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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 26 — Dec. 10, 2012
  • pp: B371–B376
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An MMI-based polarization splitter using patterned metal and tilted joint

Keisuke Kojima, Wangqing Yuan, Bingnan Wang, Toshiaki Koike-Akino, Kieran Parsons, Satoshi Nishikawa, and Eiji Yagyu  »View Author Affiliations


Optics Express, Vol. 20, Issue 26, pp. B371-B376 (2012)
http://dx.doi.org/10.1364/OE.20.00B371


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Abstract

A novel polarization splitter on an InP substrate utilizing an MMI coupler loaded with a dielectric and gold layer pad is proposed and simulated. A tilted joint is used for adjusting the phases of TE and TM modes. The MMI section is less than 540 μm. Simulations show that the device has a polarization extinction ratio over 23 dB and an insertion loss below 0.7 dB over the entire C-band for both TE and TM polarizations. The device design was optimized to maximize the wavelength range and tolerance for manufacturing variations.

© 2012 OSA

1. Introduction

Controlling polarization state of light in photonic integrated circuits (PICs) is of great importance for high-speed optical communication networks. First, the single mode fiber used in communication networks does not preserve the polarization state and many components in PICs are polarization-sensitive. This polarization dependence along with the polarization mode dispersion degrades the performances of PICs at high modulation frequency. To solve the problem, polarization diversity systems were proposed [1

1. T. Barwicz, M. R. Watts, M. A. Popovic, P. T. Rakich, L. Socci, F. X. Kartner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007). [CrossRef]

3

3. W. Bogaerts, D. Taillaert, P. Dumon, D. Van Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express 15(4), 1567–1578 (2007). [CrossRef] [PubMed]

]. Second, the polarization state can be utilized in polarization-division multiplexing (PDM) systems to double the spectral efficiency [4

4. R. Nagarajan, J. Rahn, M. Kato, J. Pleumeekers, D. Lambert, V. Lal, H. S. Tsai, A. Nilsson, A. Dentai, M. Kuntz, R. Malendevich, J. Tang, J. Zhang, T. Butrie, M. Raburn, B. Little, W. Chen, G. Goldfarb, V. Dominic, B. Taylor, M. Reffle, F. Kish, and D. Welch, “10 Channel, 45.6 Gb/s per channel, polarization-multiplexed DQPSK, InP receiver photonic integrated circuit,” J. Lightwave Technol. 29(4), 386–395 (2011). [CrossRef]

]. For the design of either the polarization diversity systems or the polarization multiplexing systems, a polarization splitter is the key component. A polarization splitter built on an Indium Phosphide (InP) substrate is especially of interest due to its capability of integration with active components including lasers, modulators, and photo-detectors [5

5. L. M. Augustin, R. Hanfoug, J. J. G. M. van der Tol, W. J. M. de Laat, and M. K. Smit, “A compact integrated polarization splitter/converter in InGaAsP-InP,” IEEE Photon. Technol. Lett. 19(17), 1286–1288 (2007). [CrossRef]

,6

6. W. Yuan, K. Kojima, B. Wang, T. Koike-Akino, K. Parsons, S. Nishikawa, and E. Yagyu, “Mode-evolution-based polarization rotator-splitter design via simple fabrication process,” Opt. Express 20(9), 10163–10169 (2012). [CrossRef] [PubMed]

].

A Multimode Interference (MMI) coupler is an excellent building block for polarization splitters due to its compactness, large bandwidth, and high fabrication tolerance [7

7. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1005).

