1. INTRODUCTION

Transmissive optical elements, such as lenses, gratings, and illumination phase diffusers (pattern generators), are commonly used in industrial and medical laser applications. These applications have strict irradiance and high-power specifications of tolerance to prolong the useful lifetime of light delivery components. The components are also distinct with respect to their spectral response, which in almost all cases is limited to very narrow wavelength linewidths. Fresnel reflectivity and angular scatter resulting from incidence on surfaces of high-power optical elements are undesirable due to operator and environmental safety, and possible system damage or instabilities caused by backpropagating radiation [1].

The most prominent technique to suppress undesired Fresnel reflections from optical component and device surfaces is use of thin film antireflection (TFAR) coatings. The simplest TFAR designs consist of a single-layer antireflection (SLAR), quarter-wave-thick ($d = {\lambda _o}/4\;n$) film coating as shown in Fig. 1(a). The SLAR optical index is equal, or as close as possible, to the geometrical average between the substrate and superstrate values, suppressing reflection efficiently at a narrow-width design-chosen wavelength ${\lambda _o}$. Alternatives to TFAR coatings are inhomogeneous film layers with a gradient refractive index (GRIN) profile, which transitions the index value gradually and monotonically from ambient to that of the substrate beyond the boundary [Fig. 1(b)] [2–4]. Although SLAR, GRIN, and other thin film coatings are widely used for high-power planar substrates, optical windows, and curved surfaces, applications on segmented-profile components, such as diffractive optics and micro-holograms, are challenging. Phase profiles of deep gratings and diffractive phase diffusers can be particularly difficult to coat anisotropically, and at times, the coatings can perturb the intended diffractive performance.

 

Fig. 1. Graphical depiction of cross sections from three types of surface structures used as AR treatments on a diffractive device. In all cases, a linear phase grating with spatial period ${\Lambda _x}$ and $\pi$-phase depth $h$ is used as the baseline optical surface; the indices are ${n_i}$ for ambient and ${n_s}$ for the substrate. (a) Surface crossection with an SLAR of index $n$ and thickness $d$ anisotropically applied to the grating surfaces. (b) GRIN AR coating of thickness $d$ and index $n(z)$ applied to the surfaces. (c) Sub-scale periodic grating $p_x$ with fill factor $f$, multiplexed 10 times across each period ${\Lambda _x}$, acting as an AR structure. (d) AR structure replaced by a distributed-feature periodic grating, with the same multiplex factor and fill factor as in (c).

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Antireflection structured surfaces (ARSSs) have been extensively studied in recent years as a practical alternative to conventional TFAR coatings for high-power laser applications, especially in wavelength bands where well-matched optical refractive index materials are unavailable, or if required laser damage thresholds are higher than TFAR can withstand [5–8]. For surface nanostructures with average subwavelength-featured transverse diameters and periodic, or random ARSSs (rARSSs), spatial surface distributions are fabricated directly on substrate surfaces to emulate TFAR and GRIN antireflective coatings [9–11]. The periodic or randomly patterned subwavelength-featured surface approximates a synthetic index of refraction, based on the averaged density between ambient and substrate materials [Figs. 1(c) and 1(d)], resulting in reduction of electromagnetic impedance with layer penetration, thus suppressing Fresnel reflectivity. In many reports, the longitudinal profiles of the nanostructured features are simulated and optimized for broadband AR performance. The resulting vertical feature profiles can vary from conical, to truncated cones, to pyramids, all organized in regular arrays or random distributions on the substrate surface. The profiles of these features help to realize different incremental refractive index transition functions with respect to depth into the substrate. For computational modeling purposes, ARSSs can be numerically approximated as homogeneous layers, with a single effective optical index value per layer. Nanostructuring of the optical surfaces can be accomplished using deposition, etching, or combined techniques [6,8,12,13]. Reports have demonstrated experimental measurements of reflectivity suppression, with enhanced transmission for normal and off-normal angles of incidence (AOIs), including polarization insensitivity for rARSSs [14,15].

