1 Introduction

High index contrast and CMOS compatibility of the silicon-on-insulator (SOI) wafers have made them a favorable platform for designing compact and cost-effective active and passive photonic devices and the realization of silicon photonic integrated circuits (PICs) [1, 2]. Despite attractive features of SOI-based PICs their performance strongly depends on the polarization of the optical waves. This limits their application in optical communication systems which usually use optical fibers with uncertain output polarization for off-chip communication [3, 4]. Application of polarization diversity scheme has been proposed as a solution to resolve this issue and realize polarization-independent (or polarization transparent) PICs [5, 6]. In this scheme, the incoming light from optical fibers is separated into two linear polarization using a polarization beam splitter (PBS) and each polarization passes through the same PIC. The two polarization will then be combined through another PBS and coupled to the output optical fiber [5, 6]. Several types of designs have been reported for PBSs over the years including symmetric and asymmetric directional couplers [7, 8], Mach–Zehnder interferometers (MZIs) [9] and multimode interference (MMI) couplers [10]. The DCs have the advantages of relatively simple structures and easy design [11, 12], but they suffer from low fabrication tolerance and limited bandwidth [13]. The MZI-based PBSs on the other offer wide operation bandwidth but have a relatively large footprint [14]. Among these structures, the MMIs are of high interest because of their significant features such as low excess loss, large fabrication tolerance and relatively wide bandwidth [15]. These PBSs usually work based on the beat-length difference between the TE and TM polarized waves. However, the beat-length of the multimode region of MMI for transverse electric (TE) and transverse magnetic (TM) polarized modes are very close together. As a result, the multimode region of MMI should be relatively long for efficient polarization splitting [16]. This, in turn, increases the overall length of the MMI-based PBS. It should be mentioned that there have also been wideband designs for TE or TM pass polarizers such as the structure of [17]. However, since these structures pass only one polarization and block the other (i.e., do not perform the polarization splitting) we here focus on the references that propose polarization splitter structures.

In this paper, instead of using the beat-length difference between TE and TM polarized waves, we use different propagation mechanisms to separate these waves. In the proposed structure the TM polarized input wave passes directly through the multimode region without coupling to higher order modes. The TE polarized input wave, on the other hand, excites higher order modes of the multimode region and the output TE wave is produced through the self-imaging process in the multimode region. This approach substantially decreases the overall length of the proposed MMI-based PBS. Furthermore, the application of sub-wavelength grating (SWG) in the multimode region of the designed MMI structure further decreases the overall length of the structure and also increases the bandwidth of the PBS. The overall footprint of the structure is 4.8 × 21 µm2 which is more than 4 times smaller than the previously reported MMI-based PBSs with the same platform [16]. Three-dimensional finite-difference-time-domain (3D FDTD) simulation results show that the designed structure has a polarization extinction ratio (PER) of more than 12 dB and an insertion loss (IL) less than 3 dB for both the TE and TM mode in the wavelength range of 1.3–1.65 µm. Here, the FDTD simulations are carried out using the trial version of the Lumerical software package [18].

The rest of this paper is organized as follows; the optical design of the PBS is proposed in Sect. 2. The properties of the designed PBS are then examined through 3D FDTD simulation in Sect. 3. The effect of inevitable fabrication errors on the performance of the proposed PBS is also modeled as variations in the duty cycle of the SWG regions and simulated using 3D FDTD in Sect. 4. Finally, a summary of the results and conclusions is provided in Sect. 5.

2 Design of polarization beam splitter

The schematic view of an MMI-based PBS is presented in Fig. 1a. The input waveguide couples TE and TM polarized waves to the multimode region, where two polarizations will be separated and each one will be coupled to one of the output waveguides. The structure is assumed to be fabricated using silicon-on-insulator (SOI) wafers with an oxide thickness of 2 µm. A cross-sectional view of the input/output waveguide is also shown in Fig. 1b. The thickness of the silicon layer; hSi and the width of the input/output waveguides; Ws are set to be 400 nm and 450 nm, respectively, so that these waveguides can support both TE and TM modes. The effective indices of the TE and TM modes of the input/output waveguides are presented in Fig. 1c. The absolute value of the electric field of the TE and TM modes of the input/output waveguides are also presented in the inset of this figure. As can be seen in the wavelength range of interest the effective index of the TE and TM modes changes from 2.96 and 2.93 to 2.60 and 2.55, respectively.

