## Abstract

A triple-output Mach-Zehnder interferometer (MZI) operating with long-range surface plasmon-polariton waves, consisting of a MZI in cascade with a triple coupler, is demonstrated at a wavelength of ~1370 nm, using the thermo-optic effect to produce phase shifting. A theoretical model for three-waveguide coupling is also proposed and was applied to compute the transfer characteristic of various designs. Dimensions for the device were selected to optimize performance, experiments were performed, and the results were compared to theory. The outputs were sinusoidally related to the thermally-induced phase shift and separated by ~2π/3 rad, as expected theoretically. Four detection schemes that take advantage of the 3 times larger dynamic range and suppress time-varying common perturbations are proposed and analyzed in order to improve the detection limit of the device. A minimum detectable phase shift ~2/3 that of a single output was obtained from a power difference scheme and a normalization scheme. The smallest minimum detectable phase shift was 7.3 mrad. The device is promising for sensing applications, including (bio)chemical sensing.

© 2014 Optical Society of America

## 1. Introduction

Surface plasmon-polaritons (SPPs) are transverse magnetic (TM) polarised optical surface waves that propagate typically along the interface of a metal and a dielectric [1,2]. Long-range surface plasmon-polaritons (LRSPPs) are SPP supermodes having significantly lower attenuation supported by a thin metal slab or stripe waveguide [3], formed as single-interface SPPs at each metal-dielectric interface couple with each other. Amongst the LRSPPs supported by the stripe, the fundamental mode exhibits a Gaussian-like field distribution [4] and thus is amenable to end-fire excitation, while the others are high-order modes exhibiting field extrema along the width of the stripe and can be cut-off by choosing appropriate stripe dimensions [5].

One application of LRSPPs is thermo-optic modulation, of which the basic mechanism is that the refractive index of a material changes with temperature. A simple model for the thermo-optic effect is the linear model given as follows:

where n is the refractive index,*T*is the temperature, and

*dn/dT*is the thermo-optic coefficient (TOC) - the ratio of the change of the refractive index of a material over that of the temperature. The linear model for the thermo-optic effect is assumed to hold over a small temperature range for constant pressure and operating wavelength. Thermo-optic modulation devices that have been designed and investigated includes straight variable optical attenuators (VOAs) based on thermally-induced anti-guiding [6] and asymmetric mode cut-off [7,8], and Mach-Zehnder interferometers (MZIs) based on thermally-induced phase shift between the sensing and the reference arms [9,10]. Except for LRSPP, there are also researches on thermo-optic modulation using dielectric-loaded SPP (DLSPP) waveguides [11,12].

MZIs are widely used for thermo-optic modulation [13,14] and other applications, e.g., electro-optic modulation [15], and biochemical sensing [16–18]. Multi-output MZIs consist of a MZI in cascade with a dual or triple coupler. Dual-output MZIs provide two complementary outputs [19] that can be used to suppress time-varying common perturbation and to obtain twice the dynamic range compared to a single-output MZI, whereas triple-output MZIs have three outputs with ~2π/3 rad phase separation [17], so at least one output is always available in the linear region (where the sensitivity is maximum). In the past, multi-output MZIs based on fibers [20,21], dielectric waveguides [22], and surface plasmon waveguides [23] were demonstrated. Theoretical research has also been conducted to apply weakly coupled theory [18,24] and strongly coupled theory [25,26] to multiple-output MZIs.

In this paper, we demonstrate thermo-optically activated triple-output MZIs based on LRSPPs, we propose a general theoretical approach based on modal analysis to model and design triple-output MZIs, and we give simple equations that model the operation of an idealised design from which perturbation cancellation and dynamic range can be easily assessed. We compare our theoretical transfer characteristics with those obtained from coupled-mode theory and with the measurements. We also investigate the suppression of common mode perturbations in the transfer characteristics and in outputs over time. In principle, thermo-optic activation can be replaced with bio-chemical interactions, so the device is relevant to biosensing.

The paper is organized as follows: Section 2 introduces the structure of interest and the general theoretical model applied, and validates the results of the model via comparison with coupled-mode theory; the design of triple-output MZIs based on LRSPPs is also discussed. Section 3 compares the measured transfer characteristics with theory, and gives a brief description of the experimental devices and setup. Section 4 investigates the dynamic range, the ability to suppress time-varying common perturbations, and the recurring availability of an output in the linear region. Section 5 concludes the paper.