More recent reports were not limited to planar optical surface applications, and have shown ARSSs and rARSSs on continuous- and segmented-profile functional surfaces such as lenses, diffractive gratings, and multiphase digital diffractive optics (beam shapers, controlled angle illuminators) [16–19]. Etching methods are more prominent on nanostructuring pre-fabricated optical phase elements because reactive-ion etching (RIE) chambers provide higher degrees of molecular-level anisotropic reactivity control compared to deposition techniques such as atomic-layer deposition (ALD), which is isotropic in nature. Moreover, structurally complicated cross sections of nanostructure profiles, such as quartic profiles [5,6] that mimic insect eyes and achieve high antireflection efficiency, are fabricated with multiple masking and etching steps. These fabrication methods are not easily applicable to pre-existing segmented-phase profile optical surfaces, especially for subwavelength periodic structures, as they have to be uniform at all layers of the segmented mesas, i.e., all phase steps of the diffractive device of application, and leave the intended optical performance as undisturbed as possible. To mitigate performance perturbation, random nanostructures are sometimes preferred as reflectivity suppressors on diffractive optical elements [19–21].

Since random distributions of nanostructures forming rARSSs are often modeled as homogeneous stratified dielectric GRIN layers, the distribution of their transverse features is globally averaged to a single “optical density fill factor” for each constituent layer. This is a direct consequence of the layered effective medium approximation, which results in loss of continuity (segmented stratification) between the GRIN layers for some models [4–6,22]. The simplest design of ordered ARSSs is a binary-phase transmission grating, with a spatial period smaller than the desired vacuum wavelength band minimum, otherwise referred to as a subwavelength binary grating (SWG), shown in Fig. 1(c). The subwavelength AR grating propagates only the axial zeroth diffracted order, and avoids angular redistribution of the transmitted power while it suppresses reflectivity by adhering to the SLAR index and thickness design conditions dictated by the effective medium theory (EMT). The duty cycle of the SWG is determined by EMT calculations to match the optimal SLAR optical index requirements to the optical density balance between the superstrate and substrate: $n = \sqrt {{n_i}{n_s}}$. This greatly simplified design methodology is augmented in practice by numerical optimization simulations, using a variety of algorithms such as rigorous coupled-wave analysis (RCWA) or finite-element propagation to achieve the desired AR effect [23,24]. As such, the GRIN-EMT model is adequate to provide an ARSS design starting point, followed by rigorous but obscured optimization computations. Expanding ARSS design beyond planar substrate applications becomes more computationally intense, as the SWG must apply anisotropically over the underlying substrate’s morphology. For example, adding ARSS to the surface of binary gratings yields complicated bi-periodic featured layers, multiplexing the diffractive performance of the SWG-ARSS to the baseline grating. The empirical design rule is to decrease the subwavelength ARSS spatial period (${p_x}$) enough to yield an increase of the AR effect, and reduce any perturbations on the baseline grating performance. This approach has practical fabrication limitations, due to challenging critical dimensions for the SWG, and because of the increased numerical simulation complexity, which becomes cumbersome to model especially for 2D SWG. As a contrasting example, we note that optical scattering analysis of rough interfaces uses multiplexed sinusoidal-SWG as the foundational model to understand relations between surface roughness scales and the incident wavelength.For scattering models, superposition of shallow near-SWGs offers a conceptual design starting point, as it explains off-axis radiance in terms of diffraction concepts [25–27]. Optical scatter can be a function of surface-roughness profile autocovariance, whereas GRIN-EMT global-index averages are capable of predicting only axial propagation intensity values and not scatter.

 

Fig. 2. RCWA simulations of the propagating diffraction order efficiency for the silica baseline binary grating, ($h = {\lambda _o}/{2}(n- {1})$) with AR structures multiplexed 14 times (${{p}_x} = {571}\;{\rm nm}$), across the base period $\Lambda_x=8.00 \mu \mathrm{m}$, with TE polarization at normal incidence and ${\lambda _o} = {633}\;{\rm nm}$. The efficiencies for the grating without the ARSS are shown as blue bars, compared to the efficiencies of the grating with ARSS in orange bars: (a) SWG in air, matched to zeroth-order EMT conditions and (b) 21-order Dammann grating. In both cases, $f = {0.406}$.