]. A polarization splitter using two MMI 3-dB couplers as a Mach-Zehnder interferometer was proposed and demonstrated [8

8. L. B. Soldano, A. I. de Vreede, M. K. Smit, B. H. Verbeek, E. G. Metaal, and F. H. Green, “Mach-Zehnder interferometer polarization splitter in InGaAsP/InP,” IEEE Photon. Technol. Lett. 6(3), 402–405 (1994). [CrossRef]

]. In this case, one of the arms is loaded with a dielectric layer and a metal layer on top, and the total device length was 3.3mm. An alternative method is to use an MMI itself as a polarization splitter, but it is very difficult to design an MMI-based polarization splitter on InP for two reasons: MMI couplers are inherently polarization-insensitive regardless of material systems, and strong polarization birefringence is difficult to achieve in InP material systems due to small index contrast between core and cladding layers compared to Silicon-On-Insulator (SOI) systems. As a result of the weak polarization birefringence, a long device length, e.g. a few millimeters long [9

9. B. M. A. Rahman, N. Somasiri, C. Themistos, and K. T. V. Grattan, “Design of optical polarization splitters in a single-section deeply etched MMI waveguide,” Appl. Phys. B 73(5), 613–618 (2001). [CrossRef]

], is required for the MMI-based polarization splitters to separate TE and TM modes. Different methods such as the quasi-state MMI coupler [10

10. J. M. Hong, H. H. Ryu, S. R. Park, J. W. Jeong, S. G. Lee, E. H. Lee, S. G. Park, D. Woo, S. Kim, and O. Beom-Hoan, “Design and fabrication of a significantly shortened multimode interference coupler for polarization splitter application,” IEEE Photon. Technol. Lett. 15(1), 72–74 (2003). [CrossRef]

] and the slot waveguide [11

11. A. Katigbak, J. F. Strother Jr, and J. Lin, “Compact silicon slot waveguide polarization splitter,” Opt. Eng. 48(8), 080503 (2009). [CrossRef]

] have been proposed to reduce the total device length. However, none of these methods can reduce the device length without compromising the device performance or inducing extra fabrication difficulties.

In this article, we propose a novel design of a polarization splitter utilizing a 1-by-2 MMI coupler. A short phase shift section loaded with dielectric and patterned metal layers is incorporated into the MMI-based polarization splitter. This creates a strong polarization birefringence between TE and TM modes, resulting in very strong separation of TE and TM modes at the output waveguides. We also introduce a tilt in MMI, to adjust the phases both for TE and TM such that they are exactly at ± π/2 with opposite signs. The device was carefully optimized to satisfy good performance over the entire C-band (1528-1567nm) and also tolerate reasonable variations in the fabrication process.

2. Principle and design

The functional block diagram of the device is shown in Fig. 1
Fig. 1 Functional block diagram of a polarization splitter based on MMI couplers. The arrows indicate the phases of the electric field.
. The input signal (TE or TM), coupled into the 1 × 2 MMI, is split into two arms with equal phase and equal power. The phase shift section is designed to add an extra –π/2 and π/2 phase shift to the TE and TM modes, respectively, in the lower arm. When the electric fields from both arms are combined via the 2 × 2 MMI coupler, the electric field in one output coming from the cross arm has an extra –π/2 phase shift compared with that from the bar arm. The interference between electric fields with different phases causes the TE polarization mode to appear at the lower output whereas the TM polarization mode appears at the upper output.

To create the birefringence needed for the phase shift, a waveguide loaded with a dielectric layer and metal layer is considered [12

12. S. C. Rashleigh, “Four-layer metal-clad thin film optical waveguides,” Opt. Quantum Electron. 8(1), 49–60 (1976). [CrossRef]

]. We conducted a 2D simulation on the effective refractive index neff of the structure consisting of 0.2 μm-thick gold layer, a SiNx dielectric layer with varying thickness, a 0.6 μm-thick InGaAsP (λ = 1.3 μm) layer as waveguide core, and an InP substrate. Figure 2
Fig. 2 2D simulation of Δneff , i.e, effective index difference with and without the gold layer, as a function of the SiNx layer thickness
shows Δneff (the difference between the neff with and without the 0.2 μm-thick gold layer) as a function of the SiNx dielectric layer thickness, for TE and TM modes. The gold layer in the phase shift section tends to push the electric field of TE mode away from it, which reduces the propagation constant for TE mode, creating negative Δneff. On the other hand, the gold layer pulls the electric field of TM mode toward it, which increases the propagation constant for TM mode. As shown in Fig. 2, by reducing the SiNx thickness, a stronger birefringence can be created. At the same time, a small change in the SiNx thickness can cause a large change in the phase shift, so there is a trade-off. We chose 0.08 μm to balance the length requirement and sensitivity to the thickness variation, as will be discussed later. The propagation losses for TE and TM mode are 9.7 cm−1 and 11.8 cm−1, respectively, when the SiNx thickness was 0.08 μm. In other words, a SiNx dielectric layer plays an important role of balancing between birefringence and absorption loss. Without it, TM mode is strongly localized at the interface as a surface plasmon polariton and absorption coefficient becomes very high.

