Abstract

A new hybrid method for the analysis of diffractive optical elements, which combines fully vectorial and scalar theories, is presented. It is suitable for use with elements of arbitrary large zone, even when the local feature size is of the order of the wavelength. To assess its applicability, we have performed cross-checking tests. The model is shown to accurately predict many optical properties of diffractive optical elements based on two-dimensional artificial dielectrics, like the useful energy diffracted into the order of interest or the deterministic loss into high diffraction orders for an illumination with a wavelength different from the design wavelength or for highly oblique incidence.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  11. J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-787 (2000).
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    [CrossRef]
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    [CrossRef]
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  15. U. Levy, H. C. Kim, C. H. Tsai, and Y. Fainman, "Near-infrared demonstration of computer-generated holograms implemented by using subwavelength gratings with space-variant orientation," Opt. Lett. 30, 2089-2091 (2005).
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    [CrossRef]
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    [CrossRef]
  23. S. Astilean, P. Lalanne, P. Chavel, E. Cambril, and H. Launois, "High efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm," Opt. Lett. 23, 552-554 (1998).
    [CrossRef]
  24. F. T. Chen and H. G. Craighead, "Diffractive lens fabricated with mostly zeroth-order gratings," Opt. Lett. 21, 177-179 (1996).
    [CrossRef] [PubMed]
  25. G. J. Swanson, "Binary optics technology: the theory and design of multilevel diffractive optical elements," MIT Technical Report 854 (MIT, 1989).
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  29. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  30. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  31. G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  32. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  33. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.
  34. M. S. L. Lee, P. Lalanne, and P. Chavel, "Blazed-binary diffractive elements with periods much larger than the wavelength," J. Opt. Soc. Am. A 17, 1250-1255 (2000).
    [CrossRef]
  35. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, "Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings," Opt. Lett. 23, 1081-1083 (1998).
    [CrossRef]

2006 (2)

H. Elfström, M. Kuittinen, T. Vallius, B. H. Kleemann, J. Ruoff, and R. Arnold, "Fabrication of blazed gratings by area-coded effective medium structures," Opt. Commun. 266, 697-703 (2006).
[CrossRef]

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, "Rigorous analysis of shadowing effects in blazed transmission gratings," Opt. Lett. 31, 3638-3640 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (2)

2003 (3)

2002 (1)

M. S. L. Lee, P. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, "Imaging with blazed-binary diffractive elements," J. Opt. A, Pure Appl. Opt. 4, 119-124 (2002).
[CrossRef]

2000 (3)

1999 (2)

1998 (2)

1997 (3)

1996 (4)

1995 (2)

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

J. M. Finlan, K. M. Flood, and R. J. Bojko, "Efficient f/1 binary-optics microlenses in fused silica designed using vector diffraction theory," Opt. Eng. 34, 3560-3564 (1995).
[CrossRef]

1993 (1)

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, "Design of a blazed grating consisting of metallic subwavelength binary grooves," Opt. Commun. 98, 5-10 (1993).
[CrossRef]

1992 (2)

1970 (1)

H. Dammann, "Phase holograms of diffuse objects," J. Opt. Soc. Am. A 60, 1635-1639 (1970).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

E. Hasman, V. Kleiner, G. Biener, and A. Niv, "Polarization dependent focusing lens by use of quantized Pancharatnam-Berry phase diffractive optics," Appl. Phys. Lett. 82, 328-360 (2003).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

M. S. L. Lee, P. Lalanne, J. C. Rodier, P. Chavel, E. Cambril, and Y. Chen, "Imaging with blazed-binary diffractive elements," J. Opt. A, Pure Appl. Opt. 4, 119-124 (2002).
[CrossRef]

J. Opt. Soc. Am. A (9)

J. Tervo, V. Kettunen, M. Honkanen, and J. Turunen, "Design of space-variant diffractive polarization elements," J. Opt. Soc. Am. A 20, 282-289 (2003).
[CrossRef]

M. S. L. Lee, P. Lalanne, and P. Chavel, "Blazed-binary diffractive elements with periods much larger than the wavelength," J. Opt. Soc. Am. A 17, 1250-1255 (2000).
[CrossRef]

H. Dammann, "Phase holograms of diffuse objects," J. Opt. Soc. Am. A 60, 1635-1639 (1970).
[CrossRef]

M. Testorf, "Perturbation theory as a unified approach to describe diffractive optical elements," J. Opt. Soc. Am. A 16, 1115-1123 (1999).
[CrossRef]

