Abstract

A compact tooth-shaped grating-assisted structure has been proposed for a multimode interference (MMI) coupler and studied theoretically with silica waveguides with silicon oxynitride cores by using a mathematical model based on the sinusoidal mode simple effective index method. The dependence of the access waveguide length and beat length of the grating-assisted MMI (GA-MMI) structure on access waveguide separation (h) is shown. The total length of the GA-MMI coupler is less than that of a grating-assisted two-mode interference (GA-TMI) coupler (h=0) for an access waveguide S bending loss of 0.2dB. It is also seen that the effect of fabrication tolerance on power imbalance of a GA-MMI coupler is almost close to that of a GA-TMI coupler.

© 2011 Optical Society of America

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References

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  1. A. Neyer, “Integrated optical multichannel wavelength multiplexer for monomode systems,” Electron. Lett. 20, 744–746 (1984).
    [CrossRef]
  2. A. K. Das and P. P. Sahu, “Compact integrated optical devices using high index contrast waveguides,” in Proceedings of the IEEE Wireless and Optical Communication Network Conference (IEEE, 2006), pp. 1–5.
  3. P. P. Sahu, “Compact multimode interference coupler with tapered waveguide geometry,” Opt. Commun. 277, 295–301(2007).
    [CrossRef]
  4. P. P. Sahu, “A tapered structure for compact multimode interference coupler,” IEEE Photon. Technol. Lett. 20, 638–640 (2008).
    [CrossRef]
  5. 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 (1995).
    [CrossRef]
  6. M. K. Chin, C. W. Lee, and S. Darrnawan, “High index contrast waveguides and devices,” Appl. Opt. 44, 3077–3086(2005).
    [CrossRef] [PubMed]
  7. P. P. Sahu, “A compact optical multiplexer using silicon nano waveguides,” IEEE J. Sel. Top. Quantum Electron. 15, 1537–1541 (2009).
    [CrossRef]
  8. T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
    [CrossRef]
  9. T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
    [CrossRef]
  10. K. S. Chiang, “Effective index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
    [CrossRef]
  11. K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
    [CrossRef]
  12. B. Deka and P. P. Sahu, “Transformation relationship of directional coupler with multimode interference coupler and two mode interference coupler,” J. Opt. 38, 75–87 (2009).
    [CrossRef]
  13. F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
    [CrossRef]
  14. M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
    [CrossRef]
  15. P. P. Sahu, “Silicon oxinitride: a material for compact waveguide device,” Indian J. Phys. 82, 265–272 (2008).
  16. H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989).
  17. P. P. Sahu, “A double S-bend geometry with lateral offset for compact two mode interference coupler,” J. Lightwave Technol. 29, 2064–2068 (2011).
    [CrossRef]
  18. R. Kashahara, M. Yanagisawa, T. Goh, A. Sugita, A. Himeno, M. Yasu, and S. Matsui, “New structures of silica-based planar light wave circuits for low power thermooptic switch and its application to 8×8 optical matrix switch,” J. Lightwave Technol. 20, 993–1000 (2002).
    [CrossRef]

2011 (1)

2009 (2)

B. Deka and P. P. Sahu, “Transformation relationship of directional coupler with multimode interference coupler and two mode interference coupler,” J. Opt. 38, 75–87 (2009).
[CrossRef]

P. P. Sahu, “A compact optical multiplexer using silicon nano waveguides,” IEEE J. Sel. Top. Quantum Electron. 15, 1537–1541 (2009).
[CrossRef]

2008 (2)

P. P. Sahu, “A tapered structure for compact multimode interference coupler,” IEEE Photon. Technol. Lett. 20, 638–640 (2008).
[CrossRef]

P. P. Sahu, “Silicon oxinitride: a material for compact waveguide device,” Indian J. Phys. 82, 265–272 (2008).

2007 (1)

P. P. Sahu, “Compact multimode interference coupler with tapered waveguide geometry,” Opt. Commun. 277, 295–301(2007).
[CrossRef]

2005 (2)

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

M. K. Chin, C. W. Lee, and S. Darrnawan, “High index contrast waveguides and devices,” Appl. Opt. 44, 3077–3086(2005).
[CrossRef] [PubMed]

2004 (1)

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

2002 (1)

2000 (1)

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

1996 (1)

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

1995 (2)

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

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 (1995).
[CrossRef]

1991 (1)

K. S. Chiang, “Effective index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

1984 (1)

A. Neyer, “Integrated optical multichannel wavelength multiplexer for monomode systems,” Electron. Lett. 20, 744–746 (1984).
[CrossRef]

Broughton, J. N.

