Abstract

We examine the operation of grating-assisted couplers by describing the fields inside the periodic coupling region in terms of a rigorous modal formulation. This approach reveals that the wave coupling process takes the form of a mode conversion mechanism. We then evaluate the effectiveness of a coupler to transfer power incident at the input waveguide to another waveguide at the output. Our results show that, in general, the optimal conditions for this power transfer may be different from those predicted by conventional coupled-mode methods. We also find that the efficiency of a coupler can be adversely affected by boundary discontinuities at the terminals of the grating region, and we suggest ways to avoid their negative impact on the desired power transfer.

© 1996 Optical Society of America

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References

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  1. R. R. R. Syms, “Optical directional coupler with a grating overlay,” Appl. Opt. 24, 717–726 (1985).
    [CrossRef] [PubMed]
  2. L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
    [CrossRef]
  3. G. Griffel, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
    [CrossRef]
  4. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 7, pp. 280–293.
    [CrossRef]
  5. W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
    [CrossRef]
  6. D. Marcuse, “Radiation loss of grating-assisted directional coupler,” IEEE J. Quantum Electron. 26, 675–684 (1990).
    [CrossRef]
  7. W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
    [CrossRef]
  8. B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” J. Lightwave Technol. 14, 188–195 (1996).
    [CrossRef]
  9. H.-P. Nolting, G. Sztefka, “Eigenmode matching and propagation theory of square meander-type couplers,” IEEE Photonics Technol. Lett. 4, 1386–1389 (1992).
    [CrossRef]
  10. W.-P. Huang, C. Xu, B. Little, “Computer-aided analysis and design in guided-wave optoelectronics,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 423–428.
  11. T. Tamir, S. Zhang, “Rigorous guided-wave solutions for planar grating structures,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 363–370.
  12. T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol 14, 914–927 (1996).
    [CrossRef]
  13. S. Zhang, T. Tamir, “Rigorous analysis of power transfer in grating-assisted couplers,” Opt. Lett. 20, 803–805 (1995).
    [CrossRef] [PubMed]
  14. S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, London, 1986), pp. 191–216.
  15. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  16. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, Philadelphia, Pa., 1991), Chap.13, pp. 519–529.
  17. T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1985), Chap. 3, pp. 84–137.
  18. S. S. Wang, R. Magnusson, J. S. Bagby, M. G. Moharam, “Guided-mode resonances in planar dielectric-layer diffraction gratings,” J. Opt. Soc. Am. A 7, 1470–1474 (1990).
    [CrossRef]
  19. S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
    [CrossRef] [PubMed]
  20. D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958), Chap. 3, pp. 66–82.
  21. B. E. Little, “Junction problem and loading effects in grating-assisted couplers,” Opt. Lett. 21, 949–951 (1996).
    [CrossRef] [PubMed]

1996 (3)

B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” J. Lightwave Technol. 14, 188–195 (1996).
[CrossRef]

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol 14, 914–927 (1996).
[CrossRef]

B. E. Little, “Junction problem and loading effects in grating-assisted couplers,” Opt. Lett. 21, 949–951 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

1993 (3)

S. S. Wang, R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32, 2606–2613 (1993).
[CrossRef] [PubMed]

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
[CrossRef]

1992 (1)

H.-P. Nolting, G. Sztefka, “Eigenmode matching and propagation theory of square meander-type couplers,” IEEE Photonics Technol. Lett. 4, 1386–1389 (1992).
[CrossRef]

1991 (1)

G. Griffel, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

1990 (2)

1985 (1)

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Alferness, R. C.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Bagby, J. S.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Buhl, L. L.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Burrus, C.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Crosignani, B.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, London, 1986), pp. 191–216.

Di Porto, P.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, London, 1986), pp. 191–216.

Griffel, G.

G. Griffel, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

Hong, J.

W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
[CrossRef]

Huang, W.-P.

W.-P. Huang, “Coupled-mode theory for optical waveguides: an overview,” J. Opt. Soc. Am. A 11, 963–983 (1994).
[CrossRef]

W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
[CrossRef]

W.-P. Huang, C. Xu, B. Little, “Computer-aided analysis and design in guided-wave optoelectronics,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 423–428.

Koch, T. L.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Koren, U.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Little, B.

