Abstract

Two-dimensional coupled-wave theory is used to analyze a volume holographic lens for conversion of plane waves into cylindrical waves at fixed frequency. The full two-dimensional solution is found to give good agreement with localized use of one-dimensional theory over a wide range of conditions. It is found that high on-Bragg efficiencies may be achieved if the thickness–focal-length ratio of the lens is small and the recording and reconstructing waves are slowly varying and that reduction in output over the lens surface under off-Bragg conditions is nonuniform.

© 1982 Optical Society of America

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References

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  1. C. B. Burkhardt, “Efficiency of a dielectric grating,” J. Opt. Soc. Am. 57, 601–603 (1967).
    [CrossRef]
  2. H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 68, 2909–2947 (1969).
    [CrossRef]
  3. L. Solymar and M. P. Jordan, “Analysis of cylindrical-to-plane-wave conversion by volume holograms,” Electron. Lett. 12, 142 (1976).
    [CrossRef]
  4. M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
    [CrossRef]
  5. J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2968–2710 (1971).
    [CrossRef]
  6. M. R. Latta and R. V. Pole, “Design techniques for forming 488 nm holographic lenses with reconstruction at 633 nm,” Appl. Opt. 18, 2418–2421 (1979).
    [CrossRef] [PubMed]
  7. J. M. Moran, “Compensation of aberrations due to a wavelength shift in holography,” Appl. Opt. 10, 1909–1913 (1971).
    [CrossRef] [PubMed]
  8. R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. 13, 415–419 (1981).
    [CrossRef]
  9. L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
    [CrossRef]
  10. G. Hatakoshi and S. Tanaka, “Grating lenses for integrated optics,” Opt. Lett. 2, 142–144 (1978).
    [CrossRef] [PubMed]
  11. G. Hatakoshi and S. Tanaka, “Coupled-mode theory for a plane and a cylindrical wave in optical waveguide Bragg grating lenses,” J. Opt. Soc. Am. 71, 40–48 (1981).
    [CrossRef]
  12. G. Hatakoshi and S. Tanaka, “Coupling of two cylindrical guided waves in optical waveguide Bragg grating lenses,” J. Opt. Soc. Am. 71, 121–123 (1981).
    [CrossRef]
  13. M. P. Jordan and L. Solymár, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
    [CrossRef]
  14. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, p. 449.
  15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 737.
  16. A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 34.

1981 (3)

1979 (1)

1978 (3)

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

M. P. Jordan and L. Solymár, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

G. Hatakoshi and S. Tanaka, “Grating lenses for integrated optics,” Opt. Lett. 2, 142–144 (1978).
[CrossRef] [PubMed]

1977 (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

1976 (1)

L. Solymar and M. P. Jordan, “Analysis of cylindrical-to-plane-wave conversion by volume holograms,” Electron. Lett. 12, 142 (1976).
[CrossRef]

1971 (2)

J. M. Moran, “Compensation of aberrations due to a wavelength shift in holography,” Appl. Opt. 10, 1909–1913 (1971).
[CrossRef] [PubMed]

J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2968–2710 (1971).
[CrossRef]

1969 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 68, 2909–2947 (1969).
[CrossRef]

1967 (1)

Burkhardt, C. B.

Courant, R.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, p. 449.

Erdélyi, A.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 34.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 737.

Hatakoshi, G.

Hilbert, D.

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, p. 449.

Jordan, M. P.

M. P. Jordan and L. Solymár, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

L. Solymar and M. P. Jordan, “Analysis of cylindrical-to-plane-wave conversion by volume holograms,” Electron. Lett. 12, 142 (1976).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 68, 2909–2947 (1969).
[CrossRef]

Latta, J. N.

J. N. Latta, “Computer-based analysis of holography using ray tracing,” Appl. Opt. 10, 2968–2710 (1971).
[CrossRef]

Latta, M. R.

Moran, J. M.

Pole, R. V.

Russell, J.

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 737.

Solymar, L.

R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. 13, 415–419 (1981).
[CrossRef]

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

L. Solymar and M. P. Jordan, “Analysis of cylindrical-to-plane-wave conversion by volume holograms,” Electron. Lett. 12, 142 (1976).
[CrossRef]

Solymár, L.

