Abstract

We present calculations of the modification of the spontaneous emission rate from a point source dipole in a Fabry-Perot microcavity containing an optically thin dielectric aperture. The dielectric aperture is described as a passive current source which is driven by the spontaneous point source. The spontaneous emission rate is shown to depend on the details of the aperture design, and shows a strong enhancement on resonance due to 3-dimensional optical confinement by the dielectric aperture.

© 1998 Optical Society of America

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  1. D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
    [CrossRef]
  2. G. Bjork, 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] [PubMed]
  3. K. Ujihara, “Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors,” Jpn. J. Appl. Phys., Part 2 30, L901–L903 (1991).
    [CrossRef]
  4. N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
    [CrossRef]
  5. D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
    [CrossRef]
  6. C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
    [CrossRef]
  7. G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. 30, 2314–2318 (1994).
    [CrossRef]
  8. C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. 75, 4668–4672 (1994).
    [CrossRef]
  9. Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A 53, 1036–1047 (1996).
    [CrossRef] [PubMed]
  10. D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
    [CrossRef]
  11. J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
    [CrossRef]
  12. D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. 67, 2494–2596 (1995).
    [CrossRef]
  13. D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 160–162 (1997).
    [CrossRef]
  14. Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33, 2319–2326 (1997).
    [CrossRef]
  15. D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
    [CrossRef]
  16. R.F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York,1961) pg. 188.

1997 (3)

D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 160–162 (1997).
[CrossRef]

Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33, 2319–2326 (1997).
[CrossRef]

D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
[CrossRef]

1996 (1)

Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A 53, 1036–1047 (1996).
[CrossRef] [PubMed]

1995 (1)

D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. 67, 2494–2596 (1995).
[CrossRef]

1994 (4)

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
[CrossRef]

G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. 30, 2314–2318 (1994).
[CrossRef]

C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. 75, 4668–4672 (1994).
[CrossRef]

1992 (1)

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

1991 (4)

D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

G. Bjork, 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] [PubMed]

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors,” Jpn. J. Appl. Phys., Part 2 30, L901–L903 (1991).
[CrossRef]

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

1990 (1)

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Bjork, G.

G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. 30, 2314–2318 (1994).
[CrossRef]

G. Bjork, 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] [PubMed]

Dallesasse, J.M.

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Deng, Q.

D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 160–162 (1997).
[CrossRef]

Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33, 2319–2326 (1997).
[CrossRef]

Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A 53, 1036–1047 (1996).
[CrossRef] [PubMed]

Deppe, D.G.

D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 160–162 (1997).
[CrossRef]

D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
[CrossRef]

Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33, 2319–2326 (1997).
[CrossRef]

Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A 53, 1036–1047 (1996).
[CrossRef] [PubMed]

D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. 67, 2494–2596 (1995).
[CrossRef]

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
[CrossRef]

C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. 75, 4668–4672 (1994).
[CrossRef]

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

Dupuis, R.D.

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

El-Zein, N.

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Harrington, R.F.

R.F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York,1961) pg. 188.

Holonyak, N.

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Honda, Y.

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

Huang, Z.

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

Huffaker, D.L.

D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
[CrossRef]

D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. 67, 2494–2596 (1995).
[CrossRef]

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

Igeta, K.

G. Bjork, 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] [PubMed]

Kumar, K.

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

Lei, C.

C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
[CrossRef]

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

Lin, C.C.

C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. 75, 4668–4672 (1994).
[CrossRef]

C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
[CrossRef]

Machida, S.

G. Bjork, 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] [PubMed]

Neff, J.G.

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

Ochi, N.

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

Oh, T.-H.

D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
[CrossRef]

Pinzone, C.J.

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

Richard, T.A.

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Rogers, T.J.

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

Shiotani, T.

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

Suemune, I.

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

Sugg, A.R.

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

Ujihara, K.

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors,” Jpn. J. Appl. Phys., Part 2 30, L901–L903 (1991).
[CrossRef]

Yamamoto, Y.

G. Bjork, 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] [PubMed]

Yaminishi, M.

