## Polarization insensitive resonance-domain blazed binary gratings

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 Si_{3}N_{4} layer on top of a SiO_{2} substrate, with period ~ 3.5*λ* 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

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]

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]

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]

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]

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 -1^{st} 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]

## 2. Structures

*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]

*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.

*M*blocks (

*m*= 1, …,

*M*) is assumed to consist of solid effective material of refractive index

*N*such that

_{m}*N*

_{1}=

*n*

_{p},

*N*>

_{m}*N*

_{m−1}, and

*N*= 1, where the maximum block index

_{M}*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).

*c*×

_{m}*c*and refractive index

_{m}*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.,

*c*= 0). The only difference between three structures is in the location of the pillars inside the blocks 2 ≤

_{M}*m*≤

*M*− 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

*c*are the same. The values

_{m}*c*for

_{m}*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 (SiO

_{2}) as the substrate material and silicon nitride (Si

_{3}N

_{4}) 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

*N*are the optimization parameters and the first-order efficiency

_{m}*η*

_{−1}averaged over TE and TM polarizations is the merit function. Each configuration is evaluated by the Fourier modal method (FMM) for linear gratings [19

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]

*N*are converted to corresponding pillar sizes

_{m}*c*using the calibration curve in Fig. 2, obtained as in Ref. [13] using FMM for crossed gratings [21

_{m}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]

*c*is performed for

_{m}*m*= 2, …,

*M*− 1. For fabrication reasons, we allow only values in the range 95 nm ≤

*c*≤ 255 nm (0.3 ≤

_{m}*c*/

_{m}*d*≤ 0.81). According to Fig. 2, this implies that 1.071 <

*N*< 1.663 for 2 ≤

_{m}*m*≤

*M*− 1. The value of grating thickness

*h*that provides full 2

*π*phase modulation at

*M*→∞ is

*h*

_{∞}=

*λ*/(

*n*

_{p}−1), and for finite

*M*one is tempted to choose

*h*=

_{M}*h*

_{∞}(

*M*− 1)/

*M*. However,

*h*must be fixed for all

*M*to facilitate binary construction of structures with spatially varying local period (such as diffractive lenses). We chose

*h*= 568 nm since films of this thickness were available.

*M*= 2, …, 14: it lists the pillar sizes

*c*after conversion from

_{m}*N*using the curve in Fig. 2. The results depart from the linear starting point

_{m}*c*=

_{m}*d*(

*M*−

*m*)/(

*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).

*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

*h*=

_{M}*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

*h*then tend to be preferable (

_{M}*h*<

_{M}*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*.

*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.

## 4. Fabrication and characterization

*M*= 7). Lithography processes commonly used in micro-optics[22] were used. The Si

_{3}N

_{4}layer was first deposited on a SiO

_{2}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 Si

_{3}N

_{4}layer in CHF

_{3}based plasma (Oxford Instruments PlasmaLab 80). The residual chromium mask was removed by wet-etching.

*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.

## 5. Conclusions

*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], 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).

*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

## References and links

1. | W. Stork, N. Streibl, H. Haidner, and P. Kipfer, “Artificial distributed-index media fabricated by zero-order gratings,” Opt. Lett. |

2. | M. W. Farn, “Binary gratings with increased efficiency,” Appl. Opt. |

3. | H. Haidner, J. T. Sheridan, and N. Streibl, “Dielectric binary blazed gratings,” Appl. Opt. |

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. |

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. |

6. | F. T. Chen and H. G. Graighead, “Diffractive phase elements based on two-dimensional artificial dielectrics,” Opt. Lett. |

7. | M. Kuittinen, J. Turunen, and P. Vahimaa, “Rigorous analysis and optimization of subwavelength-structured binary dielectric beam deflector gratings,” J. Mod. Opt. |

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. |

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 |

10. | C. Sauvan, P. Lalanne, and M.-S. L. Lee, “Broadband blazing with artificial dielectrics,” Opt. Lett. |

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. |

12. | M.-S. L. Lee, P. Lalanne, J.-C. Rodier, E. Cambril, and Y. Chen, “Imaging with blazed-binary diffractive elements,” J. Opt. A |

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

14. | E. Noponen, J. Turunen, and A. Vasara, “Parametric optimization of the multilevel diffractive optical elements by electromagnetic theory,” Appl. Opt. |

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 |

16. | N. Destouches, A. V. Tishchenko, J. C. Pommier, S. Reynard, O. Parriaux, S. Tochev, and M. Abdou Ahmed, “99% efficiency measured in the -1 |

17. | J. Pietarinen, T. Vallius, and J. Turunen, “Wideband four-level transmission gratings with flattened spectral efficiency,” Opt. Express |

18. | J. Pietarinen and T. Vallius, “Double groove broadband gratings,” Opt. Express |

19. | L. Li, “Formulation and comparison of two recursive matrix algorithm for modeling layered diffraction gratings,” J. Opt. Soc. Am. A |

20. | P. Lalanne and G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

21. | L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A |

22. | P. Rai-Choudhury, |

**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

- 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]
- M. W. Farn, "Binary gratings with increased efficiency," Appl. Opt. 31, 4453-4458 (1992). [CrossRef] [PubMed]
- H. Haidner, J. T. Sheridan, and N. Streibl, "Dielectric binary blazed gratings," Appl. Opt. 32, 4276-4278 (1993). [CrossRef] [PubMed]
- 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]
- 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]
- F. T. Chen and H. G. Graighead, "Diffractive phase elements based on two-dimensional artificial dielectrics," Opt. Lett. 20, 121-123 (1995). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- C. Sauvan, P. Lalanne, and M.-S. L. Lee, "Broadband blazing with artificial dielectrics," Opt. Lett. 29, 1593-1595 (2004). [CrossRef] [PubMed]
- 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]
- 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).
- 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).
- 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]
- 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]
- 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]
- J. Pietarinen, T. Vallius, and J. Turunen, "Wideband four-level transmission gratings with flattened spectral efficiency," Opt. Express 14, 2583-2588 (2006). [CrossRef] [PubMed]
- J. Pietarinen and T. Vallius, "Double groove broadband gratings," Opt. Express 16, 13824-13830 (2008). [CrossRef] [PubMed]
- 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]
- 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]
- 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]
- P. Rai-Choudhury, Handbook of Microlithography, Micromachining, and Microfabrication: Volume 1: Microlithography, (SPIE-The International Society for Optical Engineering, 1997)

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