M. P. Jordan and L. Solymár, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

St, P.

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

Syms, R. R. A.

R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. 13, 415–419 (1981).
[CrossRef]

Tanaka, S.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

L. Solymar, “A general two-dimensional theory for volume holograms,” Appl. Phys. Lett. 31, 820–822 (1977).
[CrossRef]

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 68, 2909–2947 (1969).
[CrossRef]

Electron. Lett. (2)

L. Solymar and M. P. Jordan, “Analysis of cylindrical-to-plane-wave conversion by volume holograms,” Electron. Lett. 12, 142 (1976).
[CrossRef]

M. P. Jordan and L. Solymár, “A note on volume holograms,” Electron. Lett. 14, 271–272 (1978).
[CrossRef]

J. Opt. Soc. Am. (3)

Microwaves Opt. Acoust. (1)

M. P. Jordan, L. Solymar, P. St, and J. Russell, “Wavefront conversion by volume holograms between cylindrical and plane waves,” Microwaves Opt. Acoust. 2, 156–162 (1978).
[CrossRef]

Opt. Lett. (1)

Opt. Quantum Electron. (1)

R. R. A. Syms and L. Solymar, “Localized one-dimensional theory for volume holograms,” Opt. Quantum Electron. 13, 415–419 (1981).
[CrossRef]

Other (3)

R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, New York, 1962), Vol. II, p. 449.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 4th ed. (Academic, New York, 1965), p. 737.

A. Erdélyi, Asymptotic Expansions (Dover, New York, 1956), p. 34.

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Figures (10)

Fig. 1
Fig. 1

General recording geometry for the off-axis lens.

Fig. 2
Fig. 2

General reconstruction geometry for the off-axis lens.

Fig. 3
Fig. 3

Hologram geometry for the fully one-dimensional case.

Fig. 4
Fig. 4

Recording geometry for the localized one-dimensional off-axis lens model.

Fig. 5
Fig. 5

Reconstruction geometry used in numerical examples, showing a finite plane wave of top-hat amplitude distribution incident upon the lens.

Fig. 6
Fig. 6

Variation of the undiffracted wave amplitude A1 and the normalized diffracted wave amplitude A 2 in the central region of the lens with normalized thickness ν, for various values of d/r0, under on-Bragg reconstruction by a finite plane wave.

Fig. 7
Fig. 7

Amplitude profiles for A1 and A2 at the output surface of the lens, under on-Bragg reconstruction by a finite plane wave (d/r0 = 0.1, 2L/r0 = 0.4, ψ = 30°, ν = 1.532).

Fig. 8
Fig. 8

Variation of the normalized diffracted wave amplitude A 2 , and the phase of A 2 additional to the term −1, in the central region of the lens with the off-Bragg parameter χ for ν = π/2 and various values of d/r0, under off-Bragg reconstruction by a finite plane wave.

Fig. 9
Fig. 9

Comparison of approximate and exact solutions for the amplitude profile of A2 at the output surface of the lens under off-Bragg reconstruction by a finite plane wave, for various angular deviations Δθ from Bragg (r0 = 10 cm, d = 10 μm, 2L = 4 cm, λfs = 0.5145 μm, n = 1.5, ψ = 30°, ν = π/2).

Fig. 10
Fig. 10

Comparison of approximate and exact solutions for the amplitude profile of A2 at the output surface of the lens under off-Bragg reconstruction by a finite plane wave, for various angular deviations Δθ from Bragg (r0 = 10 cm, d = 1 cm, 2L = 4 cm, λfs = 0.5145 μm, n = 1.5, ψ = 30°, ν = π/2).

