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Abstract
In this paper, we investigate the singular multi-wavelength and multi-waveband transparencies generated by $\mathcal {PT}$-symmetric dumbbell optical waveguide networks composed of two materials, and obtain the number regularity for the transparency wavelengths of one-unit-cell system and the general relationships for the transmission and reflection coefficients of multi-unit-cell systems. Consequently, three types of exact transparencies produced by multi-unit-cell systems are found based on the aforementioned formulas: (i)exact multi-wavelength unidirectional or bidirectional transparency as the same as those of one-unit-cell system; (ii)exact multi-wavelength bidirectional transparency at which one-unit-cell system cannot produce exact transparency, generated by adjusting the number of unit cells; (iii)exact multi-wavelength bidirectional transparency at which one-unit-cell system produces exact transparency, also generated by adjusting the number of unit cells. It provides theoretical foundations for developing highly sensitive and multi-wavelength optical filters. On the other hand, we also discover that multi-unit-cell systems can create approximate multi-waveband bidirectional transparencies by adjusting the number of unit cells, which provides scientific support for developing high-performance optical stealth devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In 1998, Bender et al. [1] first proposed the concept of parity-time-symmetry ($\mathcal {PT}$- symmetry) in the field of quantum mechanics. Subsequently, El-Ganainy et al. [2] introduced the concept of $\mathcal {PT}$-symmetry into the field of optics by using a refractive index $n(x)={{n}^{*}}(-x)$ distribution. This special refractive index distribution has attracted widespread attention [3–28]. It has been found that optical systems that satisfy this refractive index distribution can produce many unique optical phenomena, such as double refraction [3], coherent perfect absorbers and lasers (CPA lasers) [7,10,22], invisibility [9,21], transparency [21,23,25–27], and so on.
The study of non-reflective transparency [21,23,25–27,29,30] is currently a highly focused area of interest. Many reports have been published on the transparency of $\mathcal {PT}$-symmetric optical waveguide networks [21,23,25–27]. However, these studies typically focus only on the transparency of a system with a specific number of cells at a single wavelength. There have been no reports on whether $\mathcal {PT}$-symmetric systems can produce transparency at multi-wavelength, or whether they can produce transparency at a single or even multiple wavebands.
To address these two issues, we designed an interesting one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network with two materials and studied its transparency in depth. While network structures similar to this configuration have been reported [22], the property and focus of our study differ from prior research. We have found that there exists a certain regularity in the wavelengths at which one-unit-cell system produces transparency. For multi-unit-cell systems, we have derived the general relationships for the transmittance and reflectance coefficients using the generalized eigenfunction method [31,32] and transfer matrix method [7,11,17,22], and have discovered that by adjusting the one-unit-cell structure and/or the number of unit cells, the $\mathcal {PT}$-symmetric optical waveguide network can produce exact unidirectional or bidirectional transparency at multi-wavelength, which provides a theoretical basis for the development of high-sensitivity multi-wavelength optical filters. In addition, by adjusting the number of unit cells, the $\mathcal {PT}$-symmetric optical waveguide network can also produce approximate bidirectional transparency at a single or even multiple wavebands, with the reflection ${{R}_{N}}\le {{10}^{-8}}$ and transmission ${{T}_{N}}\to 1$. It provides a scientific basis for the development of high-performance optical stealth devices.Finally, we also investigated the robustness of the one-unit-cell and found that it exhibits good tolerance to small fluctuations in inter-arm length and material refractive index.
The paper is organized as follows: In Section 2, we introduce our designed one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network with two materials, as well as the generalized eigenfunction method and transfer matrix method used to study singular transparency. In Section 3, we first study the exact transparency of one-unit-cell system, then study the exact transparency and approximate transparency of multi-unit-cell systems, and finally study the robustness of the one-unit-cell system. Section 4 is the conclusion of this paper.
2. Model and theory
2.1 Model
The one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network with two materials was designed in this paper. The transparency of this network in multi-frequency and multi-frequency bands was investigated. The schematic diagram of the structure is shown in Fig. 1, where the black, red, blue, and green solid lines represent one-dimensional waveguide lines with refractive indices for ${{n}_{0}}=2$, ${{n}_{1}}=a-0.2i$, ${{n}_{2}}=a+0.2i$ and ${{n}_{l}}=2$, respectively. The length for the upper and lower arms of the ring is $d$, and the length for the inter-ring arm is $l$. The inter-ring arm length $l=d$ and the real part of the refractive index for the ring material $a=2$ are the main focus of this study. ${{E}_{\text {I}}}$, ${{E}_{\text {R}}}$ and ${{E}_{\text {T}}}$ represent the electric field intensity for the incident, reflected, and transmitted electromagnetic waves, respectively. 1, 2,…, $N+4$ represent the node numbers of the network, where nodes 1-4 represent one unit cell. It can be seen that the refractive index distribution of each waveguide and unit cell in the network satisfies $n\left ( x \right )={{n}^{*}}\left ( -x \right )$, so the waveguides, unit cells and the entire system of the network are all $\mathcal {PT}$-symmetry.
