Framework for tunable polarization state generation using Berry&#x2019;s phase in silicon waveguides

Ryan J. Patton; Ronald M. Reano

doi:10.1364/OE.384543

1. Introduction

Manipulating the polarization of light has applications in numerous branches of science and engineering. Polarimetry is utilized, for example, in ellipsometry, remote sensing, astrophysics, and medical imaging [1–4]. In communications, polarization division multiplexing can increase channel capacity and polarization dependent characteristics can impact link performance [5–7]. Control of optical polarization is also important for quantum applications such as coherent interconnects and quantum key distribution [8,9]. At the chip-scale, integrated waveguides become increasingly polarization sensitive due to high confinement and high index contrast. As a result, most devices operate only in either the fundamental quasi-transverse electric (TE) mode or the quasi-transverse magnetic (TM) waveguide mode. Accessing intermediate polarization states would add new degrees of freedom to photonic integrated circuits.

There has been extensive work on passive polarization handling components, such as polarization rotators, splitters, and combined polarization splitter-rotators [10–15]. To date, however, there have been few demonstrations of active polarization controllers in the silicon-on-insulator (SOI) platform. Polarization controllers based on adiabatic mode converters combined with Mach-Zehnder interferometers, or trench-based rotators with in-line phase shifters have been demonstrated [12,16]. The need to cascade different components together, such as polarization splitters, rotators, and 3 dB couplers leads to additional loss mechanisms and millimeter scale devices, resulting in total insertion losses of 2 to 3 dB and increased wavelength sensitivity.

In this work, we present an integrated polarization state generator that achieves ±30 dB polarization extinction ratio (PER) between TE and TM modes. Any intermediate polarization state can be generated. Our approach utilizes a single continuous waveguide such that insertion loss is primarily due to propagation loss. We use a graphically inspired approach to establish the conditions for TE to TM conversion within the entire parameter space for a passive device. The results define regions where dynamic tuning of the output polarization is possible for active devices. We use numerical modeling to show ±30 dB tuning of the output PER for a range of devices.

The approach uses silicon strip waveguides that rotate optical polarization using Berry’s phase. One manifestation of Berry’s phase is the rotation of optical polarization when light travels along a curvilinear trajectory such as a helix, which has been demonstrated in optical fiber, bulk optics, and integrated waveguides [17–22]. The polarization rotation is due to a topologically induced phase shift between the right- and left-hand circular polarizations [18,21,23]. Berry’s phase can be implemented using a single continuous waveguide and is inherently insensitive to wavelength due to its topological origins [22,23].

Waveguides in the silicon-on-insulator (SOI) platform can be released from the substrate using nano-fabrication processes such that they deflect out-of-plane and exhibit Berry’s phase [22,24,25]. Such a waveguide is shown schematically in Fig. 1(a). The light guiding core is silicon, the cladding is a SiO₂ bilayer consisting of the thermally grown buried oxide layer of the SOI wafer, and a top cladding deposited by plasma enhanced chemical vapor deposition (PECVD). Residual stress in the bilayer cladding causes out-of-plane deflection [22,24,25]. For TE and TM modes with effective indices n_TE and n_TM, the polarization rotation is maximum when n_TE = n_TM. We refer to this condition as a perfectly square waveguide. In this case, a waveguide with deflection angle θ will rotate optical polarization by angle 2θ [22,23]. In theory, a perfectly square waveguide deflected by 45° would achieve 90° TE-to-TM polarization rotation. Fabricated waveguides, however, are limited to modest deflection angles, and are not perfectly square, which limits the amount of polarization rotation. Incorporating the out-of-plane waveguide into a resonator can compensate by allowing polarization rotation to accumulate over multiple round trips, but introduces wavelength dependence [20,22]. A non-resonant approach is to use a cascade of multiple out-of-plane waveguides, connected by in-plane sections, as shown in Fig. 1(b). We note that only the 180° bend portions of the out-of-plane sections contribute to polarization rotation. The S-bend that brings the waveguide back to the chip surface does not contribute to polarization rotation, and is functionally considered as part of the in-plane waveguides.

Fig. 1. (a) Out-of-plane waveguide that rotates optical polarization due to Berry’s phase. The field plots of |E|² are calculated via three dimensional finite difference time domain computations using Δn_lin = 0, θ = 22.5°, R = 15 μm, and λ = 1550 nm. (b) Schematic of a six period device consisting of cascaded in- and out-of-plane waveguides. (c) Block diagram of the device, including relevant design parameters.

