Tunable Near-Infrared Transparent Bands Based on Cascaded Fabry–Perot Cavities Containing Phase Change Materials

Abstract

In this paper, we construct a near-infrared Fabry–Perot cavity composed of two sodium (Na) layers and an antimony trisulfide (Sb₂S₃) layer. By cascading two Fabry–Perot cavities, the transmittance peak splits into two transmittance peaks due to the coupling between two Fabry–Perot modes. We utilize a coupled oscillator model to describe the mode coupling and obtain a Rabi splitting of 60.0 meV. By cascading four Fabry–Perot cavities, the transmittance peak splits into four transmittance peaks, leading to a near-infrared transparent band. The near-infrared transparent band can be flexibly tuned by the crystalline fraction of the Sb₂S₃ layers. In addition, the effects of the layer thickness and incident angle on the near-infrared transparent band and the mode coupling are investigated. As the thickness of the Na layer increases, the coupling strength between the Fabry–Perot modes becomes weaker, leading to a narrower transparent band. As the thickness of the Sb₂S₃ layer increases, the round-trip propagating of the Sb₂S₃ layer increases, leading to the redshift of the transparent band. As the incident angle increases, the round-trip propagating of the Sb₂S₃ layer decreases, leading to the blueshift of the transparent band. This work not only provides a viable route to achieving tunable near-infrared transparent bands, but also possesses potential applications in high-performance display, filtering, and sensing.

1. Introduction

Optical cavities play a fundamental role in photonics due to their superior ability to confine light in small spatial regions [1,2,3]. As a category of optical cavities, Fabry–Perot cavities have been widely utilized in interferometers [4,5,6], filters [7,8,9,10,11], absorbers [12,13], and polarization manipulation [14,15]. It is known that sandwich structures composed of two metal layers and a dielectric layer can be treated as Fabry–Perot cavities since two metal layers act as two optical mirrors [16,17,18,19,20,21,22,23,24,25,26,27,28]. These sandwich structures support Fabry–Perot resonances, which induce discrete transmittance peaks in transmittance spectra [13,14,15,16]. In 1998, M. Scalora et al. [29] and M. J. Bloemer et al. [30] achieved transparent bands at visible wavelengths by cascading Fabry–Perot cavities. These cascaded Fabry–Perot cavities are also called one-dimensional metal-dielectric photonic crystals [29,30]. To date, transparent bands at visible wavelengths have been confirmed by various cascaded Fabry–Perot cavities [29,30,31,32,33,34]. In Refs. [29,30,31,32,33,34], researchers did not discuss the formation of the transparent bands from the perspective of the mode coupling. The transparent bands originate from the coupling between the Fabry–Perot modes. In addition, the maximum transmittances of the transparent bands in Refs. [29,30,31,32,33,34] are lower than 0.8. Owing to the coupling between the Fabry–Perot modes, a transmittance peak will split into several transmittance peaks, giving rise to a transparent band. In 2023, V. Belyaev et al. systemically discussed transparent bands in one-dimensional photonic crystals composed of weak conductive material layers and dielectric layers [35].

The near-infrared band is an important wavelength band in photonics due to its extensive applications in absorption [36,37,38,39,40], sensing [41,42,43], and polarization manipulation [44,45,46]. To the best of our knowledge, no work has reported near-infrared transparent bands based on cascaded Fabry–Perot cavities until now. In this paper, we construct a near-infrared Fabry–Perot cavity composed of two sodium (Na) layers and an antimony trisulfide (Sb₂S₃) layer. The reason we selected Na is because Na possesses ultra-low optical loss at near-infrared wavelengths [47,48]. The reason we selected Sb₂S₃ is because Sb₂S₃ is a kind of lossless phase-change material at near-infrared wavelengths [49]. Its refractive index can be flexibly tuned by the crystalline fraction [50,51]. By cascading two Fabry–Perot cavities, the transmittance peak splits into two transmittance peaks due to the coupling between two Fabry–Perot modes. We utilize a coupled oscillator model to describe the coupling between two Fabry–Perot modes and obtain a Rabi splitting of 60.0 meV. By cascading four Fabry–Perot cavities, the transmittance peak splits into four transmittance peaks, giving rise to a near-infrared transparent band. By changing the crystalline fraction of the Sb₂S₃ layers, the near-infrared transparent band can be flexibly tuned. The maximum transmittance of the near-infrared transparent band reaches 0.843. In addition, we investigate the effects of the layer thickness and incident angle on the near-infrared transparent band and the mode coupling. This work not only provides a viable route to achieving tunable near-infrared transparent bands but also possesses potential applications in high-performance display, filtering, and sensing.

