Abstract

The local plane-interface approximation (LPIA) is a method for propagating electromagnetic fields through the inhomogeneous regions (e.g., elements) of an optical system. The LPIA is the superclass of all approximations that replace the usually curved optical interfaces with local tangential planes. Therefore the LPIA is restricted to smooth optical surfaces. A maximum radius of curvature of the optical interface of the order of a few wavelengths is a rough estimate for the validity of the LPIA. Two important approximation levels of the LPIA are the thin-element approximation (TEA) and a geometric-optical version of the LPIA (LPIAray). The latter combines the wave-optical propagation of an electromagnetic field in the homogeneous region of an optical system with a ray-tracing step in the inhomogeneous region. We discuss the regions of validity of the LPIA in general and the approximation levels LPIAray and TEA in detail.

© 2000 Optical Society of America

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References

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1998 (1)

1997 (2)

1995 (5)

1994 (2)

D. A. Pommet, M. G. Moharam, E. B. Grann, “Limits of scalar diffraction theory for diffractive phase elements,” J. Opt. Soc. Am. A 11, 1827–1834 (1994).
[CrossRef]

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

1993 (1)

1990 (1)

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Aagedal, H.

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

Beckmann, P.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, pp. 53–69.
[CrossRef]

Blough, C. G.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

Blough, G.

G. Blough, M. Morris, “Hybrid lenses offer high performance at low cost,” Laser Focus World 31(11) , 67–74 (1995).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

Bronstein, I. N.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1989).

Daly, D.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Davies, N.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Faklis, D.

Fleming, M. B.

Gale, M.

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Gale, M. T.

M. T. Gale, “Direct writing of continuous-relief micro-optics,” in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 87–126.

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie Verlag, Berlin, 1997), pp. 103–145.

Goodman, J. W.

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

Grann, E. B.

Heier, H.

Herzig, H. P.

Hutley, M. C.

M. B. Fleming, M. C. Hutley, “Blazed diffractive optics,” Appl. Opt. 36, 4635–4643 (1997).
[CrossRef] [PubMed]

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Jackson, J. D.

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

Kley, E.-B.

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kunz, R. E.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Maystre, D.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

Moharam, M. G.

Morris, G. M.

Morris, M.

G. Blough, M. Morris, “Hybrid lenses offer high performance at low cost,” Laser Focus World 31(11) , 67–74 (1995).

Noponen, E.

Pedersen, J.

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Pfeil, A. v.

A. v. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. (to be published).

Pommet, D. A.

Popov, E. K.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

Raguin, D. H.

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

Rossi, M.

M. Rossi, R. E. Kunz, H. P. Herzig, “Refractive and diffractive properties of planar micro-optical elements,” Appl. Opt. 34, 5996–6007 (1995).
[CrossRef] [PubMed]

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie Verlag, Berlin, 1997), pp. 103–145.

Sales, T. R.

Schütz, H.

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Semendjajew, K. A.

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1989).

Sinzinger, S.

Sommargren, G. E.

Stamnes, J. J.

J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. 37, 3612–3622 (1998).
[CrossRef]

J. J. Stamnes, Waves in Focal Regions. Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, Bristol, UK, 1986).

Stevens, R. F.

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

Swanson, G. J.

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

Sweeney, D. W.

Testorf, M.

Thoma, F.

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

Turunen, J.

Vasara, A.

Wittig, L.

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

Wyrowski, F.

A. v. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. (to be published).

Zeitner, U.

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

Appl. Opt. (7)

J. Meas. Sci. Technol. (1)

D. Daly, R. F. Stevens, M. C. Hutley, N. Davies, “The manufacture of microlenses by melting photoresists,” J. Meas. Sci. Technol. 1, 759–766 (1990).
[CrossRef]

J. Opt. Soc. Am. A (2)

Laser Focus World (1)

G. Blough, M. Morris, “Hybrid lenses offer high performance at low cost,” Laser Focus World 31(11) , 67–74 (1995).

Opt. Eng. (1)

M. Gale, M. Rossi, J. Pedersen, H. Schütz, “Fabrication of continuous-relief micro-optical elements by direct laser writing in photoresist,” Opt. Eng. 33, 3556–3566 (1994).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (13)

M. T. Gale, M. Rossi, “Continuous-relief diffractive lenses and microlens arrays,” in Diffractive Optics for Industrial and Commercial Applications, J. Turunen, F. Wyrowski, eds. (Akademie Verlag, Berlin, 1997), pp. 103–145.

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

A. v. Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. (to be published).

I. N. Bronstein, K. A. Semendjajew, Taschenbuch der Mathematik (Verlag Harri Deutsch, Thun und Frankfurt/Main, Germany, 1989).

M. T. Gale, “Direct writing of continuous-relief micro-optics,” in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 87–126.

P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. 6, pp. 53–69.
[CrossRef]

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

M. Rossi, C. G. Blough, D. H. Raguin, E. K. Popov, D. Maystre, “Diffraction efficiency of high-NA continuous-relief diffractive lenses,” in Diffractive Optics and Microoptics, Vol. 5 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 233–236.

J. J. Stamnes, Waves in Focal Regions. Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Adam Hilger, Bristol, UK, 1986).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[CrossRef]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1991).

