1. Introduction

Silicon-on-insulator (SOI) photonics integrated circuits platforms use the crystalline silicon device layer to form waveguide cores and the underlying buried silicon dioxide (SiO₂) layer as the cladding [1]. In these platforms, the contrast between the refractive indices of the silicon as the waveguide and SiO₂ as the cladding, is high enough to ensure the confinement of the optical mode inside the waveguide. The advantage of this confinement is the possibility of having waveguide bends with small radii and more compact devices. However, time nm-scale roughness, mainly at the waveguide sidewalls, allow for higher scattering losses in case of higher index contrast [2]. One solution to this is to replace silicon nitride (SiN) with sillicon as the waveguide material. Beacouse of the relatively lower refractive index contrast between the SiN and the (SiO₂), SiN waveguides are less vulnerable to scattering losses. Also, the propagation losses are an order of magnitude lower for SiN waveguides and provide higher tolerances to fabrication variations [3,4]. However, the thermo-optical coefficient of SiN is at least one order of magnitude lower than silicon ($2.5 \times 10^{-5} K^{-1}$ [5] in comparison to $1.86 \times 10^{-4} K^{-1}$ [6]). This means that SiN is less efficient to implement devices such as thermo-optical switches [4]. Nevertheless, careful engineering can compensate for the effects of the low thermal coefficient.

The lower refractive index contrast between SiN and SiO₂ poses other challenges in integrated devices. In a SiN waveguide the fundamental mode is not as confined as Si waveguides which means lowering the bending radius results in significant radiation loss of the optical power [7]. So, waveguides on SiN platforms cannot have sharp bends which in turn results in large footprints for the devices and higher propagation losses. This reduces the integration density achievable in SiN platforms in comparison to silicon photonics, which can impact the size and cost of optical systems. Different approaches have been demonstrated to further miniaturize integrated optical devices, including the use of plasmonics [8], creating multifunctional devices [9] that can replace two components, or re-arranging the devices on the chip to minimize the space between them [10]. Nevertheless, all of these solutions have important drawbacks such as higher propagation losses, limited efficiency, and higher crosstalk.

In this article, to overcome the trade-off between size and bending losses in an unbalanced Mach-Zehnder interferometer (MZI), we demonstrated an MZI structure which uses the effective index difference between the two arms instead of using different lengths to generate the phase difference. The use of different waveguide widths has been introduced in other works, such as [11], to improve the control over the temperature related shifts in a silicon MZI. However, here we show that this concept can be used in a SiN based MZI to achieve a reduction in total device footprint of a factor of at least 40 times in comparison with the traditional approach. The same concept can be used in any other photonic platforms where the refractive index contrast between the core and cladding limits the minimum bending radius that can be used. For instance, this can be an issue in InP photonic circuits designed with shallow etched waveguides [12].

The article is organized as follows: Section 2 discusses the theoretical background and an analytical design methodology for the proposed structure while section 3 presents the simulations, fabrication results, and addresses the practicality of the device. Section 4 investigates the thermal tunability of the interferometer with the perspectives of compensating its sensitivity to fabrication tolerances and possibly implementing optical switching. Section 5 presents the application of the proposed design in a tunable integrated mirror. Finally, the conclusions are presented in section 6.

2. Theoretical analysis and design

The phase accumulated by an optical wave propagating through a waveguide is given by the expression $e^{-j\beta l}$, where $j = \sqrt {-1}$ is the imaginary number, $l$ is the propagation length. In this equation, $\beta$ is the propagation constant of the mode which is defined as $\beta = (2\pi n_{eff})/\lambda$, considering that $\lambda$ is the wavelength of the light. $n_{eff}$ shows the effective refractive index of the optical mode. the value of this term, and consequently the value of the propagation constant, is determined by the material, dimensions, and shape of the waveguide. the required phase shift in an conventional unbalanced MZI, is obtained by the change imposed between the length of the interferometer arms. In this case, only the length of one waveguide is altered which does not change thier effective refractive indices and thus the phase shift is equal to $\beta \Delta L$. An alternative way to achieve this phase difference is to alter the refractive indices or propagation constants of the MZI arms while having equal propagation lengths. In this former case, the phase shift is equal to $\Delta \beta L$. changing the width of the SiN waveguide, without changing the thickness, will directly affect the refractive index. For instance, enlarging the waveguide will increase the refractive and group index of the fundamental mode. Therefore, to impose a phase difference between two MZI arms that have equal lengths, one arm can be made wider than the other.