The rest of this paper is organized as follows. In Section 2, we construct a near-infrared Fabry–Perot cavity composed of two Na layers and a Sb₂S₃ layer. Then, we achieve two near-infrared transmittance peaks by cascading two Fabry–Perot cavities. A coupled oscillator model is utilized to describe the coupling between two Fabry–Perot modes. Next, we achieve a near-infrared transparent band by cascading four Fabry–Perot cavities. The near-infrared transparent band can be flexibly tuned by the crystalline fraction of the Sb₂S₃ layers. In Section 3, we investigate the effects of the layer thickness and incident angle on the near-infrared transparent band and the mode coupling. Eventually, the conclusions are given in Section 4.

2. Tunable Near-Infrared Transparent Bands Based on Cascaded Fabry–Perot Cavities

In this section, we achieve a near-infrared transparent band by cascading Fabry–Perot cavities. It is known that Na possesses ultra-low optical loss at near-infrared wavelengths [47,48]. Recently, researchers fabricated stable Na films using the spin-coating technique [47]. Hence, we choose Na to construct the Fabry–Perot cavities. Before investigating the Fabry–Perot cavities, we demonstrate the near-infrared reflection property of Na. At near-infrared wavelengths, the relative permittivity of Na can be well described by the Drude-Lorentz model [47]:

$${\epsilon }_{\mathrm{M}}={\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{f}}-\frac{{\omega }_{\mathrm{p}}^2}{{\omega }^2+i\omega {\gamma }_{\mathrm{p}}}+C_1\frac{{\omega }_1^2}{{\omega }_1^2-{\omega }^2-i\omega {\gamma }_1}$$

(1)

where ${\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{f}}$, ${\omega }_{\mathrm{p}}$, $\omega$, ${\gamma }_{\mathrm{p}}$, $C_1$, ${\omega }_1$, and ${\gamma }_1$ denote the high-frequency permittivity, the plasma angular frequency, the angular frequency, the damping angular frequency, the amplitude of the inter-band transition, the resonant angular frequency of the inter-band transition, and the inter-band damping rate, respectively. By fitting the measured reflectance spectrum [47], the values of the parameters of the relative permittivity of Na can be obtained, i.e., ${\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{f}}=0.5$, ${\hslash \omega }_p=5.414 \mathrm{e}\mathrm{V}$, $\hslash {\gamma }_{\mathrm{p}}=0.010 \mathrm{e}\mathrm{V}$, $C_1=0.28$, ${\hslash \omega }_1=2.945 \mathrm{e}\mathrm{V}$, and $\hslash {\gamma }_1=2.706 \mathrm{e}\mathrm{V}$. Figure 1a gives the real part of the relative permittivity of Na as a function of the wavelength. Obviously, the real part of the relative permittivity of Na remains negative in the wavelength range from 1200 to 1500 nm, which indicates that Na can be treated as a plasmonic material at near-infrared wavelengths. Specifically, as the wavelength increases from 1200 to 1500 nm, the real part of the relative permittivity of Na decreases from $-26.56$ to $-41.94$. Figure 1b gives the imaginary part of the relative permittivity of Na as a function of the wavelength. Obviously, the imaginary part of the relative permittivity of Na remains at a low level in the wavelength range from 1200 to 1500 nm, which indicates that Na possesses ultra-low optical loss at near-infrared wavelengths. Specifically, as the wavelength increases from 1200 to 1500 nm, the imaginary part of the relative permittivity of Na increases from 0.37 to 0.59. According to the transfer matrix approach [52], we calculate the reflectance spectrum of a Na layer with a thickness of 25 nm at normal incidence, as shown in Figure 1c. The incident and exit media are both airs. As demonstrated, the reflectance of the Na layer remains at a high level in the wavelength range from 1200 to 1500 nm. Specifically, as the wavelength increases from 1200 to 1500 nm, the reflectance of the Na layer increases from 0.782 to 0.846. In other words, the Na layer can be treated as an optical mirror at near-infrared wavelengths.

