Abstract

We develop a semiclassical theory of passively mode-locked surface plasmon polariton (SPP) lasers based on a SPP Bragg resonator with a metal film deposited on a polymer host and adjacent layers of a slow saturable absorber and a slow saturable gain medium. The mode-locked laser dynamics is studied for the case that both the gain medium and the saturable absorber are solid-state dyes. The SPP laser pulse parameters are calculated in dependence on layer thicknesses of the metal film and pump parameters. We predict the possibility of SPP pulse generation with ∼ 100 fs pulse duration.

© 2011 OSA

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  7. P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35, 1197–1199 (2010).
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    [CrossRef]
  37. P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20, 395–431 (1988).
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    [CrossRef]
  40. A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron. 24, 87–172 (1994).
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  41. B. H. Soffer and B. B. McFarland, “Continuously tuable, narrow band organic dye lasers,” Appl. Phys. Lett. 10, 266–267 (1967).
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  42. A. Costela, I. Garcia-Moreno, and C. Gomez, “Efficient and stable dye laser action from modified dipyrromethene BF2 complexes,” Appl. Phys. Lett. 79, 305–307 (2001).
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  43. P. Runge and R. Rosenberg, “Unconfined flowing-dye films for CW dye lasers,” IEEE J. Quantum Electron. 8, 910–911 (1972).
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  44. A. Costela, I. Garcia-Moreno, R. Sastre, D. W. Coutts, and C. E. Webb, “High repetition- rate polymeric solid-state dye lasers pumped by a copper-vapor laser,” Appl. Phys. Lett. 79, 452–454 (2001).
    [CrossRef]
  45. I. G. Kytina, V. G. Kitin, and K. Lips, “High power polymer dye laser with improved stability,” Appl. Phys. Lett. 84, 4092–4904 (2004).
    [CrossRef]
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    [CrossRef]

2011 (2)

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11, 2231–2235 (2011).
[CrossRef] [PubMed]

D. Yu. Fedyanin and A. V. Arsenin, “Surface plasmon polariton amplification in metal-semiconductor structures,” Opt. Express 19, 12524–12531 (2011).
[CrossRef] [PubMed]

2010 (4)

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35, 1197–1199 (2010).
[CrossRef] [PubMed]

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

M. C. Gather, K. Meerholz, N. Danz, and K. Lesson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nature Photon. 4, 457–461 (2010).
[CrossRef]

M. I. Stockman, “The spaser as a nanoscale quantum generator and ultrafast amplifier,” J. Opt. 12, 024004 (2010).
[CrossRef]

2009 (6)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

M. T. Hill, M. Marell, E. S. P. Leong, B. Smalbrugge, Y. Zhu, M. Sun, P. J. van Veldhoven, E. J. Geluk, F. Karouta, Y.-S. Oei, R. Noetzel, C.-Z. Ning, and M. K. Smit, “Lasing in metal-insulator-metal sub-wavelength plasmonic waveguides,” Opt. Express 17, 11107–11112 (2009).
[CrossRef] [PubMed]

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

A. Fang, T. Koschny, M. Wegener, and C. M. Soukoulis, “Self-consistent calculation of metamaterials with gain,” Phys. Rev. B 79, 241104 (2009).
[CrossRef]

P. Berini, “Long-range surface plasmon polaritons,” Advances in Optics and Photonics 1, 484–588 (2009).
[CrossRef]

I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express 17, 20191–20202 (2009).
[CrossRef] [PubMed]

2008 (7)

I. D. Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401 (2008).
[CrossRef]

M. A. Noginov, G. Zhu, M. Mayy, B. A. Ritzo, N. Noginova, and V. A. Podolskiy, “Stimulated emission of surface plasmon polaritons,” Phys. Rev. Lett. 101, 226806 (2008).
[CrossRef] [PubMed]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Resonance amplification of left-handed transmission at optical frequencies by stimulated emission of radiation in active metamaterials,” Opt. Express 16, 20974–20980 (2008).
[CrossRef] [PubMed]