Fig. 1
figure 1

a Schematic view of an MMI-based beam splitter, b cross-sectional view of the strip waveguide, c effective refractive index of the TE and TM modes of the input/output waveguides (inset presents the electric field profile of the TE/TM modes)

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Using 3D FDTD simulations we have calculated the electric field distribution in the MMI region of Fig. 1a, at the wavelength of λ = 1.55μm for both TE and TM polarizations. Figure 2a shows the absolute value of the electric field in the y–z plane when the structure is excited with TE polarized wave. Here the MMI width is set to WMMI = 4.8μm and the input waveguide is located at; y = + WMMI/6. As can be seen, when the input waveguide is placed at y = + WMMI/6, odd- and even-numbered self-images of the input field will be formed at y = − WMMI/6 and y = + WMMI/6, respectively. Self-imaging is a phenomenon in multimode waveguides, where single or multiple images of the input field profile are formed at periodic intervals along the propagation direction [19,20,21,22]. The distance between the input and first self-image (which is equal to the distance between successive self-images) is called beat-length. For a uniform MMI coupler the beat-length can be calculated as [23]

$$L_{\pi } = 4n_{\text{r}} W_{\text{e}}^{2} /3\lambda ,$$
(1)

where nr denotes the effective refractive index of the MMI region which is effective index of the corresponding (TE/TM) mode of the slab waveguide with the thickness of hSi, and We are the effective width of the multimode region which can be approximated by the physical width of this region; We ≈ WMMI. λ is the free-space wavelength and is considered to be λ = 1.55 µm in our design. For hSi = 400 nm, the effective refractive index for TE and TM waves at λ = 1.55 µm are found to be nr,TE= 3.185 and nr,TM= 2.960, respectively. The formation of the first self-images in the electric field profile for the TE and TM inputs is also presented in Fig. 2b which shows the beat-length of 61 μm and 59 μm for these polarizations, respectively. Using Eq. (1) results in the following theoretical values for beat-length of TE and TM waves; Lπ,TE= 63.1 μm, Lπ,TM~ 58.7 μm, which are very close to the values calculated from the FDTD simulations.

Fig. 2
figure 2

Electric field profile in the MMI of Fig. 1a when the input waveguide is placed at y = + WMMI/6. a Formation of first and second self-images for TE input, b formation of the first self-images for TE and TM inputs and their corresponding beat-length values

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Since the beat-length for TE and TM polarizations are different and the odd- and even-numbered self-images will be formed at y = − WMMI/6 and y = + WMMI/6, respectively [23], it is possible to find an MMI length; LMMI for which an even-numbered TM self-image and an odd-numbered TE self-image appear at the same z-point. This way the TM and TE waves can be separated by placing two waveguides at z = LMMI and y = ±WMMI/6 (see Fig. 1a). The required MMI length can be calculated by finding the integer number, m for which the above condition holds, i.e., LMMI = (2 m − 1)Lπ,TE= (2 m)Lπ,TM. However, because of the small effective index difference between the TE/TM waves (nr,TE= 3.185 and nr,TM= 2.960) the beat-length for two polarization are relatively close; Lπ,TE~ 61μm, Lπ,TM~ 59μm (see Fig. 2a), and therefore, the required MMI length will be relatively large; LMMI~ 1770μm. According to Eq. (1), the beat-length can be decreased by reducing MMI width or effective index; nr. Reducing the width of the multimode region increases the cross talk between two output waveguides and degrades the PBS performance. The reduction of the effective index can be done by the implementation of an SWG in the MMI region. SWGs are periodic structures, where the grating pitch (Λ) is much smaller than the operating wavelength (Λ ≪ λ) and, therefore, do not diffract optical waves [24]. Thus, a silicon SWG acts as a homogenous medium with an effective refractive index lower than the refractive index of the original silicon region. Application of the SWG in the multimode region also reduces the wavelength dependency of the beat-length and enhances the operating bandwidth [25]. In addition to nr reduction, the presence of the SWG increases the difference between the TE and TM effective indices of the MMI region and consequently increases the beat-length difference between these waves. This approach has been previously employed to realized compact MMI-based PBS with the overall length of ~ 100 μm [16].