It is known that Sb₂S₃ is a kind of lossless phase-change material at near-infrared wavelengths [49]. The refractive index of Sb₂S₃ can be flexibly tuned by its crystalline fraction [50,51]. By controlling the power density of a continuous-wave laser, the crystalline fraction can be changed from 0% to 100% [50]. According to the experimental data in Ref. [49], the permittivities of Sb₂S₃ under amorphous and crystalline phases in the wavelength range of interest are ${\epsilon }_{\mathrm{a}\mathrm{m}\mathrm{p}}={2.75}^2=7.5625$ and ${\epsilon }_{\mathrm{c}\mathrm{r}\mathrm{y}}={3.35}^2=11.2225$, respectively. In the intermediate phase, Sb₂S₃ can be regarded as a mixture of amorphous and crystalline molecules with different proportions. Hence, the relative permittivity of Sb₂S₃ under intermediate phase ${\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{t}}$ can be calculated by the effective medium approach, i.e., [50,51]:

$$\frac{{\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{t}}-1}{{\epsilon }_{\mathrm{i}\mathrm{n}\mathrm{t}}+2}=m\frac{{\epsilon }_{\mathrm{c}\mathrm{r}\mathrm{y}}-1}{{\epsilon }_{\mathrm{c}\mathrm{r}\mathrm{y}}+2}+(1-m)\frac{{\epsilon }_{\mathrm{a}\mathrm{m}\mathrm{p}}-1}{{\epsilon }_{\mathrm{a}\mathrm{m}\mathrm{p}}+2}$$

(2)

where $m$ denotes the crystalline fraction of Sb₂S₃. Figure 2 gives the relative permittivity of Sb₂S₃ as a function of the crystalline fraction. As the crystalline fraction increases from 0% to 100%, the relative permittivity of Sb₂S₃ can be flexibly tuned from 7.5625 to 11.2225. The tunability of the relative permittivity of Sb₂S₃ allows us to achieve a tunable near-infrared transparent band.

Then, we constructed a metal–dielectric–metal (MDM) cavity composed of two Na layers and an Sb₂S₃ layer, as schematically shown in Figure 3a. The MDM cavity can be treated as a near-infrared Fabry–Perot cavity. The thicknesses of the Na and Sb₂S₃ layers are selected to be $d_{\mathrm{M}}=25 \mathrm{n}\mathrm{m}$ and $d_{\mathrm{D}}=890 \mathrm{n}\mathrm{m}$, respectively. Initially, the crystalline fraction of Sb₂S₃ is set to be $m=0\%$. Figure 3b demonstrates the transmittance spectrum of the Fabry–Perot cavity MDM at normal incidence. A transmittance peak occurs at the wavelength of 1364.8 nm. The transmittance of the transmittance peak reaches 0.883. The quality (Q) factor of the transmittance peak can be calculated as $Q={\omega }_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}/∆\omega =1381.05 \mathrm{T}\mathrm{H}\mathrm{z}/42.66 \mathrm{T}\mathrm{H}\mathrm{z}=32.4$, where ${\omega }_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}$ denotes the angular frequency of the transmittance peak and $∆\omega$ denotes the full width at half-maximum. This transmittance peak originates from the Fabry–Perot resonance. The Fabry–Perot resonant condition can be expressed as:

$${\phi }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}={\phi }_{\mathrm{M},\mathrm{L}\mathrm{e}\mathrm{f}\mathrm{t}}+2{\phi }_{\mathrm{D}}+{\phi }_{\mathrm{M},\mathrm{R}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}=2p\pi \left(p=0, 1, 2, \dots \right)$$

