1 Introduction

Photonic crystals (PhCs), a type of new artificial material with periodic structures, have attracted worldwide interest during the past decades [1–6]. They have many unique characteristics, such as photonic band gaps (PBGs), localized modes, and surface states, which provide an effective way of manipulating light [3, 4]. Many devices have been fabricated based on PhC structures [5, 6].

Polarization devices, including polarizer analyzers, polarization selectors, polarization splitters, and polarization logic devices, play an important role in integrated optical circuits for optical communications [7–12]. However, during signal processing using such polarization devices, cross talk between polarizations from different channels will occur because different polarizations will interact during propagation and couple to neighboring channels. This type of unwanted coupling decreases the polarization extinction ratio (PER), degree of polarization (DOP), and isolation. Unwanted coupling can also promote cross talk in polarization devices for communication and signal processing. Moreover, it may destroy the logic relation for polarization logic devices. The polarization optical bridge (POB), which can effectively overcome the unwanted coupling of polarized waves between different channels, is a key technical issue in the development of optical integrated circuits.

In this paper, two types of POBs have been proposed by combining two-dimensional (2D) square-lattice PhC cross-waveguides and polarization-selective defects in the PhC waveguides. The proposed structures include an input TE channel, an input TM channel, an output TE channel, an output TM channel, and a bridge region, i.e., crossing region. A very low polarization cross talk is obtained in the proposed POBs. A TE (TM) wave can only transfer from the input TE (TM) channel to the output TE (TM) channel, avoiding any transmission from the input TE (TM) channel to the output TM (TE) channel. At the same time, the input TE (TM) channel will block TM (TE) waves, i.e., also function as a polarizer. The Bragg effect plays an important role in the POBs. The finite element method (FEM) is used to demonstrate that both of the structures perform well in preventing unwanted coupling of polarizations between different channels and also function as polarizers.

2 Physical model

Schematics of the two POB structures are shown in Fig. 1. Both of the structures are based on 2D square-lattice PhCs formed by the background rods with the radii r T and lattice constant a. Figure 1a presents structure 1, which features the crossing of two input signals in perpendicular directions. Figure 1b presents structure 2, which features the crossing of two input signals in parallel directions.

Fig. 1
figure 1

Schematics of the two POBs. Structure 1 a is a POB for two perpendicular input and output channels. Structure 2 b is a POB for two parallel input and output channels

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Structure 1 is formed by removing two straight lines of background rods in the X and Y directions to form two waveguides that are identical but with different directions of waveguide axes. Without defects, the two waveguides can be channels for both TE and TM waves. When different defects are introduced, however, they can be regarded as TE-only or TM-only channels. The defects introduced into the waveguide regions include: two arrays of 3 × 1 circular defect rods (subsequently referred to as TM defects) with radii r in the middle of the TM channel and two arrays of 2 × 2 square defect rods (subsequently referred to as TE defects) with side lengths of d in the middle of the TE channel. Structure 2 is formed by removing three oblique lines of background rods at 45° to the X- and Y-axes to form the cross-waveguide. TM and TE defects are introduced, resulting in formation of the TM and TE channels, respectively.

Structures 1 and 2 serve as POBs for two perpendicular channels and two parallel channels, respectively. Although a POB for two parallel channels can also be built based on structure 1, structure 2 is more compact. Compared with structure 1, both of the channels and defects around the cross-center in structure 2 are rotated 45°. As a result, the width of the waveguide changes from 3a in structure 1 to 2√2a in structure 2, as illustrated in Fig. 1. The input and output ports, as well as some necessary structure parameters, are shown in Fig. 1. It should be pointed out that r and d will take different values for the two structures, which will be discussed in the following section.

