1. Introduction

Periodic structures have attracted abundant research interest due to their capabilities in tailoring optical responses [1, 2 ]. Absorptance (A), emittance, reflectance (R), and transmittance (T) can be manipulated to improve optoelectronics [3–5 ], energy harvesting [6, 7 ], thermal imaging [8], liquid crystal displays [9, 10 ], and sensing [11, 12 ]. The reflectance from metallic gratings, for example, can be tuned using grating profiles or materials for various applications. These gratings are usually covered with one or multiple thin (thickness d smaller than light wavelength λ) films to prevent oxidation or contamination [13]. Films may also reduce R within a spectral range to further improve the performance of gratings [14, 15 ]. Thick (d >> λ) windows are seldom employed because they are not cost-effective for these purposes. But attention paid to a semi-transparent window in front of gratings is now rapidly growing because of innovative in photovoltaic cells and photon collectors [16]. Many of them have binary electrodes at bottom, such that their power throughput and generation of unwanted heat largely depend on R [17, 18 ].

Obtaining the R value for the structure (window/gratings) is not difficult at normal incidence; however, at oblique incidence, the problem becomes very complicated. First, R is influenced by the refraction of light in the window. Second, the oblique propagation of light brings about asymmetric wave patterns after light diffraction from gratings. Accordingly, the interference and total internal reflection of some diffracted light become difficult to trace. Third, a long optical path significantly enhances the intrinsic absorption inside the window. Moreover, these influences on R change with the incidence polarization, grating profile complexity, and dimensions of the window as well as gratings. A valid approximation of R and other optical responses at a reasonable computation cost is thus desirable.

The present work is a preliminarily study on the impacts of a semi-transparent window on the reflectance from metallic gratings at oblique incidence. Three gratings with various levels of profile complexity and periods are employed as representative samples. The R value for a 508-µm-thick Si window with Au/Ti gratings attached is determined at the incidence of transverse electric (TE) and transverse magnetic (TM) waves individually. The wavelength λ ranges from 2.5 to 25 µm, and thus Si is semi-transparent within the whole region. Programs based on the rigorous coupled-wave analysis (RCWA) [19] and Fourier transform infrared (FTIR) spectroscopy [20] are used to determine the R value numerically and experimentally, respectively. The validity of a well-known approximation, effective medium theory (EMT) [21], is evaluated for the three samples.

2. Light interactions with window/gratings

Figure 1 shows the schematic sketch of linearly polarized light incident on a semi-transparent window with metallic gratings attached. The incident light, comprising plane waves from free space, has wavevector k_inc of magnitude k_inc = 2n_fπ/λ, where n_f is the refractive index of free space. The plane of incidence is perpendicular to the orientation of the gratings. The zenith angle of incident light θ_inc is fixed at 30° throughout this work. Diffraction happens when the incident light penetrates the window and meets the gratings. Some orders of diffracted light are propagating electromagnetic (EM) waves, while others are evanescent. In Fig. 1(a), the incident light is TM waves. Their λ is sufficiently smaller than the grating period (Λ), such that there are multiple propagating diffracted waves for both transmitted waves (T) as well as reflected ones (R). The subscript of each wave specifies the order of diffraction. Diffracted waves from gratings have different propagation directions, which can be predicted using the well-known grating equation [22]:

 

Fig. 1 Diffraction from gratings at backside of semi-transparent Si substrate at oblique (θ_inc = 30°) incidence of (a) TM and (b) TE waves. Grating period to wavelength ratio (Λ/λ) is smaller in (a) than in (b). (R) and (T) are reflected and transmitted diffracted waves, respectively. (E) and (H) are electric and magnetic field vectors of a wave, respectively.

