G.

Nee, S. F.

Nee, S.-M. F.

T.-W. Nee, S.-M. F. Nee, M. Kleinschmit, S. Shahriar, “Polarization of holographic grating diffraction. II. Experiment,” J. Opt. Soc. Am. A 21, 532–539 (2004).
[CrossRef]

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), pp. 60.1–60.24.

Nee, T. W.

T. W. Nee, “Surface-irregularity-enhanced subband resonance of seminconductos. I. General theory,” Phys. Rev. B 29, 3225–3238 (1984).
[CrossRef]

Nee, T.-W.

Pommet, D. A.

Riccobono, J.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

M. S. Shahriar, J. Riccobono, W. Weathers, “Holographic beam combiner,” in Proceedings of the IEEE International Conference on Microwaves and Optics (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 10–14.

M. S. Shahriar, J. Riccobono, W. Weathers, “Highly Bragg selective holographic laser beam combiner,” presented at the Solid State and Diode Laser Technology Review, Albuquerque, N.M., June 5–8, 2000.

Shahriar, M. S.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

M. S. Shahriar, J. Riccobono, W. Weathers, “Highly Bragg selective holographic laser beam combiner,” presented at the Solid State and Diode Laser Technology Review, Albuquerque, N.M., June 5–8, 2000.

M. S. Shahriar, J. Riccobono, W. Weathers, “Holographic beam combiner,” in Proceedings of the IEEE International Conference on Microwaves and Optics (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 10–14.

Shahriar, S.

Shen, J. T.

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

van de Hulst, M. C.

M. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 44.

Weathers, W.

M. S. Shahriar, J. Riccobono, W. Weathers, “Holographic beam combiner,” in Proceedings of the IEEE International Conference on Microwaves and Optics (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 10–14.

M. S. Shahriar, J. Riccobono, W. Weathers, “Highly Bragg selective holographic laser beam combiner,” presented at the Solid State and Diode Laser Technology Review, Albuquerque, N.M., June 5–8, 2000.

Appl. Opt. (2)

Appl. Phys. (1)

R. Alferness, “Analysis of optical propagation in thick holographic gratings,” Appl. Phys. 7, 29–33 (1975).
[CrossRef]

Bell Syst. Tech. J. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Opt. Commun. (1)

M. S. Shahriar, J. Riccobono, M. Kleinschmit, J. T. Shen, “Coherent and incoherent beam combination using hologram substrates,” Opt. Commun. 220/1–3, 75–83 (2003).
[CrossRef]

Opt. Quantum Electron. (1)

S. Kessler, R. Kowarschik, “Diffraction efficiency of volume holograms,” Opt. Quantum Electron. 7, 1–14 (1975).
[CrossRef]

Phys. Rev. B (1)

T. W. Nee, “Surface-irregularity-enhanced subband resonance of seminconductos. I. General theory,” Phys. Rev. B 29, 3225–3238 (1984).
[CrossRef]

Other (8)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962).

R. A. Chipman, “Polarimetry,” in Handbook of Optics, M. Bass, E. W. Van Stryland, D. R. Williams, W. L. Wolfe, eds. (McGraw Hill, New York, New York, 1995), Vol. II, pp. 22.1–22.37.

S.-M. F. Nee, “Polarization measurement,” in The Measurement, Instrumentation and Sensors Handbook, J. G. Webster, ed. (CRC Press, Boca Raton, Fla., 1999), pp. 60.1–60.24.

M. C. van de Hulst, Scattering of Light by Small Particles (Wiley, New York, 1957), p. 44.

R. M. A. Azzam, N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, Amsterdam, 1977), p. 149.

Digital Optical Technologies, Inc., “Holographic beam combiner for ladars, printers, fiber amplifiers and cancer treatment,” MDA SBIR Phases I & II program 1999–2003.

M. S. Shahriar, J. Riccobono, W. Weathers, “Holographic beam combiner,” in Proceedings of the IEEE International Conference on Microwaves and Optics (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 10–14.

M. S. Shahriar, J. Riccobono, W. Weathers, “Highly Bragg selective holographic laser beam combiner,” presented at the Solid State and Diode Laser Technology Review, Albuquerque, N.M., June 5–8, 2000.

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Figures (8)

Fig. 1
Fig. 1

Reflected, transmitted, and diffracted beams of the holographic grating sample.

Fig. 2
Fig. 2

Angular selectivity of E1 and E2 coupling effects. For n=1.5, ϕK=Φ=0°, λ/x0=1, and m=-1, (θK, θ, Ө) relations are shown.

Fig. 3
Fig. 3

Angular selectivity of E2 coupling. For n=1.5, ϕK=Φ=0°, λ/x0=0.5, 1.0, 1.5 and m=-1, (θK, θ, Ө) relations are shown.

