Abstract

We present a theoretical analysis of lasing action in photonic crystal surface-emitting lasers (PCSELs). The semiclassical laser equations for such structures are simulated with three different theoretical techniques: exact finite-difference time-domain calculations, an steady-state ab-initio laser theory and a semi-analytical coupled-mode formalism. Our simulations show that, for an exemplary four-level gain model, the excitation of dark Fano resonances featuring arbitrarily large quality factors can lead to a significant reduction of the lasing threshold of PCSELs with respect to conventional vertical-cavity surface-emitting lasers. Our calculations also suggest that at the onset of lasing action, most of the laser power generated by finite-size PCSELs is emitted in the photonic crystal plane rather than the vertical direction. In addition to their fundamental interest, these findings may affect further engineering of active devices based on photonic crystal slabs.

© 2011 OSA

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2010 (4)

Y. Kurosaka, S. Iwahashi, Y. Liang, K. Sakai, E. Miyai, W. Kunishi, D. Ohnishi, S. Noda, “On-chip beam-steering photonic-crystal lasers,” Nat. Photonics 4, 447–450 (2010).
[CrossRef]

M. P. Nezhad, A. Simic, O. Bondarenko, B. Slutsky, A. Mizrahi, L. Feng, V. Lomakin, Y. Fainman, “Room-temperature subwavelength metallo-dielectric lasers,” Nat. Photonics 4, 395–399 (2010).
[CrossRef]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 81, 687–702 (2010).
[CrossRef]

J. Bravo-Abad, A. W. Rodriguez, J. D. Joannopoulos, P. T. Rakich, S. G. Johnson, M. Soljac̆ić, “Efficient low-power terahertz generation via on-chip triply-resonant nonlinear frequency mixing,” Appl. Phys. Lett. 96, 101110 (2010).
[CrossRef]

2009 (4)

J. Bravo-Abad, E. P. Ippen, M. Soljac̆ić, “Ultrafast photodetection in an all-silicon chip enabled by two-photon absorption,” Appl. Phys. Lett. 94, 241103 (2009).
[CrossRef]

H. Hashemi, A. W. Rodriguez, J. D. Joannopoulos, M. Soljac̆ić, S. G. Johnson, “Nonlinear harmonic generation and devices in doubly-resonant Kerr cavities,” Phys. Rev. A 79, 013812 (2009).
[CrossRef]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629 (2009).
[CrossRef] [PubMed]

2008 (7)

H. E. Türeci, L. Ge, S. Rotter, A. D. Stone, “Strong interactions in multimode random lasers,” Science 320, 643–646 (2008).
[CrossRef] [PubMed]

H. Matsubara, S. Yoshimoto, H. Saito, Y. Jianglin, Y. Tanaka, S. Noda, “GaN photonic crystal surface-emitting laser at blue-violet wavelengths,” Science 319, 445–447 (2008).
[CrossRef]

S. Gottardo, R. Sapienza, P. D. García, A. Blanco, D. S. Wiersma, C. López, “Resonance-driven random lasing,” Nat. Photonics 2, 429–432 (2008).
[CrossRef]

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, V. A. Fedotov, “Lasing spaser,” Nat. Photonics 2, 351–354 (2008).
[CrossRef]

T. Lu, C. Kao, H. Kuo, G. Huang, S. Wang, “CW lasing of current injection blue GaN-based vertical cavity surface emitting laser,” Appl. Phys. Lett. 92, 141102 (2008).
[CrossRef]

L. Ge, R. Tandy, A. D. Stone, H. E. Türeci, “Quantitative Verification of Ab Initio Self-Consistent Laser Theory,” Opt. Express 16, 16895 (2008).
[CrossRef] [PubMed]

R. E. Hamam, M. Ibanescu, E. J. Reed, P. Bermel, S. G. Johnson, E. Ippen, J. D. Joannopoulos, M. Soljac̆ić, “Purcell effect in nonlinear photonic structures: A coupled mode theory analysis,” Opt. Express 16, 12523–12537 (2008).
[CrossRef] [PubMed]

2007 (3)

J. Bravo-Abad, A. Rodriguez, P. Bermel, S. G. Johnson, J. D. Joannopoulos, M. Soljac̆ić, “Enhanced non-linear optics in photonic-crystal microcavities,” Opt. Express 15, 16161–16176 (2007).
[CrossRef] [PubMed]