P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, "Design and fabrication of blazed-binary diffractive elements with sampling periods smaller than the structural cutoff," J. Opt. Soc. Am. A 16, 1143-1156 (1999).
[CrossRef]

L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[CrossRef]

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet and B. Guizal, "Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

J. Vac. Sci. Technol. B (1)

J. R. Wendt, G. A. Vawter, R. E. Smith, and M. E. Warren, "Subwavelength, binary lenses at infrared wavelengths," J. Vac. Sci. Technol. B 15, 2946-2949 (1997).
[CrossRef]

Opt. Commun. (2)

H. Elfström, M. Kuittinen, T. Vallius, B. H. Kleemann, J. Ruoff, and R. Arnold, "Fabrication of blazed gratings by area-coded effective medium structures," Opt. Commun. 266, 697-703 (2006).
[CrossRef]

H. Haidner, J. T. Sheridan, J. Schwider, and N. Streibl, "Design of a blazed grating consisting of metallic subwavelength binary grooves," Opt. Commun. 98, 5-10 (1993).
[CrossRef]

Opt. Eng. (1)

J. M. Finlan, K. M. Flood, and R. J. Bojko, "Efficient f/1 binary-optics microlenses in fused silica designed using vector diffraction theory," Opt. Eng. 34, 3560-3564 (1995).
[CrossRef]

Opt. Express (2)

Opt. Lett. (9)

B. H. Kleemann, J. Ruoff, and R. Arnold, "Area coded effective medium (ACE) structures, a new type of grating design," Opt. Lett. 30, 1617-1619 (2005).
[CrossRef] [PubMed]

U. Levy, H. C. Kim, C. H. Tsai, and Y. Fainman, "Near-infrared demonstration of computer-generated holograms implemented by using subwavelength gratings with space-variant orientation," Opt. Lett. 30, 2089-2091 (2005).
[CrossRef] [PubMed]

O. Sandfuchs, R. Brunner, D. Pätz, S. Sinzinger, and J. Ruoff, "Rigorous analysis of shadowing effects in blazed transmission gratings," Opt. Lett. 31, 3638-3640 (2006).
[CrossRef] [PubMed]

C. Sauvan, P. Lalanne, and M. S. L. Lee, "Broadband blazing with artificial dielectrics," Opt. Lett. 29, 1593-1595 (2004).
[CrossRef]

J. Tervo and J. Turunen, "Paraxial-domain diffractive elements with 100% efficiency based on polarization gratings," Opt. Lett. 25, 785-787 (2000).
[CrossRef]

F. T. Chen and H. G. Craighead, "Diffractive lens fabricated with mostly zeroth-order gratings," Opt. Lett. 21, 177-179 (1996).
[CrossRef] [PubMed]

B. Layet and M. 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]

S. Astilean, P. Lalanne, P. Chavel, E. Cambril, and H. Launois, "High efficiency subwavelength diffractive element patterned in a high-refractive-index material for 633 nm," Opt. Lett. 23, 552-554 (1998).
[CrossRef]

P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, "Blazed-binary subwavelength gratings with efficiencies larger than those of conventional échelette gratings," Opt. Lett. 23, 1081-1083 (1998).
[CrossRef]

Other (4)

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

G. J. Swanson, "Binary optics technology: the theory and design of multilevel diffractive optical elements," MIT Technical Report 854 (MIT, 1989).

J.-P. Hugonin and P. Lalanne, "Reticolo software for grating analysis," trademark of the Institut d'Optique (2005).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Cambridge U. Press, 1989), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Different calibration functions, F : g = F ( ψ ) , corresponding to different geometries g. (a) Sketch of the 2 π periodicity of F. (b) Examples of DOEs that can be analyzed with the model. From top to bottom: échelette-type DOEs replicated with their moth-eye-type antireflection coating [20], blazed-binary element, and blazed area-coded effective-medium structure [15]. (c) Corresponding local geometries g.

Fig. 2
Fig. 2

Scattering of a DOE with a slowly varying unwrapped phase function. The DOE is illuminated by an unpolarized plane wave incident from medium 1 at arbitrary azimuthal ( δ ) and incidence ( θ ) angles. The transmitted wave field is weakly divergent and is contained in a small solid angle Ω centered around the wave vector k t of the weakly modulated plane wave propagating in medium 2. Similarly, the reflected wave field is contained in a small solid angle Ω centred around the wave vector k r .