In this paper, the background rods, TM defects and TE defects were selected to be appropriate for the anisotropic material tellurium [13–15]. Anisotropic materials have proven to be more efficient than isotropic materials for obtaining the absolute PBG [14], which is important for polarization devices. Moreover, tellurium has a very low loss in the mid- and far-infrared wavelength range, which has important military and medical applications [15]. In practical applications, material dispersions are unavoidable. Therefore, the ordinary refractive index n o and the extraordinary refractive index n e of tellurium are considered to be dispersive as follows [15]:

$$n_{\text{o}} = \sqrt {18.5346 + 4.3289\lambda^{2} (\lambda^{2} - 3.9810)^{ - 1} + 3.7800\lambda^{2} (\lambda^{2} - 11813)^{ - 1} } ,$$
(1)
$$n_{\text{e}} = \sqrt {29.5222 + 9.3068\lambda^{2} (\lambda^{2} - 2.5766)^{ - 1} + 9.2350\lambda^{2} (\lambda^{2} - 13521)^{ - 1} } ,$$
(2)

where λ is the operating wavelength measured in μm. Here, the extraordinary axis (e-axis) of the background rods and the TM defects in Fig. 1a, b is chosen to be parallel to the Z-axis. The e-axis of the TE defects in Fig. 1a is parallel to the X-axis and the e-axis of the TE defects in Fig. 1b is anticlockwise 45° to the Y-axis, i.e., perpendicular to the oblique section of the TE channel. For this work, tellurium is considered non-magnetic (μ = 1). The dielectric tensor of the background rods and the TM defects can be written as:

$$\varepsilon_{1} = \left( {\begin{array}{*{20}c} {\varepsilon_{\text{o}} } & 0 & 0 \\ 0 & {\varepsilon_{\text{o}} } & 0 \\ 0 & 0 & {\varepsilon_{\text{e}} } \\ \end{array} } \right),$$
(3)

where \(\varepsilon_{\text{o}} = n_{\text{o}}^{2} ,{\text{ and }}\varepsilon_{\text{e}} = n_{\text{e}}^{2}\). For the TE defects in Fig. 1a, the dielectric tensor can be written as:

$$\varepsilon_{2} = \left( {\begin{array}{*{20}c} {\varepsilon_{\text{e}} } & 0 & 0 \\ 0 & {\varepsilon_{\text{o}} } & 0 \\ 0 & 0 & {\varepsilon_{\text{o}} } \\ \end{array} } \right).$$
(4)

For the TE defects in Fig. 1b, using the coordinate rotation matrix [16, 17], the dielectric tensor is written as:

$$\varepsilon_{3} = \left( {\begin{array}{*{20}c} {\varepsilon_{\text{av}} } & {\varepsilon_{\text{dh}} } & 0 \\ {\varepsilon_{\text{dh}} } & {\varepsilon_{\text{av}} } & 0 \\ 0 & 0 & {\varepsilon_{\text{o}} } \\ \end{array} } \right),$$
(5)

where ɛ av = (ɛ o + ɛ e)/2 and ɛ dh = (ɛ o − ɛ e)/2.

We note that the electric field for TE polarization is parallel to the Z-axis, while for TM polarization, the electric field is parallel to the X–Y plane. Specifically, the electric field for TM waves propagating in the x direction is parallel to the Y-axis. Since the extraordinary refractive index is higher than the ordinary refractive index, the above arrangement of the e-axes for the defects results in the TE (TM) wave strongly interacting with the TM (TE) defects and weakly interacting with the TE (TM) defects, so that the TE (TM) wave cannot enter the TM (TE) channel.

It is worth mentioning that Maxwell’s equations become nonlinear near the frequency used for dispersive materials. Therefore, the conventional scaling relation between frequency and geometrical dimension [18] is no longer valid for dispersive materials.

