Abstract

A low-loss and low-crosstalk multimode waveguide bend is proposed and demonstrated for mode-division-multiplexed optical interconnects. The proposed 90°-bend is composed of two identical 45°-bends, which are defined as modified Euler curves. For the designed 90° Euler-bend with a core width of 2.36 μm for supporting four TM-polarization modes, it is allowed to achieve an effective radius as small as 45 μm, which is about 1/4 of the radius (~175 μm) for a regular 90° arc-bend. In theory, this proposed 90° Euler-bend has very low excess losses (<0.1 dB) and very low inter-mode crosstalks (<−25 dB) over a broad wavelength-band. A silicon photonic integrated circuit is designed, fabricated and characterized by integrating a pair of mode (de)multiplexers and a multimode bus waveguide with a Euler S-bend consisting of two cascaded 90° Euler-bends. The measurement results show that the fabricated Euler S-bend has low excess losses of <0.5 dB and low inter-mode crosstalks of <−20 dB over a broad band from 1520 nm to 1610 nm for all the 4 mode-channels of TM polarization.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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2018 (1)

H. Xu and Y. Shi, “Ultra sharp multimode waveguide bending assisted with metamaterial based mode converters,” Laser Photonics Rev. 12(3), 1700240 (2018).
[Crossref]

2017 (4)

2015 (1)

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

2014 (4)

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

D. Dai and J. E. Bowers, “Silicon-based on-chip multiplexing technologies and devices for Peta-bit optical interconnects,” Nanophotonics 3(4–5), 283–311 (2014).

D. Dai, “Multimode optical waveguide enabling microbends with low inter-mode crosstalk for mode-multiplexed optical interconnects,” Opt. Express 22(22), 27524–27534 (2014).
[Crossref] [PubMed]

B. A. Dorin and W. N. Ye, “Two-mode division multiplexing in a silicon-on-insulator ring resonator,” Opt. Express 22(4), 4547–4558 (2014).
[Crossref] [PubMed]

2013 (4)

D. Dai, J. Wang, and Y. Shi, “Silicon mode (de)multiplexer enabling high capacity photonic networks-on-chip with a single-wavelength-carrier light,” Opt. Lett. 38(9), 1422–1424 (2013).
[Crossref] [PubMed]

M. Cherchi, S. Ylinen, M. Harjanne, M. Kapulainen, and T. Aalto, “Dramatic size reduction of waveguide bends on a micron-scale silicon photonic platform,” Opt. Express 21(15), 17814–17823 (2013).
[Crossref] [PubMed]

D. X. Dai, J. Wang, and S. L. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagnetics Res. 143, 773–819 (2013).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

2012 (3)

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

T. Chen, H. Lee, J. Li, and K. J. Vahala, “A general design algorithm for low optical loss adiabatic connections in waveguides,” Opt. Express 20(20), 22819–22829 (2012).
[Crossref] [PubMed]

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

2011 (1)

W. Bogaerts and S. K. Selvaraja, “Compact single-mode silicon hybrid rib/strip waveguide with adiabatic bends,” IEEE Photonics J. 3(3), 422–432 (2011).
[Crossref]

2007 (2)

2005 (1)

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

2004 (1)

Aalto, T.

Baets, R.

Bai, N.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

Bernier, E.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Bogaerts, W.

W. Bogaerts and S. K. Selvaraja, “Compact single-mode silicon hybrid rib/strip waveguide with adiabatic bends,” IEEE Photonics J. 3(3), 422–432 (2011).
[Crossref]

W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Low-loss, low-cross-talk crossings for silicon-on-insulator nanophotonic waveguides,” Opt. Lett. 32(19), 2801–2803 (2007).
[Crossref] [PubMed]

Bowers, J. E.

D. Dai and J. E. Bowers, “Silicon-based on-chip multiplexing technologies and devices for Peta-bit optical interconnects,” Nanophotonics 3(4–5), 283–311 (2014).

Chen, G.

Chen, H. D.

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

Chen, S. W.

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

Chen, T.

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

T. Chen, H. Lee, J. Li, and K. J. Vahala, “A general design algorithm for low optical loss adiabatic connections in waveguides,” Opt. Express 20(20), 22819–22829 (2012).
[Crossref] [PubMed]

Chen, Y. Y.

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

Cherchi, M.

Dai, D.

Dai, D. X.

D. X. Dai, “Silicon nanophotonic integrated devices for on-chip multiplexing and switching,” J. Lightwave Technol. 35(4), 572–587 (2017).
[Crossref]

D. X. Dai, J. Wang, and S. L. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagnetics Res. 143, 773–819 (2013).
[Crossref]

Dorin, B. A.

Dumais, P.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Dumon, P.

Fini, J. M.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Fu, H.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Fujisawa, T.

Gabrielli, L. H.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

Geng, D.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Goodwill, D.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Harjanne, M.

He, S.

He, S. L.

D. X. Dai, J. Wang, and S. L. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagnetics Res. 143, 773–819 (2013).
[Crossref]

Huang, B. J.

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

Jiang, J.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Johnson, S. G.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

Kapulainen, M.

