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. 2022 Jun 30;7(27):23353–23361. doi: 10.1021/acsomega.2c01537

Reflection Mechanism of Dielectric Corner Reflectors: The Role of the Diffraction of Evanescent Waves and the Goos–Hänchen Shift

Kishin Matsumori †,*, Ryushi Fujimura ‡, Markus Retsch †
PMCID: PMC9280947  PMID: 35847333

Abstract

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Nano- and microstructures have been developed for asymmetric light transmission (ALT) filters operating in a wide wavelength range. One of the most straightforward structures with ALT properties is a dielectric corner reflector (DCR) comprising a one-dimensional grating of a triangular shape on one surface. The DCR possesses strong reflection only for one-way light illumination due to multiple total internal reflections (TIRs) inside the triangular grating. For triangular structures being much larger than the wavelength of light, the reflection properties are expected to be fully described by geometrical optics. However, geometrical optics do not account for the Goos–Hänchen (GH) shift, which is caused by the evanescent wave of the TIR. In this work, the reflection mechanism of DCRs is elucidated using the finite element method and a quantitative model built by considering the GH shift. The reduction in reflection of the DCR is dominated by diffraction of the evanescent wave at the corner of the triangular structure. Our model is based on simple mathematics and can optimize the DCR geometry for applications addressing a wide wavelength range such as radiative cooling.

1. Introduction

Asymmetric light transmission (ALT) is an optical phenomenon in which transmission differs for forward and backward illumination of a device. ALT has been mainly investigated for optical communications, but it may be applied to advance energy applications, such as radiative cooling. Radiative cooling can passively cool down an object below the ambient temperature by emitting thermal radiation in the mid-infrared range (MIR) to the cold outer space.1 However, under high humidity, it is known that the cooling performance is severely degraded since the thermal emission to outer space is attenuated, and an additional heat transfer between the emitter and the atmosphere needs to be considered.2,3 Optical elements acting like ALT filters in the MIR range have recently been controversially discussed to counteract the cooling performance degradation under high humidity conditions. The fundamental idea of such a filter is to reflect the incoming atmospheric radiation while retaining the transmission of the outgoing thermal radiation.4,5 An experimental proof of the feasibility of such a filter has not been shown, and a complete theoretical analysis is also missing, especially with a full description of the mode-to-mode conversion.6,7 Further progress in this direction requires a thorough understanding of the optical properties of a suitable ALT filter, which, furthermore, should cover a broad wavelength range and work independently of the light polarization. Ultimately, for transitioning into a potential application, such an ALT filter needed to be simple and scalable to manufacture.

ALT can be achieved by optical non-reciprocity,8−11 planar chiral structures,12,13 photonic crystals,14,15 and hyperbolic metamaterials.16 Those structures can show a high performance in ALT. However, they mainly operate only at a specific wavelength. Widening the operational wavelength of ALT has been achieved by designing grating structures.17−19 The ALT of grating structures is attributed to a difference in diffraction modes for the forward and backward illuminations. In this case, the intensity of zeroth-order transmission is the same for both illuminations, limiting ALT performance. In addition, their structure requires precise control of the periodicity of the grating, which makes it challenging to produce a scalable device. A dielectric microsphere whose top is partially covered by a perfect light absorber has been proposed as an ALT structure with a more straightforward fabrication process.20 The microsphere acts as a ball lens, which focuses the incoming light on one point. This focused light is absorbed by the perfect absorber, resulting in low transmission only for one way. The operational wavelength range of ALT can be controlled by adjusting the size of the microsphere. This ALT is not attributed to a diffraction mode at all. Therefore, the periodicity of the microspheres is not essential, providing a considerable fabrication tolerance. However, light absorption is not desirable for radiative cooling because light absorption increases the temperature of a system, which causes degradation in cooling performance.