Fig. 1. Schematic structure of a one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network with two materials.
2.2 Generalized eigenfunction method
The transmission and reflection for the one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network with two materials can be calculated using the generalized eigenfunction method [31,32], as shown in Fig. 1. The network equation for a one-dimensional $\mathcal {PT}$-symmetric optical waveguide network with two materials is given by:
The ${{k}_{y}}$ and ${{k}_{x}}$ represent the wave vector of the electromagnetic wave propagating in the waveguide lines directly or indirectly connected to node $j(j=1,2,3\cdots )$, ${{d}_{y}}$ is the length of the sub-waveguide line directly connected to node $j$, and ${{d}_{ji}}$ is the length for the waveguide line between nodes $j$ and $i$. The ${{\psi }_{j}}$ and ${{\psi }_{i}}$ represent the wave function at nodes $j$ and $i$, respectively. By using Eq. (1), the transmittance and reflectance coefficients for the one-dimensional $\mathcal {PT}$-symmetric dumbbell unit cell network with two materials can be calculated. Combined with the transfer matrix method introduced in Section 2.3, the transmission and reflection of multi-unit-cell systems can be calculated easily and quickly.
2.3 Transfer matrix method
In this section, we use the transfer matrix method to calculate the transmittance and reflectance coefficients for the one-dimensional $\mathcal {PT}$-symmetric dumbbell unit cell network with two materials, for an arbitrary number of unit cells ($N\ge 2$). The transfer matrix of a $\mathcal {PT}$-symmetric unit cell is known from Refs. [7,11,17,22]:
The relationship between the Bloch phase $\phi$ for the system and the eigenvalues ${{\lambda }_{1}}$ and ${{\lambda }_{2}}$ of the transfer matrix ${{M}_{1}}$ of one-unit-cell system is as follows:
By substituting Eq. (3) into Eq. (4), the relationship between the total transmission matrix of multi-unit-cell systems and the reflection coefficients, transmission coefficients, and Bloch phase of one-unit-cell system can be obtained:
3. Singular transparency
This Section mainly investigates the exact transparency generated by one-unit-cell system, as well as the exact and approximate transparency generated by multi-unit-cell systems.
3.1 Exact transparency of one-unit-cell system
Using the generalized eigenfunction method, we have calculated the transmission and reflection spectra of a one-dimensional $\mathcal {PT}$-symmetric dumbbell unit-cell network with two materials shown in Fig. 1. The results show that the system exhibits unidirectional or bidirectional transparency in multi-frequency. We have found that the number of its transparency:
As an example of one-unit-cell system that generates exact transparency, we choose a system with $l=d$ and $a=2$, and calculate its transmission and reflection spectra, as shown in Fig. 2. It is evident that the system exhibits exact leftward transparency at frequencies ${{I}_{\mathrm {L}}}$, ${{II}_{\mathrm {L}}}, \ldots, {{XI}_{\mathrm {L}}}$ and exact rightward transparency at frequencies ${{I}_{\mathrm {R}}}$, ${{II}_{\mathrm {R}}}, \ldots, {{XI}_{\mathrm {R}}}$ . The number of exact leftward and rightward transparent frequencies is in complete agreement with the calculation in Eq. (8).
Fig. 2. The transmission and reflection spectra for a one-dimensional $\mathcal {PT}$-symmetric dumbbell unit cell photonic waveguide network with two materials that exhibits exact transparency, where $i=\mathrm {L,R}$.
