Abstract

A transfer-matrix technique is used to calculate the lasing thresholds of second-order circular-grating polymer lasers operating at 630 nm. By use of poly[2-methoxy-5-(2-ethyl-hexyloxy)-p-phenylenevinylene] as an example polymer material, it is also shown how known optical properties of polymeric materials may be incorporated into the analysis of both the transverse waveguiding and the distributed feedback in circular-grating distributed-feedback polymer lasers.

© 2004 Optical Society of America

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References

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  1. V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
    [CrossRef]
  2. M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
    [CrossRef]
  3. M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
    [CrossRef]
  4. G. F. Barlow and K. A. Shore, “Analysis of waveguide properties of organic semiconductor lasers,” IEE Proc.: Optoelectron. 146, 15–20 (1999).
  5. C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
    [CrossRef]
  6. C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
    [CrossRef]
  7. C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
    [CrossRef]
  8. G. F. Barlow and K. A. Shore, “Application of the argument principle method to the calculation of DFB laser diode modes,” Int. J. Numer. Model. 14, 291–302 (2001).
    [CrossRef]
  9. P. L. Greene and D. G. Hall, “Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations,” IEEE J. Quantum Electron. 37, 353–364 (2001).
    [CrossRef]
  10. A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
    [CrossRef]
  11. T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
    [CrossRef]

2001 (2)

G. F. Barlow and K. A. Shore, “Application of the argument principle method to the calculation of DFB laser diode modes,” Int. J. Numer. Model. 14, 291–302 (2001).
[CrossRef]

P. L. Greene and D. G. Hall, “Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations,” IEEE J. Quantum Electron. 37, 353–364 (2001).
[CrossRef]

2000 (2)

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
[CrossRef]

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

1999 (1)

G. F. Barlow and K. A. Shore, “Analysis of waveguide properties of organic semiconductor lasers,” IEE Proc.: Optoelectron. 146, 15–20 (1999).

1998 (2)

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

1993 (1)

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

1992 (1)

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

1991 (1)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

1990 (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Barlow, G. F.

G. F. Barlow and K. A. Shore, “Application of the argument principle method to the calculation of DFB laser diode modes,” Int. J. Numer. Model. 14, 291–302 (2001).
[CrossRef]

G. F. Barlow and K. A. Shore, “Analysis of waveguide properties of organic semiconductor lasers,” IEE Proc.: Optoelectron. 146, 15–20 (1999).

Berggren, M.

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

Burrows, P. E.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Chou, S. Y.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Diaz-Garcia, M. A.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Dodabalapur, A.

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

Erdogan, T.

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Fallahi, M.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Forrest, S. R.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Glinski, J.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

Greene, P. L.

P. L. Greene and D. G. Hall, “Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations,” IEEE J. Quantum Electron. 37, 353–364 (2001).
[CrossRef]

Gupta, R.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Hall, D. G.

P. L. Greene and D. G. Hall, “Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations,” IEEE J. Quantum Electron. 37, 353–364 (2001).
[CrossRef]

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

Hide, F.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Huang, W. P.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
[CrossRef]

Khalfin, V. B.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Kozlov, V. G.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Li, X.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
[CrossRef]

Maciejko, R.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

Makino, T.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

McGhehee, M. D.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Miller, E. K.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Moses, D.

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

Najafi, S. I.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

Nalamasu, O.

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

Parthasarathy, G.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Shams-Zadeh-Amiri, A. M.

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
[CrossRef]

Shore, K. A.

G. F. Barlow and K. A. Shore, “Application of the argument principle method to the calculation of DFB laser diode modes,” Int. J. Numer. Model. 14, 291–302 (2001).
[CrossRef]

G. F. Barlow and K. A. Shore, “Analysis of waveguide properties of organic semiconductor lasers,” IEE Proc.: Optoelectron. 146, 15–20 (1999).

Slusher, R. E.

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

Svilans, M.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

Timko, A.

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

Wang, J.

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Wu, C.