W.-P. Huang, C. Xu, B. Little, “Computer-aided analysis and design in guided-wave optoelectronics,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 423–428.

Little, B. E.

B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” J. Lightwave Technol. 14, 188–195 (1996).
[CrossRef]

B. E. Little, “Junction problem and loading effects in grating-assisted couplers,” Opt. Lett. 21, 949–951 (1996).
[CrossRef] [PubMed]

Magnusson, R.

Mao, Z. M.

W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
[CrossRef]

Marcuse, D.

D. Marcuse, “Radiation loss of grating-assisted directional coupler,” IEEE J. Quantum Electron. 26, 675–684 (1990).
[CrossRef]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 7, pp. 280–293.
[CrossRef]

Miller, B. I.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Moharam, M. G.

Nolting, H.-P.

H.-P. Nolting, G. Sztefka, “Eigenmode matching and propagation theory of square meander-type couplers,” IEEE Photonics Technol. Lett. 4, 1386–1389 (1992).
[CrossRef]

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Raybon, G.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Solimeno, S.

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, London, 1986), pp. 191–216.

Syms, R. R. R.

Sztefka, G.

H.-P. Nolting, G. Sztefka, “Eigenmode matching and propagation theory of square meander-type couplers,” IEEE Photonics Technol. Lett. 4, 1386–1389 (1992).
[CrossRef]

Tamir, T.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol 14, 914–927 (1996).
[CrossRef]

S. Zhang, T. Tamir, “Rigorous analysis of power transfer in grating-assisted couplers,” Opt. Lett. 20, 803–805 (1995).
[CrossRef] [PubMed]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1985), Chap. 3, pp. 84–137.

T. Tamir, S. Zhang, “Rigorous guided-wave solutions for planar grating structures,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 363–370.

Wang, S. S.

Watkins, D. A.

D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958), Chap. 3, pp. 66–82.

Xu, C.

W.-P. Huang, C. Xu, B. Little, “Computer-aided analysis and design in guided-wave optoelectronics,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 423–428.

Yariv, A.

G. Griffel, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, Philadelphia, Pa., 1991), Chap.13, pp. 519–529.

Young, M. G.

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

Zhang, S.

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol 14, 914–927 (1996).
[CrossRef]

S. Zhang, T. Tamir, “Rigorous analysis of power transfer in grating-assisted couplers,” Opt. Lett. 20, 803–805 (1995).
[CrossRef] [PubMed]

T. Tamir, S. Zhang, “Rigorous guided-wave solutions for planar grating structures,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 363–370.

Appl. Opt. (2)

Electron. Lett. (1)

L. L. Buhl, R. C. Alferness, U. Koren, B. I. Miller, M. G. Young, T. L. Koch, C. Burrus, G. Raybon, “Grating-assisted vertical coupler/filter for extended tuning range,” Electron. Lett. 29, 81–82 (1993).
[CrossRef]

IEEE J. Quantum Electron. (3)

G. Griffel, A. Yariv, “Frequency response and tunability of grating-assisted directional couplers,” IEEE J. Quantum Electron. 27, 1115–1118 (1991).
[CrossRef]

D. Marcuse, “Radiation loss of grating-assisted directional coupler,” IEEE J. Quantum Electron. 26, 675–684 (1990).
[CrossRef]

W.-P. Huang, J. Hong, Z. M. Mao, “Improved coupled-mode formulation based on composite modes for parallel grating-assisted co-directional couplers,” IEEE J. Quantum Electron. 29, 2805–2812 (1993).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

H.-P. Nolting, G. Sztefka, “Eigenmode matching and propagation theory of square meander-type couplers,” IEEE Photonics Technol. Lett. 4, 1386–1389 (1992).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

J. Lightwave Technol (1)

T. Tamir, S. Zhang, “Modal transmission-line theory of multilayered grating structures,” J. Lightwave Technol 14, 914–927 (1996).
[CrossRef]

J. Lightwave Technol. (1)

B. E. Little, “A variational coupled-mode theory including radiation loss for grating-assisted couplers,” J. Lightwave Technol. 14, 188–195 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Other (7)

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, Philadelphia, Pa., 1991), Chap.13, pp. 519–529.