For exact frequency positions of exact transparency in $\mathcal {PT}$-symmetric systems, they can be confirmed by the abrupt change in reflected light phase [21,23,25–27]. In this paper, we utilize this method to confirm the exact frequency positions and directions of exact transparency by the abrupt change in reflected light phase $\pi$. Based on the properties of $\mathcal {PT}$-symmetric systems [10,22] and the relationship between transmission and reflection coefficients in the transfer matrix method, we can derive that:
where ${{r}_{i-1}}=\sqrt {{{R}_{i-1}}}{{e}^{i{{\phi }_{{{r}_{i}}-1}}}},i=\mathrm {L,R}$, it can be seen from Eq. (9) that the phase difference between the left and right reflected light can only be $\pi$ or 0. (i) When the phase difference between the left and right reflected light undergoes an abrupt change of $\pi$, the transmission ${{T}_{1}}\to 1$, and the reflection ${{R}_{i-1}}\to 0$. This results in exact unidirectional transparency, with the direction of transparency being the same as the direction of the reflected light with the phase change. (ii) When the phase difference between the left and right reflected light remains unchanged, but each phase undergoes an abrupt change of $\pi$, the transmission ${{T}_{1}}\to 1$, and the reflection ${{R}_{\mathrm {L}-1}}({{R}_{\mathrm {R}-1}})\to 0$. This results in exact bidirectional transparency.As examples of exact unidirectional and bidirectional transparency, the frequencies for ${{III}_{\mathrm {L}}}$ and ${{V}_{i}}\text { }\left ( i=\text {L,R} \right )$ in Fig. 2 are ${{\omega }_{1}}=0.449921\pi c/d$ and ${{\omega }_{2}}=0.885635\pi c/d$, respectively. Figure 3 shows the corresponding phase spectra of the reflected light. From Fig. 3(a), it can be seen that the system undergoes an abrupt phase change of $\pi$ in the left reflected light at ${{\omega }_{1}}=0.449921\pi c/d$, while the phase of the right reflected light remains unchanged. Moreover, as shown in Fig. 2, the system exhibits exact unidirectional transparency with ${{T}_{1}}\approx 1$, ${{R}_{\mathrm {R-1}}}\ne 0$ and ${{R}_{\mathrm {L-1}}}<{{10}^{-9}}$ at this frequency. From Fig. 3(b), it can be seen that at ${{\omega }_{2}}=0.885635\pi c/d$, both the left and right reflected light undergo a phase change of $\pi$. Furthermore, as shown in Fig. 2, the system exhibits exact bidirectional transparency with ${{T}_{1}}\approx 1$, ${{R}_{\mathrm {R-1}}}<{{10}^{-9}}$ and ${{R}_{\mathrm {L-1}}}<{{10}^{-9}}$ at this frequency.
Fig. 3. Relationship between transparency property and an abrupt phase change of $\pi$ in the reflected light at the frequency where exact transparency occurs in one-unit-cell system. (a) At ${{{\omega }_{1}}=0.449921\pi c/d}$, the system yields exact leftward transparency. (b) At ${{{\omega }_{2}}=0.885635\pi c/d}$, the system yields exact bidirectional transparency.
Therefore, the relationship between the abrupt phase change in the reflected light and the exact transparency can be utilized to verify the transparency of the device in different frequencies and directions. Moreover, by using Eq. (8), the exact transparency for the one-dimensional $\mathcal {PT}$-symmetric dumbbell unit cell waveguide network studied in this paper can be flexibly adjusted, providing a scientific basis for the fabrication for multi-wavelength and multi-property transparency devices.
3.2 Exact transparency of multi-unit-cell systems
3.2.1 General relationship between transmittance and reflectance coefficients
Exact transparency of one-unit-cell system was studied in Section 3.1, and the regularity of the exact transparency of one-unit-cell system was derived. Although the designed optical waveguide network in this paper is of finite periodicity, with an increase in the number of periods, there is a close connection between the transparency of multi-unit-cell systems and that of one-unit-cell system. Moreover, new transparency features not present in one-unit-cell system can be observed in multi-unit-cell systems. Therefore, in Section 3.2, we investigated in detail the exact transparency of multi-unit-cell systems, and using the transfer matrix method introduced in Section 2.3, we derived the following general relationship between the transmission and reflection coefficients of multi-unit-cell systems:
To verify the accuracy of Eq. (10) and demonstrate the consistency for the generalized eigenfunction method and transfer matrix method in calculating the optical properties of the designed network, we take a four-unit-cell system as an example and calculate its transmission and reflection using both the generalized eigenfunction method combined with the transfer matrix method and the pure generalized eigenfunction method separately. The results are shown in Fig. 4. It can be observed that the transmission and reflection curves of the three types, represented by the blue solid line ${{R}_{{\mathrm {L}}-4}}$ and the red dashed line $R_{{\mathrm {L}}-4}^{'}$, the green solid line ${{R}_{\mathrm {R-4}}}$ and the orange dashed line $R_{{\mathrm {R}}-4}^{'}$, and the cyan solid line ${{T}_{4}}$ and the black dashed line ${{T}_{4}}^{'}$, completely overlap with each other across the entire frequency range in Fig. 4. This demonstrates the complete consistency of the results obtained from the generalized eigenfunction method combined with the transfer matrix method and the pure generalized eigenfunction method in calculating the transmission and reflection of the system, indicating that these two methods are equivalent. As stated in Refs. [32,34], the generalized eigenfunction method is an accurate approach to calculate the transmission and reflection of the optical waveguide network. Thus, the results obtained from the generalized eigenfunction method combined with the transfer matrix method are correct. Furthermore, since Eq. (10) was derived from the transfer matrix method, it confirms that the transmission and reflection coefficients of the system calculated using Eq. (10) are also correct.
