Abstract

An asymmetrically loaded photonic crystal based polarization rotator has been introduced, designed and simulated. The polarization rotator structure consists of a single defect line photonic crystal slab waveguide with asymmetrically etched upper layer. To continue the rotation from a given input polarization to the desired output polarization the upper layer is alternated on either side of the defect line, periodically. Coupled mode theory based on semi-vectorial modes and plane wave expansion methods are employed to design the polarization rotator structure around a particular frequency band of interest. The 3D-FDTD simulation results agree with the coupled mode analysis around the region of interest specified during the design. Complete polarization rotation is achieved over the propagation length of 12λ. For this length, the coupling efficiency higher than 90% is achieved within the normalized frequency band of 0.258–0.262.

© 2009 Optical Society of America

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References

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  1. E. Yablonovitch, "Photonic band-gap structures," J. Opt. Soc. Am. B 10, 283-295 (1993).
    [CrossRef]
  2. W. Bogaerts, R. R. Baets, P. Dumon, V. Wiaux, S. Beckx, D. Taillaert, B. Luyssaert, J. V. Campnhout, P. Bienstman, and D. V. Thourhout, "Nanophotonic waveguides in silicon-on-insulator fabricated with CMOS technology," J. Lightwave Technol. 23, 401-421 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
    [CrossRef]
  6. B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, "Design and Characterization of Compact Single-Section Passive Polarization Rotator," J. Lightwave Technol. 19, 512-519 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  10. S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
    [CrossRef]
  11. M. Palamaru and Ph. Lalanne, "Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-1468 (2001).
    [CrossRef]
  12. M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
    [CrossRef]
  13. S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
    [CrossRef]
  14. S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
    [CrossRef]
  15. W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
    [CrossRef]
  16. D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004).
    [CrossRef]
  17. K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, "Polarization and thickness dependent guiding in the photonic crystal slab waveguide," Opt. Express 15, 8391-8400 (2007).
    [CrossRef] [PubMed]

2007 (1)

2005 (2)

2004 (3)

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. V. Campenhout, P. Bienstman, D. V. Thourhout, R. Baets, V. Wiaux, and S. Beckx, "Basic structures for photonic integrated circuits in Silicon-on-insulator," Opt. Express 12, 1583-1591 (2004).
[CrossRef] [PubMed]

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004).
[CrossRef]

2002 (1)

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

2001 (2)

2000 (1)

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
[CrossRef]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

1993 (1)

1992 (2)

W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

W. Huang and Z. M. Mao, "Polarization rotation in periodic loaded rib waveguides," IEEE J. Lightwave Technol. 10, 1825-1831 (1992).
[CrossRef]

1991 (1)

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

1982 (1)

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
[CrossRef]

1981 (1)

R. C. Alferness, "Electrooptic guided-wave devices for general polarization transformations," IEEE J. Quantum Electron. 17, 965-969 (1981).
[CrossRef]

Alferness, R.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Alferness, R. C.

R. C. Alferness, "Electrooptic guided-wave devices for general polarization transformations," IEEE J. Quantum Electron. 17, 965-969 (1981).
[CrossRef]

Baets, R.

Baets, R. R.

Bayat, K.

Beckx, S.

Bienstman, P.

Bogaerts, W.

Campenhout, J. V.

Campnhout, J. V.

Chaudhuri, S. K.

K. Bayat, S. K. Chaudhuri, and S. Safavi-Naeini, "Polarization and thickness dependent guiding in the photonic crystal slab waveguide," Opt. Express 15, 8391-8400 (2007).
[CrossRef] [PubMed]

W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

Chu, S. T.

W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

Dumon, P.

El-Mikathi, H. A.

El-Mikati, H. A.

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
[CrossRef]

Fan, S.

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Grattan, K. T. V.

Haung, W. P.

W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

Haus, H. A.

Hickmann, J. M.

D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004).
[CrossRef]

Huang, W.

W. Huang and Z. M. Mao, "Polarization rotation in periodic loaded rib waveguides," IEEE J. Lightwave Technol. 10, 1825-1831 (1992).
[CrossRef]

Ibanescu, M.

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Joannopoulos, J. D.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Johnson, S. G.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Koch, T.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Koren, U.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Lalanne, Ph.

M. Palamaru and Ph. Lalanne, "Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-1468 (2001).
[CrossRef]

Lidorikis, E.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Logan, R. A.

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
[CrossRef]

Luyssaert, B.

Mao, Z. M.

W. Huang and Z. M. Mao, "Polarization rotation in periodic loaded rib waveguides," IEEE J. Lightwave Technol. 10, 1825-1831 (1992).
[CrossRef]

Miller, B. I.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Obayya, S. S. A.

B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, "Design and Characterization of Compact Single-Section Passive Polarization Rotator," J. Lightwave Technol. 19, 512-519 (2001).
[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
[CrossRef]

Oron, M.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Palamaru, M.

