One-dimensional (1D) photonic crystals (PCs) that consist of an alternative stack of different material films provide an opportunity to manipulate the dispersion relation [1] and have found numerous optical and thermal applications [2–7]. Particularly, angular selectivity is among the important functionalities of PCs and has been extensively investigated [8,9]. In Ref. [10], an angular filter for $p$ polarization has been experimentally realized in the full visible spectrum by utilizing Brewster modes, providing an angular transmission window of 8° in an oblique direction of 55°. In Refs. [11,12], a transmission window for $p$ polarization has been set to the normal direction through the use of anisotropic layers. In Ref. [13], an ultrabroadband plasmonic Brewster transmission for $p$ polarization has been presented by utilizing a metallic grating. In general, a polarization-independent angular transmission window in the normal direction is desirable for many application scenarios, such as optical sensor systems [14] and communication systems [15]. However, to the best of our knowledge, there has been no work on a polarization-independent angular filter having a high-throughput transmission window in the normal direction.

PCs exhibit polarization-independent omnidirectional reflection when operating in the bandgap of the dispersion diagram [16]. Using such bandgap operation, a defect layer in PCs allows light transmission at a specific frequency as a Fabry–Perot resonator [1,17–21], and the angular transmission window is very narrow due to the intrinsic nature of resonance. PCs inherently possess a nearly polarization-independent angular transmission window that includes the normal direction when operating near the band edge in the dispersion diagram. However, the incident light is mostly reflected back at the PC–air interface due to large impedance mismatch. The wave impedance of PCs is dependent on local observation points, and, thus, the antireflection in the extremely small wavevector regime is challenging.

In this Letter, we show analytically and numerically that the reflection of the incident light is sufficiently suppressed in the extremely small wavevector regime by utilizing an antireflection PC having a different pitch from that of the host PC, resulting in a high-throughput angular transmission filter. The underlying mechanism is that the antireflection PC needs to have the interfaces with the host PC and the air structure such that the transverse impedance at each interface between two semi-infinite media has a real value, which allows us to design the antireflection PC by using the general methodology of impedance matching. In Ref. [22], we presented an antireflection structure for a two-dimensional PC in the extremely small wavevector regime. However, it showed only a sharp angular and frequency response.

We start by considering the dispersion diagram of a PC, as shown in Figs. 1(a) and 1(b), obtained by using the plane wave expansion (PWE) method [23]. The PWE method is widely used for analyses of PCs in which Maxwell’s equations are exactly solved by expanding the electromagnetic fields with a plane-wave basis set, and, thus, the wavevectors and electromagnetic fields are accurately evaluated. Two material films are alternatively and infinitely stacked in the PC [inset of Fig. 1(a)]. We assume ${\mathrm{TiO}}_2$ and ${\mathrm{SiO}}_2$ for two materials with refractive indices of $n_A=2.4$ and $n_B=1.45$, respectively. The thicknesses of the two materials are selected at $t_A=214\mathrm{nm}$ and $t_{B,h}=90\mathrm{nm}$ such that the PC has an angular transmission window of $2{\theta }_d=12°$ [oblique black dashed lines closer to the vertical axis in Fig. 1(a)] for $s$ and $p$ polarizations at the design frequency of $f_d=500\mathrm{THz}$ (horizontal pink solid line). We note that extended modes that are allowed to propagate exist within angles of less than ${\theta }_d=6°$ (shaded region) for both polarizations at $f_d$. In other angles at $f_d$, no modes are allowed to propagate (white region) except for large angles $\theta >70°$ in $p$ polarization. Therefore, we expect that the PC may work as an angular filter having a nearly polarization-independent transmission window of $2{\theta }_d$ around the normal direction, and a reflection in other angles. The effect of the leakage of electromagnetic waves in such large angles ($\theta >70°$) in $p$ polarization may be reduced by optimizing the refractive indices and the thicknesses of materials.

