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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 13 — Jun. 21, 2010
  • pp: 13444–13450
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Polarization insensitive resonance-domain blazed binary gratings

Heikki J. Hyvärinen, Petri Karvinen, and Jari Turunen  »View Author Affiliations


Optics Express, Vol. 18, Issue 13, pp. 13444-13450 (2010)
http://dx.doi.org/10.1364/OE.18.013444


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Abstract

Three variants of binary blazed gratings with subwavelength features are considered, which have high first-order efficiencies in the non-paraxial domain for arbitrarily polarized light. A combination of effective medium theory and further parametric optimization with the Fourier modal method are used in design. Experimental demonstration is provided by electron beam lithography on a structure etched in a Si3N4 layer on top of a SiO2 substrate, with period ~ 3.5λ at λ = 633 nm. The measured efficiency (81% for TE and 85% for TM polarization) agrees well with the calculated value, 84%.

© 2010 Optical Society of America

1. Introduction

Blazed binary gratings were first introduced in the form of subwavelength stripes with varying widths over each period, simulating an effective graded-index-material [1

1. W. Stork, N. Streibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991). [CrossRef] [PubMed]

, 2

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

, 3

3. H. Haidner, J. T. Sheridan, and N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993). [CrossRef] [PubMed]

, 4

4. M. E. Warren, R. E. Smith, G. A. Vawter, and J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm,” Opt. Lett. 20, 1441–1443 (1995). [CrossRef] [PubMed]

, 5

5. 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]

]. These one-dimensionally modulated structures exhibit polarization dependence caused by form birefringence. Therefore, two-dimensional binary blazed gratings consisting of subwavelength pillars were introduced [6

6. F. T. Chen and H. G. Graighead, “Diffractive phase elements based on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995). [CrossRef] [PubMed]

, 7

7. M. Kuittinen, J. Turunen, and P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998). [CrossRef]

, 8

8. 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]

, 9

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

], and structures containing both pillars and holes were proposed for broadband use [10

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

]. In the resonance domain, where the grating period is only a few wavelengths, blazed binary gratings can have first-order efficiencies exceeding those of conventional échelette gratings significantly [8

8. 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]

, 9

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

]. Moreover, the dependence of the efficiency on the angle of incidence is weaker and the binary blazed gratings work well also in conical illumination [11

11. M.-S. L. Lee, P. Lalanne, J.-C. Rodier, and E. Cambril, “Wide-field-angle behavior of blazed-binary gratings in the resonance domain,” Opt. Lett. 25, 1690–1692 (2000). [CrossRef]

, 12

12. M.-S. L. Lee, P. Lalanne, J.-C. Rodier, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A 4, S119–S124 (2002).

]. It can also be argued that the fabrication of resonance-domain blazed binary gratings is easier than fabrication of échelette (or multilevel) gratings with similar periods.

In the resonance domain, also pillar-structured blazed binary gratings become polarization sensitive because of non-paraxial propagation of the first diffraction order. Furthermore, the efficiency is reduced. To avoid these effects, the sizes and positions of the pillars must be optimized separately for each grating period [7

7. M. Kuittinen, J. Turunen, and P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998). [CrossRef]

, 13

13. H. J. Hyvärinen, J. Turunen, and P. Saarikko, “Efficiency optimization of blazed effective-medium gratings in the resonance domain,” J. Opt. A 10, 055005 (2008).

]. The design approach based on parametric optimization of the profile is thus analogous to those employed for other types of high-efficiency grating in the resonance-domain: see, e.g., Refs. [14

14. E. Noponen, J. Turunen, and A. Vasara, “Parametric optimization of the multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992). [CrossRef] [PubMed]

, 15

15. E. Popov, J. Hoose, B. Frankel, C. Keast, M. Fritze, T. y. Fan, D. Yost, and S. Rabe, “Low polarization dependent diffraction grating for wavelength demultiplexing,” Opt. Express 12, 269–275 (2004). [CrossRef] [PubMed]

, 16

16. N. Destouches, A. V. Tishchenko, J. C. Pommier, S. Reynard, O. Parriaux, S. Tochev, and M. Abdou Ahmed, “99% efficiency measured in the -1st order of a resonant grating,” Opt. Express 13, 3230–3235 (2005). [CrossRef] [PubMed]

, 17

17. J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Opt. Express 14, 2583–2588 (2006). [CrossRef] [PubMed]

, 18

18. J. Pietarinen and T. Vallius, “Double groove broadband gratings,” Opt. Express 16, 13824–13830 (2008). [CrossRef] [PubMed]

] and references cited therein.

