Abstract

Polarization-dependent spontaneous emission (SE) from multiple quantum wells (MQWs) sandwiched between a high-reflectivity distributed Bragg reflector (DBR) and a low-reflectivity surface of about 30% reflectance was investigated. In photoluminescence spectra, a split of the p- and s-polarized emission peaks was observed at the DBR band edges, while emission was completely suppressed in its bandgap. Theoretical analysis of the SE rate, based on quantum electrodynamics, explains well the experimental observations; that is, SE enhancement ratios of polarized light can be varied drastically by shifting the position of the low-reflectivity surface that modifies the nodal vacuum fluctuations in front of the high reflector.

© 2014 Optical Society of America

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2012

2010

2009

M. Liscidini, M. Galli, M. Shi, G. Dacarro, M. Patrini, D. Bajoni, and J. E. Sipe, “Strong modification of light emission from a dye monolayer via Bloch surface waves,” Opt. Lett. 34, 2318–2320 (2009).
[CrossRef]

A. Tandaechanurat, S. Ishida, K. Aoki, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Demonstration of high-Q (>8600) three-dimensional photonic crystal nanocavity embedding quantum dots,” Appl. Phys. Lett. 94, 171115 (2009).
[CrossRef]

2008

M. Makarova, V. Sih, J. Warga, R. Li, L. D. Negro, and J. Vučković, “Enhanced light emission in photonic crystal nanocavities with erbium-doped silicon nanocrystals,” Appl. Phys. Lett. 92, 161107 (2008).
[CrossRef]

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16, 426–434 (2008).
[CrossRef]

2005

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6, 3–9 (2005).
[CrossRef]

2004

J. Dong, J. Teng, S. Chua, B. Foo, Y. Wang, L. Zhang, H. Yuan, and S. Yuan, “Continuous-wave operation of AlGaInP/GaInP quantum-well lasers grown by metalorganic chemical vapor deposition using tertiary butylphosphine,” J. Appl. Phys. 95, 5252–5254 (2004).
[CrossRef]

J. P. Reithmaier, G. Sęk, A. Löffler, C. Hofmann, S. Kuhn, S. Reitzenstein, L. V. Keldysh, V. D. Kulakovskii, T. L. Reinecke, and A. Forchel, “Strong coupling in a single quantum dot–semiconductor microcavity system,” Nature 432, 197–200 (2004).
[CrossRef]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

2003

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67, 033806 (2003).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

K. J. Vahala, “Optical microcavities,” Nature 424, 839–846 (2003).
[CrossRef]

T. Baba and D. Sano, “Low-threshold lasing and Purcell effect in microdisk lasers at room temperature,” IEEE J. Sel. Topics Quantum Electron. 9, 1340–1346 (2003).
[CrossRef]

2001

J. R. Lakowicz, “Radiative decay engineering: biophysical and biomedical applications,” Anal. Biochem. 298, 1–24 (2001).
[CrossRef]

2000

Y. Xu, R. K. Lee, and A. Yariv, “Quantum analysis and the classical analysis of spontaneous emission in a microcavity,” Phys. Rev. A 61, 033807 (2000).
[CrossRef]

C. Begon, H. Rigneault, P. Jonsson, and J. G. Rarity, “Spontaneous emission control with planar dielectric structures: an asset for ultrasensitive fluorescence analysis,” Single Mol. 1, 207–214 (2000).
[CrossRef]

1999

Y. Xu, J. S. Vučković, R. K. Lee, O. J. Painter, A. Scherer, and A. Yariv, “Finite-difference time-domain calculation of spontaneous emission lifetime in a microcavity,” J. Opt. Soc. Am. B 16, 465–474 (1999).
[CrossRef]

T. M. Ritter, B. A. Weinstein, R. E. Viturro, and D. P. Bour, “Energy level alignments in strained-layer GaInP/AlGaInP laser diodes: model solid theory analysis of pressure-photoluminescence experiments,” Phys. Status Solidi B 211, 869–883 (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, 11564–11567 (1999).

