Table of Contents Author Guidelines Submit a Manuscript
International Journal of Optics
Volume 2019, Article ID 4270612, 10 pages
https://doi.org/10.1155/2019/4270612
Research Article

The Critical Adiabatic Linear Tapered Waveguide Combined with a Multimode Waveguide Coupler on an SOI Chip

Department of Electronic Engineering, National Kaohsiung University of Science and Technology, No. 415 Jiangong Road, Kaohsiung 807, Taiwan

Correspondence should be addressed to C. L. Chiu; wt.ude.tsukn@uihclc

Received 10 August 2019; Revised 12 October 2019; Accepted 17 October 2019; Published 11 November 2019

Guest Editor: Cheng-Mu Tsai

Copyright © 2019 C. L. Chiu and Yen-Hsun Liao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A multimode waveguide interference (MMI) coupler is combined with a critical linear tapered waveguide on a silicon-on-insulator (SOI) chip. When the TE0 mode is a critical adiabatic mode conversion from a single-mode waveguide to an extreme linear tapered waveguide combined with an MMI, this linear tapered waveguide is achieved to the maximum divergence angle (i.e., the shortest length). The maximum divergence angle is expressed by θ ≤ 2 tan−1[(0.35Wmmi − Ws)/(0.172Lmmi)] under a 1 × 1 MMI combined with this critical linear tapered waveguide. The expression formula is demonstrated by three different widths of a 1 × 1 MMI of 4 μm/8 μm/12 μm combined with the critical linear tapered waveguide. So, the maximum divergence angle is obtained at θ = 16°/14°/8°, with respect to this linear tapered waveguide loss of 0.022 dB/0.172 dB/0.158 dB, and this linear tapper length is reduced by 93.7%/92.9%/87.5% than the divergence angle θ = 1°. The output power of a 1 × 1 MMI combined with a critical linear tapered waveguide is enhanced at least 1.5 times under 0.95 above condition.

1. Introduction

In the last few years, there have been numerous advances in silicon photonics. Photonic devices on a silicon-on-insulator (SOI) chip with high-index contrast have high integration density. The main advantage of the optoelectronic component on an SOI structure is its good compatibilities [1]. Couplers and power dividers in photonic integrated circuits (PICs) are often implemented with multimode interference couplers (MMIs) for easy fabrication and broad bandwidth. The SOI platform is an area of interest in integrated optics at present and enables a size reduction of PICs. Therefore, their CMOS compatibility can provide optoelectronic integration on a chip in future applications [2]. MMIs are based on the expansion of a fundamental mode of the access waveguide into multiple modes of the wider width of a multimode waveguide, which interfere as they propagate and form images of the excitation. Ridge waveguides are widely used in SOI, as they offer a single-mode behaviour at micrometre scale [3]. MMIs depend on multimode waveguides, utilizing bends for higher order mode filtering. They generally propagate well, and the weak lateral confinement of narrow ridge waveguides makes it difficult to achieve high-performance devices [4].

In 2010, Thomson et al. proposed a method to achieve a reduction of optical loss through the use of linear tapers with input and output ports. The taper loss is reduced to below 1 dB without affecting static extinction [5]. In 2012, Sheng et al. proposed the compact and low-loss MMI coupler fabricated with CMOS technology. This tapered waveguide with a divergence angle θ of 1° combined with an MMI is fabricated on SOI with 0.13 μm CMOS technology to obtain an excess loss of only 0.06 dB [6]. Researchers have adopted the widest and longest linear tapered waveguide to be combined with an MMI coupler on an SOI chip in recent devices. This critical problem will increase manufacturing costs, so it is necessary to design an adiabatic tapered waveguide.

The losses inherent to a mode propagating waveguide must be reduced on the cross-sectional boundary between the single-mode waveguide and multimode waveguide. Because a tapered waveguide can change the spot size and the shape of the optical mode to achieve high coupling efficiency in the cross section boundary [7], a tapered waveguide is necessary to achieve an adiabatic state [710]. That is, as the TE0 mode from a single-mode waveguide is transmitted to the tapered waveguide, the other higher order TE modes are reduced to excited modes [1117].

In this article, we propose an expression formula to design a terminal linear tapered waveguide to enhance the coupling efficiency output power of an MMI coupler to two times above. The low-loss and maximum divergence angle linear tapered waveguide combined with a 1 × 1 MMI on an SOI chip is achieved to an output power of above 0.95. The TE0 mode component ratio is necessary to be above 97.85% in order to achieve a critical adiabatic mode conversion.

