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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 17 — Aug. 16, 2010
  • pp: 17974–17982
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Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms

Victorien Raulot, Philippe Gérard, Bruno Serio, Manuel Flury, Bernard Kress, and Patrick Meyrueis  »View Author Affiliations


Optics Express, Vol. 18, Issue 17, pp. 17974-17982 (2010)
http://dx.doi.org/10.1364/OE.18.017974


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Abstract

A new rigorous vector-based design and analysis approach of diffractive lenses is presented. It combines the use of two methods: the Finite-Difference Time-Domain for the study in the near field, and the Radiation Spectrum Method for the propagation in the far field. This approach is proposed to design and optimize effective medium cylindrical diffractive lenses for high efficiency structured light illumination systems. These lenses are realised with binary subwavelength features that cannot be designed using the standard scalar theory. Furthermore, because of their finite and high frequencies characteristics, such devices prevent the use of coupled wave theory. The proposed approach is presented to determine the angular tolerance in the cases of binary subwavelength cylindrical lenses by calculating the diffraction efficiency as a function of the incidence angle.

© 2010 OSA

1. Introduction

Subwavelength diffractive optical elements (SWDOE) refer to diffractive optical elements (DOE) that have features sizes smaller than the wavelength of the illumination light. These elements have been studied for several years [1

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]

]. They have a growing interest because they can achieve high diffraction efficiency by using only two levels for fabrication [1

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]

]. Their “superwavelength” counterparts are diffractive elements that have lateral features larger than the wavelength of illumination. To validate our approach that we propose, we present here the simulation of a subwavelength diffractive cylindrical lens. We demonstrate its ability to focus an incident beam light as a function of the incidence angle.

A great interest of binary SWDOE is accounted for their fabrication method. Indeed, one single step of photolithography is required, avoiding the alignment process between masks while maintaining very good diffractive efficiencies [4

4. 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(5), 1143–1156 (1999). [CrossRef]

9

9. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. 38(5), 870–878 (1999). [CrossRef]

]. Prism-like gratings [4

4. 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(5), 1143–1156 (1999). [CrossRef]

], antireflective surfaces [5

5. A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films 351(1-2), 73–78 (1999). [CrossRef]

], computer generated holograms [6

6. W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. 39(20), 3531–3536 (2000). [CrossRef]

], array illuminator [7

7. J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21(17), 1399–1402 (1996). [CrossRef] [PubMed]

] or diffractive lenses [8

8. F. T. Chen and H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 23, 1081–1083 (1996).

10

10. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A 16(5), 1157–1167 (1999). [CrossRef]

] designed by effective medium theory have been introduced in the past years. These elements are made up of subwavelength structures that define an effective medium. By varying the lateral feature size, the index of refraction of the effective medium, defined by the subwavelength structures, varies and creates a SWDOE corresponding to a multilevel element. In this case, light “sees” a medium containing an effective modulation of the refractive index. This modulation corresponds to a zeroth order grating which has a high efficiency in the zeroth order, and changes the phase of the incident light [11

11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996). [CrossRef]

]. The fill factors of the structures, composing the effective medium, induce a change of the phase of the incoming light. One dimensional [1

1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]

,9

9. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. 38(5), 870–878 (1999). [CrossRef]

] and two dimensional [4

4. 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(5), 1143–1156 (1999). [CrossRef]

,6

6. W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. 39(20), 3531–3536 (2000). [CrossRef]

,12

12. 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(14), 1081–1083 (1998). [CrossRef]

] surface modulations type have been successfully proposed in the literature in order to create different fill factors and, therefore, different phase modulations.

This paper focuses on a method that proceeds with one dimensional subwavelength DOEs simulations. One of their interests is to avoid the optical alignments of the successive photolithographic masks, while keeping high diffraction efficiency. Our novel approach to simulate subwavelength elements is based on a rigorous propagation of the light through a dielectric structure, using what we called a combined “Finite Difference Time Domain” – “Radiation Spectrum Method” (FDTD – RSM method) algorithms [13

13. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, 3rd Edition, 2005).