B. Gayral, J. M. Gérard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

1998

H. P. Urbach and G. L. J. A. Rikken, “Spontaneous emission from a dielectric slab,” Phys. Rev. A 57, 3913–3930 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction-Part I: basic concepts and analytical trends,” IEEE J. Quantum Electron. 34, 1612–1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction–Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632–1643 (1998).
[CrossRef]

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
[CrossRef]

1996

H. Rigneault and S. Monneret, “Modal analysis of spontaneous emission in a planar microcavity,” Phys. Rev. A 54, 2356–2368 (1996).
[CrossRef]

1995

J. J. Baumberg, T. A. Kelf, Y. Sugawara, S. Cintra, M. E. Abdelsalam, P. N. Bartlett, and A. E. Russell, “Angle-resolved surface-enhanced Raman scattering on metallic nanostructured plasmonic crystals,” Nano Lett. 5, 2262–2267 (1995).

1994

1992

H. Khosravi and R. Loudon, “Vacuum field fluctuations and spontaneous emission in a dielectric slab,” Proc. R. Soc. A 436, 373–389 (1992).

1991

G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669–681 (1991).
[CrossRef]

H. Khosravi and R. Loudon, “Vacuum field fluctuations and spontaneous emission in the vicinity of a dielectric surface,” Proc. R. Soc. A 433, 337–352 (1991).
[CrossRef]

1988

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

1984

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

1978

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

1970

H. Kuhn, “Classical aspects of energy transfer in molecular systems,” J. Chem. Phys. 53, 101–108 (1970).
[CrossRef]

1946

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681 (1946).
[CrossRef]

Abdelsalam, M. E.

J. J. Baumberg, T. A. Kelf, Y. Sugawara, S. Cintra, M. E. Abdelsalam, P. N. Bartlett, and A. E. Russell, “Angle-resolved surface-enhanced Raman scattering on metallic nanostructured plasmonic crystals,” Nano Lett. 5, 2262–2267 (1995).

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Aoki, K.

A. Tandaechanurat, S. Ishida, K. Aoki, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Demonstration of high-Q (>8600) three-dimensional photonic crystal nanocavity embedding quantum dots,” Appl. Phys. Lett. 94, 171115 (2009).
[CrossRef]

Arakawa, Y.

A. Tandaechanurat, S. Ishida, K. Aoki, D. Guimard, M. Nomura, S. Iwamoto, and Y. Arakawa, “Demonstration of high-Q (>8600) three-dimensional photonic crystal nanocavity embedding quantum dots,” Appl. Phys. Lett. 94, 171115 (2009).
[CrossRef]

Asano, T.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6, 3–9 (2005).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Baba, T.

T. Baba and D. Sano, “Low-threshold lasing and Purcell effect in microdisk lasers at room temperature,” IEEE J. Sel. Topics Quantum Electron. 9, 1340–1346 (2003).
[CrossRef]

Bajoni, D.

Barnes, W. L.

W. L. Barnes, “Fluorescence near interfaces: the role of photonic mode density,” J. Mod. Opt. 45, 661–699 (1998).
[CrossRef]

Bartlett, P. N.

J. J. Baumberg, T. A. Kelf, Y. Sugawara, S. Cintra, M. E. Abdelsalam, P. N. Bartlett, and A. E. Russell, “Angle-resolved surface-enhanced Raman scattering on metallic nanostructured plasmonic crystals,” Nano Lett. 5, 2262–2267 (1995).

Baumberg, J. J.

J. J. Baumberg, T. A. Kelf, Y. Sugawara, S. Cintra, M. E. Abdelsalam, P. N. Bartlett, and A. E. Russell, “Angle-resolved surface-enhanced Raman scattering on metallic nanostructured plasmonic crystals,” Nano Lett. 5, 2262–2267 (1995).

Begon, C.

C. Begon, H. Rigneault, P. Jonsson, and J. G. Rarity, “Spontaneous emission control with planar dielectric structures: an asset for ultrasensitive fluorescence analysis,” Single Mol. 1, 207–214 (2000).
[CrossRef]

Benisty, H.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction-Part I: basic concepts and analytical trends,” IEEE J. Quantum Electron. 34, 1612–1631 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction–Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632–1643 (1998).
[CrossRef]

Bhat, R.

