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.

© Optical Society of America

1. Introduction

Low loss, planar Fabry-Perot microcavities can be realized using epitaxial crystal growth to form highly reflecting distributed Bragg reflectors. These semiconductor microcavities have been studied extensively to determine how they might be used to control spontaneous emission. [1–6] Past studies have shown that while the spontaneous radiation pattern from a dipole confined in the planar microcavity can be modified, the spontaneous emission rate cannot be increased too significantly. For idealized cavity systems based on single, highly reflecting interfaces, the largest increase that can be achieved in the planar system is found for a half-wave cavity spacer and is about a factor of two. [7] The limitation in lifetime control for such a system stems from coupling to waveguide modes that decrease the amount of optical feedback due to the cavity. [6–9]

Recently, Fabry-Perot microcavities with current and optical confining dielectric apertures [10] formed by “wet” oxidation [11] have been demonstrated to have interesting effects on spontaneous emission. [12] The similar kind of structure has attracted a great deal of attention because of the many improvements it has made in the performance of vertical-cavity surface-emitting lasers. [10] For spontaneous emission control, the dielectrically apertured Fabry-Perot microcavity is interesting because of the 3-dimensional confinement exerted on the optical mode. Recently we showed that this system can be analyzed in its ideal form by treating the apertured region as a passive current source driven by a gain region, and we have presented a detailed study on the lasing eigenmode in such a system including the boundary condition of the active gain region. [13,14] The self-consistent analysis shows that a resonance shift associated with the aperture leads to cutoff of lateral loss that would occur due to waveguide modes, and provides the 3-dimensional confinement in the otherwise planar microcavity. [15] Because of the rigorous inclusion of the active source’s electromagnetic boundary conditions, a similar approach can be used to evaluate spontaneous emission from a spectrally narrow-band point source in such a system. We present such an analysis for a non-zero loss cavity below, and show that a sizable change over the “open-space” spontaneous emission rate is possible. Both spontaneous enhancement and inhibition effects are calculated.

2. Calculation

We consider an idealized Fabry-Perot microcavity with an optically thin dielectric aperture as shown is Fig. 1. The coordinate system chosen is also plotted in the figure. The cavity consists of two metallic reflectors with equal field reflectivity ρ that are separated by the cavity length L. There is a thin dielectric aperture of thickness ΔzR in the center of the cavity. The dielectric constant inside and outside the cavity is taken as that of free space, which simplifies the notation. However, we note that the same treatment can be applied to the semiconductor dielectric microcavity, although the complexity of the math is increased.

The spontaneous point source emitter can be written as

Jsp(r,t)=aJΔ3rnδ(rrn)Jsp(t)

where the time dependence Jsp(t) sets the frequency extent. In general, a range of frequencies should be used to study the spontaneous emission. However, for a spectrally narrow-band source, a single frequency accurately predicts the lifetime modification due to the cavity, and more clearly treats the transverse mode coupling that exists due to the nonunity mirror reflectivity and the dielectric aperture. Multifrequency emission can also be treated by expansion into a summation over the single frequency emissions. To the first order approximation, the spontaneous point source is considered as unchanged by the cavity, even though it interacts with its emitted field and the incoming vacuum field. If the point source is in free space, we know that the radiated field can be described by spherical waves, and that the spherical waves can be expanded into plane wave modes. [16] When the spontaneous point source is placed in a microcavity, the radiated field will be modified by both the cavity mirrors [9] and the dielectric aperture, and the modified field is most conveniently treated in terms of the plane wave modes as well. The spontaneous point source acts as an excitation source to the dielectric aperture (Fig. 1), which is described by a real susceptibility function χR(x,y,ω) whose z dependence is removed in the limit that the aperture is optically thin. The induced polarization current due to the aperture is related to the total electric field at the aperture’s position and the real susceptibility χR (x, y, ω) by

Jd(x,y,ω)=εoχR(x,y,ω)ΔzRδ(z)E(x,y,0,ω).

