Polarization gratings are space-variant subwavelength-structured photonic devices that control electromagnetic wave propagation by local modulation of the state of polarization of light. Using electron beam lithography, we have fabricated such devices in the form of dielectric and metallic surface-relief profiles for operation in the visible wavelength region, where structural features with dimensions on the order of 100 nm are required. We provide experimental demonstrations of various laser-beam splitting elements with diffraction efficiencies exceeding values that could be achieved by diffractive elements operating in the framework of scalar optics.
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Diffractive optics [1, 2] provides a wide range of versatile methods for splitting and shaping light fields. A key factor in such applications is the diffraction efficiency η of the element, i.e., the portion of incident light that is directed to the controllable part of the shaped field (the rest being noise that is directed outside the spatial region of interest). Conventional diffractive elements with thin (wavelength-scale) surface-relief profiles with substantially larger-scale transverse variations can be well described using paraxial scalar diffraction theory since all polarization components of the incident field are transformed in the same way . In this case the diffraction efficiency η has an upper bound ηu, which depends on the type of transformation desired and on fabrication-related restrictions on element structure . For all transformations other than simple beam deflection, ηu < 100% even if no restrictions are placed on element structure.
Efficiencies η > ηu can be achieved using subwavelength-structured diffractive elements . A particularly versatile method is based on so-called polarization gratings , which can be implemented by using subwavelength-period gratings with spatially variable fringe orientation. These elements are capable of controlling the state of polarization of incident light, thus acting as spatially varying wave plates. The additional degree of freedom provided by polarization allows in many cases efficiencies η = 100% if dielectric structures are used [6, 7]. In fact, it appears that the polarization freedom invariably ensures η ≥ ηu. The improvement in η compared to scalar designs with the same signal distribution can be quite decisive  and broadband operation is also possible [9, 10].
There are several approaches to practical realization of polarization gratings. Some of the designs presented in  were soon demonstrated by liquid-crystal spatial light modulators [11, 12]. Along with holographic gratings exposed in azobenzene-containing films [13, 14, 15], these devices have the attractive feature of being reconfigurable. Alternatively, polarization gratings can be fabricated in the form of surface-relief structures, which have a fixed optical function but are nevertheless attractive because they can be replicated in large quantities by techniques described in, e.g., Chapt. 6 of  and Chapt. 5 of . Because subwavelength periods are required, nanoscale patterning is needed in fabrication of surface-relief type polarization gratings for visible light. These devices have been demonstrated mainly for mid-infrared  and recently also for the wavelength λ = 1064 nm . In this paper we present (to our knowledge) the first demonstrations for visible light with λ = 532 nm. In particular, we present metallic gratings designed to operate as polarimeters [5, 18] and binary dielectric structures acting as beam splitters [6, 11].
2. Theoretical background
Assuming paraxial propagation in the positive z-direction throughout this paper, we describe the incident electric field by a Jones vector with components Ex and Ey. We consider polarization gratings of the type discussed by Gori , which ideally have a linearly rotating orientation of the subwavelength-period fringes of period Λ. Thus the local angle of the fringes with respect to the x-axis is taken to be of the form ϕ(x) = πx/d. Here d denotes the full rotation period, which defines another grating as illustrated in Fig. 1. The period of the latter grating is assumed to satisfy d ≫ λ, which ensures that the central diffraction orders propagate paraxially. In fact, only three central orders have non-vanishing diffraction efficiencies for a linear polarization grating, i.e., all transmitted light is distributed among these orders [5, 6].
The subwavelength carrier grating with period Λ ≪ λ of the rotating fringes does not generate diffracted orders. However, it modulates the polarization state of the incident light in a way that depends on its physical structure. Following the notation of , we denote by A and B the complex transmission coefficients of this local grating for TE- and TM-polarized light, respectively (here TE means that the incident electric field vector is parallel to the fringes and TM means that it is normal to the fringes). The diffraction efficiencies ηm of the three central orders, labeled by m = −1, m = 0, and m = 1, are then given by Eq. (2), the efficiencies η±1 depend not only on the properties of the grating (the coefficients A and B) but also on the polarization state of the incident field. This is indeed the new degree of freedom (polarization freedom) provided by the electromagnetic approach, which allows η > ηu.