## 2. Theoretical

#### 2.1 Structure

Figure 1 shows a sketch of the structure of interest in this paper. The triple-output MZI consists of a symmetric MZI in cascade with a triple coupler.

Our fabricated structure is comprised of waveguides consisting of Au stripes cladded with Cytop, all stripes designed to be 35 nm thick and 5 μm wide, and operating in a single LRSPP mode (*ss _{b}^{0}* [5]) at the operating free-space wavelength of interest (

*λ*~1310 nm). The MZI has a Y-junction splitter with a 1 μm inner waveguide separation at the split and a separation of 140 μm between its two arms at the widest point. All curved sections have the same radius of curvature of 5.5 mm. The dimensions for these elements were selected based on the modelling and design results summarised in [4,27]. The triple coupler, shown in expanded view in Fig. 1(b), has a coupling length of 828.57 μm, a separation of 2 μm between the waveguides, and a flared-out separation of 46 μm between its three outputs at the end of the structure. This coupler was modelled and designed as described in the subsequent sub-sections. The linear length of the full structure (MZI and coupler) is 4.8 mm; the optical path length is slightly longer.

_{0}Additional features for thermo-optic control includes 2 μm long gaps as well as 40 μm x 40 μm metal pads that are placed 20 μm away from the waveguides and connected to them with contact arms of the same width and thickness as the waveguides. The dimensions are chosen such that the features remain optically non-invasive. The pads are used to attach metal probes to inject electrical current generating heat in the active region of the device, and the gaps isolate the current to that region. As the active region is heated, the refractive index of the surrounding Cytop claddings change, introducing a phase shift between the LRSPP modes travelling in the two arms of the MZI, and so the optical powers of the three outputs of the coupler change as well. The designed operating wavelength of the device is 1310 nm (free-space).

#### 2.2 Coupling model

We model the performance of the triple coupler using modal analysis and overlap integrals, following [4,8], as illustrated in Fig. 1(b), despite the existing strongly coupled theory discussed in [17,25], because the modal approach is direct, straight forward, and only requires an accurate mode solver. The inputs to the triple coupler are connected to the MZI, as shown in Fig. 1(b), and excited by the fundamental LRSPP mode emerging from the arms of the MZI. Thus, at the inputs of the triple coupler (*z* = 0), the input field distribution is taken as the fundamental LRSPP mode supported by a single waveguide (*ss _{b}^{0}*) centered on one of the outside waveguides of the coupler, superimposing the same mode field distribution centered on the other outside waveguide of the coupler multiplied by a factor

*e*representing the thermally induced phase shift, as expressed as follows:

^{jϕ}here *s* is the coupler separation and *E _{y,s}* is the mode field of the

*ss*mode supported by a single waveguide, as shown in Fig. 2(a). As the input field enters the triple coupler, it redistributes among the three supermodes supported by the coupler, namely mode (1 1 1), mode (−1 0 1), and mode (1 −1 1), for which the field distributions are shown in Figs. 2(b)–2(d), respectively, and forward propagating radiative modes. The latter are neglected (but could be included by discretizing the continuum of radiative modes into a finite set of box modes, following [8]). The input overlap factors

_{b}^{0}*C*into each supermode at the input of the coupler are given as follows [8,28]:

_{i}here *A _{∞}* is the area of the entire computational domain at the transverse plane where the modes meet, and

*E*is the field of one of the three super modes supported by the coupler (with

_{y,i}*i*= (1 1 1), (−1 0 1), or (1 −1 1)). It was checked that $\sum _{i}{\left|{C}_{i}\right|}^{2}}\approx 1$, showing that most of the forward propagating power is coupled into the supermodes of the coupler and justifying the neglect of the forward propagating radiative modes. After travelling along the coupler and reaching its end (

*z = L*), the super modes superimpose to yield the output field distribution as:

here the factor ${e}^{-\left({\alpha}_{i}+j{\beta}_{i}\right)L}$ represents the attenuation and propagation along the coupler and *L* is the coupler length. The *α _{i}* and

*β*of each supermode must be used as they are different. The output field redistributes again among the single modes centered on the individual waveguides connected to the output of the coupler, and the output overlap factors

_{i}*C*are calculated similarly as in Eq.(2.2):

_{m}here *C _{2}* relates to the middle waveguide while

*C*and

_{1}*C*to the outside two. The normalized optical powers in the individual output waveguides are then calculated as

_{3}*P*= |

_{m}*C*|

_{m}^{2}with

*m*= 1, 2, or 3. We neglect the effects of the s-bends before and after the coupler on the mode field distribution of a single waveguide

*E*as the radii of curvature are large [27].