The schematic of this MZI is presented in Fig. 1(a). In a SiN waveguide with a core thickness of 440 $nm$, in order to minimize the bending losses, both from mode mismatch and radiation losses, the bending radii of at least 100 $\mu m$ are usually required [13]. Compared to a conventional MZI which provides the same free spectral range (FSR), the proposed differential MZI has a reduction in footprint by a factor of 46. This is mainly because, at least four 90 degrees bends are required to create the length difference $\Delta l$ as shown in Fig. 1(b). As a result, the couplers forming the MZI must be at least 400 $\mu m$ apart, imposing a minimum optical propagation length of 628 $\mu m$. Although it is possible to reduce the number of bends to two by folding the device (Fig. 1(c)), this architecture still imposes a minimum length difference between the arms equals to the separation between the waveguides at the output of the couplers multiplied by $\pi$ [14]. Thus, according to Fig. 1, the configuration with differential-width waveguides is 46 times smaller than the conventional design and 18 times smaller than the folded device. Even if a smaller bending radius of 50 $\mu m$ is considered to reduce the size of the standard MZI layout, the footprint is still reduced by a factor of 18 with the approach proposed here

 

Fig. 1. (a) The schematic and dimensions of the proposed MZI structure, (b) schematic of a conventional MZI with a bending radius of R, (c) schematic of a folded MZI design to reduce the footprint of conventional MZIs, and (d) analytical results of relative phase difference as a function of wavelength, calculated between the arms of the MZI with L_wg = 150 $\mu m$. The inset shows the normalized electric field amplitude of the fundamental TE mode across the cross section of the MZI waveguides presented in (a).

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In the proposed device, multimode interference (MMI) couplers are used instead of directional couplers. Owing to their larger dimensions, MMI couplers have lower sensitivity to fabrication variation while providing more accurate coupling ratio compared to directional couplers. However, they have a slightly higher insertion loss than directional couplers. The multimode interference region for the coupler used in this design is 76 by 10 $\mu m$ and is connected to the waveguides with a 50 $\mu m$ long taper. The wide side of the tapers is 3.2 $\mu m$ and their width decreases to match the waveguide width at the other end, allowing for the modes to evolve gradually and avoid losses. The input and output waveguides of the MZI have a width of 850 $nm$.

The thickness of the waveguide core in our lithography process is 440 $nm$ whereas the narrowest width which can be reliably fabricated is 450 $nm$. with these limitations the waveguide core is single mode in optical C and L bands with a width between 450 and 900 $nm$. These widths therefore provide the largest $\Delta \beta$ that we can implement while remaining both single mode and within the fabrication tolerances of our process. The cross section of both waveguides with the distribution of normalized amplitude of the electric field, are presented in the inset of Fig. 1(d). These fields correspond to fundamental transverse electric (TE) mode and show how the mode confinement is affected by the geometry of the waveguide.

The tapers connecting the MMIs to the waveguides forming the MZI arms end with different widths which means the rate of change for the width of the upper and lower tapers are different. So, even before the mode reaches the waveguides a phase difference is induced between the optical waves of the two waveguides. The analytical calculations of the phase difference of the MZI, should also consider the propagation along the length of the tapers. Therefore, as the first step, with an interval of 2.5 $\mu m$ along the taper lengths, the effective refractive index of the fundamental TE mode is calculated. These results are then plotted and through an interpolation, wee reached the equations which present the change in the effective index of the mode along the tapers. these equations are then used to calculate the extra accumulated phase at the end of each waveguide taper. Figure 1(d) shows teh results of these calculations for a MZI device in which 50 $\mu m$ tapers are connecting the couplers to the 80 $\mu m$ long MZI arms. The calculations are done in the optical L band and indicates that one can reach a $2\pi$ phase shift.