E. Yablonovitch, T. J. Gmitter, and R. Bhat, “Inhibited and enhanced spontaneous emission from optically thin AlGaAs/GaAs double heterostructures,” Phys. Rev. Lett. 61, 2546–2549 (1988).
[CrossRef]

Björk, G.

G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669–681 (1991).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1964).

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, 11564–11567 (1999).

Bour, D. P.

T. M. Ritter, B. A. Weinstein, R. E. Viturro, and D. P. Bour, “Energy level alignments in strained-layer GaInP/AlGaInP laser diodes: model solid theory analysis of pressure-photoluminescence experiments,” Phys. Status Solidi B 211, 869–883 (1999).
[CrossRef]

Buck, J. R.

J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67, 033806 (2003).
[CrossRef]

Burke, J. J.

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).
[CrossRef]

Chua, S.

J. Dong, J. Teng, S. Chua, B. Foo, Y. Wang, L. Zhang, H. Yuan, and S. Yuan, “Continuous-wave operation of AlGaInP/GaInP quantum-well lasers grown by metalorganic chemical vapor deposition using tertiary butylphosphine,” J. Appl. Phys. 95, 5252–5254 (2004).
[CrossRef]

Chutinan, A.

A. Chutinan, K. Ishihara, T. Asano, M. Fujita, and S. Noda, “Theoretical analysis on light-extraction efficiency of organic light-emitting diodes using FDTD and mode-expansion methods,” Org. Electron. 6, 3–9 (2005).
[CrossRef]

Cintra, S.

J. J. Baumberg, T. A. Kelf, Y. Sugawara, S. Cintra, M. E. Abdelsalam, P. N. Bartlett, and A. E. Russell, “Angle-resolved surface-enhanced Raman scattering on metallic nanostructured plasmonic crystals,” Nano Lett. 5, 2262–2267 (1995).

Coldren, L. A.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Corzine, S. W.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

Dacarro, G.

De Neve, H.

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction–Part II: selected exact simulations and role of photon recycling,” IEEE J. Quantum Electron. 34, 1632–1643 (1998).
[CrossRef]

H. Benisty, H. De Neve, and C. Weisbuch, “Impact of planar microcavity effects on light extraction-Part I: basic concepts and analytical trends,” IEEE J. Quantum Electron. 34, 1612–1631 (1998).
[CrossRef]

Delfan, A.

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, 11564–11567 (1999).

Deppe, D. G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Dodabalapur, A.

A. Dodabalapur, L. J. Rothberg, and T. M. Miller, “Color variation with electroluminescent organic semiconductors in multimode resonant cavities,” Appl. Phys. Lett. 65, 2308–2310 (1994).
[CrossRef]

Dong, J.

J. Dong, J. Teng, S. Chua, B. Foo, Y. Wang, L. Zhang, H. Yuan, and S. Yuan, “Continuous-wave operation of AlGaInP/GaInP quantum-well lasers grown by metalorganic chemical vapor deposition using tertiary butylphosphine,” J. Appl. Phys. 95, 5252–5254 (2004).
[CrossRef]

Dupuis, C.

B. Gayral, J. M. Gérard, A. Lemaître, C. Dupuis, L. Manin, and J. L. Pelouard, “High-Q wet-etched GaAs microdisks containing InAs quantum boxes,” Appl. Phys. Lett. 75, 1908–1910 (1999).
[CrossRef]

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. M. Gibbs, G. Rupper, C. Ell, O. B. Shchekin, and D. G. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Englund, D.

H. Iwase, D. Englund, and J. Vučković, “Analysis of the Purcell effect in photonic and plasmonic crystals with losses,” Opt. Express 18, 16546–16560 (2010).
[CrossRef]

H. Iwase, D. Englund, and J. Vučković, “Spontaneous emission control in high-extraction efficiency plasmonic crystals,” Opt. Express 16, 426–434 (2008).
[CrossRef]

H. Iwase, Y. Gong, D. Englund, and J. Vučković, “Spontaneous emission control in a plasmonic structure,” in Nanoscale Photonics and Optoelectronics: Lecture Notes in Nanoscale Science and Technology, Z. M. Wang and A. Neogi, eds., (Springer-Verlag, 2010).

