We present an optomechanical method to tune phase and group birefringence in parallel silicon strip waveguides. We first calculate the deformation of suspended, parallel strip waveguides due to optical forces. We optimize the frequency and polarization of the pump light to obtain a 9nm deformation for an optical power of 20mW. Widely tunable phase and group birefringence can be achieved by varying the pump power, with maximum values of 0.026 and 0.13, respectively. The giant phase birefringence allows linear to circular polarization conversion within 30µm for a pump power of 67mW. The group birefringence gives a tunable differential group delay of 6fs between orthogonal polarizations. We also evaluate the tuning performance of waveguides with different cross sections.
©2009 Optical Society of America
Optical forces have recently been investigated as a way of repositioning microphotonic elements such as waveguides [1–4] and microcavities [5–8]. In coupled waveguides or microcavities, the force can show either attractive or repulsive behavior depending on the relative phase of the incident light in each waveguide or microcavity. As a result, waveguide (or cavity) separation can be adjusted by tuning the power and/or polarization of input light.
Experiments have now demonstrated motion induced by optical forces in a variety of microphotonic systems. Eichenfield et al.  achieved micron-scale displacement of a micrometer-scale waveguide evanescently coupled to a high-Q optical microresonator at milliwatt optical powers. Li et al. demonstrated attractive and repulsive optical forces between a suspended waveguide and dielectric substrate, actuated by a pulsed light source [10–12]. Attractive and repulsive forces have also been observed in stacked microring structures , where Wiederhecker et al. achieved static displacements up to 12nm. Optomechanical modes have been analyzed  and measured  in doubly-clamped beam structures with linear arrays of etched air holes forming optically resonant cavities. As the field of optical forces in integrated photonic devices matures, it is important to map out the novel functionalities that this relatively new form of mechanical actuation allows.
In this paper, we study the use of optical forces to achieve highly tunable birefringence in a coupled waveguide system. Birefringence refers to differences in propagation between orthogonally polarized modes. Phase birefringence, the difference between the phase indices, induces a relative phase shift between orthogonal components of incident light, which may be used for polarization rotators in optical communication systems. Group birefringence, defined as the difference between the group indices of different polarizations, leads to a polarization-dependent time delay and can be exploited to realize polarization-sensitive delay lines, splitters, and multiplexers. It has previously been observed that silicon waveguides with multiple etched slots exhibit giant birefringence behavior, with a group index difference as large as 1.5 between TE and TM polarized light . The magnitude of birefringence depends on the slot width, as well as other design parameters. However, after the device is fabricated, the slot width is fixed, as is the value of the birefringence. Here, we present an optomechanically-controlled system which resembles a suspended one-slot waveguide with an adjustable slot width that depends on the optical force and exhibits highly tunable birefringence.
A variety of methods have been used to achieve tunable birefringence, including thermal and stress tuning. However, the achievable tuning range is generally small. For example, thermal tuning of polymer-filled microstructured fibers [17, 18] and photonic liquid crystal fibers  yields changes in birefringence on the order of 10−4. For microphotonic devices, Tsia et al.  demonstrated dynamic control of birefringence on the order of 10−4 in a silicon waveguide by electrically tuning the stress on the waveguide core using an integrated piezoelectric film. Recently, Kumar et al.  varied the air-core thickness in a 3D hollow waveguide via MEMS acutation to offer a large birefringence of 0.012 with a tuning range of 0.01. This result points to the relatively large changes in birefringence that can be achieved using mechanical motion to physically change the waveguide geometry.
In this work, we calculate the tunable birefringence that can be achieved by using an optical force to adjust the separation between parallel, suspended, strip silicon waveguides. The force arising from the evanescent coupling between the modes of the two waveguides deforms both waveguides, changing the air-slot width. Such behavior results in all-optically adjustable giant birefringence. In Section 2, we characterize the two-waveguide system, and optimize the frequency and polarization of input light to achieve a considerable optical force at a moderate laser power. We calculate the resultant displacement of both waveguides based on the finite-element method (FEM) . In Sections 3 and 4, we respectively evaluate the performance of the optimized optical force in tuning phase and group birefringence. We show that the optomechanical approach is characterized by widely tunable birefringence. We explore the applications of the two-waveguide system for tunable linear-to-circular polarization conversion and polarization-dependent delay.
2. Design method
Figure 1(a) shows two parallel silicon strip waveguides (refractive index n=3.45) separated by a distance d. Both ends of the waveguides rest on a SiO2 substrate, with a free-standing section of length L. Each waveguide has a cross section of dimensions w×h. We consider the transverse electric (TE) mode with electric field vector primarily parallel to the air slot (E y), and the transverse magnetic (TM) mode with electric field vector primarily perpendicular to the slot (E z). We first consider the case where the cross section of each waveguide is square (w=h=a). Figure 1(b) shows the dispersion relation calculated by the MIT Photonic Bands package  for the lowest TE and TM mode for several different separations. We only consider the guided modes lying below the light cone (yellow). The insets illustrate the E y field distribution of the TE mode and the E z field distribution of the TM mode, respectively. TM-polarized light is concentrated in the air region, satisfying the continuity condition on the normal component of the displacement field at the silicon-air interface [24, 25].
