Abstract

An extension of an approximate step-transition perturbation method is presented that permits numerically efficient diffraction analysis of pixel-structured surface profiles in the nonparaxial domain. Comparison with the rigorous diffraction theory of gratings shows that the method is reasonably accurate provided that the pixel size exceeds approximately two wavelengths even if the structure contains isolated pixels.

© 2002 Optical Society of America

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References

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  1. V. Kettunen, M. Kuittinen, J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
    [CrossRef]
  2. J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, pp. 341–387.
  3. T. Vallius, V. Kettunen, M. Kuittinen, J. Turunen, “Step-discontinuity approach for non-paraxial diffractive optics,” J. Mod. Opt. 48, 1195–1210 (2001).
    [CrossRef]
  4. 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. (to be published).
  5. B. Layet, M. R. Taghizadeh, “Analysis of gratings with large periods and small feature sizes by stitching of the electromagnetic field,” Opt. Lett. 21, 1508–1510 (1996).
    [CrossRef] [PubMed]
  6. B. Layet, M. R. Taghizadeh, “Electromagnetic analysis of fan-out gratings and diffractive cylindrical lens by field stitching,” J. Opt. Soc. Am. A 14, 1554–1561 (1997).
    [CrossRef]
  7. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  8. E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
    [CrossRef]
  9. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holo-grams, generated by computer,” Appl. Opt. 6, 1739–1748 (1967).
    [CrossRef] [PubMed]
  10. R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
    [CrossRef]
  11. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  12. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  13. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]

2001 (2)

V. Kettunen, M. Kuittinen, J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
[CrossRef]

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

1997 (2)

1996 (2)

1994 (1)

1993 (1)

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

1971 (1)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1969 (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

1967 (1)

Bräuer, R.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Bryngdahl, O.

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[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. (to be published).

Kettunen, V.

V. Kettunen, M. Kuittinen, J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
[CrossRef]

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

Kuittinen, M.

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

V. Kettunen, M. Kuittinen, J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
[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. (to be published).

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, pp. 341–387.

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. (to be published).

Layet, B.

Li, L.

Lohmann, A. W.

Marcatili, E. A. J.

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

Noponen, E.

Paris, D. P.

Taghizadeh, M. R.

Turunen, J.

V. Kettunen, M. Kuittinen, J. Turunen, “Effects of abrupt surface-profile transitions in nonparaxial diffractive optics,” J. Opt. Soc. Am. A 18, 1257–1260 (2001).
[CrossRef]

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

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, pp. 341–387.

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. (to be published).

Vallius, T.

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

Wyrowski, F.

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, pp. 341–387.

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

E. A. J. Marcatili, “Dielectric rectangular waveguide and directional coupler for integrated optics,” Bell Syst. Tech. J. 48, 2071–2102 (1969).
[CrossRef]

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

R. Bräuer, O. Bryngdahl, “Electromagnetic diffraction analysis of two-dimensional gratings,” Opt. Commun. 100, 1–5 (1993).
[CrossRef]

Opt. Lett. (1)

Other (2)

J. Turunen, M. Kuittinen, F. Wyrowski, “Diffractive optics: electromagnetic approach,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 2000), Vol. 40, pp. 341–387.

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. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Part of a general multilevel pixel-structured surface-relief profile.

Fig. 2
Fig. 2

Effect of a step transition on the field immediately above the transition if it is illuminated by a plane wave. (a) Amplitude and (b) phase of the transmitted field given by rigorous theory (TE polarization, solid lines) and by the complex-amplitude transmittance approach (dashed lines). The step height corresponds to a phase delay of π rad and the material has a refractive index n=1.45. The corresponding perturbations are shown in (c) and (d).

Fig. 3
Fig. 3

(a) Amplitude and (b) phase of a field immediately behind a single square pixel shown as gray-scale patterns.

Fig. 4
Fig. 4

Construction of the perturbation within a single period of a periodic diffractive element if the truncation parameter w exceeds the length of the grating half-period d/2. (a) Truncated perturbation with w>d/2. (b) Transformed perturbation in which the tails of the perturbations from neighboring periods have been superposed coherently with the perturbation of the transition at the origin.

Fig. 5
Fig. 5

Efficiencies of a grating with a single pixel covering a quarter of the period. Symbols, Fourier modal method; solid curves, perturbation approach.

Fig. 6
Fig. 6

(a) Diffractive cross-like pixel structure, (b) beam splitter with 15×5 equal-efficiency orders according to the thin-element approach.

Fig. 7
Fig. 7

Efficiencies of the structure in Fig. 6(a). Symbols, Fourier modal method; solid curves, perturbation approach.

Fig. 8
Fig. 8

Uniformity error of the 15×5 beam splitter in Fig. 6(b). Symbols, Fourier modal method; solid curves, perturbation approach.

Equations (19)

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

E(x, y, h)=ETEA(x, y, h)+j=1JPj(x, y),
Pj(x, y)=EjR(x, y)-EjTEA(x, y)if|Δdj|<w0elsewhere,
PxTE(y)=ExTE(y)-ExTEA(y)if|y|<w0elsewhere,
PyTM(x)=HyTM(x)-HyTEA(x)if|x|<w0elsewhere.
TpTM=n1tpkn22 SpTM,
tp=[(kn2)2-(2πp)2]1/2,
PxTM(x)=ExTM(x)-ExTEA(x)if|x|<w0elsewhere.
Dn,xTE=1dy-dy/2dy/2PxTE(y)exp(-i2πny/dy)dy,
Dmn,xTE=1dxxj,1xj,2Dn,xTEexp(-i2πnyj/dy)×exp(-i2πmx/dx)dx,
Dmn,xTE=Dn,xTE(i2πm)-1exp(-i2πnyj/dy)×[exp(-i2πmxj,1/dx)-exp(-i2πmxj,2/dx)].
Dmn,xTM=Dm,xTM(i2πn)-1exp(-i2πmxj/dx)×[exp(-i2πnyj,1/dy)-exp(-i2πnyj,2/dy)],
Dmn,yTM=Dm,yTM(i2πn)-1exp(-i2πmxj/dx)×[exp(-i2πnyj,1/dy)-exp(-i2πnyj,2/dy)].
Fmn=FmnTEA+j=1JDmnj,
FmnEx=FmnTEA,Ex+j=1JDmn,xj,
FmnHy=FmnTEA,Hy+j=1JDmn,yj.
FmnEz=n1αmkn22dx FmnHy,
ηmn=Rtmnkn1(|FmnEx|2+|FmnEz|2),
tmn=[k2n22-(2πm/dx)2-(2πn/dy)2]1/2.
E=max(ηmn)-min(ηmn)max(ηmn)+min(ηmn),

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