1. Introduction

Diffractive optics [1] offers a powerful toolbox for photon management, applicable in different regions of the electromagnetic spectrum ranging from soft X-rays to radio frequencies, provided that suitable material choices are made. In the near-infrared region semiconductor materials are convenient substrates for microstructures; the structure can either be etched directly into the substrate or fabricated on top of it using materials with sufficient infrared transmission. One of the major problems of these high-refractive-index materials is the large Fresnel reflection loss.

It is well known that subwavelength-period microstructures of different forms, etched into a semiconductor surface, can provide good antireflection (AR) properties in case of flat substrates [2]. Unfortunately it is difficult to integrate such surface-relief AR structures with the functional diffractive microstructure without degrading the performance of the latter.

The transmission can also be improved by coating the functional element with a thin dielectric layer (or a stack of layers) designed to act as an AR coating for a flat surface [3] as illustrated in Fig. 1a. However, this approach suffers from a fundamental problem: the features in the underlying microstructure are often too small for the AR designs (that assume laterally infinite films) to be reliable. In addition, there are several technical problems. First, if the microstructure is not etched directly into the substrate but instead fabricated on top of it by deposition of a different material, the same AR stack does not work ideally in both the grooves and the ridges. Second, the layer thickness on the grooves and ridges may differ significantly, especially if wavelength-scale transverse features are present. Third, some coating material will almost inevitably fall on the sidewalls of the microstructure thus distorting the desired profile.

To avoid (or at least reduce) the problems listed above we propose to fabricate the AR film stack underneath the microstructure prior to its patterning and deposition, as illustrated in Fig. 1b. This approach facilitates the fabrication of highly uniform AR layers provided that the topmost layer is chosen so as to act as an etch stop in the fabrication of the microstructure.

The proposed approach, however, offers a new challenge: now the effective film stack consists only of the AR coating in the groove regions, but contains an additional top layer in the ridge regions. Unless good AR performance is obtained for both film stacks simultaneously, undesired (binary) amplitude modulation results, which may invalidate the original design of the phase-only-modulating diffractive element and require a refinement of the design.

 

Fig. 1. Two alternative ways to combine a diffractive microstructure with a thin-film AR structure: (a) the microstructure is over-coated with a thin film and (b) the AR coating is fabricated underneath the microstructure.

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In this paper the first designs of AR structures of the type illustrated in Fig. 1b are presented and analyzed, conclusions are drawn, and some directions for future work are outlined. However, we start the discussion by considering some transmission properties of a high-index substrate coated by uniform thin-film stacks on both sides.

2. Properties of flat and nonuniform antireflection-coated high-index substrates

Let us consider a perfectly flat high-index substrate (Si with refractive index n=3.48 at λ=1.55 µm), which is either uncoated but polished on both sides, or AR-coated on both sides. In the latter case, the AR coating (Al₂O₃ with n=1.6 and thickness h ₁ and HfO₂ with n=1.9 and thickness h ₂) is designed to provide nearly perfect transmission from the substrate to air or vice versa. This can be achieved, e.g., with a number of double-layer AR structures. Here we consider three different AR structures with the parameters listed in Table 1. Design 1 is a standard structure consisting of two quarter-wave layers, but designs 2 and 3 (which employ larger layer thicknesses) have been optimized for the given refractive indices. In the case of designs 2 and 3 the standard optimization based on the Nelder-Mead simplex method has been used.

Table 1. Three AR coating designs consisting of a Al₂O₃ layer with n=1.6 and thickness h₁, and a HfO₂ layer (located next to the Si-substrate) with n=1.9 and thickness h₂.

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We analyzed the transmission of the silicon substrate alone as well as the both-sides-AR-coated substrate using the S-matrix approach [4], which remains numerically stable even though some of the layers involved are optically thick (in this case it is the Si substrate). The effect of variations of the substrate thickness over a fluctuation period in the transmission are illustrated in Fig. 2. Strong transmission fluctuations are seen not only for the uncoated Si substrate but for most of the AR-coated substrates as well, even though for all the Designs 1–3 the Si-stack-air transmission is better than 39%. These variations may be attributed to Fabry–Perot -type cavity effects. Indeed, transmission fluctuations reach quite an acceptable level in case of Design 2, indicating great tolerance against errors in the substrate thickness. The effect arises from the characteristic behavior of antireflection stacks. When the phase difference attributed to the stack coincides with the phase modulation of the optimal stack, the strong Fabry-Perot cavity effect of the substrate is reduced.

