Abstract

We report an all-silicon thermally insensitive (${-}{1.5}\;{{\rm pm/}^\circ}{\rm C}$) ${2} \times {2}$ Mach–Zehnder interferometer (MZI) over a spectral range from 1540 to 1620 nm. Additionally, the proposed MZI exhibits no imbalance in its extinction ratio with temperature. The broadband spectral range with minimal temperature sensitivity is achieved by slightly overcompensating the MZI design for temperature variations. At the same time, the uniformity in the extinction ratio is controlled using sub-wavelength grating adiabatic couplers for splitting and combining.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Silicon photonics (SiPh) is proving to be a promising platform for several technologies in key areas such as optical interconnects in data centers, bio-sensing, quantum computing, and artificial intelligence [1]. However, some bottlenecks related to temperature sensitivity and birefringence in the SiPh platform can hinder its usage [2]. The large thermal drift in SiPh-based devices is due to the high thermo-optic coefficient (TOC) of silicon. The latter has a value of ${1.86} \times {{10}^{- 4}}{/^\circ}{\rm C}$, which is an order of magnitude larger than silica-based devices [3]. This high-temperature sensitivity can be detrimental to several applications, particularly on-chip interconnects [4] and integrated optical sensors.

To overcome the thermal sensitivity of silicon-based photonic integrated circuits (PICs), thermoelectric coolers and on-chip heaters are used to either externally stabilize the PICs or to locally compensate for the temperature drift. Nevertheless, these methods increase the complexity and the power consumption of a SiPh chip. Alternatively, passive techniques can be used for thermal stabilization. One such approach employs a negative TOC upper-cladding material to cover the silicon waveguide [5]. In this method, the temperature dependence of the refractive indices of the silicon core and the cladding cancels out, resulting in a temperature-independent effective index of the waveguide. However, to increase the mode interaction with the negative TOC material, narrow waveguides with more considerable losses are used [6]. Additionally, the negative TOC materials are generally made of polymers that are not compatible with the mainstream complementary metal-oxide semiconductor (CMOS) silicon processing. Even when compatible material, such as titanium oxide, is used, it requires additional fabrication steps that increase the cost.

Though several temperature-insensitive all-silicon design techniques are reported on a Mach–Zehnder interferometer (MZI), the athermal response is limited to a narrow spectral range. Furthermore, the extinction ratio (ER) of the MZI may still vary with temperature [714]. In these approaches, the temperature dependence of phase delay in the MZI arms is balanced; thus, its wavelength response is made immune to temperature changes. This is achieved by designing the MZI arms with either different widths [79], different modes [1012], or even different configurations [13,14]. Since the optimization is done only at the 1550 nm wavelength, the temperature sensitivity increases in broadband operation. Also, these methods either employ directional couplers or Y-branches to split and combine light [713].

A Y-branch (${1} \times {2}$) theoretically has a wavelength-independent splitting ratio as the input port is located on the axis of symmetry. Even though Y-junctions are compact with a broadband splitting ratio [15], they cannot construct ${2} \times {2}$ active switches. On the other hand, in ${2} \times {2}$ couplers, the input port is not located at the symmetry axis. Therefore, their splitting ratio inevitably has wavelength dependence. In particular, the splitting ratio of a ${2} \times {2}$ directional coupler is strongly dependent on wavelength and temperature. When coupled with fabrication variability, this sensitivity can cause a high imbalance in the ER of an MZI. This is even true in thermally compensated MZIs, especially for broadband operations.

