Measuring optical rotations in materials is a useful tool in many experimental studies. Research may be limited by the ability to measure small rotations due to weak interactions. We propose a novel scheme wherein we use a coupled waveguide and ring resonator to amplify the effects of optical rotation, potentially opening new avenues for investigation. Our proposed device can increase the resulting optical rotation by up to six orders of magnitude.
© 2015 Optical Society of America
Measuring the rotation of linearly polarized light through a medium provides key data regarding the structure or substance of the material in question. In semiconductor devices, Kerr and Faraday rotation measurements can be used to measure the net spin polarization [1,2]. In chemistry, the isomer chirality ratio of a mixture can be determined from measurements of optical polarization rotation . In materials with weak interactions, polarization rotations may be too small to detect with existing equipment, but can be enhanced in an optical cavity [4,5]. Here we consider the amplification of the rotation of linearly polarized light through a medium when the medium is incorporated in a ring resonator. Waveguides coupled to ring resonators have been fabricated and used for amplification , wavelength filtering , and lasing . Polarization control in integrated waveguide devices could form the basis for isolators, circulators, modulators and switches, but the current achievable rotation using electrically-generated spin polarization is small . We find that by carefully tuning the coupling of the resonator to a linear waveguide, orders of magnitude increases in the rotation can be achieved.
In Fig. 1, we show a general schematic of the device . Linearly polarized light, a1, is injected into the waveguide. Inside the coupling region, we assume lossless coupling to the resonator, b2, and transmission to the waveguide output, b1. The light in the resonator passes through an interaction region where the polarization is rotated by some angle ϕ. The light decays due to material losses and may gain a phase offset due to the path length of the resonator. The rotated light enters the coupling region as a2, where it is out-coupled to b1 or transmitted through to b2. For a continuous wave input along a1, steady state conditions are reached between the various effects of resonator-waveguide coupling and transmission, optical rotation, decay, and phase offset. By tuning the device, the input signal, a1, can be mostly blocked from the b1 output. Measuring the polarization of light at b1 can lead to detecting much larger rotations than those of passing light once through the interaction region.
In this paper, we first consider a simple model of the device to give an intuitive understanding of how the large enhancements in rotation are produced. We then propose a model in which we explicitly study the rotation and coupling from one polarization of light to another. Finally, we consider the relevant device parameters and how the amplification of the rotation in the interaction region can be maximized.
2. Ring resonator with field injection
We consider the simple case in which we have lossless coupling between the waveguide and ring resonator for one polarization of light. Building upon previous work, we assume a unitary scattering matrix can describe the relationship between the fields entering and leaving the coupler [10,11]. Light is coupled from the waveguide to the resonator according to the coupling constant, κ, or transmitted through to the output by the transmission factor, t.
As light travels around the ring resonator, three effects take place. Damping due to material loss and phase offset can be described by the parameters α and θ, respectively. The interaction region effectively injects some field, ainj. We model the relationship between a2 and b2 based on these effects.
From this equation and our coupling matrix, we determine the output field b1 as a function of the input fields and system parameters.
Suppose we wish to measure ainj. We consider the conditions necessary to get maximal output at b1 due to ainj. For real coupling constants (κ, t) and resonance conditions (θ = 0) this occurs at critical coupling, α = t. When exactly on resonance, b1 due to a1 goes to zero, and the output of the device is linear with ainj.
We find that on resonance the greatest output due to the injected field will occur for low coupling parameter κ, corresponding to high α. Thus, a high quality resonator at critical coupling will give the greatest output due to the injected field.
3. Field rotation amplifier
We consider one example of how we could build such an interaction region and exploit it to increase our sensitivity in measuring small rotation angles. We allow vectors a1, a2, b1, and b2 to have components in the x and y directions. Without loss of generality, we send a linearly polarized signal into the coupler with the electric field oriented along the x direction. After coupling to the resonator, the signal enters the interaction region where the material of interest rotates the light toward the y axis by some angle ϕ. We expand our previous coupling matrix to include each polarization and assume there is no change in polarization in the coupling region.
A and B represent the vectors for the input and output of the coupling region, respectively. The matrix C has constraints and to satisfy lossless coupling. For the interaction region, we have a rotation between the x and y polarizations. We also include the decay term, α and the phase offset, θ. To account for our source, we add a constant term A0.
This formulation allows us to analytically solve for the output vector, B, as a function of our input vector, A0, and our system parameters.
For the case in which the input beam is polarized along the x direction, we set a1x = 1 and a1y = 0.
We consider the optimal tx and ty for comparison with the simple case presented before. Since we are now injecting field in the form of a rotation from one polarization to another, we can no longer eliminate b1x entirely. This is due to the fact that light polarized in the y direction that enters the interaction region will rotate to the x direction and then out-couple as b1x. The field rotated from the y to the x polarization is effectively ainj for the x polarization. As we wish to maximaize the polarization rotated from the x to the y direction, we find tx and ty that maximize.
For sufficiently small φ, the optimum values are near critical coupling. However, when tx≈αx, the approximation for ty,opt is no longer valid. However, as we will show later in the paper, device optimization depends more on tuning the coupling parameters for the x polarization than the y.