(3)

where ${\phi }_{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}$ denotes the total phase, ${\phi }_{\mathrm{M},\mathrm{L}\mathrm{e}\mathrm{f}\mathrm{t}}$ (${\phi }_{\mathrm{M},\mathrm{R}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}$) denotes the reflection phase from the Sb₂S₃ layer to the left (right) Na layer, and $2{\phi }_{\mathrm{D}}$ represents the round-trip propagating phase of the Sb₂S₃ layer. Under normal incidence, the round-trip propagating phase of the Sb₂S₃ layer $2{\phi }_{\mathrm{D}}$ can be given by:

$$2{\phi }_{\mathrm{D}}=2n_{\mathrm{D}}{k}_0{d}_{\mathrm{D}}$$

(4)

where $k_0=2\pi /\lambda$ denotes the wave vector in vacuum. Figure 3c demonstrates the total phase as a function of the wavelength. Specifically, as the wavelength increases from 1200 to 1500 nm, the total phase decreases from 7.06π to 5.30π. At the wavelength of 1364.9 nm, the total phase equals 6π. In other words, the Fabry–Perot resonant condition is satisfied at the wavelength of 1364.9 nm, which slightly deviates from the wavelength of the transmittance peak. Hence, we can confirm that the transmittance peak originates from the Fabry–Perot resonance.

Next, we cascaded two Fabry–Perot cavities together to construct a multilayer MDMDM, i.e., (MD)²M, as schematically shown in Figure 4a. The thicknesses of the Na and Sb₂S₃ layers remain unchanged. Figure 4b demonstrates the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)²M at normal incidence. In this case, two transmittance peaks occur. The left transmittance peak occurs at wavelength ${ \lambda }_{\mathrm{L}}=1310.9 \mathrm{n}\mathrm{m}$ and the right one occurs at wavelength ${\lambda }_{\mathrm{R}}=1399.5 \mathrm{n}\mathrm{m}$. The transmittances of two transmittance peaks are 0.884 and 0.769, respectively. The Q factor of the left transmittance peak can be calculated as $Q_{\mathrm{L}}={\omega }_{\mathrm{L}}/∆{\omega }_{\mathrm{L}}=1437.95 \mathrm{T}\mathrm{H}\mathrm{z}/23.80 \mathrm{T}\mathrm{H}\mathrm{z}=60.4$. The Q factor of the right transmittance peak can be calculated as $Q_{\mathrm{R}}={\omega }_{\mathrm{R}}/∆{\omega }_{\mathrm{R}}=1346.87 \mathrm{T}\mathrm{H}\mathrm{z}/23.12 \mathrm{T}\mathrm{H}\mathrm{z}=58.3$. Owing to the mode coupling, the Q factors of the two transmittance peaks in the cascaded Fabry–Perot cavities (MD)²M are higher than the Q factor of the transmittance peak in the single Fabry–Perot cavity MDM.

Now, we explain the formation of two transmittance peaks from the perspective of the mode coupling, as schematically shown in Figure 4c. The resonant angular frequencies of two single Fabry–Perot cavities are denoted by ${\omega }_1$ and ${\omega }_2$. In our design, two single Fabry–Perot cavities are identical, i.e., ${\omega }_1={ \omega }_2=1381.05 \mathrm{T}\mathrm{H}\mathrm{z}$. The cascaded Fabry–Perot cavities (MD)²M can be treated as a system composed of two coupled oscillators. According to the coupled oscillator model [53,54,55,56,57,58,59,60], the Hamiltonian of the system can be determined by:

$$H=\left[\begin{array}{cc}{\omega }_1+i{\mathsf{\Gamma }}_1& {\mathsf{\Omega }}_{\mathrm{C}}\\ {\mathsf{\Omega }}_{\mathrm{C}}& {\omega }_2+i{\mathsf{\Gamma }}_2\end{array}\right]$$

