Abstract

We report on a numerical analysis method for diffractive optical elements that consist of features ranging from subwavelength to more than 10λ. The essence of the method is treating local structures of the optical elements as diffraction gratings. It is shown that the method can provide accuracy of results comparable with fully electromagnetic treatments in much shorter time. The theory and results are explained assuming micro-Fresnel lenses with one-dimensional structures for investigating polarization properties.

© 2009 Optical Society of America

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References

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  1. J. Turunen and F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic Verlag, 1997), pp. 11-12.
  2. M. Lang and T. D. Milster, “Investigation of optics in the 10-200 μm regime,” Opt. Rev. 14, 189-193 (2007).
    [CrossRef]
  3. D. W. Prather, S. Shi, and J. S. Bergey, “Field stitching algorithm for the analysis of electrically large diffractive optical elements,” Opt. Lett. 24, 273-275 (1999).
    [CrossRef]
  4. A. v. Pfeil, F. Wyrowski, A. Drauschke, and H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304-3313 (1999).
    [CrossRef]
  5. T. Vallius, M. Kuittinen, J. Turunen, and Ville Kettunen, “Step-transition perturbation approach for pixel-structured nonparaxial diffractive elements,” J. Opt. Soc. Am. A 19, 1129-1135 (2002).
    [CrossRef]
  6. G. H. Spencer and M. V. R. K. Murty, “General ray-tracing procedure,” J. Opt. Soc. Am. 52, 672-678 (1962).
    [CrossRef]
  7. E. Noponen, J. Turunen, and A. Vasara, “Electromagnetic theory and design of diffractive-lens arrays,” J. Opt. Soc. Am. A 10, 434-443 (1993).
    [CrossRef]
  8. Y. Sheng, D. Feng, and S. Larochelle, “Analysis and synthesis of circular diffractive lens with local linear grating model and rigorous coupled-wave theory,” J. Opt. Soc. Am. A 14, 1562-1568 (1997).
    [CrossRef]
  9. B. H. Kleemann and R. Güther, “Zonal diffraction efficiencies and imaging of micro-Fresnel lenses,” J. Mod. Opt. 45, 1405-1420 (1998).
    [CrossRef]
  10. J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H.P.Herzig, ed. (Taylor & Francis, 1997), pp. 31-52.
  11. A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).
  12. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 55-61.
  13. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38, 304-313 (1999).
    [CrossRef]
  14. F. Montiel and M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241-3250 (1994).
    [CrossRef]
  15. B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
    [CrossRef]
  16. D. W. Prather, M. S. Mirotznik, and J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34-43 (1997).
    [CrossRef]
  17. G. Bao, Z. Chen, and H. Wu, “Adaptive finite-element method for diffraction gratings,” J. Opt. Soc. Am. A 22, 1106-1114 (2005).
    [CrossRef]

2007 (1)

M. Lang and T. D. Milster, “Investigation of optics in the 10-200 μm regime,” Opt. Rev. 14, 189-193 (2007).
[CrossRef]

2005 (1)

2002 (1)

1999 (3)

1998 (1)

B. H. Kleemann and R. Güther, “Zonal diffraction efficiencies and imaging of micro-Fresnel lenses,” J. Mod. Opt. 45, 1405-1420 (1998).
[CrossRef]

1997 (2)

1996 (1)

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

1994 (1)

1993 (1)

1962 (1)

Aagedal, H.

Bao, G.

Bergey, J. S.

Chandezon, J.

Chen, Z.

Drauschke, A.

Feng, D.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 55-61.

Granet, G.

Güther, R.

B. H. Kleemann and R. Güther, “Zonal diffraction efficiencies and imaging of micro-Fresnel lenses,” J. Mod. Opt. 45, 1405-1420 (1998).
[CrossRef]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Kettunen, Ville

Kleemann, B. H.

B. H. Kleemann and R. Güther, “Zonal diffraction efficiencies and imaging of micro-Fresnel lenses,” J. Mod. Opt. 45, 1405-1420 (1998).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Kuittinen, M.

Lang, M.

M. Lang and T. D. Milster, “Investigation of optics in the 10-200 μm regime,” Opt. Rev. 14, 189-193 (2007).
[CrossRef]

Larochelle, S.

Li, L.

Mait, J. N.

Milster, T. D.

M. Lang and T. D. Milster, “Investigation of optics in the 10-200 μm regime,” Opt. Rev. 14, 189-193 (2007).
[CrossRef]

Mirotznik, M. S.

Mitreiter, A.

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Montiel, F.

Murty, M. V. R. K.

Nevière, M.

Noponen, E.

Pfeil, A. v.

Plumey, J.-P.

Prather, D. W.

Sheng, Y.

Shi, S.

Spencer, G. H.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

Turunen, J.

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

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

J. Turunen and F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic Verlag, 1997), pp. 11-12.

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

Vallius, T.

Vasara, A.

Wu, H.

Wyrowski, F.