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A block diagram of the device is shown in Fig. 1(c), and lists the relevant design parameters. For the out-of-plane waveguides, Δn_lin is half the modal birefringence, i.e. Δn_lin = (n_TE − n_TM)/2. Parameter θ is the deflection angle, and L = πR is the length. The in-plane waveguides are characterized by the differential phase shift ρ = k2Δn′_linL′ accumulated between the TE and TM modes, where k = 2π / λ is the free-space wavenumber, and 2Δn′_lin, and L′ are the in-plane waveguide birefringence and length, respectively. In the passive case, the tuning electrodes in Fig. 1(b) are unused, and the device is periodic so that ρ_i =ρ_i+1 = ρ. In previous work, we presented a passive TE-to-TM rotator for one set of parameter values Δn_lin, θ, L, ρ, and N [23]. Here, we generalize our findings by establishing conditions for passive TE-to-TM conversion for the entire parameter space (Δn_lin, θ, L, ρ, and N). We then use the results to demonstrate dynamic control of the output PER and polarization state when electrically tuning the in-plane phase shifts ρ_i in an active device via the tuning electrodes, utilizing for example, the thermo-optic or plasma dispersion effect.

2. Analytical framework

The framework is motivated by visualizing the polarization evolution on the Poincaré sphere. We establish conditions for TE-to-TM conversion based on the Stokes vector evolution, then derive the corresponding equations. The Stokes vectors are calculated using a Jones matrix formalism (see appendix) [23]. In Fig. 2(a) we show the Stokes vector evolution on propagation through a perfectly square (300 × 300 nm Si core) out-of-plane waveguide, and a nominally square (295 × 300 nm Si core) out-of-plane waveguide. In this work, we assume TE input polarization. Stokes parameter S₁ = ±1 corresponds to TE and TM polarizations, S₃ = ±1 corresponds to right- and left-hand circular polarizations, the equator corresponds to linear polarizations, and the remainder of the sphere’s surface corresponds to elliptical polarizations. Figure 2(a) also shows the corresponding polarization eigenstates of the Jones matrices for the out-of-plane waveguides (see appendix). When the waveguide is perfectly square, it acts as an ideal polarization rotator, evolving the polarization along the linear states of the equator. The corresponding eigenstates are right- and left-hand circular polarizations, as expected from Berry’s phase [23].

Fig. 2. (a) Evolution of the Stokes vectors on the Poincaré sphere for a single out-of-plane waveguide with θ = 15°, R = 20 μm, and λ = 1550 nm. Perfectly square and nominally square cases correspond to Δn_lin = 0 (300 × 300 nm Si core) and Δn_lin = −7.2 × 10⁻³ (295 × 300 nm Si core) respectively. (b) Evolution of the Stokes vector for the nominally square case with N = 6, which achieves TE-to-TM polarization rotation. The in-plane differential phase shift is ρ = −2.444 rad.

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Any power conserving and non-depolarizing optical element, i.e. an element that can be described by a Jones matrix, can be represented in Stokes space as a transformation of the input Stokes vector by rotating it about the corresponding eigenstate. The amount of rotation in Stokes space is equal to the acquired phase shift between the eigenstates [26]. In Fig. 2(a), as the waveguide becomes less square, the circular eigenstates travel along the S₁-S₃ meridian line towards S₁ = ±1. This causes the polarization evolution to curl back on itself, and reduces the amount of polarization rotation. By cascading multiple in- and out-of-plane waveguides together as shown in Fig. 1(b), TE-to-TM polarization rotation can be achieved as shown in Fig. 2(b). The in-plane sections are designed to add an appropriate phase shift ρ between the TE and TM modes, such that the partial polarization rotations of each out-of-plane waveguide can accumulate. The eigenstates of the in-plane sections are S₁ = ±1 (TE and TM), so that the Stokes vectors indicated by the gray curves in Fig. 2(b) are parallel to the S₂-S₃ plane. Below, we establish under what conditions the scenario in Fig. 2(b) occurs, i.e. passive TE-to-TM conversion is achieved. The results are used in Section 3 to demonstrate dynamic tuning of the output polarization.