M. T. Hill, Y. -S. Oei, B. Smalbrugge, Y. Zhu, T. de Vries, P. J. van Veldhoven, F. W. M. van Otten, T. J. Eijkemans, J. P. Turkiewicz, H. de Waardt, E. J. Geluk, S. -H. Kwon, Y.- H. Lee, R. Notzel, M. K. Smit, “Small-divergence semiconductor lasers by plasmonic collimation,” Nat. Photonics 1, 589–594 (2007).
[CrossRef]

S. Noda, M. Fujita, T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

2006 (4)

P. Bermel, E. Lidorikis, Y. Fink, J. D. Joannopoulos, “Active materials embedded in photonic crystals and coupled to electromagnetic radiation,” Phys. Rev. B 73, 165125 (2006).
[CrossRef]

H. E. Türeci, A. D. Stone, B. Collier, “Self-consistent multimode lasing theory for complex or random lasing media,” Phys. Rev. A 74, 043822 (2006).
[CrossRef]

B. Bakir, C. Seassal, X. Letartre, P. Viktorovitch, M. Zussy, L. Cioccio, J. Fedeli, “Surface-emitting micro-laser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,” Appl. Phys. Lett. 88, 081113 (2006).
[CrossRef]

H. Altug, D. Englund, J. Vuckovic, “Ultra-fast photonic-crystal nanolasers,” Nat. Phys. 2, 485–488 (2006).
[CrossRef]

2004 (4)

H.- G. Park, S.- H. Kim, S.- H. Kwon, Y. -G. Ju, J.- K. Yang, J.- H. Baek, S. -B. Kim, Y.- H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85, 3889–3891 (2004).
[CrossRef]

H. Y. Ryu, M. Notomi, E. Kuramochi, T. Segawa, “Large spontaneous emission factor (> 0.1) in the photonic crystal monopole-mode laser,” Appl. Phys. Lett. 84, 1067–1069 (2004).
[CrossRef]

W. Suh, Z. Wang, S. Fan, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40, 1511–1518 (2004).
[CrossRef]

2003 (2)

D. J. Bergman, M. L. Stockman, “Surface Plasmon Amplification by Stimulated Emission of Radiation: Quantum Generation of Coherent Surface Plasmons in Nanosystems,” Phys. Rev. Lett. 90, 027402 (2003).
[CrossRef] [PubMed]

X. Duan, Y. Huang, R. Agarwal, C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
[CrossRef] [PubMed]

2002 (4)

M. Imada, A. Chutinan, S. Noda, M. Mochizuki, “Multidirectionally distributed feedback photonic crystal lasers,” Phys. Rev. B 65, 195306 (2002).
[CrossRef]

J. C. Johnson, H.- J. Choi, K. P. Knutsen, R. D. Schaller, P. Yang, R. J. Saykally, “Single gallium nitride nanowire lasers,” Nat. Mater. 1, 106–110 (2002).
[CrossRef]

S. Fan, J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65, 235112 (2002).
[CrossRef]

M. Soljac̆ić, M. Ibanescu, S. G. Johnson, Y. Fink, J. D. Joannopoulos, “Optimal bistable switching in non-linear photonic crystals,” Phys. Rev. E 66, 055601 (2002).
[CrossRef]

2001 (4)

T. Ochiai, K. Sakoda, “Dispersion relation and optical transmittance of a hexagonal photonic crystal slab,” Phys. Rev. B 63, 125107 (2001).
[CrossRef]

A. A. Erchak, D. J. Ripin, S. Fan, P. Rakich, J. D. Joannopoulos, E. P. Ippen, G. S. Petrich, L. A. Kolodziejski, “Enhanced coupling to vertical radiation using a two-dimensional photonic crystal in a semiconductor light-emitting diode,” Appl. Phys. Lett. 78, 563–565 (2001).
[CrossRef]

S. Noda, M. Yokoyama, M. Imada, A. Chutinan, M. Mochizuki, “Polarization mode control of two-dimensional photonic crystal laser by unit cell structure design,” Science 293, 1123–1125 (2001).
[CrossRef] [PubMed]

H. Cao, Y. Ling, J. Y. Xu, C. Q. Cao, P. Kumar, “Photon statistics of random lasers with resonant feedback,” Phys. Rev. Lett. 86, 4524–4527 (2001).
[CrossRef] [PubMed]

2000 (1)

X. Jiang, C. M. Soukoulis, “Time dependent theory for random lasers,” Phys. Rev. Lett. 85, 70–73 (2000).
[CrossRef] [PubMed]