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There is a gap in investigations from binary periodic SWG-ARSS to randomly distributed feature nanoscale structured surfaces. Randomness in nanoscale features is usually analyzed with respect to autocovariance or autocorrelations between the “roughness” profile of optical surfaces and their AR performance. Roughness is quantified as a measure of height variations across a surface, and can be related to transverse feature groupings; however, roughness models allow the investigation of scatter, not axial alignment of the propagating radiation. As such, scatter calculations cannot overcome Fresnel losses, due to the change of optical index when incident light transitions from ambient to substrate and vice versa. Rigorous numerical modeling is used to approximate randomness as a collection of various periodic structures by linear superpositions of their spatial frequencies. In general, these mixtures are based on some scale approximation or model assumption(s). Approaching the scaling problem using the EMT method does not offer a clear solution either. The global averaging of the effective index within the transitional layer is not sensitive to feature distributions, although it can overcome Fresnel losses. EMT is not capable of predicting scatter, as it has no mechanism to direct light off the axial direction, at least in the case of SLAR. There is an absence of a transitional model, which considers the 2D spatial redistribution of features within a chosen unit cell for a given effective medium index requirement. In this paper, we numerically investigate the AR and scatter functionality of the transitional redistribution of phase nanostructures, which lead from periodic SWG ARSS to a surface with quasi-randomized features.

To study optical performance effects of various ARSS transverse feature distributions on a pre-patterned substrate surface, we have chosen a baseline 50% duty cycle, 8.00 µm period, binary $\pi$-phase grating, with $h = {\lambda _o}/2({{n_s} - {n_i}})$, onto which SWG profiles are anisotropically superimposed to act as antireflective structures [Figs. 1(c) and 1(d)]. We used two classes of ARSS: conventional binary single-phase-transition gratings based on EMT fill factor requirements (B-series) and a selection of Dammann gratings of various orders (D-series). Dammann gratings are pseudorandom binary-phase profiles, with varying optical densities within their spatial-period unit cells, designed to generate equal intensity multiple diffraction orders (DOs) for periodicities larger than the incident wavelength. As such, Dammann gratings have deterministic profiles based on specific selection rules, which allows a methodical study of optical cross-coupling effects due to their SWG periodicity multiplexed with the baseline grating. In this study, Dammann gratings with increasing feature-distribution complexity and comparable EMT fill fractions are multiplexed on the binary baseline grating, and their performance as efficient ARSS is computed. The choice of a 50% duty cycle $\pi$-phase baseline grating is based on its simplicity and segregation of the resulting angular intensity spectrum. For all incident wavelengths smaller than the baseline grating’s spatial period, there are even and odd propagating DOs (Fig. 2). However, the baseline grating profile choice distributes energy efficiently only to the odd DOs, suppressing the even ones by about two orders of magnitude. This selection rule allows for the segregation of the transmitted intensity spatial distribution to: controlled angles (from odd orders) and scatter (from even orders). To illustrate the ARSS presence effects, we chose the binary SWG fulfilling the zeroth-order EMT criteria with a single phase-transition boundary (B-0) and directly compared the numerical results to a Dammann D-21 that has the same EMT criteria. In Fig. 2, the baseline grating even-order suppression is averaged to a normalized intensity of $2({10}^{- 3})$ as shown by the blue bars, while the controlled normalized intensity peaks for the $\pm {1}$-DOs ($\pm {1}{:}{\rm DO}$) are at 0.390. Perturbations on the baseline grating surfaces due to the presence of the SWG-ARSS, because of evanescent coupling between the two diffractive elements, can induce undesirable efficiency changes to the baseline angular intensity spectrum, including possible reduction of the odd-order and increase of even-order intensities. This is evident from Fig. 2(a) (orange bars), especially for the parasitic increase of the $\pm {14}$ diffraction orders ($\pm {14}{:}{\rm DO}$) to a level of 0.100, due to the presence of the B-0 ARSS on top of the baseline grating. The controlled $\pm {1}{:} {\rm DO}$ are reduced to 0.325, although overall light transmission is enhanced by 2% over Fresnel reflectance. In contrast, the effect of a D-21 ARSS on the same baseline grating [Fig. 2(b)] produces an intensity spectrum closer to the original, with the $\pm {1}{:} {\rm DO}$ efficiency at 0.395 and an overall transmission enhancement of 3%. For both simulation results shown in the figure, the EMT fill factor is exactly the same $f = {0.406}$, suggesting that the layer-averaging method should have the same outcome. In our study, any “parasitic” effects on the original baseline grating performance can be monitored through systematic selection of high-order Dammann gratings, including the calculation of surface-feature correlations between Dammann-SWG structural parameters, compared to the ARSS functionality.