,14

14. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. 140(1-3), 128–145 (1997). [CrossRef]

]. This method enables us to describe the vectorial behavior of light over a long distance. In particular, a f/6 subwavelength diffractive lens has been designed by pulse width modulation and has been studied by the FDTD-RSM method that we propose.

2. Proposed combined FDTD-RSM numerical method principles

In our proposed approach, we use the FDTD method to calculate the propagation of light through the dielectric structure in the near field and the RSM to simulate the propagation of the light through free space preserving the vectorial aspects of the electromagnetic fields.

The Finite Difference Time Domain is a very well known vector-based method used to simulate the propagation of light by discretising Maxwell’s equation on a grid in space and time [13

13. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, 3rd Edition, 2005).

]. To achieve our calculations, we use a FDTD solver called Meep, which is an open software package with subpixel smoothing for increased accuracy, developed at Massachusetts Institute of Technology [15

15. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

]. Since the studied geometry has only one dimension, we choose to use one longitudinal coordinate for field propagation and one transverse coordinate for the definition of the profile. An advantage of the FDTD simulation is that a large range of different geometries can be simulated, if the lattice is correctly chosen. However, this method, well suited for near field calculations, is very time-consuming. The need to calculate every component of the complex electric and magnetic fields for each space and time grid point limits its use to local scale.

Therefore, in order to make a complete simulation of a SWDOE, jointly concerning near and far fields, the modeling and simulation problem is divided into two parts with appropriate solutions. The first one uses FDTD to evaluate the electromagnetic fields near the diffractive component. The second one describes the propagation in the free space by the RSM up to the focal point of the lens, what considerably reduces the modeling and simulation computing time.

3. Binary subwavelength design

In this paper, we propose a numerical method to evaluate the phase difference introduced by a binary subwavelength grating based on FDTD. Thanks to the possibility to solve periodic boundaries conditions problems by Meep, the propagation of a monochromatic wave is calculated over a single period of the grating. From a practical point of view, we consider a TE polarized plane wave which is generated about 5 wavelengths units (roughly 3 µm) below the grating. The TE polarization stands for an electric vector that is perpendicular to the incident plane. Then, the wave propagation is computed through the grating until the steady state is reached. Finally, the phase difference is estimated for all the points located within distance ranging from 1 to 5 wavelength units away from the grating (between 0.6 and 3 µm), and is calculated by estimating the arguments of the electric complex field.

We were looking to simulate a 2N phase levels surface spatial modulation in order to design the layout of a subwavelength lens which acts as a 2N levels digital lens. For a dielectric grating in a transmission configuration, the curve giving the phase difference as a function of the fill factor presents a strictly positive slope [11

11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996). [CrossRef]

]. Then, the maximum phase difference is reached for an aspect ratio value equal to 1 (i.e. a plane surface corresponding to the height level of the grating). The height of the grating h and the maximum phase difference ϕmax are linked therefore by Eq. (1):
h=12πλn1φmax
(1)
where n is the refractive index of the medium at the wavelength λ. The diffraction efficiency of the zeroth order is defined by the ratio between the electromagnetic energy (calculated by the flux of the Poynting vector) before the grating, and the energy in the zeroth order. For simulations presented in Fig. 1
Fig. 1 a) Phase difference and b) Diffraction efficiency as a function of the period and the fill factor of a 1203.5 nm height grating in fused silica without substrate for TE polarization at 632.8 nm.
below, the considered refractive index was 1.46, corresponding to the index of the fused silica at the 632.8 nm designed wavelength that we selected (He-Ne laser). Using the FDTD method, we calculated the main parameters of the zeroth order gratings. An eight phase levels diffractive structure simulated by the effective medium method was targeted, for which the higher phase difference was 7/8 x 2π rad.

Δφ=2π(fn2+1f)n1
(2)

For periods smaller than the third of the wavelength, a good agreement between the phase difference computed by the FDTD method and the zeroth order EMT approximation is observed. But the zeroth order EMT is only valid for grating periods smaller than the wavelength.