2. Device Structure

The cross section of an SOI structure is shown in Figure 1. The thickness of the upper cladding SiO2 layer is 2 μm, and the Si layer is deposited at a height hco of 220 nm on a 2 -μm-thick buried oxide layer based on a Si substrate. The refractive indices of Si and SiO2 are nSi = 3.475 and nSiO2 = 1.444, respectively. The ridge waveguide has a depth of 2.22 μm, and the effective core refractive index nr of 2.509 and cladding refractive index nc of 2.372 at an operating wavelength λ0 of 1550 nm in a slab waveguide [18].

Figure 1: Schematic diagram showing the cross section of a ridge waveguide on an SOI structure.

MMI couplers have higher tolerance to dimensional changes in the fabrication process, an easier fabrication process than other couplers, lower inherent loss, large optical bandwidth, and low polarization dependence [19]. Multimode waveguides excite numerous modes depending on their width and depth. The width of a fixed step index multimode waveguide Wmmi is generally referred to as N ×  M MMI coupler, where N and M indicate input and output ports. For high-index contrast waveguides, the penetration depth is very small so that We ≈ Wmmi. However, the effective width We can correspond to the fundamental mode [18]:where λ0 is an operating wavelength and nr and nc are the effective core and cladding refractive indices, respectively. The term σ = 0 represents transverse electric (TE) mode and σ = 1 is for transverse magnetic (TM) mode. Lπ is defined as the beat length of the two lowest order modes [19], as follows:where β0 and β1 are individual zero-order and first-order propagation constant. The term nr is the effective core refractive index of the slab waveguide from which a 1 × 1 MMI coupler is made. We is the effective width of the MMI waveguide, and Lmmi is the exact imaging length [20]:

The geometric shape of a basic 1 × 1 MMI coupler is shown in Figure 2(a). A single-mode ridge waveguide with width Ws of 0.4 μm, length Ls of 100 μm, and a depth of 2.22 μm is calculated by the effective core refractive index nr of 2.509 and the effective cladding refractive index nc of 2.372 [17]. An MMI width adapted to 10/20/30 times the single waveguide of 0.4 um as an inspecting standard case. The widths of a 1 × 1 MMI Wmmi are choice of 4 μm/8 μm/12 μm respect to the beat lengths Lπ of 45.7 μm/159.8 μm/342.9 μm from equation (2) at an operating wavelength of λ0 = 1550 nm. Therefore, the exact image length of a 1 × 1 MMI Lmmi achieves 34.3 μm/119.8 μm/257.1 μm, respectively, by equation (3). A linear tapered waveguide combined with a 1 × 1 MMI is shown in Figure 2(b). The input/output port of this 1 × 1 MMI is a single-mode waveguide linked with a linear tapered waveguide. Wt is the width and Lt the length of a linear tapered waveguide. The divergence angle θ of a linear tapered waveguide is a taper angle. A half angle of the divergence angle is defined by the following equation:

Figure 2: (a) A basic 1 × 1 MMI combined with the input/output single-mode waveguide for width Ws = 0.4 μm and length Ls = 100 μm. (b) A linear tapered waveguide is inserted between the single-mode waveguide and MMI coupler. Width Wt, length Lt, and divergence angle θ describe the linear tapered waveguide.

The simulation analysis utilizes the film mode matching method (FMM) solver in FIMMWAVE software [21,22]. The output power of a basic 1 × 1 MMI with the exact length Lmmi is shown in Figure 3. Figure 3(a) is the length Lmmi of a 1 × 1 MMI scanning the range from 26.7 μm to 30.7 μm with a step of 0.2 μm at MMI width Wmmi = 4 μm. The maximum output power is 0.62 at Lmmi = 28.7 μm. Figure 3(b) is the length Lmmi of a 1 × 1 MMI scanning the range from 111.0 μm to 115.0 μm with a step of 0.2 μm at MMI width Wmmi = 8 μm. Here, the maximum output power is 0.51 at Lmmi = 113.0 μm. Figure 3(c) is the same method at MMI width Wmmi = 12 μm for Lmmi scanning the range from 255.2 μm to 259.2 μm. Maximum output power is 0.41 at Lmmi = 257.2 μm. Accordingly, Lmmi at MMI width Wmmi = 4 μm/8 μm/12 μm is 28.7 μm/113.0 μm/257.2 μm, respectively. The device loss of a 1 × 1 MMI is 2.08 dB/2.92 dB/3.87 dB, respectively, which is very significant.