3. Results and discussions

The proposed structure was fabricated on a silicon wafer with 3.2 $\mu m$ of SiO₂ deposited on this substrate as the bottom cladding. The top cladding of the waveguide is also SiO₂ with a thickness of 3.4 $\mu m$. The bottom cladding layer was made with low pressure chemical deposition (LPCVD) of tetraethyl orthosilicate (TEOS) whereas the top one was obtained through plasma-enhanced chemical vapor deposition. The core layer is made of LPCVD SiN. To couple the light in and out of the chip, surface grating couplers were used. An image of a fabricated device is presented in Fig. 2(a). Ports 2 and 3 in this figure are the outputs, which correspond to S₂₁ and S₃₁ respectively in Fig. 2(c). For three different length of the MZI arms, L_wg, the transmitted power at the outputs of this novel MZI structure are simulated and plotted in Fig. 2(b). The simulations for this device were carried out using the eigenmode expansion method in the MODE software from Ansys Lumerical. As expected, an increase in L_wg decreases the FSR of the interferometer. Increasing the length of both MZI arms leads to a larger phase difference at the outputs. This is equivalent to increasing $\Delta L$ (the length difference) in a conventional MZI.

 

Fig. 2. (a) Image of the fabricated device, (b) simulation of the transmitted power at port 3 of the MZI for three distinct waveguide lengths, L_wg, and (c) measurement and simulation results of the transmitted power at the outputs (port 2, and 3) of a fabricated MZI with an arm length of L_wg = 150 $\mu m$.

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Figure 2(c) shows an example of the data measured out of ports 2 and 3 for L_wg = 150 $\mu m$ after compensation of the contribution of grating couplers by normalizing with a reference structure. The results are in good agreement with the simulations. The average insertion loss of the device is 0.42 dB, as measured over six different dies from three wafers, which compares advantageously with conventional SiN-based MZI structures. Comparing the blue and red solid curves from Fig. 2(c), we note that the extinction ratio at each output of the MZIs are not equal. Here, the extinction ratio is defined as the difference between the maximum and minimum of the transmission spectrum. From our appraisal of the data from all dies (not shown) this trend seems generalized to all devices, although all ports should theoretically have the same extinction ratios. The difference in the extinction ratios of the outputs is most likely caused by a slight deviation in the splitting ratio of the MMI couplers. Indeed, standalone MMI couplers were fabricated on the same dies as the MZIs and the measurements confirm a minor offset in the splitting ratios. The average of this offset over the six measured dies is 1.38 $\%$. Measurements from these standalone couplers (data not shown) also revealed that the main source of loss in the MZI is the insertion loss of the couplers, as expected.

Any MZI, including in the proposed design, require changes in the length of the waveguides forming the arms of the interferometer to tune the FSR. However, in the proposed design the lengths of both arms must change equally to achieve the phase difference required for a given FSR, whereas in a conventional design only one of the two arms changes. Fig. 3 shows the change in FSR as the waveguide length is changed for both types of MZI. In Fig. 3(a) L_wg is the actual length of the equal-length waveguide MZI whereas in Fig. 3(b) $\Delta L$ is the length difference between the two arms of a conventional MZI. In Fig. 3(b), $\Delta L$ = 0 means that there is no phase difference between the arms (i.e., the MZI is balanced) resulting in infinite FSR.

 

Fig. 3. (a) Total MZI arms lengths required to obtain a FSR between 65 and 42 $nm$ with the proposed design compared to (b) the length difference needed to achieve the same FSR in a conventional MZI. The inset in (a) shows the corresponding FSRs for longer length of the MZI arms in the proposed design.

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The phase difference $\Delta \phi$ in the MZi with arms of equal lengths but different widths, is expressed by the multiplication of the fixed propagation constant difference by the length of the waveguides, i.e. $\Delta \phi =\Delta \beta L_{wg}$. As we limit ourselves to SiN waveguides that are single mode with a thickness of 440 $nm$, the difference between the effective indices is always less than 0.17, such that $\Delta \beta$ is always less than 6.5$\times$105 rad/m in the L band. In a conventional MZI, the phase difference is given by $\Delta \phi =\beta \Delta L$, where the propagation constant has values above $5.5\times 106$. This explains why larger changes in waveguide lengths are needed to modify the FSR in the proposed design compared to conventional MZIs. However, as explained above, the minimum waveguide length for the arms of a conventional MZI must be at least equal to four 90 degrees bends, to which the value of $\Delta L$ is added. Even with the long waveguide required to maintain the FSR below 10 $nm$, the proposed design remains more compact than conventional MZIs ( 25 times smaller in surface area). In fact, it does not matter how small the length difference is in a conventional MZI, large bending radii, required to achieve low loss, are the limiting factor leading to significant device footprints. As a good example, 150 $\mu m$ long arms with the proposed design will result in a 42.83 $nm$ FSR, whereas a length difference ($\Delta L$) of only 27.2 $\mu m$ is needed in a conventional MZI with equal waveguide width of 850 $nm$. However, as shown by Fig. 1(a) and (b), such a conventional MZI will occupy more than 46 times the floor space than the proposed design due to the large bending radii.