Fig. 4. Transmission and reflection spectra for a one-dimensional two-materials $\mathcal {PT}$-symmetric dumbbell four-unit-cell waveguide network, where solid and dashed lines represent the results calculated using the generalized eigenfunction method combined with the transfer matrix method and the pure generalized eigenfunction method, respectively.
From Eq. (10), it is evident that in order to ensure that the transmission and reflection coefficients of the multi-unit-cell systems are meaningful, it is essential to guarantee that the denominators of the Eq. (10) are non-zero. Therefore, the following constraint condition must be satisfied:
Additionally, it is well known that for $\mathcal {PT}$-symmetric systems, there exists a relationship between the transmission and reflection, as shown in Refs. [10,11]:
As shown in Eq. (12), when ${{R}_{\mathrm {L(R)}}}\to 0$, must exit $T\to 1$, indicating the system exhibits exact transparency. Therefore, based on Eq. (10)–(12), it can be concluded that multi-unit-cell systems can exhibit three types of exact transparencies: (i) when one-unit-cell system exhibits exact transparency at a certain frequency and $\sin N\phi \ne 0$, the multi-unit-cell systems exhibit the same exact transparency as the one-unit-cell system; (ii) when one-unit-cell system does not exhibit exact transparency at a certain frequency and $\sin N\phi \to 0$, the multi-cell-systems exhibit exact bidirectional transparency; (iii) when one-unit-cell system exhibits exact transparency at a certain frequency and $\sin N\phi \to 0$, the multi-unit-cell systems exhibit pure exact bidirectional transparency.
3.2.2 First type of exact transparency
Based on Eq. (9), (10), and (12), it can be deduced that, for any system with multiple unit cells, Eq. (11) is valid at the frequency where exact transparency occurs in one-unit-cell system. Therefore, at the frequency where exact transparency occurs in one-unit-cell system and $\sin N\phi \ne 0$, multi-unit-cell systems will exhibit exact transparency with the same characteristics as the one-unit-cell system.
As an example of multi-unit-cell systems that generates the first type of exact unidirectional and bidirectional transparency, the one-unit-cell system in Fig. 2 produces exact leftward and bidirectional transparency at ${{\omega }_{1}}=0.449921\pi c/d$ and ${{\omega }_{2}}=0.885635\pi c/d$, respectively. Figure 5 shows the relationship between the number of unit cells and the reflectivity of the system at these two frequencies, where the number of unit cells satisfies $\sin N\phi \ne 0$.
Fig. 5. Relationship between the number of unit cells and the reflection at the frequency position of exact transparency in one-unit-cell system. (a) Property diagram of exact leftward transparency at ${{\omega }_{1}}=0.449921\pi c/d$. (b) Property diagram of exact bidirectional transparency at ${{\omega }_{2}}=0.885635\pi c/d$.
Based on Fig. 5(a), it can be observed that as the number of unit cells $N$ increases at ${{\omega }_{1}}=0.449921\pi c/d$, the left reflection ${{R}_{\mathrm {L}-N}}\le {{10}^{-9}}$ while the right reflection ${{R}_{\mathrm {R}-N}}\ge {{10}^{-6}}$. In addition, according to Eq. (12), the transmission ${{T}_{N}}\approx 1$ at this frequency. Therefore, the system produces the same exact unidirectional transparency as the one-unit-cell system shown in Fig. 2. Moreover, Fig. 5(b) shows that at ${{\omega }_{2}}=0.885635\pi c/d$, as $N$ increases, ${{R}_{\mathrm {L}-N}}\le {{10}^{-9}}$ and ${{R}_{\mathrm {R}-N}}\le {{10}^{-9}}$, In addition, according to Eq. (12), ${{T}_{N}}\approx 1$. Thus, the system generates the same exact bidirectional transparency as the one-unit-cell system in Fig. 2.
Furthermore, if systems with different unit cell numbers always exhibit exact transparency at a certain frequency, then the one-unit-cell system will also exhibit exact transparency at that frequency. Therefore, when exact transparency at a specific frequency is desired, it is possible to fabricate a network with one unit cell instead of a complex network with multiple unit cells. This approach can effectively improve measurement precision and minimize fabrication costs.