An important observation in Fig. 2(b) is that the output Stokes vectors of each in- and out-of-plane section are evenly spaced, that is, separated by equal arc length segments, along two symmetric geodesic curves. A concatenation of Jones matrices, in this case for in- and out-of-plane waveguides, can also be represented in Stokes space as a single rotation about a new eigenvector, as opposed to multiple rotations about the eigenvectors of the individual matrices [26]. Thus, the cascaded structure can be viewed as rotating the input Stokes vector in discrete steps about a new eigenvector along the geodesic curve. In Fig. 2(b), this corresponds to the input Stokes vector at S₁ = +1 traveling along the geodesic to point 1 as the light propagates through the first unit cell in Figs. 1(b)–1(c). The second unit cell rotates the Stokes vector from point 1 to point 2, the third unit cell from point 2 to point 3, and so on. The steps are evenly spaced because the structure is periodic. There are two symmetric geodesics because the unit cell in Fig. 1(c) could be equivalently defined with the in-plane section first, followed by the out-of-plane section.

As we have observed, converting the polarization from TE to TM requires the Stokes vector to travel along a geodesic curve from S₁ = +1, to S₁ = −1. To do this, we require two criteria using the definitions in Fig. 3:

1. The angular separation between polarizations along the geodesic must be α =π / N, since TE-to-TM conversion corresponds to an angle of π, and there are N periods.
2. The differential phase must be ρ = 2θ_geo, since each in-plane section rotates the Stokes vector by 2θ_geo, where θ_geo is the angle of the geodesic relative to the equator.

Fig. 3. Parameters definitions. Angles θ_geo and ψ are in the range [−π/2, +π/2].

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Both criterion depend on the location of point A in Fig. 3, which is the Stokes vector at the output of the first out-of-plane section. The location of point A depends on the parameters Δn_lin, θ, and L. The first criterion connects these parameters to the number of periods N, and the second criterion connects them to the in-plane parameter ρ. When combined, the two criteria determine the conditions for TE to TM conversion for any combination of Δn_lin, θ, L, ρ, and N.

We use the two criteria above to derive corresponding analytical equations. We refer the reader to the appendix for the detailed derivation of the following expressions. First, we find the normalized Stokes vector at point A in Fig. 3

(1)$${{\boldsymbol s}_\textrm{A}} = \left( {\begin{array}{{c}} {\frac{{\Delta n_{\textrm{lin}}^2 - \Delta n_{\textrm{cir}}^2}}{{{\delta^2}}}{{\sin }^2}\phi + {{\cos }^2}\phi }\\ {2\frac{{\Delta {n_{\textrm{cir}}}}}{\delta }\sin \phi \cos \phi }\\ { - 2\frac{{\Delta {n_{\textrm{lin}}}\Delta {n_{\textrm{cir}}}}}{{{\delta^2}}}{{\sin }^2}\phi } \end{array}} \right),\;\;\textrm{where}\;\;\;\;\;\begin{array}{{l}} {\Delta {n_{\textrm{cir}}} = 2\theta /kL}\\ {\phi = k\delta L}\\ {{\delta ^2} = \Delta n_{\textrm{lin}}^2 + \Delta n_{\textrm{cir}}^2} \end{array}.$$

Quantity 2Δn_cir is the effective circular birefringence due to Berry’s phase in the out-of-plane waveguide. It is proportional to the solid angle subtended in k-space by the light’s trajectory, which is 2θ in this case [23]. Quantity 2δ is the birefringence between the eigenstates of the out-of-plane waveguide. When Δn_lin = 0, δ = |Δn_cir|, so the eigenstates are circular polarizations. Conversely, when Δn_cir = 0, δ = |Δn_lin|, so the eigenstates are TE and TM, and no polarization rotation occurs. To satisfy criterion 1, we take the dot product s_A · ŝ₁ = cos (α), where ŝ₁ is the unit vector along the S₁ axis, and require α =π / N, yielding

(2)$$\cos \left( {\frac{\pi }{N}} \right) = \frac{{\Delta n_{\textrm{lin}}^2 - \Delta n_{\textrm{cir}}^2}}{{{\delta ^2}}}{\sin ^2}\phi + {\cos ^2}\phi.$$

To satisfy criterion 2, we find the Stokes vectors of the geodesic curve which are given by

(3)$${{\boldsymbol s}_{\textrm{geo}}} = \left( {\begin{array}{{c}} {\cos \alpha }\\ {\cos {\theta_{\textrm{geo}}}\sin \alpha }\\ {\sin {\theta_{\textrm{geo}}}\sin \alpha } \end{array}} \right).$$