Figure 3: (a) The maximum output power is 0.62 at MMI length Lmmi = 28.7 μm with a 1 × 1 MMI width Wmmi = 4 μm. (b) Maximum output power is 0.51 at Lmmi = 113.0 μm and Wmmi = 8 μm. (c) Maximum output power is 0.41 at Lmmi = 257.2 μm and Wmmi = 12 μm.

3. Linear Tapered Waveguide Analysis

When the divergence angle of a linear tapered waveguide is set at θ = 1° with a width Wt of 4.2 μm and a length Lt of 217.7 μm as an experimental standard result from reference 6, this pair of tapered waveguide loss of almost 0.004 (0.018 dB) can be ignored. This linear tapered waveguide with a divergence angle of 1° is combined with the three different widths of 1 × 1 MMI Wmmi of 4 μm/8 μm/12 μm with respect to the exact imaging lengths Lmmi of 28.7 μm/113.0 μm/257.2 μm. When the ratio of Wt/Wmmi is increased from 0.1 to 1 at a step of 0.05, the range of the output power increases from 0.68 to 1, as shown in Figure 4. The output power of a 1 × 1 MMI combined with the linear tapered waveguide is necessary to be above 0.95 as the ratio of Wt/Wmmi is set to above 0.35.

Figure 4: This linear tapered waveguide with a divergence angle of 1° is combined with the three different widths of a 1 × 1 MMI coupler Wmmi of 4 μm/8 μm/12 μm with respect to the exact imaging lengths Lmmi of 28.7 μm/113.0 μm/257.2 μm. When Wt/Wmmi is set at above 0.35, the output power of a 1 × 1 MMI coupler combined with a linear tapered waveguide achieves above 0.95.

Figure 5 shows the effective refractive index neff for eight TE eigenmodes, including TE0, TE1, TE2, TE3, TE4, TE5, TE6, and TE7 distributed with the width of a linear tapered waveguide from 0.4 μm to 4.5 μm. As the effective refractive index neff of a slab waveguide on an SOI chip is 2.509, the eight TE eigenmodes, including TE0 to TE7, correspond to the widths of the linear tapered waveguide Wt. Three different widths of 1 × 1 MMI Wmmi of 4 μm/8 μm/12 μm obtain a minimum width for a linear tapered waveguide Wt of 1.4 μm/2.8 μm/4.2 μm, respectively. As the TE0 mode is transmitted from a single-mode waveguide with a width of 0.4 μm into a linear tapered waveguide with a width Wt of 1.4 μm, TE0 and TE1 are excited. The taper width Wt of 2.8 μm is excited for TE0, TE1, TE2, TE3, and TE4 modes. The taper width Wt of 4.2 μm is excited for TE0, TE1, TE2, TE3, TE4, TE5, and TE6 modes. As the geometric shape of the device is symmetrical structure, the odd modes are suppressed and inexistent.

Figure 5: The effective refractive index of the distributed state of eight TE modes with the width of a linear tapered waveguide is ranging from 0.4 μm to 4.5 μm. The effective refractive index neff of the slab waveguide on an SOI chip is 2.509 and the thickness of this linear tapered waveguide hco is 220 nm at an operating wavelength of 1550 nm.

The single-mode waveguide with width Ws of 0.4 μm is combined with the width of the linear tapered waveguide Wt of 1.4 μm/2.8 μm/4.2 μm, respectively. TE mode component ratio is distributed with the divergence angle of a linear tapered waveguide ranging from 1° to 45°, as shown in Figure 6. When the even modes of TE2, TE4, and TE6 and the odd modes of TE1, TE3, TE5, and TE7 except TE0 are suppressed in the linear tapered waveguide, the maximum divergence angle θ of the linear tapered waveguide is 16°/14°/8° with respect to a 1 × 1 MMI with width Wmmi of 4 μm/8 μm/12 μm. The TE0 mode component ratio obtains individual 99.13%/98.51%/97.87%. So, this linear tapered waveguide achieves the TE0 mode adiabatic mode conversion when the TE0 mode component ratio is at least 97.87% and TE2 mode and the other modes component ratio is below 2.13%.