One alternative structure for broadband ALT is a dielectric corner reflector (DCR).5,21 The DCR is a film with a one-dimensional grating of a triangular structure. The DCR is composed of a relatively simple microstructure that can be made of a low-cost polymer by a mechanical stamping,22 enabling the production of large-area devices. The DCR can have high reflectance only when the light propagates into the DCR from the flat side. This is because a round-trip reflection, in which the light is reflected multiple times and its propagation direction is redirected backward, is made inside the triangular structure. The round-trip reflection is attributed to total internal reflections (TIRs) occurring at the surfaces of the triangular structure. The ALT of the DCR does not rely on either diffraction modes of the grating or light absorption. Therefore, a high-performance ALT for radiative cooling can be achieved. From a geometrical optics point of view, it is expected that the DCR can possess nearly perfect reflection in the MIR region by designing a triangular structure much larger than the wavelength. This is because the surfaces of the triangular structure can act as a reflective surface by TIR comparable to metallic surfaces. However, the reflectance of the DCR linearly decreases with an increase in the wavelength. This decrease occurs even though the triangular structure is sufficiently larger than the wavelength. Braginsky and Vyatchanin and Tarabrin suggested that evanescent waves created by TIRs could be diffracted at the downward corner of the triangular structure.23,24 Therefore, the reflectance can decrease even within the geometrical optics regime. This diffraction loss was theoretically estimated; however, Braginsky and Vyatchanin and Tarabrin just considered the whole field around the surface of the triangular structure but did not consider directions of energy flux (EF) of the evanescent waves around the downward corner. In addition, their mathematical approach was too simplified compared to the diffraction of the evanescent wave created by the surface plasmon polaritons.25,26 Therefore, their simple analysis may lose a lot of important information describing the diffraction loss mechanisms.

In this work, the reflection mechanism of a DCR is theoretically investigated by building a quantitative model to fundamentally understand the inherent reflection reduction. We build the model by considering the following: (1) part of the incident light is shifted outside of the DCR by the Goos–Hänchen (GH) shift and (2) the energy of the incident light shifted outside is lost by diffraction of the evanescent wave. Reflection spectra of the DCR estimated using our model are in good agreement with reflection spectra obtained using the finite element method (FEM). Electric field distribution maps obtained from the FEM simulation demonstrate that the diffraction of the evanescent wave occurs at the downward corner of the triangular structure. Our model suggests that this diffraction occurs no matter how large the DCR is relative to the wavelength. Our analysis is not confined to a specific wavelength range, which enables us to contribute to high-performance DCRs not only for radiative cooling but also for other applications requiring an ALT, such as silicon solar cells27 and luminescent solar concentrators.28

2. Goos–Hänchen Shift

When the TIR reflects the light, the reflected light is laterally shifted from a point predicted by geometrical optics. This lateral shift is called the GH shift (Figure 1a). The amount of the shift was first formulated by Artmann.29 He obtained the GH shift by mathematically treating the phase difference between the incident and reflected lights. His stationary-phase (SP) method gives the GH shift for TE and TM polarizations as

2. 1
2. 2

where λ0 is the wavelength in vacuum, θ is the incident angle, n is n2/n1, and n1 and n2 are refractive indices of an optically denser and less dense medium, respectively. Equations 1 and 2 are plotted in Figure 1b. After the SP method was proposed, Renard proposed an EF method.30 This method is based on a consideration that the total EF of an evanescent wave created in an optically less dense medium (blue arrow in Figure 1a) must be conserved by the EF entering back into an optically denser medium (red arrow in Figure 1a). The EF method gives the GH shift as

2. 3
2. 4

Figure 1.

Figure 1

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(a) Schematic illustration of the GH shift. The gray dashed lines show the reflected light path, predicted by geometrical optics. (b) GH shift calculated using the SP method and the EF method. The refractive indices of the optically denser and less dense medium are n1 = 1.6 and n2 = 1, respectively. The GH shift is normalized by the wavelength in vacuum.

Equations 3 and 4 are plotted in Figure 1b. It can be observed that there is a considerable discrepancy between the SP and EF methods at the incident angle far from the critical angle θc = sin–1(n2/n1). Yasumoto pointed out that this discrepancy occurs because the EF created by the interference between the incident and reflected light (yellow arrow in Figure 1a) is not considered in the EF method.31 The GH shift calculated by considering both the evanescent wave and the interference coincides with the SP method.31,32 At the incident angle close to θc, the GH shift is dominated by the EF of the evanescent wave. In contrast, when the incident angle is far from θc, the GH shift is dominated by the EF of the interference. Based on an understanding of the GH shift, we will investigate the reflection properties of the DCR.

