Abstract

We present an approach for achieving large Kerr χ(3)–mediated thermal energy transfer at the nanoscale that exploits a general coupled-mode description of triply resonant, four-wave mixing processes. We analyze the efficiency of thermal upconversion and energy transfer from mid- to near-infrared wavelengths in planar geometries involving two slabs supporting far-apart surface plasmon polaritons and separated by a nonlinear χ(3) medium that is irradiated by externally incident light. We study multiple geometric and material configurations and different classes of intervening mediums—either bulk or nanostructured lattices of nanoparticles embedded in nonlinear materials—designed to resonantly enhance the interaction of the incident light with thermal slab resonances. We find that even when the entire system is in thermodynamic equilibrium (at room temperature) and under typical drive intensities ~ W/μm2, the resulting upconversion rates can approach and even exceed thermal flux rates achieved in typical symmetric and non-equilibrium configurations of vacuum-separated slabs. The proposed nonlinear scheme could potentially be exploited to achieve thermal cooling and refrigeration at the nanoscale, and to actively control heat transfer between materials with dramatically different resonant responses.

© 2017 Optical Society of America

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2016 (3)

H. Soo and M. Krüger, “Fluctuational electrodynamics for nonlinear media,” EPL 115, 41002 (2016).
[Crossref]

D. Ding, T. Kim, and A. J. Minnich, “Active Thermal Extraction and Temperature Sensing of Near-field Thermal Radiation,” Sci. Rep. 6, 32744 (2016).
[Crossref] [PubMed]

C. Khandekar, W. Jin, O. D. Miller, A. Pick, and A. W. Rodriguez, “Giant frequency-selective near-field energy transfer in active-passive structures,” Phys. Rev. B 94, 115402 (2016).
[Crossref]

2015 (4)

K. Chen, P. Santhanam, S. Sandhu, L. Zhu, and S. Fan, “Heat-flux control and solid-state cooling by regulating chemical potential of photons in near-field electromagnetic heat transfer,” Phys. Rev. B 91, 134301 (2015).
[Crossref]

B. Song, A. Fiorino, E. Meyhofer, and P. Reddy, “Near-field radiative thermal transport: From theory to experiment,” AIP Adv. 5, 053503 (2015).
[Crossref]

C. Khandekar, Z. Lin, and A. W. Rodriguez, “Thermal radiation from optically driven Kerr χ(3) photonic cavities,” Appl. Phys. Lett. 106, 151109 (2015).
[Crossref]

C. Khandekar, A. Pick, S. G. Johnson, and A. W. Rodriguez, “Radiative heat transfer in nonlinear Kerr media,” Phys. Rev. B 91, 115406 (2015).
[Crossref]

2014 (3)

C. R. Otey, L. Zhu, S. Sandhu, and S. Fan, “Fluctuational electrodynamics calculations of near-field heat transfer in non-planar geometries: A brief overview,” J. Quant. Spectrosc. Radiat. Transfer 1323 (2014).
[Crossref]

P. Ben-Abdallah and S. Biehs, “Near-field thermal transistor,” Phys. Rev. Lett. 112, 044301 (2014).
[Crossref] [PubMed]

Z. Lin, T. Alcorn, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities,” Phys. Rev. A 89, 05389 (2014).
[Crossref]

2013 (3)

R. Messina and P. Ben-Abdallahm, “Graphene-based photovoltaic cells for near-field thermal energy conversion,” Sci. Rep. 3, 1383 (2013).
[Crossref]

A. C. Jones, B. T. O’Callahan, H. U. Yang, and M. B. Raschke, “The thermal near-field: coherence, spectroscopy, heat-transfer, and optical forces,” Prog. Surf. Sci. 88, 349 (2013).
[Crossref]

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
[Crossref] [PubMed]

2012 (3)

U. Hohenester and A. Trügler, “MNPBEM-A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Comm. 183, 370 (2012).
[Crossref]