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When light or an EM wave propagates within a semi-transparent window, some energy is specularly reflected at the interface between free space and the window. The reflected wave R' propagates along the direction θ_{R '} = θ_inc. The penetrating wave T' experiences refraction based on Snell’s law [23], i.e., n_fsinθ_inc = n_wsinθ_{T '}. θ_{T '} is about 8° for θ_inc = 30° because a Si window is optically denser than free space. For all propagating diffracted waves, they experience interference after bouncing off from both surfaces of the window. However, the orders are incoherent with respect to each other because any two orders of diffracted waves differ in propagation direction. The wavevector magnitude of an order of diffracted wave is enlarged inside the window, such that the arrowhead length of R is 2n_wπ/λ, which is longer than that (2n_fπ/λ) of T in free space. Since the difference in the x component of these wavevectors (Δk_x) is multiples of 2π/Λ, the number of propagating R is no less than that of T.

The energy carried by propagating R may reflect specularly, travel along other directions (see Eq. (1)), or be trapped within the window due to total internal reflection. Examples corresponding to these three cases are R ₀, R _-1, and R ₊₁ in Fig. 1(b), respectively. Specifically, propagating R ₊₁ has a zenith angle between sin⁻¹(n_f/n_w) and 90°. Its energy is totally absorbed within the window in modeling, but in reality, part of it can be lost at the window sides. If the magnitude of j is sufficiently large, θ_R and θ_T become non-physical. These waves are evanescent, and their magnitudes exponentially decay from grating surfaces [24].

Although diffraction and interactions between light and the window influence R, influences are weaker in measurements than in numerical modeling. The difference is partially attributed to the light bandwidth, which is finite in measurements. R and other optical responses are actually averaged over a spectral range. In contrast, light is monochromatic in modeling, making R a spectral response. Curved window surfaces and imperfections in grating profiles can also reduce interference and other interactions. These effects were diminished here by dicing samples to have a small area (7.5 mm × 7.5 mm) and carefully fabricating gratings.

3. Profile design and sample fabrication

Figure 1 also shows the schematic sketch of the samples. Each sample is composed of a 508-µm-thick lightly doped crystalline Si and metallic gratings. The gratings are composed of 100-nm-thick Au and a 10-nm-thick Ti adhesion layer. The Au is thick enough to be opaque at 2.5 µm ≤ λ ≤ 25 µm. The grating dimensions of the three samples are listed in Table 1 . SG_I and SG_II are simple gratings (SG) because each has only one ridge in a period (Λ). SG_I has Λ = 12 µm and a lateral filling ratio of metals f = 0.42. SG_II has Λ = 3 µm and f = 0.67. In contrast to SG, a complex grating (CG) has three ridges equally spaced in a period Λ = 12 µm. The widths of three ridges are 2, 2, and 5 µm, respectively. The profile is symmetric with respect to the center of the widest ridge, making R identical for θ_inc = 30° and −30°. The total width and lateral filling ratio of metallic gratings are 9 µm and f = 0.75, respectively. CG has the largest f among the three samples.

Table 1. Grating profiles and dimensions for SG_I, SG_II, and CG.

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Figure 2 shows fabrication process and scanning electron microscopy (SEM) images of SG_I, SG_II, and CG samples. All samples were fabricated using standard lithography microfabrication techniques [25]. The fabrication process contains four steps: exposure, development, deposition, and lift-off. Before exposure, a thin layer of HMDS (hexmethyldisilane) was coated on a silicon substrate to strengthen adhesion between the substrate and photoresist. The exposure employed a positive photoresist (TMHR ip3650) and an I-line stepper (FPA-3000i5 + ). The intensity was 2000 J/m² for exposure, which was followed by a post exposure bake at 110 °C for 60 seconds to remedy standing wave effects. A sample was then dipped into a developer for 60 seconds. After being cleaned with DI water and purged with nitrogen gas, the sample was hard baked at 120 °C for 90 seconds. Metallic films were evaporated on the sample using an electron-beam evaporator (Kanagawa, VT1-10CE). The deposition rates for Ti and Au were 1 Å/sec and 0.5 Å/sec, respectively. The ample was dipped into both a positive photoresist stripper and acetone for 5 minutes each in the lift-off step. Once the sample was cleaned, it was ready for profile inspection and reflectance measurement.