In this paper, we present RCWA simulated performances of 1D, single-layer, deterministic ARSS profiles, initially optimized to fulfill zeroth- (B-0) and second-order (B-2) EMT criteria, and compare the performance of pseudo-random binary-phase-encoded profiles (Dammann gratings), multiplexed on a simple baseline, 50% duty cycle, binary grating. The ARSS optical depth is restricted to that of a conventional SLAR performing at 633 nm on a fused silica substrate to minimize the control variables exclusively to transverse feature distributions. Due to the in-plane anisotropy of the 1D baseline grating and ARSS, the analysis was performed for both TE and TM modes at normal incidence conditions. To ensure RCWA calculation accuracy and convergence, we retained up to five times the number of propagating orders required for the baseline binary grating and overlaid ARSS multiplexed SWG input. The goal is to identify parameters beyond the EMT conditions for a fixed thickness ARSS single layer, which will not affect the baseline grating performance but will improve the optical transmission efficiency through the component.

2. COMPUTATIONAL EXPERIMENT RATIONALE

Random ARSS surfaces are often modeled using zeroth-order EMT for each stratum. For SWG-ARSS profiles, with periods only two to 10 times smaller than the incident wavelength, the zeroth-order EMT approximation becomes insufficient, and the use of higher-order EMT analysis becomes necessary. Zeroth-order EMT is formulated by static approximations in the relation between the incident wavelength and the periodic boundary profile, whereas in higher-order approximations, the analysis includes non-static interactions between the incident field and the effective optical index layers. In this study, the Rytov analysis between polarized incident electromagnetic waves perpendicular ($\bot$) and parallel ($\parallel$) to the stratified-media grating vector ${\boldsymbol \Lambda}$ was used, to determine the modes of a binary-phase SWG-ARSS [28]. Polarization considerations are necessary, even at normal incidence, because of our choice to use 1D periodic profiles for both the baseline grating and multiplexed SWG AR gratings. Considering the periodicity of the AR structures, this method is similar to guided modes in a slab waveguide parallel to the substrate surface. The equations of the stratified medium’s effective permittivity to second-order coupling between the spatial period (${p_x}$) and incident wavelength ($\lambda$) are given as

In Eqs. (1) and (2) ${\varepsilon _o}$, ${\varepsilon _i}$, and ${\varepsilon _s}$ are respectively the permittivities of free space, superstrate, and substrate, where $f$ is the filling factor of the stratified layer(s), representing the volume fraction of substrate material contained within a single spatial period. The expressions of the zeroth-order EMT permittivities within the above equations are

Equations (1)–(4) represent the permittivities of any SWG for incident light with TE or TM polarization, approximated to the average values of the equivalent SLAR coating, and do not depend on the relative position of distributed features within ${p_x}$, but only on the “volume filling” factor $f$, as illustrated in Fig. 1. For a binary-phase profile SWG designed as an AR structure [Fig. 1(c)], the effective index for the layer is computed as the square root of the resulting permittivity from the equations above. For an optimal SLAR using zeroth-order EMT, the volume fill fraction is calculated to match $f_{B0}^{\,({\rm TE})} = {({1 + {n_s}/{n_i}})^{- 1}}$ and $f_{B0}^{\,({\rm TM})} = {({1 + {n_i}/{n_s}})^{- 1}}$, for TE and TM, respectively. The SWG depth was selected to be equal to the quarter-wave depth of the substrate’s wavelength value, in accordance with the SLAR condition. These parametric selections define the design of two candidate structures for the numerical experiment: a zeroth-order EMT and a second-order EMT binary-phase SWG, performing as AR structures and listed in this paper as B-0 and B-2, respectively.

To analyze variations in transverse feature distributions with respect to performance effectiveness as rARSS, volume fill fractions $f$ of a fused-silica binary grating comparable to the SLAR geometrical average were estimated using EMT, for an incident wavelength of 633 nm. The predicted fill fractions [Eqs. (1)–(4)] are for TE zeroth order: $f_{B0}^{({\rm TE})} = 0.406$, second order: $f_{B2}^{({\rm TE})} = 0.334$; and for TM zeroth order: $f_{B0}^{({\rm TM})} = 0.594$, and second order: $f_{B2}^{({\rm TM})} = 0.506$.