M. Kauranen and A. V. Zayats, “Nonlinear plasmonics,” Nat. Photon. 6, 737 (2012).
[Crossref]

B. Guha, C. Otey, C. B. Poitras, S. Fan, and M. Lipson, “Near-field radiative cooling of nanostructures,” Nano Lett. 12, 4546 (2012).
[Crossref] [PubMed]

2011 (7)

D. M. Ramirez, A. W. Rodriguez, H. Hashemi, J. D. Joannopoulos, M. Soljačič, and S. G. Johnson, “Degenerate four-wave mixing in triply resonant Kerr cavities,” Phys. Rev. A 83, 033834 (2011).
[Crossref]

H. Zhao, J. Zhang, G. Liu, and N. Tansu, “Surface plasmon dispersion engineering via double-metallic Au/Ag layers for III-nitride based light-emitting diodes,” Appl. Phys. Lett. 98, 151115 (2011).
[Crossref]

F. Kheirandish, E. Amooghorban, and M. Soltani, “Finite-temperature Casimir effect in the presence of nonlinear dielectrics,” Phys. Rev. A 83, 032507 (2011).
[Crossref]

F. H. Koppens, D. E. Chang, and F. J. G. de Abajo, “Graphene plasmonics: a platform for strong light–matter interactions,” Nano Lett. 11, 3370 (2011).
[Crossref] [PubMed]

R. G. Chaudhari and S. Paria, “Core/shell nanoparticles: classes, properties, synthesis mechanisms, characterization, and applications,” Chem. Rev. 112, 2373 (2011).
[Crossref]

Q. Quan and M. Lončar, “Deterministic design of wavelength scale, ultra-high Q photonic crystal nanobeam cavities,” Opt. Exp. 19, 18529 (2011).
[Crossref]

X. Yan, X. Zhang, S. Shi, Z. Liu, and J. Tian, “Third-order nonlinear susceptibility tensor elements of CS 2 at femtosecond time scale,” Opt. Express 19, 5559 (2011).
[Crossref] [PubMed]

2010 (4)

I. W. Frank, P. B. Deotare, M. W. McCutcheon, and M. Lončar, “Programmable photonic crystal nanobeam cavities,” Opt. Express 18, 8705 (2010).
[Crossref] [PubMed]

A. Bahabad, M. M. Murnane, and H. C. Kapteyn, “Quasi-phase-matching of momentum and energy in nonlinear optical processes,” Nat. Photon. 4, 570 (2010).
[Crossref]

P. R. West, S. Ishii, G. V. Naik, N. K. Emani, V. M. Shalaev, and A. Boltasseva, “Searching for better plasmonic materials,” Laser Photon. Rev. 4, 795 (2010).
[Crossref]

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104, 154301 (2010).
[Crossref] [PubMed]

2009 (5)

I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, “On-chip cooling by superlattice-based thin-film thermoelectrics,” Nat. Nanotechnol. 4, 235 (2009).
[Crossref] [PubMed]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33, 1203 (2009).
[Crossref]

M. G. Blaber, M. D. Arnold, and M. J. Ford, “Search for the ideal plasmonic nanoshell: the effects of surface scattering and alternatives to gold and silver,” J. Phys. Chem. C 113, 3041 (2009).
[Crossref]

C. Koos, P. Vorreau, T. Vallaitis, P. Dumon, W. Bogaerts, R. Baets, B. Esembeson, I. Biaggio, T. Michinobu, F. Diederich, W. Freude, and J. Leuthold, “All-optical high-speed signal processing with silicon-organic hybrid slot waveguides,” Nat. Photon. 3, 216 (2009).
[Crossref]

R. H. Hadfield, “Single-photon detectors for optical quantum information applications,” Nat. Photon. 3, 696 (2009).
[Crossref]

2008 (2)

S. Franzen, “Surface plasmon polaritons and screened plasma absorption in indium tin oxide compared to silver and gold,” J. Phys. Chem. C 112, 6027 (2008).
[Crossref]