M. Wegener, J. L. Garcia-Pomar, N. M. C. M. Soukoulis, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16, 19785–19798 (2008).
[CrossRef] [PubMed]

S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 024301 (2008).
[CrossRef]

M. I. Stockman, “Spasers explained,” Nature Photon. 2, 327–329 (2008).
[CrossRef]

2007 (1)

2006 (3)

G. Winter, S. Wedge, and W. L. Barnes, “Can lasing at visible wavelengths be achieved using the low-loss long-range surface plasmon-polariton mode?” New J. Phys. 8, 211102 (2006).
[CrossRef]

A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech. 24(2), 912–918 (2006).
[CrossRef]

R. Bornemann, U. Lemmer, and E. Thiel, “Continuous-wave solid-state dye laser,” Opt. Lett. 31, 1669–1671 (2006).
[CrossRef] [PubMed]

2005 (3)

J. Seidel, S. Grafstroem, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71, 115409 (2005).
[CrossRef]

I. E. Protsenko, A. V. Uskov, O. A. Zaimidoroga, V. N. Samoilov, and E. P. O’Reilly, “Dipole nanolaser,” Phys. Rev. A 71, 063812 (2005).
[CrossRef]

2004 (1)

I. G. Kytina, V. G. Kitin, and K. Lips, “High power polymer dye laser with improved stability,” Appl. Phys. Lett. 84, 4092–4904 (2004).
[CrossRef]

2003 (1)

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90027402 (2003).
[CrossRef] [PubMed]

2001 (2)

A. Costela, I. Garcia-Moreno, R. Sastre, D. W. Coutts, and C. E. Webb, “High repetition- rate polymeric solid-state dye lasers pumped by a copper-vapor laser,” Appl. Phys. Lett. 79, 452–454 (2001).
[CrossRef]

A. Costela, I. Garcia-Moreno, and C. Gomez, “Efficient and stable dye laser action from modified dipyrromethene BF2 complexes,” Appl. Phys. Lett. 79, 305–307 (2001).
[CrossRef]

1998 (1)

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

1994 (1)

A. A. Ishchenko, “Laser media based on polymethine dyes,” Quantum Electron. 24, 87–172 (1994).
[CrossRef]

1992 (1)

1988 (1)

P. Sperber, W. Spangler, B. Meier, and A. Penzkofer, “Experimental and theoretical investigation of tunable picosecond pulse generation in longitudinally pumped dye laser generators and amplifiers,” Opt. Quantum Electron. 20, 395–431 (1988).
[CrossRef]

1986 (1)

L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. 59, 3289–3291 (1986).
[CrossRef]

1984 (1)

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

1982 (2)

J. Herrmann and F. Weidner, “Theory of passively mode-locked cw dye lasers,” Appl. Phys. B 27, 105–113 (1982).
[CrossRef]

B. Kopainsky, P. Qiu, W. Kaiser, B. Sens, and K. H. Drexhage, “Lifetime, photostability, and chemical structure of IR heptamethine cyanine dyes absorbing beyond 1 mm,” Appl. Phys. B 29, 15–18 (1982).
[CrossRef]

1975 (1)

H. A. Haus, “Theory of mode locking with a slow saturable absorber,” IEEE J. Quantum Electron. QE-11, 736–746 (1975).
[CrossRef]

1972 (1)

P. Runge and R. Rosenberg, “Unconfined flowing-dye films for CW dye lasers,” IEEE J. Quantum Electron. 8, 910–911 (1972).
[CrossRef]

1967 (1)

B. H. Soffer and B. B. McFarland, “Continuously tuable, narrow band organic dye lasers,” Appl. Phys. Lett. 10, 266–267 (1967).
[CrossRef]

Adams, M. J.

M. J. Adams, An Introduction to Optical Waveguides (John Wiley and Sons, Chichester-New York-Brisbane-Toronto, 1981).

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Elsevier, Amsterdam, 2007).