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

E.-B. Kley, F. Thoma, U. Zeitner, L. Wittig, H. Aagedal, “Fabrication of micro-optical surface profiles by using gray scale masks,” in Miniaturized Systems with Micro-Optics and Micromechanics III, R. Goering, M. Motamedi, eds., Proc. SPIE3276, 254–262 (1997).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Schematic illustration of the typical inhomogenity considered in this paper. It is an arbitrarily shaped interface separated by two homogeneous dielectrics.

Fig. 2
Fig. 2

Illustration of the analysis of an optical element by two methods. In both cases the field is propagated by wave optics in the homogenous region. In the inhomogeneous region (a) the TEA (a) and (b) LPIAray are applied.

Fig. 3
Fig. 3

Curvature C of sinusoidal gratings. The gratings have a height of λ. r = 1/C = 1.3λ, λ = 632.8 nm.

Fig. 4
Fig. 4

Rigorously calculated amplitudes of fields behind the periods of sinusoidal gratings with height λ, refractive index 1.5, and grating periods (a) 2.5 µm and (b) 10 µm. The gratings are illuminated with plane waves with wavelength λ = 632.8 nm under normal incidence.

Fig. 5
Fig. 5

Rigorously calculated amplitudes of the fields behind the periods of blazed gratings with height 2λ, refractive index 1.5, and grating periods (a) 2.5 and (b) 10 µm. The gratings are illuminated with plane waves with wavelength λ = 632.8 nm under normal incidence.

Fig. 6
Fig. 6

First-order diffraction efficiencies of sinusoidal gratings as a function of period. The gratings have a height of λ = 632.8 nm and are illuminated with on-axis plane waves in TE polarization. They are analyzed by the TEA, LPIAray, or electromagnetic analysis (dotted curve).

Fig. 7
Fig. 7

First-order diffraction efficiencies of blazed gratings as a function of period. The gratings have height 2λ and are illuminated with on-axis plane waves in TE polarization with wavelength λ = 632.8 nm. They are analyzed by the TEA, LPIAray, or an electromagnetic analysis.

Fig. 8
Fig. 8

Maximum distance of propagation d max of a Gaussian intensity distribution with a 1/e 2 full width w 0, assuming that w(d max)/w 0 = 1.05. The wavelength of the Gaussian beam is λ = 632.8 nm.

Fig. 9
Fig. 9

Ray tracing through a prism with base d and height h, which results in deflection angle φ. The illuminated region behind the prism has width . s is the distance that a ray with the starting point (x, y, z I) has to be traced to hit the wave front, which runs through (x, y, z II).

Fig. 10
Fig. 10

An optical interface can be thought to be composed of approximately local prisms. The prism angle is defined by the tangential plane on the interface at the intersection of a ray r n . The height of the prism h n is the distance from the intersection of r n to zII.

Fig. 11
Fig. 11

Difference in optical path between TEA and LPIAray as a function of φ. The dashed line indicates an error in the optical path of λ/10.

Fig. 12
Fig. 12

Overview of prisms expressed by φ and the height of the prism. The locations of the elements that can be considered with the TEA or have to be analyzed with LPIAray are shown. The calculations are based on a maximum error in the optical path of λ/10.

Fig. 13
Fig. 13

Cross section of the field in the second conversion plane of the aspherical interface of a lens with a N.A. of 0.4, refractive index 1.5, and diameter 500 µm, illuminated with a plane wave with wavelength λ = 632.8 nm. The effects that can be observed with LPIAray are compared with calculations performed with the TEA. (a) OPD, (b) modulation of the intensity of the field owing to the refraction at the optical interface, and (c) modulation of the intensity of the field according to the Fresnel coefficients. In (c) the dashed and the dotted curves represent simulations with LPIAray. In (b) and (c) the intensities are scaled with the intensity value obtained with the TEA.

Fig. 14
Fig. 14

Quotient ITEA/ILPIAray of the intensities in the second conversion plane of a prism obtained with LPIAray and the TEA, calculated for two refractive indices n of the prism.

Fig. 15
Fig. 15

Transmitted intensity of a prism with a single interface. The transmission is calculated for TE and TM polarization with a refractive index of 3.5.

Fig. 16
Fig. 16

Three lenses with the same surface profile apart from the corresponding phase wrapping. (a) The phase of the modulo 2π lens is wrapped such that the phase difference between two zones is λ. (b) The modulo 4π lens has twice the height of the modulo 2π lens. (c) The lens is smooth, that is, the phase is completely unwrapped.

Fig. 17
Fig. 17

Intensity on the optical axis in the focal region of cylindrical lenses analyzed with the TEA as well as with LPIAray. The lenses are illuminated with a plane wave with a wavelength of λ = 632.8 nm in TE polarization. Intensity is plotted as a function of the distance from the second conversion plane of the lens. The lenses were designed with a TEA with a N.A. of 0.4, refractive index 1.5, and width 500 µm.

Equations (7)

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

Cx=yx1+yx23/2,
wdw0=1+4dλπw0221/2,
Δs=n2h-s  λ,
Δsλ=hn2λ1-cos φ  1
ITEAILPIAray=d˜d=1-n2 sin φn1-n2 cos φ tan φ,
ITEITM=cos φ2.
hx=f-x2+f2n1-n2,

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