The phase shift section described in Fig. 1 is achieved by using a tilted MMI coupler partially covered by a gold layer on a SiNx layer as shown in Fig. 3(a)
Fig. 3 (a) The schematic top view of the proposed MMI-based polarization splitter and (b) the cross section view of the phase shift section (S2 and S3) with metal-dielectric cladding layer.
. In this design, the phase shift section (S2 and S3) is integrated into the 1 × 2 MMI and 2 × 2 MMI coupler, which helps to reduce the total device length. The partial gold layer in the phase shift section creates an extra negative phase shift to the TE mode and an extra positive phase shift to the TM mode in the lower arm. The length of the phase shift section is adjusted to give the TE and TM phase shift difference to be π. Furthermore, the tilt angle in the MMI joint [13

13. Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, “Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers,” Electron. Lett. 32(17), 1576–1577 (1996). [CrossRef]

], which gives the same amount of phase shift both for TE and TM modes, is adjusted, so that the phase shift for TE and TM modes become –π/2 and π/2, respectively.

The geometrical parameters of the design are described as follows: the input waveguide has a width of Winput = 4.5 µm. The MMI section is composed of four sections, S1, S2, S3, and S4. The S1 and S4 sections do not contain the deposited gold layer, whereas the lower parts of the S2 and S3 sections are covered by deposited gold. The S2 and S3 sections are joined by an angled tilt of 0.45 degree. The MMI section has a width of WMMI = 10 µm and a total length of L = 538 µm. The metal (gold) layer has a width of Wm = 4.5 µm and a length of Lm = 72 µm in total. The lengths of the S1, S2, S3, and S4 sections are 1/5L-1/2Lm, 1/2Lm, 1/2Lm, and 4/5L-1/2Lm, respectively. The upper output has a width of WTM = 4.5 µm and is placed 3 µm from the center of the MMI coupler. The first section of the lower output has a width of Wcon = 4.9 µm and a length of Lcon = 47 µm; the second section of the lower output has a width of WTE = 4.45 µm. The proposed device has an InP substrate, a 0.6 μm-thick InGaAsP (λ = 1.3 μm) layer as waveguide core, and 0.08μm-thick Silicon Nitride (SiNx) layer as buffer layer. The gold layer is 0.2 μm thick. The cross section view of the phase shift section is shown in Fig. 3(b). A 10 μm-long polarizer, where a gold layer is directly deposited on InGaAsP is also included in the lower output waveguide, to absorb residual TM mode, while TE mode is passed with minimal loss. This gives about 7dB PER improvement for a TM Input. The total device length of less than 600 μm is much shorter than 1050 μm with InP MMI-based polarization splitter reported in [10

10. J. M. Hong, H. H. Ryu, S. R. Park, J. W. Jeong, S. G. Lee, E. H. Lee, S. G. Park, D. Woo, S. Kim, and O. Beom-Hoan, “Design and fabrication of a significantly shortened multimode interference coupler for polarization splitter application,” IEEE Photon. Technol. Lett. 15(1), 72–74 (2003). [CrossRef]

].