The total electric field includes the field directly radiated from the spontaneous point source, the radiated fields from the aperture and cavity mirrors as driven by the spontaneous source. Under the planar cavity boundary conditions, it is very convenient to use the plane wave modes which forms a complete and orthogonal basis. Any radiation field in the passive cavity can be expanded into the spatial plane wave mode set. In general, when a single frequency source, either passive or active, is placed inside the planar microcavity, it is included in the Fourier transformed Maxwell’s equations as

×H(r,ω)=J(r,ω)iωε(r,ω)E(r,ω)

where ε(r,ω) is the dielectric constant that satisfies the planar cavity boundary conditions. For the case of Fig. 1, J(r,ω) has two contributions which are the spontaneous point source J sp(r,ω) and the induced polarization current in the aperture J d(r, ω). The radiation field from J(r, ω) can be calculated directly from Maxwell’s equations with the precaution that reflections from the cavity mirrors must be considered. Taking the spontaneous source as having vector amplitude lying in the x-y plane, for a symmetrical system as shown in Fig 1 the z component of the radiation field in the center of the cavity is zero while the x and y components take the Fourier transformed form [13,14]

E(kx,ky,0,ω)=12ωεk2kx2ky21+ρeiLk2kx2ky21ρeiLk2kx2ky2.
{k×[k×(dkxdky(εo4π2)ΔzRχR(kx,ky,ω)E(kxkx,kyky,0,ω))+JspΔ3rn]}
 

Fig. 1. Schematic illustrating the idealized Fabry-Perot microcavity with a dielectric aperture. The chosen coordinate system is also shown in the figure.

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In solving Eq. (4) a self-consistent solution can be obtained through numerical iteration by starting with the calculated field due to the spontaneous point source without the presence of dielectric aperture to get the first order approximation of J d (r, ω) from Eq. (2). The solution is then reinserted back into the right-hand side of (4) to calculate the next higher order approximation. This process is continued through enough iterations to obtain the convergent spontaneous field in the presence of the dielectric aperture.

3. Results

We apply the analysis to calculate the spontaneous lifetime for a half wave cavity with the cavity length tuned for resonance at 1 μm wavelength. Once the single frequency, self-consistent field for a point source is found, the spontaneous emission rate is found either by summing the emission rate into each plane wave mode, [1,2] or using the strength of the field at the emitter’s position to find the radiated power from ∫d3rE *(r, t) · J sp(r, t) with the time dependence approximated as e-iωt . The two cavity mirrors have the same field reflectivity of 0.995. To obtain rapid convergence in Eq. (4), we assume that the thin dielectric disk has a Gaussian distribution for χR(x,y,ω)ΔzR. The amount of dielectric confinement is then characterized by the index step χR(0,0,ω)ΔzR at the center of the cavity and the radius of the disk is taken as wχR = 2 μm. The influence of the aperture on the spontaneous emission rate is studied by choosing different values of χR(0,0,ω)ΔzR (real values) and finding the self-consistent solutions from Eq. (4) for a range of ω Figure 2 shows the calculated results for three values of susceptibility given as (a) χR(0,0,ω)ΔzR = 0, (b) χR(0,0,ω)ΔzR=18Å, and (c) χR(0,0,ω)ΔzR =36Å. The case of (a) χR(0,0,ω)ΔzR = 0 (planar half-wave cavity), in particular, has been studied previously with the results fairly well understood. [7,8] For the purely planar cavity, waveguide modes in general play a major role in establishing the spontaneous lifetime from a point source emitter and are rather insensitive to the cavity length. [6] The emission into the angular range around the cavity normal, on the other hand, is quite sensitive to the cavity length. For the half-wave cavity the coupling to the waveguide modes both for ideal and low loss dielectric cavities is reduced, and the spontaneous emission rate is much more sensitive to small variations in the cavity length for a fixed emitter frequency, [8] or to the emitter frequency for a fixed cavity length.

 

Fig. 2. Change of spontaneous emission rate versus frequency.

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The result as shown in curve (a) is a spontaneous lifetime that, for the somewhat idealized cavity and emitter of Fig. 1, varies by nearly an order of magnitude as the emitter frequency is changed in a 2% range from below to above resonance (Fig. 2). This change in the spontaneous emission rate is caused by inhibition for the smaller frequencies as compared to resonance, and enhancement for the larger frequencies into the angular range of emission near the cavity normal.