These results are taken into account to determine eight fill factors corresponding to the required phase differences of an 8-levels diffractive structure. However, our aim is to propose a design that can fit fabrication technologies. The period of 600 nm is only considered, even if the diffraction efficiency of the zeroth order is higher in the case of a 100 nm period grating, what is hard to achieve experimentally. This period leads us to consider the following fill factors and grating sizes, reported in the following Table 1

Table 1. Fill factors and widths of the line in the gratings allowing to realize an 8 levels simulated diffractive element. The grating is 1203.5 nm high, for an incidence light having a TE polarization and a normal incidence.

table-icon
View This Table
:

4. Lens design method

The proposed method to design and simulate the behavior of SWDOE is explained in detail in this section. Our aim is to design a diffractive lens operating with subwavelength structures in order to use the coding scheme of the phase.

4.1. Diffractive lens layout design

Subwavelength diffractive lenses designed from N-levels superwavelength lens are called N-levels simulated subwavelength lens.

4.2. Electromagnetic wave propagation simulation validating the proposed approach

In order to simulate the electromagnetic behavior of the lens, the combined FDTD – RSM approach that we propose is employed, taking into account all the components of the electric and magnetic fields. Indeed, scalar simulation is not accurate enough because of the subwavelength characteristics of the element.

First, the near field simulation is done with Meep, in order to calculate the propagation of light over a few micrometers after the dielectric structure of the lens. To ensure a sufficient resolution, several samplings are tested and the results compared to each other. Once this propagation is calculated, the components of the fields after the dielectric structures are retrieved in order to become the source of the RSM program.

To calculate the propagation of light through the free space, we use the RSM method. From any electromagnetic source, the guided and radiation modes of a dielectric medium are estimated using this algorithm. In our case, only the radiation modes are considered because the propagation is done through the free space.

The diffraction efficiency η of the lens is then estimated as the flux of the Poynting vector through a surface that corresponds to the diameter of the Airy disk at the focal point. For a lens diameter D, a focal length f, a focal point located at the yf coordinate, at a working wavelength λ and at final step of the simulation tf, the diffraction efficiency is calculated by Eq. (4):
η=u=0NfE(xu,y=yf,t=tf)×H(xu,y=yf,t=tf)v=0N0E(xv,y=y0,t=t0)×H(xv,y=y0,t=t0)
(4)
xu=0.66λ.fD+u.ΔxRSM,0uNf, Nf=1.22λ.fD.ΔxRSM,
xv=D2+v.ΔxFDTD,0v<N0, N0=DΔxFDTD
where ×denotes the cross product of the electric Eand magnetic Hvector fields, xu the uth spatial discrete coordinate x, at the focal plane, and xv the vth spatial discrete coordinate x, at the incident plane, ΔxFDTD and ΔxRSM stand respectively for the sampling steps for FDTD and RSM computation. Figure 4
Fig. 4 Near electric field propagation modeling in an inhomogeneous medium by the FDTD method on the left (the subwavelength lens is located inside the dotted black line) and far field Poynting vector simulation by RSM on the right. The plane A corresponds to the incident plane for the FDTD and the plane B to the recovery plane for the FDTD and also the incident plane for RSM.
shows the principle of the “combined” simulation. Left on the Fig. 4, we present amplitude of electric field E inside the dielectric structure and the recovery plane used for the RSM simulation. The Poynting vector is then carried out, step by step, with the RSM to analyse the energy propagation. A detail in the focal plane is observed on the Fig. 4. The diameter of the lens is here 1 mm with a 6 mm focal length. Computation time length of this simulation is approximately 2 hours for FDTD and several minutes for the RSM on Dual Core Pentium(R) D CPU 3.40 GHz Computer model.