In this paper we refine the two-stage design procedure introduced in Ref. [13

13. H. J. Hyvärinen, J. Turunen, and P. Saarikko, “Efficiency optimization of blazed effective-medium gratings in the resonance domain,” J. Opt. A 10, 055005 (2008).

] by considering some relevant fabrication constraints. While constraints inevitably reduce the diffraction efficiency to some degree, they lead to grating structures that can be manufactured in a relatively straightforward way. We then demonstrate experimentally that nearly polarization-insensitive high-efficiency operation of blazed binary gratings is possible in the resonance domain.

2. Structures

We consider four different binary transmission gratings illustrated in Fig. 1, which shows the top views of binary structures of height h across one grating period. The material below the structure, from where light is assumed incident, has a refractive index n = n s, while the material above the structure has refractive index n = 1. All structures consist of M square blocks of constant size d × d, where d < λ/n to ensure operation below the structural cut-off [9

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

]. Since d is fixed, the grating period in the x direction is Md, i.e., the period of the blazed grating increases linearly with the number of subwavelength blocks M. Consequently, the propagation angle θ −1 of the (minus) first transmitted order at normal incidence is given by sin θ −1 = −λ/(Md). Our aim is to obtain equal (and high) first-order efficiency η −1 for x and y polarized incident light (TM and TE, respectively) by optimizing the lateral structure of the grating period.

Fig. 1. Schematic top view of the structures considered. (a) Effective-material solid blocks with varying refractive index. Two-dimensionally modulated gratings with square pillars located to the left side (b), in the middle (c) and to the right side (d) of each block.

In the grating illustrated in Fig. 1 (a) each of the M blocks (m = 1, …,M) is assumed to consist of solid effective material of refractive index Nm such that N 1 = n p, Nm > N m−1, and NM = 1, where the maximum block index n p may be larger than n s. Hence this grating is in fact invariant in the y direction, facilitating the use of linear-grating theory instead of crossed-grating theory that must be applied to all other structures in Fig. 1. However, it is also rather artificial in the sense that ‘solid’ effective media with indices close to N = 1 are difficult to realize (the effective indices of porous materials like sol-gels are hard to control in subwavelength scale).

The three structures in Figs. 1(b)–(d) are area-coded such that each block contains a square pillar of size cm × cm and refractive index n p. To obtain the largest possible index modulation, we assume that block m = 1 is fully filled with material of index n p so that c 1 = d, while the block m = M has no pillar (index n = 1, i.e., cM = 0). The only difference between three structures is in the location of the pillars inside the blocks 2 ≤ mM − 1, as illustrated in Figs. 1(b)–(d). From fabrication point of view the disadvantage of the structure in Fig. 1(c) is that the smallest feature size is one half of that in the others if the values of cm are the same. The values cm for m = 2, …, M − 1 are our final optimization parameters; the main motivation for considering the three pillar variants is to see the effect of varying the center positions of the pillars and thus to reduce the number of optimization parameters to one half. We choose silicon dioxide (SiO2) as the substrate material and silicon nitride (Si3N4) as the pillar material. Therefore, at the wavelength λ = 633 nm considered throughout the paper, n s = 1.457 and n p = 1.99. The block size is fixed to d = 315 nm so that d < λ/n s.

3. Design procedure and results

Fig. 2. Calibration curve for the effective refractive index N of a regular Si3N4 square pillar structure with period d × d, d = 315 nm, as a function of the pillar size c × c.

Table 1 defines the structures obtained after the first design step for M = 2, …, 14: it lists the pillar sizes cm after conversion from Nm using the curve in Fig. 2. The results depart from the linear starting point cm = d(Mm)/(M − 1) of the optimization in step 1. The efficiencies given by FMM, presented in Fig. 3, improve for all four design variants in Fig. 1 but the variant in Fig. 1(b) is inferior to the others. Considering the issue of minimum feature size, we thus chose the structure of Fig. 1(b) for step 2 of the design, where unconstrained nonlinear optimization was used (Matlab command fminsearch).

Table 1. Area-coded pillar sizes cm after step 1 for structures with h = 568 nm.