At point A, s_A = s_geo, so we can solve for θ_geo by equating the ratios of S₃/ S₂ which yields

(4)$$\tan {\theta _{\textrm{geo}}} ={-} \frac{{\Delta {n_{\textrm{lin}}}}}{\delta }\tan \phi.$$

Together, Eqs. (2) and (4) determine the conditions for TE to TM conversion, when ρ = 2θ_geo. To further generalize Eq. (2), we find the Stokes vectors of the two eigenstates of the out-of-plane waveguide

(5)$${{\boldsymbol s}_ \pm } = \frac{1}{\delta }\left( {\begin{array}{{c}} { \mp \Delta {n_{\textrm{lin}}}}\\ 0\\ { \pm \Delta {n_{\textrm{cir}}}} \end{array}} \right).$$

Then the first term on the right side of Eq. (2) can be written in terms of the angle ψ in Fig. 3 which the eigenstate makes with the S₁ axis yielding (see appendix)

(6)$$\cos \left( {\frac{\pi }{N}} \right) = \cos 2\psi {\sin ^2}\phi + {\cos ^2}\phi.$$

Equation (6) is independent of specific choice of device parameters such as L, λ, θ, and Δn_lin, but depends only on their combination which determines the orientation ψ and phase shift ϕ of the eigenstates of the Jones matrix for the out-of-plane waveguide. Equation (6) is generally applicable to any two mode coupling problem that can be described with a Jones matrix type formalism. The right hand side can be viewed as a surface with heights ranging from −1 to 1, which is a function of ψ, and ϕ. The left hand side can be viewed as a set of horizontal planes at discrete heights ranging from −1 to 1 for N = 1 to infinity. The intersection between the planes and the surface correspond to complete TE-to-TM mode conversion. We plot the intersection curves of Eq. (6) in Fig. 4(a) for N = 2 to 8 periods, forming a contour plot of the surface. Case N = 1 corresponds to single points at ψ = ±π/2, and ϕ =π/2, and 3π/2. The solution curves are symmetric about ψ = 0, and periodic with respect to ϕ. There is a gap at ϕ =π, 2π, etc. where polarization rotation is impossible, which corresponds to point A in Fig. 3 completing a circle about the eigenstate and returning to the launch state at S₁ = 1.

Fig. 4. (a) The solutions to Eq. (6) for N = 2 to N = 8. (b) The solutions to Eq. (2) plotted in terms of out-of-plane waveguide deflection angle θ, and birefringence parameter Δn_lin.

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For our purposes, we also examine the solution space in terms of the explicit device parameters. In Fig. 4(b) we plot the solution curves of Eq. (2) as a function of waveguide deflection angle θ, and birefringence parameter Δn_lin using the definitions in Eq. (1) with R = 20 μm (L = πR) and λ = 1550 nm. We overlay the curves corresponding to the gap regions, ϕ =π, 2π, and similar curves for the case ϕ =π/2, 3π/2. The latter curves correspond to point A in Fig. 3 landing on the meridian, which yields the maximum amount of polarization rotation per unit cell, and can be important for minimizing total device length. A fabricated device can be viewed as a single point landing on Figs. 4(a) or 4(b). Critically, the parameters of the out-of-plane waveguide must land on a solution curve for TE-to-TM conversion to be possible. Then, the in-plane sections must provide the appropriate differential phase shift ρ = 2θ_geo, where θ_geo is found using Eq. (4). These are challenging requirements to meet using a purely passive device. Below, we examine the capabilities of an active device to not only achieve TE-to-TM polarization rotation, but also to dynamically tune the output polarization state to any point on the Poincaré sphere.

3. Active tuning

The parameters that are feasibly tuned are the wavelength, waveguide birefringence, and deflection angle. Since the wavelength is typically fixed, we focus on tuning the birefringence, and leave tuning the deflection angle for future work. Tuning the birefringence is most practical for the in-plane sections, and corresponds to tuning the in-plane differential phase shift ρ. The out-of-plane waveguide birefringence parameter Δn_lin remains fixed.