Ambati, M.

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Arsenin, A. V.

Bakker, R.

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

Barnes, W. L.

G. Winter, S. Wedge, and W. L. Barnes, “Can lasing at visible wavelengths be achieved using the low-loss long-range surface plasmon-polariton mode?” New J. Phys. 8, 211102 (2006).
[CrossRef]

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

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

M. Ambati, S. H. Nam, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Observation of stimulated emission of surface plasmon polaritons,” Nano Lett. 8, 3998–4001 (2008).
[CrossRef] [PubMed]

Belgrave, A. M.

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

Benfey, D. P.

Bergman, D. J.

K. Li, X. Li, M. I. Stockman, and D. J. Bergman, “Surface plasmon amplification by stimulated emission in nanolenses,” Phys. Rev. B 71, 115409 (2005).
[CrossRef]

D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90027402 (2003).
[CrossRef] [PubMed]

Berini, P.

I. D. Leon and P. Berini, “Amplification of long-range surface plasmons by a dipolar gain medium,” Nature Photon. 4, 382–387 (2010).
[CrossRef]

P. Berini, “Long-range surface plasmon polaritons,” Advances in Optics and Photonics 1, 484–588 (2009).
[CrossRef]

I. D. Leon and P. Berini, “Modeling surface plasmon-polariton gain in planar metallic structures,” Opt. Express 17, 20191–20202 (2009).
[CrossRef] [PubMed]

I. D. Leon and P. Berini, “Theory of surface plasmon-polariton amplification in planar structures incorporating dipolar gain media,” Phys. Rev. B 78, 161401 (2008).
[CrossRef]

Bolger, P. M.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11, 2231–2235 (2011).
[CrossRef] [PubMed]

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35, 1197–1199 (2010).
[CrossRef] [PubMed]

Boltasseva, A.

A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech. 24(2), 912–918 (2006).
[CrossRef]

Bornemann, R.

Bozhevolnyi, S. I.

A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Light. Tech. 24(2), 912–918 (2006).
[CrossRef]

Brown, D. C.

Cao, J.-X.

Chang, S.-W.

S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 024301 (2008).
[CrossRef]

Chuang, S.-L.

S.-W. Chang, C.-Y. A. Ni, and S.-L. Chuang, “Theory for bowtie plasmonic nanolasers,” Opt. Express 16, 024301 (2008).
[CrossRef]

Costela, A.

A. Costela, I. Garcia-Moreno, and C. Gomez, “Efficient and stable dye laser action from modified dipyrromethene BF2 complexes,” Appl. Phys. Lett. 79, 305–307 (2001).
[CrossRef]

A. Costela, I. Garcia-Moreno, R. Sastre, D. W. Coutts, and C. E. Webb, “High repetition- rate polymeric solid-state dye lasers pumped by a copper-vapor laser,” Appl. Phys. Lett. 79, 452–454 (2001).
[CrossRef]

Coutts, D. W.

A. Costela, I. Garcia-Moreno, R. Sastre, D. W. Coutts, and C. E. Webb, “High repetition- rate polymeric solid-state dye lasers pumped by a copper-vapor laser,” Appl. Phys. Lett. 79, 452–454 (2001).
[CrossRef]

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[CrossRef] [PubMed]

Danz, N.

M. C. Gather, K. Meerholz, N. Danz, and K. Lesson, “Net optical gain in a plasmonic waveguide embedded in a fluorescent polymer,” Nature Photon. 4, 457–461 (2010).
[CrossRef]

Davis, S. J.

Dickson, W.