Profiles of non-binary ARSS are usually modeled as gradient-index homogeneous sequential layers, with gradual increase of effective optical index from ambient to substrate. Random ARSSs have various feature profiles and spatial distributions, resulting from differences in fabrication methods, usually initiated through random masking such as sputtering or annealing, and have been shown to have various antireflective performances [11,12]. To methodically control the numerical experiment from the ordered SWG single-transition state (binary-phase profile), to a quasi-randomized feature distribution, we chose to consider the redistribution of the periodic volume fill fraction $f$ by segmenting it and rearranging the segments, always within the net sum values for polarization dependence TE or TM. A deterministic rearrangement of the phase transition locations within the periodic structure can be achieved using Dammann’s selection rules [29].

 

Fig. 3. Top-down view of 1D, normalized N-order Dammann binary-phase grating profiles, within their periodic unit cell ${{p}_x}$. Black regions and white regions have optical index values corresponding to air and silica, respectively. (Left column) Odd-order Dammann gratings with volume fractions comparable to $f_{}^{({\rm TE})}$ requirements. (Right column) Even-order Dammann gratings with volume fractions comparable to $f_{}^{({\rm TM})}$ requirements. The last entry is a reversed-tone grating to match the TM conditions.

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Dammann gratings are deterministic binary-phase profiles, designed to distribute an incident wave to an array of equal-intensity DOs in the far field, provided the optical binary-phase depth of the profile is half-a-cycle (π) [30]. These deterministically designed gratings are ranked with respect to their ${2}\;N + {1}$ equal intensity orders, and are characterized by $N$ phase-transition points within their spatial period. The 1D-Dammann phase profile $g(x)$ can be described as [9]

In addition to the Dammann grating intensity-equalized DOs (IEDOs), there are propagating DOs at higher deflection angles, resulting in undesirable light distribution. These and the IEDOs will become evanescent as the period of the Dammann grating is reduced to subwavelength values. To characterize the transverse feature spatial distributions (complexity) of the phase profiles, we computed the normalized Dammann unit cell autocorrelations, as a measure of self-similarity of the pseudo-random profiles, shown in Fig. 4 with respect to their volume fraction values. The normalized autocorrelation lengths have a decreasing trend with increasing volume fraction values for odd-order Dammann and a decreasing trend with increasing IEDO values for even-order Dammann.

 

Fig. 4. Autocorrelation length normalized to the spatial period, with respect to Dammann order number, as a function of EMT volume fraction. Dammann gratings used in the simulations (D-XX) are represented with black solid markers, and other Dammann-order gratings are shown with open circles for comparison. EMT binary gratings for each polarization state are indicated as vertical lines labeled BX.

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To analyze the variation in performance of the AR-Dammann gratings as efficient ARSSs, designs with varying transverse feature distributions for fill fractions spanning the predicted zeroth-order and second-order EMTs for each polarization were chosen, as shown by the solid markers in Fig. 4. We numerically simulated the a 50% duty cycle, binary, $\pi$-phase depth, baseline grating profiles, with the choice Dammann grating phase profiles and binary SWG multiplexed on two-level phase mesas, for even multiplexed frequencies $k = {2},{4},{6},{8}, \ldots {20}$. The ratio of the base grating period cell to the ARSS grating periodicity defines $k = {\Lambda _x}/{p_x}$. We restricted $k$ to even numbers to have equalized ${p_x}$ AR grating cells on each phase level of base grating. The ARSS depth was restricted to perfect SLAR values for normal AOI as mentioned before.

 

Fig. 5. Total transmission enhancement of fused silica 1D binary phase, 50% duty cycle grating, with 1D Dammann gratings acting as ARS surfaces, as a function of the Dammann multiplex ratio frequency $k$. (a) TE polarization and (b) TM polarization for various volume fractions of Dammann gratings, in comparison to binary gratings with EMT zeroth- and EMT second-order approximation. IEDOs are labeled as DXX, based on their equal-order population. The post-script F indicates that the phase was reversed to satisfy the corresponding EMT fill factor upper limit for TM. Negative values indicate transmission losses due to reflection.