Tables (1)

Tables Icon

Table 1 Limits and Integration Constants for Integrals Used in the Numerical Examples

Equations (32)

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( p ) 2 = 1 ,             ( a 2 p ) = 0.
r = r 0 + r 1 a 10 a 20 cos [ ( β ( p 10 - p 20 ) ] .
2 E + β 2 { 1 + r 1 r 0 a 10 a 20 cos [ β ( p 10 - p 20 ) ] } E = 0
E = A 1 ( r ) a 10 ( r ) exp [ - j β p 1 ( r ) ] + A 2 ( r ) a 20 ( r ) exp [ - j β p 20 ( r ) ] .
A 1 = b 1 a 10 ,             A 2 = 0
p 1 A 1 + j κ 0 ( a 10 a 10 ) a 20 2 exp [ j β ( p 1 - p 10 ) ] A 2 = 0 , p 20 A 2 + j κ 0 a 10 a 10 exp [ j β ( p 10 - p 1 ) ] A 1 = 0 ,
2 A 1 ξ η + [ a 20 2 ( η p 1 ) ξ ( η p 1 a 20 2 ) + j β p 20 ( p 10 - p 1 ) ( ξ p 20 ) ] A 1 η + κ 0 2 a 10 2 a 20 2 ( η p 1 ) ( ξ p 20 ) A 1 = 0 , 2 A 2 ξ η + [ a 10 2 ( ξ p 20 ) η ( ξ p 20 a 10 2 ) + j β p 1 ( p 1 - p 10 ) ( η p 1 ) ] A 2 ξ + κ 0 2 a 10 2 a 20 2 ( η p 1 ) ( ξ p 20 ) A 2 = 0.
E = A 10 a 10 ( r ) exp [ - j β ( x cos ψ - y sin ψ ) ] + A 20 g 1 ( r ) ( r 0 / r ) 1 / 2 exp ( - j β r ) .
E = A 1 ( r ) a 1 ( r ) exp [ - j β ( y sin θ - x cos θ ) ] + A 2 ( r ) g 1 ( r ) ( r 0 / r ) 1 / 2 exp ( + j β r ) .
p 10 ( r ) = - x cos ( ψ ) + y sin ( ψ ) , p 1 ( r ) = - x cos ( θ ) + y sin ( θ ) , p 20 ( r ) = - ( x 2 + y 2 ) 1 / 2 , ξ = y cos ( θ ) + x sin ( θ ) , η = tan - 1 ( y / x ) , a 10 ( r ) = a 1 ( ξ ) , a 20 ( r ) = g 1 ( η ) ( r 0 / r ) .
2 A 1 ξ η - j β Δ θ A 1 η - κ 0 2 a 1 2 ( ξ ) g 1 2 ( η ) r 0 A 1 sin 2 ( η + θ ) = 0 , 2 A 2 ξ η + cot ( η + θ ) A 2 ξ - κ 0 2 a 1 2 ( ξ ) g 1 2 ( η ) r 0 A 2 sin 2 ( η + θ ) = 0.
A 1 = b 1 ( ξ ) a 1 ( ξ ) ,             A 2 = 0 on ξ = r 0 sin ( η + θ ) cos ( η ) .
A 1 ( P ) = b 1 ( ξ B ) a 1 ( ξ B ) + κ 0 exp ( j β Δ θ ξ B ) ξ A ξ B b 1 ( ξ Q ) a 1 ( ξ Q ) × exp ( - j β Δ θ ξ Q ) ( f 1 / f 2 ) 1 / 2 J 1 [ 2 κ 0 ( f 1 f 2 ) 1 / 2 ] d ξ Q , A 2 ( P ) = j κ 0 sin ( η P + θ ) ξ A ξ B b 1 ( ξ Q ) a 1 ( ξ Q ) × exp ( - j β Δ θ ξ Q ) J 0 [ 2 κ 0 ( f 1 f 2 ) 1 / 2 ] d ξ Q ,
f 1 = r 0 η Q η A g 1 2 ( η ) d η sin 2 ( η + θ ) ,             f 2 = ξ B ξ Q a 1 2 ( ξ ) d ξ