A. v. Pfeil, F. Wyrowski, A. Drauschke, and H. Aagedal, “Analysis of optical elements with the local plane-interface approximation,” Appl. Opt. 39, 3304-3313 (1999).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Turunen and F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic Verlag, 1997), pp. 11-12.

Appl. Opt. (2)

J. Mod. Opt. (2)

B. H. Kleemann and R. Güther, “Zonal diffraction efficiencies and imaging of micro-Fresnel lenses,” J. Mod. Opt. 45, 1405-1420 (1998).
[CrossRef]

B. H. Kleemann, A. Mitreiter, and F. Wyrowski, “Integral equation method with parametrization of grating profile theory and experiments,” J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Opt. Rev. (1)

M. Lang and T. D. Milster, “Investigation of optics in the 10-200 μm regime,” Opt. Rev. 14, 189-193 (2007).
[CrossRef]

Other (4)

J. Turunen and F. Wyrowski, “Introduction to diffractive optics,” in Diffractive Optics for Industrial and Commercial Applications, J.Turunen and F.Wyrowski, eds. (Akademic Verlag, 1997), pp. 11-12.

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

A. Taflove and S. C. Hagness, Computational Electrodynamics: the Finite-Difference Time-Domain Method (Artech House, 2000).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996), pp. 55-61.

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Figures (13)

Fig. 1
Fig. 1

Analyzed optical system. Surface relief layer h z 0 is analyzed by an electromagnetic diffraction grating theory.

Fig. 2
Fig. 2

Location of sampling points and their numbering order in the surface relief layer of a micro-Fresnel lens, for example. θ is an incidence angle.

Fig. 3
Fig. 3

Sampling point and grating depth in the case of L = 4 , as an example.

Fig. 4
Fig. 4

Definition of local period and application of the FMM.

Fig. 5
Fig. 5

Experimental setup.

Fig. 6
Fig. 6

Observed binary micro-Fresnel lens of f = 1 mm .

Fig. 7
Fig. 7

Focusing properties of a micro-Fresnel lens of f = 100 λ and r = 50.4 λ . (a) Spot profiles: Δ z = 0.7 λ for LGT, 0.6 λ for FMM, and 0.5 λ for FDTD. (b) Intensity with axial defocus.

Fig. 8
Fig. 8

Focusing properties of a micro-Fresnel lens of f = 1 mm and r = 0.75 mm . (a) Spot profiles: Δ z = 0 for both LGT and FDTD. (b) Intensity with axial defocus.

Fig. 9
Fig. 9

Focusing properties of a micro-Fresnel lens of f = 1 mm and r = 0.2 mm . (a) Spot profiles: Δ z = 4.6 μ m for both LGT and FDTD. (b) Intensity with axial defocus.

Fig. 10
Fig. 10

Comparison between LGT and experiments: power within focused spot of a micro-Fresnel lens of f = 1 mm . Solid curve, LGT for TE; dashed curve, LGT for TM. Solid circles, experiments for TE. Open circles, experiments for TM. All data are normalized with the value of r = 0.75 mm for TM.

Fig. 11
Fig. 11

Convergence study. (a) Peak intensities at x = 0 for the best focus. (b) Intensities at x = 0 with axial defocus for various numbers of truncation orders.

Fig. 12
Fig. 12

Why does the LGT work satisfactorily with a much smaller number of truncation orders than the full vectorial FMM? Case B is assumed as a model. (a) Left vertical axis, local periods of a lens; right vertical axis, required numbers of truncation orders. Three circles correspond to convergence of first-order diffraction efficiency within 10 2 , 10 3 , and 10 4 assuming a rectangular grating of fill factor 0.5 with the same period. (b) Convergence of transmitted first order for the grating mentioned in (a). The values denote the grating period in μ m .

Fig. 13
Fig. 13

Comparison of computation time between the LGT and FMM with various numbers of truncation orders.

Tables (3)

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Table 1 Parameters of Simulated Lenses

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Table 2 Parameters for Simulation

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Table 3 Relative Computation Time for the Same Quality of Computation

Equations (13)

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ϕ ( x ) = l Φ , for ( l + 1 ) Φ < mod [ ϕ 0 ( x ) , 2 π ] l Φ ,
ϕ 0 ( x ) = 2 π n 2 ( f f 2 + x 2 ) λ
M L = int [ ( f 2 + r 2 f ) λ ] + 1 ,
x m = m 2 λ 2 + 2 m L f λ L .
d ( n ) = x P + L x P ,
h ( n ) = λ | n 2 n 1 | ( int [ m 1 L ] m L + 1 ) .
h = λ L | n 2 n 1 |
u ( n , 0 ) = q T q exp ( i α q x ¯ ) ,
u ( n , 0 ) { u ( n , 0 ) n 2 n 1 , for TE u ( n , 0 ) n 1 n 2 , for TM } .
A j ( 0 ) = DFT { u ( n , 0 ) } .
A j ( z ) = A j ( 0 ) exp [ i ( 2 π n 2 λ ) z 1 ( ν j λ n 2 ) 2 ] ,
u ( n , z ) = IDFT { A j ( z ) } ,
I ( n ) = | u ( n , z ) | 2 .

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