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3. RESULTS AND DISCUSSION

Numerical RCWA computations of the total transmission enhancement ($\tau$) as a function of the SWG binary zeroth- and second-order-EMT multiplex ratios on the baseline grating, along with corresponding choices of AR-Dammann SWG with volume fractions restricted to EMT conditions, are presented in Fig. 5 for TE and TM normal-incidence polarizations. The transmission enhancement was calculated as the fraction greater than the on-axis transmittance for each DO for the baseline grating:

In our numerical experiment, at normal incidence, the left side of Eq. (7) is independent of polarization because of the summation of all DO contributions from the baseline grating. The right side is polarization independent as well, and can be expressed in terms of the SWG effective index zeroth-order EMT fill fractions for an optimum SLAR.

 

Fig. 6. Odd DO transmission enhancement (directed power) of fused silica 1D binary phase, 50% duty cycle grating, with 1D Dammann gratings acting as ARS surfaces, as a function of the Dammann multiplex ratio frequency $k$. (a) TE polarization and (b) TM polarization for various volume fractions of Dammann gratings, in comparison to binary gratings with EMT zeroth- and EMT second-order approximation. IEDOs are labeled as DXX, based on their equal-order population. The post-script F indicates that the phase was reversed to satisfy the corresponding EMT fill factor upper limit for TM. Negative values indicate transmission losses due to reflection or scatter.

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Fused-silica windows have an average normalized transmission of 0.931 across the visible spectrum, with each surface contributing to Fresnel reflectance by 3.5%. Since our simulations were limited to a single surface, the subwavelength regime multiplexed gratings ($k = {20}$) were observed to approach a total enhancement of 3.5% with little to no difference for either incident polarization. For multiplex ratios ${18} \ge k \ge {10}$, in the wavelength-scale regime, the presence of ARSS propagating orders, and cross coupling between the multiplexed gratings, resulted in a transmission enhancement decrease. This is evident for B-0 and B-2 ARSSs, along with Dammann gratings with high autocorrelation lengths (such as D-05 and D-07), while gratings with smaller autocorrelation lengths show a consistent enhancement of about 3% for both polarizations. In the super-wavelength regime, for multiplex ratios ${10} \ge k \ge {2}$, while gratings B-0 and B-2 show little to no enhancement, gratings D-05 and D-07 show average enhancement of about 1%. Gratings D-11, D-21, and D-12 (for TE) along with gratings D-21 F, D-08, and D-10 (for TM) show better performance with an average enhancement of about 2%.

Although the results above suggest an enhancement in overall transmission due to the ARSS, one should be mindful of the baseline grating’s performance, which was designed to propagate odd DOs and suppress even orders (Fig. 2). Superposition of the AR gratings perturbs the function of the baseline grating, affecting the transmission intensities of the propagating odd orders and even orders, as a function of multiplex ratios. To investigate the variations in the performance of AR gratings, net enhancement [Eq. (6)] of the odd-DO transmission is shown in Fig. 6 as a function of multiplex ratios for the AR grating choices. The directed odd DOs were considered as contributing to the design transmission $t_{B,D}^{({\rm TE,TM})}(k)$, whereas the even DO can be considered as a form of scatter $\sigma _{B,D}^{({\rm TE,TM})}(k)$:

 

Fig. 7. Transmission enhancement ($\tau$) of all DOs for a fused silica 1D binary 50% duty cycle grating, as a function of multiplex ratio $k$ with 1D ARS surface gratings. The baseline grating without ARSS is represented by $k = {0}$ and $\tau = {0}$. (a) TE polarization and (b) TM polarization for selected gratings. The arrows on TE B-0 and D-21 indicate $\tau$ for the data shown in Fig. 2.

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In Eq. (8), the summation over $q^\prime$ includes only odd integers, while the $q^{\prime\prime}$ integers are only even. The scattering term includes the zeroth DO, which incorporates any phase-step perturbation effects due to the presence of the ARSS layer thickness. In the subwavelength regime ($k = {20}$), for both incident polarizations, the transmission enhancement of the odd orders matches in value the net transmission enhancement, while a significant reduction in enhancement is observed in the wavelength-scale regime (${18} \ge k \ge {10}$). For TE polarization, directed transmission scattering losses were observed for all AR gratings with ${14} \ge k \ge {10}$, except for D-21, which had an average enhancement of about 1%, in contrast to the total transmission enhancement observed in Fig. 5(a) for all ARSS types. For TM polarization, the observed net enhancement of odd orders for all AR gratings follows closely to the trends observed in Fig. 5(b), other than B-0 and B-2. In the super-wavelength-scale regime (${8} \ge k \ge {2}$), there was negligible enhancement for TE polarization in correlation to the near-wavelength regime, whereas an average enhancement of about 1% to 1.5% for all AR gratings other than B-0 and B-2 for TM polarization.