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

Cha, M. H.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

Chen, C. C.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Chen, F. S.

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Chen, J. R.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Chiang, K. S.

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

K. S. Chiang, “Effective index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

Chin, M. K.

Darrnawan, S.

Das, A. K.

A. K. Das and P. P. Sahu, “Compact integrated optical devices using high index contrast waveguides,” in Proceedings of the IEEE Wireless and Optical Communication Network Conference (IEEE, 2006), pp. 1–5.

Deka, B.

B. Deka and P. P. Sahu, “Transformation relationship of directional coupler with multimode interference coupler and two mode interference coupler,” J. Opt. 38, 75–87 (2009).
[CrossRef]

Fang, Y. C.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

Gau, C. S.

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Goh, T.

Haruna, M.

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989).

Himeno, A.

Ho, S. T.

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

Janz, F.

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

Kashahara, R.

Keyworth, B. P.

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

Lee, C. W.

Lee, Z. C.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Matsui, S.

Neyer, A.

A. Neyer, “Integrated optical multichannel wavelength multiplexer for monomode systems,” Electron. Lett. 20, 744–746 (1984).
[CrossRef]

Nishihara, H.

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989).

Park, S.

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

Pennings, E. C. M.

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 (1995).
[CrossRef]

Piam, M. R.

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

Sahu, P. P.

P. P. Sahu, “A double S-bend geometry with lateral offset for compact two mode interference coupler,” J. Lightwave Technol. 29, 2064–2068 (2011).
[CrossRef]

B. Deka and P. P. Sahu, “Transformation relationship of directional coupler with multimode interference coupler and two mode interference coupler,” J. Opt. 38, 75–87 (2009).
[CrossRef]

P. P. Sahu, “A compact optical multiplexer using silicon nano waveguides,” IEEE J. Sel. Top. Quantum Electron. 15, 1537–1541 (2009).
[CrossRef]

P. P. Sahu, “A tapered structure for compact multimode interference coupler,” IEEE Photon. Technol. Lett. 20, 638–640 (2008).
[CrossRef]

P. P. Sahu, “Silicon oxinitride: a material for compact waveguide device,” Indian J. Phys. 82, 265–272 (2008).

P. P. Sahu, “Compact multimode interference coupler with tapered waveguide geometry,” Opt. Commun. 277, 295–301(2007).
[CrossRef]

A. K. Das and P. P. Sahu, “Compact integrated optical devices using high index contrast waveguides,” in Proceedings of the IEEE Wireless and Optical Communication Network Conference (IEEE, 2006), pp. 1–5.

Soldano, L. B.

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 (1995).
[CrossRef]

Sugita, A.

Suhara, T.

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989).

Tsai, T. Y.

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

Wang, L.

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

Yanagisawa, M.

Yasu, M.

Yong, M.

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

Appl. Opt. (1)

Electron. Lett. (1)

A. Neyer, “Integrated optical multichannel wavelength multiplexer for monomode systems,” Electron. Lett. 20, 744–746 (1984).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Y. Tsai, Z. C. Lee, J. R. Chen, C. C. Chen, Y. C. Fang, and M. H. Cha, “A novel ultra compact two-mode-interference wavelength division multiplexer for 1.5 μm operation,” IEEE J. Quantum Electron. 41, 741–746 (2005).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

P. P. Sahu, “A compact optical multiplexer using silicon nano waveguides,” IEEE J. Sel. Top. Quantum Electron. 15, 1537–1541 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