3. Simulated performance and fabrication tolerance

The performance of the polarization splitter is simulated using commercial software FIMMWAVE employing the eigenmode expansion method [14

14. D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons,” Proc. SPIE 4987, 69–82 (2003). [CrossRef]

]. The finite element mode solver is used for solving modes in all sections. Figure 4
Fig. 4 The electric field intensity in the MMI-based polarization splitter showing the wave propagation for (a) TM mode and (b) TE mode inputs.
shows the interference patterns of the MMI-based polarization splitter excited by (a) fundamental TE mode (TE0) and (b) fundamental TM mode (TM0) inputs. In this figure, the tilted MMI coupler looks like a straight waveguide, but it is because the tilt angle is small to be visible. As predicted, the single self-images for TE and TM modes build constructively near the upper and lower outputs, respectively. However, the self-image plane for TM polarization, ZTM, does not coincide with the self-image plane for TE polarization, ZTE. When the upper and lower outputs are placed at the TM self-image plane, an accurate TE self-image cannot be obtained at the lower output, which reduces the transmission. To solve this problem, a two-section TE output waveguide is designed. As shown in Fig. 4(b), the first section of the lower output waveguide functions as a mode converter, refocusing the divergent electric field of the TE mode back into a self-image, which was realized by adjusting the phase difference of the different MMI waveguide propagation modes.

Figure 5
Fig. 5 Simulation result on (a) PER and (b) Insertion loss for TE and TM modes as a function of wavelength.
shows the simulation result of polarization extinction ratio (PER) and insertion loss for TE and TM modes as functions of wavelength from 1528 to 1567 nm, covering the entire C band. For both TE and TM polarizations, the polarization splitter of this design exhibits an insertion loss smaller than 0.7 dB and PER larger than 23 dB over the wavelength range of 39 nm. This shows that one of the advantages of MMI, i.e., relative insensitivity to wavelength, is still intact.

The fabrication tolerance is also studied in this work. In the actual device fabrication, the most sensitive parameter is the SiNx buffer layer thickness, which controls the phase shift, as shown in Fig. 2. Figure 6
Fig. 6 Simulation result on (a) PER and (b) Insertion loss for TE and TM modes as a function of SiNx layer thickness error.
shows the PER and insertion loss of TE and TM modes at the wavelength of 1528 nm and 1567nm, as functions of thickness error of SiNx buffer layer. For both TE and TM polarizations, the polarization splitter of this design exhibits an insertion loss smaller than 1 dB and PER larger than 20 dB with the thickness variation of ± 6.2%, over the entire C-band. One of the method of achieving this requirement is ALE (Atomic Layer Deposition), which is capable of depositing an SiNx layer with monolayer accuracy (<0.2nm) [15

15. H. Goto, K. Shibahara, and S. Yokoyama, “Atomic layer controlled deposition of silicon nitride with self-limiting mechanism,” Appl. Phys. Lett. 68(23), 3257–3259 (1996). [CrossRef]

].

Improving the tolerance to manufacturing variations is important, and it is included in the device optimization process. All the possible combinations of tilt angle, MMI length (LMMI), gold pattern length (Lm) and width (Wm) are used for the PER and insertion loss calculations, to generate the types of curves shown in Fig. 6(a) and 6(b). Then the best combination is chosen, such that it maximizes the SiNx thickness range under which the PER is larger than 20dB and the insertion loss is less than 1dB both for TE and TM input modes at 1528 nm and 1567nm.

4. Conclusion

A novel MMI-based polarization splitter on an InP substrate is proposed. A patterned metal on a dielectric cladding layer creates birefringence, and a tilt joint of the MMI adjusts the phase such that phase shift for TE and TM mode are exactly -π/2 and π/2, resulting in separation of TE and TM modes within a short propagation distance. The total device length is less than 600 μm. Simulation shows that the device has a PER greater than 23 dB and an insertion loss below 0.7 dB over the wavelength range from 1528 to 1567 nm for both TE and TM polarizations. The fabrication tolerance is also studied, showing that for a thickness variation of ± 6% the insertion loss remains below 1 dB and the PER remains over 20 dB over the same wavelength range. The total device length is less than 600 μm. Although this device is proposed for InGaAsP/InP material systems, the device concept could also be readily applied to other material systems, such as SOI.

References and links

1.

T. Barwicz, M. R. Watts, M. A. Popovic, P. T. Rakich, L. Socci, F. X. Kartner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics 1(1), 57–60 (2007). [CrossRef]

2.

H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Silicon photonic circuit with polarization diversity,” Opt. Express 16(7), 4872–4880 (2008). [CrossRef] [PubMed]

3.