As an illustrative example, we consider using differential phase shifters to control the output PER for the cases N = 3, and N = 6. Differential phase shifters are implemented by tuning the waveguide birefringence, e.g. through waveguide heaters or PIN junctions, such that ρ is a function of applied voltage. A schematic of the N = 3 and N = 6 cases is shown in Fig. 5(a). We note that a similar architecture has been utilized with trench style polarization rotating elements [16,27]. The last phase shifter (regardless of N) does not contribute to the output PER. Thus, for N = 2, there is only one shifter to control, and the parameters must be on the N = 2 solution curve to achieve TE-to-TM conversion. For N > 2, differing phase shifts in the in-plane sections breaks the assumption of periodicity that was used to establish the design criterion above. We find, however, that the ideal periodic cases define regions where active tuning is possible. Additionally, we note that for N > 3, one could use N – 1 separate voltages to control each phase shifter independently to set the output PER. In most cases, however, only two independent controls are necessary. For example, for N = 6 in Fig. 5(a), shifters one, three, and five are connected to the same voltage and produce the same phase shift ρ₁. A third voltage can be used in the final section to tune the output polarization state via ρ₃ when working with an intermediate PER, for example to generate circular polarization. The approach enables full coverage of the Poincaré sphere, limited only by the resolution of the tuning elements. Other orderings of the phase shifters are possible, and yield qualitatively equivalent results.

Fig. 5. (a) Block diagrams of cases N = 3 and N = 6 periods. (b) Solution curves from the first quadrant of Fig. 4(b). (c) PER as a function of differential phase shifts ρ₁ and ρ₂ for the two hypothetical device points in (b) with N = 3 and N = 6.

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In Fig. 5(b) we reproduce the unique portion of the solution curves from Fig. 4(b), with the N = 3 and N = 6 curves highlighted, and two x’s marking hypothetical devices (1) and (2). We note that the markers indicate the fixed parameters Δn_lin and θ of the out-of-plane waveguides; any number of periods N, and any in-plane phase shifts ρ₁ and ρ₂ can be chosen for either device. The output PER for the two devices as a function of differential phases ρ₁ and ρ₂ is shown in Fig. 5(c) for N = 3 and N = 6. The PER is defined as |a_TM|² / |a_TE|², where a_TM and a_TE are the complex modal weights (see appendix). Device (1) can be tuned between TE and TM for both N = 3 and N = 6. Device (2) can only tune to TM for the N = 6 case. The result is explained by an examination of Fig. 5(b).

The upper left region of Fig. 5(b) consists of large deflection angles and low birefringence, corresponding to greater polarization rotation. Conversely, the lower right region corresponds to reduced polarization rotation. The curves indicate where the polarization rotation adds up to exactly 90° for a particular number of periods. Thus, device (1) has excess polarization rotation for both N = 3 and N = 6, since it is to the upper left of both curves. Active tuning allows device (1) to achieve exactly 90° of polarization rotation. Device (2) is below the N = 3 curve, and therefore when using only 3 periods, there is less than 90° of polarization rotation regardless of the tuning.

We repeat the analysis of Fig. 5 for a range of devices of varying θ and Δn_lin. The results are summarized in Fig. 6, with (a-d) corresponding to N = 3 to 6. The marker locations indicate the device parameters; the shape and color indicate maximum possible PER. Assuming TE input polarization, a red x implies that the output polarization cannot be tuned to TM. A blue o implies that the output can be tuned to TM, but not to TE. A green + implies that the device can fully tune the output PER between TE and TM, which is the most desirable scenario.

Fig. 6. The maximum PER achieved via tuning the differential phases for a range of devices with varying θ and Δn_lin. The total number of periods is (a) N = 3, (b) N = 4, (c) N = 5, and (d) N = 6. In all cases the out-of-plane waveguide radius is 20 μm, the wavelength is 1550 nm, and the input light is TE.

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In general, a device with θ and Δn_lin falling inside a solution curve can achieve tunable TE-to-TM conversion. The exception is at the extreme upper left region around θ = 45°, where full tuning may not be possible. The difficulty is that θ = 45° corresponds to 90° of polarization rotation when Δn_lin = 0. A device with an even number of periods N nominally has 180° of polarization rotation, so that the initial output is nearly TE, and it is difficult to tune to TM. This region, however, is not representative of fabricated devices, which have more modest deflection angles. Figure 6 also gives insight into robustness against fabrication errors, showing that the fabricated device parameters can lie anywhere in the green + region and still achieve arbitrary polarization state generation. Another unique feature in Fig. 6 is the additional lobe of curves around Δn_lin = 0.015, which define a second region where TE-to-TM conversion is possible. In general, there are an infinite number of lobes for increasing Δn_lin due to the periodicity with ϕ in Fig. 4. However, they require increasingly larger N to access with reasonable deflection angles.