 

Fig. 2 Fabrication process and SEM images of three samples (SG_I, SG_II, and CG). The fabrication process contains four steps: exposure, development, deposition, and lift-off. Lateral dimensions of each sample are marked on its image.

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Lateral dimensions of gratings were obtained with a 3D laser scanning microscopy (KEYENCE, VK-9710) and marked in SEM images. Those images were taken with a high resolution SEM (JEOL, JSM-7001). On the other hand, the specular reflectance R_s from samples was measured with an FTIR spectroscopy (Nicolet 6700) and its accessory (30 Spec, Pike Technologies) [26]. Two linear polarizers were utilized to purify the polarization of incident light. The optical path of the accessory only allowed reflected EM waves propagating along the specular direction. Waves along other directions could not be collected by the detector at the end of the path. As a result, the hemispherical reflectance R_h could not be obtained experimentally. The R_s from measurements is denoted as R_s (Exp) hereafter. Capabilities of our measurement set-up were validated with a double-side polished and lightly-doped silicon wafer. The R_s (Exp) at the incidence of both TM and TE waves agreed well with those from numerical modeling using the thin-film optics algorithm [27].

Sample profiles were constructed to numerically determine both R_s and R_h. The wavelength-dependent refractive index n and extinction coefficient κ of Au, Si, and Ti were adopted from previous studies [28, 29 ]. Interpolation between neighboring data was sometimes necessary to assure consistent data resolution in the spectra. Figures 3(a) and 3(b) show the n and κ values of intrinsic materials, respectively. The n value of Au monotonically increases with λ, while that of Ti has an oscillation at 2.5 µm ≤ λ ≤ 5 µm. The n value of Si is almost a constant (about 3.4) within the whole spectral range. The κ values of Au and Ti increase with λ, but that of Au is an order of magnitude higher than that of Ti. The κ value of Si is much smaller than those of the two metals, making Si semi-transparent. However, Si has some intrinsic absorption in the mid-infrared region. The n and κ values of free space are assumed to be like those of a vacuum, i.e., n = 1 and κ = 0.

 

Fig. 3 Optical constants (n, κ) of bulk Au, Si, Ti, and effective media using f = 0.42 (SG_I) and f = 0.75 (CG). (a) n and (b) κ of intrinsic materials and (c) n and (d) κ of effective media.

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Besides RCWA, the EMT was employed to obtain the R value for the window/gratings numerically. But, the demands of computation resource for EMT are much lower than for RCWA [24]. For EMT here, the Au and Ti gratings were respectively homogenized with free space to form a virtual film. The film thickness is identical to that of the original gratings, but the effective n and κ values vary with incidence polarization [24]. Obtaining effective n and κ values takes advantages of the relative dielectric function, ε = (n + iκ)², where i is the square root of (−1). The effective dielectric function (ε_EM) is determined by the filling ratio f and the dielectric functions of involved materials [21]:

4. Results and discussion

Figures 4(a) and 4(b) show R_s and R_h spectra from SG_I at the incidence of TM and TE waves, respectively. In both figures, R_s (Exp) is consistent with R_s (RCWA) within the whole spectral region. This consistency in a broad band guarantees the accuracy of measurements and numerical models both. R_s (RCWA) splits from R_h (RCWA) at λ ≤ 18.0 µm because R _-1 can penetrate free space. Its energy is lost, leading to R_s < R_h. Similarly, the energy carried by R ₊₁ contributes to the difference between R_s and R_h at λ ≤ 6.0 µm, making a wider split. Wavy patterns show up within the whole spectral region of spectra R_s (RCWA) and R_h (RCWA). Such patterns mainly result from the total internal reflection of propagating waves inside Si. For example, R _-1 and R ₊₁ are confined inside the window at 18 µm ≤ λ ≤ 47 µm and 6 µm ≤ λ ≤ 35 µm, respectively. The interference of each order ruins the smoothness of the spectra. Dense oscillations show up in the measured spectra only at λ ≈4.3 µm and λ ≈6.3 µm owing to the absorptance from CO₂ and H₂O, respectively [22]. There is no wavy pattern in the spectra from EMT because the interference within the thick Si window is weak. The free spectral range (wavelength difference between neighboring peaks) in a spectrum is on the order of λ²/2n_wd_w [27], which is too short to be observed. A deep valley at λ ≈16.7 µm shows up in every spectrum due to the intrinsic absorption of Si associated with κ enlargement. When A is increased, R decreases based on energy balance.