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

Appl. Phys. Lett. (2)

M. D. McGhehee, M. A. Diaz-Garcia, F. Hide, R. Gupta, E. K. Miller, and D. Moses, “Semiconducting polymer distributed feedback lasers,” Appl. Phys. Lett. 72, 1536–1538 (1998).
[CrossRef]

M. Berggren, A. Dodabalapur, R. E. Slusher, A. Timko, and O. Nalamasu, “Organic solid-state lasers with imprinted gratings on plastic substrates,” Appl. Phys. Lett. 72, 410–411 (1998).
[CrossRef]

IEE Proc.: Optoelectron. (1)

G. F. Barlow and K. A. Shore, “Analysis of waveguide properties of organic semiconductor lasers,” IEE Proc.: Optoelectron. 146, 15–20 (1999).

IEEE J. Quantum Electron. (4)

C. Wu, T. Makino, R. Maciejko, S. I. Najafi, M. Svilans, J. Glinski, and M. Fallahi, “Threshold gain and threshold current analysis of circular grating DFB and DBR lasers,” IEEE J. Quantum Electron. 29, 2596–2606 (1993).
[CrossRef]

P. L. Greene and D. G. Hall, “Effects of radiation on circular-grating DFB lasers. I. Coupled-mode equations,” IEEE J. Quantum Electron. 37, 353–364 (2001).
[CrossRef]

A. M. Shams-Zadeh-Amiri, X. Li, and W. P. Huang, “Above threshold analysis of second-order circular-grating DFB lasers,” IEEE J. Quantum Electron. 36, 256–267 (2000).
[CrossRef]

V. G. Kozlov, G. Parthasarathy, P. E. Burrows, V. B. Khalfin, J. Wang, S. Y. Chou, and S. R. Forrest, “Structures for organic diode lasers and optical properties of organic semiconductors under intense optical and electrical excitations,” IEEE J. Quantum Electron. 36, 2596–2606 (2000).
[CrossRef]

Int. J. Numer. Model. (1)

G. F. Barlow and K. A. Shore, “Application of the argument principle method to the calculation of DFB laser diode modes,” Int. J. Numer. Model. 14, 291–302 (2001).
[CrossRef]

J. Appl. Phys. (1)

T. Erdogan and D. G. Hall, “Circularly symmetric distributed feedback semiconductor laser: an analysis,” J. Appl. Phys. 68, 1435–1444 (1990).
[CrossRef]

J. Lightwave Technol. (2)

C. Wu, T. Makino, J. Glinski, R. Maciejko, and S. I. Najafi, “Self-consistent coupled-wave theory for circular gratings on planar dielectric waveguides,” J. Lightwave Technol. 9, 1264–1277 (1991).
[CrossRef]

C. Wu, T. Makino, R. Maciejko, and S. I. Najafi, “Simplified coupled-wave equations for cylindrical waves in circular grating planar waveguides,” J. Lightwave Technol. 10, 1575–1589 (1992).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

General CG-DFB design.

Fig. 2
Fig. 2

TE fundamental mode intensity profile at various wavelengths.

Fig. 3
Fig. 3

TM fundamental mode intensity profile at various wavelengths.

Fig. 4
Fig. 4

Dispersion of the MEH-PPV waveguide.

Fig. 5
Fig. 5

TE mode confinement versus film thickness.

Fig. 6
Fig. 6

TM mode confinement versus film thickness.

Fig. 7
Fig. 7

Effect of the duty cycle on radiation and feedback couplings in a 400-nm-period, 30-nm-depth, second-order, rectangular profile DFB grating.

Fig. 8
Fig. 8

Radiation and feedback couplings in a 400-nm-period, second-order, rectangular profile DFB grating.

Fig. 9
Fig. 9

Design profiles of (a) single-section and (b) two-section polymer CG-DFB lasers. D.C., duty cycle.

Fig. 10
Fig. 10

Lasing modes of a 100-µm-radius, single-section polymer CG-DFB laser with Ω=2π.