To study the wavelength dependence, we calculate the N = 6 solution curve for out-of-plane waveguide radius R = 20 μm and 30 μm, and wavelength λ = 1500 nm and 1600 nm. The results are shown in Fig. 7. Only chromatic dispersion is included here. We have previously observed that waveguide and material dispersion can offset chromatic dispersion, improving the overall wavelength sensitivity [23]. We find that for a fixed radius, the curves are only weakly wavelength dependent. This implies that arbitrary polarization state generation can be achieved over a large bandwidth, as long as the device parameters remain inside the curve as the wavelength changes. For a fixed tuning state, i.e. fixed values of ρ₁ and ρ₂, the change in output polarization with wavelength is dependent on the specific design. In general, however, longer devices are more wavelength sensitive. We highlight that in our approach, the differential phase shifters in the in-plane sections are expected to be the bandwidth limiting components, since the Berry’s phase in the out-of-plane sections is insensitive to wavelength.

Fig. 7. The solution curves for N = 6 periods with varying radius and wavelength.

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4. Conclusion

We presented a framework for building an arbitrary polarization state generator that uses a cascade of out-of-plane, and in-plane silicon strip waveguides. Examining the Stokes vector evolution on the Poincaré sphere provided two criteria for achieving passive TE-to-TM polarization rotation. We derived the relevant equations to satisfy these criteria, and showed that the resulting curves define the regions where active tuning of the output PER is possible. The results give concise fabrication requirements for an arbitrary polarization state generator, and are promising for applications in on-chip polarimetry, sensing, communications, and quantum optics. The analysis can be extended to other material platforms and other types of coupled mode devices.

Appendix

Here we present the details of our Jones matrix formalism, and the derivations of Eqs. (1)–(6). Further information can be found in [23] and references therein. The electric field in the waveguide is written as a superposition of the fundamental waveguide modes as

(7)$${\textbf E} = {a_{\textrm{TE}}}{{\textbf E}_{\textrm{TE}}} + {a_{\textrm{TM}}}{{\textbf E}_{\textrm{TM}}}, $$

where a_TE, a_TM, E_TE, and E_TM are the complex modal weights and field distributions. We assume a normalization such that |a_TE|² + |a_TM|² = 1. The modal weights are used to construct a Jones vector

(8)$${\textbf J} = \left( {\begin{array}{{c}} {{a_{\textrm{TE}}}}\\ {{a_{\textrm{TM}}}} \end{array}} \right). $$

On propagation the Jones vector is governed by the differential equation [28]

(9)$$\frac{{d{\textbf J}}}{{ds}} = T{\textbf J}(s), $$

where s is the position along the waveguide and T is the differential Jones Matrix. If T is constant with respect to s, Eq. (9) has the analytical solution

(10)$${\textbf J}(s = L) = \exp (TL){\textbf J}(0), $$

where J(0) is the input polarization, L is the length of the waveguide, and exp(TL) is the traditional Jones matrix [28]. A conventional in-plane waveguide acts like a bulk wave plate with its axes aligned to the lab frame, which adds a phase delay between the TE and TM modes. Assuming lossless propagation, neglecting global phase accumulation, and using the phase convention exp(jkz - jωt) for forward propagating waves, the differential Jones matrix is given by

(11)$${T_{\textrm{in}}} = jk\left( {\begin{array}{{cc}} {\Delta {n_{\textrm{lin}}}}&0\\ 0&{ - \Delta {n_{\textrm{lin}}}} \end{array}} \right), $$

where Δn_lin = (n_TE − n_TM)/2 is half the modal birefringence, and k = 2π / λ is the wavenumber. A perfectly square waveguide has Δn_lin = 0. Since T_in is diagonal, the matrix exponential is simply

(12)$${M_{\textrm{in}}} = \exp ({T_{\textrm{in}}}L) = \left( {\begin{array}{{cc}} {\exp (jk\Delta {n_{\textrm{lin}}}L)}&0\\ 0&{\exp ( - jk\Delta {n_{\textrm{lin}}}L)} \end{array}} \right) = \left( {\begin{array}{{cc}} {\exp (j\rho /2)}&0\\ 0&{\exp ( - j\rho /2)} \end{array}} \right).$$

Equation (12) is the Jones matrix for the in-plane waveguides which impart differential phase shift ρ.