TE and TM modes have different propagation characteristics. The group birefringence is obtained by taking the difference between the group indices , where and are the group indices of the TE and TM modes, respectively, at a fixed frequency ω. The group index is calculated as and is proportional to the reciprocal of the slope of the dispersion curve in Fig. 1(b). Phase birefringence is , where and respectively denote the phase indices of the TE and TM modes at a fixed frequency ω. The phase index is calculated by . For small separation (d=0.2a), the optical fields in the two strips are strongly coupled. The bands for TE and TM light are significantly different. It can be inferred from the figure that group and phase birefringence are relatively large in some frequency ranges. If the separation is increased (d=0.4a or d=0.6a), the coupling between the strips becomes weaker, and the dispersion curves of TE and TM modes are closer to one another. Group and phase birefringence are reduced. For infinite waveguide separation (d=∞), no coupling occurs. Modes propagate independently in each single-strip waveguide. The bands of the TE and TM modes overlap, and the birefringence vanishes.
Coupling between the strips gives rise to an optical force on the waveguides . The force depends on the frequency of light propagating in the waveguides (pump frequency), the incident power, and the waveguide separation. For fixed incident power and waveguide separation, we optimize the force magnitude by selecting an appropriate frequency and polarization of pump light. As in previous work , we calculate the force as a function of frequency using the Maxwell Stress Tensor formulation , which gives the force as an integral expression over the electric and magnetic fields of the waveguides. We solve for the full vectorial fields using the MIT Photonic Bands package . Figure 2(a) shows the force magnitude as a function of the pump frequency for a waveguide separation d = 0.35a. We focus on the dimensionless frequency (ωa/2πc) range from 0.15 to 0.21. For lower frequencies, the bands are closer to the light line, and the waveguide modes spread out in the surrounding air. For frequencies larger than 0.21, there are at least two symmetric TE (or TM) modes for a single frequency, resulting in multimodal coupling. We compare the force induced by TM (red) and TE (black) modes. The strongest force is obtained for TM-polarized light at frequency ωpa/2πc=0.165. According to Eq. (1) in ,Fig. 1(b), we see that for a fixed wave vector k, if the waveguide separation is reduced, the frequency of the TM mode changes more than that of the TE mode. Therefore, the TM mode induces a bigger force. In the remainder of the paper, we will assume that TM pump light is used to exert force on the waveguides and deform their shape. We will consider the effects of the deformation on the propagation of both TE and TM signal light.
On the left and bottom axes of Fig. 2(b), we plot the dimensionless force per unit area (F/Lh)(a 2 c/P) as a function of the normalized separation d/a at the optimized frequency ωpa/2πc=0.165. The symbol c represents the speed of light in vacuum, and P denotes the incident power. Substituting the values P=20mW, w=h=a=263.5nm, and pump wavelength λp=1597nm, we plot the force per unit area versus separation on the right and top axes in Fig. 2(b).
We perform FEM numerical simulations to calculate the deformation of waveguides caused by the attractive force using COMSOL Multiphysics software . We simulate the full three-dimensional structure consisting of two silicon waveguides, each with a square cross section of 263.5×263.5nm2 and a length of 30µm. The ends of each waveguide are fixed. We assume an initial waveguide separation d=0.35a=92.2nm. The waveguide ends are fixed at this separation. The optical force pulls the waveguides toward one another. As the waveguides deform, the magnitude of the force increases. We use a second-order polynomial to approximate the force density for waveguide separation d between 0.2a and 0.35a, as shown by the solid blue line in Fig. 2(b). For the mechanical parameters of Si, we take the mass density ρ=2330 kg/m3 and Young’s modulus E=131GPa. Figure 3 shows the waveguide deformation as a function of position along the waveguides with L=30µm, w=h=a=263.5nm, d=92.2nm, and P=20mW. The biggest displacement (9 nm) is obtained at the waveguide center. The smallest distance between the two waveguides is 74.2nm. Following , we estimate the van der Waals force per unit area to be approximately 25N/m2 between two parallel infinitely extended Si slabs of thickness 263.5nm and separation 74.2nm. For this separation, the van der Waals force is several orders of magnitude less than the optically-induced force and can therefore be neglected. It is interesting to note that experiments have been done in a similar coupled system, showing that pulsed light can obtain a comparable displacement at a lower power of a few milliwatts .