W. Bogaerts, D. Taillaert, P. Dumon, D. Van Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express 15(4), 1567–1578 (2007). [CrossRef] [PubMed]

4.

R. Nagarajan, J. Rahn, M. Kato, J. Pleumeekers, D. Lambert, V. Lal, H. S. Tsai, A. Nilsson, A. Dentai, M. Kuntz, R. Malendevich, J. Tang, J. Zhang, T. Butrie, M. Raburn, B. Little, W. Chen, G. Goldfarb, V. Dominic, B. Taylor, M. Reffle, F. Kish, and D. Welch, “10 Channel, 45.6 Gb/s per channel, polarization-multiplexed DQPSK, InP receiver photonic integrated circuit,” J. Lightwave Technol. 29(4), 386–395 (2011). [CrossRef]

5.

L. M. Augustin, R. Hanfoug, J. J. G. M. van der Tol, W. J. M. de Laat, and M. K. Smit, “A compact integrated polarization splitter/converter in InGaAsP-InP,” IEEE Photon. Technol. Lett. 19(17), 1286–1288 (2007). [CrossRef]

6.

W. Yuan, K. Kojima, B. Wang, T. Koike-Akino, K. Parsons, S. Nishikawa, and E. Yagyu, “Mode-evolution-based polarization rotator-splitter design via simple fabrication process,” Opt. Express 20(9), 10163–10169 (2012). [CrossRef] [PubMed]

7.

L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1005).

8.

L. B. Soldano, A. I. de Vreede, M. K. Smit, B. H. Verbeek, E. G. Metaal, and F. H. Green, “Mach-Zehnder interferometer polarization splitter in InGaAsP/InP,” IEEE Photon. Technol. Lett. 6(3), 402–405 (1994). [CrossRef]

9.

B. M. A. Rahman, N. Somasiri, C. Themistos, and K. T. V. Grattan, “Design of optical polarization splitters in a single-section deeply etched MMI waveguide,” Appl. Phys. B 73(5), 613–618 (2001). [CrossRef]

10.

J. M. Hong, H. H. Ryu, S. R. Park, J. W. Jeong, S. G. Lee, E. H. Lee, S. G. Park, D. Woo, S. Kim, and O. Beom-Hoan, “Design and fabrication of a significantly shortened multimode interference coupler for polarization splitter application,” IEEE Photon. Technol. Lett. 15(1), 72–74 (2003). [CrossRef]

11.

A. Katigbak, J. F. Strother Jr, and J. Lin, “Compact silicon slot waveguide polarization splitter,” Opt. Eng. 48(8), 080503 (2009). [CrossRef]

12.

S. C. Rashleigh, “Four-layer metal-clad thin film optical waveguides,” Opt. Quantum Electron. 8(1), 49–60 (1976). [CrossRef]

13.

Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, “Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers,” Electron. Lett. 32(17), 1576–1577 (1996). [CrossRef]

14.

D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons,” Proc. SPIE 4987, 69–82 (2003). [CrossRef]

15.

H. Goto, K. Shibahara, and S. Yokoyama, “Atomic layer controlled deposition of silicon nitride with self-limiting mechanism,” Appl. Phys. Lett. 68(23), 3257–3259 (1996). [CrossRef]

OCIS Codes
(060.2340) Fiber optics and optical communications : Fiber optics components
(130.0130) Integrated optics : Integrated optics
(130.5440) Integrated optics : Polarization-selective devices

ToC Category:
Waveguide and Optoelectronic Devices

History
Original Manuscript: October 2, 2012
Revised Manuscript: November 10, 2012
Manuscript Accepted: November 11, 2012
Published: November 29, 2012

Virtual Issues
European Conference on Optical Communication 2012 (2012) Optics Express

Citation
Keisuke Kojima, Wangqing Yuan, Bingnan Wang, Toshiaki Koike-Akino, Kieran Parsons, Satoshi Nishikawa, and Eiji Yagyu, "An MMI-based polarization splitter using patterned metal and tilted joint," Opt. Express 20, B371-B376 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-26-B371