1999 (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819–1821 (1999).
[CrossRef] [PubMed]

1998 (2)

A. S. Nagra, R. A. York, “FDTD analysis of wave propagation in nonlinear absorbing and gain media,” IEEE Trans. Antennas Propag. 46, 334–340 (1998).
[CrossRef]

C. Gmachl, F. Capasso, E.E. Narimanov, J.U. Nöckel, A. D. Stone, J. Faist, D. Sivco, A. Cho, “High power directional emission from lasers with chaotic resonators,” Science 280, 1556–64 (1998).
[CrossRef] [PubMed]

1996 (2)

D. S. Wiersma, A. Lagendijk, “Light diffusion with gain and random lasers,” Phys. Rev. E 54, 4256–4265 (1996).
[CrossRef]

S. D. Glauber, “An anisotropic perfectly matched layer absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antennas Propag. 44, 1630–1639 (1996).
[CrossRef]

1994 (1)

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[CrossRef]

1992 (1)

H. Yokoyama, “Physics and device applications of optical microcavities,” Science 256, 66–70 (1992).
[CrossRef] [PubMed]

1989 (1)

S. Haroche, D. Kleppner, “Cavity quantum electrodynamics,” Phys. Today 42, 24–30 (1989).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302–307 (1966).
[CrossRef]

1960 (1)

T. H. Maiman, “Stimulated optical radiation in ruby,” Nature 187, 493–494 (1960).
[CrossRef]

Agarwal, R.

X. Duan, Y. Huang, R. Agarwal, C. M. Lieber, “Single-nanowire electrically driven lasers,” Nature 421, 241–245 (2003).
[CrossRef] [PubMed]

Altug, H.

H. Altug, D. Englund, J. Vuckovic, “Ultra-fast photonic-crystal nanolasers,” Nat. Phys. 2, 485–488 (2006).
[CrossRef]

Asano, T.

S. Noda, M. Fujita, T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

Baba, T.

T. Baba, D. Sano, K. Nozaki, K. Inoshita, Y. Kuroki, F. Koyama, “Observation of fast spontaneous emission decay in GaInAsP photonic crystal point defect nanocavity at room temperature,” Appl. Phys. Lett. 85, 3889–3891 (2004).
[CrossRef]

Baek, J.- H.

H.- G. Park, S.- H. Kim, S.- H. Kwon, Y. -G. Ju, J.- K. Yang, J.- H. Baek, S. -B. Kim, Y.- H. Lee, “Electrically driven single-cell photonic crystal laser,” Science 305, 1444–1447 (2004).
[CrossRef] [PubMed]

Bakir, B.

B. Bakir, C. Seassal, X. Letartre, P. Viktorovitch, M. Zussy, L. Cioccio, J. Fedeli, “Surface-emitting micro-laser combining two-dimensional photonic crystal membrane and vertical Bragg mirror,” Appl. Phys. Lett. 88, 081113 (2006).
[CrossRef]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[CrossRef] [PubMed]

Balachandran, R. M.

N. M. Lawandy, R. M. Balachandran, A. S. L. Gomes, E. Sauvain, “Laser action in strongly scattering media,” Nature 368, 436–438 (1994).
[CrossRef]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629 (2009).
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Figures (9)

Fig. 1
Fig. 1

Band diagrams of 1D, 2D, 3D systems, illustrating zero group velocity at k|| = 0(2π/a). The light lines ω = ck|| (red) separate the modes that are oscillatory (ω > ck||) in the air regions from those that are evanescent (ω < ck||) in air. (a) TM band diagram of a 1D system: Cavity enclosed by 25 and 30 bilayers (on top and below, respectively) of quarter-wave distributed Bragg reflectors. Pink shaded region represents a continuum of bands corresponding to the guided modes in the DBRs. Green line is the fundamental mode guided via total internal refraction while blue line is the mode guided within the band gap of the DBRs. Only modes with electric field oriented along z direction are considered. Inset shows the VCSEL structure extending uniformly to infinity in the x and z directions, with a 1–λ thick n = 3.55 cavity layer (green). Alternate red and blue layers of the reflectors correspond to n = 3.17 and n = 3.51 respectively. (b) Band diagram of a 2D system: n = 3.17 slab of height 0.3a with 1D periodic grooves that are 0.15a deep along y and 0.1a wide along x. Blue lines are the photonic bands. Only modes with electric field oriented along z direction are considered. Inset shows the structure, which is periodic in the x direction and extends uniformly in the z direction. (c) Band diagram of a 3D system: n = 3.17 slab of eight 0.3a with square lattices of circular air cylinders whose depth and radius are 0.25a. Blue lines are the photonic bands. Only TE-like modes are considered. Inset shows the slab structure, which is periodic in x and y directions.