Intensity transmission enhancement for each propagating DO $\tau _{B,D,m}^{({\rm TE,TM})}(k)$, as a function of multiplexed AR grating frequency for gratings B-0, B-2, D-7, D-10, and D-21, is plotted in Fig. 7. For TE polarization in the subwavelength regime ($k = {20}$), a uniform enhancement of the angular intensity spectrum is observed for D-21, while slight perturbations in the higher DOs for B-0, B-2, and D-07 are prominent. In the wavelength-scale and super-wavelength regime (${14} \ge k \ge {2}$), even though a uniform transmission enhancement of the low DO is observed for odd and even orders, perturbations of higher orders has increased considerably from the nominal baseline performance values. This is a direct consequence of the higher DO generated by AR gratings, which are propagating for smaller $k$ choices, cross coupling into the baseline grating angular spectrum, and causing the odd and even DOs opposite performance in Figs. 5 and 6. TM polarization has similar trends with decreasing multiplex ratios; in contrast, the DO perturbations in intensity are weaker compared to TE for all AR gratings simulated. Variations in DO intensity distributions signify some transverse feature distribution dependence beyond the EMT volume fraction value. It is also noted that transverse feature distribution dependence of multiplexed AR gratings with lower autocorrelation lengths, such as D-21, results in weaker scatter contributions compared to their larger autocorrelation length counterparts, such as D-7 or D-10. Other than the numerical results presented, there are no qualitative indications as to why these trends are as such.

 

Fig. 8. Simulated intensity distribution in the plane of incidence, near the baseline-grating surface for (a) TE from B-2; (b) TE from D-21; (c) TM from B-2, and (d) TM from D-21. In all cases, $k = {20}$. For comparison, the top-down phase profile of the AR gratings is shown next to the enumeration. The baseline-grating profile begins at $z = {0.5}\;{\unicode{x00B5}{\rm m}}$ and extends into the substrate for higher values of $z$. The color bar is in normalized arbitrary units.

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To quantify the feature distribution effects on the directed transmission enhancement $t_{B,D}^{({\rm TE,TM})}(k)$, we simulated the intensity distribution for each polarization state near the baseline grating’s surface. The results for two cases, B-2 and D-21, are presented in Fig. 8, corresponding to the schematic shown in Figs. 1(c) and 1(d). Both AR gratings have a subwavelength periodicity ($k = {20}$), and a single period of the baseline grating is shown (${\Lambda _x} = {8}\;{\unicode{x00B5}{\rm m}}$). We note that both AR gratings have a directional transmission enhancement $t_{}^{({\rm TE,TM})}(k) \ge 0.03$, as seen in Figs. 5 and 6. Figures 8(a) (TE) and 8(c) (TM) show the formation of high-intensity nodes within the B-2 AR-grating layer. The reflected intensity has low contrast fringes, whereas the transmitted intensity has high-contrast regions within the substrate medium, indicating the flow of diffracted energy into the substrate. The intensity within the D-21 AR-grating layer, in Fig. 8(b) (TE), shows considerable reduction in intensity localization within the ARS layer, with lower fringe contrast in reflection. The TM polarization for D-21 [Fig. 8(d)] shows localization of intensity within the AR grating’s layer, with a doubled spatial frequency distribution compared to Fig. 8(c). These results indicate that AR-grating profile feature distributions on the surface of the baseline grating have noticeable effects in the performance of the multiplexed device. More importantly, we note that if second-order EMT is used for the initial AR-grating design, B-2 is a conventionally better criterion match, although D-21 (which matches the zeroth-order EMT criterion) is actually performing at the same or higher AR efficiency and with less transmission scatter.