5. Angular tolerance analysis in a specific case

Two different implementations of subwavelength lenses were designed and their optical tolerancings were studied. The first one uses 8 phase levels-simulated structures (considering the phase function previously presented in Fig. 2). The second one is a 4 levels simulated lens for which another phase function was calculated, because the maximum desired phase difference was not the same (3π/2 vs 7π/4). As a consequence, the high level value of the gratings composing this 4 levels simulated lens dropped from 1203.5 nm to 1031.6 nm.

In order to determine the angular tolerance of the proposed lenses, the diffraction efficiency was considered as a function of the incident angle of illumination. Figure 5
Fig. 5 Angular tolerance of subwavelength diffractive lenses
presents the comparison results:

Figure 5 confirms that the diffraction efficiency of the 8 levels simulated lens is higher than the 4 levels simulated one. Thanks to the scalar theory, it is known that diffraction efficiency is a function of the number of levels. This relation is also verified, because the subwavelength structures are defined thanks to the complete classical diffractive profiles.

The diffraction efficiencies follow the same curve for the angular tolerance whereas the number of phase levels has changed. 65% and higher diffraction efficiencies are obtained for angles smaller than 25°. For angles higher than 40°, diffraction efficiency drop to less than 30%. As the angle increases, no significant differences are observed between the 8 levels and 4 levels cases. So increasing the number of levels is not a rewarding strategy, because the phase sampling is not improved in the case of subwavelength structures. Indeed, at worst, a 8 levels simulated lens requires 1.2 µm high gratings whose line width can be 54 nm thick (see Table 1) although a 4 level simulated lens needs only 1 µm with a 150 nm line thick. That means an aspect ratio of about 22 in the first case versus about 7 in the second case. Therefore, 4 levels seem to be a good compromise between technological constraints and optical performances with angular tolerancing.

6. Conclusion

SWDOEs have a great interest since they offer the possibility to reduce the fabrication process steps to one single photolithographic step, avoiding optical alignment issues. Thanks to the strong improvement of nanotechnology processes, SWDOEs could be used into micro-electro-opto-mechanical-systems (MEOMS). Therefore, specific methodologies are necessary to design on demand optic functions based on their specificities.

This paper presents a new rigorous vector analysis approach fitting these requirements. The proposed algorithm enables to simulate the propagation of light over subwavelength, aperiodic and finite optical elements. The Finite Difference Time Domain algorithm is used to simulate the propagation in the near field condition, and the Radiation Spectrum Method enables the computation of the propagation in the far field. This approach has been used to study a subwavelength diffractive cylindrical lens design, implemented to simulate 4 and 8 levels of phase.

Nevertheless, because 8-levels simulated lens is more difficult to fabricate (because of a higher aspect ratio of the subwavelength gratings for which the width of the line can be smaller), 4-levels simulated lens seems to present the best compromise between good diffraction efficiency and technological constraints.

References and links

1.

M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]

2.

B. Kress, and P. Meyrueis, Digital Diffractive Optics: An introduction to planar diffractive optics and related technologies (John Wiley and Sons, 2000).

3.

S. Sinzinger, and J. Jahns, “Diffractive Optics,” in Microoptics, (Wiley-VCH, 2nd edition, 2005), pp. 138–141.

4.

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(5), 1143–1156 (1999). [CrossRef]

5.

A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films 351(1-2), 73–78 (1999). [CrossRef]

6.

W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. 39(20), 3531–3536 (2000). [CrossRef]

7.

J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21(17), 1399–1402 (1996). [CrossRef] [PubMed]

8.

F. T. Chen and H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 23, 1081–1083 (1996).

9.

D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. 38(5), 870–878 (1999). [CrossRef]

10.

J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A 16(5), 1157–1167 (1999). [CrossRef]

11.

P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996). [CrossRef]

12.

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(14), 1081–1083 (1998). [CrossRef]

13.

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, 3rd Edition, 2005).

14.

P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. 140(1-3), 128–145 (1997). [CrossRef]

15.

A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]

16.

D. Khalil, “Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide,” Opt. Quantum Electron. 35(8), 801–809 (2003). [CrossRef]

17.