_{y,s}#### 2.3 Validation

To validate the theoretical model, Fig. 4(a) of Ref [17] was reproduced and the calculated results were compared to the original ones obtained by applying the strongly coupled theory. The structure considered consisted of a slab dielectric waveguide symmetric triple coupler having a coupler separation *s* = 1 μm and coupler length *L* = 0.75*L _{c}*.

*L*is defined as the coupling length where maximum power is transferred from a single input at one outside waveguide to the other outside waveguide. The dielectric coupler has claddings of refractive index of 1.6 and cores of refractive index 1.646, operating at a free-space wavelength of 786 nm, and the core width was 1 μm. The modeling work was conducted using the Finite Element Method (FEM) and COMSOL. As shown in Fig. 3, the results coincide very well, providing verification of our theoretical model.

_{c}#### 2.4 Design

The model was implemented to the structure of interest in this paper, at the operation wavelength *λ _{0}* = 1310 nm, for which the refractive index of Cytop is

*n*= 1.3348 [29] and the relative permittivity of Au is

*ε*= −86.08 - j8.322 [30] (e

_{r}^{+}

*time harmonic form implied). The*

^{jωt}*β/β*and the mode power attenuation (MPA) of the three supermodes of interest in the triple coupler (Figs. 2(b)–2(d)) were computed for various stripe separations and plotted in Figs. 4(a) and 4(b), respectively. It is observed that as the separation increases, both the

_{0}*β/β*and the MPA of the three supermodes merge to those of the

_{0}*ss*mode, becoming approximately equal at s ~16 μm. Herein

_{b}^{0}*β/β*is the real part of the effective index and is denoted

_{0}*n*, and the MPA is related to the attenuation constant

_{eff}*α*in m

^{−1}, as:

From these calculations, the transfer characteristic of the triple-output MZI was computed and its dimensions optimized. Figures 5(a), 5(b), and 5(c) plot the normalized optical power of the three outputs as a function of the phase shift applied between the two arms of the MZI (*ϕ*) for coupler separations *s* = 1, 2 and 3μm, respectively. All three normalized powers are sinusoidal with the phase shift. The corresponding coupler lengths *L* were chosen as 457 μm, 828.57 μm and 1530 μm in order that the individual sinusoidal responses separate to ~2π/3 rad from each other. The wider the separation the longer the required coupler length. The total output power is also plotted and is observed to be approximately constant with *ϕ* in all cases, justifying our neglect of the forward propagating radiative modes. Comparing Figs. 5(a), 5(b) and 5(c), the total output power becomes increasingly constant as the separation increases, but at the expense of a decrease in the dynamic range, so there is a trade-off between these two quantities. Another consideration is that a shorter device is more compact and profits integration. The devices used in the experiments are designed to have a coupler separation of *s* = 2 μm and coupler length *L* = 828.57 μm, providing a flatter total output power response, with good dynamic range, and an adequate device length.

## 3. Experimental

#### 3.1 Device and setup

Figure 6(a) shows a microscope image of the triple-output MZI device used in the experiments and Fig. 6(b) shows a higher magnification image of the triple coupler portion of the device. The devices were fabricated on a 450 μm thick silicon substrate by sequentially depositing a 10 μm thick Cytop layer as the bottom cladding, a 35 nm thick Au layer patterned using lift-off lithography, and another 10 μm thick Cytop layer as the top cladding that maintains the optical symmetry of the device [31,32]. All devices originate from the same wafer (identified internally as ND II). Due to fabrication most of the waveguides suffer some extent of deformation and the operation wavelength of the devices changes from the designed wavelength of 1310 nm to ~1370 nm (according to the experimental results). Other possible defects may include slightly thicker and wider waveguides, unbalanced Y-junction splitters and MZI arms [27], a wider coupler separation, and a slightly decentralized triple coupler.