Fig. 4
Fig. 4

Angular selectivity of E2 coupling. For n=1.5, ϕK=20°, λ/x0=0.5, 1.0, 1.5, and m=-1, (θ, Ө, Φ) relations are shown.

Fig. 5
Fig. 5

One-dimensional holographic-grating lattice profile function. x0=1, u0=0.1 and 0.2821 µm. The profile from Eq. (45) is shown by the dashed curve.

Fig. 6
Fig. 6

Diffracted angle-dependent beam profiles of the diffracted transmittances (m=-1) with (solid curve) u0=0.1 and (dashed curve) 0.2821 µm. Nx=500 and (θ, Ө, Φ)=(40°, 20.6042°, -180°) are chosen.

Fig. 7
Fig. 7

Transmittance (top) and diffraction efficiency (bottom) of transmitted and diffracted beams (Ta and Tb). n=1.5, (θK, ϕK)=(5.902773°, 0), (θ, Ө, Φ)=(40°, 20.6042°, -180°), λ=x0=1 µm, and sample thickness=2 mm are chosen.

Fig. 8
Fig. 8

Ellipsometric parameters ψ and Δ of transmitted and diffracted beams. Parameters are the same as in Fig. 7.

Tables (1)

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Table 1 Calculated Mueller Matrix Propertiesa

Equations (72)