Near critical coupling, the output from the device would be polarized nearly entirely in the y direction. This would occur for nearly all small values of ϕ, as shown in Fig. 2. For any α, the device has a constant output polarization. The magnitude of b1y would change due to loss, and measuring angles of rotation has the advantage of being independent of the input beam intensity and thus less sensitive to intensity fluctuations. Therefore, our analysis presents the polarization angle of the output rather than the intensity.
Figure 2 shows the output polarization angle of b1 as measured from the x axis. For small ϕ, the output polarization is π/2, or 90°, corresponding to complete conversion from polarization along x to polarization along y. This is a desirable regime for polarization conversion, but it is not ideal if the application of this device is to measure the rotation in the interaction region. For larger ϕ, the rotation in the interaction region is sufficiently large to rotate some non-negligible field from the y polarization back into the original x polarization. The rotated field would be equivalent to an injected x polarization, which would be efficiently out coupled to the waveguide. This leads to a non-constant output polarization for large rotations in the interaction region.
4. Device optimization
An interesting phenomenon occurs when the system is off critical coupling. We define the amplification as the ratio of the polarization angle from the x axis of the output of the device to the polarization change in a single pass through the interaction region, ϕ. In Figs. 3(a) and 3(b), we show the amplification of the polarization for different combinations of parameters α and t. We find that for a large range of interaction region strengths, we have a constant amplification. Constant amplification is ideal for using this device to amplify small rotation signals.
In Fig. 3(a), we show the amplification of a device far off critical coupling. A promising feature of this graph is that it demonstrates the constant amplification over many orders of magnitude of rotation. For example, for α = 0.9 and t = 0.85, the polarization amplification is 21.25 from ϕ = 10−2 to 10−6 rad. In Fig. 3(b), we consider the effects very close to critical coupling. For this graph, α = 0.9, although the results are similar for a wide range of α, as will be demonstrated later. Amplification strongly depends on the offset from critical coupling, δ = α-t.
Figure 4 shows the amplification as a function of critical coupling offset for ϕ = 10−6 rad. At low offset, the amplification converges on a constant value. As offset decreases we approach the critical coupling case. In the low δ regime, the b1x output goes to 0 leading to a constant output polarization of π/2 for all b1y. This leads to an amplification of 1.57 × 106, as shown in Fig. 4. Since the amplification is similar for vastly different values of α, we find the amplification is only weakly dependant on the quality of the resonator. Higher α does correspond to higher amplification, as expected. However, a much stronger effect is the offset from critical coupling, δ.
In Fig. 5(a) we see the effects of δ to the off resonant behavior of the device. Here we rescale the polarization amplification relative to the amplification on resonance. As δ decreases, the width of the off-resonance curve decreases significantly. In Fig. 5(b), we plot the full width at half maximum of the relative amplification curve as a function of the critical coupling offset for different values of α. The slope of the curves varies from 1.00 when α = 0.2 to 0.94 when α = 0.999, showing an approximately linear relationship between the width of the amplification curve and the offset from critical coupling. The width of the amplification curve is also dependent on α, with lower α producing greater bandwidths. In practice, the device could be on resonance for one polarization direction while being off resonance for the other due to a polarization dependent refractive index. This would complicate the polarization amplification considered in this paper and is worthy of more investigation. For the proposed application of measuring optical rotation, however, it is possible to design a resonator to have the same resonance position for both polarizations.
Figures 6(a) and 6(b) shows the polarization amplification as a function of the resonator decay parameter, α. Except for the region of high loss (α < 0.05), the amplification varies weakly with α, in very good agreement with the earlier results shown in Fig. 4. In Fig. 6(b), we consider the low loss regime. Again, we see a slowly increasing amplification with α (decreasing with 1-α), but a strong dependence on the offset from critical coupling, δ.
In general, the coupling between the resonator and the waveguide may differ for the two polarizations . Additionally, the decay parameters for the different polarizations can vary based on the device geometry. We now consider the effects of different coupling and decay parameters for each polarization. We first consider a device with a fixed coupling coefficient αy = 0.999 and δy = 10−3. In Fig. 7(a), we show the amplification as a function of the transmission coefficient for the x polarization, tx. The result matches previous studies and our earlier analysis showing that maximal amplification will occur at critical coupling for the x polarization, tx = αx. In Fig. 7(b), we hold the x parameters constant at αx = 0.999 and δx = 10−3. Maximal amplifications occur when ty = αy. We also find that for any ty≠1 we get some polarization amplification. This occurs because we have a field injected into the y polarization in the interaction region. Any ty≠1 will couple that injected field to the waveguide output, creating an output rotation.
It may be difficult to build a device in which the coupling constants for the different polarizations can be tuned independently, although this can be realized over a finite range using a tunable coupler based on Mach-Zender interferometry . Based on the results shown in Figs. 7(a) and 7(b), tuning the coupling for the x polarization will have the greatest effect on the amplification.
We have demonstrated how integrating a device which rotates the polarization of light with a coupled waveguide and ring resonator can be used to amplify the polarization rotation in a specific interaction region. Amplification factors are strongly dependent on the ability to tune coupling parameters. By varying the device coupling parameters near critical coupling, amplification factors of 104-106 may be achieved. This could potentially allow for studying weak phenomena in materials that were previously unattainable.
This material is based upon work supported by the National Science Foundation under Grant No. ECCS-0844908, the Office of Naval Research, the Air Force Office of Scientific Research, and the Defense Threat Reduction Agency, Basic Research Award #HDTRA1-13-1-0013.
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