3.2.3 Second type of exact transparency
For the second type of exact transparency in multi-unit-cell systems, we can adjust the number of unit cells $N$ to make $\sin N\phi$ strictly approach 0. At this point, according to Eq. (10) and (12), the $N$ unit cell system will exhibit exact bidirectional transparency at the same position where the one-unit-cell system does not exhibit exact transparency. Furthermore, the number of unit cells $N$ must satisfy the following relationship:
where the determination of the unit cell number $N$ is related to the Bloch phase $\phi$, and only when the Bloch phase $\phi$ is a pure real number, the unit cell number $N$ calculated by Eq. (13) is meaningful. Therefore, Eq. (11) can be further simplified as follows:At the frequencies where the reflection coefficient of one-unit-cell system ${{r}_{{\mathrm {L}}-1}}({{r}_{{\mathrm {R}}-1}})\ne 0$ and the Bloch phase $\phi$ is a pure real number, we can use Eq. (10) to show that the variation of the reflection of multi-unit-cell systems is mainly caused by the number of cells $N$. Therefore, at these frequencies, the reflection of the multi-unit-cell systems changes synchronously with the number of cells $N$, and the system will exhibit exact bidirectional transparency at these two frequency positions when $N$ satisfies Eq. (13).
At the frequencies ${{\omega }_{5}}$ and ${{\omega }_{6}}$ where the reflection coefficient ${{r}_{{\mathrm {L}}-1}}({{r}_{{\mathrm {R}}-1}})\ne 0$ in one-unit-cell system and the pure real Bloch phases ${{\phi }_{5}}\ne {{\phi }_{6}}$. According to Eq. (13), the system satisfies the following relation when the number of cells $N$:
As an example of multi-unit-cell systems producing the second exact transparency, Fig. 6 shows the relation between the number of cells and reflection at the frequencies ${{\omega }_{3}}=0.23623283\pi c/d$ and ${{\omega }_{4}}=1.09661550\pi c/d$ where the one-unit-cell system does not produce exact transparency when the pure real Bloch phase is the same, and at the frequencies ${{\omega }_{5}}=1.23590062\pi c/d$ and ${{\omega }_{6}}=1.75438769\pi c/d$ where the one-unit-cell system does not produce exact transparency when the pure real Bloch phase is different. From Fig. 6(a), it can be observed that when the pure real Bloch phase $\phi$ is the same, the red (blue) triangular dotted lines represent the variation of the number of cells $N$ and the reflection $R_{\mathrm {L-}N}^{'}(R_{\mathrm {R-}N}^{'})$ at the frequency ${{\omega }_{3}}=0.23623283\pi c/d$, while the green (black) square dotted lines represent the variation of the number of cells $N$ and the reflection $R_{\mathrm {L-}N}^{'}(R_{\mathrm {R-}N}^{'})$ at the frequency ${{\omega }_{4}}=1.09661550\pi c/d$. These two types of curves change synchronously. When the number of cells $N=58$, $\sin N\phi \approx 0$ can be calculated from Eq. (13), and the multi-unit-cell systems will exhibit the second type of exact transparency at these two frequency positions. It is also evident from Fig. 6(a) that the left and right reflection of the system at these two frequency positions are less than ${{10}^{-9}}$. Additionally, from Eq. (12), it can be concluded that the transmission ${{T}_{58}}\approx 1$ at these positions. Therefore, the system exhibits exact bidirectional transparency at these two frequency positions. From Fig. 6(b) and 6(c), it can be observed that when the pure real Bloch phase $\phi$ is not the same, the variation in the number of cells $N$ and the reflectance curve at ${{\omega }_{5}}=1.23590062\pi c/d$ and ${{\omega }_{6}}=1.75438769\pi c/d$ is not synchronized. When the number of cells $N=86$, using Eq. (13) and (15), $sin N{{\phi }_{5}}\approx \sin N{{\phi }_{6}}\approx 0$ can be calculated, and the multi-unit-cell systems produce the second type of exact transparency at these two frequencies. From Fig. 6(b)and 6(c), it can also be observed that the left and right reflection of the multi-unit-cell systems at these two frequencies are less than ${{10}^{-9}}$. Furthermore, according to Eq. (12), ${{T}_{86}}\approx 1$ at these frequencies. Therefore, the system exhibits exact bidirectional transparency at these two frequency positions.