Lee, H.

T. Chen, H. Lee, J. Li, and K. J. Vahala, “A general design algorithm for low optical loss adiabatic connections in waveguides,” Opt. Express 20(20), 22819–22829 (2012).
[Crossref] [PubMed]

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

Li, G.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

Li, J.

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

T. Chen, H. Lee, J. Li, and K. J. Vahala, “A general design algorithm for low optical loss adiabatic connections in waveguides,” Opt. Express 20(20), 22819–22829 (2012).
[Crossref] [PubMed]

Li, M.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Lipson, M.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

Liu, D.

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

Makino, S.

Mao, L. H.

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

McNab, S.

Miyamoto, Y.

T. Mizuno and Y. Miyamoto, “High-capacity dense space division multiplexing transmission,” Opt. Fiber Technol. 35, 108–117 (2017).
[Crossref]

Mizuno, T.

T. Mizuno and Y. Miyamoto, “High-capacity dense space division multiplexing transmission,” Opt. Fiber Technol. 35, 108–117 (2017).
[Crossref]

Nelson, L. E.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Painter, O.

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

Richardson, D. J.

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Saitoh, K.

Sato, T.

Selvaraja, S. K.

W. Bogaerts and S. K. Selvaraja, “Compact single-mode silicon hybrid rib/strip waveguide with adiabatic bends,” IEEE Photonics J. 3(3), 422–432 (2011).
[Crossref]

Shi, Y.

Sun, C.

Tu, X.

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

Vahala, K. J.

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

T. Chen, H. Lee, J. Li, and K. J. Vahala, “A general design algorithm for low optical loss adiabatic connections in waveguides,” Opt. Express 20(20), 22819–22829 (2012).
[Crossref] [PubMed]

Van Thourhout, D.

Vlasov, Y.

Wang, J.

D. Dai, J. Wang, and Y. Shi, “Silicon mode (de)multiplexer enabling high capacity photonic networks-on-chip with a single-wavelength-carrier light,” Opt. Lett. 38(9), 1422–1424 (2013).
[Crossref] [PubMed]

D. X. Dai, J. Wang, and S. L. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagnetics Res. 143, 773–819 (2013).
[Crossref]

Xia, C.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

Xu, H.

H. Xu and Y. Shi, “Ultra sharp multimode waveguide bending assisted with metamaterial based mode converters,” Laser Photonics Rev. 12(3), 1700240 (2018).
[Crossref]

Yan, Q. F.

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

Ye, W. N.

Ylinen, S.

Yu, J. Z.

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

Yu, Y.

Zhang, X.

Zhang, Z.

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

Zhang, Z. Y.

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

Zhao, N.

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

Adv. Opt. Photonics (1)

G. Li, N. Bai, N. Zhao, and C. Xia, “Space-division multiplexing: the next frontier in optical communication,” Adv. Opt. Photonics 6(4), 413–487 (2014).
[Crossref]

Appl. Opt. (1)

IEEE Photonics J. (1)

W. Bogaerts and S. K. Selvaraja, “Compact single-mode silicon hybrid rib/strip waveguide with adiabatic bends,” IEEE Photonics J. 3(3), 422–432 (2011).
[Crossref]

J. Lightwave Technol. (1)

J. Semicond. (2)

Y. Y. Chen, J. Z. Yu, Q. F. Yan, and S. W. Chen, “Analysis on Influencing Factors of Bend Loss of Silicon-on-Insulator Waveguides,” J. Semicond. 26(13), 216 (2005).

H. D. Chen, Z. Zhang, B. J. Huang, L. H. Mao, and Z. Y. Zhang, “Progress in complementary metal-oxide-semiconductor silicon photonics and optoelectronic integrated circuits,” J. Semicond. 36(12), 121001 (2015).
[Crossref]

Laser Photonics Rev. (1)

H. Xu and Y. Shi, “Ultra sharp multimode waveguide bending assisted with metamaterial based mode converters,” Laser Photonics Rev. 12(3), 1700240 (2018).
[Crossref]

Nanophotonics (1)

D. Dai and J. E. Bowers, “Silicon-based on-chip multiplexing technologies and devices for Peta-bit optical interconnects,” Nanophotonics 3(4–5), 283–311 (2014).

Nat. Commun. (2)

L. H. Gabrielli, D. Liu, S. G. Johnson, and M. Lipson, “On-chip transformation optics for multimode waveguide bends,” Nat. Commun. 3(1), 1217 (2012).
[Crossref] [PubMed]

H. Lee, T. Chen, J. Li, O. Painter, and K. J. Vahala, “Ultra-low-loss optical delay line on a silicon chip,” Nat. Commun. 3(1), 867 (2012).
[Crossref] [PubMed]

Nat. Photonics (1)

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7(5), 354–362 (2013).
[Crossref]

Opt. Express (6)

Opt. Fiber Technol. (1)

T. Mizuno and Y. Miyamoto, “High-capacity dense space division multiplexing transmission,” Opt. Fiber Technol. 35, 108–117 (2017).
[Crossref]

Opt. Lett. (3)

Prog. Electromagnetics Res. (1)

D. X. Dai, J. Wang, and S. L. He, “Silicon multimode photonic integrated devices for on-chip mode-division-multiplexed optical interconnects,” Prog. Electromagnetics Res. 143, 773–819 (2013).
[Crossref]

Other (3)

D. J. Goodwill and J. Jiang, “Apparatus and method for a waveguide polarizer comprising a series of bends,” U.S. patent 9690045 B2 (June 27, 2017).