Fig. 11
Fig. 11

Lasing modes of a 100-µm-radius, single-section polymer CG-DFB laser with Ω=π/2.

Fig. 12
Fig. 12

Lasing modes of a 100-µm-radius, two-section polymer CG-DFB laser with Ω=2π.

Fig. 13
Fig. 13

Near-field intensity of the first four lasing modes in a 100-µm-radius, single-section polymer CG-DFB laser with Ω=2π.

Fig. 14
Fig. 14

Near-field intensity of the first four lasing modes in a 100-µm-radius, single-section polymer CG-DFB laser with Ω=2π.

Fig. 15
Fig. 15

Far-field intensity of the first four lasing modes in a 100-µm-radius, single-section polymer CG-DFB laser with Ω=2π.

Fig. 16
Fig. 16

Far-field intensity of the first four lasing modes in a 100-µm-radius, two-section polymer CG-DFB laser with Ω=2π.

Equations (34)

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

no(λ)=3.3+0.1481-(552/λ)2-9.7×10-7λ21/2.
2x2E(x)+[n2(x)-neff2]ko2E(x)=0,
n(x)=no(x)-jαo(x)2ko,
neff=ne-jαe2ko,
2x2E(x)+[no2(x)-ne2]ko2E(x)
-j[no(x)αo(x)-neαe]koE(x)=0.
neαeE(x)=no(x)α(x)E(x).
αo(x)=αs exp(-αpx),0<x<d,
αe=αsΓe,
Γe=x=0dE2(x)exp(-2αpx)dxE2(x)dx,
1rrrr+1ρ22ϕ2+2x2Hx
+k02(x, r, ϕ)Hx=0.
Hx(r, ϕ, x)=m=0Hxm(r, x)exp(imϕ).
Hxm(r, x)=[Am(r)Hm(1)(βrρ)exp(-iβrr0)+Bm(r)Hm(2)(βrr)exp(iβrr0)X(x)+ΔHxm(r, x),
Ω=lπΛ(Λ+2r0),
ΔHxm(r, x)=Am(r)Hm(1)(βrr)g1(x)exp(-iβrr)+Bm(r)Hm(2)(βrr)g-1(x)exp(iβrr),
Δβp=Δβp+iΓeγ=βr-pπΛ+iΓeγ.
ddxAm(r)=-(-1)miκfBm(r)-iΔβ2Am(r)-κr[Am(r)+(-1)mBm(r)],
ddxBm(z)=-(-1)miκfAm(r)+iΔβ2Bm(r)+κr[(-1)mAm(r)+Bm(r)],
κf=ko2Δ2βrV d|E(x)|2C2(x)dx,
κr=ko4Δ24βrβx(r)V dE(x)C1(x)exp(jβx(r)x)dx,
Cl=sin(πlΛ/Λ)πl,
Am(r)=af exp(iqr)+ab exp(-iqr),
BM(r)=bf exp(iqr)+bb exp(-iqr),
q(Δβ2)=±[Δβ22-kf2+2jκr(κf-Δβ2)]1/2.
Rm(q)=(-1)m -(q+Δβ2-iκr)-iκr+κf=(-1)m -iκr+κfq-Δβ2+iκr,
Am(r)=af exp(iqr)+Rm(q)bb exp(-iqr),
Bm(r)=Rm(q)af exp(iqr)+bb exp(-iqr).
Am(r)Bm(r)=exp(iqr)R exp(-iqr)R exp(iqr)exp(-iqr)afbb=m(r)afbb.
A(R1)B(R1)=m(R1)m(R2)-1A(R2)B(R2)=M(R1, R2)A(R2)B(R2),
Bm(R2)=0,
Am(0)=exp(iω)Bm(0),
M(r1, r2)2,1M(r1, r2)2,2+exp(iϕ)M(r1, r2)2,2=0.
F(r)=A(r)2iπβ0r1/2i-m+B(r)2iπβ0r1/2im.

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