In out-of-plane waveguides with deflection angle θ, the Berry’s phase is modeled as an effective circular birefringence 2Δn_cir, defined as Δn_cir = (n_L − n_R) / 2 = 2θ / kL, where n_L and n_R are the effective indices of the left- and right-hand circular polarizations, and 2θ is the solid angle subtended by the light’s trajectory in k-space. Light propagating in an out-of-plane waveguide experiences simultaneous linear and circular birefringence which is described by the following differential Jones matrix [28,29]

(13)$${T_{\textrm{out}}} = jk\left( {\begin{array}{{cc}} {\Delta {n_{\textrm{lin}}}}&{j\Delta {n_{\textrm{cir}}}}\\ { - j\Delta {n_{\textrm{cir}}}}&{ - \Delta {n_{\textrm{lin}}}} \end{array}} \right). $$

The off diagonal terms add coupling between the TE and TM modes due to Berry’s phase. The matrix exponential of Eq. (13) is non-trivial, but can be calculated using known formula, which yields [30]

(14)$${M_{\textrm{out}}} = \exp ({T_{\textrm{out}}}L) = \left( {\begin{array}{{cc}} {\cos \phi + j\frac{{\Delta {n_{\textrm{lin}}}}}{\delta }\sin \phi }&{ - \frac{{\Delta {n_{\textrm{cir}}}}}{\delta }\sin \phi }\\ {\frac{{\Delta {n_{\textrm{cir}}}}}{\delta }\sin \phi }&{\cos \phi - j\frac{{\Delta {n_{\textrm{lin}}}}}{\delta }\sin \phi } \end{array}} \right), $$

where δ² = Δn²_lin + Δn²_cir, and ϕ = kδL. We note that Eqs. (13)-(14) use the values Δn_lin and L of the out-of-plane waveguides, whereas Eqs. (11)-(12) use the values Δn_lin and L of the in-plane waveguides. An out-of-plane waveguide with TE input yields the output Jones vector

(15)$${\textbf J}(L) = {M_{\textrm{out}}}\left( {\begin{array}{{c}} 1\\ 0 \end{array}} \right) = \left( {\begin{array}{{c}} {\cos \phi + j\frac{{\Delta {n_{\textrm{lin}}}}}{\delta }\sin \phi }\\ {\frac{{\Delta {n_{\textrm{cir}}}}}{\delta }\sin \phi } \end{array}} \right). $$

The Stokes parameters are calculated using the following definitions [31]

(16)$$\begin{array}{{l}} {{S_0} = {J_1}J_1^\ast{+} {J_2}J_2^\ast }\\ {{S_1} = {J_1}J_1^\ast{-} {J_2}J_2^\ast }\\ {{S_2} = {J_1}J_2^\ast{+} {J_2}J_1^\ast }\\ {{S_3} = j({J_1}J_2^\ast{-} {J_2}J_1^\ast )} \end{array}. $$

where J_i corresponds to the elements of the Jones vector. Stokes parameter S₀ = 1 because of the initial normalization and lossless propagation. The normalized Stokes vector s_A in Eq. (1) is found by substituting the Jones vector from Eq. (15) into Eq. (16) which yields

(17)$${{\boldsymbol s}_\textrm{A}} = \frac{1}{{{S_0}}}\left( {\begin{array}{{l}} {{S_1}}\\ {{S_2}}\\ {{S_3}} \end{array}} \right) = \left( {\begin{array}{{c}} {\frac{{\Delta n_{\textrm{lin}}^2 - \Delta n_{\textrm{cir}}^2}}{{{\delta^2}}}{{\sin }^2}\phi + {{\cos }^2}\phi }\\ {2\frac{{\Delta {n_{\textrm{cir}}}}}{\delta }\sin \phi \cos \phi }\\ { - 2\frac{{\Delta {n_{\textrm{lin}}}\Delta {n_{\textrm{cir}}}}}{{{\delta^2}}}{{\sin }^2}\phi } \end{array}} \right). $$

The Stokes parameters for the geodesic curve in Eq. (3) are found by taking the curve along the equator and rotating it about the S₁ axis by angle θ_geo. The Stokes vectors along the equator and the rotation matrix are given by