M. Born, and E. Wolf, “Optics of crystals,” in Principles of Optics, (Cambridge University Press, Cambridge, 7th ed., 2005), 837–840.

OCIS Codes
(050.1965) Diffraction and gratings : Diffractive lenses
(050.2065) Diffraction and gratings : Effective medium theory
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: December 1, 2009
Revised Manuscript: January 19, 2010
Manuscript Accepted: January 19, 2010
Published: August 6, 2010

Citation
Victorien Raulot, Philippe Gérard, Bruno Serio, Manuel Flury, Bernard Kress, and Patrick Meyrueis, "Modeling of the angular tolerancing of an effective medium diffractive lens using combined finite difference time domain and radiation spectrum method algorithms," Opt. Express 18, 17974-17982 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-17-17974


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References

  1. M. W. Farn, “Binary Grating with increased efficiency,” Appl. Opt. 31(22), 4453–4458 (1992). [CrossRef] [PubMed]
  2. B. Kress, and P. Meyrueis, Digital Diffractive Optics: An introduction to planar diffractive optics and related technologies (John Wiley and Sons, 2000).
  3. S. Sinzinger, and J. Jahns, “Diffractive Optics,” in Microoptics, (Wiley-VCH, 2nd edition, 2005), pp. 138–141.
  4. 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(5), 1143–1156 (1999). [CrossRef]
  5. A. Gombert, W. Glaubitt, K. Rose, J. Dreibholz, B. Blasi, A. Heinzel, D. Sporn, W. Doll, and V. Wittwer, “Subwavelength-structured antireflective surfaces on glass,” Thin Solid Films 351(1-2), 73–78 (1999). [CrossRef]
  6. W. Yu, K. Takahara, T. Konishi, T. Yotsuya, and Y. Ichioka, “Fabrication of multilevel phase computer-generated hologram elements based on effective medium theory,” Appl. Opt. 39(20), 3531–3536 (2000). [CrossRef]
  7. J. M. Miller, N. de Beaucoudrey, P. Chavel, E. Cambril, and H. Launois, “Synthesis of a subwavelength-pulse-width spatially modulated array illuminator for 0.633 µm,” Opt. Lett. 21(17), 1399–1402 (1996). [CrossRef] [PubMed]
  8. F. T. Chen and H. G. Craighead, “Diffractive lens fabricated with mostly zeroth-order gratings,” Opt. Lett. 23, 1081–1083 (1996).
  9. D. W. Prather, “Design and application of subwavelength diffractive lenses for integration with infrared photodetectors,” Opt. Eng. 38(5), 870–878 (1999). [CrossRef]
  10. J. N. Mait, D. W. Prather, and M. S. Mirotznik, “Design of binary Subwavelength diffractive lenses by use of zeroth-order effective-medium theory,” J. Opt. Soc. Am. A 16(5), 1157–1167 (1999). [CrossRef]
  11. P. Lalanne and D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structure,” J. Mod. Opt. 43, 2063–2085 (1996). [CrossRef]
  12. 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(14), 1081–1083 (1998). [CrossRef]
  13. A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, (Artech, 3rd Edition, 2005).
  14. P. Gerard, P. Benech, D. Khalil, R. Rimet, and S. Tedjini, “Towards a full vectorial and modal technique for the analysis of integrated optics structures: the Radiation Spectrum Method,” Opt. Commun. 140(1-3), 128–145 (1997). [CrossRef]
  15. A. Farjadpour, D. Roundy, A. Rodriguez, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, and G. Burr, “Improving accuracy by subpixel smoothing in FDTD,” Opt. Lett. 31, 2972–2974 (2006). [CrossRef] [PubMed]
  16. D. Khalil, “Extension of the radiation spectrum method for the reflection calculation at the end of a strongly guiding optical waveguide,” Opt. Quantum Electron. 35(8), 801–809 (2003). [CrossRef]
  17. M. Born, and E. Wolf, “Optics of crystals,” in Principles of Optics, (Cambridge University Press, Cambridge, 7th ed., 2005), 837–840.

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