To investigate the performance of the proposed structures, we calculate the DOP, PER, and IL in the next section according to the following definitions:

$$\text{DOP}{ = }\left| {\left( {I_{\text{TE}} - I_{\text{TM}} } \right)/\left( {I_{\text{TE}} + I_{\text{TM}} } \right)} \right|,$$
(6)

which is valid for both the TE and TM channels. I TE and I TM are the wave intensity in the TE and TM channels, respectively;

$${\text{PER}}_{\text{TE}} = 10 \times \log_{10} \left( {I_{\text{TE}} /I_{\text{TM}} } \right),$$
(7)
$${\text{PER}}_{\text{TM}} = 10 \times \log_{10} \left( {I_{\text{TM}} /I_{\text{TE}} } \right),$$
(8)

which are the PERs for the TE and TM channels, respectively, and

$${\text{IL}} = 10 \times \log_{10} \left( {P_{\text{IN}} /P_{\text{OUT}} } \right),$$
(9)

which is valid for both the TE and TM channels. P IN and P OUT are the input and output power, respectively, for TE (TM) waves at the TE (TM) channel.

Absolute or complete PBG is crucial to keep both of the polarization waves confined and transmitting inside the waveguides in the POBs. The largest absolute PBG in perfect 2D square-lattice PhC that has previously been obtained uses r T = 0.3431a [6]. The range of the largest absolute band gap obtained was λ = (3.893a–4.223a). Here, the simulated band gap map or band structure is omitted as it can be found in [6]. When the lattice constant a is selected to be 1 μm, the range of operating wavelengths is from 3.893 to 4.223 μm, which is located in the mid infrared band. In this range of wavelength, losses due to the tellurium can be ignored [15]. It should be pointed out that although the scaling law is no longer valid in PhCs made of dispersive materials, the same method for designing such POB structures can be applied to other operating wavelength ranges.

3 Numerical results and discussions

First, we investigate the performance of structure 1, as shown in Fig. 1a. The DOPs and PERs versus radius r of the TM defects and side length d of the TE defects for TM and TE channels are shown in Fig. 2a, b, respectively. In these figures, the red and blue lines represent the PERs and DOPs, respectively. The operating wavelength was selected to be 4.058a, i.e., the center wavelength of the absolute PBG.

Fig. 2
figure 2

PERs and DOPs versus r for the TM channel (a) and that versus d for the TE channel (b), where the red and blue lines represent the PERs and DOPs, respectively. Here the radius r T of background rods and the operating wavelength are 0.3431a and 4.058a, respectively

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From Fig. 2, we can see that for the TM channel with r = 0.1954a, the PER is over 60 dB and the DOP is as high as almost 1. For the TE channel with d = 0.543a, the PER is >45 dB and the DOP is almost 1. The performance of the TM channel is slightly better than the TE channel, but the PER and DOP for both the TE and TM channels can be very high. That is to say, with proper values of r and d, TM waves can only transmit in the TM channel; the function of the TM defects is like a door that opens for TM waves but closes for TE waves. The situation is reversed for the TE channel. This demonstrates that the e-axis of the defect rods has been selected to realize the POB functions.

We can also analyze the effective refractive index of the defect rods in the channels. For TE (TM) defects in the TE (TM) channels in Fig. 1, with the e-axis chosen as described previously, we can calculate the effective index of each defect rod according to the following formulae [16]:

$$n_{\text{eff, TE}}^{\text{TM}} = \sqrt {\varepsilon_{\text{eff, TE}}^{\text{TM}} } ,\quad n_{\text{eff, TM}}^{\text{TM}} = \sqrt {\varepsilon_{\text{eff, TM}}^{\text{TM}} } ,\quad n_{\text{eff, TE}}^{\text{TE}} = \sqrt {\varepsilon_{\text{eff, TE}}^{\text{TE}} } ,\;\quad n_{\text{eff, TM}}^{\text{TE}} = \sqrt {\varepsilon_{\text{eff, TM}}^{\text{TE}} } ,$$
(10)

where the subscript ‘eff’ denotes ‘effective’, the superscript TE (TM) denotes the TE (TM) channel, the subscript TE (TM) denotes TE (TM) waves, and