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K=2πnK/x0=[Kx, Ky, Kz],
nK=[cos θK cos ϕK, cos θK sin ϕK, sin θK],
Ei(x)=[Eip cos θ, Eis, Eip sin θ]exp[ik(x sin θ-z cos θ)].
E=E0+Edf,
P(x)=αE(x)ρg(x),
ρg(x)=1AuΣlf(nK·x-lx0), l=, -2, -1, 0, 1, 2,.
f(u)=1LuΣpF(p)exp(ipu),
E(x)=E0=E1(x)+E2(x),0<z<a,
E1(x)=E1exp[ik(x sin θ+z cos θ)]=-E1p cos θE1sE1p sin θexp[ik(x sin θ+z cos θ)],
E2(x)=E2exp[ik(x sin θ-z cos θ)]=E2p cos θE2sE2p sin θexp[ik(x sin θ-z cos θ)],
sin θ=n sin θ,
P(x, y, z)=Σqx exp[iqxx]Σqy exp[iqyy]δqy tanϕK(qx-k sin θ)×W(qx)[E1exp(iQ1z)+E2exp(-iQ2z)],
Q1=k cos θ+tan θK(qx-k sin θ)/cos ϕK=k cos θ+mKz,
Q2=k cos θ-tan θK(qx-k sin θ)/cos ϕK=k cos θ-mKz
W(qx)=ApolFqx-k sin θcos θK cos ϕKg(qx-k sin θ)x0cos θK cos ϕK,
g(β)=Σl exp[-ilβ]/Nx=sin NxβNx sin β.
Apol=αNx/(AuLu)
β=(qx-k sin θ)x0cos θK cos ϕK=2πm,m=-1, 0, 1,,
qx=k sin θ+mKx=k sin θ+mKx.
qy=tanϕK(qx-k sin θ)=mKy.
Edf(x)=Σqx exp[iqxx]Σqy exp[iqyy]Edf(qx, qy, z).
[d2/dz2+k2-qx2-qy2]Edf(qx, qy, z)=SE(qx, qy, z),
SE(qx, qy, z)=δqy tan ϕK(qx-k sin θ)W(qx)×[S1 exp(iQ1z)+S2 exp(-iQ2z)].
S1=4πE1p[(k2-qx2)cos θ+qxQ1 sin θ]+E1sqxqyE1pqy(-qx cos θ+Q1 sin θ)+E1s(qy2-k2)E1p[(Q12-k2)sin θ-qxQ1 cos θ]+E1sqyQ1,
S2=4πE2p[(-k2+qx2)cos θ-qxQ2 sin θ]+E2sqxqyE2pqy(qx cos θ-Q2 sin θ)+E2s(qy2-k2)E2p[(Q22-k2)sin θ-qxQ2 cos θ]-E2sqyQ2,
E=Er=-Erp cos Ө cos Φ-Ers sin Φ-Erp cos Ө sin Φ+Ers cos ΦErp sin Ө×exp[ik(x sin Ө cos Φ+y sin Ө sin Φ+z cos Ө)].
E=Et=Etp cos Ө cos Φ-Ets sin ΦEtp cos Ө sin Φ+Ets cos ΦEtp sin Өexp[ik(x sin Ө cos Φ+y sin Ө sin Φ-z cos Ө)].
E=E1+E2,
E1=-E1p cos Ө cos Φ-E1s sin Φ-E1p cos Ө sin Φ+E1s cos ΦE1p sin Ө+Ax(z)Ay(z)Az(z)×exp[ik(x sin Ө cos Φ+y sin Ө sin Φ+z cos Ө)],
E2=E2p cos Ө cos Φ-E2s sin ΦE2p cos Ө sin Φ+E2s cos ΦE2p sin Ө+Bx(z)By(z)Bz(z)×exp[ik(x sin Ө cos Φ+y sin Ө sin Φ-z cos Ө)],
A(z)=12ik cos Ө0zdz exp(-ikz cos Ө)×SE(qx, qy, z),
B(z)=12ik cos Өzadz exp(ikz cos Ө)×SE(qx, qy, z).
k sin Ө=k sin Ө.
γ±(x)=exp(±ix)-1±ix
k cos θ+mKz=k cos Ө,
k cos θ-mKz=k cos Ө.
E=E(z)exp[ik(x sin Ө cos Φ+y sin Ө sin Φ-z cos Ө)],
E(z)=Ep(z)cos Ө cos Φ-Es(z)sin ΦEp(z)cos Ө sin Φ+Es(z)cos ΦEp(z)sin Ө.
ddzE(z)=i2k cos Өexp(ikz cos Ө)SE(qx, qy, z),
SE(qx, qy, z)=δqy,tan ϕK(qx-k sin θ)W(qx)S2 exp(-iQ2z),
S2=4πEp(z)[(-k2+qx2)cos θ-qxQ2 sin θ]+Es(z)qxqyEp(z)qy(qx cos θ-Q2 sin θ)+Es(z)(qy2-k2)Ep(z)[(Q22-k2)sin θ-qxQ2 cos θ]-Es(z)qyQ2.
Ea(z)=Eap(z)cos θEas(z)Eap(z)sin θ,
Eb(z)=Ebp(z)cos Ө cos Φ-Ebs(z)sin ΦEbp(z)cos Ө sin Φ+Ebs(z)cos ΦEbp(z)sin Ө.
ddzEb(z)=iW2k cos Өexp(ikz cos θ)×[WbaSba(Ө, Φ, θ, 0, z)+WbbSbb(Ө, Φ, Ө, Φ, z)],
ddzEa(z)=iW2k cos Өexp(ikz cos θ)×[WaaSaa(θ, 0, θ, 0, z)+WabSab(θ, 0, Ө, Φ, z)],
ddzEap(z)Ebp(z)Eas(z)Ebs(z)=χEap(z)Ebp(z)Eas(z)Ebs(z),
Ebs(a)=Ebp(a)=0,
Eas(a)=ts(θ, n)Eis,
Eap(a)=tp(θ, n)Eip.
Ej=tj(θ, n)Ej(0),j=ap,as,bp,bs.
Ej=Ej(p)Eip+Ej(s)Eis,j=ap,as,bp,bs.
T1=M11(1)(1=a,b).
χ=-ih01/cos θη cos(θ-Ө)/cos θ00η cos(θ-Ө)/cos Ө1/cos Ө0000cos θη cos θ00η cos Өcos Ө,
h0=2πknApolg(qx-k sin θ)x0cos θK cos ϕK,
η=Fqx-k sin θcos θK cos ϕK.
ddzEaj(z)Ebj(z)=χa(j)χb(j)χb(j)χa(j)Eaj(z)Ebj(z),j=s,p.
ka(u)=(sin θ, 0, -cos θ),
kb(u)=(sin Ө cos Φ, sin Ө sin Φ, -cos Ө).
χ=-ih01/cos θuη/cos θu00η/cos θu1/cos θu0000cos θuη cos θu00η cos θucos θu.
cos(2θu)=ka(u)·kb(u)=sin θ sin Ө cos Φ+cos θ cos Ө.
Eap(u)(0)=exp(ih0a/cos θu)cos(h0aη/cos θu)Eap(u)(a),
Ebp(u)(0)=i exp(ih0a/cos θu)sin(h0aη/cos θu)Eap(u)(a),
Eas(u)(0)=exp(ih0a cos θu)cos(h0aη cos θu)Eas(u)(a),
Ebs(u)(0)=i exp(ih0a cos θu)sin(h0aη cos θu)Eas(u)(a).
P(Ө, Φ)=W2|γ±|2.
f(u)=1u0πexp-u2u02.
f(u)=2x0cos2πux0
η=Tb/(Ta+Tb)
M(j)=R(j)1Px(j)00Px(j)1-2Dv0000Py(j)Pz(j)00-Pz(j)Py(j)
(j=a,b),
M(a)=0.04451-0.7860.3610.026-0.5130.390±0.5840.1480.696±0.8360.0960.1670.031±0.136-0.178-0.450,
M(b)=0.706810.0340.0940.0290.0430.9100.0320.4030.0900.0800.979-0.1790.032±0.3940.1980.892.

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