The polarimeter for measurement of the Stokes parameters of light beams  is obtained if we choose A = 0 and B = 1. This means that the TE-polarized components of light is completely blocked, while the TM-polarized component is fully transmitted. These are the transmission amplitudes of an ideal linear polarizer, i.e., the basic polarization grating can be realized locally as a subwavelength-period metal-wire grid . Now the diffraction efficiencies of the central orders have the following properties: η0 = 1/4 and the sum of η±1 equals 1/4 and one half of the incident light is transmitted, irrespective of the incident polarization state, because TE polarization is locally blocked and the fringes rotate a full circle within each period d. The diffraction efficiencies η±1 are equal if the phase difference Δθ = 0, i.e. if the incident polarization state is linear. With other values of the Δθ the diffraction efficiencies η±1 are generally unequal.
Several different distributions of light among orders m = −1, 0, +1 were considered in  assuming that |A| = |B| = 1, which implies that the grating transmits 100% of incident light. The simplest example is the one-point signal, i.e., one of the orders has a unit diffraction efficiency. In the paraxial domain, this ‘blazing effect’ is usually achieved with triangular-profile dielectric gratings. With polarizations gratings, the same effect is obtained for order m = +1 with left-circularly polarized (LCP) incident light and for order m = −1 for right-circularly-polarized (RCP) light if we choose A = 1, B = −1, and Δθ = ±π/2 in Eq. (2). Linearly polarized incident light produces a duplicator with η−1 = η+1 = 1/2. A triplicator with η−1 = η0 = η+1 = 1/3 is achieved for linearly polarized light by the choices A = 1 and B = exp(iθ) with cosθ = −1/3. The upper bound of diffraction efficiency for a triplicator in the scalar regime is ηu = 0.938 and the highest efficiency that is obtainable within scalar optics is η = 0.926 .
3. Structural design
A continuous variation of ϕ(x) would lead to a spatially varying grating period if it was generated rigorously, and therefore we employ the approach of Biener et al.  by quantizing the continuously varying fringe orientation into segments in which the fringe orientation is constant (see Fig. 1). If the number of segments within d is Q, so that the width of each segment is d/Q, the local fringe orientation in q:th segment d(q – 1)/Q < x < dq/Q (q = 1,..., Q) is ϕq = π(q – 1)/Q. As a result of quantization, the efficiency of the element is reduced negligibly for Q ≥ 20 . However, we can use a constant local grating period Λ across the full grating area. Thus we are left with the problem of deciding the optimum local structural factors: the ratio Λ/λ, the normalized ridge height h/λ, and the fill factor f = c/Λ of the profile. Only rectangular ridges are considered in the designs since it is a good approximation of the profile using the optimized fabrication processes. Also other types of profiles as well as buried structures could be applied with careful design.
The local structural designs of all elements was carried out by the Fourier Modal Method (FMM) for linear gratings . In all cases we assumed incidence from the substrate side, which is SiO2 with refractive index ns ≈ 1.46 for λ = 532 nm. The ridges of the dielectric gratings in Figs. 2(a) and 2(b) were made of Si3N4, which has a high refractive index n ≈ 2 and is also easy to process . Two different layer thicknesses were available: SEM measurement from dummy substrates gave h = 840 nm and h = 550 nm. Hence, in the design of dielectric gratings, Λ/λ and c/Λ are the parameters to be chosen for either one of these ridge thicknesses.The grating material of the polarimeter in Fig. 2(c) was chosen to be Aluminum, which has a suitable complex refractive index n̂ ≈ 0.9241 +6.4788i @ λ =532 nm and is a relatively simple material to process using dry etching methods .
Figure 3 shows a tolerance analysis for the blazed grating and triplicator in the case of a circular and linear polarization, respectively, from which the designs can be obtained. The efficiency of a blazed grating is plotted in Fig. 3(a) as a function of the fill factor f and the local period Λ/λ if the ridge thickness is h = 840 nm. Only the zero (reflected) order propagates in the substrate at normal incidence if Λ < λ/ns ≈ 364 nm. We therefore choose Λ = 350 nm to ensure that the grating works also at non-normal angles of incidence and, taking f = 0.4, we obtain the efficiency η−1 = 0.978.
We designed triplicators for different periods Λ/λ for both available film thicknesses h by plotting the efficiencies η0 and η±1 as illustrated in Fig. 3(b), where the first crossing of the two curves represent an ideal design with η0 = η±1. We found it advantageous to use h = 550 nm, and to reduce Λ/λ well below the cutoff 364 nm to obtain a less strict tolerance of error in f. Considering fabrication, we thus chose Λ = 220 nm, which gives η0 = η±1 = 0.327 and η = 0.981.