A simple way to overcome this limitation is to use an ideal broadband 3 dB coupler that is wavelength and temperature insensitive. The nonuniformity in the MZI ER can be significantly reduced with this approach. In literature, several classes of broadband ${2} \times {2}$ couplers are reported, such as adiabatic, multimode interference (MMI), sub-wavelength grating (SWG) adiabatic, and broadband directional couplers (BDC). However, their performance under temperature variations is not discussed. In order to find the most thermally stable and broadband 3 dB coupler, we experimentally study the effect of temperature on the splitting ratio of these couplers. For quantitative evaluation, different coupler designs are integrated first with a standard uncompensated MZI. After finalizing the coupler, it is finally combined with the proposed MZI design for broadband athermal ${2} \times {2}$ operation. All of the broadband 3 dB couplers are relatively larger than a conventional directional coupler, which has the drawback of increased sensitivity to the phase errors in its output ports. From Ref. [16], one can say that the higher the power splitting accuracy, the higher the worst-case phase error. Consequently, point mirror configuration needs to be used while designing to eliminate phase errors [16].

The proposed MZI is shown in Fig. 1(a), and the couplers are shown in Figs. 1(b)–1(e). The MZI consists of two arms coupled using a 3 dB splitter and combiner at its input and output.

 figure: Fig. 1.

Fig. 1. (a) Proposed MZI, (b) adiabatic coupler, (c) MMI, (d) broadband directional coupler (BDC), and (e) sub-wavelength grating (SWG) adiabatic coupler.

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One of the arms propagates a length of ${{L}^*}$ horizontally and ${{L}_1}$ vertically with a constant width of 0.5 µm. The second arm has a length ${{L}_2}$ with a waveguide width of 0.5 µm (${{W}_0}$) and is tapered down to 0.2 µm (${{W}_1}$) for length ${{L}^*}$. Adiabatic tapers (10 µm) are used to make transitions from 0.5 µm to 0.2 µm. The path length difference (${\Delta}L)$ is given by ${{L}_2} - {{L}_1}$. Furthermore, arm 1 has a constant width of ${{W}_0}$, while arm 2 has regions of a narrower width of ${{W}_1}$.

The MZI matching condition gives the maximum transmission wavelength (${\lambda _0}$ in a vacuum):

$$m{\lambda _0} = {n_{\text{eff}}}{\Delta}L + {\Delta}{n_{\text{eff}}}{L^*},$$
where ${\Delta}{n_{\text{eff}}} = {n_{\text{eff}}}({{W_1}}) - {n_{\text{eff}}}({{W_0}}),$ and $m$ is the interference order. The order is set to half-integer for destructive interference. Due to the waveguide dispersion, the order is modified to M [Eq. (2)], and the temperature sensitivity is modeled by Eq. (3) [17]:
$$M = m - {\Delta}L\frac{{\partial {n_{\text{eff}}}}}{{\partial \lambda}} - {L^*}\frac{{\partial {\Delta}{n_{\text{eff}}}}}{{\partial \lambda}},$$
$$\frac{{{\Delta}{\lambda _0}}}{{{\rm \Delta T}}} = \frac{1}{M}\left[{{\Delta}L\frac{{\partial {n_{\text{eff}}}}}{{\partial T}} + {L^*}\frac{{\partial {\Delta}{n_{\text{eff}}}}}{{\partial T}}} \right]\!.$$

In the above equation, $\frac{{\partial {{n}_{\text{eff}}}}}{{\partial {T}}}$ is always positive. However, $\frac{{\partial {\Delta}{{n}_{\text{eff}}}}}{{\partial {T}}}$ is made negative by selecting ${{W}_0} = {0.5}$ µm and ${{W}_2} = {0.2}$ µm. If the values of ${\Delta}L$ and ${L^*}$ are appropriately chosen, then Eq. (3) can be brought to zero. The free spectral range (FSR) of this MZI is given by

$${\rm FSR} = \frac{{{\lambda _0}}}{M}.$$

The design following Eq. (3) can have zero spectral shift with the temperature only at ${\lambda _0}$. The wavelength shift (red shift) with temperature increases very fast as the wavelength diverges from ${\lambda _0}$. In our proposed design, the design is slightly overcompensated at 1550 nm to average out the red shift at other wavelengths, which increases bandwidth. However, it should be noted that if the design is overcompensated beyond a specific limit, then the device experiences a very strong blue shift, detuning it from athermal behavior. Simulations are performed in Lumerical Mode and Interconnect to avoid this detuning.