T. Y. Tsai, Z. C. Lee, C. S. Gau, F. S. Chen, J. R. Chen, and C. C. Chen, “A novel wavelength-division multiplexer using grating assisted two-mode interference,” IEEE Photon. Technol. Lett. 16, 2251–2253 (2004).
[CrossRef]

P. P. Sahu, “A tapered structure for compact multimode interference coupler,” IEEE Photon. Technol. Lett. 20, 638–640 (2008).
[CrossRef]

F. Janz, M. R. Piam, B. P. Keyworth, and J. N. Broughton, “Bent waveguide couplers for demultiplexing of arbitrary broadly separated wavelengths using two mode interference,” IEEE Photon. Technol. Lett. 7, 1037–1039 (1995).
[CrossRef]

M. Yong, S. Park, L. Wang, and S. T. Ho, “Ultra compact multimode interference 3 dB coupler with strong lateral confinement by deep dry etching,” IEEE Photon. Technol. Lett. 12, 492–494 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

K. S. Chiang, “Analysis of the effective-index method for the vector modes of rectangular-core dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 692–700 (1996).
[CrossRef]

Indian J. Phys. (1)

P. P. Sahu, “Silicon oxinitride: a material for compact waveguide device,” Indian J. Phys. 82, 265–272 (2008).

J. Lightwave Technol. (4)

P. P. Sahu, “A double S-bend geometry with lateral offset for compact two mode interference coupler,” J. Lightwave Technol. 29, 2064–2068 (2011).
[CrossRef]

R. Kashahara, M. Yanagisawa, T. Goh, A. Sugita, A. Himeno, M. Yasu, and S. Matsui, “New structures of silica-based planar light wave circuits for low power thermooptic switch and its application to 8×8 optical matrix switch,” J. Lightwave Technol. 20, 993–1000 (2002).
[CrossRef]

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 (1995).
[CrossRef]

K. S. Chiang, “Effective index method for the analysis of optical waveguide couplers and arrays: an asymptotic theory,” J. Lightwave Technol. 9, 62–72 (1991).
[CrossRef]

J. Opt. (1)

B. Deka and P. P. Sahu, “Transformation relationship of directional coupler with multimode interference coupler and two mode interference coupler,” J. Opt. 38, 75–87 (2009).
[CrossRef]

Opt. Commun. (1)

P. P. Sahu, “Compact multimode interference coupler with tapered waveguide geometry,” Opt. Commun. 277, 295–301(2007).
[CrossRef]

Other (2)

H. Nishihara, M. Haruna, and T. Suhara, Optical Integrated Circuits (McGraw-Hill, 1989).

A. K. Das and P. P. Sahu, “Compact integrated optical devices using high index contrast waveguides,” in Proceedings of the IEEE Wireless and Optical Communication Network Conference (IEEE, 2006), pp. 1–5.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of a 2 × 2 tooth-shaped GA-MMI coupler. (a) Three-dimensional top view. (b) 2D cross-sectional view with the x and z axes.

Fig. 2
Fig. 2

Normalized coupling power distribution of tooth-shaped GA-MMI couplers for h = 0.0 , 3.0, 4.0, 5.0, and 6.0 μm with Δ W = 0.05 μm , a = 1.5 μm , b = 1.5 μm , cladding index 1.45 , Δ n = 5 % , and λ 1.55 μm .

Fig. 3
Fig. 3

Beam propagation results at a cross-coupling point obtained by using OptiBPM software for (a) a tooth-shaped GA-TMI coupler, (b) a conventional TMI coupler, (c) a tooth-shaped GA-MMI coupler, and (d) a conventional MMI coupler with a = 1.5 μm , b = 1.5 μm , cladding index 1.45 , Δ n = 5 % , and λ 1.55 μm .

Fig. 4
Fig. 4

Beat length ( L π ) versus index contrast ( Δ n ) of tooth-shaped GA-MMI couplers with Δ W = 0.05 , 0.1, and 0.25 μm , and a conventional MMI coupler ( Δ W = 0 μm ).