The metallic polarimeter grating was designed by plotting the spectral TM transmittance and extinction ratio (TTM/TTE) of a wire-grid polarizer as a function of h for different values of Λ. Figure 4 illustrates such results for Λ = 150 nm, f = 0.5 and h = 160 nm. These parameters provide satisfactory performance over a wavelength region exceeding the visible spectrum, from 380 nm to 780 nm.
Figure 5 illustrates the fabrication of both dielectric and metallic polarization gratings. Silicon nitride layers were grown using plasma enhanced chemical vapor deposition (PECVD), and the aluminum layer was vacuum deposited with an electron gun. Electron-beam exposure (Vistec EBPG 5000+ ES HR) of the fringe pattern was followed by development and fabrication of the etching mask by either Chromium lift-off for Si3N4 and SiO2 dry etching for aluminum gratings (Plasmalab 80+). Finally the structures were transferred to the final grating materials by reactive ion etching. The etching chemistries for Si3N4 and aluminum were CHF3/O2 and Cl/BCl3, respectively. We used Q = 20 discrete segments for the S3N4 gratings and Q = 36 segments for the polarimeter grating. The periods were d = 100 μm and d = 108 μm, respectively, and the physical size of each elements was 4 × 4 mm. The e-beam exposure could be replaced by nanoimprinting enabling mass-production.
The elements were characterized by illuminating them with a DPSS laser with λ = 532 nm. The 1/e2 half-width of the illumination beam was approximately 2 mm, so that several grating periods were illuminated and therefore the diffraction orders of the gratings were clearly separated in the far field. We used a CCD line detector to obtain far-field intensity profiles at a distance of ∼ 1 m from the grating without a focusing lens; such results are presented in Fig. 6 for the three types of grating. In these figures the three central orders can be seen as separate spots: orders m = −1, m = 0, and m = +1 appear from left to right. The vertical scale is normalized to the maximum intensity in the observed distribution. We also measured the absolute diffraction efficiency of each diffraction order using a large-area square-law detector. Such measurements were compared to the intensity of the incident beam and corrected by removing (mathematically) the effect of Fresnel’s reflection (∼ 4%) at the flat air-substrate interface. After this correction the results can be compared directly with the theoretical FMM calculations, where light was assumed incident from an infinitely thick substrate.
Figure 6(a) illustrates the intensities of different illumination polarization conditions. As predicted by theory, incident LCP polarization concentrates light in order m = +1, RCP concentrates it in order m = −1, and linear polarization leads to a duplicator. Ideally, we should have η0 = 0 in each case, but due to fabrication errors we measured η0 ≈ 2%. In LCP and RCL cases the efficiency of the blazed order is ∼ 93%, which is also the efficiency of the duplicator. These results should be compared with the FMM prediction of ∼ 98% for a grating with ideal profile.
The intensity distribution for a triplicator (with linearly polarized incident light) is illustrated in Fig. 6(b). The uniformity of the three central orders is nearly perfect and the efficiency η = 97% almost matches the FMM prediction 98%. The polarimeter grating considered in Fig. 6(c) should ideally have η0 = 1/4 and η±1 = 1/8. We measured η0 = 17.8%, η+1 = 9.4% and η−1 = 8.9%. The values predicted by FMM are η0 = 20.3% and η±1 = 10.5%.
We finally measured the efficiencies of orders m = −1, m = 0, and m = +1 of the triplicator and polarimeter gratings using linearly polarized incident light and rotating a linear polarizer in front of the detector. According to Eqs. (1) and (2), order m = −1 should ideally be LCP, order m = 0 should be linearly polarized, and order m = +1 should be RCP. Thus the efficiencies m = ±1 should not be affected by the angle of the linear polarizer, but η0 should vary cosinusoidally. This is indeed the case, to a good approximation, as seen from Fig. 7.
6. Final remarks
We have demonstrated the fabrication and characterization of sub-wavelength space-variant gratings that locally modulate the polarization of light. To our knowledge, these are the first results of a surface relief polarization gratings for the visible light. The measured diffraction efficiencies exceeded the scalar limits as was predicted by the rigorous analysis. The structures demonstrated here represent the simplest cases of the polarization gratings, i.e. the only free parameters are fringe period, groove depth and width and fringe rotation. In the most general polarization gratings, the grooves stand on a continuous surface profile. This enables us to design lossless beam splitters with more complex dimensions . In addition, some far field propagation induced changes in the degree of polarization in the case of partially coherent sources behind the polarization gratings predicted by Piquero et al.  should be investigated.
The authors would like to acknowledge J. Viheriälä from Tampere University of Technology for providing the Si3N4 films. This work was supported by the Academy of Finland (projects 118951, 129155, and 209806).
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