Fig. 2
Fig. 2

(a) Variation of Q as a function of frequency for the lowest two bands above the light line for the infinite slab structure illustrated in Fig. 1 (b), as well as two other similar designs where the depth of the grooves are reduced to 0.05a and 0.1a. (b) Variation of Q as a function of frequency for the infinite slab (red lines), and slabs that are finite in the x direction (but remain uniform and infinite in the z direction) with length Lx. Depth of the grooves is 0.05a for all slabs considered in (b) and (c). (c) The photonic crystal slab is outlined in green and electric field pointing into the page is depicted with positive (negative) values in red (blue). First two insets illustrate the mode profiles of the lower and upper bands respectively, of the 2D infinite slabs at the band edges. Only a period, a, of the slab in the x-y plane is shown. The lower band edge mode is anti-symmetric about the groove while the upper band edge mode is symmetric. Corresponding to the band edges of the bottom line plotted in (b), the two insets on the right show the modes of the 20a finite slabs. The top (bottom) profile resembles the infinite slab’s lower (upper) band edge mode where near their centers, they share the same symmetry relative to the groove.

Fig. 3
Fig. 3

Total Q of the two band-edge modes for the finite PhC slab punctured with 0.05a deep grooves, and having total lateral size, Lx, ranging from 20 to 320 unit cells. Green lines are the fitted curves using the relationships described in the text and the horizontal line indicates Q value of the corresponding infinite slab (for the symmetric mode case).

Fig. 4
Fig. 4

Output power versus Rp relationships of the 2D infinite slab described in Fig. 1(b), for three depths of the air grooves at 0.05a, 0.1a, and 0.15a (width remains at 0.1a), with corresponding Q values 1964, 451, and 230 respectively. Both semi-analytic predictions from CMT (solid lines) and FDTD (filled circles) steady-state calculations are plotted for the upper band-edge mode at kx = 0(2π/a). There is good agreement between the semi-analytic and calculated values. The threshold is higher for the lower-Q PhC slab which clearly suggests that higher pumping rates are needed to overcome systems with higher losses.

Fig. 5
Fig. 5

Output power versus Rp relationships of the 2D finite slabs described in Fig. 2(b) for three dimensions of Lx at 20a, 40a, and 80a. Size of the grooves is fixed at 0.05a ×0.1a. (a) Higher-frequency symmetric modes with corresponding Q values 179, 413, and 925. (b) Lower-frequency anti-symmetric modes with corresponding Q values 231, 749, and 3243. Both semi-analytic predictions from CMT (solid lines) and FDTD (filled circles) steady-state calculations are shown. Insets plot the same data in linear scale for Rp values near threshold. In addition, (b) also shows the SALT (dashed lines) results for the 20a and 40a slabs. Good agreements between the three methods are observed. Slope of the lines changes with Lx (see text) while the right plot confirms that the anti-symmetric mode has the largest Q in the finite system that is available for lasing.

Fig. 6
Fig. 6

CF eigenvalue spectrum ηn for PCSELs with lateral sizes Lx ranging from 20a to 30a. Im[ηn] represents the gain needed to reach threshold, hence eigenvalues with the smallest magnitude of the imaginary part and nearest to the gain center correspond to the first lasing mode. For each structure, we choose the gain center ωm to minimize the threshold for the anti-symmetric Fano mode, and compute {ηn(ω)} at the resulting lasing frequency ω = ωL. Inset: threshold lasing mode for the Lx = 20a structure.

Fig. 7
Fig. 7

(Left) Lasing frequency ωL and optimal gain center ωm, (Center) the lasing threshold R p th , (Right) and the power slope (dP/dRp)/L z , as a function of lateral size Lx, computed using the self-consistent ab-initio laser theory (SALT). The inset shows the modal gain of the first and second modes (a mode turns on when its modal gain reaches unity [25]), indicating that the second mode does not turn on at any Rp for this choice of gain medium.