Fig. 6. Relationship between cell number and reflection at frequencies where one-unit-cell system do not exhibit exact transparency. (a) Red (blue) triangle dashed line and green (black) square dashed line represent the reflection of the multi-unit-cell systems at ${{\omega }_{3}}=0.23623283\pi c/d$ and ${{\omega }_{4}}=1.09661550\pi c/d$, respectively. (b) Reflection of the multi-unit-cell systems at ${{\omega }_{5}}=1.23590062\pi c/d$. (c) Reflection of the multi-unit-cell systems at ${{\omega }_{6}}=1.75438769\pi c/d$.
Through the study of the second type of exact transparency in multi-unit-cell systems, we have discovered that by adjusting the number of cells in the system, exact bidirectional transparency can be achieved at frequencies where the one-unit-cell system cannot exhibit exact transparency. This finding provides us with more options for developing bidirectional transparent devices.
3.2.4 Third type of exact transparency
In our study of the third type of exact transparency in the multi-unit-cell systems, we discovered two interesting cases: (i) At the frequency where the one-unit-cell system exhibits exact unidirectional transparency, by adjusting the number of cells $N$ to satisfy $\sin N\phi \to 0$, an $N$-cell system can achieve exact bidirectional transparency at that frequency; (ii) At the frequency where the one-unit-cell system exhibits exact bidirectional transparency, regardless of the number of cells, the multi-unit-cell systems will always exhibit exact bidirectional transparency at that frequency.
As an example of the multi-unit-cell systems that generates the third type of exact transparency, Fig. 2 shows that the one-unit-cell system exhibits exact unidirectional and bidirectional transparency at frequencies ${{\omega }_{1}}=0.449921\pi c/d$ and ${{\omega }_{2}}=0.885635\pi c/d$, respectively. Figure 7 presents the relationship between the number of cells and the reflection of the multi-unit-cell systems at these two frequencies. Fig. 7(a) shows that when $\sin N\phi \ne 0$, the left reflection ${{R}_{\mathrm {L}-N}}\le {{10}^{-9}}$, and the right reflection ${{R}_{\mathrm {R}-N}}\ge {{10}^{-6}}$ as the number of cells $N$ increases at frequency ${{\omega }_{1}}=0.449921\pi c/d$. In addition, according to Eq. (12), the transmission ${{T}_{N}}\approx 1$ in this case. Therefore, the system exhibits the same exact unidirectional transparency as the one-unit-cell system at this frequency position. When the number of cells is $N=3127$, $\sin N\phi \approx 0$, Fig. 7(a) shows that ${{R}_{\mathrm {L}-3127}}\le {{10}^{-9}}$ and ${{R}_{\mathrm {R}-3127}}\le {{10}^{-9}}$. Moreover, from Eq. (12), we know that ${{T}_{3127}}\approx 1$ in this case. Therefore, the system exhibits exact bidirectional transparency at this frequency. Fig. 7(b) shows that when $\sin N\phi \ne 0$, ${{R}_{\mathrm {L}-N}}\le {{10}^{-9}}$ and ${{R}_{\mathrm {R}-N}}\le {{10}^{-9}}$ as the number of cells $N$ increases at frequency ${{\omega }_{2}}=0.885635\pi c/d$. In addition, according to Eq. (12), the transmission ${{T}_{N}}\approx 1$ in this case. Therefore, the system exhibits exact bidirectional transparency at this frequency, which is the same as the one-unit-cell system shown in Fig. 2. When the number of unit cells is $N=188$, $\sin N\phi \approx 0$, Fig. 7(b) shows that ${{R}_{\mathrm {L}-188}}\le {{10}^{-9}}$ and ${{R}_{\mathrm {R}-188}}\le {{10}^{-9}}$, and Eq. (12) confirms that ${{T}_{188}}\approx 1$. Thus, the system exhibits exact bidirectional transparency at this frequency, consistent with the behavior of the one-unit-cell system as shown in Fig. 2.
Fig. 7. Relationship between the number of cells and reflection at the frequency where exact transparency is achieved in the one-unit-cell system. (a) Property diagram of exact bidirectional transparency at frequency ${{\omega }_{1}}=0.449921\pi c/d$. (b) Property diagram of exact bidirectional transparency at frequency ${{\omega }_{2}}=0.885635\pi c/d$.
Furthermore, we can use the principles of the third type of exact transparency in multi-unit-cell systems to design and develop optical devices that can switch between exact unidirectional and bidirectional transparency.
