Abstract

We analyze high-order harmonic generation (HHG) in a disordered semiconductor within the context of the Anderson model of disorder. Employing the theoretical methods pioneered for the study of disordered metals, we show that disorder is a source of ultrafast dephasing of the HHG signal in semiconductors. Furthermore, it is shown that the dephasing effect induced by disorder on HHG spectra depends on both strength and correlation length of the disorder and very weakly on the frequency and intensity of the laser. Our results suggest that HHG has the potential to be a new spectroscopic tool for the analysis of disordered solids.

© 2020 Optical Society of America

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    [Crossref]
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  8. G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534, 520–524 (2016).
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  38. J. Ma, C. Zhang, H. Cui, Z. Ma, and X. Miao, “Theoretical investigation of the electron dynamics in high-order harmonic generation process from the doped periodic potential,” Chem. Phys. Lett. 744, 137207 (2020).
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  46. T. Lukes, “On the electronic structure of disordered systems,” Philos. Mag. 12(118), 719–724 (1965).
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    [Crossref]
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    [Crossref]

2020 (4)

J. Ma, C. Zhang, H. Cui, Z. Ma, and X. Miao, “Theoretical investigation of the electron dynamics in high-order harmonic generation process from the doped periodic potential,” Chem. Phys. Lett. 744, 137207 (2020).
[Crossref]

K. Chinzei and T. N. Ikeda, “Disorder effects on the origin of high-order harmonic generation in solids,” Phys. Rev. Res. 2, 013033 (2020).
[Crossref]

A. Pattanayak, M. S. Mrudul, and G. Dixit, “Influence of vacancy defects in solid high-order harmonic generation,” Phys. Rev. A 101, 013404 (2020).
[Crossref]

M. S. Mrudul, N. Tancogne-Dejean, A. Rubio, and G. Dixit, “High-harmonic generation by spin-polarized defects in solids,” Comput. Mater. 6, 10 (2020).
[Crossref]

2019 (6)

Y. W. Kim, T.-J. Shao, H. Kim, S. Han, S. Kim, M. Ciappina, X.-B. Bian, and S.-W. Kim, “Spectral interference in high harmonic generation from solids,” ACS Photon. 6, 851–857 (2019).
[Crossref]

N. Tsatrafyllis, S. Kühn, M. Dumergue, P. Foldi, S. Kahaly, E. Cormier, I. A. Gonoskov, B. Kiss, K. Varju, S. Varro, and P. Tzallas, “Quantum optical signatures in a strong laser pulse after interaction with semiconductors,” Phys. Rev. Lett. 122, 193602 (2019).
[Crossref]

G. Orlando, T.-S. Ho, and S.-I. Chu, “Macroscopic effects in high-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 36, 1873–1880 (2019).
[Crossref]

C. Yu, K. K. Hansen, and L. B. Madsen, “Enhanced high-order harmonic generation in donor-doped band-gap materials,” Phys. Rev. A 99, 013435 (2019).
[Crossref]

C. Yu, K. K. Hansen, and L. B. Madsen, “High-order harmonic generation in imperfect crystals,” Phys. Rev. A 99, 063408 (2019).
[Crossref]

S. Y. Kruchinin, “Non-Markovian pure dephasing in a dielectric excited by a few-cycle laser pulse,” Phys. Rev. A 100, 043839 (2019).
[Crossref]

2018 (10)

S. Almalki, A. M. Parks, G. Bart, P. B. Corkum, T. Brabec, and C. R. McDonald, “High harmonic generation tomography of impurities in solids: conceptual analysis,” Phys. Rev. B 98, 144307 (2018).
[Crossref]

G. Orlando, C.-M. Wang, T.-S. Ho, and S.-I. Chu, “High-order harmonic generation in disordered semiconductors,” J. Opt. Soc. Am. B 35, 680–688 (2018).
[Crossref]

T. Ikemachi, Y. Shinohara, T. Sato, J. Yumoto, M. Kuwata-Gonokami, and K. L. Ishikawa, “Time-dependent Hartree-Fock study of electron-hole interactions effects on high-order harmonic generation from periodic crystals,” Phys. Rev. A 98, 023415 (2018).
[Crossref]

S. Jiang, J. Chen, H. Wei, C. Yu, R. Lu, and C. D. Lin, “Role of the transition dipole amplitude and phase on the generation of odd and even high-order harmonics in crystals,” Phys. Rev. Lett. 120, 253201 (2018).
[Crossref]

P. Xia, C. Kim, F. Lu, T. Kanai, H. Akiyama, J. Itatani, and N. Ishii, “Nonlinear propagation effects in high-harmonic generation in reflection and transmission from gallium arsenide,” Opt. Express 26, 29393–29400 (2018).
[Crossref]

M. Garg, H. Y. Kim, and E. Goulielmakis, “Ultimate waveform reproducibility of extreme ultraviolet pulse by high-harmonic generation in quartz,” Nat. Photonics 12, 291–296 (2018).
[Crossref]

G. P. Zhang, M. S. Si, M. Murakami, Y. H. Baiand, and T. F. George, “Generating high-order optical and spin harmonics from ferromagnetic monolayers,” Nat. Commun. 9, 3031 (2018).
[Crossref]

Y. Murakami, M. Eckstein, and P. Werner, “High-harmonic generation in Mott insulators,” Phys. Rev. Lett. 121, 057405 (2018).
[Crossref]

D. Bauer and K. K. Hansen, “High-harmonic generation in solid with and without topological edge states,” Phys. Rev. Lett. 120, 177401 (2018).
[Crossref]

I. Floss, C. Lemell, G. Wachter, M. Pickem, J. Burgdörfer, X.-M. Tong, K. Yabana, and S. A. Sato, “Ab initio simulation of high-order harmonic generation in solids,” Phys. Rev. A 97, 011401 (2018).
[Crossref]

2017 (9)

N. Tancogne-Dejean, O. D. Mücke, F. X. