Abstract

The boundary diffraction method (BDM) is an approximate method that permits the derivation of analytic solutions for the output beams, both forward and backward propagating, that arise from the fundamental nature of holographic gratings. The method is based on the assumption that the volume scatter inside the grating can be supplemented by boundary diffraction coefficients. The boundary diffraction method is used for analysis of thick transmission geometry gratings in a unified way that deals with both the slanted and the unslanted cases. During the analysis, evidence emerges for the superiority of the first-order two-wave beta-value method over the Kogelnik k-vector closure method. The BDM is then further generalized to the case of a volume transmission grating, index matched to its surroundings, and replayed normally on-Bragg, i.e., satisfying the Bragg condition for normal incidence. The analytic equations derived are compared with results calculated with the rigorous coupled-wave method.

© 1994 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating sprurious beams produced by volume gratings,” Electron. Lett. 26, 1840–1841 (1990).
    [CrossRef]
  2. J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
    [CrossRef]
  3. J. T. Sheridan, L. Solymar, “Spurious beams in reflection gratings: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
    [CrossRef]
  4. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).
  5. N. Uchido, “Calculation of the diffraction efficiency in hologram gratings attenuated along the direction perpendicular to the grating vector,”J. Opt. Soc. Am. 63, 280–387 (1973).
    [CrossRef]
  6. A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Optoelectronics 5, 606–614 (1990).
  7. G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).
  8. M. P. Owen, A. A. Ward, L. Solymar, “Internal reflections in bleached reflection holograms,” Appl. Opt. 22, 159–163 (1983).
    [CrossRef] [PubMed]
  9. T. K. Gaylord, M. G. Moharam, “Analysis and application of optical diffraction by gratings,” Proc. Inst. Electr. Eng. 73, 894–937 (1985).
    [CrossRef]
  10. J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
    [CrossRef]
  11. H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
    [CrossRef]
  12. L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).
  13. R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).
  14. J. T. Sheridan, “Stacked volume holographic gratings. Part I: transmission gratings in series,” Optik (to be published).
  15. H. J. Gerritsen, “Dispersion effects in relief holograms immersed in near index matched liquids or Christiansen revisited,” Appl. Opt. 25, 2382–2385 (1986).
    [CrossRef] [PubMed]
  16. H. J. Gerritsen, D. K. Thornton, S. R. Bolton, “Application of Kogelnik’s two-wave theory to deep, slanted, highly efficient, relief transmission gratings,” Appl. Opt. 30, 807–814 (1991).
    [CrossRef] [PubMed]
  17. J. Maser, G. Schmal, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
    [CrossRef]

1993 (1)

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

1992 (4)

J. Maser, G. Schmal, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in reflection gratings: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

1991 (1)

1990 (2)

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Optoelectronics 5, 606–614 (1990).

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating sprurious beams produced by volume gratings,” Electron. Lett. 26, 1840–1841 (1990).
[CrossRef]

1986 (1)

1985 (1)

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

1983 (1)

1973 (1)

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

Bolton, S. R.

Cooke, D. J.

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Gaylord, T. K.

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

Gerritsen, H. J.

Gluch, E.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

James, G. L.

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).

Kobolla, H.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

Maser, J.

J. Maser, G. Schmal, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

Moharam, M. G.

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

Owen, M. P.

Ramsbottom, A.

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Optoelectronics 5, 606–614 (1990).

Schmal, G.

J. Maser, G. Schmal, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

Schmidt, J.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

Schwider, J.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

Sheridan, J. T.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in reflection gratings: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating sprurious beams produced by volume gratings,” Electron. Lett. 26, 1840–1841 (1990).
[CrossRef]

J. T. Sheridan, “Stacked volume holographic gratings. Part I: transmission gratings in series,” Optik (to be published).

Solymar, L.

J. T. Sheridan, L. Solymar, “Diffraction by volume gratings: approximate solution in terms of boundary diffraction coefficients,” J. Opt. Soc. Am. A 9, 1586–1591 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Spurious beams in reflection gratings: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating sprurious beams produced by volume gratings,” Electron. Lett. 26, 1840–1841 (1990).
[CrossRef]

M. P. Owen, A. A. Ward, L. Solymar, “Internal reflections in bleached reflection holograms,” Appl. Opt. 22, 159–163 (1983).
[CrossRef] [PubMed]

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

Streibl, N.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

Thornton, D. K.

Uchido, N.

Voelkel, R.

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

Ward, A. A.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2945 (1969).

Electron. Lett. (1)

J. T. Sheridan, L. Solymar, “Boundary diffraction coefficients for calculating sprurious beams produced by volume gratings,” Electron. Lett. 26, 1840–1841 (1990).
[CrossRef]

J. Mod. Opt. (2)

J. T. Sheridan, “A comparison of diffraction theories for off-Bragg replay,” J. Mod. Opt. 39, 1709–1718 (1992).
[CrossRef]

H. Kobolla, J. T. Sheridan, E. Gluch, J. Schmidt, R. Voelkel, J. Schwider, N. Streibl, “Holographic 2-D mixed polarization deflection elements,” J. Mod. Opt. 40, 613–624 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Commun. (2)

J. T. Sheridan, L. Solymar, “Spurious beams in reflection gratings: a solution in terms of boundary diffraction coefficients,” Opt. Commun. 94, 8–12 (1992).
[CrossRef]

J. Maser, G. Schmal, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[CrossRef]

Optoelectronics (1)

A. Ramsbottom, “Characteristics and properties of powered reflection holographic optical elements,” Optoelectronics 5, 606–614 (1990).

Proc. Inst. Electr. Eng. (1)

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

Other (4)

G. L. James, Geometrical Theory of Diffraction for Electromagnetic Waves, IEE Electromagnetic Wave Series (Peregrinus, London, 1976).

L. Solymar, D. J. Cooke, Volume Holography and Volume Gratings (Academic, London, 1981).

R. R. A. Syms, Practical Volume Holography (Clarendon, Oxford, 1990).

J. T. Sheridan, “Stacked volume holographic gratings. Part I: transmission gratings in series,” Optik (to be published).

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 (10)

Fig. 1
Fig. 1

Two-wave asymmetric transmission geometry grating. The volume grating is index matched to the surrounding material and has an average index of ɛ2 = 1.

Fig. 2
Fig. 2

Ewald diagram, BVM representation, for the asymmetric transmission case.

Fig. 3
Fig. 3

z = 0 boundary. All the primary and spurious waves that are incident are shown. The wave vectors of the beams and the grating vector K are also shown.

Fig. 4
Fig. 4

z = d boundary; other information as in Fig. 3.

Fig. 5
Fig. 5

(a) Comparison of the BDM–BVM result (dashed curve) with the RCWM result (solid curve) for the spurious reflection order, I0/0 with θ1 = 30°, θ2 = 26°, and ϕg = 92°. Normalized diffraction efficiency as a function of the grating thickness is shown. (b) The BDM–KVCM result (dashed curve) is compared with the RCWM result (solid curve) for the same grating as in (a). The agreement is not so good as in (a).

Fig. 6
Fig. 6

Transmission grating, ϕg > 90°, and its Ewald diagram. Thick lines, primary beams; thin dashed lines, spurious boundary diffracted waves. This is the case examined in Fig. 5.

Fig. 7
Fig. 7

Three-wave transmission case. The grating is index matched to its surroundings, and the input beam is incident normally on-Bragg.

Fig. 8
Fig. 8

Ewald diagram for Fig. 7. All six BVM wave vectors are shown. The −1/d forward order is produced by off-Bragg replay of the volume grating. θ+1 = θ−1 = θ.

Fig. 9
Fig. 9

(a) Three contributions to T−1: the off-Bragg boundary diffraction at both boundaries and the off-Bragg volume contribution. (b) Three contributions to R−1: an off-Bragg scatter from the z = 0 boundary and two spurious beams resulting from scatter at the z = d boundary, one from an off-Bragg boundary scatter and the other from an on-Bragg scatter. Both of these spurious beams replay the volume grating on-Bragg. (c) Contributions to R0. Both are due to the on-Bragg replay of the volume grating by two spurious waves, one on-Bragg boundary scatter and the other off-Bragg, arising at z = d. (d) Contributions to R+1: the on-Bragg boundary scatter arising at z = 0 and the on-Bragg boundary scatter arising at z = d, which replays the volume grating off-Bragg.

Fig. 10
Fig. 10

I−1/0(z). The solid curve is the approximate curve from Eq. (34). The dashed curve is the RCWM curve. The slant angle ϕg = 35°; the diffracton angle is θ2 = 70°. (b) I+1/0(z). The solid curve is the approximate curve from expression (36). Other information is as in (a). (c) I−1/d(z). The solid curve is the approximate curve from Eq. (29). Other information is as in (a). (d) I−1/d(z) for a grating with ϕg = 15° and θ2 = 30°. The solid curve is the approximate curve from Eq. (29). The dashed curve is the RCWM curve.

Equations (43)

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

2 E y ( x , z ) + β 2 [ ɛ 2 + ɛ 21 cos ( K · r ) ] E y ( x , z ) = 0 ,
E y ( x , z ) = A 0 ( z ) exp ( - j ρ 0 · r ) + A + 1 ( z ) exp ( - j ρ + 1 · r ) + a 0 ( z ) exp [ - j σ 0 · ( r - d i z ) ] + a + 1 ( z ) exp [ - j σ + 1 · ( r - d i z ) ] .
cos ( θ 0 ) d A 0 d z + j κ A + 1 = 0 , cos ( θ + 1 ) d A + 1 d z + j κ A 0 = 0 ,
- cos ( θ 0 ) d a 0 d z + j κ a + 1 exp [ - j ( ψ z - K z d ) ] = 0 ,
- cos ( θ + 1 ) d a + 1 d z + j κ a 0 exp [ + j ( ψ z - K z d ) ] = 0.
1 + R 0 = A 0 ( 0 ) + a 0 ( 0 ) exp ( + j σ 0 z d ) ,
R + 1 = A + 1 ( 0 ) + a + 1 ( 0 ) exp ( + j σ + 1 z d ) .
- j ρ 0 z ( R 0 - 1 ) = [ - j ρ 0 z A 0 ( 0 ) + d A 0 d z | z = 0 ] + [ - j σ 0 z a 0 ( 0 ) + d a 0 d z | z = 0 ] exp ( + j σ 0 z d ) ,
+ j ρ + 1 z R + 1 = [ - j ρ + 1 z A + 1 ( 0 ) + d A + 1 d z | z = 0 ] + [ - j σ + 1 z a + 1 ( 0 ) + d a + 1 d z | z = 0 ] exp ( + j σ + 1 z d ) .
d A n ( z ) d z | z = ζ = - j κ cos ( θ n ) A m ( ζ )
a i ( z ) ɛ 21 ( i = 1 , 2 ) d a i d z ɛ 21 2 0.
A + 1 ( 0 ) ɛ 21 R 0 - a 0 ( 0 ) exp ( + j σ 0 z d ) .
R + 1 = a + 1 ( 0 ) exp ( + j σ + 1 z d ) - κ 2 cos ( θ + 1 ) ρ + 1 z A 0 ( 0 ) ,
a 0 ( d ) = κ 2 cos ( θ 0 ) ρ 0 z A + 1 ( d ) exp ( - j ρ 0 z d ) = ɛ 21 8 cos 2 ( θ 0 ) j sin ( ν ) exp [ - j β cos ( θ 0 ) d ] ,
a + 1 ( d ) = κ 2 cos ( θ + 1 ) ρ + 1 z A 0 ( d ) exp ( - j ρ + 1 z d ) = ɛ 21 8 cos 2 ( θ + 1 ) cos ( ν ) exp [ - j β cos ( θ + 1 ) d ] ,
S 0 ( z ) = a 0 ( z ) exp [ + j ( ψ z - K z d ) / 2 ] : S + 1 ( z ) = a + 1 ( z ) exp [ - j ( ψ z - K z d ) / 2 ] .
S 0 ( z ) = 1 2 Φ cos ( θ 0 ) { [ cos ( θ 0 ) ( Φ + ψ 2 ) S 0 ( d ) + κ S + 1 ( d ) ] × exp [ + j Φ ( z - d ) ] - [ κ S + 1 ( d ) - cos ( θ 0 ) ( Φ - ψ 2 ) S 0 ( d ) ] × exp [ - j Φ ( z - d ) ] } ,
S + 1 ( z ) = 1 2 Φ κ { ( Φ - ψ 2 ) × [ cos ( θ 0 ) ( Φ + ψ 2 ) S 0 ( d ) + κ S + 1 ( d ) ] × exp [ + j Φ ( z - d ) ] - ( Φ + ψ 2 ) × [ κ S + 1 ( d ) - cos ( θ 0 ) ( Φ - ψ 2 ) S 0 ( d ) ] × exp [ - j Φ ( z - d ) ] } ,
ψ ν Φ ψ / 2 = K z .
d 2 A - 1 ( z ) d z 2 - 2 j β cos ( θ ) d A - 1 ( z ) d z + 2 β κ A 0 ( z ) exp ( - j ψ - 1 z ) = 0 ,
d A 0 ( z ) d z + j κ [ A + 1 ( z ) + A - 1 ( z ) exp ( + j ψ - 1 z ) ] = 0 ,
cos ( θ ) d A + 1 ( z ) d z + j κ A 0 ( z ) = 0.
ψ - 1 = 2 β [ 1 - cos ( θ - 1 ) ] ,             ν = κ cos ( θ + 1 ) .
R - 1 = A - 1 ( 0 ) + A - 1 backscatter ,
+ j β cos ( θ ) R - 1 = d A - 1 ( z ) d z | z = 0 - j β cos ( θ ) A - 1 ( 0 ) + j β cos ( θ ) A - 1 backscatter .
d A - 1 ( z ) d z | z = 0 = + j 2 β cos ( θ ) A - 1 ( 0 ) .
A - 1 ( 0 ) = - κ 2 β cos 2 ( θ ) - 1 4 β 2 cos 2 ( θ ) d 2 A - 1 ( z ) d z 2 | z = 0 ,
A - 1 ( z ) = A - 1 ( 0 ) exp [ + j 2 β cos ( θ ) z ] - 2 β κ { - cos ( ν z ) exp ( - j ψ - 1 z ) ψ - 1 [ 2 β cos ( θ ) + ψ - 1 ] + 1 2 β cos ( θ ) ψ - 1 - exp [ + j 2 β cos ( θ ) z ] 2 β cos ( θ ) [ 2 β cos ( θ ) + ψ - 1 ] } .
T - 1 = A - 1 ( d ) exp [ - j β cos ( θ ) d ] - A - 1 ( 0 ) cos ( ν d ) exp { - j β [ 2 - cos ( θ ) ] d } ,
[ - j β cos ( θ ) ] A - 1 ( d ) exp [ - j β cos ( θ ) d ] + d A - 1 ( z ) d z | z = d exp [ - j β cos ( θ ) d ] - [ + j β cos ( θ ) ] A - 1 ( 0 ) cos ( ν d ) exp { - j β [ 2 - cos ( θ ) ] d } = [ - j β cos ( θ ) ] T - 1 .
d A - 1 ( z ) d z | z = d = [ + j 2 β cos ( θ ) ] A - 1 ( 0 ) cos ( ν d ) exp ( - j ψ - 1 d ) .
A - 1 ( 0 ) = - κ cos ( θ ) [ 2 β cos ( θ ) + ψ - 1 ] = - ɛ 21 8 cos ( θ ) .
T - 1 exp [ + j β cos ( θ ) d ] = { - ɛ 21 8 1 cos ( θ ) [ 1 - cos ( θ ) } [ 1 - cos ( ν d ) exp ( - j ψ - 1 d ) ] ,
d 2 A - 1 ( z ) d z 2 | z = 0 = - κ ψ - 1 .
d 2 A - 1 ( z ) d z 2 | z = 0 = - 2 β κ .
A 0 ( 0 ) cos ( ν d ) exp ( - j β d ) ,
- j A 0 ( 0 ) ( c r c s ) 1 / 2 sin ( ν d ) exp [ - j β cos ( θ ) d ] ,
ϕ ( d ) = 2 π d Λ / cos ( ϕ ) = K z d = β [ 1 - cos ( θ ) ] d .
- A - 1 ( 0 ) exp { - j β [ 1 - cos ( θ ) ] d } .
R - 1 = ɛ 21 8 cos ( θ ) ( 1 + [ cos 2 ( ν d ) + sin 2 ( ν d ) ] × exp { - j β [ 1 + cos ( θ ) ] d + j ϕ ( d ) } ) ,
R 0 ɛ 21 8 sin ( 2 ν d ) cos ( θ ) ,
R + 1 ɛ 21 8 cos 2 ( θ ) { 1 + cos ( ν d ) exp [ - j 2 β cos ( θ ) d ] } ,
ɛ 21 8 cos 2 ( θ ) = ɛ 21 8 [ 1 - cos ( θ ) ] cos ( θ ) cos ( θ ) = 1 2 .

Metrics