Figure 6: The width of a linear tapered waveguide (a) Wt = 1.4 μm, Wmmi = 4 μm; (b) Wt = 2.8 μm, Wmmi = 8 μm; (c) Wt = 4.2 μm, Wmmi = 12 μm; with divergence angle θ of a linear tapered waveguide ranging from 1° to 45°. The maximum divergence angle for linear tapered waveguide is achieved at θ = 16°/14°/8° for Wt = 1.4 μm/2.8 μm/4.2 μm, respectively.

For a standard adiabatic mode conversion analysis, a 1 × 1 MMI in width of 12 μm and in length of 257.2 μm is combined with the divergence angle θ = 1° of linear tapered waveguide in width Wt of 4.2 μm and in length Lt of 217.7 μm. When the location of input/output linear tapered waveguide is z/Lt of 0/0.25/0.5/0.75/1, the fundamental mode TE0 shape of input/output port is simulated to change the mode shape size from smaller to larger mode shape under TE0 adiabatic mode conversion as shown in Figure 7. The coupling efficiency of this device between the single-mode waveguide and the multimode waveguide is enhanced from 0.41 to 0.95.

Figure 7: A 1 × 1 MMI in width of 12 μm and in length of 257.2 μm is combined with the linear tapered waveguide in width Wt of 4.2 μm and in length Lt of 217.7 μm. When the location of input/output linear tapered waveguide is z/Lt of 0/0.25/0.5/0.75/1, the fundamental mode TE0 shape of input/output port is simulated to change the mode shape size from small to larger mode shape under TE0 adiabatic mode conversion.

The 1 × 1 MMI with width Wmmi of 4 μm/8 μm/12 μm is combined with the width of the linear tapered waveguide Wt of 1.4 μm/2.8 μm/4.2 μm, respectively, as the input and output port with the divergence angle θ of a linear tapered waveguide scanning the range from 1° to 45° at a step of 1°. The maximum divergence angle θ is achieved at 16°/14°/8°, respectively, under the condition of output power of a 1 × 1 MMI combined with a linear tapered waveguide of at least 0.95, as shown in Figure 8. Figure 9 shows that the spectral responses are insensitivity for the wavelength from 1546 to 1554 nm with the step of 1 nm under the condition of this linear tapered waveguide with a maximum divergence angle θ of 16°/14°/8° combined with the three different widths of a 1 × 1 MMI of 4 μm/8 μm/12 μm, respectively. The output power of three different widths of a 1 × 1 MMI linked with the maximum divergence angle θ of 16°/14°/8° of a linear tapered waveguide is 0.95 when the ratio of Wt/Wmmi is equal to 0.35. Three different divergence angles θ of the linear tapered waveguide of 16°/14°/8° with respect to three different widths of a 1 × 1 MMI width Wmmi of 4 μm/8 μm/12 μm are taken into equation (4). The length of a linear tapered waveguide Lt is calculated by 3.6 μm/9.8 μm/27.2 μm, respectively. The ratio of the length of a linear tapered waveguide to the length of a 1 × 1 MMI is expressed as Lt/Lmmi ≧ 0.086.

Figure 8: When Wt/Wmmi is set at 0.35, the 1 × 1 MMI coupler with Wmmi = 4 μm/8 μm/12 μm achieves a minimum width of linear tapered waveguide at Wt = 1.4 μm/2.8 μm/4.2 μm, respectively. As the divergence angle θ of a linear tapered waveguide is scanning the range from 1° to 45° at a step of 1°, a maximum divergence angle θ = 16°/14°/8° is obtained under the constraint of  ≥ 0.95, respectively.
Figure 9: The spectral responses are insensitivity for the wavelength from 1546 to 1554 nm with the step of 1 nm under the condition of this linear tapered waveguide with a maximum divergence angle θ of 16°/14°/8° combined with the three different widths of a 1 × 1 MMI of 4 μm/8 μm/12 μm, respectively.

The expressions of equations (5) and (6) are demonstrated under three different widths of a 1 × 1 MMI coupler combined with a designed linear tapered waveguide:where Wt is the width of the linear tapered waveguide, Lt is the length of the linear tapered waveguide, and Wmmi is the width of a 1 × 1 MMI and Lmmi of the exact imaging length of a 1 × 1 MMI. When the width of the single-mode waveguide Ws of 0.4 μm and equations (5) and (6) are taken into equation (4), the maximum divergence angle, θ, is expressed as equation (7).