Fig. 3
Fig. 3

(a) DOE composed of tiny pillars and holes etched into a Si 3 N 4 substrate ( n = 2.1 ) for a nominal operation at λ 0 = 0.8 μ m . The sampling period is Λ s = 0.5 λ 0 , and the etching depth is h = 1.9 λ 0 . (b) Real (solid curve) and imaginary (dashed-dotted curve) parts of the transmitted zero-order coefficient t = t 11 = t 22 as a function of the unwrapped phase ψ. The vertical dashed line shows the transition between the pillars ( ψ < 0.25 π ) and the holes ( ψ > 0.25 π ) . The calculation has been performed for a normally incident plane wave at λ = 0.62 λ 0 .

Fig. 4
Fig. 4

Illustration of the accuracy achieved by the numerical technique developed to calculate the diffraction efficiencies into high orders. The bold horizontal lines, calculated with the Parseval relation, represent the total energy diffracted into all orders. The thin-solid curves, computed with Eqs. (10, 11), represent the energy η ( N ) diffracted into the N first orders, from N to + N . (a) Reflectance. (b) Transmittance. All results are obtained for the pillar–hole geometry of Fig. 3 at λ = 0.62 λ 0 .

Fig. 5
Fig. 5

Optical properties of blazed-binary DOE with subwavelength holes and square pillars etched in a Si 3 N 4 substrate ( n = 2.1 ) , as compared with échelette-type DOEs (dashed curves). (a) Wavelength dependence of the first-order diffraction efficiency (solid curve) for normal incidence and for unpolarized light. (b) First-order diffraction efficiency as a function of the incidence angle θ in Si 3 N 4 for a null azimuthal angle ( δ = 0 ) at λ = λ 0 . Squares, TE polarization, triangles, TM polarization. (c), (d) Energy scattered into the background orders for normal incidence and λ 0.584 λ 0 (c) and for oblique incidence ( θ = 18 ° in the substrate) and λ = λ 0 (d). All curves are normalized to the total energy diffracted in all transmitted orders, and all results hold for an incident illumination impinging on the DOE from the substrate side.

Fig. 6
Fig. 6

Equivalence in the long-period limit between area-coded effective medium gratings and blazed-binary gratings composed of subwavelength ridges. (a), (c) Area-coded effective-medium grating and the associated local geometry. (b), (d) Blazed-binary grating with subwavelength ridges and the associated local geometry. Note that the subwavelength transverse period Λ s in (a) is equal to the sampling period of the blazed-binary element in (b). The local geometries in (c) and (d) are identical, except for a π 2 rotation marked by the azimuthal angle δ. (e) Three different area-coded gratings with identical optical properties in the long-period limit. Note that the local fraction of high-index material f is the same in all figures.

Equations (12)

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F : ψ g = F ( ψ ) ( 2 π ) ,
( ν μ ) = T ( x , y ) ( α β ) .
( ν μ ) = T ( ψ ) ( α β ) = ( t 11 ( ψ ) t 12 ( ψ ) t 21 ( ψ ) t 22 ( ψ ) ) ( α β ) ,
T ( ψ ) = m = m = C m exp ( j m ψ ) ,
C m = ( c 11 m c 12 m c 21 m c 22 m ) = 1 2 π 0 2 π T ( ψ ) exp ( j m ψ ) d ψ .
η m TE = ( c 21 m 2 + c 11 m 2 ) ,
η m TM = ( c 22 m 2 + c 12 m 2 )
m η m TE = 1 2 π 0 2 π ( t 21 ( ψ ) 2 + t 11 ( ψ ) 2 ) d ψ
c i , j m = 1 2 π 0 2 π t i , j ( ψ ) exp ( j m ψ ) d ψ .
0 2 π t i , j ( ψ ) exp ( κ ψ ) d ψ = p = 1 M G p ( ψ p ) exp ( κ ψ p ) G p ( ψ p 1 ) exp ( κ ψ p 1 ) ,
G p ( ψ ) = ( a p κ ) 2 ψ 2 + ( b p κ 2 a p κ 2 ) ψ + ( 2 a p κ 3 b p κ 2 + c p κ ) .
η m = sinc 2 ( λ 0 λ m ) ,

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