T. Tamir, “Beam and waveguide couplers,” in Integrated Optics, T. Tamir, ed. (Springer-Verlag, New York, 1985), Chap. 3, pp. 84–137.

D. A. Watkins, Topics in Electromagnetic Theory (Wiley, New York, 1958), Chap. 3, pp. 66–82.

W.-P. Huang, C. Xu, B. Little, “Computer-aided analysis and design in guided-wave optoelectronics,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 423–428.

T. Tamir, S. Zhang, “Rigorous guided-wave solutions for planar grating structures,” in Guided-Wave Optoelectronics: Device Characterization, Analysis and Design, T. Tamir, G. Griffel, H. L. Bertoni, eds. (Plenum, New York, 1995), pp. 363–370.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, 1991), Chap. 7, pp. 280–293.
[CrossRef]

S. Solimeno, B. Crosignani, P. Di Porto, Guiding, Diffraction and Confinement of Optical Radiation (Academic, London, 1986), pp. 191–216.

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Figures (10)

Fig. 1
Fig. 1

Periodic configurations: (a) typical grating-assisted coupler, (b) general geometry of a rectangular grating layer forming the periodic part of the coupler. Unless otherwise stated, the numerical examples given in this paper assume that 0 = 2 = p2 = 3.02, 1 = 3.52, 4 = p1 = 3.22, 5 = 1.0, t1 = 0.22 μm, t2 = 0.55 μm, t3 = tp = 0.05 μm, t4 = 0.45 μm, and Λ1 = Λ2 = Λ/2 = 5.374 μm.

Fig. 2
Fig. 2

TE-mode fields in structures having two guiding layers: (a) asymmetric case corresponding to the region 0 z L in Fig. 1(a) but with the grating replaced by a homogeneous layer with permittivity p = (2 + 4)/2 = 9.62, (b) symmetric case having t1 = t3 = 0.5 μm,t2 = 0.8 μm, 0 = 2 = 4 = 3.02, and 1 = 3 = 3.22. The locations of interface boundaries are indicated by dashed vertical lines.

Fig. 3
Fig. 3

Typical dispersion curves for periodic structures having two guiding layers.

Fig. 4
Fig. 4

Dispersion curves for the first two propagating TE modes in the grating region of Fig. 1: (a) variation of β/ko versus Λ/λ., (b) variation of αλ versus λ.

Fig. 5
Fig. 5

Dispersion behavior of the first two propagating TE modes in the grating region of a coupler as in Fig. 1, except that the grating is on top of the upper guiding layer, i.e., p = (3 + 5)/2 = (3.22 + 1.02)/2 = 5.62: (a) geometry of the coupler, (b) variation of β/ko versus Λ/λ.

Fig. 6
Fig. 6

Variation of Er(x,z) for the mode given by κr in Fig. 4 and Δ = 0. In (c) the two regions 0 z / Λ 0.5 and 0.5 z / Λ 1 are separated for clarity.

Fig. 7
Fig. 7

Transverse TE-mode variation of Er,El, and Es fields in the grating region at z = 0 and B in Fig. 1(a) with Δ = 0 and λ = λgap = 1.49775 μm. The × signs outline the fields Eod and Eev (for δ = 0) as in Fig. 2(a). Layer boundaries are indicated by vertical lines, which are solid for the grating and dashed for the other interfaces.

Fig. 8
Fig. 8

Power transfer in a coupler with weak terminal discontinuities. (a) Geometry and power flow in a coupler having the same parameters inside 0 < z < L as those in Fig. 1(a); outside that region the grating layer continues as a homogeneous layer with permittivity p = (2 + 4)/2 = 9.62. (b) Power ratios versus L for λ = λgap and λph. The curves in (b) are shown for Δ = 0, but they are not discernibly different if Δ ≠ 0.

Fig. 9
Fig. 9

Power transfer in the realistic coupler of Fig. 1: (a) geometry showing power flow, (b) variation of efficiency ηtb with L when power Pt is transferred from the top guide to Pb in the bottom guide.

Fig. 10
Fig. 10

Schemes for avoiding deterioration in power-conversion efficiency.