Here we will use a new approach which will result in a significant reduction of the overall PBS length. Instead of using the beat-length difference between the TE/TM waves, these waves will be separated by breaking the symmetry of the SWG-based MMI region in a way that the one polarization transfer to its corresponding output waveguide through direct guidance and the other through the self-imaging process. Figure 3 shows the top view of the proposed PBS. In this design, a thin horizontal strip is included in the MMI region at y = + WMMI/6 (see the inset of Fig. 3). By choosing the proper width of the interconnecting waveguide; Wi, it will pass the TM polarized wave directly to the corresponding output waveguide. The TE wave will reach its corresponding output waveguide through the self-imaging process, and therefore, LMMI should be selected based on the beat-length of the TE wave; LMMI = Lπ,TE.

Fig. 3
figure 3

Schematic view of the proposed PBS

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In our design, the width of the multimode region is set to be WMMI = 4.8 µm which leads to output waveguide separation of 1.6 µm and negligible cross talk between the output TE/TM waveguides. The SWG is assumed to be formed by completely etching the silicon layer in a portion of each grating pitch. This means that the multimode region is a periodic arrangement of silicon rods with the width of WMMI, the height of hSi, length of a, and period of \(\varLambda\). The design parameters of the structure of Fig. 3 are the period and duty cycle of the SWG (\(\varLambda\) and a/\(\varLambda\) × 100, respectively), which mainly affect the transmission of the TE waves, and the width of the thin silicon strip; Wi, which control the transmission of the TM waves. In addition, proper taper regions should be designed to minimize the coupling loss between the multimode region and input/output waveguides. In what follows we first design the SWG region based on the goal of reducing the beat-length of the MMI region for the TE polarization. Then the width of the interconnecting waveguide; Wi will be selected in a way to support the propagation of a TM mode with an effective index value close to that of the input/output strip waveguides and also, does not affect the self-imaging process of the TE waves significantly. As it will be shown, a large Wi will improve the effective index matching and reduce the IL of the TM waves and at the same time increase the IL of the TE waves by breaking the symmetry of the MMI region and changing the beat-length of TE waves. In other words, Wi should be selected based on a tradeoff between the IL and PER of the PBS.

For operation in the sub-wavelength regime, the grating pitch; \(\varLambda\) must be smaller than Bragg period; \(\varLambda\)Bragg= λ/2nr,SWG. In this equation, λ is the free-space wavelength and nr, SWG is the effective refractive index of the SWG region which can be estimated as [24]

$$n_{\text{r,SWG}} = \sqrt {\frac{a}{\varLambda }n_{\text{r}}^{2} + \frac{\varLambda - a}{\varLambda }n_{\text{clad}}^{2} } ,$$
(2)

where nr and nclad are the effective refractive indices of the multimode region and the clad region [for the value of nr see the explanation below Eq. (1)]. The effective refractive index of the multimode region (400 nm thick silicon slab) for both TE and TM polarization is a function of wavelength and its highest value in the wavelength range of interest (1.3 µm < λ < 1.65 µm) is ~ 3.3. Considering a duty cycle of 100% (i.e., a/\(\varLambda\) = 1.0) and setting nr,SWG = nr = 3.3 for an upper estimation, results in \(\varLambda\)Bragg = 197 nm at λ = 1.3 µm. For higher wavelengths, the Bragg period is larger than the abovementioned value.