Figure 7 illustrates the block diagram of the experimental setup. Light from a tunable laser is butt-coupled into the triple-output MZI device through a polarisation maintaining fiber (PMF) and the three outputs are collimated by a microscope objective and captured by a CCD camera simultaneously, constituting the optical path. Electrical current from a power supply is injected into the active region of one arm of the MZI through metal probes, heating the waveguide to apply the thermo-optic phase modulation, and measured by a multi-meter, forming the electrical circuit. The power supply, the multi-meter, and the camera are controlled by LabVIEW programs to change the applied voltages, to read the currents, and to take pictures of output mode images every time step (0.5 s), respectively. The mode images are captured in real time using frame grabber software (LBA-710PC, Ophir Spiricon), and post-processed using Matlab to determine the power (in arbitrary units) within three pre-set software apertures that cover the three outputs of the coupler. The power is computed by numerical integration of the pixel values within each aperture (as a diamond sum) and is related to the absolute power via a constant that can be determined via calibration. In this method the powers of all three outputs are captured simultaneously so that the suppression of time-varying common perturbations can be carried out.

#### 3.2 Results

With the above-mentioned device and setup, experiments were conducted by increasing the applied voltage in a step of 0.01 V every 0.5 s. The dissipated electrical powers corresponding to the voltages were calculated. The operating wavelength of the laser was tuned to 1364 nm, selected to optimize the performance of the devices. The output optical power versus dissipated electrical power of the individual outputs and the sum of the optical powers are plotted in Fig. 8(a), revealing the transfer characteristic of the device. The measured optical powers are given in arbitrary units (A.U.). A mosaic showing the outputs for three dissipated electrical powers that clearly shows the power switching among the three outputs is given as the inset.

As observed from Fig. 8(a), the curves of the individual outputs are sinusoidal and with ~2π/3 rad phase separation, but the two curves of the outside outputs are not as symmetric as in the ideal case of Fig. 5. This could be caused by asymmetry in the coupling structure induced by fabrication imperfections. In order to compare the measurements with theory, computations were conducted where the middle stripe of the triple coupler was decentralized (but maintained parallel) causing asymmetry relative to the two outside stripes. By optimising the fit between theory and the measurements, the left and right coupler separations increased from the designed 2 μm, to 2.2 μm and 2.5 μm, respectively. Also, the waveguide thickness decreases from the designed 35 nm to 31 nm (which is consistent with the results in [27]). Both the measured and the simulated results are normalized by dividing each by the sum of the optical powers, to exclude possible errors caused by the drifting of the powers in the experiments and to make them comparable. The measured results are re-plotted as a function of thermally induced phase shift in the same way that the computed results are plotted. The conversion from dissipated electrical power to thermally induced phase shift is discussed in [9] and [23]. Figure 8(b) shows the comparison of the measured and computed results, from which the generally good fit shows that decentralizing the middle waveguide of the triple coupler is a reasonable explanation for the difference in dynamic range between the two outside output powers.

## 4. Discussion

#### 4.1 Operation with maximum sensitivity

There are three main advantages for the triple-output MZI device compared to a single-output MZI. The first one is that an output near maximum sensitivity is always available. It is observed from Figs. 5 and 8 that as the three output curves are separated by ~2π/3 rad in phase, at any given point there is at least one output in the linear region of a sinusoidal curve, providing the fastest change in optical power in response to a phase shift, and thus maximum MZI sensitivity.

#### 4.2 Dynamic range

Secondly, the triple-output MZI has another advantage: In the ideal case (*e.g.*, Fig. 5(c)), the three outputs can be combined appropriately to obtain a response having a dynamic range three times larger than that of a single output, without significantly increasing the time-varying perturbations. Ideally, the dynamic range of the three individual outputs are nearly equal, the phase separation between them is very close to ~2π/3, and the total power can be taken as constant. Thus the three sinusoidal outputs *P _{1}*,

*P*and

_{2}*P*can be modeled mathematically using the following equations in terms of the induced phase shift

_{3}*ϕ*and the input optical power

*P*, with time-varying common perturbation terms (

_{in}*p*,

_{i}*p*) included:

_{o}here *a* is the amplitude and *b _{1}*,

*b*and

_{2}*b*are the constant offsets of the sinusoidal curves.