Curve (b) of Fig. 2 shows the spontaneous emission rate dependence on frequency when a thin dielectric aperture is introduced into the cavity, and the horizontal emitter is placed at the aperture center, with χR(0,0,ω)ΔzR=18Å. The cut-off of the waveguide modes due to the aperture leads to 3-dimensional confinement, and we see a peak form in the spontaneous emission rate close to resonance. As expected for a 3-dimensionally confined mode, a reduction in the spontaneous emission rate is obtained for frequencies either too far above or below resonance due to detuning. For a 3-dimensionally confined mode, a loss rate dependence is expected for the resonant frequency and this is also observed. The 3-dimensionally confined mode suffers loss due to both mirror transmission and waveguide propagation. [9] Increasing the aperture susceptibility decreases the loss rate due to waveguide propagation, and both sharpens the peak in the spontaneous emission rate and increases the peak value at resonance. This is seen in Fig. 2(c) as compared to (b) for the increased aperture susceptibility of χR(0,0,ω)ΔzR=36Å.

We note that two additional regimes can also be considered. Dipole dephasing generally occurs at a high rate in semiconductors, and leads to the spontaneous emission occurring into a range of frequencies. The dipole dephasing can occur due to collisions such as phonon scattering or carrier-carrier scattering, or due to the spontaneous emission itself that leads to amplitude decay of the spontaneous current source. Therefore the actual emission rate from a spontaneous source will have a spectral bandwidth set by a Lorentzian lineshape, and the total spontaneous emission rate will be due to a summation over of the rates for each frequency in the Lorentzian. In the limit that dipole dephasing tends to zero, the nonlinear response due to Rabi oscillations must be included.

 

Fig. 3 Calculated plots of the spontaneous near field profiles E(x,y,0,ω) of Eq. (4).

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The normalized self-consistent field profiles calculated from Eq. (4) for the resonant frequencies and three radii of susceptibilities are shown in Fig. 3. The actual calculated amplitudes can be related to the spontaneous source amplitude that generates the field. Of interest in the curves of Fig. 3 is how the field profiles depend on the radius of the dielectric aperture. The solid curve shows the case for wχR->∞, and like χR(0,0,ω)ΔzR =0, describes a fully planar half-wave cavity. One might expect the field profile to increase in spatial extent as the aperture radius is increased. However, with respect to the complete and orthogonal set of modes of the cavity, the field generated by the point source is in general a multi-spatial mode field. As the aperture size is increased, numerous transverse modes are excited in phase by the point source, and lead to the field being strongly peaked at the position of the source. Reducing the aperture size restricts the number of transverse modes that can be excited. The longer dashed curve of Fig. 3 shows the calculated field profile for an aperture radius of wχR=2 μm, which excites predominantly a single transverse mode set by the aperture size, and is broadened over the planar cavity case. Once the aperture size is reduced to what is required to enter the single mode regime, we then see that the field profile does indeed reduce with reducing aperture size over a range of sizes. In Fig. 3 the field profile has a smaller spatial extent for the 1 μm radius for wχR(short dashed curve) as compared to the 2 μm radius.

4. Summary

Although the planar Fabry-Perot microcavity suffers a significant drawback in optical confinement due to the continuous range of waveguide modes, we show above that the dielectrically-apertured Fabry-Perot microcavity can in large part correct this problem. Recent results on ultra-low threshold oxide-confined vertical-cavity surface-emitting lasers show that such systems are readily fabricated from III-V semiconductors. Although dipole dephasing must be controlled to fully realize the benefits of the 3-dimensional mode confinement, new epitaxial growth techniques to realize quantum dot emitters as well as excitons can yield narrow spectral linewidth semiconductor sources, and might be used to demonstrate novel cavity effects in apertured microcavities. We further note that while the analysis above treats rather idealized cavities, the plane wave mode basis set is also suitable for analyzing systems based on distributed Bragg reflectors. [14]

Acknowledgments

This work is supported by the Air Force Office of Sponsored Research under Grant No. F49620-96-1-0336, and the Joint Services Electronics Program under contract No. F49620-95-C-0045.

References and links

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 Jr., 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.

References

  • View by:
  • |

  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 coupling 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, Jr., 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.

Other (16)

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]

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]

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]

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

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, Jr., 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]

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]

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)

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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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