Abstract

Super luminescent Diode (SLD) with a new structure is proposed in which light is guided by surface plasmon waveguide (SPWG) rather than by the conventional dielectric waveguide. This results in a great increase of the spontaneous emission coupling. Other parameters important to the device operation such as the confinement factor, waveguide loss and waveguide facets reflectivities are also considered. It is shown that the new design outperforms the conventional ones using dielectric waveguides in both the output power and optical spectral width.

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2007

2006

2005

J. W. Park, X. Li, and W.-P. Huang, “Comparative study on mixed frequency–time–domain models of semiconductor laser optical amplifiers,” Optoelectronics 152(3), 151–159 (2005).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

2000

M. J. Hamp and D. T. Cassidy, “Critical design parameters for engineering broadly tunable asymmetric multiple-quantum-well lasers,” J. Quantum Electron. 36(8), 978–983 (2000).
[CrossRef]

C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express 7(8), 260–272 (2000).
[CrossRef] [PubMed]

1999

W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from the emissive devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999).
[CrossRef]

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

1998

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

1997

C.-F. Lin and B.-L. Lee, “Extremely broadband AlGaAs/GaAs superluminescent diodes,” Appl. Phys. Lett. 71(12), 1598–1600 (1997).
[CrossRef]

1992

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10(10), 1344–1351 (1992).
[CrossRef]

1991

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

1988

D. Ahn and S. L. Chuang, “A field-effect quantum-well laser with lateral current injection,” J. Appl. Phys. 64(1), 440–442 (1988).
[CrossRef]

1986

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal film,” Phys. Rev. B 33(8), 5186–5201 (1986).
[CrossRef]

1984

1983

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Surface polariton reflection and radiation at end facets,” Appl. Phys. Lett. 42(9), 764–766 (1983).
[CrossRef]

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1(1), 98–105 (1983).
[CrossRef]

1978

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978) (and references therein).
[CrossRef]

Ahn, D.

D. Ahn and S. L. Chuang, “A field-effect quantum-well laser with lateral current injection,” J. Appl. Phys. 64(1), 440–442 (1988).
[CrossRef]

Aitchison, J. S.

Alam, M. Z.

Anemogiannis, E.

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10(10), 1344–1351 (1992).
[CrossRef]

Barnes, W. L.

Berini, P.

Boroditsky, M.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal film,” Phys. Rev. B 33(8), 5186–5201 (1986).
[CrossRef]

Burns, W. K.

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1(1), 98–105 (1983).
[CrossRef]

Cassidy, D. T.

M. J. Hamp and D. T. Cassidy, “Critical design parameters for engineering broadly tunable asymmetric multiple-quantum-well lasers,” J. Quantum Electron. 36(8), 978–983 (2000).
[CrossRef]

Chance, R. R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978) (and references therein).
[CrossRef]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, C.

Chen, C.-L.

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1(1), 98–105 (1983).
[CrossRef]

Chilwell, J.

Chuang, S. L.

D. Ahn and S. L. Chuang, “A field-effect quantum-well laser with lateral current injection,” J. Appl. Phys. 64(1), 440–442 (1988).
[CrossRef]

DenBaars, S. P.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Deveaud, B.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Dupertuis, M.-A.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Feng, D.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Ford, G. W.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113(4), 195–287 (1984).
[CrossRef]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Glytsis, E. N.

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10(10), 1344–1351 (1992).
[CrossRef]

Gontijo, I.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Haacke, S.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Hamp, M. J.

M. J. Hamp and D. T. Cassidy, “Critical design parameters for engineering broadly tunable asymmetric multiple-quantum-well lasers,” J. Quantum Electron. 36(8), 978–983 (2000).
[CrossRef]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Hessler, T.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Hodgkinson, I.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, W.-P.

J. W. Park, X. Li, and W.-P. Huang, “Comparative study on mixed frequency–time–domain models of semiconductor laser optical amplifiers,” Optoelectronics 152(3), 151–159 (2005).
[CrossRef]

Keller, S.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Lee, B.-L.

C.-F. Lin and B.-L. Lee, “Extremely broadband AlGaAs/GaAs superluminescent diodes,” Appl. Phys. Lett. 71(12), 1598–1600 (1997).
[CrossRef]

Li, X.