Fig. 8
Fig. 8

(a) Magnetic and electric field profiles of the first singly-degenerate mode at Γ in a unit cell of the 0.3a-thick 3D PhC slab structure shown in Fig. 1(c), partially punctured with square lattice of air cylinders having height and radius 0.15a. The PhC slab is outlined in green. Top panels depict the lateral cuts along the xy-plane with magnetic field pointing into the page and electric field pointing to the left, where positive (negative) values are in red (blue). Bottom panels are cuts along the yz-plane with magnetic field pointing downwards and electric field pointing into the page, where positive (negative) values are in red (blue). These results are only for TE-like modes. Ey (not shown) has the same profile as Ex, except rotated 90° about z-axis. (b) Same as in (a) but for the second singly-degenerate mode. (c) Same as in (a) but for the doubly-degenerate mode. Its counterpart at the same frequency has the magnetic field profile rotated 90° about z-axis. (d) Output power versus Rp relationships retrieved from CMT is also plotted for air cylinders with radius 0.3a, 0.4a, and 0.5a, for the doubly-degenerate mode presented in (c). Their respective frequencies are 0.449, 0.457, and 0.466(2πc/a) with Qs equal 764, 263, and 126.

Fig. 9
Fig. 9

Top: Magnetic and electric field profiles corresponding to the singly-degenerate mode in finite 15a ×15a PhC slab structure described in Fig. 8(a). The PhC slab is outlined in green. Top two panels depict the lateral cuts along the xy-plane with magnetic field pointing into the page and electric field pointing to the left, where positive (negative) values are in red (blue). Bottom panels are cuts along the yz-plane with magnetic field pointing downwards and electric field pointing into the page, where positive (negative) values are in red (blue). These results are only for TE-like modes. Ey (not shown) has the same profile as Ex, except rotated 90° about z-axis. Bottom: Output power versus Rp relationships retrieved from CMT. Frequencies and Q values of the 15a ×15a, 25a ×25a, and 35a ×35a PCSEL structures are 0.431, 0.433, 0.435(2πc/a), and 64, 178, 385, respectively.

Equations (40)