3. DCRs in Geometrical Optics and Nano-Optics

3.1. Round-Trip Condition

Figure 2a shows a schematic of the DCR. We considered that the grating and triangular structures are infinitely large in the x- and z-direction, respectively. The forward incidence is that the light illuminates on the structured side. In this case, the DCR has high transmission. For the backward incidence, in which the light illuminates on the DCR from the flat side, the DCR can have a high reflectance by the round-trip reflection. Figure 2b shows the cross-section of one grating period, consisting of an isosceles triangular structure with a width W and a base angle θb. In Figure 2b, two light paths inside the triangular structure are shown for backward incidence at the incident angle θin. For Path1, the light is reflected at Interface1 and goes to Interface2. Then, the light is reflected again at Interface2 and goes back to the flat side of the DCR. Path2 is in the opposite direction to Path1. We first consider the case of normal incidence, where θin = 0°, to find the round-trip condition. In this case, Path1 and Path2 are now identical (θ11 = θ22 = θb), meaning that only Path1 needs to be considered. The round-trip condition can be found using Snell’s law and taking into account the incident angles at each interface, which are θ11 and θ12 (S1, Supporting Information).

3.1. 5

Figure 2.

Figure 2

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(a) Illustration of the DCR. (b) Schematic illustration of the cross-section of the DCR. The red and blue lines indicate the light paths inside the triangular structure. (c) Round-trip condition at oblique incidence for n1 = 1.6. (d) Schematic illustration of the diffraction loss (DL) model. The red shaded area shows the light traveling along Path1 in (b). The dark red, yellow, and orange arrows indicate that the incident light strikes on Interface1, the light reflected at Interface1 and traveling to Interface2, and the light reflected at Interface2, respectively. The gray shaded area shows an imaginary path of the reflected light indicated by the yellow arrows if Interface1 was infinitely long and if there was no reflection at Interface2. (e) DL model for oblique incidence.

From the round-trip condition, a refractive index of the DCR n1 should be larger than 1.42 when the surrounding of the DCR is the air. The minimum of the condition is determined by θc. For our study, n1 = 1.6 is considered, which conventional polymers, such as polystyrene, possess.33,34 In this case, θb can be in an angle range from 38.7 to 47.1° to satisfy the round-trip condition (Figure S1b, Supporting Information).

When the light is incident on the DCR at an oblique angle, we have to find round-trip conditions for Path1 and Path2 separately. Figure 2c shows the round-trip condition for the oblique incidence. The yellow and blue shaded areas indicate the round-trip conditions for Path1 and Path2, respectively. The round-trip condition for oblique incidence can be satisfied in the area where those two areas overlap each other (S1, Supporting Information). When θb is 45°, the round-trip reflection can occur at an incident angle up to about 6° for n1 = 1.6. In the experimental system, the refraction at the flat surface of the DCR has to be taken into account. In this case, the condition can be satisfied up to the incident angle of 10° (Figure S3b, Supporting Information). For simplicity, the flat surface of the DCR is ignored, and this refraction is not considered in further theoretical investigations. The propagation direction of the light after the round-trip reflection, denoted by θout in Figure 2b, can be found in S1, Supporting Information. θout is equal to θin when θb is 45°, meaning that the DCR can act as a retroreflector.35,36