The finite-difference-Eigenmode (FDE) solver was used to calculate the effective index and the values of $\frac{{\partial {{n}_{\text{eff}}}}}{{\partial {T}}}$ (Fig. 2).

 figure: Fig. 2.

Fig. 2. (a) Calculated effective and group indices for different widths. (b) Temperature dependence of effective index at 1550 nm wavelength.

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The TOC of silicon (device layer) and oxide (box and cladding) are set to ${1.86} \times {{10}^{- 4}}/{\rm K}$ and ${0.8} \times {{10}^{- 5}}/{\rm K}$, respectively. The lengths ${{L}^*}$ and ${\Delta}L$ are selected to be 76 µm and 64 µm, respectively. The device is optimized for the fundamental transverse electric (TE) mode of operation.

Figure 3 depicts the simulation results (Lumerical Interconnect) of the proposed MZI with a broadband 3 dB coupler [Fig. 3(a)] and a conventional directional coupler [Fig. 3(b)]. The wavelength sensitivity in the case of a directional coupler is clearly observed for a ${2} \times {2}$ operation. Also, as shown in Fig. 4(a), if the broadband 3 dB coupler is temperature insensitive, then the ER is uniform, and performance is robust. On the other hand, with a conventional ${2} \times {2}$ directional coupler, a high imbalance in the ER is observed with temperature variations [see Fig. 4(b)]. This is due to the strong wavelength and temperature dependence in the splitter. The relation between ERs of the MZI spectrum and splitting ratio of 3 dB couplers ($\eta$) is given by

$$\eta = \frac{1}{2} \pm \frac{1}{2}\sqrt {\frac{1}{{{{10}^{{\rm ERs}/10}}}}} .$$
 figure: Fig. 3.

Fig. 3. Simulated ${2} \times {2}$ response of the proposed MZI with (a) a broadband 3 dB coupler with a slight deviation from 50/50 coupling and (b) a standard directional coupler.

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 figure: Fig. 4.

Fig. 4. Response of the proposed MZI with simulated temperature variations using (a) a broadband 3 dB coupler (assuming athermal) and (b) a directional coupler.

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It is challenging to fabricate an identical splitter and combiner in an MZI circuit. Since the splitting ratio of a directional coupler has strong wavelength/fabrication dependence, the changes in splitting ratio with temperature can differ in both the splitter and combiner, causing a high imbalance. All of the simulations are done in the wavelength range of 1540–1620 nm, and the spectrum shift is recorded to be ${-}{0.8}\;{{\rm pm/}^\circ}{\rm C}$.

Therefore, to have a robust response in a ${2} \times {2}$ MZI, its input and output couplers need to be wavelength and temperature insensitive. It is computationally very intensive to simulate the temperature sensitivity of the couplers and to account for the fabrication variations. For example, in a planar directional coupler formed by two symmetric waveguides, the coupling length as a function of temperature is given by

$${L_c}\left(T \right) = \frac{{\left[{\pi {k_z}\left(T \right)\left({\frac{w}{2}} \right)\left({1 + \frac{2}{{\left({{\alpha _x}\left(T \right)w} \right)}}} \right)\left({\alpha _x^2\left(T \right) + k_x^2\left(T \right)} \right)} \right]}}{{\left[{2{\alpha _x}\left(T \right)k_x^2\left(T \right){e^{- {\alpha _x}\left(T \right)g}}} \right]}},$$
where ${k_z}(T)$ is the effective propagation constant, ${\alpha _x}(T)$ is the substrate transversal propagation constant, ${k_x}(T)$ is the core waveguide transversal propagation constant, $w$ is the waveguide width, and $g$ is the gap between two waveguides.

Hence, to study the impact of the input–output splitter and combiner, various broadband fabrication tolerant coupling types [Figs. 1(b)–1(e)] were first experimentally evaluated using a standard uncompensated MZI structure. By observing the changes in ERs with temperature, we can extract the thermal effect on the power splitting ratio ($\eta)$ of the couplers.