 

Fig. 2. Transmission of a perfectly flat Si substrate (dashed line) as a function of its thickness and the influence of AR-coatings on both sides. Green line: Design 1. Red line: Design 2. Blue line: design 3.

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The importance of the observed tolerance of relatively thick AR layers against substrate thickness variations should not be underestimated in diffractive optics. We did assume a perfectly flat substrate, being aware of the fact that this assumption is incorrect: the specified thicknesses (and flatnesses) of commercially available Si wafers [5, 6] are substantially larger than the quarter-wave range considered in our analysis presented in Fig. 2. Thus it is more than likely that both extremes (minimum and maximum transmission) are encountered within the area covered by a typical diffractive element and therefore some average transmission far from the ideal value would be observed. Moreover, even a minimal rotation of the element with respect to the optical axis could affect the results considerably, especially in case of elements with spatially variant optical properties such as lens arrays or free-space optical interconnects [7]: in the worst-case scenario, two neighboring elements might differ in transmission efficiency to a factor approaching three even though they would have identical surface profiles.

3. Simulation and optimization of microstructures on top of AR layers

We now proceed to consider thin, low-index (SiO₂ with n=1.6 at λ=1.55 µm) diffractive structures fabricated on top of uncoated and AR-coated high-index (Si with thickness 575 µm)) substrates keeping in mind that the AR coating should be of the type of Design 2 to operate satisfactorily in the transmission mode. In particular, we consider two different binary beam splitters and a four-level ‘blazed’ grating, assuming in all cases that period d is far greater than vacuum wavelength λ of the incident light. In this case the thin element approximation is valid. However, for the sake of reliability, we resort to electromagnetic theory [2].

Let us first analyze 1→2 beam splitters, which employ diffraction orders m=-1 and m=+1 of a binary surface-relief grating with the thickness of the SiO₂ layer denoted by h. The diffraction efficiency of this beam splitter is defined as η=η _-1+η ₊₁. If a binary profile of thickness h=λ/2(n _Si-1)=313 nm and the fill factor f=1/2 is etched directly into the Si substrate, the efficiency is η=0.413. If the microstructured region is made of SiO₂ instead of Si, f=1/2, and h=λ/2 $(n_{\mathrm{Si}{O}_2}-1)=1292\mathrm{nm}$, we obtain η=0.658. The efficiencies η_m of several central diffraction orders are illustrates for the latter case by the blue bars in figure 3 using rigorous theory (100 truncation orders included) and assuming that d=50λ. Hence non-vanishing values of η_m are obtained also for even-valued m.

 

Fig. 3. Diffraction efficiencies of central transmitted orders of the 1→2 beam-splitter. Substrate is AR-coated on the both sides. Blue bars: Grating without AR-layers. Green bars: Optimal AR stack with a standard grating. Red bars: Simultaneously optimized AR and grating layer thicknesses.

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Considering then a structure similar to the previous one, but with an AR coating ($h_{\mathrm{Hf}{O}_2}=320\mathrm{nm}$ and $h_{{\mathrm{Al}}_2{O}_3}=316\mathrm{nm}$) beneath it, we obtain η=0.746. The results shown by the green bars show considerably increased performance, which may be expected because now the Si-AR-air intensity transmission coefficient T has been maximized. However, this is not true for the Si-AR-SiO₂-air stack, for which the transmission coefficient is now only T=0.846. Hence, we have a binary grating that modulates both the amplitude and the phase. Hence we are faced with the following optimization problem: Find two high-transmittance stacks generating a mutual phase delay such that the efficiency η reaches its maximum value. We do not attempt a complete numerical solution to this problem here, let alone an analytical solution. However, we show numerically that such solutions can be found. This is illustrated by the red bars in figure 3. The optimized layer thicknesses are now $h_{\mathrm{Hf}{O}_2}=382\mathrm{nm}$, $h_{{\mathrm{Al}}_2{O}_3}=263\mathrm{nm}$, and $h_{\mathrm{Si}{O}_2}=1399\mathrm{nm}$. In this case the theory of stratified media gives intensity transmission coefficients of T=0.995 and T=0.985 for the groove and ridge regions, respectively. In this case η=0.793. Let us now shift the attention to a beam splitter with a somewhat more complicated structure, designed to generate five equal-efficiency orders m=-2,…,+2. Binary gratings capable of doing this are often called Dammann gratings [9]. Figure 4 illustrates the numerical simulations for the 1→5 binary beam splitters, characterized by transition points x_j, j=1,…,4. Transition points x ₁=0.0692d, x ₂=0.3733d, x ₃=0.4147d, and x ₄=0.8056d provide good array uniformity and a diffraction efficiency of η=0.598 if the grating were fabricated as a surface-relief structure in SiO₂. In rigorous analysis the period was chosen to be d=350λ to ensure that the smallest feature size is larger than 20λ so that the grating operates in the paraxial domain. In this case, 120 truncation orders are included.