M. Francoeur, M. P. Mengüc, and R. Vaillon, “Near-field radiative heat transfer enhancement via surface phonon polaritons coupling in thin films,” Appl. Phys. Lett. 93, 043109 (2008).
[Crossref]

2007 (3)

A. W. Rodriguez, M. Soljačič, J. D. Joannopoulos, and S. G. Johnson, “χ(2) and χ(3) harmonic generation at a critical power in inhomogeneous doubly resonant cavities,” Opt. Exp. 15, 7303 (2007).
[Crossref]

J. Greffet and C. Henkel, “Coherent thermal radiation,” Contemp. Phys. 48, 183 (2007).
[Crossref]

R. Kitamura, L. Pilon, and M. Jonasz, “Optical constants of silica glass from extreme ultraviolet to far infrared at near room temperature,” Appl. Opt. 46, 8118 (2007).
[Crossref] [PubMed]

2006 (2)

A. H. Sihvola, “Peculiarities in the dielectric response of negative-permittivity scatterers,” Prog. Electromagn. Res. 66191 (2006).
[Crossref]

Y. de Wilde, F. Formanek, R. Carminati, B. Gralak, P. Lemoine, K. Joulain, J. Mulet, Y. Chen, and J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nat. 444, 740 (2006).
[Crossref]

2005 (1)

K. Joulain, J. Mulet, F. Marquier, R. Carminati, and J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59 (2005).
[Crossref]

2004 (2)

M. Soljačič and J. D. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater. 3, 211 (2004).
[Crossref]

J. B. Pendry, L. Martin-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305, 847 (2004).
[Crossref] [PubMed]

2003 (1)

A. Zakery and S. R. Elliott, “Optical properties and applications of chalcogenide glasses: a review,” J. Non-Cryst. Solids 330, 1 (2003).
[Crossref]

2002 (1)

1999 (2)

J. A. Porto, F. J. Garcia-Vidal, and J. B. Pendry, “Transmission resonances on metallic gratings with very narrow slits,” Phys. Rev. Lett. 83, 2845 (1999).
[Crossref]

D. Cotter, R. J. Manning, K. J. Blow, A. D. Ellis, A. E. Kelly, D. Nesset, I. D. Phillips, A. J. Poustie, and D. C. Rogers, “Nonlinear optics for high-speed digital information processing,” Science 286, 1523 (1999).
[Crossref] [PubMed]

1998 (1)

S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “Nanoengineering of optical resonances,” Chem. Phys. Lett. 288, 243 (1998).
[Crossref]

1991 (1)

J. R. Sambles, G. W. Bradbery, and F. Yang, “Optical excitation of surface plasmons: an introduction,” Contemp. Phys. 32, 173 (1991).
[Crossref]

1985 (1)

C. G. Granqvist, “Spectrally selective coatings for energy efficiency and solar applications,” Phys. Scripta 32, 401 (1985).
[Crossref]

1984 (1)

W. Eckhardt, “Macroscopic theory of electromagnetic fluctuations and stationary radiative heat transfer,” Phys. Rev. A 29, 1991 (1984).
[Crossref]

1978 (1)

L. G. Aio, A. M. Efimov, and V. F. Kokorina, “Refractive index of chalcogenide glasses over a wide range of compositions,” J. Non-Cryst. Solids 27, 299 (1978).
[Crossref]

1975 (1)

M. I. Dykman, “Theory of nonlinear nonequilibrium oscillators interacting with a medium,” Zh. Eksp. Theor. Fiz 68, 2082 (1975).

1971 (1)

D. Polder and M. Van Hove, “Theory of radiative heat transfer between closely spaced bodies,” Phys. Rev. B 4, 3303 (1971).
[Crossref]

Aggarwal, I. D.

Aio, L. G.

L. G. Aio, A. M. Efimov, and V. F. Kokorina, “Refractive index of chalcogenide glasses over a wide range of compositions,” J. Non-Cryst. Solids 27, 299 (1978).
[Crossref]

Ajayan, P. M.

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
[Crossref] [PubMed]

Alcorn, T.

