Abstract

Diffraction of plane waves, obliquely incident on a (waveguide) grating with infinitely long grooves, is analyzed with the four-wave coupled-mode theory (CMT). Unlike the ordinary analysis that considers only coupling between pairs (i.e., TE+TE-, TE+TM-, TM+TE-, or TM+TM-), the simultaneous interaction among these waves is properly accounted for. The field variation along the grating and the impulse response representation of the grating (reflectivity and transmission as functions of the plane-wave incidence angle) are calculated. Significant differences between the ordinary (two-wave) CMT and the four-wave CMT are observed over a wide range of incident angles.

© 1998 Optical Society of America

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References

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  1. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).
  2. A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).
  3. H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
  4. L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
    [CrossRef]
  5. D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
    [CrossRef]
  6. K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
    [CrossRef]
  7. J. Van Roey, P. E. Lagasse, “Coupled wave analysis of obliquely incident waves in thin film gratings,” Appl. Opt. 20, 423–429 (1981).
    [CrossRef] [PubMed]
  8. A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
    [CrossRef]
  9. K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
    [CrossRef]
  10. S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.
  11. A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.
  12. L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM coupling in waveguide gratings,” Opt. Lett. 12, 756–758 (1987).
    [CrossRef] [PubMed]
  13. H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
    [CrossRef]
  14. A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
    [CrossRef]
  15. W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).
  16. A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
    [CrossRef]

1993 (1)

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

1991 (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

1988 (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

1987 (1)

1986 (1)

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

1984 (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

1981 (1)

1979 (1)

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
[CrossRef]

1977 (1)

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

1972 (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Boyce, W. E.

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

DeMars, S. D.

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

Diprima, R. C.

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

Dzurko, K. M.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

Hall, D. G.

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM coupling in waveguide gratings,” Opt. Lett. 12, 756–758 (1987).
[CrossRef] [PubMed]

Hardy, A.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

Kogelnik, H.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

Lagasse, P. E.

Lang, R. J.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

Nakamura, M.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

Oliner, A. A.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

Peng, S. T.

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

Saito, S.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
[CrossRef]

Sakaki, H.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
[CrossRef]

Schoenfelder, A.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

Scifres, D. R.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

Shank, C. V.

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

Streifer, W.

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

Van Roey, J.

Waarts, R. G.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

Wagatsuma, K.

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
[CrossRef]

Welch, D. F.

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

Weller-Brophy, L. A.

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

L. A. Weller-Brophy, D. G. Hall, “Measured TM–TM coupling in waveguide gratings,” Opt. Lett. 12, 756–758 (1987).
[CrossRef] [PubMed]

Yariv, A.

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

Appl. Opt. (1)

Appl. Sci. Res. (1)

A. A. Oliner, S. T. Peng, “New physical effects on periodically grooved optical planar waveguides,” Appl. Sci. Res. 41, 271–274 (1984).
[CrossRef]

IEEE J. Quantum Electron. (4)

K. M. Dzurko, A. Hardy, D. R. Scifres, D. F. Welch, R. G. Waarts, R. J. Lang, “Distributed Bragg reflector ring oscillators: a large aperture source of high single-mode optical power,” IEEE J. Quantum Electron. 29, 1895–1905 (1993).
[CrossRef]

K. Wagatsuma, H. Sakaki, S. Saito, “Mode conversion and optical filtering of obliquely incident waves in corrugated waveguide filters,” IEEE J. Quantum Electron. 15, 632–637 (1979).
[CrossRef]

A. Yariv, M. Nakamura, “Periodic structures for integrated optics,” IEEE J. Quantum Electron. 13, 233–253 (1977).
[CrossRef]

A. Hardy, W. Streifer, “Coupled mode solutions of multiwaveguide systems,” IEEE J. Quantum Electron. 22, 528–534 (1986).
[CrossRef]

J. Appl. Phys. (1)

H. Kogelnik, C. V. Shank, “Coupled-wave theory of distributed feedback lasers,” J. Appl. Phys. 43, 2327–2335 (1972).
[CrossRef]

J. Lightwave Technol. (1)

L. A. Weller-Brophy, D. G. Hall, “Local normal mode analysis of guided mode interactions with waveguide gratings,” J. Lightwave Technol. 6, 1069–1082 (1988).
[CrossRef]

Opt. Lett. (1)

Prog. Opt. (1)

D. G. Hall, “Optical waveguide diffraction gratings: coupling between guided modes,” Prog. Opt. 29, 1–63 (1991).
[CrossRef]

Other (6)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1991).

A. Yariv, Optical Electronics (Saunders, Philadelphia, Pa., 1991).

H. Kogelnik, “Theory of optical waveguides,” in Guided Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Angled grating distributed feedback laser with 1 W cw single-mode, diffraction-limited output at 980 nm,” in Conference on Lasers and Electron-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 77–78.

A. Schoenfelder, S. D. DeMars, K. M. Dzurko, R. J. Lang, D. F. Welch, D. R. Scifres, A. Hardy, “Ultra-high brightness, high power broad area laser diode arrays,” in Conference on Lasers and Electro-Optics, Vol. 9 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper No. 7.