3.3 Approximate bidirectional transparency of multi-unit-cell systems
According to the phase transition rule of reflected light introduced in Section 3.1 and the general relationship between transmission and reflection coefficients for the unit- and multi-unit-cell systems described in Section 3.2, we can determine the precise locations where a system produces either single-frequency or multi-frequency transparency. However, there are currently no reports on the study of $\mathcal {PT}$-symmetric optical waveguide networks in terms of multi-frequency bands transparency. Additionally, in the development of some optical instruments, people are not only concerned with determining the precise frequency locations of transparency but also with optimizing the system to achieve low reflection and high transmission across frequency bands. Therefore, we introduce the concept of approximate transparency to better study the transparency characteristics of $\mathcal {PT}$-symmetric optical waveguide networks across frequency bands. We define the production of approximate transparency at a given frequency band as a system’s reflection being ${{R}_{N}}\le {{10}^{-8}}$ and its transmission being ${{T}_{N}}\to 1$.
From Eq. (10), it can be seen that $\left | \sin N\phi \right |\le 1$. For a one-unit-cell system, there exists a frequency band at the exact bidirectional transparency frequency, satisfying Eq. (11), where the multi-unit-cell systems will always produce approximate bidirectional transparency. As the reflection of the one-unit-cell system increases, certain multi-unit-cell systems will no longer be able to maintain approximate bidirectional transparency at these frequencies. However, as shown in Eq. (10), by adjusting the number of unit cells, it is possible to maintain approximate bidirectional transparency for a specific number of cells at these frequencies. Therefore, different multi-unit-cell systems have different frequency band widths for achieving overall approximate bidirectional transparency.
As an example of the multi-unit-cell systems producing approximate bidirectional transparency, we select the frequency range $\Delta {{\omega }_{1}}=(0.88555\sim 0.88570)\pi c/d$ at which the one-unit-cell system produces exact bidirectional transparency, as shown in Fig. 2. Figure 8 illustrates the transmission and reflection spectra of the multi-unit-cell systems at this frequency range. It can be observed from Fig. 8(a) and 8(b) that the frequency range at which the multi-unit-cell systems has ${{R}_{\mathrm {L}-N}}({{R}_{\mathrm {R}-N}})\le {{10}^{-8}}$ and ${{T}_{N}}\ge 1-{{10}^{-8}}$. The green solid line in Fig. 8 indicates the range in which any multi-unit-cell systems has ${{R}_{\mathrm {L}-N}}({{R}_{\mathrm {R}-N}})\le {{10}^{-8}}$ and ${{T}_{N}}\ge 1-{{10}^{-8}}$ at frequency range $\Delta {{\omega }_{2}}=(0.88563\sim 0.88564)\pi c/d$, indicating that the system always has approximate bidirectional transparency in this frequency range. When $N=35$, the system has ${{R}_{\mathrm {L}-35}}({{R}_{\mathrm {R}-35}})\le {{10}^{-8}}$ and ${{T}_{35}}\ge 1-{{10}^{-8}}$ in the frequency ranges $\Delta {{\omega }_{3}}=(0.88556\sim 0.88563)\pi c/d$ and $\Delta {{\omega }_{4}}=(0.88564\sim 0.88569)\pi c/d$. Therefore, the multi-unit-cell systems with this number of unit cells can still achieve approximate bidirectional transparency at these frequency ranges. The red dashed line in Fig. 8 marks the total frequency range $\Delta {{\omega }_{2}}+\Delta {{\omega }_{3}}+\Delta {{\omega }_{4}}=(0.88556\sim 0.88569)\pi c/d$ in which this multi-unit-cell systems can achieve approximate bidirectional transparency.
Fig. 8. Reflectance and transmittance spectra of the multi-unit-cell systems that reveal the frequency band of approximate bidirectional transparency, where, (a) the reflection spectrum of the multi-unit-cell systems, (b) the transmission spectrum of the multi-unit-cell systems.
Section 3.1 demonstrates that the number of exact transparency frequencies of a one-unit-cell system can be controlled by adjusting the inter-ring arm length or the real part of the refractive index of the ring material. This principle can be applied to regulate the number of approximate bidirectional transparency frequency bands of a multi-unit-cell systems, providing a new approach for developing optical stealth devices with tunable frequency bands. The resulting optical devices could achieve stealth effects across multi-frequency bands, thereby effectively enhancing their stealth performance.