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References

  1. T. Barwicz, M. R. Watts, M. A. Popovic, P. T. Rakich, L. Socci, F. X. Kartner, E. P. Ippen, and H. I. Smith, “Polarization-transparent microphotonic devices in the strong confinement limit,” Nat. Photonics1(1), 57–60 (2007). [CrossRef]
  2. H. Fukuda, K. Yamada, T. Tsuchizawa, T. Watanabe, H. Shinojima, and S. Itabashi, “Silicon photonic circuit with polarization diversity,” Opt. Express16(7), 4872–4880 (2008). [CrossRef] [PubMed]
  3. W. Bogaerts, D. Taillaert, P. Dumon, D. Van Thourhout, R. Baets, and E. Pluk, “A polarization-diversity wavelength duplexer circuit in silicon-on-insulator photonic wires,” Opt. Express15(4), 1567–1578 (2007). [CrossRef] [PubMed]
  4. R. Nagarajan, J. Rahn, M. Kato, J. Pleumeekers, D. Lambert, V. Lal, H. S. Tsai, A. Nilsson, A. Dentai, M. Kuntz, R. Malendevich, J. Tang, J. Zhang, T. Butrie, M. Raburn, B. Little, W. Chen, G. Goldfarb, V. Dominic, B. Taylor, M. Reffle, F. Kish, and D. Welch, “10 Channel, 45.6 Gb/s per channel, polarization-multiplexed DQPSK, InP receiver photonic integrated circuit,” J. Lightwave Technol.29(4), 386–395 (2011). [CrossRef]
  5. L. M. Augustin, R. Hanfoug, J. J. G. M. van der Tol, W. J. M. de Laat, and M. K. Smit, “A compact integrated polarization splitter/converter in InGaAsP-InP,” IEEE Photon. Technol. Lett.19(17), 1286–1288 (2007). [CrossRef]
  6. W. Yuan, K. Kojima, B. Wang, T. Koike-Akino, K. Parsons, S. Nishikawa, and E. Yagyu, “Mode-evolution-based polarization rotator-splitter design via simple fabrication process,” Opt. Express20(9), 10163–10169 (2012). [CrossRef] [PubMed]
  7. L. B. Soldano and E. C. M. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. 13, 615–627 (1005).
  8. L. B. Soldano, A. I. de Vreede, M. K. Smit, B. H. Verbeek, E. G. Metaal, and F. H. Green, “Mach-Zehnder interferometer polarization splitter in InGaAsP/InP,” IEEE Photon. Technol. Lett.6(3), 402–405 (1994). [CrossRef]
  9. B. M. A. Rahman, N. Somasiri, C. Themistos, and K. T. V. Grattan, “Design of optical polarization splitters in a single-section deeply etched MMI waveguide,” Appl. Phys. B73(5), 613–618 (2001). [CrossRef]
  10. J. M. Hong, H. H. Ryu, S. R. Park, J. W. Jeong, S. G. Lee, E. H. Lee, S. G. Park, D. Woo, S. Kim, and O. Beom-Hoan, “Design and fabrication of a significantly shortened multimode interference coupler for polarization splitter application,” IEEE Photon. Technol. Lett.15(1), 72–74 (2003). [CrossRef]
  11. A. Katigbak, J. F. Strother, and J. Lin, “Compact silicon slot waveguide polarization splitter,” Opt. Eng.48(8), 080503 (2009). [CrossRef]
  12. S. C. Rashleigh, “Four-layer metal-clad thin film optical waveguides,” Opt. Quantum Electron.8(1), 49–60 (1976). [CrossRef]
  13. Q. Lai, M. Bachmann, W. Hunziker, P. A. Besse, and H. Melchior, “Arbitrary ratio power splitters using angled silica on silicon multimode interference couplers,” Electron. Lett.32(17), 1576–1577 (1996). [CrossRef]
  14. D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics: pros and cons,” Proc. SPIE4987, 69–82 (2003). [CrossRef]
  15. H. Goto, K. Shibahara, and S. Yokoyama, “Atomic layer controlled deposition of silicon nitride with self-limiting mechanism,” Appl. Phys. Lett.68(23), 3257–3259 (1996). [CrossRef]

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