Foo, B.

J. Dong, J. Teng, S. Chua, B. Foo, Y. Wang, L. Zhang, H. Yuan, and S. Yuan, “Continuous-wave operation of AlGaInP/GaInP quantum-well lasers grown by metalorganic chemical vapor deposition using tertiary butylphosphine,” J. Appl. Phys. 95, 5252–5254 (2004).
[CrossRef]

Forchel, A.

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Figures (11)

Fig. 1.
Fig. 1.

a) MQW sandwiched between a DBR and an output plane. b) PL spectra from the MQW without the DBR. The emission angles θ are set to be 0° (top) and 60° (bottom). The dashed lines in the top plots show Gaussian fits to the spectrum, which, respectively, correspond to the HH and LH transitions.

Fig. 2.
Fig. 2.

a) Reflection and emission properties of Structure A. b) Reflection and emission properties of Structure B. From top to bottom are plotted the reflectance, the PL spectrum at θ=0°, and the PL spectra of s- and p-polarized emission at θ=60°.

Fig. 3.
Fig. 3.

a) Reflection and emission properties of Structure C. From top to bottom are plotted the reflectance, the PL spectrum at θ=0°, and the PL spectra of s- and p-polarized emission at θ=60°. b) Magnified plots of PL spectra of s-polarized light (dashed line) and p-polarized light (solid line) at θ=60°.

Fig. 4.
Fig. 4.

a) Geometry of a single HR. The filled arrows and open arrows represent light incident from the +z-direction and from the z-direction, respectively. The unit vectors es,β,θ and es,β,θ, respectively, represent the directions of the electric fields of incident and reflected light for s-polarization, and ep,β,θ and ep,β,θ represent the corresponding directions for p-polarization. The symbols r± and t±, respectively, represent the reflection and transmission coefficients. b) Plots of fs,(β,ν) with β=0 for emitters located at z=0.67λH/n (solid line) and at z=(0.67+0.25)λH/n (dashed line).

Fig. 5.
Fig. 5.

a) Geometry of a slab sandwiched between a HR and an output plane. The symbols r± and t±, respectively, represent the reflection and transmission coefficients at the HR, and R± and T± represent the corresponding coefficients at the output plane. b) Directions of electric fields eβm and propagation vectors β for TE modes (left) and TM modes (right). Unit vectors e and e show the direction of an electric dipole μ; μ=μe+μe. B is a magnetic field of TM modes, parallel to the slab.

Fig. 6.
Fig. 6.

a) Plots of fs,(β,ν) for Structure A. b) Plots of fp,(β,ν) for Structure A. The values are plotted only over the light line.

Fig. 7.
Fig. 7.

a) PL spectra of s-polarized light for several angles from 0° to 60° in Structure B. b) Plot of fs,(β,ν) for Structure B. The values are plotted only over the light line. The dashed line across the plots in (a) represents the frequencies corresponding to the trough of fs,(β,ν).

Fig. 8.
Fig. 8.

a) PL spectra of p-polarized light for several angles from 0° to 60° in Structure B. b) Plot of fp,(β,ν) for Structure B. c) Plot of fp,(β,ν) for Structure B. The values are plotted only over the light line. The dashed line across the plots in (a) represents the frequencies corresponding to the trough of fp,(β,ν).

Fig. 9.
Fig. 9.

a) PL spectra of s-polarized light for several angles from 0° to 60° in Structure C. b) Plot of fs,(β,ν) for Structure C. The values are plotted only over the light line.

Fig. 10.
Fig. 10.

a) PL spectra of p-polarized light for several angles from 0° to 60° in Structure C. b) Plot of fp,(β,ν) for Structure C. c) Plot of fp,(β,ν) for Structure C. The values are plotted only over the light line.