MMI couplers offer broadband wavelength-insensitive performance. They operate on the principle of self-imaging; the input field can be reproduced in single or multiple images at periodic intervals along the field propagation direction [18]. MMIs have relaxed fabrication tolerance and a wider spectral range compared to conventional directional couplers. We designed a ${2} \times {2}$ MMI splitter using paired interference with the width and length of the multimode section as 7 µm and 150 µm, respectively [18]. Adiabatic couplers are essentially wavelength-independent and less sensitive to fabrication variations compared to MMI couplers [19]. In an adiabatic 3 dB coupler, only one mode (even/odd) is excited, and all of the power stays in the excited mode, even with dimensional changes [20]. The only drawback with an adiabatic coupler is its large footprint. The adiabatic coupler in this study is based on Ref. [21]. We have also adopted a BDC design based on Ref. [22]. The coupler has 50/50 splitting in a wavelength range of 1540–1620 nm and uses a phase control element made from asymmetric waveguides between two symmetric regions. Additionally, the SWG can reduce the footprint of adiabatic couplers. Therefore, we design and study an adiabatic coupler with a SWG [23].

The devices are fabricated using electron-beam lithography with plasma etching at Applied Nanotools (Canada). In the test measurement setup, a fiber array of four TE-mode polarization-maintaining fibers and a pitch of 127 µm was used, which matches the silicon chip grating coupler spacing. A tunable laser source, Agilent 81600B, and optical detector sensor, Agilent 81635A, are used to characterize the optical transmission response of the MZI. Finally, the chip is mounted on a stage with a temperature controller.

Here, in Fig. 5, we show and analyze the measurement results of various broadband couplers. Since a standard unbalanced MZI is used, the spectrum shift with temperature is around 80 pm/°C. To quantify the thermal sensitivity of 3 dB couplers, the changes in the ER are observed with temperature. The larger change in ER signifies a greater change in splitting ratio with temperature. Our results show that among all of the couplers, the SWG adiabatic coupler is most thermally robust. In the SWG waveguides, the propagating optical modes interact less with the silicon and more with the cladding oxide, which is an order of magnitude less sensitive to temperature.

 figure: Fig. 5.

Fig. 5. Measured transmission of uncompensated MZI with different coupler types using (a) MMI, (b) a BDC, (c) an adiabatic coupler, and (d) an SWG adiabatic coupler.

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On the other hand, the MMI coupler is found to be most temperature sensitive. The thermal sensitivity of broadband 3 dB couplers follows the order ${\rm MMI} \gt {\rm BDC}$ (phase control) > adiabatic > SWG adiabatic. Table 1 summarizes the performance metrics of each coupler type.

Tables Icon

Table 1. Coupler Performance Metrics

Finally, we fabricated the proposed athermal MZI with the SWG adiabatic coupler. For the sake of comparison, the proposed MZI design is fabricated with an MMI coupler as well. Figure 6 shows the scanning electron microscope (SEM) images of the proposed MZI with (a) MMI coupler and (b) SWG adiabatic coupler.

 figure: Fig. 6.

Fig. 6. SEM image of proposed athermal MZI with (a) MMI and (b) an SWG adiabatic coupler.

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Figure 7 shows the transmission response of the proposed MZI with (a) MMI coupler and (b) SWG adiabatic coupler. The nonuniformity in ER with temperature is seen when the MMI coupler is used in the design. In contrast, using the SWG adiabatic coupler, the overall robustness is greatly improved. It additionally shows a near-zero temperature sensitivity of less than ${-}{1.5}\;{{\rm pm/}^\circ}{\rm C}$ over a wide wavelength range from 1540 nm to 1620 nm. The measured sensitivity is less than 50 fold the value of a standard uncompensated MZI design (Fig. 5).

 figure: Fig. 7.

Fig. 7. Measured transmission of proposed athermal MZI with (a) MMI and (b) an SWG adiabatic coupler.