_{3}*p*represents a perturbation on the input side, such as a drift in the input optical power or input coupling conditions, and

_{i}*p*represents a perturbation received on the output side, for instance, due to background light. We define three power difference terms based on the individual powers:

_{o}*D*, where

_{i}= 2P_{i}- P_{j}-P_{k}*i*is one of 1, 2, 3 while

*j*and

*k*are the other two. In the case of

*D*for example, by multiplying Eq. (4.1.b) by 2 and subtracting Eqs. (4.1.a) and (4.1.b), the following expression is obtained:

_{2}Compared to *P _{2}*, the input common perturbation

*p*of

_{i}*D*remains the same whereas the output common perturbation

_{2}*p*is cancelled and the amplitude of the sinusoidal term triples. It is easy to calculate

_{o}*D*and

_{1}*D*and observe that they behave the same way as

_{3}*D*. Thus the difference terms provide a signal with 3 times larger dynamic range compared to the individual ones without increasing the level of any time-varying perturbations. Figure 9 shows

_{2}*D*,

_{1}*D*and

_{2}*D*obtained from the measurements of Fig. 8(a). It is observed that they have 3 times larger dynamic ranges, in agreement with Eq. (4.2).

_{3}#### 4.3 Suppression of common perturbations

The third advantage is the ability to suppress common input perturbations. Suppose there is no output perturbation *p _{o}*, then the input perturbation

*p*can be removed by normalization. Take the normalized

_{i}*P*for instance; it has the following expression as obtained from Eqs. (4.1.a)-(4.1.c):

_{2}with no output perturbation (*p _{o}* = 0) the above simplifies to:

As observed from Eq. (4.4) the input perturbation *p _{i}* is removed. If the output perturbation

*p*exists, the term

_{o}*P*cannot be completely cancelled but normalization can at least suppress to some extent the time-varying perturbation.

_{in}+ p_{i}The perturbation suppression obtained by normalization was observed in experiments. Figure 10(a) shows the unnormalized output optical powers and power differences versus dissipated electrical power. It is obvious that strong perturbations prevail in the curves. The amplitudes of the perturbations are proportional to the power of the signal and so the power differences cannot suppress them, therefore they are assumed to be input perturbations *p _{i}*. Figure 10(b) shows the same measurements but normalised, revealing significant suppression of the time-varying perturbations.

Normalization can also be applied to the power difference terms. From Eqs. (4.1.a)-(4.1.c) the normalized *D _{2}* is calculated as:

This produces a similar effect as normalizing individual powers, as is easily observed setting *p _{o}* = 0 into the above, yielding:

Thus there are four possible detection schemes: an individual power (*i.e. P _{2}*), a power difference (

*i.e. D*), a normalized individual power (

_{2}*i.e. P*(

_{2}/*P*)), and a normalized power difference (

_{1}+ P_{2}+ P_{3}*i.e. D*(

_{2}/*P*)). A statistical comparison between the four detection schemes is necessary. One way of doing this is to conduct a time tracing experiment where the applied voltage (dissipated electrical power, phase difference between arms) is kept constant and the output optical power is recorded over time. The standard deviation

_{1}+ P_{2}+ P_{3}*σ*of each scheme is then calculated, however, those of the normalized group cannot be directly compared to those of the unnormalized group (the units are different). Therefore the minimum detectable phase shift Δ

*ϕ*is introduced and used to compare all schemes directly; it is defined as [23]:

_{min}*P*is taken as the corresponding

_{min}*σ*times a factor

*k*, say

*k*= 2. Rewriting Eq. (4.1.b) by removing all the perturbation terms, we obtain:from which ∂

*P*/∂

_{2}*ϕ*is obtained as:substituting Eq. (4.9) into Eq. (4.7) yields the minimum detectable phase shift based on

*P*:

_{2}Similarly, the Δ*ϕ _{min}* for the other individual powers

*P*and

_{1}*P*are obtained, and so are those for the power differences, the normalized individual powers and the normalized power differences:

_{3}For maximum sensitivity, the device operates in the linear region where the cos*ϕ* term in Eq. (4.8) is very close to 0, so sin*ϕ* ~1. Substituting this and *k* = 2 into Eqs. (4.10)-(4.13), with our experimental value for *P _{in}* and the standard deviations obtained from a time-tracing experiment, the minimum detectable phase shifts of the four schemes can be calculated and compared directly to determine which scheme provides the best performance. Table 1 gives the

*σ*and the Δ

*ϕ*for three time-tracing experiments (nominally M 1, M 2 and M 3 in Table 1), each taken on the same triple-output MZI device with

_{min}*P*= 0.7 mW (the absolute power was determined by calibration), and each with a different voltage applied so that a different output operates in the linear region (

_{in}*i.e.*, M 1 has

*P*in the linear region, M 2 has

_{1}*P*in the linear region and M3 has

_{2}*P*in the linear region). The parameters

_{3}*a*= 0.22 and

*b*= 1.01 are chosen by best fitting with the transfer characteristic.