Fig. 11.
Fig. 11.

a) PL spectra of s-polarized light (dashed line) and of p-polarized light (solid line) emitted at an angle of 60° in Structure C. The white and black arrows, respectively, show the emission peaks of s- and p-polarized light. b) Theoretical reflectance (top) and parallel electric field intensity |E|2 (bottom) at the position of the emitter. The white and black arrows indicate reflectance dips and peaks of |E|2, respectively, corresponding to the PL peaks of s- and p-polarized light.

Equations (26)

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Ej,β,ω+(r)={E0ei(β·xqz)ej,β,θ+rj,β,ω+E0ei(β·x+qz)ej,β,θz0tj,β,ω+E0ei(β·xqz)ej,β,θzΔ,
Ej,β,ω(r)={tj,β,ωE0ei(β·x+qz)ej,β,θz0E0ei(β·x+qz)ej,β,θ+rj,β,ωE0ei(β·xqz)ej,β,θzΔ,
Ψj,β,ω+(r)=12{Ej,β,ω+(r)+ieiφEj,β,ω(r)},
Ψj,β,ω(r)=12{[Ej,β,ω+(r)]*+ieiφ[Ej,β,ω(r)]*},
Γ(ν)=j=s,pβγj(β,ν),
γj(β,ν)=qπω[|μ·Ψj,β,ω+(re)|2+|μ·Ψj,β,ω(re)|2]δ(ων),
F(ν)ΓnΓ0=(μμ)2[Fs,(ν)+Fp,(ν)]+(μμ)2Fp,(ν),
Fj,a(ν)=2πβfj,a(β,ν)dβ.
2πβfs,(β,ν)=34uk1u2+34Re[rs,β,ν+ei2qduk1u2],
2πβfp,(β,ν)=34(1u2)uk1u2+34Re[rp,β,ν+ei2qd(1u2)uk1u2],
2πβfp,(β,ν)=32u3k1u232Re[rp,β,ν+ei2qdu3k1u2],
F(ν)=(μμ)2[Fs,(ν)+Fp,(ν)]+(μμ)2Fp,(ν)+Θ(ν),
2πβfs,(β,ν)=34Re[G(rs,β,ν+,Rs,β,ν)uk1u2],
2πβfp,(β,ν)=34Re[G(rp,β,ν+,Rp,β,ν)(1u2)uk1u2],
2πβfp,(β,ν)=32Re[G(rp,β,ν+,Rp,β,ν)u3k1u2],
G(x,y)=(1+xei2qd)(1+yei2q(ld))1xyei2ql.
Θm(ν)=3πc34nν2βvgm(μμ)2|Eβm(d)|2Lβm,
Θm(ν)=3πc34nν2βvgm[(μμ)2|Eβm(d)cosθβm|2Lβm+2(μμ)2|Eβm(d)sinθβm|2Lβm],
Es,β,ω+(r)=T+eiqz(1+r+ei2qz)1Rr+ei2qleiβ·xe1,
Es,β,ω(r)=teiqz(1+Rei2q(lz))1Rr+ei2qleiβ·xe1,
Ep,β,ω+(r)=T+eiqz(1+r+ei2qz)1Rr+ei2qleiβ·x1u2e2+T+eiqz(1r+ei2qz)1Rr+ei2qleiβ·xue3,
Ep,β,ω(r)=teiqz(1+Rei2q(lz))1Rr+ei2qleiβ·x1u2e2+teiqz(1Rei2q(lz))1Rr+ei2qleiβ·xu(e3).
βqδ(ων)1(2π)3dβdφdq[βδ(ων)],
γm=βπωβm|μ·Ψβm(re)|2δ(ωβmν)(12π)2πν02πdϕ[β|μ·Ψβm(re)|21vgm]ωβm=ν,
|μ·Ψβm(re)|2=(μ)2|Eβm(d)|2sin2ϕ/ε0Lβm,
|μ·Ψβm(re)|2=(μ)2|Eβm(d)cosθβm|2cos2ϕ/ε0Lβm+(μ)2|Eβm(d)sinθβm|2/ε0Lβm,

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