3.2. DL Model

By considering the GH shift and the geometry of the triangular structure, we investigate the reflection mechanism of the DCR in detail using Figure 2d. First, we only consider Path1 for normal incidence and assume that the plane wave strikes only one-half of the triangular structure (red arrows in Figure 2d). As mentioned earlier, the incident angle at Interface1 θ11 is equal to θb when θin = 0°. The center of the reflected light is laterally shifted on Interface1 by the GH shift dGH11. However, Interface1 is not infinite. Therefore, the amount of the reflected light’s energy, which is determined by the GH shift dGH11, cannot completely come back into the structure (blue arrows in Figure 2d). According to the physical mechanism of the GH shift mentioned earlier, this energy is related to the evanescent wave propagating in the optically less dense medium. From the research on propagating surface plasmon, it has been known that a propagating evanescent wave can be diffracted at the corner of the metal film.25,26 A similar diffraction scenario is expected for the DCR because the evanescent wave still has an EF at the downward corner of the triangular structure. In addition, the EF created by the interference between the incident and reflected lights is also attributed to the GH shift. This amount of energy may also be diffracted at the downward corner. The rest of the incident light reaches Interface2 (yellow arrows in Figure 2d) and is reflected at Interface2 by the TIR (orange arrows in Figure 2d). The GH shift dGH12 occurs at Interface2, but the EF of either the evanescent wave or the interference is not toward the downward corner. Therefore, all the energy of the light reflected at Interface2 can go back into the triangular structure. Consequently, we expect a fraction of the incident energy to be lost by the diffraction governed by dGH11 during one cycle of the round-trip reflection. The same process happens for Path2. From this diffraction loss (DL) model, the reflectance of the DCR can be estimated by using the ratio of dGH11 to the side length of the triangular structure L. The DL model gives the reflectance of the DCR for normal incidence as

3.2. 6

For oblique incidence, a different amount of energy is lost during the round-trip reflection in Path1 and Path2 because the incident angles θ11 = θb + θin and θ22 = θb – θin are different, resulting in dGH11 ≠ dGH22. When light is normally incident on the DCR, the energy of the incident light on Interface1 is equal to that on Interface2. However, for oblique incidence, the energy of the incident light striking on Interface1 and Interface2 is different (Figure 2e). This has to be taken into account in the DL model by considering the cross-section of the incident light entering a single triangular structure. The cross-section of the incident light can be expressed as s = Wcosθin. The cross-section of the incident light that goes to Interface1 is s1 = Lcos(θb + θin), and the rest goes to Interface2. Therefore, the reflectance of the DCR for oblique incidence can be estimated by

3.2. 7

where Ro1 and Ro2 are the reflectance for Path1 and Path2, respectively. Equation 7 coincides with eq 6 when θin = 0°. The DL model shows that a lossless reflection cannot be obtained because the GH shift cannot be zero even if the triangular structure was sufficiently larger than the wavelength. It has to be mentioned that, for Path2, part of the light reflected at Interface2 (R22) does not strike on Interface1. However, R22 points to the flat top surface of the DCR. Therefore, R22 is not lost and is counted in the total reflectance of the DCR (S1, Supporting Information).

4. Results and Discussion

4.1. DL Model with SP and EF Methods

We investigate the reflection properties of the DCR for backward incidence with TM polarization using the FEM (see Section 6). In this investigation, reflection and refraction occurring at the flat top surface of the DCR are ignored. We also use the DL model to gain insight into the reflection mechanism of the DCR. Reflection spectra are plotted as a function of the normalized unit of λ0/L.

First, we compare the DL model (eq 6) with the SP method (eq 2) and that with the EF method (eq 4) to understand which method can properly describe the reflection of the DCR. Figure 3a shows the reflection spectra of the DCR with n1 = 1.6 and θb = 45° at the incident angle of θin = 0°. The dashed lines are spectra calculated by the FEM simulation. The black and red solid lines are calculated using the DL model with the SP and EF methods, respectively. There is a difference in spectral shape between the FEM and the DL model because the FEM considers the grating effect,21 but the DL model does not. In terms of how reflectance decreases with an increase in the wavelength, it can be found that the DL model with the SP method is in good agreement with the FEM simulation. When the EF method is used in the DL model, the reflectance is slightly overestimated, indicating that the evanescent wave’s energy and the energy created by the interference are lost by diffraction. From this result, we use only the SP method for further investigations.

Figure 3.

