Abstract

A new local elementary interface approximation is introduced for the modeling of wave propagation through interfaces between homogeneous media. The incident wave and the surface profile are approximated locally by a spherical wave and a spherical surface, respectively. The wave field travels through the modulated structure according to the laws of geometrical optics, being refracted by the surface and propagating to the output plane locally as a geometric spherical wave. Diffraction theory is applied to propagate the field from the output plane onwards. We provide comparisons of the method with the thin-element approximation, the local plane-wave and interface approach, and rigorous diffraction theory using a sinusoidal surface-relief grating as an example. We illustrate the power of the new method by applying it to the analysis of a diffractive beam splitter.

© 2003 Optical Society of America

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References

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  1. J. Turunen, F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).
  2. H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. P. Beckmann, “Scattering of light by rough surfaces,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1967), Vol. VI, pp. 53–69.
    [CrossRef]
  5. G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).
  6. E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
    [CrossRef]
  7. M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.
  8. J. J. Stamnes, H. Heier, “Scalar and electromagnetic diffraction point-spread functions,” Appl. Opt. 37, 3612–3622 (1998).
    [CrossRef]
  9. A. von Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
    [CrossRef]
  10. A. von Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).
  11. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  12. R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).
  13. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
    [CrossRef]
  14. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  15. H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
    [CrossRef]
  16. J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
    [CrossRef]
  17. T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).
  18. T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).
  19. T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
    [CrossRef]

2002 (1)

2001 (2)

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

2000 (2)

A. von Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

A. von Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

1998 (1)

1993 (2)

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

1990 (1)

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
[CrossRef]

Aagedal, H.

Beckmann, P.

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

Blough, C. G.

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Dändliker, R.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
[CrossRef]

Drauschke, A.

Goodman, J. W.

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

Heier, H.

Herzig, H. P.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
[CrossRef]

Jefimovs, K.

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

Kettunen, V.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
[CrossRef]

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

Kingslake, R.

R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).

Kuittinen, M.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
[CrossRef]

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

Laakkonen, P.

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

Mandel, L.

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

Maystre, D.

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

Miller, J. M.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Noponen, E.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

Popov, E. K.

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

Prongué, D.

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
[CrossRef]

Ragwin, D. H.

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

Ross, N.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Rossi, M.

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

Stamnes, J. J.

Swanson, G. J.

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).

Taghizadeh, M. R.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

Turunen, J.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
[CrossRef]

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

Vallius, T.

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129–1135 (2002).
[CrossRef]

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

Vasara, A.

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

E. Noponen, J. Turunen, A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434–443 (1993).
[CrossRef]

von Pfeil, A.

A. von Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

A. von Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

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

Wyrowski, F.

A. von Pfeil, F. Wyrowski, A. Drauschke, H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304–3313 (2000).
[CrossRef]

A. von Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

Appl. Opt. (2)

J. Mod. Opt. (4)

A. von Pfeil, F. Wyrowski, “Wave-optical structure design with the local plane-interface approximation,” J. Mod. Opt. 47, 2335–2350 (2000).

J. M. Miller, M. R. Taghizadeh, J. Turunen, N. Ross, E. Noponen, A. Vasara, “Kinoform array illuminators in fused silica,” J. Mod. Opt. 40, 723–732 (1993).
[CrossRef]

T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).

T. Vallius, K. Jefimovs, V. Kettunen, M. Kuittinen, P. Laakkonen, J. Turunen, “Design of non-paraxial array illuminators by step-transition perturbation approach,” J. Mod. Opt. 48, 1869–1879 (2001).

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

Jpn. J. Appl. Phys. (1)

H. P. Herzig, D. Prongué, R. Dändliker, “Design and fabrication of high efficient fan-out elements,” Jpn. J. Appl. Phys. 29, 1307–1309 (1990).
[CrossRef]

Other (10)

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

R. Kingslake, Lens Design Fundamentals (Academic, San Diego, Calif., 1978).

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

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. Turunen, F. Wyrowski, eds., Diffractive Optics for Industrial and Commercial Applications (Akademie Verlag, Berlin, 1997).

