1. Introduction

Silicon-on-insulator is quickly becoming a major platform for photonic integration, offering CMOS-compatible fabrication and a strong index contrast that enables high device density [1]. This index contrast also induces significant birefringence between the horizontal (TE) and vertical (TM) polarization states, so that devices for polarization management become of critical importance [2]. Among these, polarizers suppress one polarization without affecting the other. Specifically, TM-pass polarizers can be used to provide enhanced polarization purity in a variety of applications, including biosensing, where the TM mode offers a stronger interaction with the sensing medium than the TE mode [3], and high-speed optical communications where information is encoded independently in the TE and TM polarization states [4]. For specialized silicon thicknesses of $300\,\mathrm {nm}$ and $400\,\mathrm {nm}$, TM-pass polarizers can be implemented by designing waveguides in which the fundamental TE mode is cut-off, while the fundamental TM mode remains guided [5,6]. Conversely, in standard $220\,\mathrm {nm}$ thick silicon waveguides, TE-pass polarizers can exploit the weaker confinement of the TM mode, while preserving the more strongly confined TE mode [7–9]. For this standard thickness, TM-pass polarizers are generally more challenging to implement and often rely on more sophisticated waveguide structures to absorb the TE-polarized mode, such as plasmonic waveguides [10,11], graphene-loaded waveguides [12,13], or multilayer structures [14]. Bragg gratings or hyperuniform disordered structures that reflect the TE mode while transmitting the TM mode can be realized with simple waveguide structures and achieve excellent performance [15–19]. However, this approach reflects most of the unwanted TE polarized light directly back into the waveguide, which is undesirable for applications that are sensitive to spurious reflections.

Several ingenious solutions have been proposed to implement TM-pass polarizers on $220\,\mathrm {nm}$ thick silicon that avoid undesired back-reflection without complicating the waveguiding structure. In [20] a polarization splitting mechanism is exploited to achieve a simulated extinction ratio of $\sim 30\,\mathrm {dB}$ and insertion losses below $1.7\,\mathrm {dB}$ over a $200\,\mathrm {nm}$ bandwidth, albeit using an air cladding which can hamper further integration. A Bragg grating with narrow slots is proposed in [21], with a simulated extinction ratio of $\sim 40\,\mathrm {dB}$, insertion losses below $0.5\,\mathrm {dB}$ and back-reflections below $-10\,\mathrm {dB}$ over a $100\;\mathrm {nm}$ bandwidth; however, the narrow $50\,\mathrm {nm}$ slots can prove difficult to fabricate. Using a combination of silicon and silicon nitride waveguides, a TM-pass polarizer with an operational bandwidth of $110\,\mathrm {nm}$ is reported in [22].

Here we present a TM-pass polarizer for standard $220\,\mathrm {nm}$ silicon that can be fabricated with a single etch-step and achieves an extinction ratio in excess of $20\,\mathrm {dB}$, insertion losses below $0.5\mathrm {dB}$ and back-reflections of the order of $-20\,\mathrm {dB}$ in a bandwidth of $150\,\mathrm {nm}$. We achieve this by leveraging the flexibility of subwavelength metamaterials [23,24] to engineer a Bragg grating that reflects the fundamental TE mode into the first order TE mode, which is readily radiated in a waveguide bend, without affecting the fundamental TM mode.

2. Device structure and operating principle

As illustrated in Fig. 1, our polarizer is based on a subwavelength grating (SWG) structure, in which every second element is tilted by an angle $\theta$. The center-to-center pitch ($\Lambda _\mathrm {SWG}$) is chosen such that for $\theta = 0\,^\circ$ [see Fig. 1(c)] the structure operates in the subwavelength regime for the fundamental TE and TM mode, thus transmitting light with negligible losses. A non-zero tilt angle has virtually no effect on the TM mode but introduces an antisymmetric perturbation for the TE mode [25]. This perturbation, shown in Fig. 1(d) and (e), has a period $\Lambda _\mathrm {B}=2\Lambda _\mathrm {SWG}$ and induces coupling between the forwarding propagating fundamental TE mode and the backward propagating first order TE mode. TE polarized light is thus reflected back and simultaneously converted into the first order TE mode [26], which can be readily eliminated using a simple curve whose design is shown in section 3.

 

Fig. 1. (a) Schematic of the TM-pass polarizer. For TE polarization the tilted segments form a Bragg reflector that converts the forward propagating fundamental mode into the backward propagating first order mode, whereas for TM polarization the structure behaves as an homogeneous subwavelength metamaterial. (b) Detail of the taper section. (c-e) Geometry of the unit cell of the periodic section for a tilt of (c) $\theta =0\,^\circ$, (d) $\theta =8\,^\circ$ and (e) $\theta =15\,^\circ$, respectively. For the sake of clarity we only represent a small number of periods in the Bragg and the taper sections.