A. V. Krasavin, T. P. Vo, W. Dickson, P. M. Bolger, and A. V. Zayats, “All-plasmonic modulation via stimulated emission of copropagating surface plasmon polaritons on a substrate with gain,” Nano Lett. 11, 2231–2235 (2011).
[CrossRef] [PubMed]

P. M. Bolger, W. Dickson, A. V. Krasavin, L. Liebscher, S. G. Hickey, D. V. Skryabin, and A. V. Zayats, “Amplified spontaneous emission of surface plasmon polaritons and limitations on the increase of their propagation length,” Opt. Lett. 35, 1197–1199 (2010).
[CrossRef] [PubMed]

Diels, J.C.

J.C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena, 2nd ed. (Academic Press, San Diego, 2006).

Dong, Z.-G.

Drexhage, K. H.

B. Kopainsky, P. Qiu, W. Kaiser, B. Sens, and K. H. Drexhage, “Lifetime, photostability, and chemical structure of IR heptamethine cyanine dyes absorbing beyond 1 mm,” Appl. Phys. B 29, 15–18 (1982).
[CrossRef]

Eng, L.

J. Seidel, S. Grafstroem, and L. Eng, “Stimulated emission of surface plasmons at the interface between a silver film and an optically pumped dye solution,” Phys. Rev. Lett. 94, 177401 (2005).
[CrossRef] [PubMed]

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Nature (2)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R.-M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
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Figures (5)

Fig. 1
Fig. 1

(Color online) Configuration of the SPP laser. The layer c is a dielectric layer with gain (green). For mode locking, we add a saturable absorber layer (gray-colored) into the layer c adjacent to the metal film layer b. The absorber and gain media are doped in the same host material with permittivity ɛc. For the feedback we take a Bragg reflectors of SPPs as the resonator mirrors.

Fig. 2
Fig. 2

(Color online) Long range SPP intensity profile normalized by its maximum (a) and the dependence of attenuation γ0 and linear gain g0 on the metal film thickness d (b). Incident pump intensities 5, 10 and 15 MW/cm2 were considered. Resonator length is 1 cm, Styryl-9 concentration is Ng = 2.5 × 1018 cm−3, lasing wavelength is λL = 900 nm, the thicknesses of gain and absorber sublayers are Dq = 400 nm and Dg = 5 μm. The permittivities of the layers are ɛa (λL) = ɛc (λL) = 2.20 (PMMA) and ɛb (λL) = −35.99 + 2.20i (silver), respectively.

Fig. 3
Fig. 3

Pump intensity distribution Ip (x) in the gain sublayer (a) and normalized lifetime of the upper state for both gain and saturable absorber sublayers. In (a), the intensity of incident pump beam is Ip0 = 15 MW/cm2. In (b) doted line shows the interface between gain (right) and absorber sublayers; the thickness of the Ag layer b and the saturable absorber sublayer are d = 30 nm and Dq = 400 nm, respectively.

Fig. 4
Fig. 4

(Color online) Evolution of gain gi (dashed blue line) and total loss qi + l0 (dash-dot green line) (a) and their behavior near the pulse maximum (b). The pulse shape is presented at the position adjacent to the Ag film in the dielectric layers a and c. The concentrations of gain and absorber molecules are Ng = 2.5 × 1018 cm−3 and Nq = 1 × 1017 cm−3, respectively. The thicknesses of Ag film, gain and absorber sublayers are d = 30 nm, Dq = 400 nm and Dg = 5 μm, respectively. The intensity of the incident pump beam is Ip0 = 10.26 MW/cm2.

Fig. 5
Fig. 5

(Color online) Dependence of maximum pulse fluence Fmax (at the position adjacent to the Ag film) and pulse duration τ on the pump intensity Ip. The other parameters are the same as in Fig. 4.