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While most of the previous athermal designs utilize ${2} \times {2}$ directional couplers, the temperature-insensitive response is reported only for one of the output ports [812]. Table 2 summarizes the performance of various temperature-insensitive designs based on directional couplers. When comparing the performance of the proposed athermal MZI (SWG adiabatic coupler) with previously reported ones, this work exhibits excellent athermal performance over an extended range of 80 nm, and a complete ${2} \times {2}$ response is recorded.

Tables Icon

Table 2. Bandwidth Comparison

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. B. Jalali and S. Fathpour, J. Lightwave Technol. 24, 4600 (2006). [CrossRef]  

2. Z. Mohammed, B. Paredes, and M. Rasras, Appl. Sci. 11, 2366 (2021). [CrossRef]  

3. R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007). [CrossRef]  

4. K. Padmaraju, J. Chan, L. Chen, M. Lipson, and K. Bergman, Opt. Express 20, 27999 (2012). [CrossRef]  

5. Y. Zhang and Y. Shi, Opt. Lett. 40, 264 (2015). [CrossRef]  

6. B. Guha, J. Cardenas, and M. Lipson, Opt. Express 21, 26557 (2013). [CrossRef]  

7. M. Uenuma and T. Moooka, Opt. Lett. 34, 599 (2009). [CrossRef]  

8. B. Guha, A. Gondarenko, and M. Lipson, Opt. Express 18, 1879 (2010). [CrossRef]  

9. H. Yang, J. Zhang, Y. Zhu, X. Zhou, S. He, and L. Liu, Opt. Lett. 42, 615 (2017). [CrossRef]  

10. P. Xing and J. Viegas, Opt. Express 23, 24098 (2015). [CrossRef]  

11. X. Guan and L. H. Frandsen, Opt. Express 27, 753 (2019). [CrossRef]  

12. S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013). [CrossRef]  

13. P. Xing and J. Viegas, Proc. SPIE 9752, 97520U (2016). [CrossRef]  

14. T. Hiraki, H. Fukuda, K. Yamada, and T. Yamamoto, Front. Mater. 2, 1 (2015). [CrossRef]  

15. Y. Zhang, S. Yang, A. Lim, G. Lo, C. Galland, T. Jones, and M. Hochberg, Opt. Express 21, 1310 (2013). [CrossRef]  

16. C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005). [CrossRef]  

17. Z. Mohammed, B. Paredes, and M. Rasras, in IEEE Photonics Conference (IPC) (2020), pp. 1–2.

18. L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995). [CrossRef]  

19. K. Suzuki, H. C. Nguyen, T. Tamanuki, F. Shinobu, Y. Saito, Y. Sakai, and T. Baba, Opt. Express 20, 4796 (2012). [CrossRef]  

20. J. D. Domenech, J. S. Fandino, B. Gargallo, and P. Munoz, J. Lightwave Technol. 32, 2536 (2014). [CrossRef]  

21. H. Yun, W. Shi, Y. Wang, L. Chrostowski, and N. A. F. Jaeger, Proc. SPIE 8915, 89150V (2013). [CrossRef]  

22. Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, Opt. Express 23, 3795 (2015). [CrossRef]  

23. H. Yun, Y. Wang, F. Zhang, Z. Lu, S. Lin, L. Chrostowski, and N. A. F. Jaeger, Opt. Lett. 41, 3041 (2016). [CrossRef]  

References

  • View by:

  1. B. Jalali and S. Fathpour, J. Lightwave Technol. 24, 4600 (2006).
    [Crossref]
  2. Z. Mohammed, B. Paredes, and M. Rasras, Appl. Sci. 11, 2366 (2021).
    [Crossref]
  3. R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007).
    [Crossref]
  4. K. Padmaraju, J. Chan, L. Chen, M. Lipson, and K. Bergman, Opt. Express 20, 27999 (2012).
    [Crossref]
  5. Y. Zhang and Y. Shi, Opt. Lett. 40, 264 (2015).
    [Crossref]
  6. B. Guha, J. Cardenas, and M. Lipson, Opt. Express 21, 26557 (2013).
    [Crossref]
  7. M. Uenuma and T. Moooka, Opt. Lett. 34, 599 (2009).
    [Crossref]
  8. B. Guha, A. Gondarenko, and M. Lipson, Opt. Express 18, 1879 (2010).
    [Crossref]
  9. H. Yang, J. Zhang, Y. Zhu, X. Zhou, S. He, and L. Liu, Opt. Lett. 42, 615 (2017).
    [Crossref]
  10. P. Xing and J. Viegas, Opt. Express 23, 24098 (2015).
    [Crossref]
  11. X. Guan and L. H. Frandsen, Opt. Express 27, 753 (2019).
    [Crossref]
  12. S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013).
    [Crossref]
  13. P. Xing and J. Viegas, Proc. SPIE 9752, 97520U (2016).
    [Crossref]
  14. T. Hiraki, H. Fukuda, K. Yamada, and T. Yamamoto, Front. Mater. 2, 1 (2015).
    [Crossref]
  15. Y. Zhang, S. Yang, A. Lim, G. Lo, C. Galland, T. Jones, and M. Hochberg, Opt. Express 21, 1310 (2013).
    [Crossref]
  16. C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
    [Crossref]
  17. Z. Mohammed, B. Paredes, and M. Rasras, in IEEE Photonics Conference (IPC) (2020), pp. 1–2.
  18. L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
    [Crossref]
  19. K. Suzuki, H. C. Nguyen, T. Tamanuki, F. Shinobu, Y. Saito, Y. Sakai, and T. Baba, Opt. Express 20, 4796 (2012).
    [Crossref]
  20. J. D. Domenech, J. S. Fandino, B. Gargallo, and P. Munoz, J. Lightwave Technol. 32, 2536 (2014).
    [Crossref]
  21. H. Yun, W. Shi, Y. Wang, L. Chrostowski, and N. A. F. Jaeger, Proc. SPIE 8915, 89150V (2013).
    [Crossref]
  22. Z. Lu, H. Yun, Y. Wang, Z. Chen, F. Zhang, N. A. F. Jaeger, and L. Chrostowski, Opt. Express 23, 3795 (2015).
    [Crossref]
  23. H. Yun, Y. Wang, F. Zhang, Z. Lu, S. Lin, L. Chrostowski, and N. A. F. Jaeger, Opt. Lett. 41, 3041 (2016).
    [Crossref]

2021 (1)

Z. Mohammed, B. Paredes, and M. Rasras, Appl. Sci. 11, 2366 (2021).
[Crossref]

2019 (1)

2017 (1)

2016 (2)

2015 (4)

2014 (1)

2013 (4)

H. Yun, W. Shi, Y. Wang, L. Chrostowski, and N. A. F. Jaeger, Proc. SPIE 8915, 89150V (2013).
[Crossref]

B. Guha, J. Cardenas, and M. Lipson, Opt. Express 21, 26557 (2013).
[Crossref]

Y. Zhang, S. Yang, A. Lim, G. Lo, C. Galland, T. Jones, and M. Hochberg, Opt. Express 21, 1310 (2013).
[Crossref]

S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013).
[Crossref]

2012 (2)

2010 (1)

2009 (1)

2007 (1)

R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007).
[Crossref]

2006 (1)

2005 (1)

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

1995 (1)

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
[Crossref]

Baba, T.

Bergman, K.

Bogaerts, W.

S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013).
[Crossref]

Buhl, L.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Cappuzzo, M.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Cardenas, J.

Chan, J.

Chen, E.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Chen, L.

Chen, Z.

Chrostowski, L.

Dekker, R.

R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007).
[Crossref]

Dheer, H.

S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013).
[Crossref]

Doerr, C. R.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Domenech, J. D.

Driessen, A.

R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007).
[Crossref]

Dwivedi, S.

S. Dwivedi, H. Dheer, and W. Bogaerts, IEEE Photon. Technol. Lett. 25, 2167 (2013).
[Crossref]

Fandino, J. S.

Fathpour, S.

Forst, M.