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For our design we consider a $220\,\mathrm {nm}$ thick silicon layer, with a $2\,\mathrm{\mu}$m-thick $\mathrm {SiO}_2$ buried layer and cladding. Referring to Fig. 1(c), which shows the structure without any tilting, we use a typical duty-cycle ($a/\Lambda _\mathrm {SWG}$) of $50\,\%$ for the SWG waveguide, and set the width to $w=1.1\,\mathrm{\mu} \mathrm {m}$ so that both the fundamental TE and TM modes, as well as the first order TE mode are well guided. The period of the tilted structure [see Fig. 1(d) and (c)] is chosen as $\Lambda _\mathrm {B}=2\Lambda _\mathrm {SWG}=440\,\mathrm {nm}$, which centers the polarizer response at the design wavelength $\lambda = 1.55\,\mathrm{\mu} \mathrm {m}$, as discussed below.

In order to rigorously analyze how the structure reflects the fundamental TE mode into the first order TE mode we use band diagrams obtained with the MPB software package [27]. Figure 2 shows the first Brillouin zone of the structure with a zero tilt angle, i.e. without any perturbation. This diagram shows the normalized wavenumbers $k\Lambda _\mathrm {B}/(2\pi )$ of the TE modes versus wavelength, with $k=n_{\mathrm {eff}}2\pi /\lambda$. For this zero tilt configuration, the structure operates in the subwavelength regime, so that no bandgaps appear: since the horizontal axis of the band diagram is expressed in terms of $\Lambda _\mathrm {B} = 2\Lambda _\mathrm {SWG}$ the bandgap of this subwavelength structure would only appear at $k\Lambda _\mathrm {B}/(2\pi )=2k\Lambda _\mathrm {SWG}/(2\pi )=1$. The different curves that form the diagram correspond to the fundamental TE mode travelling in the forward and backward directions ($\mathrm {TE}_0^+$ and $\mathrm {TE}_0^-$), and the first order TE mode travelling in the forward and backward directions ($\mathrm {TE}_1^+$ and $\mathrm {TE}_1^-$). To help visualize this, representative mode profiles are shown in the insets. At the intersection marked with a star in Fig. 2, the phase matching condition between the $\mathrm {TE}_0^+$ mode and the $\mathrm {TE}_1^-$ mode is met, and hence there is a potential bandgap for the desired backwards mode conversion. To center the polarizer response at the desired wavelength we note that since $k_0^-=2\pi /\Lambda _\mathrm {B} - k_0^+$, the phase matching condition $k_0^-= k_1^+$ can be rewritten as $k_0^+ + k_1^+ = 2\pi /\Lambda _\mathrm {B}$, from where the adequate pitch is readily obtained [28]. Finally we note that because of symmetry considerations all the information about the structure is contained in the right part of Fig. 2, so in the following we only display the band diagrams for positive values of $k$.

 

Fig. 2. Dispersion diagram of the TE modes calculated for one period of the structure with a pitch of $\Lambda _\mathrm {B}=440\,\mathrm {nm}$ and a width of the segments of $w= 1.1\,\mathrm{\mu} \mathrm {m}$ when no tilt is applied ($\theta = 0\,^\circ$). The star marks the operating point of the device.

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Referring to Fig. 3(a), an $\theta = 8^\circ$ tilt introduces the necessary asymmetric perturbation for power transfer between the $\mathrm {TE}_0^+$ and $\mathrm {TE}_1^-$ modes to take place, so that the desired bandgap materializes with a $95\,\mathrm {nm}$ bandwidth. From Fig. 3(b) we note that an increase in the tilt angle to $\theta = 15^\circ$ results in broader bandgap, with a $150\,\mathrm {nm}$ bandwidth. We choose $\theta = 15^\circ$ as our design value, since further increasing the tilt angle results in an appreciable $\mathrm {TE}_0^+$ to $\mathrm {TE}_0^-$ bandgap, which results in undesired back-reflections into the fundamental mode. The insets in Fig. 3(b) show the Bloch-Floquet mode profiles at the edge of the bandgaps. Although not shown in Figs. 3(a) and (b), the bandgaps for the TM modes appear only at wavelengths shorter than $1.4\,\mathrm{\mu} \mathrm {m}$, so that these modes are can potentially travel through the device without losses.