3. Tunable phase birefringence
The phase birefringence is a function of the separation between the strips. We consider the birefringence experienced by light at a signal frequency ω. We assume that the signal power is much weaker than the pump power, such that the optical force due to the signal can be neglected. We calculate the phase birefringence Δnp as a function of dimensionless signal frequency ωa/2πc for different separations from 0.2a to 0.35a, as shown in Fig. 4(a) . The arrow indicates that at signal frequency ωsa/2πc=0.17 (a=263.5nm, λs=1550 nm), the absolute value of Δnp increases as the waveguide separation decreases.
When the waveguides are deformed by an attractive optical force, the separation varies along the waveguide length as shown in Fig. 3. The propagating signal experiences a spatially-varying birefringence. Figure 4(b) shows the phase birefringence as a function of position along waveguides for a CW pump at frequency ωpa/2πc=0.165 with power P=20mW. For zero pump power (zero force), the waveguides are parallel and the birefringence is ≈0.067 throughout the whole free-standing section. With the pump on, the birefringence is increased. The maximum value is approximately 0.093 at the waveguide center. The difference between the maximum and the minimum of is 0.026.
3.1 Tunable relative phase-shift
A potential application of the system is to dynamically adjust the relative phase shift between TE and TM modes. The total phase shift between TE and TM modes is calculated as:
The phase shift in two parallel waveguides with a separation of 92.2 nm is estimated to be 2π×30µm×0.067/1.55 µm ≈2.59π, equal to 0.59π modulo 2π. When we input CW pump light with a power of 20 mW, the waveguides deform and the phase shift is changed to 3.11π, or 1.11π modulo 2π. The tuning range is about 0.5π. If we reduce the waveguide length to 23.05 µm, the total relative phase shift of a linear signal is 2π at the output of the un-deformed device. We plot the relative phase difference (phase difference modulo 2π) as a function of input pump power, as shown by black squares in Fig. 5 . By increasing the pump power, the phase difference can be adjusted from 0.12π to 0.51π. Taking the pump power to be about 67 mW, we obtain a total relative phase shift of 0.5π. Thus, an incoming signal of linear light whose E-field is at 0.25π to the y (or z) axes will emerge from the device circularly polarized.
We note that a variety of approaches exist for polarization control in fiber  and on-chip microphotonic systems [29–32]. In microphotonic systems in particular, polarization rotators have been designed based on tapered waveguide cores  and asymmetric waveguides [30–32]. Ref . presented a linear-to-circular polarization converter based on a high-Q dielectric microring resonator. In contrast to these approaches, we take advantage of mechanical motion to yield a widely tunable phase shift that is also broadband (non-resonant) in response. For a device with fixed length, the phase birefringence and relative phase shift can be dynamically adjusted by varying pump light power. Unlike other polarization control schemes using semiconductor optical amplifiers [34, 35] or electrooptic materials, our approach is fully compatible with SOI materials systems.
3.2 Other tuning configurations
The dependence of phase birefringence on separation can be changed by altering the waveguide cross section. Figure 6 (a) depicts the birefringence versus signal frequency in two a×2a waveguides. The arrow indicates that the value of Δnp decreases with decreasing separation for frequencies from 0.11 to 0.16. The tuning trend is opposite from that in the system with two square waveguides. We can understand the trend by considering two extreme cases. For zero separation, light propagates in a square waveguide of cross section 2a×2a, which has no birefringence due to spatial symmetry. For infinite separation, light propagates in a single rectangular waveguide asymmetric with respect to the y and z axes, which has large birefringence. Figure 6 (b) shows another waveguide structure with a cross section a×0.5a. The effective index of the TE mode is less than that of the TM mode. The absolute value of Δnp increases with decreasing separation for frequencies from 0.20 to 0.25.
4. Tunable group birefringence
The group birefringence also depends on waveguide separation and can be tuned by deforming the waveguides via an optical force. We plot the group birefringence Δng as a function of dimensionless signal frequency ωa/2πc, for different separations from 0.2a to 0.35a in Fig. 7(a) . The birefringence increases with decreasing separation. The arrow indicates that the group birefringence Δng increases fastest at a frequency ωsa/2πc=0.17 (a=263.5nm, λs=1550 nm). We consider deformation due to an attractive optical force for CW pump light at frequency ωpa/2πc=0.165 and power P=20 mW, as above. Figure 7(b) shows the group birefringence of signal light as a function of length along the waveguides for L=30μm. The initial value 1.09 corresponds to the birefringence of the undeformed waveguides. The peak of the birefringence is approximately 1.22 at the waveguide center. The difference between the maximum and the minimum of │Δnp│ is 0.13.
The system can be used to tune the polarization-selective delay between TE and TM pulses. The delay is tuned by adjusting the pump power. The differential time delay between TE and TM pulses is calculated as:
For two parallel waveguides without deformation, the differential time delay is 1.09×30µm/3×108=109fs. By inputting CW pump light at a power of 20mW, we increase the differential time delay to 115fs.