(5)

where ${\omega }_1$ and ${\omega }_2$ denote the resonant angular frequencies of two single Fabry–Perot cavities, ${\mathsf{\Gamma }}_1$ and ${\mathsf{\Gamma }}_2$ denote the half widths at half-maximum of two single Fabry–Perot cavities, and ${\mathsf{\Omega }}_{\mathrm{C}}$ denotes the coupling strength between two Fabry–Perot modes. In our design, ${\omega }_1={\omega }_2={\omega }_0=1381.05 \mathrm{T}\mathrm{H}\mathrm{z}$ and ${\mathsf{\Gamma }}_1={\mathsf{\Gamma }}_2={\mathsf{\Gamma }}_0=21.33 \mathrm{T}\mathrm{H}\mathrm{z}$. Therefore, Equation (5) reduces to:

$$H=\left[\begin{array}{cc}{\omega }_1+i{\mathsf{\Gamma }}_1& {\mathsf{\Omega }}_{\mathrm{C}}\\ {\mathsf{\Omega }}_{\mathrm{C}}& {\omega }_2+i{\mathsf{\Gamma }}_2\end{array}\right]$$

(6)

Two eigen-angular frequencies can be given by:

$${\omega }_{\mathrm{L}}={\omega }_0+{\mathsf{\Omega }}_{\mathrm{C}}+i{\mathsf{\Gamma }}_0$$

(7)

$${\omega }_{\mathrm{R}}={\omega }_0-{\mathsf{\Omega }}_{\mathrm{C}}+i{\mathsf{\Gamma }}_0$$

(8)

Hence, the Rabi splitting can be calculated by [53,54,55,56,57,58,59,60]:

$$\hslash {\mathsf{\Omega }}_{\mathrm{R}\mathrm{a}\mathrm{b}\mathrm{i}}=\hslash \left({\omega }_{\mathrm{L}}-{\omega }_{\mathrm{R}}\right)=2\hslash {\mathsf{\Omega }}_{\mathrm{C}}$$

(9)

According to Equations (7) and (8), the angular frequency of the Fabry–Perot mode splits into two angular frequencies owing to the mode coupling. Hence, the transmittance peak splits into two transmittance peaks. Substituting ${\omega }_{\mathrm{L}}=1437.95 \mathrm{T}\mathrm{H}\mathrm{z}$ and ${\omega }_{\mathrm{R}}=1346.87 \mathrm{T}\mathrm{H}\mathrm{z}$ into Equation (9), the Rabi splitting can be obtained as ${\mathrm{\hslash }\mathsf{\Omega }}_{\mathrm{R}\mathrm{a}\mathrm{b}\mathrm{i}}=60.0 \mathrm{m}\mathrm{e}\mathrm{V}$. According to Equation (9), the two angular frequencies of two hybrid modes are symmetrical with respect to the angular frequency of the single Fabry–Perot mode. However, from Figure 3c, the two angular frequencies of two hybrid modes exhibit a weak asymmetry with respect to the angular frequency of the single Fabry–Perot mode. The underlying reason is that the Q factor of the single Fabry–Perot mode $Q=32.4$ is not very high due to the optical loss of the Na layers. Hence, the resonant strength of the Fabry–Perot mode is not too strong, giving rise to a weak inaccuracy of the coupled oscillator model.

To achieve a near-infrared transparent band, we cascaded four Fabry–Perot cavities together to construct a multilayer (MD)⁴M, as schematically shown in Figure 5a. Figure 5b demonstrates the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M at normal incidence. Now, four transmittance peaks occur owing to the coupling between five Fabry–Perot modes. Specifically, four transmittance peaks occur at the wavelengths of 1277.6, 1324.8, 1378.1, and 1420.8 nm. The bandwidth of the near-infrared transparent band is 143.2 nm. The average wavelength difference between the four transmittance peaks is 47.8 nm. The transmittances of the four transmittance peaks are 0.843, 0.812, 0.740, and 0.503. By cascading four Fabry–Perot cavities, we achieve a near-infrared transparent band. Compared with the reported transparent bands based on cascaded Fabry–Perot cavities [29,30,31,32,33,34], the transmittance of the transparent band in this work is higher.