$$\varepsilon_{\text{eff, TE}}^{\text{TM}} = \frac{{\int_{\varOmega } {\varepsilon_{\text{e}} E_{\text{z}}^{2} (x,y)d\varOmega } }}{{\int_{\varOmega } {E_{\text{z}}^{2} (x,y)d\varOmega } }},\quad \varepsilon_{\text{eff, TM}}^{\text{TM}} = \frac{{\int_{\varOmega } {\varepsilon_{o} \left( {E_{\text{x}}^{2} (x,y) + E_{\text{y}}^{2} (x,y)} \right)d\varOmega } }}{{\int_{\varOmega } {\left( {E_{\text{x}}^{2} (x,y) + E_{\text{y}}^{2} (x,y)} \right)d\varOmega } }},$$
(11)
$$\varepsilon_{\text{eff, TE}}^{\text{TE}} = \frac{{\int_{\varOmega } {\varepsilon_{\text{o}} E_{\text{z}}^{2} (x,y)d\varOmega } }}{{\int_{\varOmega } {E_{\text{z}}^{2} (x,y)d\varOmega } }},\quad \varepsilon_{\text{eff, TM}}^{\text{TE}} = \frac{{\int_{\varOmega } {\left( {\varepsilon_{\text{e}} E_{1}^{2} (x,y) + \varepsilon_{\text{o}} E_{2}^{2} (x,y)} \right)d\varOmega } }}{{\int_{\varOmega } {\left( {E_{\text{x}}^{2} (x,y) + E_{\text{y}}^{2} (x,y)} \right)d\varOmega } }},$$
(12)

where E 1(x, y) and E 2(x, y) are the electric field components parallel and perpendicular to the e-axis, respectively, E x(x, y) and E y(x, y) are the electric field components parallel to the X- and Y-axes, respectively, and Ω denotes the cross-sectional area of the defect rod. Equations (11) and (12) are obtained by assuming that energy is conserved in the defect rod area when an anisotropic defect rod is replaced by its equivalent isotropic rod.

For the TM channel in structure 1, the TE wave has only E z, while the TM wave has only E y. In structure 2, the TE wave has only E z, while the TM wave has both E x and E y. We can see from Eq. (11) that the effective refractive index for TE waves is greater than that for TM waves, so that a TE wave with proper frequency trying to transfer to the TM channel will be rejected by the defect rods because of the strong Bragg reflection effect. Because of the difference in the effective refractive index of the defect rods for TE and TM polarizations, the Bragg reflection effect is very weak for the TM wave with the same frequency because the frequency for strong TM Bragg reflection is different from that for TE waves. Similarly, from Eq. (12), we can understand the mechanism behind the function of the TE channel. This explains the operating mechanism of the POBs.

To further understand the structure, the field distributions are shown in Fig. 3. Figure 3 demonstrates that the TE (TM) wave is transmitted in the TE (TM) channel and is blocked by the TM (TE) channel. Although the different polarizations share a common region in the center of the structure, they do not couple with each other. The TM channel is the so-called TM bridge, while the TE channel is the so-called TE bridge. Therefore, the device structure can be thought of as two crossing bridges, with only a certain polarization being allowed to pass along a certain bridge to the corresponding output port.

Fig. 3
figure 3

Field distributions in structure 1 for TM input from port 1 (a), for TE input from port 2 (b), for TM inputs from port 1 and port 2 (c), and for TE inputs from port 1 and port 2 (d). Here, the radius r of TM defects and the side length d of TE defects are 0.1954a and 0.543a, respectively, while other parameters are the same with those of Fig. 2

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To further investigate the performance of structure 1 at the wavelength range of absolute PBG, a wavelength scan is performed on the PERs, DOPs, and ILs, as shown in Fig. 4, where r and d are 0.1954a and 0.543a, respectively. A stable bandwidth is obtained from 3.9647a to 4.122a, with PER over 32 dB, DOP above 0.9985, and IL under 0.3 dB, i.e., the POB is demonstrated to be excellent in a wide wavelength band, much better than the other polarization devices proposed by Refs. [9–12].