If we consider applications where the pulse width is comparable to the tunable delay range, we can take the pulse width T=6fs. We can estimate the dispersion length as , where the dispersion coefficient is 2.2×10−25 s2/m. LD is around 164 micrometers, much larger than the waveguide length. We can thus assume that minimal distortion occurs due to propagation through the 30µm long suspended section.
The tuning range can be increased by increasing the pump power. The tuning range can also be increased by fabricating longer suspended sections, which deform more easily for fixed power, or by fabricating multiple suspended sections in series.
Note that this analysis also applies to vibrating waveguides actuated by pulsed pump light . The dynamic system experiences similar displacement as the static system for much smaller pump powers. While the waveguide beam vibrates in a period of a few hundred nanoseconds, the transmission time of a signal pulse through the 30µm waveguide is on the order of a femtosecond. The beam can thus be treated as static in the calculation of time delay, provided that the signal pulse is much shorter than the vibration period in time.
5. Thermal effects
In addition to optical forces, thermal effects may also contribute to waveguide deformation. The waveguides absorb propagating light, experience thermal expansion, and are mechanically deformed. We performed FEM (finite element method) simulations to calculate the displacement caused by the photo-thermo-mechanical effect using COMSOL Multiphysics software. We considered two silicon waveguides with identical cross section 263.5×263.5nm2 and separation 92.2nm sitting on top of a silicon dioxide substrate, as shown in Fig. 1(a). The free-section length of the beams is 30µm. We used a similar simulation method to that introduced in the supplementary methods of . The beams are assumed to be thermally isolated, except where they contact the substrate. The boundaries of the silicon dioxide substrate are assumed to be of constant temperature 300K. The absorption constant of silicon α is estimated to be 0.0079cm−1 at 1.597µm pump wavelength by using the equation α=4πk/λ and the value of the extinction coefficient k in . With 20mW incident optical power, the absorbed power is approximately 0.474µW. After using the RF module to solve for the pump mode profile, we define a resistive heating power of 0.474µW as the heating source in the thermal module. We obtain a calculated temperature profile that shows a peak temperature T-300=0.097K near the center of the beams. The thermal simulation is followed by a mechanical simulation to solve for the mechanical stress that results from the thermal load. We are primarily concerned with the mechanical displacement in the z direction, which may introduce inaccuracy into the birefringence tuning. A peak z-displacement of 0.392pm is reached at the waveguide center. The peak displacement in the y-direction was <5pm. The thermally induced displacement is thus negligible compared with the displacement due to the optical force of 9nm.
Considering that the thermo-optical coefficient ∂n/∂T=1.86×10−4K−1 for silicon, we estimate that the refractive index change due to our calculated temperature increase is on the order of 10−5, which is far smaller than the index change caused by optomechanical deformation.
We conclude that thermal effects can be neglected in the system studied here, in agreement with the conclusions found in  for a similar experimental system.
In this work, we present an optomechanically-controlled system which resembles a suspended one-slot waveguide with adjustable slot width depending on the optical force. The change in slot width corresponds to a change in the symmetry of the cross-section of the device, and therefore tunes both phase and group birefringence. We optimize the polarization and frequency of continuous pump light to obtain a considerable force at low operating power. The deformed 30µm waveguide shows a maximum displacement of about 9 nm at the center for a power of 20mW. Correspondingly, the phase (group) birefringence at the waveguide center is different from that at the clamped ends by 0.026 (0.13), which is larger than previous records. When used to tune the relative phase of two orthogonally polarized light signals, the device has a tuning range of around π/2. If the orthogonally polarized components have identical amplitudes, it is possible to rotate the polarization state from linear to circular in a distance of 30μm. We show further that the dependence of phase birefringence on separation can be changed by altering the waveguide cross section. When used as a polarization-dependent tunable delay element, the two-waveguide system has a tuning range of 6fs. We note that the birefringence tuning trends studied here apply not only to a two-beam system deformed by an optical force, but also by other actuation methods, including static electrostatic forces used in micro-electro-mechanical systems (MEMS).
In the future, the potential to tune birefringence via an optical force may be applied to design novel microring resonator systems. Waveguide-coupled microrings exhibit a strong (static) birefringence, which can be used to realize polarization converters and polarization-sensitive delay devices [38–40]. Recent experiments have shown that the separation between vertically-stacked microrings can be adjusted via optical forces . This capability may in the future be exploited to design highly-tunable birefringent devices along similar lines as studied here.
Computing resources were provided by the USC Center for High Performance Computing and Communications. Jing Ma was funded by a Viterbi Fellowship. Michelle Povinelli was funded by USC WiSE Jr. Gabilan and Powell Awards.
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