Here we choose \(\varLambda\)= 170 nm to ensure the sub-wavelength operation condition. According to Eqs. (1) and (2), a smaller value of a (smaller duty cycle) results in lower nr,SWG, which in turn decreases the beat-length of the MMI and the overall length of the proposed PBS. However, the lowest achievable structural dimensions are restricted by the resolution of the fabrication processes. Furthermore, according to our simulations, the IL of the PBS raises above 3 dB for duty cycle values lower than 50% (a/L < 0.5) which is corresponding to a < 85 nm. Based on these considerations we have set a = 100 nm in our design.

The top view of the designed MMI region and the electric field distribution in the MMI for TE polarized input is, respectively, shown in Fig. 4a, b. As can be seen, the TE polarized input light is coupled to the multimode region and it excites different optical modes of these regions. After propagation length of Lπ, the combination of the excited modes form a mirrored replica of the input field at the cross port. According to Fig. 4, the beat-length for TE polarization is ~ 15μm. Therefore, the geometrical parameters of the MMI region can be summarized as; WMMI = 4.8 µm, Λ = 170 nm, a = 100 nm, and LMMI = Lπ = 88\(\varLambda\) = 14.96 µm.

Fig. 4
figure 4

a Top view and geometrical parameters of the SWG-based MMI. b Electric field distribution and formation of the first self-image in the MMI region for TE polarized input

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To match the effective refractive index of input/output strip waveguides with the lower effective refractive index of the multimode region, the tapered regions in Fig. 4 are realized by implementing an SWG structure with the same periodicity and duty cycle as that of the multimode region. In addition, to improve the field profile matching between the TE/TM waves of the strip waveguides and the multimode region the width of the input/output waveguide is linearly increased to Wa and its core is narrowed to Wi in the length of Ltaper. According to simulation results, setting Ltaper and Wa to 20\(\varLambda\) = 3.4 µm and 1.2 µm, respectively, results in a good tradeoff between the overall length of proposed PBS and IL of the TE/TM polarized waves.

As mentioned, the transmission of the TM waves from input to output waveguides is through the interconnecting waveguide which is a periodic waveguide with the period of \(\varLambda\) (see the inset of Fig. 3). The top view of the interconnecting waveguide and its geometrical parameters are presented in Fig. 5a. According to FDTD simulation for Wi> 40 nm this waveguide supports a TM mode whose effective index increases for larger values of Wi. The dominant electric field component of the TM mode of interconnecting waveguide (i.e., Ex) is shown in Fig. 5b. The field profile of this mode in the x–y plane (transverse plane) is similar to the filed profile of the TM mode of the input waveguide, and therefore, input TM wave can be successfully coupled to the interconnecting waveguide and transferred to the output waveguide if (partial or exact) effective index matching condition can be fulfilled. The effective index of the interconnecting waveguide for Wi = 50, 60, and 70 nm, is plotted in Fig. 5c. A comparison between this figure and the effective index of the TM mode of the input/output waveguide in Fig. 1c reveals the effective index mismatch. For the input/output waveguide the effective index at λ = 1.55μm is ~ 2.65 and for the interconnecting waveguide with Wi= 50, 60 and 70 nm, respectively, it is 1.85, 1.91, and 1.96. As can be seen in Fig. 5c the effective index of the interconnecting waveguide mode can be increased by increasing Wi. This will improve the effective index matching and reduce the insertion loss of the TM waves. However, a thick interconnecting waveguide (large Wi value) will disturb the self-imaging process for the TE polarized waves by breaking the symmetry of the multimode region (see Fig. 6a) and deteriorate the PER of the proposed PBS. On the other hand, a thin interconnecting waveguide increases the insertion loss of the TM polarized waves and for Wi< 40 nm the interconnecting strip does not support any propagating modes. Therefore, Wi should be selected by considering the tradeoff between the IL and PER for both polarizations. We defined the following figure of merit (FOM) to consider the effect of the interconnecting waveguide on PER and IL of TE/TM polarization and find the optimum value of Wi:

Fig. 5
figure 5

a Top view of the (periodic) interconnecting waveguide. b Dominant component of the electric field of the TM mode (i.e., Ex) for Wi= 50 nm, at λ = 1.55μm. c Effective index of the TM mode of the interconnecting waveguide for Wi = 50, 60 and 70 nm in the wavelength range of 1.45–1.65μm

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Fig. 6
figure 6

Absolute value of the electric field distribution (i.e., |E(x,y)|) in the designed PBS, when excited by a TE polarized light, b TM polarized light

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$${\text{FOM}} = \frac{{{\text{Min}}\left\{ {\left. {\text{PER}} \right|_{\text{TE}} ,\left. {_{{}} {\text{PER}}} \right|_{\text{TM}} } \right\}}}{{{\text{Max}}\left\{ {\left. {\text{IL}} \right|_{\text{TE}} ,_{{}} \left. {\text{IL}} \right|_{\text{TM}} } \right\}}}.$$
(3)

In this equation PER|TE and PER|TM, respectively, are the polarization extinction ratio for TM and TE input (definitions will be provided shortly), and IL|TE and IL|TM, respectively, are the insertion loss for TE and TM output ports, at λ = 1.55μm. Here the maximum of the IL and minimum of the PER are taken into account to consider the worst-case scenario. According to the 3D FDTD simulation results, the largest FOM for the proposed structure is 16 and it can be achieved with Wi= 60 nm which will keep the IL below 1 dB and result in PER values higher than 16 dB (at λ = 1.55 µm). It should be mentioned that with Wi= 60 nm, there will be a refractive index mismatch of Δneff % = (2.65–1.91)/2.65 × 100 ≈ 28% (at λ = 1.55μm) between the input/output strip waveguides and the interconnecting waveguide which would result in large IL of 3 dB. But the application of the taper regions will also improve the index matching between the strip and interconnecting waveguides and reduce the IL below 1 dB. Using these values, the parameters of the proposed PBS in Fig. 3 can be summarized as: \(\varLambda\) = 170 nm, a = 100 nm, Ltaper = 20Λ, LMMI = 88Λ, Ws = 450 nm, Wa = 1.2 µm, Wi = 60 nm and WMMI = 4.8 µm.

In the next section, we will use 3D FDTD simulations to evaluate the performance parameters of the proposed PBS.

3 Simulation results and discussion

As mentioned, the performance of the designed PBS will be examined through 3D FDTD simulations in this Section. To test the operation of the proposed PBS in separating input TE/TM polarized waves, the structure is first illuminated with a TE input wave and then with a TM wave both at the design wavelength of 1.55 µm. The absolute value of the electric field distribution (i.e., |E(x,y)|) of the proposed PBS in these two cases are presented in Fig. 6a, b, respectively. As expected, for the TE polarized light, constructive interference of excited modes forms a mirrored image of the input wave at the cross port (output TE waveguide). The destructive effect of the interconnecting waveguide can also be seen in Fig. 6a. According to the discussions of the previous Section, the damaging effect of the interconnecting waveguide on the TE wave is reduced by minimizing the width of this region (i.e., Wi).

For the TM polarization, the input light is directly guided to the bar port (output TM waveguide) through the interconnecting waveguide. Figure 6b shows the ability of the 60 nm thick interconnecting region in guiding confined TM polarized waves.

The transmission spectra of bar and cross ports of the PBS when it is illuminated with input TE and TM waves are shown in Fig. 7a b, respectively. According to these figures, for the TE input, the cross-port power level is − 1.25 dB at λ = 1.55 µm and the minimum bar port power level which occurs at λ = 1.55 µm is − 18.1 dB. In addition, in the wavelength range of 1.3 µm < λ < 1.65 µm the cross and bar port power levels are, respectively, above − 2.94 dB and below − 12.6 dB. For the TM polarized input, the bar and cross port power levels are, respectively, − 0.91 dB and − 19.2 dB at λ = 1.55 µm and in the wavelength range of 1.3 µm < λ < 1.65 µm these powers are above − 1.2 dB and below − 14 dB, respectively.