Next, we changed the crystalline fraction of the Sb₂S₃ layers to investigate the tunability of the near-infrared transparent band. Figure 6a gives the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M at normal incidence for $m=50\%$. Four transmittance peaks occur at the wavelengths of 1401.3, 1453.2, 1511.6, and 1559.3 nm. The bandwidth of the near-infrared transparent band is 157.0 nm. The average wavelength difference between the four transmittance peaks is 52.3 nm. The transmittances of the four transmittance peaks are 0.828, 0.794, 0.714, and 0.466. Figure 6b gives the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M at normal incidence for $m=100\%$. Four transmittance peaks occur at the wavelengths of 1556.3, 1613.7, 1678.8, and 1730.5 nm. The bandwidth of the near-infrared transparent band is 174.2 nm. The average wavelength difference between the four transmittance peaks is 58.1 nm. The transmittances of the four transmittance peaks are 0.805, 0.766, 0.679, and 0.418. The near-infrared transparent band can be flexibly tuned by the crystalline fraction of the Sb₂S₃ layers. As the crystalline fraction of the Sb₂S₃ layers changes from 0% to 100%, the short-wavelength edge of the near-infrared transparent band can be tuned from 1277.6 to 1556.3 nm and the long-wavelength edge of the near-infrared transparent band can be tuned from 1420.8 to 1730.5 nm.

Finally, we discuss the fabrication scheme of the proposed cascaded Fabry–Perot cavities. The proposed cascaded Fabry–Perot cavities are multilayers composed of Na and Sb₂S₃ films. It is known that both Na films [47] and Sb₂S₃ films [61] can be fabricated by the spin-coating technique. In addition, the crystalline fraction of Sb₂S₃ film can be flexibly controlled by the power density of a continuous-wave laser [50]. Therefore, the tunable near-infrared transparent band would be achieved using the spin-coating and continuous-wave lasing techniques.

3. Effects of Layer Thickness and Incident Angle on Near-Infrared Transparent Bands and Mode Coupling

According to the Fabry–Perot resonant condition (Equation (3)) and the coupled oscillators model, both the layer thickness and incident angle will affect the near-infrared transparent band and the mode coupling. In this section, the crystalline fraction of the Sb₂S₃ layers is kept as $m=0\%$. Now, we investigate the effect of the thickness of the Na layer on the near-infrared transparent band and the mode coupling in Figure 7. Figure 7a gives the dependence of the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M at normal incidence on the thickness of the Na layer. The thickness of the Na layer is changed from 15 to 35 nm while the thickness of the Sb₂S₃ layer and the incident angle are the same as those in Section 2. As the thickness of the Na layer increases, the four transmittance peaks become closer. In other words, the bandwidth of the transparent band becomes smaller. Specifically, when the thickness of the Na layer is 15 nm, the transparent band ranges from 1276.9 to 1470.0 nm. The bandwidth of the transparent band is 193.1 nm. When the thickness of the Na layer increases to 35 nm, the transparent band ranges from 1282.4 to 1389.5 nm. The bandwidth of the transparent band is 107.1 nm. Figure 7b gives the dependence of the average wavelength difference between the four transmittance peaks $∆\lambda$ on the thickness of the Na layer. As the thickness of the Na layer increases from 15 to 35 nm, the average wavelength difference between the four transmittance peaks decreases from 64.3 to 35.7 nm. The underlying reason can be explained as follows. As the thickness of the Na layer increases, the reflectance of the Na layer increases. Consequently, the electric field of the Fabry–Perot cavity becomes more localized, leading to a smaller coupling strength between different Fabry–Perot modes. According to Equation (9), the Rabi splitting becomes smaller. In addition, we give the dependences of the transmittances of the four transmittance peaks on the thickness of the Na layer, as shown in Figure 7c. As the thickness of the Na layer increases from 15 to 35 nm, the transmittances of the four transmittance peaks decrease. Specifically, the transmittance of transmittance peak 1 decreases from 0.912 to 0.743. The transmittance of transmittance peak 2 decreases from 0.893 to 0.712. The transmittance of transmittance peak 3 decreases from 0.861 to 0.600. The transmittance of transmittance peak 4 decreases from 0.743 to 0.298.