W. E. Boyce, R. C. Diprima, Elementary Differential Equations and Boundary Value Problems (Wiley, New York, 1986).

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Figures (10)

Fig. 1
Fig. 1

Schematic illustration of the waveguide diffraction grating with obliquely TE+ incident plane wave.

Fig. 2
Fig. 2

Bragg condition diagram. (a) Top view of the waveguide grating and the four wave vectors. (b) Schematic diagram of Bragg condition and its deviations.

Fig. 3
Fig. 3

Two-wave TE–TE coupling (waveguide grating I). (a) Intensity variation through the grating at θi=θe=60°. (b) TE- power reflectivity coefficient and TE+ power transmission coefficient as functions of the plane-wave incident angle.

Fig. 4
Fig. 4

Two-wave TE–TM coupling (waveguide grating I). (a) Intensity variation through the grating at θi=θe=60°. (b) TM- power reflectivity coefficient and the TE+ power transmission coefficient as a functions of the plane-wave incident angle.

Fig. 5
Fig. 5

Four-wave coupling for TE+ incident plane wave at θi=θe=60° (waveguide grating I). TE+ (|R(z)|2), TE- (|S(z)|2), TM+ (|T(z)|2), and TM- (|U(z)|2), intensity variations along the grating.

Fig. 6
Fig. 6

Four-wave coupling (waveguide grating I) for TE+ incident plane wave. (a) Reflection coefficients, Re-(θi), Rm-(θi). (b) Transmission coefficients, Te+(θi), Tm+(θi).

Fig. 7
Fig. 7

Difference θm-θe as a function of θi=θe (waveguide grating I).

Fig. 8
Fig. 8

Four-wave coupling (waveguide grating II). (a) Reflection coefficient, TE-, Re-(θi). (b) Reflection coefficient, TM-, Rm-(θi). (c) Transmission coefficient, TE+, Te+(θi). (d) Transmission coefficient, TM+, Tm+(θi).

Fig. 9
Fig. 9

Four-wave coupling for TM+ incident plane wave at θi=θm=60° (waveguide grating I). TE+ |R(z)|2), TE- (|S(z)|2), TM+ (|T(z)|2), and TM- (|U(z)|2) intensity variations along the grating.

Fig. 10
Fig. 10

Four-wave coupling (waveguide grating I) for TM+ incident plane wave. (a) Reflection coefficients, Re-(θi), Rm-(θi). (b) Transmission coefficients, Te+(θi), Tm+(θi).

Tables (1)

Tables Icon

Table 1 Waveguides (WG. I, II) and Their Gratings (First Order q=1)

Equations (32)

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

βi+qG=βd,
βi cos θi+βd cos θd=q 2πΛ
βi sin θi=βd sin θd.
2δab=βa cos θa+βb cos θb-q 2πΛ.
θm=sin-1βeβmsin(θe).
δee=βe cos θe-(qπ/Λ),
δmm=βm cos θm-(qπ/Λ),
δem=12βe cos θe+βm cos θm-q2πΛ=δee+δmm2.
E(r)=Et(r)+Ez(r),
Et(r)ae+(y, z)Ete(+)(x)+ae-(y, z)Ete(-)(x)+am+(y, z)Etm(+)(x)+am-(y, z)Etm(-)(x),
aj±(y, z)Aj±(z)exp[-i(βjy sin θj±βjz cos θj)],
βe cos(θe)2ωμ-|E0(x)|2dx=1,TEmodes
βm cos(θm)2ω0-|H0(x)|2/n2(x)dx=1,TMmodes
ddzAe+(z)Ae-(z)Am+(z)Am-(z)=QAe+(z)Ae-(z)Am+(z)Am-(z),
Q0κee exp(i2δeez)0κem exp(i2δemz)κee* exp(-i2δeez)0κme* exp(-i2δemz)00κme exp(i2δemz)0κmm exp(i2δmmz)κem* exp(-i2δemz)0κmm* exp(-i2δmmz)0,
R(z)Ae+(z)exp(-iδeez),
S(z)Ae-(z)exp(iδeez),
T(z)Am+(z)exp(-iδmmz),
U(z)Am-(z)exp(iδmmz),
ddzR(z)S(z)T(z)U(z)=MR(z)S(z)T(z)U(z),
M-iδeeκee0κemκee*iδeeκme*00κme-iδmmκmmκem*0κmm*iδmm.
dR(z)dz=-iδR(z)+κS(z),
-dS(z)dz=-iδS(z)-κ*R(z),
R(z)=i=14CiV1(λi)exp(λiz),
S(z)=i=14CiV2(λi)exp(λiz),
T(z)=i=14CiV3(λi)exp(λiz),
U(z)=i=14CiV4(λi)exp(λiz),
R(0)=Ai=1S(L)=T(0)=U(L)=0,
Re-|S(0)/R(0)|2,TEr-/TEi+,
Rm-|U(0)/R(0)|2,TMr-/TEi+,
Te+|R(L)/R(0)|2,TEt+/TEi+,
Tm+|T(L)/R(0)|2,TMt+/TEi+.

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