Fig. 7
figure 7

Transmission spectra of the bar and cross ports of the designed PBS for: a TE polarized, b TM polarized input light

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The most important criteria for PBS evaluation are polarization extinction ratio (PER) and insertion loss (IL) which for the proposed structure of Fig. 3 can be defined as

$$\begin{aligned} \left. {\text{PER}} \right|_{\text{TE}} = 10\log \left( {\frac{{P_{\text{I}} }}{{P_{\text{cross}} }}} \right)_{{{\text{for}}_{{}} {\text{TM}}_{{}} {\text{input}}}} \hfill \\ \left. {\text{PER}} \right|_{\text{TM}} = 10\log \left( {\frac{{P_{\text{I}} }}{{P_{\text{bar}} }}} \right)_{{{\text{for}}_{{}} {\text{TE}}_{{}} {\text{input}}}} \hfill \\ \end{aligned}$$
(4)
$$\begin{aligned} \left. {\text{IL}} \right|_{\text{TE}} = 10\log \left( {\frac{{P_{\text{I}} }}{{P_{\text{cross}} }}} \right)_{{{\text{for}}_{{}} {\text{TE}}_{{}} {\text{input}}}} \hfill \\ \left. {\text{IL}} \right|_{\text{TM}} = 10\log \left( {\frac{{P_{\text{I}} }}{{P_{\text{bar}} }}} \right)_{{{\text{for}}_{{}} {\text{TM}}_{{}} {\text{input}}}} . \hfill \\ \end{aligned}$$
(5)

In these equations, PI is the input power, and Pcross and Pbar denote the output power of cross and bar ports of the designed PBS, respectively. The PER and the IL of the proposed PBS in the wavelength range of 1.3μm < λ < 1.65μm are plotted in Fig. 8a, b, respectively. According to these figures, at the wavelength of 1.55μm, the PER for TE and TM polarizations are 18 dB and 17.3 dB, respectively, and the IL of TE and TM polarizations are, respectively, 1.25 dB and 0.9 dB. The simulation results also show the PER of larger than 12 dB over the ultrawide wavelength range of 350 nm (1.38–1.63 μm) for both TE and TM polarizations and IL of smaller than 1.2 dB over the entire 350 nm bandwidth for the TM and less than 3 dB for TE polarization.

Fig. 8
figure 8

a IL spectra and b PER spectra, of the designed PBS for TE and TM polarized input light

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As can be seen in Figs. 7 and 8, there are some fluctuations in the output spectrums of the designed PBS. Similar behavior has been observed in the transmission spectrum of the MMI-based BPSs proposed in Ref. [16, 26, 27]. These fluctuations can be attributed to the Fabry–Perot (FP) oscillations caused by the reflections from input/output ends of the MMI region. The free spectral range (FSR) of the FP oscillations at the wavelength of λ is related to the length of the MMI region (length of the FP resonator) and the group index as [27]

$${\text{FSR}} = \frac{{\lambda^{2} }}{{2n_{\text{g}} L_{\text{MMI}} }}.$$
(6)

Using ng = 3.3 as the group index of the MMI region the FSR of FP oscillations at λ = 1.55μm is calculated to be; FSR= 24.3 nm. According to Fig. 8, the FSR of oscillations at λ = 1.55μm is 23 nm which in close agreement with the values calculated based on the FP model.

To reduce the amplitude of the FP oscillation, the reflection from the input/output ends of the MMI region should be reduced by improving the matching between the input/output waveguides and the MMI region (both effective index and field profile matching) with the application of longer tapers. To examine the effect of the taper length on the FP oscillation amplitude, we have increased the length of the taper regions from 3.5 to 6 μm and calculated the IL and PER for both TE and TM polarizations. The IL and PER spectrums for TM polarization, respectively, are presented in Fig. 9a, b. As can be seen, while the FSR of the FP oscillation is unchanged (since the properties of the MMI are not changed), the amplitude of the oscillation is decreased because of lower reflection from the input/output ends of the MMI region.