Equations (36)

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F ( x , z ) = exp ( i κ z ) n W n ( κ ; x ) exp ( 2 i n π z / Λ ) ,
E j = n { f j n exp ( i k j n x j ) + g j n × exp [ i k j n ( t j x j ) ] } exp ( i k z n z ) ,
H j = n y j n { f j n exp ( i k j n x j ) g j n × exp [ i k j n ( t j x j ) ] } exp ( i k z n z ) ,
k z n = κ + 2 n π / Λ , with κ = β + i α ,
k j n = k j n + i k j n = k o 2 j k z n 2 for all j p ,
y j n = { k j n / ω μ o ( TE ) ω o j / k j n ( TM ) ,
k j n > 0 if | β n | Re ( k o j ) , or k j n > 0 otherwise ,
E p = m { f p m exp ( i k p m x p ) + g p m × exp [ i k p m ( t p x p ) ] } n a n m exp ( i k z n z ) ,
H p = m y p m { f p m exp ( i k p m x p ) g p m × exp [ i k p m ( t p x p ) ] } n b n m exp ( i k z n z ) ,
y p m = { k p m / ω μ o ( TE ) ω o / k p m γ p ( TM ) ,
E j f j + g j = f j + 1 + E j + 1 g j + 1 ,
Y j ( E j f j g j ) = Y j + 1 ( f j + 1 E j + 1 g j + 1 ) ,
f 0 = g J + 1 = 0 .
E p 1 f p 1 + g p 1 = A ( f p + E p g p ) ,
Y p 1 ( E p 1 f j g p 1 ) = B Y p ( f p E p g p ) ,
A ( E p f p + g p ) = f j + 1 + E j + 1 g j + 1 ,
B Y p ( E p f p g p ) = Y p + 1 ( f p + 1 E p + 1 g p + 1 ) ,
M g 0 = 0 ,
| M | = 0 .
( z ) = p + δ p ( z ) ,
D = N ev N od = λ ph / Λ .
E r ( λ ) = E r ( λ ; x , z ) A r 0 ( λ ) w ev ( x ) exp ( i κ r z ) + A r , 1 ( λ ) w od ( x ) exp [ i ( κ r 2 π / Λ ) z ] ,
| w ev ( x ) | 2 d x = | w od ( x ) | 2 d x = 1 .
E r ( λ gap ) A r 0 ( λ gap ) [ w ev ( x ) exp ( i π z / Λ ) + w od ( x ) exp ( i π z / Λ ) ] exp [ i ( κ r π / Λ ) z ] .
E l ( λ ) A l 0 ( λ ) w ev ( x ) exp ( i κ l z ) + A l , 1 ( λ ) w od ( x ) × exp [ i ( κ l 2 π / Λ ) z ] ,
E l ( λ gap ) A l 0 ( λ gap ) [ w ev ( x ) exp ( i π z / Λ ) w od ( x ) exp ( i π z / Λ ) ] exp [ i ( κ l π / Λ ) z ] .
E s ( λ gap ) = E r ( λ gap ) + E l ( λ gap ) 2 A 0 [ w od ( x ) exp ( i π z / Λ ) cos ( π z / 2 B ) + i w ev ( x ) exp ( i π z / Λ ) sin ( π z / 2 B ) ] × exp [ i ( κ av π / Λ ) z ] ,
κ av = ( κ r + κ l ) / 2 ,
B = π / ( β r β l ) .
E in ( x , 0 ) = w od ( x ) C r E r ( x , 0 ) + C l E l ( x , 0 ) = E s ( x , 0 ) ,
C r = C r ( Δ ) w od ( x ) E r * ( x , 0 ) d x | E r ( x , 0 ) | 2 d x ,
C l = C l ( Δ ) w od ( x ) E l * ( x , 0 ) d x | E l ( x , 0 ) | 2 d x .
E s ( x , L ) = C r E r ( x , L ) + C l E l ( x , L ) C ev w ev ( x ) + C od w od ( x ) = E out ( x ; Δ , L ) ,
C ev = C ev ( Δ , L ) = E s ( x , L ) w ev * ( x ) d x ,
C od = C ev ( Δ , L ) = E s ( x , L ) w od * ( x ) d x .
η od ev ( Δ , L ) = P ev ( L ) P od ( 0 ) = N ev | C ev w ev ( x ) | 2 d x N od | w od ( x ) | 2 d x = N ev N od | C ev | 2 ,

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