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2 P ( r , t ) t 2 + Γ m P ( r , t ) t + ω m 2 P ( r , t ) = σ m Δ N ( r , t ) E ( r , t )
N 3 ( r , t ) t = R p N 0 ( r , t ) N 3 ( r , t ) τ 32
N 2 ( r , t ) t = 1 h ¯ ω m E ( r , t ) P ( r , t ) t + N 3 ( r , t ) τ 32 N 2 ( r , t ) τ 21
N 1 ( r , t ) t = 1 h ¯ ω m E ( r , t ) P ( r , t ) t + N 2 ( r , t ) τ 21 N 1 ( r , t ) τ 10
N 0 ( r , t ) t = R p N 0 ( r , t ) + N 1 ( r , t ) τ 10 ,
B y n + 1 / 2 ( i + 1 2 ) = B y n 1 / 2 ( i + 1 2 ) Δ t Δ z [ E x n ( i + 1 ) E x n ( i ) ]
H y n + 1 / 2 ( i + 1 2 ) = 1 μ o B y n + 1 / 2 ( i + 1 2 )
P x n + 1 ( i ) = ( 1 + Γ m Δ t / 2 ) 1 { ( 2 ω m 2 Δ t 2 ) P x n ( i ) + ( Γ m Δ t / 2 1 ) P x n 1 ( i ) Δ t 2 σ m [ N 2 n ( i ) N 1 n ( i ) ] E x n ( i ) }
D x n + 1 ( i ) = D x n ( i ) Δ t Δ z [ H y n + 1 / 2 ( i + 1 2 ) H y n + 1 / 2 ( i 1 2 ) ]
E x n + 1 ( i ) = 1 ɛ ( i ) ɛ o [ D x n + 1 ( i ) P x n + 1 ( i ) ]
A ˜ N n + 1 ( i ) = B ˜ N n ( i ) + C
N n + 1 ( i ) = [ N 0 n + 1 ( i ) N 1 n + 1 ( i ) N 2 n + 1 ( i ) N 3 n + 1 ( i ) ] , N n ( i ) = [ N 0 n ( i ) N 1 n ( i ) N 2 n ( i ) N 3 n ( i ) ] , C = [ 0 E P E P 0 ] ,
E P = ( 2 h ¯ ω m ) 1 { [ E x n + 1 ( i ) + E x n ( i ) ] × [ P x n + 1 ( i ) P x n ( i ) ] } ,
A ˜ = [ 1 + e 1 e 2 0 0 0 1 + e 2 e 3 0 0 0 1 + e 3 e 4 e 1 0 0 1 + e 4 ] , B ˜ = [ 1 e 1 e 2 0 0 0 1 e 2 e 3 0 0 0 1 e 3 e 4 e 1 0 0 1 e 4 ] ,
e 1 = Δ t R p 2 , e 2 = Δ t 2 τ 10 , e 3 = Δ t 2 τ 21 , e 4 = Δ t 2 τ 32 .
P in = 1 2 Re { d 3 r [ P ( r , t ) t ] E * ( r , t ) }
d a ( t ) d t = ( 1 τ I O + 1 τ e x ) a ( t ) + ξ 1 [ i ω m P ( t ) d P ( t ) d t ] ,
d P ( t ) d t + Γ m 2 P ( t ) = i σ m 2 ω m a ( t ) Δ N ( t )
Δ N ( t ) = A d 3 r | E 0 ( r ) | 2 Δ N ( r , t ) A d 3 r | E 0 ( r ) | 2
d N 3 ( t ) d t = R p N 0 ( t ) N 3 ( t ) τ 32
d N 2 ( t ) d t = 1 4 h ¯ ξ 2 { a ( t ) [ i P * ( t ) + 1 ω m d P * ( t ) d t ] + c . c . } + N 3 ( t ) τ 32 N 2 ( t ) τ 21
d N 1 ( t ) d t = 1 4 h ¯ ξ 2 { a ( t ) [ i P * ( t ) + 1 ω m d P * ( t ) d t ] + c . c . } + N 2 ( t ) τ 21 N 1 ( t ) τ 10
d N 0 ( t ) d t = R p N 0 ( t ) + N 1 ( t ) τ 10
Δ N th = Γ m σ m τ tot ξ 1
R p th = Γ m σ m τ tot ξ 1 τ 21 N tot
d P e d R p = 4 h ¯ ω m N tot ξ 1 ξ 2 = N tot h ¯ ω m ( A d 3 r | E 0 ( r ) | 2 ) 2 A d 3 r | E 0 ( r ) | 4
E ( r , t ) = Ψ ( r ) e i ω L t + c . c . , P ( r , t ) = p ( r ) e i ω L t + c . c . .
2 h ¯ ω m E P ˙ + α ( R p ) [ δ 0 ( R p ) N tot ( r ) Δ N ] = 0 ,
α ( R p ) = 2 τ 21 τ 10 τ 21 + τ 10 β ( R p ) 1 + β ( R p ) 2 τ 21
β ( R p ) = 1 + τ 32 τ 10 + 1 τ 10 R p 1 τ 10 R p
δ 0 ( R p ) = τ 21 τ 10 τ 21 + τ 10 β ( R p ) ( τ 21 τ 10 ) R p .
[ 2 + ( ɛ ( r ) + μ 0 c 2 σ m / 2 ω m ω L ω m + i Γ m / 2 δ 0 ( R p ) N tot ( r ) 1 + h ( r , R p ) ) ( ω L c ) 2 ] Ψ ( r ) = 0
h ( r , R p ) = 2 σ m h ¯ ω m 2 α ( R p ) ω L Γ m / 2 ( ω L ω m ) 2 + ( Γ m / 2 ) 2 | Ψ ( r ) | 2 .
[ 2 + ( ɛ ( r ) + η n ( ω ) N tot ( r ) ) ( ω c ) 2 ] u n ( r ; ω ) = 0 ,
d 2 r N tot ( r ) u n ( r , ω ) u n ( r , ω ) δ n n .
η n ( ω ) = μ 0 c 2 σ m / 2 ω m ω ω m + i Γ m / 2 δ 0 , δ 0 + .
Ψ ( r ) I ( R p ) u L ( r ; ω L ) ,
I A 1 [ δ 0 ( R p ) δ 0 ( R p th ) 1 ]
A = σ m h ¯ ω m 2 α ( R p ) ω L Γ m ( ω L ω m ) 2 + ( Γ m / 2 ) 2 d 2 r N tot ( r ) u L 2 ( r ) | u L ( r ) | 2 d 2 r N tot ( r ) u L 2 ( r ) .
P = N tot h ¯ ω m L z ( 1 τ 10 τ 21 ) | ( A d 2 r | u L | 2 ) ( A d 2 r u L 2 ) ( A d 2 r u L 2 | u L | 2 ) | ( R p R p th ) .

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