Z. Lin, T. Alcorn, M. Loncar, S. G. Johnson, and A. W. Rodriguez, “High-efficiency degenerate four-wave mixing in triply resonant nanobeam cavities,” Phys. Rev. A 89, 05389 (2014).
[Crossref]

Alley, R.

I. Chowdhury, R. Prasher, K. Lofgreen, G. Chrysler, S. Narasimhan, R. Mahajan, D. Koester, R. Alley, and R. Venkatasubramanian, “On-chip cooling by superlattice-based thin-film thermoelectrics,” Nat. Nanotechnol. 4, 235 (2009).
[Crossref] [PubMed]

Amooghorban, E.

F. Kheirandish, E. Amooghorban, and M. Soltani, “Finite-temperature Casimir effect in the presence of nonlinear dielectrics,” Phys. Rev. A 83, 032507 (2011).
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ACS Nano (1)

Z. Fang, S. Thongrattanasiri, A. Schlather, Z. Liu, L. Ma, Y. Wang, P. M. Ajayan, P. Nordlander, N. J. Halas, and F. J. G. de Abajo, “Gated tunability and hybridization of localized plasmons in nanostructured graphene,” ACS Nano 7, 2388 (2013).
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Figures (5)

Fig. 1
Fig. 1

(a) Schematic of three equal-temperature T resonators supporting modes at ωj, each with decay rate γj stemming from internal dissipation γjd and/or radiation into an external channel γjc, with j = {1, 2, 3}. The thermal modes 1 and 3 are strongly coupled to one another through a four-wave mixing process involving a Kerr χ(3) medium excited by externally incident light of power P from a waveguide that resonantly couples to the mode at ω2 = 1/2(ω3ω1). Nonlinear mixing between modes 1 and 3 can be described by an effective, linear but power-dependent coupling κ [Eq. (7)] obtained via a spatial overlap between the three modes, given in Eq. (4). (b) Thermal emission spectra Pj (ω) (blue and red curves) normalized by kBT, for a choice of far-apart frequencies ω1 = kBT/ħ and ω3 = 10ω1, and decay rates γjd = γjc = 0.01ω1, as a function of the dimensionless frequency ħω/kBT. Emission at any ω is bounded above by the Planck distribution Θ(ω, T) (black curve) in the absence of the pump κ = 0 (blue curves), but is exponentially enhanced under finite κ > 0 (red curves). (c) Thermal emission P1(ω) (solid lines) and heat-transfer Pex(ω) (dotted lines) spectra near ω1 as a function of the dimensionless frequency (ωω1)/γ1, demonstrating splitting of the resonances into Stokes (red-shifted) and anti-Stokes (blue-shifted) peaks, which grow apart with increasing κ and further enhance emission.

Fig. 2
Fig. 2

(a) Schematic of two plates consisting of different materials held at temperatures T1 and T3 and which support surface plasmon polaritons (SPPs) at far-apart resonant frequencies ω1 and ω3. Also shown is a schematic of the corresponding SPP dispersions ωj (k) (blue and red). Thermal upconversion and transfer of energy from ω1 to ω3 is facilitated by a mediator mode at ω 2 ~ ( ω 3 ω 1 ) 2 (green). (b) Table summarizing three different geometric configurations that result in significant thermal upconversion, along with various possible choices of emitting, absorbing, and nonlinear materials. The main difference between configurations is the choice of intervening medium, which consist of either a (i) lattice of nanoparticles embedded in the nonlinear medium or (ii) bulk nonlinear thin film, and serves as a tunable means to enhance the incident light.

