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, 2, 3, 4, 5]. 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, 7, 8, 9], and structures containing both pillars and holes were proposed for broadband use [10]. 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, 9]. 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, 12]. 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, 13]. 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, 15, 16, 17, 18] and references cited therein.

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

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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 N_m such that N ₁ = n _p, N_m > N _m−1, and N_M = 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 c_m × c_m 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., c_M = 0). The only difference between three structures is in the location of the pillars inside the blocks 2 ≤ 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_m are the same. The values c_m 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

We follow the two-step design procedure of our earlier paper [13] with some modifications that arise from fabrication considerations. In the first step the structure of Fig. 1(a) is considered (effective structure in short). The effective indices N_m are the optimization parameters and the first-order efficiency 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, 20]. In the second step the optimized values N_m are converted to corresponding pillar sizes c_m using the calibration curve in Fig. 2, obtained as in Ref. [13] using FMM for crossed gratings [21]. Then the final parametric optimization of c_m is performed for m = 2, …, M − 1. For fabrication reasons, we allow only values in the range 95 nm ≤ c_m ≤ 255 nm (0.3 ≤ c_m/d ≤ 0.81). According to Fig. 2, this implies that 1.071 < N_m < 1.663 for 2 ≤ m ≤ M − 1. The value of grating thickness h that provides full 2π 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.

 

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.

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Table 1 defines the structures obtained after the first design step for M = 2, …, 14: it lists the pillar sizes c_m after conversion from N_m using the curve in Fig. 2. The results depart from the linear starting point 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).

Table 1. Area-coded pillar sizes c_m 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).

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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 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_M then tend to be preferable (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.

Table 2. Area-coded pillar sizes c_m 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 h_M (red line with squares). (b) Theoretical spectral first order transmission (black line) and experimental results for three different wavelengths (blue and red circles).

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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] 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.

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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], 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.