J. W. Park and X. Li, “Theoretical and numerical analysis of superluminescent diodes,” J. Lightwave Technol. 24(6), 2473–2480 (2006).
[CrossRef]

J. W. Park, X. Li, and W.-P. Huang, “Comparative study on mixed frequency–time–domain models of semiconductor laser optical amplifiers,” Optoelectronics 152(3), 151–159 (2005).
[CrossRef]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Lin, C.-F.

C.-F. Lin and B.-L. Lee, “Extremely broadband AlGaAs/GaAs superluminescent diodes,” Appl. Phys. Lett. 71(12), 1598–1600 (1997).
[CrossRef]

Maradudin, A. A.

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Surface polariton reflection and radiation at end facets,” Appl. Phys. Lett. 42(9), 764–766 (1983).
[CrossRef]

Meier, J.

Mishra, U. K.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Moeller, R. P.

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1(1), 98–105 (1983).
[CrossRef]

Mojahedi, M.

Park, J. W.

J. W. Park and X. Li, “Theoretical and numerical analysis of superluminescent diodes,” J. Lightwave Technol. 24(6), 2473–2480 (2006).
[CrossRef]

J. W. Park, X. Li, and W.-P. Huang, “Comparative study on mixed frequency–time–domain models of semiconductor laser optical amplifiers,” Optoelectronics 152(3), 151–159 (2005).
[CrossRef]

Pleumeekers, J. L.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Prock, A.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978) (and references therein).
[CrossRef]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Selbmann, P. E.

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Silbey, R.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978) (and references therein).
[CrossRef]

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal film,” Phys. Rev. B 33(8), 5186–5201 (1986).
[CrossRef]

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Surface polariton reflection and radiation at end facets,” Appl. Phys. Lett. 42(9), 764–766 (1983).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal film,” Phys. Rev. B 33(8), 5186–5201 (1986).
[CrossRef]

Tanev, S.

Tzolov, V.

Wallis, R. F.

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Surface polariton reflection and radiation at end facets,” Appl. Phys. Lett. 42(9), 764–766 (1983).
[CrossRef]

Weber, W. H.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113(4), 195–287 (1984).
[CrossRef]

Yablonovitch, E.

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

Adv. Chem. Phys.

R. R. Chance, A. Prock, and R. Silbey, “Molecular fluorescence and energy transfer near interfaces,” Adv. Chem. Phys. 37, 1–65 (1978) (and references therein).
[CrossRef]

Appl. Phys. Lett.

R. F. Wallis, A. A. Maradudin, and G. I. Stegeman, “Surface polariton reflection and radiation at end facets,” Appl. Phys. Lett. 42(9), 764–766 (1983).
[CrossRef]

C.-F. Lin and B.-L. Lee, “Extremely broadband AlGaAs/GaAs superluminescent diodes,” Appl. Phys. Lett. 71(12), 1598–1600 (1997).
[CrossRef]

J. Appl. Phys.

D. Ahn and S. L. Chuang, “A field-effect quantum-well laser with lateral current injection,” J. Appl. Phys. 64(1), 440–442 (1988).
[CrossRef]

J. Lightwave Technol.

W. L. Barnes, “Electromagnetic crystals for surface plasmon polaritons and the extraction of light from the emissive devices,” J. Lightwave Technol. 17(11), 2170–2182 (1999).
[CrossRef]

J. W. Park and X. Li, “Theoretical and numerical analysis of superluminescent diodes,” J. Lightwave Technol. 24(6), 2473–2480 (2006).
[CrossRef]

E. Anemogiannis and E. N. Glytsis, “Multilayer waveguides: efficient numerical analysis of general structures,” J. Lightwave Technol. 10(10), 1344–1351 (1992).
[CrossRef]

W. K. Burns, C.-L. Chen, and R. P. Moeller, “Fiber-optic gyroscopes with broad-band sources,” J. Lightwave Technol. 1(1), 98–105 (1983).
[CrossRef]