X. Tu, M. Li, J. Jiang, D. Goodwill, P. Dumais, E. Bernier, H. Fu, and D. Geng, “Compact low-loss adiabatic bends in silicon shallow etched waveguides,” in Proceedings of IEEE International Conference on Group IV Photonics (IEEE, 2016), pp. 48–49.
[Crossref]

X. Wu, W. Zhou, D. Huang, Z. Zhang, Y. Wang, J. Bowers, and H. K. Tsang, “Low Crosstalk Bent Multimode Waveguide for On-chip Mode-Division Multiplexing Interconnects,” in Conference on Lasers and Electro-Optics, OSA Terchnical Digest (online) (Optical Society of America, 2018), paper JW2A.66.
[Crossref]

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Figures (10)

Fig. 1
Fig. 1 Schematic configurations for 90°-bends. (a) The 90° Euler-bend composed of a pair of 45° modified Euler-curves with a maximum curvature radius Rmax at the starting point (0, 0) and a minimum curvature radius Rmin at the end point (xE, yE); (b) A regular 90° arc-bend with a constant radius R = Reff (here the modified Euler curve is also shown by the dashed curve to give a comparison).
Fig. 2
Fig. 2 Calculated mode excitation ratio ξij from the i-th mode launched at the input SWG to the j-th mode in a BWG with a constant curvature radius Rmax varies for the cases with i = 0 (a), 1 (b), 2 (c), and 3 (d), respectively.
Fig. 3
Fig. 3 Calculated transmissions in the whole structure consisting of an input SWG, a 90° modified Euler-bend, and an output SWG when the TM0 mode is launched at the input SWG. The 90° Euler bend is designed with Rmin = 15 μm (a), 20 μm (b), 25 μm (c), and 30 μm (d). Here Rmax = 600 μm.
Fig. 4
Fig. 4 Calculated transmissions Tij from the i-th mode launched at the input SWG to the j-th TM mode in the output SWG for the cases with i = 0 (a), 1 (b), 2 (c), and 3 (d), respectively. Here the modified 90° Euler-bend is with Rmax = 600 μm, Rmin = 25 μm and Reff = 45 μm.
Fig. 5
Fig. 5 Calculated transmissions Tij from the i-th mode launched at the input SWG to the j-th TM mode in the output SWG for the cases with i = 0 (a), 1 (b), 2 (c), and 3 (d), respectively. Here the 90° arc-bend is with R = 45 μm.
Fig. 6
Fig. 6 Simulated light propagation in the designed 90° Euler-bend with Reff = 45 μm (i.e., Rmax = 600 μm, and Rmin = 25 μm) (a-d) and the regular 90° arc-bend with R = 45 μm (e-h), when launching the TMi mode-channel from the input SWG. Here i = 0 (a, e), 1 (b, f), 2 (c, g), 3 (4, h).
Fig. 7
Fig. 7 Microscope mage of the fabricated silicon PIC consists of a 4-channel mode multiplexer (ITM1, ITM2, ITM3, and ITM4), a multimode bus waveguide with an S-bend, and 4-channel mode demultiplexer (OTM1, OTM2, OTM3, and OTM4); (b) Scanning electron microscopic (SEM) images of the S-bends with two 90° Euler-bends; (c) SEM image of a 90° arc-bends.
Fig. 8
Fig. 8 Measured spectral responses of the transmissions for the fabricated silicon PIC consists of a 4-channel mode multiplexer, a fabricated S-bend with the 90° Euler-bends, and 4-channel mode demultiplexer when launching the TM0 (a), TM1 (b), TM2 (c) and TM3 (d) modes, respectively.
Fig. 9
Fig. 9 Measured spectral responses of the transmissions for the silicon PIC consists of a 4-channel mode multiplexer, a straight multimode bus waveguide, and 4-channel mode demultiplexer when launching the TM0 (a), TM1 (b), TM2 (c) and TM3 (d) modes, respectively.
Fig. 10
Fig. 10 Measured spectral responses of the transmissions for the PIC consists of a 4-channel mode multiplexer, a fabricated S-bend with the 90° arc-bends, and 4-channel mode demultiplexer when launching TM0 (a), TM1 (b), TM2 (c) and TM3 (d) modes, respectively.

Equations (4)

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dθ dL = 1 R = L A 2 + 1 R max ,
A= [ L 0 /( 1/ R min 1/ R max ) ] 1/2 ,
x=A 0 L/A sin( θ 2 2 + Aθ R max )dθ ,
y=A 0 L/A cos( θ 2 2 + Aθ R max )dθ .

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