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

Appl. Phys. Lett. (6)

N. Ochi, T. Shiotani, M. Yaminishi, Y. Honda, and I. Suemune, “Controllable enhancement of excitonic spontaneous emission in quantum microcavities,” Appl. Phys. Lett. 58, 2735–2737 (1991).
[CrossRef]

D.L. Huffaker, Z. Huang, C. Lei, D.G. Deppe, C.J. Pinzone, J.G. Neff, and R.D. Dupuis, “Controlled spontaneous emission in room temperature semiconductor microcavities,” Appl. Phys. Lett. 60, 3202–3205 (1992).
[CrossRef]

D.L. Huffaker, D.G. Deppe, K. Kumar, and T.J. Rogers, “Native-oxide defined ring contact for low threshold vertical-cavity lasers,” Appl. Phys. Lett. 64, 97–99 (1994).
[CrossRef]

J.M. Dallesasse, N. Holonyak, A.R. Sugg, T.A. Richard, and N. El-Zein, “Hydrolization oxidation of AlGaAs-AlAs-GaAs quantum well heterostructures,” Appl. Phys. Lett. 57, 2844–2846 (1990).
[CrossRef]

D.L. Huffaker and D.G. Deppe , “Spontaneous coupling to planar and index-confined quasimodes of Fabry-Perot microcavities,” Appl. Phys. Lett. 67, 2494–2596 (1995).
[CrossRef]

D.G. Deppe and Q. Deng, “Eigenmode analysis of the dielectrically-apertured Fabry-Perot microcavity and its relation to self-focusing in the vertical-cavity surface-emitting laser,” Appl. Phys. Lett. 71, 160–162 (1997).
[CrossRef]

IEEE J. Quantum Electron. (3)

Q. Deng and D.G. Deppe, “Self-consistent calculation of the lasing eigenmode of the dielectrically-apertured Fabry-Perot microcavity with idealized or distributed Bragg reflectors,” IEEE J. Quantum Electron. 33, 2319–2326 (1997).
[CrossRef]

C.C. Lin, D.G. Deppe, and C. Lei, “Role of waveguide light emission in planar microcavities,” IEEE J. Quantum Electron. 30, 2304–2313 (1994).
[CrossRef]

G. Bjork, “On the spontaneous lifetime change in an ideal planar microcavity - transition from a mode continuum to quantized modes,” IEEE J. Quantum Electron. 30, 2314–2318 (1994).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

D.G. Deppe, T.-H. Oh, and D.L. Huffaker, “Eigenmode confinement in the dielectrically apertured Fabry-Perot microcavity,” IEEE Photonics Technol. Lett. 9, 713–715 (1997).
[CrossRef]

J. Appl. Phys. (2)

C.C. Lin and D.G. Deppe, “Calculation of lifetime dependence of Er3+ on cavity length in dielectric half-wave and full-wave microcavities,” J. Appl. Phys. 75, 4668–4672 (1994).
[CrossRef]

D.G. Deppe and C. Lei, “Spontaneous emission from a dipole in a semiconductor microcavity,” J. Appl. Phys. 70, 3443–3448 (1991).
[CrossRef]

Jpn. J. Appl. Phys., Part 2 (1)

K. Ujihara, “Spontaneous emission and the concept of effective area in a very short cavity with plane parallel dielectric mirrors,” Jpn. J. Appl. Phys., Part 2 30, L901–L903 (1991).
[CrossRef]

Phys. Rev. A (2)

G. Bjork, 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] [PubMed]

Q. Deng and D.G. Deppe, “Spontaneous-emission upling from multiemitters to the quasimode of a Fabry-Perot microcavity,” Phys. Rev. A 53, 1036–1047 (1996).
[CrossRef] [PubMed]

Other (1)

R.F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York,1961) pg. 188.

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Figures (3)

Fig. 1.
Fig. 1.

Schematic illustrating the idealized Fabry-Perot microcavity with a dielectric aperture. The chosen coordinate system is also shown in the figure.

Fig. 2.
Fig. 2.

Change of spontaneous emission rate versus frequency.

Fig. 3
Fig. 3

Calculated plots of the spontaneous near field profiles E(x,y,0,ω) of Eq. (4).

Equations (5)

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

J sp ( r , t ) = a J Δ 3 r n δ ( r r n ) J sp ( t )
J d ( x , y , ω ) = ε o χ R ( x , y , ω ) Δ z R δ ( z ) E ( x , y , 0 , ω ) .
× H ( r , ω ) = J ( r , ω ) iωε ( r , ω ) E ( r , ω )
E ( k x , k y , 0 , ω ) = 1 2 ωε k 2 k x 2 k y 2 1 + ρ e iL k 2 k x 2 k y 2 1 ρ e iL k 2 k x 2 k y 2 .
{ k × [ k × ( dk x dk y ( ε o 4 π 2 ) Δ z R χ R ( k x , k y , ω ) E ( k x k x , k y k y , 0 , ω ) ) + J sp Δ 3 r n ] }

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