ξ A = r 0 sin ( η P + θ ) cos ( θ ) ,             ξ B = ( r 0 - d ) sin ( η P + θ ) cos ( θ ) , η A = tan - 1 { [ ξ A - r 0 sin ( θ ) ] r 0 cos ( θ ) } , η Q = tan - 1 { [ ξ Q - r 0 sin ( θ ) ] r 0 cos ( θ ) } .
A 2 ( P ) = - j ϕ 1 b 1 ( ξ p ) g 1 ( η p ) [ cos ( θ ) ] 1 / 2 cos ( η p ) ν 0 1 Re ( ϕ 2 ) J 0 [ ν ( 1 - α 2 ) 1 / 2 ] d α ,
ϕ 1 = exp [ - j β Δ θ ( r 0 - d 2 ) sin ( η p + θ ) cos ( η p ) ] , ϕ 2 = exp [ j β Δ θ α d 2 sin ( η p + θ ) cos ( η p ) ] ,
ν = κ 0 a 1 ( ξ p ) g 1 ( η p ) d [ cos ( θ ) ] 1 / 2 = κ p d [ cos ( θ ) cos ( η p ) ] 1 / 2 .
A 2 ( P ) = - j ϕ 1 b 1 ( ξ p ) g 1 ( η p ) [ cos ( θ ) ] 1 / 2 cos ( η p ) × sin { ν [ 1 + ( χ ν ) 2 ] 1 / 2 } / [ 1 + ( χ ν ) 2 ] 1 / 2 ,
χ = β Δ θ d sin ( η p + θ ) cos ( η p ) .
A 1 ( P ) = b 1 ( ξ p ) a 1 ( ξ p ) exp ( - j χ ) ( cos { ν [ 1 + ( χ ν ) 2 ] 1 / 2 } + j ( χ ν ) sin { ν [ 1 + ( χ ν ) 2 ] 1 / 2 } / [ 1 + ( χ ν ) 2 ] 1 / 2 ) .
A 2 ( P ) = - j b 1 ( ξ p ) g 1 ( η p ) [ cos ( ψ ) ] 1 / 2 cos ( η p ) .
E 2 = S exp [ - j ( ρ - k ) · r ] ,
S = - j ( C R / C S ) 1 / 2 exp ( - j ξ ) × sin { ν [ 1 + ( ξ 2 / ν 2 ) ] 1 / 2 } / [ 1 + ( ξ 2 / ν 2 ) ] 1 / 2
ν = κ 0 d ( C R C S ) 1 / 2 , ξ = [ β 2 - ( ρ - k ) 2 ] d 2 C s 2 β , C R = cos ( θ ) , C S = cos ( θ ) - k β cos ( ϕ ) .
K p = 2 β sin ( η p + ψ ) 2 ,             ϕ p = π 2 + ( ψ - η p ) 2 ,             κ P = κ 0 [ cos ( η p ) ] 1 / 2 g 1 ( η p ) a 1 ( ξ p ) .
S = b 1 ( ξ p ) S g 1 ( η p ) [ cos ( η p ) ] 1 / 2 exp [ - j β ( r 0 - d ) sin ( η p + θ ) cos ( η p ) ] .
S = - j ϕ 1 b 1 ( ξ p ) [ cos ( θ ) ] 1 / 2 g 1 ( η p ) cos ( η p ) × sin { ν [ 1 + ( χ ν ) 2 ] 1 / 2 } / [ 1 + ( χ ν ) 2 ] 1 / 2 ,
A 1 ( P ) = b 1 - ϕ 3 ν 2 α A α B ϕ 2 { 1 + α ( 1 - α ) [ 1 - ( 1 + α ) d r 0 ] } 1 / 2 × J 1 ( ν { 1 - α 2 [ 1 - ( 1 + α ) d r 0 ] } 1 / 2 ) d α , A 2 ( P ) = - j ϕ 1 [ cos ( θ ) ] 1 / 2 cos ( η p ) ν 2 α A α B ϕ 2 × J 0 ( ν { 1 - α 2 [ 1 - ( 1 + α ) d r 0 ] } 1 / 2 ) d α ,
ϕ 3 = exp [ - j β Δ θ d 2 sin ( η p + θ ) cos ( η p ) ] ,             ν is as Eq . ( 18 ) ,
output surface A 2 g 1 2 r 0 d η ÷ input surface A 1 a 1 2 d ξ ,
A 2 = - j [ cos ( θ ) ] 1 / 2 cos ( η p ) sin ( χ ) χ .