R. Dekker, N. Usechak, M. Forst, and A. Driessen, J. Phys. D 40, R249 (2007).
[Crossref]

Foy, A. W.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Frandsen, L. H.

Fukuda, H.

T. Hiraki, H. Fukuda, K. Yamada, and T. Yamamoto, Front. Mater. 2, 1 (2015).
[Crossref]

Galland, C.

Gargallo, B.

Gomez, L.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Gondarenko, A.

Griffin, A.

C. R. Doerr, M. Cappuzzo, E. Chen, A. W. Foy, L. Gomez, A. Griffin, and L. Buhl, IEEE Photon. Technol. Lett. 17, 1211 (2005).
[Crossref]

Guan, X.

Guha, B.

He, S.

Hiraki, T.

T. Hiraki, H. Fukuda, K. Yamada, and T. Yamamoto, Front. Mater. 2, 1 (2015).
[Crossref]

Hochberg, M.

Jaeger, N. A. F.

Jalali, B.

Jones, T.

Lim, A.

Lin, S.

Lipson, M.

Liu, L.

Lo, G.

Lu, Z.

Mohammed, Z.

Z. Mohammed, B. Paredes, and M. Rasras, Appl. Sci. 11, 2366 (2021).
[Crossref]

Z. Mohammed, B. Paredes, and M. Rasras, in IEEE Photonics Conference (IPC) (2020), pp. 1–2.

Moooka, T.

Munoz, P.

Nguyen, H. C.

Padmaraju, K.

Paredes, B.

Z. Mohammed, B. Paredes, and M. Rasras, Appl. Sci. 11, 2366 (2021).
[Crossref]

Z. Mohammed, B. Paredes, and M. Rasras, in IEEE Photonics Conference (IPC) (2020), pp. 1–2.

Pennings, E. C. M.

L. B. Soldano and E. C. M. Pennings, J. Lightwave Technol. 13, 615 (1995).
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P. Xing and J. Viegas, Proc. SPIE 9752, 97520U (2016).
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Data Availability

Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)

Fig. 1.
Fig. 1. (a) Proposed MZI, (b) adiabatic coupler, (c) MMI, (d) broadband directional coupler (BDC), and (e) sub-wavelength grating (SWG) adiabatic coupler.
Fig. 2.
Fig. 2. (a) Calculated effective and group indices for different widths. (b) Temperature dependence of effective index at 1550 nm wavelength.
Fig. 3.
Fig. 3. Simulated ${2} \times {2}$ response of the proposed MZI with (a) a broadband 3 dB coupler with a slight deviation from 50/50 coupling and (b) a standard directional coupler.
Fig. 4.
Fig. 4. Response of the proposed MZI with simulated temperature variations using (a) a broadband 3 dB coupler (assuming athermal) and (b) a directional coupler.
Fig. 5.
Fig. 5. Measured transmission of uncompensated MZI with different coupler types using (a) MMI, (b) a BDC, (c) an adiabatic coupler, and (d) an SWG adiabatic coupler.
Fig. 6.
Fig. 6. SEM image of proposed athermal MZI with (a) MMI and (b) an SWG adiabatic coupler.
Fig. 7.
Fig. 7. Measured transmission of proposed athermal MZI with (a) MMI and (b) an SWG adiabatic coupler.

Tables (2)

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Table 1. Coupler Performance Metrics

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Table 2. Bandwidth Comparison

Equations (6)

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m λ 0 = n eff Δ L + Δ n eff L ,
M = m Δ L n eff λ L Δ n eff λ ,
Δ λ 0 Δ T = 1 M [ Δ L n eff T + L Δ n eff T ] .
F S R = λ 0 M .
η = 1 2 ± 1 2 1 10 E R s / 10 .
L c ( T ) = [ π k z ( T ) ( w 2 ) ( 1 + 2 ( α x ( T ) w ) ) ( α x 2 ( T ) + k x 2 ( T ) ) ] [ 2 α x ( T ) k x 2 ( T ) e α x ( T ) g ] ,

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