(18)$${{\boldsymbol s}_{\textrm{eq}}} = \left( {\begin{array}{{c}} {\cos \alpha }\\ {\sin \alpha }\\ 0 \end{array}} \right),\;\;\;{R_{{S_1}}} = \left( {\begin{array}{{ccc}} 1&0&0\\ 0&{\cos {\theta_{\textrm{geo}}}}&{ - \sin {\theta_{\textrm{geo}}}}\\ 0&{\sin {\theta_{\textrm{geo}}}}&{\cos {\theta_{\textrm{geo}}}} \end{array}} \right), $$

where α ranges from −π to π, and θ_geo ranges from –π/2 to +π/2. The stokes parameters of a general geodesic curve passing through S₁ = ±1 are given by

(19)$${{\boldsymbol s}_{\textrm{geo}}} = {R_{{S_1}}}{{\boldsymbol s}_{\textrm{eq}}} = \left( {\begin{array}{{c}} {\cos \alpha }\\ {\cos {\theta_{\textrm{geo}}}\sin \alpha }\\ {\sin {\theta_{\textrm{geo}}}\sin \alpha } \end{array}} \right). $$

Below we find the Stokes vectors of the eigenstates in Figs. 2–3 and Eq. (5). We note that the eigenvalues of T_out and M_out are ± jkδ and exp(±jkδL) respectively. It can be shown that any matrix A and its exponentiation exp(A) have identical eigenvectors. The eigenvectors (not normalized) of T_outL and M_out are given by

(20)$$v = jkL\left( {\begin{array}{{c}} { - j\Delta {n_{\textrm{cir}}}}\\ {\Delta {n_{\textrm{lin}}} \mp \delta } \end{array}} \right). $$

Substituting Eq. (20) into the definitions in Eq. (16), the stokes parameters are

(21)$$\begin{array}{{l}} {{S_0} = 2\delta {{(kL)}^2}({\delta \mp \Delta {n_{\textrm{lin}}}} )}\\ {{S_1} ={\pm} 2\Delta {n_{\textrm{lin}}}{{(kL)}^2}({\delta \mp \Delta {n_{\textrm{lin}}}} )}\\ {{S_2} = 0}\\ {{S_3} ={\mp} 2\Delta {n_{\textrm{cir}}}{{(kL)}^2}({\delta \mp \Delta {n_{\textrm{lin}}}} )} \end{array}. $$

Then the normalized stokes vectors for the eigenstates are

(22)$${{\boldsymbol s}_ \pm } = \frac{1}{{{S_0}}}\left( {\begin{array}{{l}} {{S_1}}\\ {{S_2}}\\ {{S_3}} \end{array}} \right) = \frac{1}{\delta }\left( {\begin{array}{{c}} { \pm \Delta {n_{\textrm{lin}}}}\\ 0\\ { \mp \Delta {n_{\textrm{cir}}}} \end{array}} \right). $$

The angle ψ in Fig. 3 which the eigenstate makes with the S₁ axis can be expressed in terms of trigonometric functions as

(23)$$\cos \psi = \frac{{{S_1}}}{{{S_0}}} = \pm \frac{{{\Delta }{n_{\textrm{lin}}}}}{\delta },\;\;\sin \psi = \frac{{{S_3}}}{{{S_0}}} = \mp \frac{{{\Delta }{n_{\textrm{cir}}}}}{\delta },\;\;\tan \psi = - \frac{{{\Delta }{n_{\textrm{cir}}}}}{{{\Delta }{n_{\textrm{lin}}}}}, $$

where the appropriate signs are chosen depending on the state of interest. For the case depicted in Fig. 3, s₋ is the eigenstate shown. The generalization of Eq. (2) to Eq. (6) is accomplished by using Eq. (23) to write

(24)$$\frac{{\Delta n_{\textrm{lin}}^2 - \Delta n_{\textrm{cir}}^2}}{{{\delta ^2}}} = {\cos ^2}\psi - {\sin ^2}\psi = \cos 2\psi. $$

Funding

National Science Foundation (1610797).

Disclosures

The authors declare no conflicts of interest.

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Framework for tunable polarization state generation using Berry’s phase in silicon waveguides

Abstract

1. Introduction

2. Analytical framework

3. Active tuning

4. Conclusion

Appendix

Funding

Disclosures

References

Cited By

Figures (7)

Equations (24)

Optics Express