 

Fig. 1. (a) Dispersion diagram of a 1D PC in $s$ and $p$ polarizations. Extended modes exist in the shaded regions. The oblique black dashed lines represent the design angle of ${\theta }_d=6°$ and light lines, respectively. The horizontal pink solid line represents the design frequency of $f_d=500\mathrm{THz}$. The diagram is enlarged around $f_d$ in (b). (c) Configuration of a PC having angular selectivity. (d) Two-dimensional map of transmission coefficients obtained numerically by using Eqs. (1a)–(1d) and the transfer matrix method. Angular dependence of transmission coefficients at (e) 500 THz [horizontal pink solid line in (d)] and at (f) 515 THz [horizontal blue dashed line in (d)]. [blue solid lines, numerical results, Eqs. (1a)–(1d); pink dots, analytical results, Eqs. (3a)–(3d)] (parameters: $n_A=2.4$, $n_B=1.45$, $t_A=214\mathrm{nm}$, $t_{B,h}=90\mathrm{nm}$, $a_h=t_A+t_{B,h}=304\mathrm{nm}$, $t_{B,\mathrm{ar}}=93\mathrm{nm}$, $N_{AB}=97$, $m=15$).

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Figure 1(c) shows the configuration of a PC having the angular selectivity function. Two material films with refractive indices of $n_A$ and $n_B$ are alternatively stacked. The films are numbered from $j=1$ to $N_{AB}$, where $N_{AB}$ is the total number of layers. From the viewpoint of the underlying mechanism, the heterostructure consists of the host PC ($j=m$ to $N_{AB}-m+1$) and the antireflection PCs at both sides ($j=1$ to $m$ and $j=N_{AB}-m+1$ to $N_{AB}$), where each of two layers $j=m$ and $N_{AB}-m+1$ has the interface between the antireflection PC and the host PC positioned at the center of the thickness of the layer [indicated by red dotted lines $F$ in Fig. 1(c)] and is counted for the layer numbers of both the host and antireflection PCs. From the geometrical point of view, material $A$ has the same thickness $t_A=214\mathrm{nm}$ in both the host PC and the antireflection PC except for the first layer $j=1$ and the last layer $j=N_{AB}$. The two layers at both ends have the half-thickness $t_A/2$ such that the antireflection PC has the same impedance at the interface with the air structure and at the interface with the host PC [indicated by red dotted lines $F$ in Fig. 1(c)]. Material $B$, however, has two different thicknesses in the entire structure: $t_{B,h}=90\mathrm{nm}$ for the host PC and $t_{B,ar}=93\mathrm{nm}$ for the antireflection PC.

The monochromatic electromagnetic waves with $s$ polarization or $p$ polarization are impinged on the entire PC from the air, where the electric field or the magnetic field is perpendicular to the incident plane, respectively. The Fresnel reflection and transmission coefficients of the electric field for $s$ polarization and the magnetic field for $p$ polarization at the interface between layers $j$ and $j+1$ are given by [24]

Figure 1(d) shows the two-dimensional map of the transmission coefficients of the entire structure of Fig. 1(c) with variations of the incident angle and the operating frequency. Due to the properly designed antireflection structure, high transmission (bright-color region) is obtained for both polarizations in the propagation-allowed region that is characterized by the dispersion diagram (shaded regions) in Figs. 1(a) and 1(b). It should be emphasized that the reflection is sufficiently suppressed in a wide range of frequency in the propagation-allowed region; our structure provides not only a transmission window of $2{\theta }_d$ at $f_d=500\mathrm{THz}$ but also wide angular transmission window at high frequency. When $\theta >70°$ for $p$ polarization around $f_d=500\mathrm{THz}$ in Fig. 1(d), the incident light is leaked out, as observed from the dispersion diagram.

We plot the transmission coefficient as a function of the incident angle at $f_d=500\mathrm{THz}$ in Fig. 1(e). We clearly observe that our structure has efficient angular selectivity; light passes through around the normal direction and is reflected over other angles. The half-power angular width $2\theta$ is 11° for $s$ polarization and 13° for $p$ polarization, respectively. In addition, at a frequency of 515 THz in Fig. 1(f), a wide transmission window appears: 57° for $s$- polarization and 69° for $p$ polarization. Interestingly, sufficient antireflection is maintained when the frequency goes up. This wideband characteristic of the sufficient antireflection comes from the fact that the antireflection structure consists of a PC; the antireflection structure has a trend in the frequency response similar to that of the host medium.

We next elucidate how sufficient antireflection, as discussed above, is obtained. The underlying mechanism can be understood in the impedance behavior. Consider the impedance behavior at $f_d=500\mathrm{THz}$ in Fig. 2. The transverse impedance for $s$ or $p$ polarization, respectively, is defined as

 

Fig. 2. Transverse impedance of the host PC at $f_d=500\mathrm{THz}$. The transverse impedance is normalized with respect to the free-space intrinsic impedance $Z_0$. (a) Locally dependent distribution of the real part (blue solid line) and the imaginary part (pink dashed line) along the $x$ axis at normal incidence. (b) Angular dependence of the real part of the transverse impedance $Z_h$ of the host PC (blue solid line) at the location $F$. $Z_{\mathrm{ar}}^2/Z_{\text{air}}$, which comes from the antireflection condition, is plotted as the green dashed line. The imaginary parts of the impedances $Z_h$, $Z_{\mathrm{ar}}$, and $Z_{\text{air}}$ are zero (not presented).