Fig. 3
Fig. 3

(a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a rectangular lattice of nanospheres with unit-cell size Λx × Λy embedded in a χ(3) nonlinear medium (ChG). The SiO2/Au core/shell nanoparticles have core/shell radii of 15nm and 20nm, respectively, and the dimensions of the unit cell are Λx = 5R and Λy = πc/ω2. The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to dipolar particle resonances at ω 2 ~ 1 2 ( ω 3 ω 1 ) excited by a laterally incident, monochromatic, z ^ -polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

Fig. 4
Fig. 4

(a) Schematic of a planar system of SiC and ITO slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with a square lattice of doped graphene nanodisks with unit-cell size Λ × Λ embedded in a χ(3) nonlinear medium (ChG). The graphene nanodisks have radius R = 20nm, Fermi energy EF = 0.7eV and are placed at a distance of 10nm from ITO slab. The SiC and ITO slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to nanodisk resonances at ω 2 ~ 1 2 ( ω 3 ω 1 ) excited by a normally incident, monochromatic, x ^ + i y ^ -polarized planewave of frequency ω2 and intensity I. (b) xy and yz cross sections of the particle resonances |E2|2, with yellow/black denoting maximum/zero amplitude. (c) Normalized slab mode-profiles at a representative wavenumber kd = 1, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the ITO slab. (e) Frequency-integrated heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) also show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

Fig. 5
Fig. 5

(a) Schematic of a planar system of SiC and K slabs at thermal equilibrium (room temperature) separated by a gap of size d = 60nm that is filled with χ(3) nonlinear medium (ChG). The SiC and K slabs support SPPs at frequencies ω1 and ω3, respectively, which couple nonlinearly to a SPP resonance at ω 2 ~ 1 2 ( ω 3 ω 1 ) that is excited via a grating (of period Λ) by monochromatic light of frequency ω2, intensity I, and wavevector k2y incident at an angle θinc with respect to the z ^ direction. (b) Schematic illustration of the coupling of SPP resonances of wavevectors k1 and k3; the polar plot shows the directional dependence of the nonlinear coupling coefficient κ(k, θ), where k1 = (k, θ) is expressed in polar coordinates, with θ denoting the angle extended by k1 with respect to y ^. (c) Normalized slab mode-profiles at a representative wavenumber kd = 2 and angle θ = 0, with the shaded region indicating the intervening medium. (d) Variations in the nonlinear coupling κ, frequency mismatch Δω, and frequency ω1 with respect to kd, normalized by the corresponding dissipation rate γ3 of the K slab. (e) Frequency-integrated, angle-averaged heat-extraction spectrum Pex(k), normalized by P0 = 2γ1Θ(ω1, T), as a function of kd and for multiple incident intensities I. Also shown are the (f) associated spectral density Φex(ω) and (g) net extracted and upconverted flux rates, Hex and Hup, as a function of I. For comparison, (e–g) show the heat-transfer rates associated with two vacuum-separated SiC slabs held at either 300K (light blue) or 1K (dark blue) temperature differences.

Equations (20)