Figure 3

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For (a–c), the dashed lines are reflection spectra calculated by the FEM simulation. The solid lines are calculated by the DL model (eq 6) using the SP method (eq 2). The incident light is TM polarized. Only backward incidence is considered. (a) Reflectance of the DCR with n1 = 1.6 and θb = 45° at θin = 0°. The red solid line is calculated by the DL model using the EF method. (b) n1 dependence. θb and θin are 45 and 0°, respectively. The red, green, black, and blue lines are for n1 = 2, 1.8, 1.6, and 1.5, respectively. (c) θb dependence for the DCR with n1 = 1.6 and θin = 0°. L is fixed and W changes according to θb. The red, black, green, and blue lines are for θb = 47, 45, 41, and 39°, respectively. The blue dashed–dotted line is calculated using the DL model with eq 9. w0 is determined so that w0/L becomes about 1.7. (d) Schematic illustration of the plane wave diffracted at the triangular structure’s upward corners. (e) GH shift at around θc, which is given by eqs 8 and 9 for TE and TM polarizations. The blue and red dashed lines are the SP method, found in Figure 1b. (f) Reflection spectra for n1 = 1.6, θb = 45°, and θin = 6°. The reflection spectra obtained from the DL model for Path1 (Ro1) and Path2 (Ro2) are the red and blue lines, respectively. Ro1 and Ro2 are calculated using eq 7 with eq 2 and with eq 9, respectively. The black solid is the sum of the red and blue solid lines.

4.2. λ0/L and n1 Dependencies

The reflectance of the FEM approximately linearly decreases with an increase in the wavelength for a fixed L (Figure 3a). Our DL model indicates that this linear decrease in reflectance is attributed to the GH shift being proportional to the wavelength. For a fixed λ0, the reflectance can be higher for a large L since the reflection reduction of the DCR is inversely proportional to L (eq 6). We are interested in the high reflectance of the DCR; therefore, we focus on a region of λ0/L in which the reflectance can exceed 80%. In Figure 3b, the n1 dependence of the reflection of the DCR is examined by changing n1 in a range where the round-trip condition is satisfied. For those spectra, other parameters are fixed as θb = 45° and θin = 0°. The DL model shows a good agreement with the FEM for all n1. Since the GH shift is small for large n1 and less energy is lost by the diffraction, the reflectance is higher when n1 is larger. From the λ0/L and n1 dependencies, we conclude that a DCR with large L and large n1 can possess strong reflection. However, an increase in L increases the size of the structure, which is undesirable for some applications requiring a compact system. In addition, increasing n1 is not straightforward in experimental systems.37 Therefore, it is required to find an optimum combination of L and n1 to achieve a high-performance DCR.

4.3. θb Dependence

Figure 3c shows reflection spectra calculated for different θb with n1 = 1.6 and θin = 0°. For this calculation, L is fixed, and W is changed with the change in θb. The GH shift decreases with an increase in the incident angle (Figure 1b) so that the DCR with higher θb can possess a higher reflectance. However, the round-trip condition can be satisfied in a smaller incident angle range for larger θb (Figure 2c); therefore, there is a trade-off between higher reflectance and the incident angle insensitivity. This trade-off can be compensated just by increasing L. When θb is larger, the DL model shows good agreement with the FEM. However, the DL model cannot reproduce the results from the FEM when θb is close to θc. This is because neither the SP nor the EF methods can explain the GH shift around θc.

4.4. Influence of the GH Shift around θc

The GH shifts derived by eqs 1–4 diverge at θc (Figure 1b). However, when the incident light is a beam, such as a Gaussian beam, the GH shift has a maximum value at slightly greater than θc because of angular distribution in wave vector components of the incident beam. In the case of the DCR, we consider a plane wave as the incident light. In this case, diffraction occurs when the light strikes on the upward corners of the DCR (Figure 3d).38 Therefore, the incident light on the triangular structure’s surfaces is no longer a plane wave. The GH shift around θc was formulated using the Weber function39,40 or modified Bessel function41 for the Gaussian beam. Using the modified Bessel function,41 the GH shift for TE and TM polarizations is

4.4. 8
4.4. 9

where

4.4.
4.4.