 

Fig. 4 Specular reflectance (R_s) and hemispherical reflectance (R_h) from SG_I at incidence of (a) TM and (b) TE waves. Text inside parentheses specifies origin of spectrum (experiments (Exp), rigorous coupled-wave analysis (RCWA), and effective medium theory (EMT)).

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Incidence polarization causes three differences between the spectra in Fig. 4(a) and their counterparts in Fig. 4(b). First, the R_s and R_h values are a little higher at the incidence of TE waves than at the incidence of TM waves. The R dependence on polarization for the structure (window/gratings) is identical to that for a bulk window only. Second, the R_s and R_h (EMT) values are much higher at the incidence of TE waves than at the incidence of TM waves. The difference between spectra is larger than 0.5 at 2.5 µm ≤ λ ≤ 7.5 µm and 22 µm ≤ λ ≤ 25 µm. The effective n and κ are responsible for the difference because they are close to those of metals at the incidence of TE waves. In contrast, they are similar to those of a dielectric at the incidence of TM waves. For example, κ_EM in Fig. 3(d) is on the same order as κ_m based on Eq. (2a), but it becomes almost negligible based on Eq. (2b). As a result, the two films allow the transmission of TM waves, but they act as a mirror at the TE wave incidence. Third, the EMT gives a good approximation (absolute error of less than 0.1) of R_s (Exp) at 6 µm ≤ λ ≤ 20 µm when the incidence is TM waves. The validity results from the small filling ratio (f = 0.42). Gratings allow transmission like virtual dielectric films. However, the EMT fails in approximating R_h within most of the spectral region. The approximation is even worse at the TE wave incidence.

Both spectra from RCWA show a jump at λ = 6.0 µm of the TM wave incidence, but the jump gets trivial at the TE wave incidence. Such a jump is attributed to Rayleigh anomalies [30], which occur when an order of diffracted waves propagates at the grazing angle (90°). The waves are between propagating and evanescent, such that their energy is re-distributed among other diffracted waves. The occurrence wavelength of a Rayleigh anomaly and its responsible diffracted wave can be obtained from Eq. (1). Table 2 lists the occurrence wavelength λ (in µm) for Rayleigh anomalies and their responsible diffracted waves. Corresponding marks are utilized in Figs. 4–7. The occurrence wavelengths are consistent for SG_I and CG because they have identical grating periods (Λ). A diffracted wave is propagating and evanescent at wavelengths shorter and longer than the specified wavelength, respectively. For example, R _-1 is propagating at λ < 47.0 µm for SG_1. One should be cautious as the energy carried by R _-1 cannot always contribute to R_s or R_h. R _-1 is confined inside the substrate due to total internal reflection. Similarly, R ₊₁ is propagating within the window at λ < 35.0 µm, but it cannot contribute to R_h until λ < 6.0 µm.

Table 2. Occurrence wavelength λ (in µm) of Rayleigh anomalies and responsible diffracted wave for three samples. Transition marks are utilized in Figs. 4-7.