Comparison of basic 1 × 1 MMI device loss with a 1 × 1 MMI combined with a linear tapered waveguide device loss is shown in Tables 1 and 2. When 1 × 1 MMI with width Wmmi of 4 μm/8 μm/12 μm is combined with a maximum divergence angle θ = 16°/14°/8° in a linear tapered waveguide with a width Wt of 1.4 μm/2.8 μm/4.2 μm and length Lt of 3.6 μm/9.8 μm/27.2 μm, the loss of this linear tapered waveguide is 0.022 dB/0.172 dB/0.158 dB. The length of a maximum divergence angle θ = 16°/14°/8° in this linear tapered waveguide is reduced to 93.7%/92.9%/87.5% than the length of the divergence angle θ = 1° combined a 1 × 1 MMI with width of 4 μm/8 μm/12 μm. The output power of a 1 × 1 MMI combined with the maximum divergence angle of a linear tapered waveguide is 0.95 (0.22 dB). A 1 × 1 MMI device loss with a linear tapered waveguide reduces 1.86 dB/2.70 dB/3.65 dB than a 1 × 1 MMI device loss without a linear tapered waveguide. This device loss represents a significant reduction.

Table 1: A basic 1 × 1 MMI device loss.
Table 2: A 1 × 1 MMI combined with a linear tapered waveguide device loss.

4. Conclusion

A 1 × 1 MMI is combined with a symmetrical linear tapered waveguide on an SOI chip. When TE0 mode from a single-mode waveguide is transmitted to this critical linear tapered waveguide linked with a 1 × 1 MMI, the TE0 mode component ratio is necessary to be at least 97.87% and the TE2 mode and the other modes’ component ratios are to be below 2.13%. So, the TE0 mode presents a critical adiabatic mode conversion. The designed linear tapered waveguide is achieved to the shortest length and the maximum divergence angle.

Under the condition of a 1 × 1 MMI coupler combined with the designed linear tapered waveguide, the maximum divergence angle is demonstrated by θ ≤ 2 tan−1 [(0.35Wmmi − Ws)/(0.172 Lmmi)]. When the width of a 1 × 1 MMI Wmmi is 4 μm/8 μm/12 μm with respect to the length Lmmi of 28.7 μm/113.0 μm/257.2 μm, the maximum divergence angle θ is achieved to 16°/14°/8°, respectively.

A 1 × 1 MMI width Wmmi of 4 μm/8 μm/12 μm combined with a maximum divergence angle θ = 16°/14°/8° linear tapered waveguide to a 1 × 1 MMI without linear tapered waveguide. The simulation result shows that the device loss is reduced by 1.86 dB/2.70 dB/3.65 dB, respectively, with respect to an extreme linear tapered waveguide loss of 0.022 dB/0.172 dB/0.158 dB. The length of a maximum divergence angle θ = 16°/14°/8° linear tapered waveguide is reduced to 93.7%/92.9%/87.5% than the length of the divergence angle θ = 1° linear tapered waveguide combined with a 1 × 1 MMI with width of 4 μm/8 μm/12 μm. The output power of a 1 × 1 MMI combined with a critical linear tapered waveguide is at least 0.95, which enhanced the coupling efficiency by 1.5 times.

Data Availability

The data used to support the findings of this study are included within the article files.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported in part by the Ministry of Science and Technology, Taiwan, Republic of China, under the grant no. MOST 108-2221-E-992-080.