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d a 1 d t = ( i ω 1 γ 1 ) a 1 i β ω 1 a 3 ( a 2 * ) 2 + 2 γ 1 d ξ 1
d a 2 d t = ( i ω 2 γ 2 ) a 2 i β ω 2 a 3 a 1 * a 2 * + 2 γ 2 d ξ 2 + 2 γ 2 c s in
d a 3 d t = ( i ω 3 γ 3 ) a 3 i β * ω 3 a 1 a 2 2 + 2 γ 3 d ξ 3
β = d V χ i j k l ( 3 ) E 1 i E 2 j E 2 k E 3 l * 2 ε 0 d V ( ω ε ) ω E 1 i * E 1 i d V ( ω ε ) ω E 3 i * E 3 i d V ( ω ε ) ω E 2 i * E 2 i ,
d a 2 d t = ( i ω 2 γ 2 ) a 2 + 2 γ 2 c s in .
d a 1 d t = ( i ω 1 γ 1 ) a 1 i κ e 2 i ω 2 t a 3 + 2 γ 1 d ξ 1
d a 3 d t = ( i ω 3 γ 3 ) a 3 i ω 3 ω 1 κ * e 2 i ω 2 t a 1 + 2 γ 3 d ξ 3 ,
κ = 2 β ω 1 γ 2 c P γ 2 2 .
P ex ( ω ) = 4 | κ | 2 D 1 ( ω ) [ γ 1 γ 3 d Θ ( ω + 2 ω 2 , T 3 ) + γ 1 d γ 3 ( ω 3 / ω 1 ) Θ ( ω , T 1 ) ]
P 1 ( ω ) = 4 γ 1 c D 1 ( ω ) [ γ 1 d | i ( ω ω 1 Δ ω ) + γ 3 | 2 Θ ( ω , T 1 ) + γ 3 d | κ | 2 Θ ( ω + 2 ω 2 , T 3 ) ]
P 3 ( ω ) = 4 γ 3 c D 3 ( ω ) [ γ 3 d | i ( ω ω 3 Δ ω ) + γ 1 | 2 Θ ( ω , T 3 ) + γ 1 d ( ω 3 / ω 1 ) | κ | 2 Θ ( ω 2 ω 2 , T 1 ) ]
ω 1 ± = ω 1 + Δ ω ± Δ ω 2 + 4 ( γ 1 γ 3 + ω 3 / ω 1 | κ | 2 ) 2 ω 3 ± = ω 1 ± ( 1 3 , Δ ω Δ ω )
e 2 i κ z 2 d = ( ϵ 1 κ z 2 ϵ 2 κ z 1 ) ( ϵ 3 κ z 2 ϵ 2 κ z 3 ) ( ϵ 2 κ z 3 + ϵ 3 κ z 2 ) ( ϵ 1 κ z 2 + ϵ 2 κ z 1 )
β ( k 1 , k 3 ) = p d V χ i j k ( 3 ) e i ( k 3 k 1 ) . x E 1 i ( z ) E 2 j ( x x p , z ) E 2 k ( x x p , z ) E 3 * ( z ) 2 ϵ 0 ( p d V ϵ ω ω | E 2 ( x x p , z ) | 2 ) ( d V ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d V ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
H = 0 d ω 2 π 0 d ω 2 π k P ex ( ω , k ) Φ ex ( ω )
κ ( k ) = ω 1 σ abs I cell χ i j k ( 3 ) E 1 i ( z ) E 2 j ( x x p , z ) E 2 k ( x x p , z ) E 3 * ( z ) 4 ϵ 0 γ 2 d Λ x Λ y ( cell ϵ ω ω | E 2 ( x x p , z ) | 2 ) ( d z ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d z ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
H ( 0 ) = 0 d ω 2 π [ Θ ( ω , T 1 ) Θ ( ω , T 3 ) ] 0 d k k 2 Im { r 21 p } Im { r 23 p } e 2 Im { k z 2 } d π | 1 r 21 p r 23 p e 2 Im { k z 2 } d | 2 Φ ( 0 ) ω )
ϵ 2 ω 2 sin ( θ inc ) c + 2 π Λ = k 2 y ,
β ( k 1 , k 3 ) = d V χ i j k ( 3 ) e i ( k 3 k 1 2 k 2 y y ^ ) . x E 1 i ( z ) E 2 j ( ω 2 , z ) E 2 k ( ω 2 , z ) E 3 * ( z ) 2 ϵ 0 ( d V ϵ ω ω | E 2 ( ω 2 , z ) | 2 ) ( d V ϵ ω ω | E 1 ( z ) | 2 ) 1 / 2 ( d V ϵ ω ω | E 3 ( z ) | 2 ) 1 / 2
κ ( k , θ ) = ω 1 γ 2 c I d z χ i j k ( 3 ) E 1 i ( ω 1 , z ) E 2 j ( ω 2 , z ) E 2 k ( ω 2 , z ) E 3 * ( ω 3 , z ) ϵ 0 γ 2 2 d z ϵ ω ω | E 2 ( ω 2 , z ) | 2 ( d z ϵ ω ω | E 1 ( ω 1 , z ) | 2 ) 1 / 2 ( d z ϵ ω ω | E 3 ( ω 3 , z ) | 2 ) 1 / 2

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