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Fig. 3. Efficiencies η −1 for (a) linearly coded gratings and (b) designs after step 1. Black/circles refer to the structure in Fig. 1(a), green/squares to Fig. 1(b), red/triangles to Fig. 1(c), and blue/diamonds to Fig. 1(d).

The pillar sizes after step 2 are listed in Table 2 and the optimized η −1 are given by the black line in Fig. 4(a). The results are improved by as much as several percentage points in step 2. We show, for comparison, also the efficiencies for structures with hM = h (M − 1)/M (red line). For large M the latter are slightly superior to those with fixed h = 568 nm, as one might expect. However, the opposite is true if M < 9, which indicates that structures deeper than hM then tend to be preferable (hM < h if M < 9). We also tested direct parametric optimization (without step 1) using an 8-core computer with 24GB of memory. Results up to M = 14 could still be found rather easily, though much longer computation times were required. Nevertheless, the two-step procedure gave better first order efficiencies for large values of M than the direct approach (the same solutions were found only up to M = 6). The most probable reason is the stagnation of direct optimization into local minima of the merit function; the number of such minima increases quickly with increasing M.

Table 2. Area-coded pillar sizes cm after step 2 for structures in Fig. 1(d) with h = 568 nm.

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The black solid line in Fig. 4(b) illustrates the spectral performance of the grating. As expected, the efficiency is highest at the design wavelength λ = 633 nm and reduces especially towards small wavelengths. However, the high-efficiency spectral band is considerably wide, with η −1 > 80% for a spectral region from ~ 550 nm to ~ 670 nm.

Fig. 4. (a) First order efficiencies after step 2 for structures in Fig. 1(d) with fixed height h = 568 nm (black line with circles) and variable height hM (red line with squares). (b) Theoretical spectral first order transmission (black line) and experimental results for three different wavelengths (blue and red circles).

4. Fabrication and characterization

We verified the theoretical results by fabricating a blazed binary structure with 6 pillars (M = 7). Lithography processes commonly used in micro-optics[22

22. P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Volume 1: Microlithography, (SPIE-The International Society for Optical Engineering, 1997)

] were used. The Si3N4 layer was first deposited on a SiO2 substrate by chemical vapor deposition (CVD). Then the sample was spin-coated with PMMA resist and exposed by an electron beam pattern generator (Vistec EBPG 5000+ESHR). The sample was developed and the resulting resist structure was then used in a lift-off process. The resulting chromium structure was used as a mask for reactive ion etching (RIE) of the Si3N4 layer in CHF3 based plasma (Oxford Instruments PlasmaLab 80). The residual chromium mask was removed by wet-etching.

The blazed binary grating, patterned over an area of 5 mm × 5 mm area, was measured to have 81% and 85% first order efficiencies for TE and TM polarized incident He-Ne laser beams, respectively, leading to a polarization-averaged 83% first order efficiency at normal incidence; see the blue circle in Fig. 4(b). This corresponds well to the theoretically calculated efficiency, which is 84%. Two SEM images of the fabricated structure are shown in Fig. 5. The pillar sizes estimated from these SEM corresponds are close to the designed pillar sizes. As seen in Fig. 5(b), especially the smallest pillar is more like a circle than a square. However, Because of the small size of the pillar compared to wavelength, its exact shape does not have an appreciable effect in the functionality of the grating: what matters most is the fill factor.

We estimated that local variations of the pilar boundary positions were less than 1 nm across the grating. This causes only slight (virtually unmeasurable) local variations of diffraction efficiency, which average out if the incident laser beam illuminates an appreciable portion of the grating. In particular, these fabrication errors in subwavelength-scale features do not cause appreciable stray light since the periodicity of the grating is highly regular.

The metal coating required for SEM inspection of the structure was dissolved after taking pictures such as those presented in Fig. 5. We then measured polarization-averaged efficiencies η 1 at the design wavelength λ = 532 nm and, in addition, at λ = 532 nm and λ = 488 nm. The results are shown by the red circles in Fig. 4(b). The efficiency at λ = 633 nm had reduced to η −1 ≈ 78%, most probably because the metal coating had not been dissolved completely. Similarly, the efficiencies at shorter wavelengths lie somewhat below the theoretical predictions.

Fig. 5. SEM images of the fabricated structure. Scale bar = 1 µm.