Fig. 9
figure 9

a IL spectra, and b PER spectra, of the TM polarization for 3.5μm and 6μm input/output tapers

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A comparison between the properties of the proposed PBS and those of the previously reported MMI-based PBSs is provided in Table 1. As can be seen, the proposed PBS provides much wider bandwidth (350 nm compared to the 84 nm bandwidth of Ref. [16]) and its overall length is smaller than the smallest reported MMI-based PBS with SOI platform by a factor of four. However, despite its interesting features, the performance of the proposed PBS is subject to variations because of the fabrication imperfections (as any other optical element). To provide an estimate of the changes in the properties of the designed PBS in response to the fabrication errors, we have performed a set of simulations with structures that their geometrical parameters have been varied from the original design. The results of these simulations are presented in the next section.

Table 1 Comparison of the properties of the proposed PBS against those of previously reported MMI-based PBS
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It worth mentioning that, there have been wideband designs for TE or TM pass polarizers such as [17] which cover the entire optical communication band. Compared to the proposed PBS the structure of [17] has larger bandwidth but it requires a much-complicated fabrication process. Since such structures do not perform polarization splitting they are not included in Table 1.

In recent years inverse design and optimization methods have beam used to design ultra-compact PBSs and power splitters with improved performances such as [28, 29], where the device is formed by selective etching of complex patterns in an SOI-based MMI. The etching pattern is a combination of pixels with a small area of < 100 nm × 100 nm and the device performance is highly sensitive to fabrication errors which can induce variations in the lateral size and line-edge roughness of each pixel [28]. Furthermore, the pixel combination is optimized for operation in a specific wavelength range and the whole design steps should be repeated to achieve the same operation in adjacent bands. In other words, the geometrical parameters are extracted from optimization which provides no design insight for extension of the device operation to other wavelength bands. In addition, according to simulation results the proposed PBS much wider operation bandwidth and lower sensitivity to the fabrication errors. For example, the bandwidth of the proposed PBS is 350 nm, while the bandwidth of the PBS of [28] is 32 nm.

4 The sensitivity of the proposed polarization beam splitter to the fabrication errors

As mentioned, MMI-based devices are more robust to fabrication errors compared to other passive optical elements such as DCs and MZIs. However, properties of the SWGs are sensitive to the variation of their geometrical parameters, and therefore, the overall performance of the proposed PBS is susceptible to the fabrication imperfections.

Implementation of the SWG structures on SOI wafers is usually realized through high precision lithography techniques such as electron-beam lithography (EBL) and then etching of the unwanted silicon regions [30]. Although the EBL method provides very high resolution, the final structure may still suffer slight size variations in the order of 10 nm. In addition, over or under etching of the unwanted silicon regions can be another source of geometrical variation of the proposed structure [31]. Both these errors (lithography and etching errors) can be modeled as a variation of the duty cycle of the SWGs. Fabrication errors can also change other geometrical parameters of the proposed PBS (such as WMMI, Wa, Wi) but based on our simulations the variations in the duty cycle of the SWG have the most profound effect on the overall performance of the PBS.

In what follows we have considered two cases, where the length of some of the silicon rods of the SWG regions (both in the multimode region and in the input/output tapers) are changed by ± 5% or ± 10% (i.e., a = 100 ± 5 nm or a = 100 ± 10 nm, respectively). In each case the length of the silicon strip of each SWG period; a is randomly changed by the amount of + Δa or − Δa (Δa is 5 nm and 10 nm in the first and second case, respectively) and the transmission spectrum of the structure for input TE and TM polarization is calculated through 3D FDTD simulations. This process is repeated five times for both Δa = 5 nm and Δa = 10 nm cases. The results of the simulations with Δa = 5 nm for TE and TM polarized input waves are presented in Fig. 10a, b, respectively.