Then, we investigated the effect of the thickness of the Sb₂S₃ layer on the near-infrared transparent band and mode coupling, as shown in Figure 8. Figure 8a gives the dependence of the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M at normal incidence on the thickness of the Sb₂S₃ layer. The thickness of the Sb₂S₃ layer is changed from 850 to 950 nm while the thickness of the Na layer and the incident angle are the same as those in Section 2. As the thickness of the Sb₂S₃ layer increases, five transmittance peaks shift towards longer wavelengths (i.e., redshift). Specifically, when the thickness of the Sb₂S₃ layer is 850 nm, the transparent band ranges from 1222.2 to 1362.0 nm. The bandwidth of the transparent band is 139.8 nm. When the thickness of the Sb₂S₃ layer is 950 nm, the transparent band ranges from 1360.6 to 1508.6 nm. The bandwidth of the transparent band is 148.0 nm. The underlying reason can be explained as follows. As the thickness of the Sb₂S₃ layer increases, the round-trip propagating phase of the Sb₂S₃ layer for a fixed wavelength increases. As a result, the wavelength satisfying the Fabry–Perot resonant condition (Equation (3)) becomes larger, giving rise to the redshift of the transparent band. Figure 8b gives the dependence of the average wavelength difference between four transmittance peaks $∆\lambda$ on the thickness of the Sb₂S₃ layer. As the thickness of the Sb₂S₃ layer increases from 850 to 950 nm, the average wavelength difference between the four transmittance peaks increases from 46.6 to 49.4 nm. In addition, we give the dependences of the transmittances of the four transmittance peaks on the thickness of the Sb₂S₃ layer, as shown in Figure 8c. As the thickness of the Sb₂S₃ layer increases from 850 to 950 nm, the transmittances of the four transmittance peaks slightly decrease. Specifically, the transmittance of transmittance peak 1 decreases from 0.848 to 0.834. The transmittance of transmittance peak 2 decreases from 0.820 to 0.799. The transmittance of transmittance peak 3 decreases from 0.753 to 0.718. The transmittance of transmittance peak 4 decreases from 0.527 to 0.466.

Finally, we investigated the effect of the incident angle on the near-infrared transparent band and the coupling. The thicknesses of the Na and Sb₂S₃ layers are the same as those in Section 2. At normal incidence, the transmittance spectra of the cascaded Fabry–Perot cavities (MD)⁴M under transverse magnetic (TM) and transverse electric (TE) polarizations are identical due to the axial symmetry. Nevertheless, under oblique incidence, the transmittance spectra of the cascaded Fabry–Perot cavities (MD)⁴M under TM and TE polarizations are different since the axial symmetry is broken. Hence, we should discuss TM and TE polarizations separately. Figure 9a gives the dependence of the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M on the incident angle under TM polarization. As the incident angle increases, four transmittance peaks shift towards shorter wavelengths (i.e., blueshift) under TM polarization. Specifically, when the incident angle is 0°, the transparent band ranges from 1277.6 to 1420.8 nm. The bandwidth of the transparent band is 143.2 nm. When the incident angle increases to ~90°, the transparent band under TM polarization ranges from 1182.9 to 1326.1 nm. The bandwidth of the transparent band is 143.2 nm. The underlying reason can be explained as follows. As the incident angle increases, the round-trip propagating phase of the Sb₂S₃ layer for a fixed wavelength decreases. As a result, the wavelength satisfying the Fabry–Perot resonant condition (Equation (3)) becomes smaller, giving rise to the blueshift of the transparent band. Figure 9b gives the dependence of the average wavelength difference between four transmittance peaks $∆\lambda$ on the incident angle under TM polarization. As the incident angle increases from 0° to ~90°, the average wavelength difference between the four transmittance peaks first decreases from 47.7 to 45.5 nm and then increases to 47.8 nm. In addition, we give the dependences of the transmittances of the four transmittance peaks on the incident angle under TM polarization, as shown in Figure 9c. As the incident angle increases from 0° to ~90°, the transmittances of the four transmittance peaks first increase and then decrease. Specifically, the transmittance of transmittance peak 1 first increases from 0.843 to 0.962, and then decreases to 0. The transmittance of transmittance peak 2 first increases from 0.812 to 0.936, and then decreases to 0. The transmittance of transmittance peak 3 first increases from 0.740 to 0.887, and then decreases to 0. The transmittance of transmittance peak 4 first increases from 0.503 to 0.734, and then decreases to 0.