 

Fig. 3. Dispersion diagrams of the TE modes calculated for one period of the structure with a pitch of $\Lambda _\mathrm {B}=440\,\mathrm {nm}$ and a width of the segments of $w= 1.1\,\mathrm{\mu} \mathrm {m}$. As the tilt angle is increased from (a) $\theta = 8\,^\circ$ to (b) $\theta = 15\,^\circ$, the bandgap increases.

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3. Polarizer design

Referring to Fig. 1(a), to quantify the performance of the device we define $p_\mathrm {in/out}^{\mathrm {TX}n\pm }$ as the power in the input/output port of the device, within the forward($+$)/backward($-$) propagating $n$-th order mode with polarization TX, when the fundamental TX mode is launched into the input port with $1\,\mathrm {W}$ power. The key parameters of the polarizer can then be expressed as: insertion losses $\mathrm {IL} [\mathrm {dB}] = -10\log _{10}\left (p_\mathrm {out}^{\mathrm {TM}0+}\right )$, extinction ratio $\mathrm {ER} [\mathrm {dB}] = 10\log _{10}\left (p_\mathrm {out}^{\mathrm {TM}0+}/ p_\mathrm {out}^{\mathrm {TE}0+}\right )$, back reflection into the fundamental mode $\mathrm {BR0} [\mathrm {dB}] = 10\log _{10}\left (p_\mathrm {in}^{\mathrm {TE}0-}\right )$, and back reflection into the first order mode $\mathrm {BR1} [\mathrm {dB}] = 10\log _{10}\left (p_\mathrm {in}^{\mathrm {TE}1-}\right )$. Our goal is to minimize insertion losses, while maximizing the extinction ratio and minimizing the back-reflection into the fundamental mode.

Using MEEP, an open-source 3D-FDTD simulator [29], we first explore the qualitative behavior of the device without the tapered section shown in Fig. 1. As expected from Fig. 3(a) and (b), increasing the tilt angle $\theta$ yields a broader operational bandwidth due to the wider bandgap. However, for TE polarization this also reinforces the local index perturbation, thus increasing the undesired back-reflection into the fundamental TE mode. Increasing the waveguide width has a similar effect: the stronger confinement enhances the perturbation, which is beneficial in terms of bandwidth, but detrimental in terms of undesired back-reflections. The pitch governs the central wavelength of the bandgap and has to be adjusted together with the width and the tilt angle to achieve the desired central wavelengths.

Excluding the tapered section shown in Fig. 1, we obtained the wavelength response of the device with the following structural parameters: width $w= 1.1\,\mathrm{\mu} \mathrm {m}$, tilt angle $\theta = 15\,^\circ$, pitch $\Lambda _\mathrm {B} = 440\,\mathrm {nm}$, and a length of $N = 20\,\mathrm {periods}$. As shown in Fig. 4(a) with these parameters the extinction ratio (ER) of the device is above $20\,\mathrm {dB}$ in a $150\,\mathrm {nm}$ bandwidth. However, the insertion losses for TM polarization (IL) are comparatively high, around $2\,\mathrm {dB}$, and while most of the TE polarized light is reflected into the higher order mode (BR1), the reflection into the fundamental mode (BR0) is still around $-10\,\mathrm {dB}$.

 

Fig. 4. 3D FDTD simulation of the full structure (a) without and (b) with the tapers. The dashed line represents the $\mathrm {TM}_0$ insertion losses (IL) corresponding to the right axis. The continuous lines, corresponding to the left axis, represent the TE back reflection into both the fundamental (BR0) and first order (BR1) modes, and the extinction ratio (ER).