M. Palamaru and Ph. Lalanne, "Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-1468 (2001).
[CrossRef]

Povinelli, M. L.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

Rahman, B. M. A.

B. M. A. Rahman, S. S. A. Obayya, N. Somasiri, M. Rajarajan, K. T. V. Grattan, and H. A. El-Mikathi, "Design and Characterization of Compact Single-Section Passive Polarization Rotator," J. Lightwave Technol. 19, 512-519 (2001).
[CrossRef]

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
[CrossRef]

Rajarajan, M.

Reinhart, F. K.

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
[CrossRef]

Safavi-Naeini, S.

Shani, Y.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Sinclair, W. R.

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
[CrossRef]

Skorobogati, M. A.

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

Soljacic, M.

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

Solli, D. R.

D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004).
[CrossRef]

Somasiri, N.

Taillaert, D.

Thourhout, D. V.

Villeneuve, P. R.

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Watts, M. R.

Wiaux, V.

Yablonovitch, E.

Young, M. G.

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Appl. Phys. Lett. (2)

M. Palamaru and Ph. Lalanne, "Photonic crystal waveguides: out-of-plane losses and adiabatic modal conversion," Appl. Phys. Lett. 78, 1466-1468 (2001).
[CrossRef]

Y. Shani, R. Alferness, T. Koch, U. Koren, M. Oron, B. I. Miller, and M. G. Young, "Polarization rotation in asymmetric periodic loaded rib waveguides," Appl. Phys. Lett. 59, 1278-1280 (1991).
[CrossRef]

Appl. Phys.Lett. (1)

M. L. Povinelli, S. G. Johnson, E. Lidorikis, J. D. Joannopoulos, and M. Soljacic, "Effect of a photonic band gap on scattering from waveguide disorder," Appl. Phys.Lett. 84, 3639-3641 (2004).
[CrossRef]

IEEE J. Lightwave Technol. (1)

W. Huang and Z. M. Mao, "Polarization rotation in periodic loaded rib waveguides," IEEE J. Lightwave Technol. 10, 1825-1831 (1992).
[CrossRef]

IEEE J. Quantum Electron. (3)

W. P. Haung, S. T. Chu, and S. K. Chaudhuri, "Scalar coupled-mode theory with vector correction," IEEE J. Quantum Electron. 28, 184-193 (1992).
[CrossRef]

R. C. Alferness, "Electrooptic guided-wave devices for general polarization transformations," IEEE J. Quantum Electron. 17, 965-969 (1981).
[CrossRef]

F. K. Reinhart, R. A. Logan, and W. R. Sinclair, "Electrooptic polarization modulation multielectrode AlxGa1-xAs rib waveguides," IEEE J. Quantum Electron. 18, 763-766 (1982).
[CrossRef]

IEEE Photon. Tech. Lett. (1)

S. S. A. Obayya, B. M. A. Rahman, and H. A. El-Mikati, "Vector beam propagation analysis of polarization conversion in periodically loaded waveguides," IEEE Photon. Tech. Lett. 12, 1346-1348 (2000).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

D. R. Solli and J. M. Hickmann, "Periodic crystal based polarization control devices," J. Phys. D 37, R263-R268 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, "Guided modes in photonic crystal slabs," Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Phys. Rev. E (1)

S. G. Johnson, P. Bienstman, M. A. Skorobogati, M. Ibanescu, E. Lidorikis, and J. D. Joannopoulos, "Adiabatic theorem and continuous coupled-mode theory for efficient taper transitions in photonic crystals," Phys. Rev. E 66, 066608 (2002).
[CrossRef]

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Figures (10)

Fig. 1.
Fig. 1.

The sketch of (a) periodic asymmetric loaded triangular PC slab waveguide (b) asymmetric loaded PC slab waveguide.

Fig. 2.
Fig. 2.

Top view of the asymmetrically loaded PC based polarization rotator. The top cover layer is marked by the dark solid line in the figure. κ1 and κ2 represent the cross-coupling coefficient for regions 1 and 2 inside a unit cell.

Fig. 3.
Fig. 3.

(a) The supercell of the asymmetric loaded PC slab waveguide for PWEM analysis. (b) The band diagram for the asymmetric loaded PC slab waveguide obtained by PWEM.

Fig. 4.
Fig. 4.

The distribution of (a) Hy and (b) Hx at (x-y) plane and (c) Hy and (d) Hx at y=0 plane in asymmetric loaded PC slab waveguide with w=0.6a, t=0.8a, tup=0.2a, a=132.5 μm and a/λ=0.245; input has launched at z=0.

Fig. 5.
Fig. 5.

The profile of (a) Ex and (b)Ey components of x-polarized and y-polarized modes of the structure shown in Fig. 1(b) obtained by semi-vectorial 3D BPM analysis (t=0.8a, tup =0.2a, w=0.6a, a=132.5 μm, nsi =3.48 and λ=500 μm).

Fig. 6.
Fig. 6.