_{1}+ b_{2}+ b_{3}Comparing the standard deviations of *P _{i}* and

*D*for all three measurements in Table 1, suggests that there was very little time-varying output perturbation (

_{i}*p*, see Eqs. (4.1.a)-(4.1.c)) in the experiments to be cancelled using the power difference scheme, because all power differences have ~2 times larger standard deviation than the individual powers. However, comparing their corresponding Δ

_{o}*ϕ*it is observed that the power difference scheme provides an reduction of ~3/2 which originates from the 3 times larger dynamic range and the 2 times worse standard deviation (compare Eq. (4.10) with Eq. (4.11)). If the standard deviation of the power difference scheme remains comparable to that of a single power then Δ

_{min}*ϕ*would be reduced by a factor of 3.

_{min}Although the cases seem to be similar considering Eqs. (4.12) and (4.13), the normalized power difference scheme does not benefit from the larger dynamic range, because there is a fixed relation between the standard deviations of both. According to the definition of *D _{i}*,

*D*/(

_{i}*P*+

_{1}*P*+

_{2}*P*) can be expressed as 3[

_{3}*P*/(

_{i}*P*+

_{1}*P*+

_{2}*P*)]-1 and thus always has a 3 times larger standard deviation than

_{3}*P*/(

_{i}*P*+

_{1}*P*+

_{2}*P*). Substituting this relation into Eqs. (4.12) and (4.13) it is observed that the 3 times larger dynamic range of

_{3}*D*/(

_{i}*P*+

_{1}*P*+

_{2}*P*) is cancelled and the Δ

_{3}*ϕ*remains the same. Therefore the normalized individual power and the normalized power difference can actually be regarded as the same scheme. This is verified experimentally by comparing Δ

_{min}*ϕ*of

_{min}*P*/(

_{i}*P*+

_{1}*P*+

_{2}*P*) and

_{3}*D*/(

_{i}*P*+

_{1}*P*+

_{2}*P*) in Table 1.

_{3}A comparison of Δ*ϕ _{min}* of the power difference scheme and the normalized schemes reveals very similar values though not exactly equal, However, both are smaller compared to the individual power scheme, thus both are good schemes. The smallest value of Δ

*ϕ*obtained in these experiments was 7.3 mrad compared to 3.1 mrad in the dual-output case [23]. The length of the active region of both MZIs is similar so this is not the cause of the larger Δ

_{min}*ϕ*in the triple-output case. Considering that the two cases were tested under different experimental conditions, and the perturbation levels were neither controllable nor comparable, the larger detection limit in the triple-output case is attributed to larger perturbations that were not as easily suppressed. Relative to the dual-output, the triple-output eliminates the sensitivity fading and directional ambiguity in the former (there is always one output in a linear region).

_{min}## 5. Conclusions

In the interest of both thermal-optic and bio-chemical sensing applications, LRSPP triple-output MZIs were thermo-optically characterized at a free-space operating wavelength of ~1370 nm. The device consists of a MZI and a triple coupler in cascade, implemented with Au stripe waveguides 5 µm wide and 35 nm thick cladded with Cytop. The thermo-optic phase modulation was applied to one arm of the MZI, heated by injecting electrical currents thereon through metal probes. A theoretical model for three-waveguide coupling was proposed, implemented and validated by comparisons to literature results and to the measurements. The approach is based on modal analysis and overlap integrals, and is easy to implement. Dimensions for the triple coupler were chosen by optimizing the performance. As expected, the three output optical powers were sinusoidally related to the thermally induced phase shift and separated ~2π/3 rad. Four detection schemes were proposed and analyzed, and applied to improve the detection limit. Improvements are obtained from a 3 times larger dynamic range as well as the suppression of time-varying common perturbations, relative to a single output. The power difference scheme and the normalization schemes were found to be equally effective. A minimum detectable phase shift (Δ*ϕ _{min}*) ~2/3 that of a single output was obtained. The smallest value of Δ

*ϕ*obtained was 7.3 mrad. The device is of interest for sensing applications.

_{min}## Acknowledgments

The authors are grateful to Robin Buckley for helpful discussions and assistance with the modeling work, and to Ewa Lisicka-Skrzek and Hamoudi Asiri for sample preparation and fabrication, respectively.

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