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These effects can be further mitigated by introducing a smooth transition between the access waveguide and the subwavelength structure. We employ a two stage taper, as shown in Fig. 1. First the waveguide width is decreased from $w_\mathrm {acc}=0.5\,\mathrm{\mu} \mathrm {m}$ to $w_\mathrm {taper}=0.3\,\mathrm{\mu} \mathrm {m}$ over a length of $L_t=1\,\mathrm{\mu} \mathrm {m}$, to decrease the effective index of the input mode. A second section is then used to gradually increase the width of the periodic structure, while eliminating the bridging elements; this section is $N_{\mathrm {taper}}=8\,\mathrm {periods}$ long. The complete device, with both the input and output tapers has a total length of $17.84\,\mathrm{\mu} \mathrm {m}$. Figure 4(b) shows the improved performance of the device when the tapers are included: the extinction ratio and insertion losses are better than $20\,\mathrm {dB}$ and $0.4\,\mathrm {dB}$, respectively in a bandwidth of $150\,\mathrm {nm}$. The undesired back-reflections are kept below $-20\,\mathrm {dB}$ for most of the operational bandwidth for both polarizations: for TE polarization virtually all the reflected power is coupled into the first order mode, whereas for TM virtually all the power is transmitted forward. A simple curve with a $14\,\mathrm{\mu} \mathrm {m}$ radius and a waveguide width of $400\,\mathrm {nm}$ can be used to radiate the power in the backward propagating first order TE mode without affecting the fundamental TE and TM modes, with insertion losses for the $\mathrm {TE}_1$ of above 18 dB in the operational bandwidth, and for the $\mathrm {TM}_0$ and $\mathrm {TE}_0$ below 0.15 and 0.03 dB, respectively. Indeed, such a curve is most probably already present for routing purposes in the photonic circuit of interest, so that no extra space is required for the elimination of the higher order mode. Another alternative is to use a single SWG waveguide segment – a 500nm wide SWG waveguide with 220 nm thickness, 220 nm pitch, 50% duty cycle and 40 periods, results in an extinction of the unwanted higher-order TE mode of 20dB. Leakage losses due to weak confinement of the TM mode were studied by incorporating the effect of finite $2\,\mathrm{\mu} \mathrm {m}$ buried oxide layer, showing negligible losses of at most $0.1\,\mathrm {dB}$ in the operational bandwidth.

The polarizer’s behaviour is shown in Fig. 5, which represents the propagation when the $\mathrm {TE}_0$ or $\mathrm {TM}_0$ are excited at the operation wavelength of $1.55\,\mathrm{\mu} \mathrm {m}$ calculated via 3D FDTD simulation. In Fig. 5(a) the $\mathrm {TE}_0$ mode is excited, being almost totally reflected into the $\mathrm {TE}_1$ mode, with no transmission at all. In Fig. 5(b), the $\mathrm {TM}_0$ mode travels through the structure with almost no losses.

 

Fig. 5. Propagation through the proposed structure when the (a) $\mathrm {TE}_0$ and (b) $\mathrm {TM}_0$ modes are excited, representing the $|E_x|$ and $|E_y|$ fields respectively, obtained with 3D FDTD computations. The white line indicates the position of the source.

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In terms of fabrication tolerances, the most critical parameters are the duty-cycle of the subwavelength structure and the thickness of the silicon layer. We assume a $\pm 5\,\%$ error in the duty-cycle of the subwavelength structure, which corresponds to a physical variation of $11\,\mathrm {nm}$ in the length, $a$, of the silicon strips [see Fig. 1(c)]. 3D FDTD simulations shown in Fig. 6(a) and (b) reveal that such a change in duty-cycle causes a significant change in the polarizer’s central operation wavelength of approximately $40\,\mathrm {nm}$. On the other hand, variation in silicon layer thickness only have a minor impact on device performance, as shown in Figs. 6(c) and 6(d). When a −10 nm thickness variation results in a blue shift of the response by approximately 15 nm, while +10 nm thickness variation yields a red shift of approximately 10 nm. Furthermore, incomplete filling of the gaps with the top cladding has been observed in several subwavelength structures, see for example [30]. In our design the most critical regions are the inner corners of the “N” shaped waveguide with an acute angle of 15 degrees [see Fig. 1(e)]. We have assumed that within these inner corners a triangular region that is 100 nm wide and 100 nm ${\times }$ tan(15$^{\circ }$) = 27 nm long region remains completely void. This results in only a minor 7 nm blue shift of the bandgap. However, the performance and broadband operation is maintained, so that the device could potentially be thermally re-tuned. Alternatively bricked subwavelength structures [31], which exhibit a rectangular shape that is more amenable to fabrication, could be investigated.

 

Fig. 6. 3D FDTD simulation of the full structure considering a duty-cycle error of (a) $-5\,\%$ and (b) $+5\,\%$, as well as silicon layer thickness variations of (c) −10 nm and (d) +10 nm around the nominal thickness of h = 220nm.

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4. Conclusions

We have proposed a novel approach to design TM-pass polarizers, based on tilted subwavelength gratings. Leveraging the asymmetry of the structure, the fundamental TE mode is reflected as a higher order mode which is readily eliminated to avoid spurious back-reflections. The TM mode passes through the structure with virtually no losses by virtue of optimized tapers at the input and output of the periodic structure. Full 3D FDTD simulations predict insertion losses below $0.4\,\mathrm {dB}$ and an extinction ratio in excess of $20\,\mathrm {dB}$, while keeping unwanted reflection below $-20\,\mathrm {dB}$. The device could be fabricated with a single etch step, thus opening new venues for the implementation of CMOS compatible TM polarizers.