Power exchange between the x-polarized and y-polarized wave versus the propagation length (t=0.8a, tup =0.2a, w=0.6a, a=132.5 μm, nsi =3.48) for a/λ=0.255, 0.265 and 0.275 obtained by coupled-mode analysis.

Fig. 7.
Fig. 7.

The contour plot of the cross section of (a) Ey, (b) Hx, (c) Ex and (d) Hy components of electromagnetic field propagating in asymmetric loaded PC slab waveguide at the input plane and (e) Ex and (f) Hy at z=5.5 mm (w=0.6a, t=0.8a, tup=0.2a, a/λ=0.265 and λ=500 μm).

Fig. 8.
Fig. 8.

Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.265 obtained by 3D-FDTD simulation (t=0.8a, tup =0.2a, w=0.6a, a=132.5 μm, nsi =3.48, λ=500 μm).

Fig. 9.
Fig. 9.

Power exchange between the x-polarized and y-polarized waves versus frequency for both coupled-mode analysis and 3D-FDTD simulations ((t=0.8a, tup =0.2a, w=0.6a, a=132.5 μm, nsi =3.48).

Fig. 10.
Fig. 10.

Power exchange between the x-polarized and y-polarized wave versus the propagation length for a/λ=0.275 obtained by 3D-FDTD simulation (t=0.8a, tup =0.2a, w=0.6a, a=132.5 μm, nsi =3.48).

Equations (29)

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

2 E x z 2 + t 2 E x + n 2 k 2 E x = x ( E x 1 n 2 n 2 x ) x ( E y 1 n 2 n 2 y )
2 E y z 2 + t 2 E y + n 2 k 2 E y = y ( E x 1 n 2 n 2 x ) y ( E y 1 n 2 n 2 y )
t 2 = 2 x 2 + 2 y 2
E = E x x ̂ + E y y ̂ = a x ( z ) e x ( x , y ) e j β x z + a y ( z ) e y ( x , y ) e j β y z ,
j 2 β x e x d a x ( z ) d z + a x ( z ) t 2 e x + n 2 k 2 a x ( z ) e x β x 2 a x ( z ) e x =
a x ( z ) x ( e x 1 n 2 n 2 x ) a y ( z ) e j Δ z x ( e y 1 n 2 n 2 y )
j 2 β y e y d a y ( z ) d z + a y ( z ) t 2 e y + n 2 k 2 a y ( z ) e y β y 2 a y ( z ) e y =
a y ( z ) y ( e y 1 n 2 n 2 y ) a x ( z ) e j Δ z y ( e x 1 n 2 n 2 x )
t 2 e x + ( n 2 k 2 β ave 2 ) e x = 0
t 2 e y + ( n 2 k 2 β ave 2 ) e y = 0 ,
j 2 β x e x d a x ( z ) d z + ( β ave 2 β x 2 ) a x ( z ) e x =
a x ( z ) x ( e x 1 n 2 n 2 x ) a y ( z ) e j Δ z x ( e y 1 n 2 n 2 y )
j 2 β y e y d a y ( z ) d z + ( β ave 2 β x 2 ) a y ( z ) e y =
a y ( z ) y ( e y 1 n 2 n 2 y ) a x ( z ) e j Δ z y ( e x 1 n 2 n 2 x )
d a x ( z ) d z = j κ xx a x ( z ) j κ xy a y ( z )
d a y ( z ) d z = j κ yy a y ( z ) j κ yx a x ( z )
κ xx = ( β ave 2 β x 2 ) ∫∫ e x * · e x dxdy + ∫∫ e x * · y ( e y 1 n 2 n 2 y ) dxdy 2 β x ∫∫ e x * · e x dxdy
κ xy = e j Δ z ∫∫ e x * · x ( e y 1 n 2 n 2 y ) dxdy 2 β x ∫∫ e x * · e x dxdy
κ yy = ( β ave 2 β y 2 ) ∫∫ e y * · e y dxdy + ∫∫ e y * · y ( e y 1 n 2 n 2 y ) dxdy 2 β y ∫∫ e y * · e y dxdy
κ yx = e j Δ z ∫∫ e y * · y ( e x 1 n 2 n 2 x ) dxdy 2 β y ∫∫ e y * · e y dxdy
M i ± = ( cos ( Ω i z i ) j cos ( φ i / 2 ) sin ( Ω i z i ) j sin ( φ i / 2 ) sin ( Ω i z i ) j sin ( φ i / 2 ) sin ( Ω i z i ) cos ( Ω i z i ) + j cos ( φ i / 2 ) sin ( Ω i z i ) ) i = 1,2
Ω i = δ i 2 + κ i 2
δ i = κ xxi κ yyi 2
tan ( φ i / 2 ) = κ i δ i
z 1 = a w
z 2 = w
M ± = M 1 ± M 2 ± ,
Λ π Ω 1 + Ω 2 ,
P . C . E = P TM P TM + P TE × 100 = a x 2 a x 2 + a y 2 × 100 ,

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