Abstract

First-order perturbation theory is used to describe diffractive optical elements. This method provides an extension of Kirchhoff’s thin element approximation. In particular, the perturbation approximation considers propagation effects due to a finite depth of diffractive structures. The perturbation method is explicitly applied to various problems in diffractive optics, mostly related to the analysis of surface-relief structures. As part of this investigation this approach is compared with alternative extensions of the thin element model. This comparison illustrates that perturbation theory allows a consistent unified treatment of many diffraction phenomena, preserving the simplicity of Fourier optics.

© 1999 Optical Society of America

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References

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  5. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 2, Chap. 13, pp. 1283–1302.
  6. A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1992), Vol. 2, Chap. 17, pp. 722–761 and Chap. 19, pp. 801–815.
  7. M. V. Berry, The Diffraction of Light by Ultrasound (Academic, London, 1966), Chap. 4, pp. 26–30.
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1998 (3)

1997 (1)

1996 (1)

A. W. Lohmann, V. Arrizón, “Computer-generated pseudo-deep holograms,” J. Mod. Opt. 43, 2381–2402 (1996).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

1992 (1)

Y. N. Denisyuk, N. M. Ganzherli, “Pseudodeep holograms: their properties and applications,” Opt. Eng. 31, 731–738 (1992).
[CrossRef]

1989 (1)

1981 (1)

1978 (1)

1968 (1)

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[CrossRef]

1967 (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1967).
[CrossRef]

Arrizón, V.

A. W. Lohmann, V. Arrizón, “Computer-generated pseudo-deep holograms,” J. Mod. Opt. 43, 2381–2402 (1996).
[CrossRef]

Berry, M. V.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, London, 1966), Chap. 4, pp. 26–30.

Brenner, K.-H.

Cho, M. H.

Chugui, Y.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 2, Chap. 13, pp. 1283–1302.

Denisyuk, Y. N.

Y. N. Denisyuk, N. M. Ganzherli, “Pseudodeep holograms: their properties and applications,” Opt. Eng. 31, 731–738 (1992).
[CrossRef]

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 2, Chap. 13, pp. 1283–1302.

Feit, M. D.

Fleck, J. A.

Gabor, D.

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[CrossRef]

Ganzherli, N. M.

Y. N. Denisyuk, N. M. Ganzherli, “Pseudodeep holograms: their properties and applications,” Opt. Eng. 31, 731–738 (1992).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Grann, E. B.

Huang, A.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1992), Chap. 9.8, pp. 427–432.

Jahns, J.

Kim, P. S.

Kim, Y. S.

Kogelnik, H.

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1967).
[CrossRef]

Koronkevitch, V. P.

Krivenkov, B. E.

Laloe, F.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 2, Chap. 13, pp. 1283–1302.

Lohmann, A.

A. Lohmann, Optical Information Processing, available through A. Lohmann, Univ. of Erlangen, Institute of Physics (Erlangen, Germany, 1986), Chap. 16, pp. 78–83.

Lohmann, A. W.

A. W. Lohmann, V. Arrizón, “Computer-generated pseudo-deep holograms,” J. Mod. Opt. 43, 2381–2402 (1996).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Orlando, Fla., 1974), Chap. 3, pp. 95–131.

Messiah, A.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1992), Vol. 2, Chap. 17, pp. 722–761 and Chap. 19, pp. 801–815.

Mikhlyaev, S. V.

Moharam, M. G.

Oh, C. H.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 234–250 and Chap. 3, p. 76.

Park, S.

Pommet, D. A.

Rohrbach, A.

Singer, W.

Song, S. H.

Stroke, G. W.

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[CrossRef]

Swanson, G.

G. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

Testorf, M.

Tiziani, H.

Turunen, J.

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52 and references therein.

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

H. Kogelnik, “Coupled wave theory for thick hologram gratings,” Bell Syst. Tech. J. 48, 2909–2947 (1967).
[CrossRef]

J. Mod. Opt. (1)

A. W. Lohmann, V. Arrizón, “Computer-generated pseudo-deep holograms,” J. Mod. Opt. 43, 2381–2402 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (1)

Y. N. Denisyuk, N. M. Ganzherli, “Pseudodeep holograms: their properties and applications,” Opt. Eng. 31, 731–738 (1992).
[CrossRef]

Opt. Lett. (1)

Proc. R. Soc. London Ser. A (1)

D. Gabor, G. W. Stroke, “The theory of deep holograms,” Proc. R. Soc. London Ser. A 304, 275–289 (1968).
[CrossRef]

Other (10)

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H.-P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52 and references therein.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1992), Chap. 9.8, pp. 427–432.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, New York, 1977), Vol. 2, Chap. 13, pp. 1283–1302.

A. Messiah, Quantum Mechanics (North-Holland, Amsterdam, 1992), Vol. 2, Chap. 17, pp. 722–761 and Chap. 19, pp. 801–815.

M. V. Berry, The Diffraction of Light by Ultrasound (Academic, London, 1966), Chap. 4, pp. 26–30.

A. Lohmann, Optical Information Processing, available through A. Lohmann, Univ. of Erlangen, Institute of Physics (Erlangen, Germany, 1986), Chap. 16, pp. 78–83.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Orlando, Fla., 1974), Chap. 3, pp. 95–131.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7, pp. 234–250 and Chap. 3, p. 76.

G. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” (MIT, Cambridge, Mass., 1991).

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Figures (6)

Fig. 1
Fig. 1

Surface-relief DOE. One surface of a transparent substrate with homogeneous refractive index n is modulated, and the shape of the surface determines the response of the DOE.

Fig. 2
Fig. 2

Diffraction at a transparent wedge with a homogeneous refractive index n: (a) refraction of a plane wave according to Snell’s law, (b) diffraction as predicted by first-order perturbation theory.

Fig. 3
Fig. 3

Diffraction at a linear blazed grating. The periodic structure yields a finite set of discrete diffraction order.

Fig. 4
Fig. 4

Comparison of the perturbation approximation (PA) and the thin element approximation (TEA) with a rigorous coupled-wave analysis (RCWT). The diffraction efficiency of the second diffraction order of a linear blazed grating (d=4λ, n=1.5) is computed with respect to the grating height. The incident plane wave is propagating on axis (β0=n/λ).

Fig. 5
Fig. 5

2 f setup for reconstruction of pseudodeep holograms. The desired response of the tilted DOE is selected with a slit in the Fraunhofer diffraction plane. The width of the dashed rectangle corresponds to the effective thickness of the DOE.

Fig. 6
Fig. 6

DOE as used in planar-integrated micro-optics: (a) A wave front incident along the tilted optical axis is diffracted by the reflective surface relief. (b) Unfolded equivalent setup; the optical element is tilted with respect to the optical axis.

Equations (36)

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[H0+Hpt]u(x, y, z)=-2z2u(x, y, z),
H0=2x2+2y2+k02,
Hpt(x, y, z)=k02[n2(x, y, z)-1].
u(x, y, z)=--a(ν, μ)exp[i2π(νx+μy)]×exp(i2πβz)dνdμ.
ν2+μ2+β2=1λ2,
-2z2a(ν, μ, z)-i4πβ za(ν, μ, z)
=--a(ν, μ, z)H˜pt(ν-ν, μ-μ, z)
×exp[-i2π(β-β)z]dνdμ,
H˜pt(ν, μ, z)=--Hpt(x, y, z)×exp[-i2π(νx+μy)]dxdy.
a(ν, μ, z)=i4πβ--a(ν, μ, 0)×exp[-i2π(β-β)z]dzdνdμ+a(ν, μ, 0).
Hpt(x, y, z)=k02(n2-1)×{pheav(z)-pheav[z-h(x, y)]},
pheav(x)=0:x<01:x0.
a(ν, μ)=k02(n2-1)2(2π)2β(β-β0)δ (ν-ν0, μ-μ0)+asurf (ν, μ)+δ (ν-ν0, μ-μ0),
asurf (ν, μ)=-k02(n2-1)2(2π)2β(β-β0)×--exp{-i2π(β-β0)h(x, y)-i2π[(ν-ν0)x+(μ-μ0)y]}dxdy.
{(β-β0)h(x, y)-[(ν-ν0)x+(μ-μ0)y]}=0.
ν=nλ-1λ2-ν21/2tan α.
h=λn-[1-(λ/d)2]1/2.
asurf (ν, μ)=-- expi 2πλ(n-1)h(x, y)-i2π[x(ν-ν0)+y(μ-μ0)]dxdy,
Hpt(x, y, z)=-i4πβt(x, y)δ(z)
Hpt(x, y, z)=-i4πβt(x, y)δ (x-z tan ϑ),
a(0, μ)=--t(z/tan ϑ, y)×exp{-i2π[(β-β0)z+μy]}dydz.
Hpt(x, y, z)=-i4πβδ[z-h(x, y)],
a(ν, μ)=-- exp{-i2π(β-β0)h(x, y)-i2π[(ν-ν0)x+(μ-μ0)y]}d xd y.
νl=ν0+ld,l=0,±1,±2,.
al=1d-d/2d/2 exp[i2π(β-β0)x tan α-i2πxl/d]dx
=sinc[d(β-β0)tan α-l].
h=lβ-β0=-lλcos2 ϑ-2l sin ϑ λd-l2 λ2d21/2+1.
Hpt,1(x, y, z)=k02(n2-1)pheav[z-h1(x, y)],
Hpt,2(x, y, z)=-k02(n2-1)pheav[z-h2(x, y)].
a(ν, μ)=--a2(ν, μ|ν, μ)×a1(ν, μ|ν0, μ0)dνdμ.
Hpt,1(x, z)=-i4πβt1(x)δ(z+h),
Hpt,2(x, z)=-i4πβt2(x)δ(z).
a(ν)=-t˜1(ν-ν0)×exp[i2π(β-β0)h]t˜2(ν-ν)dν,
a(ν)=exp(iπλhν02)--t1(x)t2(x)×exp[-i2π(νx-ν0x)]×-exp[-iπλhν2+i2π(x-x)ν]×dνdxdx.
a(ν)=exp(iπλhν02)--t1(x)exp(i2πν0x)×expiπλh(x-x)2dxt2(x)exp(-i2πνx)dx.
a(ν)=exp(-iπhλν02)-t1x-hλν2×t2x+hλν2exp(-i2πxν)dx.

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