4. Tunability

Sensitivity to fabrication variations/tolerances is a challenge in any type of integrated circuits, including integrated photonics circuits. Using robust components, e.g. MMI couplers instead of directional couplers as we did, can indeed help to mitigate this. However, because the MZI arms are not of equal widths in the proposed design, the resulting devices are more sensitive to fabrication variations. Assuming dimensional changes are uniformly/randomly spread over a fabrication area, slight changes in width in a conventional MZI with equal waveguides widths, will most likely modify the propagation constant equally on average in each arm. This will have a limited effect on the phase difference at the output of the MZI. In the proposed design, a variation in the dimensions (width) changes the propagation constant of each arm at a different rate, which can affect the phase difference accumulated through the device. Fortunately, variations caused by the fabrication process can be compensated by implementing a tuning mechanism. Although the thermo-optic coefficient of SiN is lower than that of Si, it can still be leveraged to fine tune the response of the MZI. As will be demonstrated below, having waveguides with different dimensions can simplify the design of thermal actuators.

For this study, we consider a simple and easily fabricated 15 $\mu m$ wide aluminum (Al) local heater, located on top of the MZI arms. Changes in effective index due to thermo-optic effects in the waveguides were calculated with the Finite-Difference Eigenmode (FDE) method and are presented in Fig. 4. The plots show ng, neff, and the corresponding change in accumulated phase for 100 $\mu m$ long waveguides caused by variations in temperature. Because the temperature is controlled by changing the power dissipated by the heater, the results are plotted as a function of applied electrical power

 

Fig. 4. Modeling of thermal effects on the effective index (shown in blue) and group index (shown in red) of a waveguide with a width of (a) 450 $nm$, and (b) 900 $nm$. (c) Change in the accumulated phase between the narrow and wide waveguides after propagation over a length of 100 $\mu m$ as a function of power dissipated by the heater. The insets show the temperature at the center of the waveguides versus heater power and the heat map around a 450 $nm$ waveguide when P = 100 $mW$. (d) Spectrum of the MZI when the Al thermal heater dissipate a power of 100 $mW$. The dashed lines show the spectrum of the MZI when the local heater is not working.

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Since the MZI is very compact, its arms are separated by less than 5 $\mu m$ , which corresponds to the separation between the outputs of the MMIs. Note that the different propagation constants in each of the MZI arms ensure that there is no evanescent coupling between them despite the small separation. Since the 15 $\mu m$ wide Al heater covers both waveguides, they should experience the same temperature change as the power dissipated by the heater increases. Yet as shown in Fig. 4, the change in the accumulated phase vs power is significantly higher in the 900 $nm$ wide waveguide compared to the 450 nm wide waveguide. Therefore, it is possible to effectively tune the response of this compact MZI with a single heater. Figure 4(d) shows that a 5.47 nm shift ($\sim 12.5$ $\%$) is expected in the output spectrum of the MZI for 100 $mW$ of heater power. Fig. 5 shows a simulation of fabrication variations compensation with such a local heater. In this example, the 7.33 $nm$ shift in transmission spectrum incurred by a 10 $nm$ increase in waveguides width can be corrected by increasing the temperature of the MZI arms by $67.86^o C$, which corresponds to a power dissipation of 118 $mW$.

 

Fig. 5. Changes in the output of the one of the MZI ports imposed by 10 $nm$ deviation in the width of the waveguides and the results of the thermally tuning

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While this appears rather efficient, the compact design does have limitations with regards to tuning power efficiency in comparison to conventional MZIs. Indeed, in the latter the large bent waveguides make it possible to thermally isolate each arm such that large temperature gradients can easily be created between them by heating only one of them. This typically results in lager phase differences per unit power. Coming back to the conventional MZI design with 850 $nm$ wide waveguide and a 27.2 $\mu m$ $\Delta L$ corresponding to a FSR of 42 $nm$, 100 $mW$ of heater power applied to only one arm will shift the spectrum by 11.5 nm. Therefore, for the same amount of power, the conventional MZI provides a $57\%$ larger spectral shift over the compact MZI configuration.