4. CONCLUSIONS

In this paper, we presented results from a numerical experiment that was designed to explore the effects of nanostructured surfaces acting as anti-reflective layers, added on the surface of a 1D, 50% duty cycle, baseline binary-phase grating. The goal was to determine the degree of agreement between ordered periodic ARSS profiles designed based on EMT refractive index criteria and, pseudo-randomly distributed binary-phase encoded profiles (Dammann gratings) acting as transmission enhancers. To differentiate between the simple binary AR-SWG and the Dammann grating profiles, we used the EMT layer fill fraction and the ARSS profile autocorrelation length to categorize the AR structures. We restricted the depth of the AR layer to one-fourth of the light wavelength at normal incidence, to investigate solution possibilities for thin-layered ARSSs, which would preserve the form and function of the baseline diffractive element. By restricting the depth and phase profiles of the AR-SWG and Dammann gratings to the simplest possible set of binary-stepped surface structuring solutions, the number of fabrication processing steps to add the ARSS is reduced without compromising the baseline component’s efficiency of operation at the design narrow wavelength band. It also confines the AR-structure parameter space to a narrow EMT subset, forcing a single structural layer, which is very sensitive to the redistribution of constituent features. We simulated both incident polarization states and collected the transmission efficiency of all propagating orders from the baseline grating, segregating the angular spectrum in directed and scattered DOs.

The results show that for simple profile AR gratings with large autocorrelation lengths, or just a single phase transition within their periodic cells, overall Fresnel reflectivity can be suppressed from the structured surface, although perturbative effects on the directed transmission efficiency increase relative scatter. In contrast, complex grating profiles with short autocorrelation lengths tend to preserve the directed power transmission distribution, have significantly lower scatter, and are equally effective in reflectivity reduction. These effects are strongly dependent on the ratio of the ARSS unit cell period to the incident wavelength value, improving antireflectivity as the cells become subwavelength in size. Off-direction scatter is not reduced just by decreasing the size of the ARSS periodic cell, as it showed strong dependence on the nanostructure-distribution organization within the cell.

The two foundational models used for light interacting with an optical surface are the EMT and superposition of near-SWGs. The former can match AR conditions using homogeneous-layer optical index stratification of boundaries, whereas the latter accounts for scattering by superposing a large collection of “thin” gratings. We note that these models are complementary in concept, as EMT cannot predict scatter and SWGM cannot suppress Fresnel reflectivity. Our results show that even at near-wavelength periodic scales and thin single-layer restrictions, a redistribution of the ARSS features can result in high values of light total transmission and directionality. The effects are sensitive to feature distribution and organization, not just to the feature number or density. Based on the tested pseudo-random Dammann grating profiles, we observe that increasing Dammann-order ranks have more stability and consistency in results, as the periodic cell length is reduced from wavelength scale to subwavelength. This realization suggests that deterministic pseudo-random scatterers can “cooperate,” reducing the optical boundary electromagnetic impedance discontinuity. The last observation is supported by RCWA simulations of the wavefront intensity near the surface. As shown in Fig. 8, simple AR-SWG structures added on a diffractive optical surface can induce high-intensity localization within the ARS layer for both incident polarization states. This is due to the confined superposition of incident and reflected wavefronts, which continue to propagate in the transmission and reflection directions, eventually forming, respectively, forward-diffracted orders and incident-reflected destructive interference conditions. The redistribution of AR-grating features has distinct effects for each incident polarization state. In the case of incident TE polarization [Fig. 8(b)], the effect is equivalent to an optical impedance match, resulting in transparency of the diffractive boundary, without disturbing the baseline grating performance. For TM polarization [Fig. 8(d)], since the polarization vector is in the plane of the AR-grating grooves, a bi-periodic intensity distribution is observed, indicating that there are two counteracting field distributions, responsible for the generation of destructive interference in the reflected direction and constructive superposition in the transmitted direction. For both polarization cases, Fresnel reflectivity is suppressed and directional transmission is enhanced with unaltered directionality. We note that there is no strict subwavelength scale criterion satisfied for cases of high-order Dammann ARSSs, as they are very effective for $k$ values as low as 14, which for our experiment is equivalent to a 571 nm periodic cell, and therefore in the wavelength-scale range. The long-correlation-length ARSSs fail to compete successfully, as for similar periodic scales, they scatter strongly and have lower overall transmission enhancement. An underlying feature complexity indicator could possibly quantify the optical impedance matching process with some predictability. Such scale transition studies could bridge understanding of rARSSs, which have multi-periodic features and short surface autocorrelation lengths.