Fig. 5
Fig. 5

Transition length ( L T ) and beat length ( L π ) versus waveguide separation gap (h) variation of tooth-shaped a GA-MMI coupler (solid line) and a conventional MMI coupler (dotted line) with a = 1.5 μm , b = 1.5 μm , index contrast 5 % , and cladding index 1.45 .

Fig. 6
Fig. 6

Bending loss ( T S ) and beat length ( L π ) versus waveguide separation gap (h) variation for tooth-shaped a GA-MMI coupler with a = 1.5 μm , b = 1.5 μm , index contrast 5 % , and cladding index 1.45 .

Fig. 7
Fig. 7

Power imbalance characteristics versus width tolerances ( δ w ) for a tooth-shaped GA-MMI coupler, a tooth-shaped GA-TMI coupler, and a conventional MMI coupler with index contrast 5 % , cladding index 1.45 , h 3.0 μm , a = 1.5 μm , b = 1.5 μm , and λ 1.55 μm .

Fig. 8
Fig. 8

Power imbalance characteristics versus wavelength variation for a tooth-shaped GA-MMI coupler (solid line), a tooth-shaped GA-TMI coupler (dashed line), and a conventional MMI coupler (dotted line) with a = 1.5 μm , b = 1.5 μm , h 3.0 μm , index contrast 5 % , and cladding index 1.45 .

Equations (16)

Equations on this page are rendered with MathJax. Learn more.

L π = [ ( N + 1 ) l m + N l g ] = π [ ( β 00 m β 01 m ) + ( β 00 g β 01 g ) ] ,
H ( x , 0 ) = i = 0 r 1 b i H i ( x ) ,
H ( x , z ) = i = 0 r 1 H i ( x , z ) = i = 0 r 1 b i H i ( x ) exp [ j ( β 0 β i ) z ] ,
H i ( x ) = sin [ ( i + 1 ) π x W g ] .
H 3 ( x , L ) = i = 0 r 1 H 3 , i ( x , L ) = i = 0 r 1 c 3 , i H i ( x ) exp [ j ( β 0 β i ) L ] ,
C 3 , i C 0 π 2 16 b 2 k 2 ( n 1 2 n 2 2 ) × exp { h k ( n eff 2 n 2 2 ) 1 / 2 } × [ exp { b k ( n 1 2 n 2 2 ) 1 / 2 } exp { b k ( n 1 2 n 2 2 ) 1 / 2 } ] ,
C 0 = 0.4 F C × ( n 1 2 n eff 2 ) n eff 2 n 2 2 n eff ( n 1 2 n 3 2 ) [ W m + 2 k 0 n eff 2 n 2 2 ] ,
F c = 3 ( 1 + 0.2 h ) { 13.5 + 185 ( β m 00 β m i ) } h ,
n eff = β i K ( λ 2 π ) , K = m , g .
P 3 i = | H 3 , i ( x , L ) | 2 .
P 3 , i ( x , L ) P 1 , i ( x , o ) = | i = 0 r 1 H 3 , i ( x , L ) | 2 | i = 0 r 1 H 1 , i ( x , 0 ) | 2 i = 0 r 1 C 3 , i H i 2 ( x ) + i = 0 r 1 j = 1 + i r 1 [ 2 C 3 , i C 3 , j H i ( x ) H j ( x ) × cos { i = 0 , j = i + 1 r 1 K = m , g [ ( N + q K ) ( β i K β j K ) l K ] } ] ,
l K = λ 4 n eff ( j , K ) , K = m , g .
P 4 , i ( x , L ) P 1 , i ( x , o ) = | i = 0 r 1 H 4 , i ( x , L ) | 2 | i = 0 r 1 H 1 , i ( x , 0 ) | 2 i = 0 r 1 C 4 , i H i 2 ( x ) + i = 0 r 1 j = 1 + i r 1 [ 2 C 4 , i C 4 , j H i ( x ) H j ( x ) × cos { i = 0 , j = i + 1 r 1 K = m , g [ ( N + q K ) ( β i K β j K ) l K ] } ] ,
L T = ( H h 2 ) [ 4 R + h 2 H ] ,
T S = 4.343 α S ,
S = 2 R cos 1 [ 1 ( H h / 2 ) 2 R ] .

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