J. Opt. Soc. Am. A

J. Quantum Electron.

M. J. Hamp and D. T. Cassidy, “Critical design parameters for engineering broadly tunable asymmetric multiple-quantum-well lasers,” J. Quantum Electron. 36(8), 978–983 (2000).
[CrossRef]

J. L. Pleumeekers, M.-A. Dupertuis, T. Hessler, P. E. Selbmann, S. Haacke, and B. Deveaud, “Longitudinal spatial hole burning and associated nonlinear gain in gain-clamped semiconductor optical amplifiers,” J. Quantum Electron. 34(5), 879–886 (1998).
[CrossRef]

Nature

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[CrossRef] [PubMed]

Opt. Express

Optoelectronics

J. W. Park, X. Li, and W.-P. Huang, “Comparative study on mixed frequency–time–domain models of semiconductor laser optical amplifiers,” Optoelectronics 152(3), 151–159 (2005).
[CrossRef]

Phys. Rep.

G. W. Ford and W. H. Weber, “Electromagnetic interactions of molecules with metal surfaces,” Phys. Rep. 113(4), 195–287 (1984).
[CrossRef]

Phys. Rev. B

I. Gontijo, M. Boroditsky, E. Yablonovitch, S. Keller, U. K. Mishra, and S. P. DenBaars, “Coupling of InGaN quantum-well photoluminescence to silver surface plasmons,” Phys. Rev. B 60(16), 11564–11567 (1999).
[CrossRef]

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal film,” Phys. Rev. B 33(8), 5186–5201 (1986).
[CrossRef]

Science

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[CrossRef] [PubMed]

Other

X. Li and M. Ranjbaran, “Performance enhancement of superluminescent light emitting diode built on surface plasmonic waveguide,” Invited paper, Photonics and Optoelectronics Meetings (POEM2009), Wuhan, P. R. China (Aug. 2009).

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).

C. T. Tai, Dyadic Green's Function in Electromagnetic Theory (Oxford University Press, 1996).

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[CrossRef]

G. P. Agrawal, and N. K. Dutta, Semiconductor Lasers (Springer, 1993).

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Figures (7)

Fig. 2
Fig. 2

Maximum allowable asymmetry vs. metal thickness for the structure of Fig. 1 to support the sb mode. ε 3 = 11.2 while ε 1 is reduced to make the SPWG asymmetric.

Fig. 3
Fig. 3

Minimum superstrate layer thickness required to support the sb mode as a function of film thickness. ε 1 = ε 3 = 11.2 and λ = 1.55 μ m ; inset: structure with superstrate layer (not to be scaled) and the profile of Hy component of the symmetric mode.

Fig. 1
Fig. 1

(a) Real and (b) imaginary parts of the propagation constants of bounded symmetric (sb) and antisymmetric (ab) modes of a silver slab SPWG. λ = 1.55 μ m , . ε 2 = 116.38 + i 11.1 ., ε 1 = ε 3 = 11.2 .

Fig. 4
Fig. 4

Normalized nonradiative decay rate of an isotropic dipole near a silver film as a function of normalized distance (film thickness divided by the dipole distance from the film).

Fig. 5
Fig. 5

Coupling percentage versus dipole distance, dashed line: percentage of dipole power coupled to surface plasmon modes; Dotted line: percentage of surface plasmon coupled energy that goes to the long-range mode and solid line: the product of the previous two curves for a 5 nm thick silver film at 1.55 μ m embedded in dielectric with ε = 11.2 .

Fig. 6
Fig. 6

Schematic cross section of the proposed SLD structure with SPWG and lateral current injection

Fig. 7
Fig. 7

(a)-(c), respectively, facet power, spectral width and their product in an SLD with dielectric waveguide for two values of loss (cm−1); (d)-(f), corresponding graphs for a device with SPWG with different values of loss.

Equations (3)

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P p = 2 β n s p h c 2 λ p 3 exp { α L + G ( N ) }
tanh ( S 2 h ) ( ε 1 ε 3 S 2 2 + ε 2 2 S 1 S 3 ) + ε 2 S 2 ( ε 1 S 3 + ε 3 S 1 ) = 0
Γ       =         Re ( A . R . E × H * z ^     d x     ) Re (         A . S . E × H * z ^     d x )

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