3.4 Robustness analysis
According to Eq. (10), changes in the transmission and reflection of a one-unit-cell system can also affect those of the multi-unit-cell systems. In practical applications, parameters such as the length of the waveguide and the refractive index of the material in optical waveguide networks may be subject to random and systematic errors. Therefore, it is crucial to investigate the robustness of a one-unit-cell system. Because the transparency frequency and frequency band of the multi-unit-cell systems are closely related to the reflection valleys of the one-unit-cell system, the number and position of the reflection valleys in the one-unit-cell system can be used as an important indicator to measure the robustness of the system. We use the reflection spectrum of the one-unit-cell system shown in Fig. 2 as a standard and calculate the variations in the number and position of the system’s reflection valleys when errors occur in the lengths of the upper and lower arms of the ring waveguides (represented by ${{d}_{1}},{{d}_{2}}$, respectively), the length of the inter-arm Section (represented by $l$), the refractive index of the material, and the ratio of the waveguide widths on the upper and lower arms. We summarized the changes in the reflection spectrum of the one-unit-cell system caused by deviations in waveguide length and material refractive index due to random and systematic errors in Table 1.
Table 1. Reflection spectrum variations due to slight deviations in waveguide length and refractive index in a one-dimensional $\mathcal {PT}$-symmetric dumbbell one-unit-cell network with two materials.
As an example of quantitative analysis of the robustness of a single protocell system, we consider the system inter-ring arm length, refractive index, and the change of the left reflection spectrum of the system when the upper and lower arms of the ring are broken, respectively, as shown in Fig. 9. We observe that the inter-ring arm $l$ is comparatively long, the position of reflection valleys is shifted to the left in Fig. 9(a); $\operatorname {Re}{{n}_{1}}$ is comparatively large, the position of reflection valleys is shifted to the left in Fig. 9(b); $\operatorname {Im}{{n}_{1}}$ is comparatively large, the partial slight shift of the reflection valley in Fig. 9(c); ${{n}_{l}}$ is comparatively large, the leftward slight shift of the reflection valley in Fig. 9(d); ${{n}_{0}}$ comparatively is large, the position of some reflection valleys is shifted in Fig. 9(e); and when there is an error in the ratio of the waveguide of the ring’s upper and lower arms, the number and the position of the reflection valleys are changed in Fig. 9(f). It is obviously consistent with the results summarized in Table 1.
Fig. 9. Left reflection spectra reveal the robustness of single protocell systems, where, (a)$l$, (b)$\operatorname {Re}{{n}_{1}}$, (c)$\operatorname {Im}{{n}_{1}}$, (d)${{n}_{l}}$, (e)${{n}_{0}}$, (f)$\Lambda d$ slight deviations result in changes in the number and position of reflection valleys.
In summary, when there are small deviations in the inter-arm length and refractive index of the materials, the positions and number of the reflection valleys in the system’s reflection spectrum exhibit small changes, indicating system stability. However, when there are breaks in the ratio of the upper and lower arm waveguides in the one-unit-cell system, both the number and position of the reflection valleys change significantly, which can greatly affect the system’s stability. Therefore, when fabricating a one-dimensional $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network, great attention must be paid to the accuracy of the ratio of the upper and lower arm waveguides in the ring waveguide, in order to improve the accuracy of the system’s reflection spectrum measurements.
4. Conclusion
In this paper, we design an interesting one-dimensional two-material $\mathcal {PT}$-symmetric dumbbell finite periodic optical waveguide network and investigate its singular transparency. We found that by adjusting the inter-ring arm length and/or the real part of the ring material refractive index, the number of exact transparency frequencies in the one-unit-cell system can be flexibly tuned. We also derived general relations for the transmission and reflection coefficients of the multi-unit-cell systems. Based on these results, we discovered three types of precise transparencies: (i) exact multi-frequency transparency, which shares the same properties as the one-unit-cell system; (ii) exact multi-frequency bidirectional transparency, which can be achieved at frequencies where the one-unit-cell system does not produce exact transparency by adjusting the number of unit cells; (iii) exact multi-frequency bidirectional transparency, which can be achieved at frequencies where the one-unit-cell system produces exact transparency by adjusting the number of unit cells. It provides a theoretical basis for developing high-sensitivity optical filters. At the same time, we have discovered that by adjusting the number of unit cells in a one-unit-cell system at the frequency where exact transparency occurs, we can achieve approximate multi-frequency bands transparency with the reflection ${{R}_{\mathrm {L}-N}}({{R}_{\mathrm {R}-N}})\le {{10}^{-8}}$ and the transmission ${{T}_{N}}\to 1$. It provides a scientific basis for developing high-performance optical stealth devices. Additionally, we found that the one-unit-cell system has good fault tolerance for small fluctuations in inter-arm length and material refractive index. These findings deepen our understanding of the transparency of $\mathcal {PT}$-symmetric optical waveguide networks. In the future, further exploration can be conducted on the transparency characteristics of $\mathcal {PT}$-symmetric optical waveguide networks with different structures and materials, as well as their applications in other fields.
Funding
National Natural Science Foundation of China (11674107, 11775083, 61475049, 61771205, 61774062).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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