Fig. 10
figure 10

Transmission spectra of the bar and cross ports of the proposed PBS for the size variation of ∆a = 5 nm, a for TE polarized light, and b for TM

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In these figures, the black lines represent the transmission spectra of the original structure (with no fabrication errors) and other lines represent the transmission of the five perturbed structures. According to Fig. 10, in the worst case at the wavelength of λ = 1.55μm for TE polarized input the output powers of the cross and bar ports are, respectively, − 1.5 dB and − 17.8 dB (i.e., IL = 1.5 dB and PER = 17.8 dB), and for TM polarized input the output powers of the cross and bar ports are, respectively, − 19.1 dB and − 1 dB (i.e., IL = 1 dB and PER = 19.1 dB). Furthermore, the variation of ∆a = 5 nm TE polarization extinction ratio is still more than 12 dB over the wide wavelength range of 220 nm from 1.39μm to 1.61μm and it is over 14 dB for the TM polarized light in the wavelength range of 1.42–1.63 μm.

The simulation results for the case of Δa = 10 nm for TE and TM input waves are shown in Fig. 11a, b, respectively. Again, the black lines represent the transmission spectra of the original structure and other lines represent the transmission of the five perturbed structures. It can be observed that in the worst case at the wavelength of λ = 1.55μm for TE input the output powers of the cross and bar ports are, respectively, − 1.9 dB and − 16.1 dB (i.e., IL = 1.9 dB and PER = 16.1 dB), and for TM input the output powers of the cross and bar ports are, respectively, − 18.3 dB and − 1.3 dB (i.e., IL = 1.3 dB and PER = 18.3 dB). In addition, for the variation of ∆a = 10 nm the proposed PBS exhibits PER of better than 12 dB for the TM polarized light over the wide wavelength range of 1.38–1.65 μm and higher than 11 dB for the TE polarization in the range of 1.43–1.58 μm. It can be inferred from these simulation results that, the designed PBS retains its performance even in the presence of fabrication imperfections.

Fig. 11
figure 11

Transmission spectra of the bar and cross ports of the proposed PBS for the size variation of ∆a = 10 nm, a for TE polarized light, and b for TM

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The sensitivity of the PBS performance to the fabrication errors have also been considered in the previous works [28, 32, 33]. It is mentioned that the active tuning methods such as the implementation of p-i-n diodes [32] and heating structures [33] can also be used to compensate for the effect of fabrication errors on device performance. Although the designed PBS is fabrication tolerant, its sensitivity to the fabrication errors can be further reduced by implementing a heating structure beneath or around the MMI region.

5 Conclusion

In this In this paper, we have designed and characterized a compact and ultra-wideband SOI PBS using an SWG-based MMI coupler. Birefringence engineering of MMI coupler is performed by employing the SWG to achieve a compact and wideband device. The length of the multimode region is properly optimized so that the first single mirrored image of the input field is formed at the cross port for the TE polarization, while the TM polarized light is directly transmitted to the bar port through the interconnecting strip waveguide. The simulation results exhibit the extinction ratio of more than 17 dB for both the TE and TM polarization at the wavelength of 1.55 μm. Besides, the PERs of the proposed PBS is better than 15 dB for the wavelength ranging from 1.38 to 1.63 μm and larger than 12 dB for broad operational bandwidth of 350 nm (1.3–1.65 μm) for the TE and TM polarization, respectively. In addition, the calculated IL is less than 1.2 dB in the range of 1.38–1.54 μm and 1.3–1.65 μm for the TE and TM polarizations, respectively. The overall footprint of the designed PBS is 4.8 × 21 µm2 and is compatible with the CMOS fabrication process. These values show that the proposed PBS operates in the ultra-wide wavelength range of 1.3–1.65 μm and its overall length is smaller than the smallest reported MMI-based PBS with SOI platform by a factor of four. Moreover, our simulations suggest that the designed structure can retain its performance even in the presence of fabrication imperfections.