Figure 10a gives the dependence of the transmittance spectrum of the cascaded Fabry–Perot cavities (MD)⁴M on the incident angle under TE polarization. As the incident angle increases, four transmittance peaks shift towards shorter wavelengths (i.e., blueshift) under TE polarization. Specifically, when the incident angle is 0°, the transparent band ranges from 1277.6 to 1420.8 nm. The bandwidth of the transparent band is 143.2 nm. When the incident angle increases to ~90°, the transparent band under TE polarization ranges from 1189.8 to 1322.5 nm. The bandwidth of the transparent band is 132.7 nm. The underlying reason can be explained as follows. As the incident angle increases, the round-trip propagating phase of the Sb₂S₃ layer for a fixed wavelength decreases. As a result, the wavelength satisfying the Fabry–Perot resonant condition (Equation (3)) becomes smaller, giving rise to the blueshift of the transparent band. Figure 10b gives the dependence of the average wavelength difference between four transmittance peaks $∆\lambda$ on the incident angle under TE polarization. As the incident angle increases from 0° to ~90°, the average wavelength difference between the four transmittance peaks decreases from 47.7 to 44.2 nm. In addition, we give the dependences of the transmittances of four transmittance peaks on the incident angle under TE polarization, as shown in Figure 10c. As the incident angle increases from 0° to ~90°, the transmittances of the four transmittance peaks gradually decrease. Specifically, the transmittance of transmittance peak 1 decreases from 0.843 to 0. The transmittance of transmittance peak 2 decreases from 0.812 to 0. The transmittance of transmittance peak 3 decreases from 0.740 to 0. The transmittance of transmittance peak 4 decreases from 0.503 to 0.

4. Conclusions

In summary, we realized a near-infrared transparent band by cascading four Na/Sb₂S₃/Na Fabry–Perot cavities. The near-infrared transparent band originates from the mode coupling between four Fabry–Perot modes. By changing the crystalline fraction of the Sb₂S₃ layers, the near-infrared transparent band can be flexibly tuned. As the crystalline fraction of the Sb₂S₃ layers changes from 0% to 100%, the short-wavelength edge of the near-infrared transparent band can be tuned from 1277.6 to 1556.3 nm and the long-wavelength edge of the near-infrared transparent band can be tuned from 1420.8 to 1730.5 nm. In addition, we studied the effects of layer thickness and incident angle on the near-infrared transparent band and mode coupling. As the thickness of the Na layer increases, the bandwidth of the near-infrared transparent band becomes smaller. As the thickness of the Sb₂S₃ layer increases, the near-infrared transparent band shifts towards longer wavelengths. As the incident angle increases, the near-infrared transparent band shifts towards shorter wavelengths. These results not only provide a viable route to achieving tunable near-infrared transparent bands but also possess potential applications in high-performance display, filtering, and sensing.