5. Applications

MZIs are among the most used devices in integrated photonics. The compact device demonstrated above can drastically reduce the total chip area required in complex or redundant systems such as wavelength division multiplexers (WDMs). In this section, we study the use of the proposed design in a tunable mirror, which can find applications in integrated lasers and transmitters [15,16], for instance. To implement this tunable mirror, a Sagnac loop mirror (SLM) is added at the output of a MZI to reflect the optical energy towards the inputs of the coupler, as shown in Fig. 6(a). The device fabricated based on this schematic is presented in Fig. 6(b), while the corresponding experimental results are presented in Fig. 6(c). The radius of the SLM is 100 $\mu m$ to ensure low propagation losses and it is made of a 850 $nm$-wide SiN waveguide. For the MZI, we have used the same configuration as the one presented in Fig. 1(a). Although the ultimate use case is meant to be tunable, the device presented in Fig. 6 was not fitted with a heater and is therefore not thermally tunable.

 

Fig. 6. (a) Schematics of a tunable mirror using the proposed MZI and (b) image of the fabricated devices. (c) Measurement results for the fabricated structure using a waveguide length of 150 $\mu m$. The dashed line shows the simulation results for the same structure.

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SLMs can provide high reflections if they have a radius of curvature large enough to minimize losses. Also, the propagation length inside the SLM is rigorously the same for both inputs since they propagate through the exact same length of waveguide but in opposite directions. Therefore, the addition of the SLM does no change the phase difference created by the MZI. The response of the mirror is completely controlled by the MZI since it sets the amount of light that is launched in the upper and lower branches of the SLM. For example, if the light at a given wavelength is split equally between the two ports of the SLM, then it will be completely transmitted to the output port. On the other hand, if all the light is incident at the upper branch of the SLM, then it will be reflected towards the input port in Fig. 6(a).

Moreover, the light traversing this device goes through the MZI twice. Hence, to switch the state of the tunable mirror from complete reflection of the light towards the input to total transmission toward the output requires only half of the phase change needed in a conventional MZI switch [17]. As demonstrated in the previous section, this relative phase change can easily be generated by adding a local heater over the arms of the MZI.

Figure 6(c) shows that the device as fabricated resulted in a FSR of 20.16 $nm$. This is close to half the FSR of a comparable standalone MZI, (42.83 $nm$) because the interferometer is in a folded configuration. Moreover, since the mode travels twice through each MMI coupler, the insertion loss increased to 2.56 $dB$. Simulation results, overlaid as a dashed line predict a slightly larger FSR (21.39 $nm$) than what is measured from the fabricated device. This difference can be explained by variations of the effective and group index caused by the fabrication process which were not thermally compensated in this device.

6. Conclusion

We proposed and demonstrated a compact MZI for SiN platforms with differential waveguide widths. This device was fabricated and experimentally characterized in the C and L bands. We showed that by adding a simple local heater it is possible to overcome the fabrication sensitivity. We also showed that we can take advantage of the different change in effective refractive index between waveguides of different widths to effectively tune the response of the MZI with a single simple and compact heater. The experimental measurements are in good agreement with simulations and analytical calculations.

A larger index contrast could be obtained with a fabrication process that combines both silicon and silicon nitride waveguides [18], which would results in even smaller devices. Moreover, the larger thermo-optic coefficient of silicon would improve the tuning efficiency. However, few commercial foundries currently offer this capability and the transition regions required to couple between the SiN and Si waveguides could create small losses that would have a significant impact on the extinction ratio of the MZI since they would affect the power ratio between the arms

We also demonstrated how the proposed device can be used as an integrated mirror by adding a Sagnac loop at one MZI end. The reduced footprint of our proposed interferometer will significantly increase device density on SiN chips by decreasing the footprint of MZI by a factor of a least 46, at the cost of a slightly lower tuning power efficiency. This could help reduce the cost of low-loss high-performance photonic integrated circuits not only in telecommunications but also in other areas such biosensors since SiN waveguides have a wide transparency widow.