w0 is the beam waist of the Gaussian beam, k is the angular wavenumber expressed as 2π/λ0, and Iv(x) is the modified Bessel function of the first kind. Equations 8 and 9 hold in a limitation of kw0 > > 1. Equations 8 and 9 are plotted in Figure 3e for different ratios of w0/λ0. Compared to the SP method, those equations have a different line shape around θc but coincide with the SP method at the incident angle far from θc. With a decrease in w0/λ0, the maximum of the GH shift decreases and slightly shifts to a larger θ. Since the GH shift around θc is influenced by the angular distribution of wave vector components, the diffraction pattern at the upward corners of the DCR has to be carefully investigated for detailed analysis. However, for our phenomenological investigation, we use eqs 8 and 9 because the GH shift at around θc shows a similar line shape for different angular distributions, for example, for the case of the single-slit diffraction beam.40,41 The beam waist determines the angular distribution of wave vector components in the Gaussian beam, so we use w0 as a parameter that determines the diffraction pattern of the upward corners of the triangular structure.

In Figure 3c, the blue dashed–dotted line shows the reflection spectra estimated by the DL model using eq 9 with w0 determined by w0/L ≈ 1.7. In this case, the DL model shows a similar trend to the FEM. We can understand why the reflectance does not linearly decrease for θb = 39° by considering the wavelength dependence of plane wave diffraction at the upward corners. Diffraction is weaker for smaller wavelengths, resulting in a larger GH shift. However, diffraction becomes stronger for longer wavelengths, and the GH shift becomes smaller. Therefore, the slope of the reflection spectrum is steeper in the shorter wavelength range, but the slope becomes gradually less steep with an increase in the wavelength.

4.5. Oblique Incidence

Considering the GH shift around θc, the reflection properties of the DCR at oblique incidence are investigated using the DL model in eq 7. Figure 3f shows the reflection spectrum of the DCR with n1 = 1.6 and θb = 45°. The incident angle is θin = 6°, which is the upper limit of the incident angle that can satisfy the round-trip condition (see Figure 2c). In this case, the incident light’s energy on Interface1 and Interface2 is different, determined as s1/s ≈ 0.45 and s2/s ≈ 0.55, respectively. In addition, the incident angles at Interface1 for Path1 and at Interface2 for Path2 are different. The incident angle θ11 is 51° at Interface1 for Path1. θ11 is much larger than θc. Therefore, the SP method can be used for dGH11 in the DL model. The incident angle θ22 is 39° at Interface2 for Path2. θ22 is now close to θc, so dGH22 should be expressed using eq 9. w0 is used again as a fitting parameter. When w0/L ≈ 1.7, the DL model shows good agreement with the FEM. For Path1, the reflectance linearly and slowly decreases with an increase in the wavelength because the GH shift is small for the incident angle of 51° (Figure 1b). For Path2, the reflectance decreases nonlinearly with an increase in the wavelength because θ22 is close to θc. The reflectance for Path1 is higher than that for Path2 even though s1/s < s2/s because dGH11 is much smaller than dGH22. Overall, our proposed DL model is robust for any combinations of the structural parameters of the DCR, and the DL model can consider the incident angle dependence of the DCR.

4.6. Polarization Dependence and Transmission for Forward Incidence

The same investigations are made for TE polarization (Figure S5, Supporting Information). The DL model also shows good agreement with the FEM for TE polarization. The DCR has the same reflection properties for TE and TM polarizations. The GH shift for TE polarization is smaller than that for TM polarization when the incident angle is smaller than about 48° (Figure 1b). Therefore, the reflectance is higher for TE polarization for normal incidence. For the DCR with θb close to θc, the same fitting made in Figure 3c was employed for TE polarization using w0 as a fitting parameter. The fitting resulted in w0/L ≈ 3.5, which is significantly different from that for TM polarization. To understand the physical meaning of those values, the polarization dependence of the diffraction pattern created by upward corners of the DCR has to be investigated in detail.38

Compared with TE polarization, the FEM and the DL model difference is larger for TM polarization for all reflection spectra. The diffraction at the upward corner may contribute to this discrepancy. Our analysis suggested that the value of w0 can be smaller for TM polarization, meaning that the diffraction at the upward corner is more substantial for TM polarization. If the diffraction is strong, more incident light may go outside the DCR (blue arrows in Figure 3d), which the DL model does not consider.