5. Conclusions

We have demonstrated that blazed binary gratings are capable of nearly polarization-independent operation deep inside the resonance domain. The two-stage design approach proved superior to a direct optimization because a good starting configuration was achieved in the first, numerically highly efficient step. First-order efficiencies in the range 80–85% were obtained by the two-step design procedure, with fixed grating thickness, down to M = 6 pillars even though we employed a rather severe constraint on minimum feature size. This constraint is responsible for the slight decrease of efficiency for M > 9, seen in Figs. 3 and 4. As discussed in Ref. [13

13. H. J. Hyvärinen, J. Turunen, and P. Saarikko, “Efficiency optimization of blazed effective-medium gratings in the resonance domain,” J. Opt. A 10, 055005 (2008).

], the efficiencies for such large periods can exceed 90–95% without this constraint. Indeed, relaxing the minimum-feature constraint is the most effective method to improve the results further (more effective than, e.g., using non-rectangular features).

An experimental demonstration was given with M = 7 pillars. With the parameters assumed here, this corresponds to a first-order propagation angle θ −1 = 16.7°, i.e., the grating operates in the non-paraxial domain. The measured polarization-averaged efficiency η ≈ 83% at the design wavelength is nearly equal to the theoretical prediction.

Acknowledgments

This work was funded by Academy of Finland and the Graduate School of Modern Optics and Photonics.

References and links

1.

W. Stork, N. Streibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. 16, 1921–1923 (1991). [CrossRef] [PubMed]

2.

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

3.

H. Haidner, J. T. Sheridan, and N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. 32, 4276–4278 (1993). [CrossRef] [PubMed]

4.

M. E. Warren, R. E. Smith, G. A. Vawter, and J. R. Wendt, “High-efficiency subwavelength diffractive optical element in GaAs for 975 nm,” Opt. Lett. 20, 1441–1443 (1995). [CrossRef] [PubMed]

5.

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]

6.

F. T. Chen and H. G. Graighead, “Diffractive phase elements based on two-dimensional artificial dielectrics,” Opt. Lett. 20, 121–123 (1995). [CrossRef] [PubMed]

7.

M. Kuittinen, J. Turunen, and P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. 45, 133–142 (1998). [CrossRef]

8.

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]

9.

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

10.

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

11.

M.-S. L. Lee, P. Lalanne, J.-C. Rodier, and E. Cambril, “Wide-field-angle behavior of blazed-binary gratings in the resonance domain,” Opt. Lett. 25, 1690–1692 (2000). [CrossRef]

12.

M.-S. L. Lee, P. Lalanne, J.-C. Rodier, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A 4, S119–S124 (2002).

13.

H. J. Hyvärinen, J. Turunen, and P. Saarikko, “Efficiency optimization of blazed effective-medium gratings in the resonance domain,” J. Opt. A 10, 055005 (2008).

14.

E. Noponen, J. Turunen, and A. Vasara, “Parametric optimization of the multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. 31, 5910–5912 (1992). [CrossRef] [PubMed]

15.

E. Popov, J. Hoose, B. Frankel, C. Keast, M. Fritze, T. y. Fan, D. Yost, and S. Rabe, “Low polarization dependent diffraction grating for wavelength demultiplexing,” Opt. Express 12, 269–275 (2004). [CrossRef] [PubMed]

16.

N. Destouches, A. V. Tishchenko, J. C. Pommier, S. Reynard, O. Parriaux, S. Tochev, and M. Abdou Ahmed, “99% efficiency measured in the -1st order of a resonant grating,” Opt. Express 13, 3230–3235 (2005). [CrossRef] [PubMed]

17.

J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Opt. Express 14, 2583–2588 (2006). [CrossRef] [PubMed]

18.

J. Pietarinen and T. Vallius, “Double groove broadband gratings,” Opt. Express 16, 13824–13830 (2008). [CrossRef] [PubMed]

19.

L. Li, “Formulation and comparison of two recursive matrix algorithm for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]

20.

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]

21.

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

22.