Fig. 4
figure 4

Wavelength scan on PERs (a), DOPs (b), and ILs (c) for structure 1, where the blue and red lines are calculated for the TE and TM polarization, respectively. Here, the parameters are the same with those of Fig. 3

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Structure 1 is suitable for input signals from vertical and horizontal directions. In practical applications, we often need to cross two input signals in parallel directions or interchange their spatial position. Therefore, it is necessary to investigate the characteristics of structure 2, shown in Fig. 1b.

The PERs and DOPs were recalculated at the operating wavelength 4.058a for the TM and TE channels. The best r and d obtained for structure 2 were 0.165a and 0.5711a, respectively. These parameters are different from those of structure 1. The calculated field distributions are shown in Fig. 5. These results demonstrate that different polarized waves are transmitted through their own channel but are blocked by the other channel, and the different polarizations do not couple between channels. The field distributions in structure 2 are similar to those in structure 1.

Fig. 5
figure 5

Field distributions in structure 2 for TM input from port 1 (a), for TE input from port 2 (b), for TM inputs from port 1 and port 2 (c), and for TE inputs from port 1 and port 2 (d). Here, the radius r of TM defects, the side length d of TE defects, and the operating wavelength are 0.165a, 0.5711a and 4.058a, respectively

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To obtain the stable bandwidth for structure 2, the PERs, DOPs, and ILs versus wavelength within the absolute PBG range were also studied, as shown in Fig. 6. These results demonstrate that there is a stable bandwidth from 3.9933a to 4.1137a, at which the PER is over 35 dB, the DOP is above 0.9992, and the IL is below 0.3 dB. The stable bandwidth of structure 2 is slightly narrower than that of structure 1. The difference in performance between the two structures is caused by the difference in waveguide width and distance from the polarization-selective defects to the corresponding channels. However, both of the POB structures are of high quality.

Fig. 6
figure 6

Wavelength scan on PERs (a), DOPs (b), and ILs (c) for structure 2, where the blue and red lines are calculated for the TE and TM polarization, respectively. Here, the parameters are the same with those of Fig. 5

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The material used in this study can only work for infrared bands above 5 μm. However, the same concepts and structures can be applied to optical bridges designs in other wave bands.

Finally, we will discuss how to maintain structural robustness and 3D realization for practical applications. First, holes having the same radii as those of the cylinders consisting of the PhCs should be drilled in a block of low-refractive index material, e.g., foamed plastic, which has a refractive index of approximately 1. The dielectric cylinders should be placed into the holes to form the PhCs. Within the wave-guiding area, no holes should be drilled in the foamed material. Holes should be drilled for holding the cylinders of waveguide defects. Such a structure would be strong and stable. The foamed material would act as a moisture barrier so that the structure would be stable in different external conditions. It should also be noted that the defect rods should be cut along different directions from a tellurium crystal because the optical axes of some defect rods are different from the background rods. Further discussion for 3D realization can be found in the literature, e.g., Refs. [6, 19].

4 Conclusion

In summary, we have proposed and demonstrated two types of POB structures based on waveguides in 2D square-lattice PhCs with anisotropic materials and Bragg defects in the waveguides. The Bragg defects provide polarization selection. Structure 1 is suitable for two input signals in perpendicular directions, while structure 2 is suitable for two input signals in parallel directions. In these structures, a very low polarization cross talk can be obtained: the TE (TM) wave can only transmit in the TE (TM) channel, and different polarizations do not couple between channels. The resulting PERs are over 32 dB, DOPs are almost 1, and ILs are below 0.3 dB in a wide wavelength band. Such structures are useful for separating different polarization modes, keeping the independence of different polarizations and interchanging the spatial position of different polarization modes in large-scale optical integrated devices.