The transmission properties of the DCR for the forward incidence can be found in Figure S7, Supporting Information. The transmission was calculated without taking the reflection and refraction at the flat surface of the DCR into account. The transmission is >90% for normal and oblique incidence for TM and TE polarizations. Since the DCR has a high reflectance for backward incidence, it can act as a high-performance broadband ALT filter in a specific incident angle range.

4.7. Diffraction of Evanescent Waves

For a better insight into the reflection mechanisms of the DCR, electric field distribution maps for TM and TE polarizations were obtained using the FEM. The top row of Figure 4 shows the field maps of the DCR with n1 = 1.6 for TM polarization at λ0/L ≈ 0.07. For the DCR with θb = 45° and θin = 0° (Figure 4a), the electric field is confined inside the DCR, which attributes to the high reflectance. The GH shift for θb = 45° is not large, so the diffraction of the evanescent wave at the downward corner is not significant. Figure 4b shows the field map of the DCR with θb = 39°. The incident angle is 0°. Compared to θb = 45°, the electric field at the interfaces of the triangular structure is stronger because the penetration depth of the evanescent wave is larger for the incident angle close to θc. Interestingly, strong electric fields in the air region lower than the downward corner can be found. These fields are attributed to the diffraction of the evanescent wave. Figure 4c shows the field map for the DCR with θb = 45° and θin = 6°. In this configuration, the incident angles at Interface1 and Interface2 are 51 and 39°, respectively. At interface1, the incident angle is much higher than θc, resulting in a weak evanescent wave. On the other hand, the incident angle at Interface2 is close to θc, so the evanescent wave is strong. Therefore, the diffraction of evanescent waves can be observed only along Interface2. This field map confirms why the reflectance is smaller for Path2, as shown in Figure 3f. The bottom row of Figure 4 shows the field maps for TE polarization, which are similar to those for TM polarization. This is because, as mentioned earlier, the DCR has almost the same reflection properties for both polarizations. The field maps at large λ0/L can be found in S2, Supporting Information. In all cases, more energy of the incident light reaches outside of the triangular structure because the reflection reduction of the DCR is proportional to λ0/L.

Figure 4.

Figure 4

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Color maps of the absolute value of electric field distribution at λ0/L ≈ 0.07 for (a) θb = 45° and θin = 0°, (b) θb = 39° and θin = 0°, and (c) θb = 45° and θin = 6°. n1 and n2 are 1.6 and 1, respectively. The incident light is (top row) TM and (bottom row) TE polarized.

We must mention the diffraction of the EF created by the interference between incident and reflected lights, which cannot be observed in the field maps. Since the contribution of the interference on the GH shift is small compared to the evanescent wave (Figure 1b), we conclude that the diffraction of the evanescent wave dominates the reduction in reflection of the DCR. To understand the mechanism of diffraction loss by the GH shift, a more detailed mathematical analysis is required by considering the EF of electromagnetic waves inside and outside the triangular structure.42

4.8. Optimization of the DCR

Next, we discuss how to optimize the DCR for high reflectance in a specific wavelength range using the DL model. As mentioned earlier, the reflectance of the DCR linearly decreases with increasing wavelength. Therefore, we only have to consider the reflectance at the maximum of the wavelength range λmax to find an optimum size of the DCR. For example, if the DCR has to have a reflectance higher than 90% in a wavelength range shorter than λmax for normal incidence and unpolarized light, we can find an optimum W of the DCR with arbitrary n1 and θb using eq 6

4.8. 10

Since the GH shift is large at around θc, a DCR with θb ≈ θc cannot possess a high reflectance. Also, the expression of the GH shift around θc is relatively complicated. Therefore, for simplicity, θb close to θc is not considered, and only the SP method is used in eq 10. From eq 10, when n2 = 1, W should be