Equations (28)

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( ɛ a α b + ɛ b α a ) ( ɛ b α c + ɛ c α b ) + ( ɛ a α b ɛ b α a ) ( ɛ b α c ɛ c α b ) e 2 α b d = 0 ,
U n = A a n q a n e α a n x ( > x 0 ) U n = A b n + q b n + e α b n x + A b n q b n e α b n x ( 0 > x d ) U n = A c n q c n + e α c n ( x + d ) ( d > x ) ,
A a n = ɛ c α b n + ɛ b α c n ɛ a α b n ɛ b α a n e α b n d A c n ,
A b n ± = ( ɛ c α b n ± ɛ b α c n ) 2 ɛ b α b n e ± α b n d A c n .
K n 2 = α j n 2 ɛ j ( Ω n ) c 2 ( γ n + i Ω n ) 2 ,
E c = 1 2 n A n ( t ) { x ^ sin ( K n z ) + z ^ ( α c n / K n ) cos ( K n z ) } e α c n ( x + d ) i ω n t + c . c . ,
A ˙ n + γ n A n = σ g ɛ c 𝒟 n Γ g A n M n 0 L D g d g d g | U n | 2 N g d z d x ,
g n = β n σ g D g d g d g N ¯ g ( x , t ) e κ ( x + d ) d x ,
T R A T = [ g ( τ ) q ( τ ) γ 0 ] A + δ 1 A τ + δ 2 2 A τ 2 ,
g l τ = g l g 0 ( x ) τ 0 g g l | E c ( x , τ ) | 2 A s g 2 τ 0 g ,
q l τ = q l q 0 ( x ) τ 0 q q l | E c ( x , τ ) | 2 A s q 2 τ 0 q ,
E c = E c + + E c ,
E c + = 1 4 n A n ( t ) e i ω n t + K n z ( i x ^ + α c n K n z ^ ) e α c n ( x + d ) + c . c . ,
E c = 1 4 n A n ( t ) e i ω n t K n z ( i x ^ + α c n K n z ^ ) e α c n ( x + d ) + c . c .
E c + ( x , z , t ) = 1 4 ( i x ^ + α c K L z ^ ) e α c ( x + d ) n A n ( t ) e i ω n t + K n z + c . c .
A ( t , z ) = n A n ( t ) e i ( ω n ω L ) t + ( K n K L ) z .
A ( t , z ) = A ( t , k ) e i δ ω t + i k z d k ,
k = K K L = n eff c ( ω ω L ) ,
A ( t , z ) t = [ A ( t , k ) t i δ ω A ( t , k ) ] e δ ω t + i k z d k .
δ ω = v g k + 1 2 2 ω k 2 | k = 0 k 2 +
A ( t , k ) t = [ γ ( k ) + g ( t , k ) ] A ( t , k ) ,
γ ( k ) = γ 0 + γ k | k = 0 k + 1 2 2 γ k 2 | k = 0 k 2 +
g ( t , k ) = g ( t ) 𝒟 n Γ g ,
g ( t ) = β σ g D g d g d g N ¯ g ( x , t ) e κ ( x + d ) d z d x ,
g ( t , k ) = g ( t ) [ 1 + i δ ω Γ g ( δ ω Γ g ) 2 + ] .
g ( t , k ) = g ( t ) [ 1 + i v g Γ g k ( i 2 Γ g 2 ω k 2 | k = 0 v g 2 Γ g 2 ) k 2 + ] .
A ( t , z ) t = { ( g γ 0 ) i ( i v g γ k | k = 0 g Γ g + 1 ) ( v g k ) + [ 1 2 v g 2 γ k | k = 0 g ( i 2 Γ g v g 2 2 ω k 2 | k = 0 1 Γ g 2 ) i 2 v g 2 2 ω k 2 | k = 0 ] ( v g k ) 2 } × A ( t , k ) e i δ ω t + i k z d k .
T R A ( t , z ) t = { ( g q γ 0 ) i ( i v g γ k | k = 0 g Γ g + q Γ q + T R ) ( v g k ) + ( g Γ g 2 q Γ q 2 + 1 2 v g 2 [ i ( T R + g Γ g q Γ q ) 2 ω k 2 + 2 γ k 2 ] | k = 0 ) ( v g k ) 2 } × A ( t , k ) e i δ ω t + i k z d k ,

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