References

  1. R. Soref, “The past, present, and future of silicon photonics,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 12, no. 6, pp. 1678–1687, 2006. View at Publisher · View at Google Scholar · View at Scopus
  2. M. Lipson, “Guiding, modulating, and emitting light on silicon-challenges and opportunities,” Journal of Lightwave Technology, vol. 23, no. 12, pp. 4222–4238, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. K. Kruse and C. T. Middlebrook, “Polymer taper bridge for silicon waveguide to single mode waveguide coupling,” Optics Communications, vol. 362, pp. 87–95, 2016. View at Publisher · View at Google Scholar · View at Scopus
  4. J. Guo and Y. Zhao, “Analysis of mode hybridization in tapered waveguides,” IEEE Photonics Technology Letters, vol. 27, no. 23, pp. 2441–2444, 2015. View at Publisher · View at Google Scholar · View at Scopus
  5. D. J. Thomson, Y. Hu, G. T. Reed, and J.-M. Fedeli, “Low loss MMI couplers for high performance MZI modulators,” IEEE Photonics Technology Letters, vol. 22, no. 20, pp. 1485–1487, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. Z. Sheng, Z. Zhiqi Wang, C. Chao Qiu et al., “A compact and low-loss MMI coupler fabricated with CMOS technology,” IEEE Photonics Journal, vol. 4, no. 6, pp. 2272–2277, 2012. View at Publisher · View at Google Scholar · View at Scopus
  7. P. Sethi, A. Haldar, and S. K. Selvaraja, “Ultra-compact low-loss broadband waveguide taper in silicon-on-insulator,” Optics Express, vol. 25, no. 9, pp. 10196–10203, 2017. View at Publisher · View at Google Scholar
  8. Y. Liu, W. Sun, H. Xie et al., “Adiabatic and ultra-compact waveguide tapers based on digital metamaterials,” IEEE Journal of Selected Topics in Quantum Electronics, vol. 25, no. 3, pp. 1–6, 2018. View at Publisher · View at Google Scholar · View at Scopus
  9. J. Zhang, J. Yang, H. Xin, J. Huang, D. Chen, and Z. Zhaojian, “Ultrashort and efficient adiabatic waveguide taper based on thin flat focusing lenses,” Optics Express, vol. 25, no. 17, pp. 19894–19903, 2017. View at Publisher · View at Google Scholar
  10. Y. Fu, T. Ye, W. Tang, and T. Chu, “Efficient adiabatic silicon-on-insulator waveguide taper,” Photonics Research, vol. 2, no. 3, pp. A41–A44, 2014. View at Publisher · View at Google Scholar · View at Scopus
  11. J. Wang, M. Qi, Y. Xuan et al., “Proposal for fabrication-tolerant SOI polarization splitter-rotator based on cascaded MMI couplers and an assisted bi-level taper,” Optics Express, vol. 22, no. 23, pp. 27869–27879, 2014. View at Publisher · View at Google Scholar · View at Scopus
  12. Y. Zhang, S. Yang, A. E.-J. Lim et al., “A CMOS-compatible, low-loss, and low-crosstalk silicon waveguide crossing,” IEEE Photonics Technology Letters, vol. 25, no. 5, pp. 422–425, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. D. Dai, Y. Tang, and J. E. Bowers, “Mode conversion in tapered submicron silicon ridge optical waveguides,” Optics Express, vol. 20, no. 12, pp. 13425–13439, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. L. He, Y. He, A. Pomerene et al., “Ultrathin silicon-on-insulator grating couplers,” IEEE Photonics Technology Letters, vol. 24, no. 24, pp. 2247–2249, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. C.-H. Chen and C.-H. Chiu, “Taper-integrated multimode-interference based waveguide crossing design,” IEEE Journal of Quantum Electronics, vol. 46, no. 11, pp. 1656–1661, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. W. Bogaerts, P. Dumon, D. V. Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Optics Letters, vol. 32, no. 19, pp. 2801–2803, 2007. View at Publisher · View at Google Scholar · View at Scopus
  17. J. J. Wu, B. R. Shi, and M. Kong, “Exponentially tapered multimode interference couplers,” Chinese Optics Letters, vol. 4, no. 3, pp. 167–169, 2006. View at Google Scholar
  18. P. K. Bhattacharya, Semiconductor Optoelectronic Devices, Prentice-Hall, Englewood Cliffs, NJ, USA, 1998.
  19. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” Journal of Lightwave Technology, vol. 13, no. 4, pp. 615–627, 1995. View at Publisher · View at Google Scholar · View at Scopus
  20. M. Bachmann, P. A. Besse, and H. Melchior, “Overlapping-image multimode interference couplers with a reduced number of self-images for uniform and nonuniform power splitting,” Applied Optics, vol. 34, no. 30, pp. 6898–6910, 1995. View at Publisher · View at Google Scholar · View at Scopus
  21. A. S. Sudbo, “Film mode matching: a versatile numerical method for vector mode field calculations in dielectric waveguides,” Pure and Applied Optics: Journal of the European Optical Society Part A, vol. 2, no. 3, pp. 211–233, 1993. View at Publisher · View at Google Scholar · View at Scopus
  22. Integrated Optics Software FIMMWAVE 5.2, Photon Design, Oxford, U.K, 2015, http://www.photond.com.