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Figures 5(a) and 5(b) show R_s and R_h spectra from SG_II at the incidence of TM and TE waves, respectively. Since the period (Λ = 3 µm) is smaller than both λ and the period of SG_I, three unique features appear in the spectra. The first feature is the absence of wavy patterns at λ ≥ 11.8 µm for the spectra from RCWA. T ₀ and R ₀ are the only orders of propagating transmitted and reflected diffracted waves, respectively. Although R _-1 becomes propagating at λ < 11.8 µm, it experiences total internal reflection until λ < 4.5 µm. Accordingly, the interference of R _-1 causes wavy patterns at 4.5 µm ≤ λ ≤ 11.8 µm. The interference of other orders can enlarge the wavy patterns, for example, R ₊₁ contributes wavy patterns at 1.5 µm ≤ λ ≤ 8.8 µm. Second, the spectrum R_s (Exp) splits from R_s (RCWA) at 4.5 µm ≤ λ ≤ 11.8 µm. The reason, the total internal reflection of R _-1, is the same as that for wavy patterns. After multiple bounces, the energy carried by R _-1 is expected to partially contribute to R_s in modeling. However, it is actually lost at the side walls of the small sample, making R_s (RCWA) > R_s (Exp). The difference between R_s (RCWA) and R_s (Exp) diminishes once R _-1 can directly penetrate free space after the first bounce from the gratings. Third, the EMT works well at the incidence of TE waves with a long wavelength. The good approximation is due to the large f (0.67) and small period (Λ = 3 µm) of SG_II, which behaves like a mirror at the TE wave incidence. At the TM wave incidence, the approximation of two dielectric films for the gratings underestimates the reflection from SG_II, and thus R_s and R_h (EMT) are lower than R_s (RCWA) in Fig. 5(a).

 

Fig. 5 R_s and R_h from SG_II at incidence of (a) TM and (b) TE waves.

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Figures 6(a) and 6(b) show the R_s and R_h values from CG at the incidence of TM waves and TE waves, respectively. As the complexity of the grating profile increases, two new features appear in the spectra. First, wavy patterns in the spectra from RCWA appear within the whole spectral range at the incidence of TM waves. However, they are quite trivial at the incidence of TE waves of λ > 15 µm. The polarization dependence is explained by the dominance of R ₀ at the incidence of TE waves. The energy carried by other orders, R _-1 and R ₊₁ for example, is too weak to generate wavy patterns from their interference. Wavy patterns become significant only when the dominance of R ₀ is reduced. For instance, the energy carried by R ₀ is relatively low at the incidence of TM waves. Wavy patterns then exhibit in the whole spectral range. Second, there exists a wavelength region (16 µm ≤ λ ≤ 18 µm) in which EMT works well at the incidence of both TE and TM waves. All spectra have a dip in the region, and the dip results from the intrinsic absorption within Si.

 

Fig. 6 R_s and R_h from CG at incidence of (a) TM and (b) TE waves.

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Figures 7(a) and 7(b) show measured R_s from the three samples at the incidence of TM and TE waves, respectively. The spectra from prior figures are shown here to study the grating profile impact on R_s. There are four interesting features. First, the R_s values of SG_II and CG are higher than that of SG_I because the f of SG_I is lower than those of the other two gratings. Second, the spectrum of SG_II is almost identical to that of CG due to profile similarity. The profile of CG can be generated easily from that of SG_II by putting a grating ridge inside every one out of three trenches periodically. This way, any optical response from both structures is expected to be close even though their grating periods are different. Third, Rayleigh anomalies in the spectra of SG_I and CG are unclear, whereas they can be easily identified for SG_II. Fourth, abrupt changes in the spectra of SG_II associated with Rayleigh anomalies appear in the spectra of CG. This co-existence of anomalies is also attributed to profile similarity.

 

Fig. 7 Measured R_s from SG_I, SG_II, and CG. Predicted wavelengths of Rayleigh anomalies and their responsible diffracted waves are marked.

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5. Conclusion

Both the specular and hemispherical reflectance from three window/gratings structures were investigated at the oblique incidence of both TE and TM waves. Impacts from the light itself (polarization and wavelength), light interaction with a subject (refraction, diffraction, interference, and total internal reflection), gratings (profile complexity dimensions and materials), and the window (material) were determined. Reflectance spectra can be significantly tailored using any one or a combination of these factors for specific applications. The EMT is able to obtain optical responses from a window/gratings structure under certain conditions, such as for gratings with a large f at the TE wave incidence and for gratings with a small f at the TM wave incidence. Since the profile complexity is not a dominant factor to the validity of EMT, the EMT can still provide good approximations to optical responses from window/CG.