H. P. Herzig, ed., Micro-Optics: Elements, Systems and Applications (Taylor & Francis, London, 1997).

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

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

G. J. Swanson, “Binary optics technology: theoretical limits on the diffraction efficiency of multilevel diffractive optical elements,” Tech. Rep. 914 (Massachusetts Institute of Technology, Cambridge, Mass., 1991).

M. Rossi, C. G. Blough, D. H. Ragwin, 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), p. 233.

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Figures (6)

Fig. 1
Fig. 1

Definitions of the sections and the control positions with the help of the principal rays in different parts of the structure.

Fig. 2
Fig. 2

Convergence of the transmitted first-order diffraction efficiency η1 of the sinusoidal grating for different period-depth combinations computed with LSIA (solid curve) and LPIA (dashed curve).

Fig. 3
Fig. 3

First-order diffraction efficiency η1 for the sinusoidal grating with h = λ as a function of the period computed with FMM (solid curves), LSIA (dashed curves), and TEA (dotted lines).

Fig. 4
Fig. 4

Phase and amplitude of the transmitted field for the sinusoidal grating with h = 5λ and d = 20λ (upper plots) and the corresponding diffraction efficiencies (lower plot) computed with FMM (solid curves), LSIA (dashed curves), and TEA (dotted curves).

Fig. 5
Fig. 5

Arrays produced by a 1 → 9 analog grating beam splitter with a period d = 15λ predicted with FMM, LSIA, and TEA.

Fig. 6
Fig. 6

Definition of the segments Δx j i,h and Δx j t,h . The reflected part and the segment Δx j r,h are not shown for clarity.

Equations (30)

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

Uix, 0=|Uix, 0|expiϕix, 0,
12xji+xj-1i<x<12xji+xj+1i,
0<x<12x1i+x2i,
12xji+xJ-1i<x<D.
sin θji=1kn1ddx ϕx, 0|x=xji,
n2 sin ϑjt=n1 sin ϑji
Rji0=1+ϕx, 0|x=xji23/2ϕx, 0|x=xji,
Rjihxjh=Rji0+Δsji,
Δsji=xjh-xji2+h2xjh1/2
n2 cos2 ϑjtRjthxjh=n1 cos2 ϑjiRjihxjh+n1 cos ϑji-n2 cos ϑjtRj.
RjtH=Rjthxjh+Δsjt,
Δsjt=xjt-xjh2+H-hxjh21/2
|Ujtx, H|=tϑji|Ujxji, 0|cos ϑjt cos θjiΔxjicos ϑji cos θjtΔxjt1/2,
ϕtxjt, H=ϕixji, 0+k0n1Δsji+k0n2Δsjt,
ϕtx, H=ϕtxjt, H±k0n2x-xjc2+H-zjc21/2RjtH,
-cos ϑjiRjrhxjh=cos ϑjiRjihxjh+2Rj.
Rjr0=Rjrhxjh-Δsjr,
Δsjr=xjr-xjh2+hxjh21/2.
ϕrx, 0=ϕrxjr, 0±k0n1x-xjc2+zjc21/2Rjr0,
ϕrxjr, 0=ϕixji, 0+k0n1Δsji+Δsjr,
|Ujrx, 0|=rϑji|Ujxji, 0|cos θjiΔxjicos θjrΔxjr1/2,
hx=h21+sin2πx/d
Tm=1d0d Ux, Hexp-i2πmx/ddx,
Δxjt,hΔxji,h=cos ϑjt cos θjicos ϑji cos θjt.
Δxjr,hΔxji,h=cos θjicos θjr.
Δxji|Ujixji, 0|2=Δxji,h|Ujixjh, hxjh|2.
Δxjt|Ujtxjt, H|2=ΔxjiΔxjt,hΔxji,h |tϑji|2|Ujixji, 0|2,
Δxjr|Ujrxjr, 0|2=ΔxjiΔxjr,hΔxji,h |rϑji|2|Ujixji, 0|2.
|Ujtxjt, H|2=|tϑji|2|Ujixji, 0|2cos ϑjt cos θjiΔxjicos ϑji cos θjtΔxjt,
|Ujrxjt, 0|2=|rϑji|2|Ujixji, 0|2cos θjiΔxjicos θjrΔxjr,

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