P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Volume 1: Microlithography, (SPIE-The International Society for Optical Engineering, 1997)

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.5745) Diffraction and gratings : Resonance domain
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: March 18, 2010
Revised Manuscript: May 10, 2010
Manuscript Accepted: May 10, 2010
Published: June 8, 2010

Citation
Heikki J. Hyvärinen, Petri Karvinen, and Jari Turunen, "Polarization insensitive resonance-domain blazed binary gratings," Opt. Express 18, 13444-13450 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-13-13444


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References

  1. W. Stork, N. Streibl, H. Haidner, and P. Kipfer, "Artificial distributed-index media fabricated by zero-order gratings," Opt. Lett. 16, 1921-1923 (1991). [CrossRef] [PubMed]
  2. M. W. Farn, "Binary gratings with increased efficiency," Appl. Opt. 31, 4453-4458 (1992). [CrossRef] [PubMed]
  3. H. Haidner, J. T. Sheridan, and N. Streibl, "Dielectric binary blazed gratings," Appl. Opt. 32, 4276-4278 (1993). [CrossRef] [PubMed]
  4. M. E. Warren, R. E. Smith, G. A. Vawter, and J. R. Wendt, "High-efficiency subwavelength diffractive optical element in GaAs for 975 nm," Opt. Lett. 20, 1441-1443 (1995). [CrossRef] [PubMed]
  5. 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]
  6. F. T. Chen and H. G. Graighead, "Diffractive phase elements based on two-dimensional artificial dielectrics," Opt. Lett. 20, 121-123 (1995). [CrossRef] [PubMed]
  7. M. Kuittinen, J. Turunen, and P. Vahimaa, "Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings," J. Mod. Opt. 45, 133-142 (1998). [CrossRef]
  8. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, "Blazed binary subwavelength gratings with efficiencies larger than those of conventional ´echelette gratings," Opt. Lett. 23, 1081-1083 (1998). [CrossRef]
  9. Q1. P. Lalanne, S. Astilean, P. Chavel, E. Cambril, and H. Launois, "Design and fabrication of the blazed binary diffractive elements with sampling periods smaller than the structural cutoff," J. Opt. Soc. Am. A 16, 1143-1156 (1999). [CrossRef]
  10. C. Sauvan, P. Lalanne, and M.-S. L. Lee, "Broadband blazing with artificial dielectrics," Opt. Lett. 29, 1593-1595 (2004). [CrossRef] [PubMed]
  11. M.-S. L. Lee, P. Lalanne, J.-C. Rodier, and E. Cambril, "Wide-field-angle behavior of blazed-binary gratings in the resonance domain," Opt. Lett. 25, 1690-1692 (2000). [CrossRef]
  12. Q2. M.-S. L. Lee, P. Lalanne, J.-C. Rodier, E. Cambril, and Y. Chen, "Imaging with blazed-binary diffractive elements," J. Opt. A 4, S119-S124 (2002).
  13. Q3. H. J. Hyv¨arinen, J. Turunen, and P. Saarikko, "Efficiency optimization of blazed effective-medium gratings in the resonance domain," J. Opt. A 10, 055005 (2008).
  14. E. Noponen, J. Turunen, and A. Vasara, "Parametric optimization of the multilevel diffractive optical elements by electromagnetic theory," Appl. Opt. 31, 5910-5912 (1992). [CrossRef] [PubMed]
  15. E. Popov, J. Hoose, B. Frankel, C. Keast, M. Fritze, T. y. Fan, D. Yost, and S. Rabe, "Low polarization dependent diffraction grating for wavelength demultiplexing," Opt. Express 12, 269-275 (2004). [CrossRef] [PubMed]
  16. N. Destouches, A. V. Tishchenko, J. C. Pommier, S. Reynard, O. Parriaux, S. Tochev, and M. Abdou Ahmed, "99% efficiency measured in the −1st order of a resonant grating," Opt. Express 13, 3230-3235 (2005). [CrossRef] [PubMed]
  17. J. Pietarinen, T. Vallius, and J. Turunen, "Wideband four-level transmission gratings with flattened spectral efficiency," Opt. Express 14, 2583-2588 (2006). [CrossRef] [PubMed]
  18. J. Pietarinen and T. Vallius, "Double groove broadband gratings," Opt. Express 16, 13824-13830 (2008). [CrossRef] [PubMed]
  19. Q4. L. Li, "Formulation and comparison of two recursive matrix algorithm for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996). [CrossRef]
  20. Q5. 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]
  21. Q6. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997). [CrossRef]
  22. P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Volume 1: Microlithography, (SPIE-The International Society for Optical Engineering, 1997)

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