The blue bars illustrate the performance of the above-defined SiO₂ structure if it is fabricated on top of a Si substrate. Clearly, the efficiency is reduced as could be expected, but also the array uniformity is decreased because of the amplitude modulation. The green bars show the efficiencies if the same AR structure is used on both sides of the substrate as in case of the 1→2 beam splitter ($h_{\mathrm{Hf}{O}_2}=320\mathrm{nm}$ and $h_{{\mathrm{Al}}_2{O}_3}=316\mathrm{nm}$). The efficiency is now improved significantly from η=0.598 to η=0.672, but the zero-order efficiency differs rather substantially from the efficiencies of other beams in the array. The situation can be improved by using the AR structure in which the AR-layer and SiO₂ layer thicknesses are optimized simultaneously (red bars) and even further if the transition points are also taken as free parameters and modified slightly.

 

Fig. 4. Diffraction efficiencies η_m of the central orders of 1→5 beam splitters. Substrate is AR-coated on the both sides. Blue bars: SiO₂ grating on a Si substrate without AR-layers. Green bars: Optimal AR-layer stack with a standard grating. Red bars: Simultaneously optimized AR and grating layer thicknesses.

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As the final example, we consider a four-level staircase grating, in which each step height is 646 nm and the period is d=50λ. Now we face the task of finding an AR structure that would perform well for four different SiO₂ layer thicknesses, which however are at least in principle adjustable, as are the transition points. In this case, rigorous diffraction theory with 100 truncation orders is used. Figure 5 illustrates some of the results we have obtained so far. Again it is clear that the efficiency η ₁ increases if we add AR layers on both sides of the substrate and even further if we design simultaneously the AR layer and grating-layer thicknesses ($h_{\mathrm{Hf}{O}_2}=327\mathrm{nm}$, $h_{{\mathrm{Al}}_2{O}_3}=183\mathrm{nm}$, step heights of $h_{\mathrm{Si}{O}_2}=$ [0, 561, 822, 547] nm). Further improvements could be expected if all grating-layer thicknesses and transition points were treated as free optimization parameters.

As an example we calculated also wavelength and incident angle tolerance analysis of the designed blaze grating. Numerical simulation showed that transmission of the first diffraction order is changing only slightly when the incident angle varies from -2 to +2 degrees. Variations in wavelength is more critical and only ±0.2 nm tolerance is allowed.

 

Fig. 5. Transmitted diffraction efficiencies of 4-level staircase gratings. Substrate is AR-coated on the both sides. Blue: Grating without AR-layers. Green: Optimized AR stacks. Red: AR-layers and grating thickness scale optimized simultaneously.

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4. Discussion and conclusions

In this communication we have introduced a novel antireflection-coating method for diffractive elements manufactured on high-index substrates that are of interest in, e.g., sophisticated optical communication components. Some first analysis and designs were presented, which illustrate the importance of cavity effects and the possibility to greatly improve the performance of the diffractive elements by simultaneous design of the AR layers and the diffractive structure. Nevertheless, we also showed that it is possible to design reasonably universal AR layers which perform in an acceptable manner at least for binary structures. Slight structural modification improved the results, but this procedure can be implemented in general design methods of diffractive elements, such as the Iterative Fourier-Transform Algorithm, rather easily; for multilevel structures one just need to calculate (and tabulate) the entire stack’s complex-amplitude transmission properties as a function of the thickness of the modulated layer and use this information in the optimization of both the vertical and the lateral surface profile.

Here we have considered only diffractive elements with feature sizes that are comparatively large with respect to the vacuum wavelength of the illuminating beam. Some analysis has already been made (using rigorous theory) without this assumption, and it appears possible to obtain excellent diffraction efficiencies, in some case even without AR stacks. Such resonance-domain effects are, however, a subject of further study.