4.8. 11

L = W/2cosθb was used to derive eq 11. Figure 5 shows eq 11 in the round-trip condition for normal incidence (eq 5). The SP method diverges at θc, so eq 11 was calculated for θb larger than θc + 1°. W/λmax is smaller for larger n1 and larger θb because the GH shift becomes smaller. The DCR can be optimized efficiently by determining n1 and θb depending on applications and finding W from Figure 5. For radiative cooling, high reflectance in a wide incident angle range is expected so that 45° is an optimum angle for θb (Figure 2c). For example, in the case of n1 = 1.6 and θb = 45°, W/λmax is 9.7. Since many polymers have a refractive index lower than 1.6,33,34 it can be found that the DCR made of a polymer is required to be W/λmax > 9.7 for reflectance higher than 90%. It should be mentioned that, as can be seen in Figure 3, the DL model slightly overestimates the reflectance of the DCR. Therefore, the DCR has to be slightly larger than the W estimated by eq 11.

Figure 5.

Figure 5

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Size optimization of the DCR for reflectance higher than 90% in a wavelength range shorter than λmax.

Besides the structural optimization shown in Figure 5, a challenge for using such DCR structures as filters for radiative cooling applications is their sensitivity to the incident angle (Figure 2c). As outlined in the Section 1, the feasibility of their ALT properties is also controversially debated until now.4−7 Our improved understanding of the reflection mechanism of a DCR enabled us to reassess their contribution to radiative cooling. Our simplified mode-to-mode conversion analysis indicates a potential positive effect (S3, Supporting Information). However, it also requires further analysis to take all contributions into account.

5. Conclusions

We have revealed the reflection mechanism of a DCR by using the DL model. From the DL model, we found that the reflectance of the DCR can be estimated just by considering the GH shift and the geometry of the triangular structure. The reflection spectra of the DCR were calculated using the FEM for different structural parameter combinations, which were compared with the spectra estimated by the DL model. Even though the DL model does not require any complicated mathematics, the DL model showed good agreement with the FEM. From the comparison, we revealed that the reflectance linearly decreases with an increase in the wavelength because the GH shift is proportional to the wavelength. We obtained the electric field distribution maps using the FEM and demonstrated that the diffraction of the evanescent wave occurs at the downward corner of the triangular structure. Since the GH shift does not become zero even though the triangular structure is significantly larger than the wavelength, this diffraction causes the intrinsic reflection reduction of the DCR.

In addition to radiative cooling, the DL model is useful to optimize the DCR efficiently for its potential applications, such as silicon solar cells,27 luminescent solar concentrators,28 non-contact temperature sensors,22 and free-space optical communications.43,44 Furthermore, our work may be essential to better understand light–matter interactions in other systems. For example, recently, coloration has been achieved by hemispherical dielectric microstructures.45−47 The coloration is attributed to the round-trip reflection by TIRs and light interference occurring inside the microstructure. Geometrical optics explained this coloration mechanism. However, since TIR plays a pivotal role in this coloration mechanism, the GH shift and the diffraction of evanescent waves may also contribute to the reflected spectrum. Therefore, we expect that our DL model provides a deeper insight into the coloration mechanism of those microstructures.

6. Method

6.1. FEM Simulation

The optical properties of the DCR for the forward and backward incidence were investigated using COMSOL multiphysics, a commercial software package based on the FEM. The simulation model of the DCR was built in two dimensions. Figure 2b is a unit cell of the simulation model. The periodic boundary condition was applied to the x-direction to consider the infinitely long grating. The periodicity is the same as the width of the triangular structure. The film part of the DCR, which is on top of the triangular structure, was considered infinitely thick to remove noise from the Fabry–Perot resonance. The perfectly matched layers were applied on the top and bottom of the model in the y-direction. To obtain total reflectance and transmittance, not only zeroth-order but also all higher-order diffraction modes were taken into account. The incident light is a plane wave with either TE (the electric field is along the z-direction) or TM (the magnetic field is along the z-direction) polarization.

Acknowledgments

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 714968).

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.2c01537.

  • Round-trip reflection for normal and oblique incidence; optical properties of the DCR; and mode-to-mode conversion of the DCR (PDF)

The authors declare no competing financial interest.

Supplementary Material

ao2c01537_si_001.pdf (2.4MB, pdf)

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Supplementary Materials

ao2c01537_si_001.pdf (2.4MB, pdf)

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