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We assume infinite PCs for impedance analysis throughout this Letter and thus obtain each transverse impedance in Eqs. (2a) and (2b) by the PWE method. Figure 2(a) shows the locally dependent transverse impedance of the host PC at normal incidence. The transverse impedance is normalized with respect to the free-space intrinsic impedance $Z_0$. We point out that the imaginary part of the impedance is zero at the center of the thickness of each film [indicated by red dotted lines $F$ and $H$ in Fig. 2(a)]. This allows us to obtain sufficient antireflection by using the general methodology of impedance matching. However, at interface $G$, which is a regular interface between layers, the impedance has a non-zero value in the imaginary part, and, thus, antireflection is in general difficult. In our case, we have selected interface $F$ since the real part of the impedance at interface $F$ is smaller than that at interface $H$.

Figure 2(b) shows the angular dependence of the transverse impedance $Z_h$ (blue solid line) of the host PC at interface $F$. The real part of the transverse impedance is presented since the imaginary part is zero even for oblique incidence. The impedance has a slowly varying characteristic within $\pm 5°$ for both polarizations. In addition, we calculate the transverse impedance $Z_{\mathrm{ar}}$ of the antireflection PC that has a different thickness of $t_{B,\mathrm{ar}}=93\mathrm{nm}$ for ${\mathrm{SiO}}_2$ films with the same thickness of $t_A=214\mathrm{nm}$ for ${\mathrm{TiO}}_2$ films and plot $Z_{\mathrm{ar}}^2/Z_{\text{air}}$ in the same figure, where $Z_{\text{air}}=Z_0/\cos (\theta )$ for $s$ polarization and $Z_{\text{air}}=Z_0\cos (\theta )$ for $p$ polarization. We verify that the general antireflection condition $Z_{\mathrm{ar}}^2\approx Z_h{Z}_{\text{air}}$ [25] is satisfied within $\pm 5°$ in Fig. 2(b). The length of the antireflection PC has been selected at a quarter-guided-wavelength (number of layers: $m=15$) at $f_d$ according to the general antireflection condition [25]. Therefore, the reflection at the interface of the entire structure has been sufficiently suppressed, as shown in the high transmission coefficient of Fig. 1(e).

Based on the underlying mechanism above, we now develop an analytical model, as shown in Fig. 3. The heterostructure consists of three layers; the host PC having the antireflection PCs at both sides in air. The reflection and transmission coefficients at the interfaces of the analytical model of Fig. 3 are given by

 

Fig. 3. Analytical model of the heterostructure of Fig. 1(c). Wavevectors and transverse impedances are obtained by the PWE method. The lengths of the equivalent media are $t_{\mathrm{ar}}=\frac{(m-1)(t_A+t_{B,\mathrm{ar}})}{2}=2149\mathrm{nm}$ and $t_h=\frac{(N_{AB}-2m+1)a_h}{2}=\mathrm{10,336}\mathrm{nm}$, respectively.

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We have used a number of $N_{AB}=97$ layers in the entire structure. The number of layers can be reduced; only the number of layers in the host PC is reduced, while that in the antireflection PCs ($m=15$) is maintained. Reducing the number of layers gives gradual slope in the angular response.

In conclusion, we have explored a 1D PC that operates in the extremely small wavevector regime and, therefore, has an angular transmission window around the normal direction. Based on the analysis of the locally dependent impedance, the interfaces of an antireflection PC with the host PC and the air structure have been selected at the center of the thickness of a material film. This allowed us to use the general method of impedance matching. Consequently, the reflection at the PC–air interface was sufficiently suppressed. In addition, numerical results have revealed that our PC structure provides a high-throughput wide angular transmission window. The current nanofabrication technology for multilayers can be used for our structure. Therefore, our results nicely extend the range of potential applications of 1D PCs. For example, the signal to interference and noise ratio in optical sensor systems and communication systems can be enhanced by suppressing noise and unwanted signals coming from other directions. Based on our antireflection mechanism, one can optimize a desirable angular and frequency response for desired applications.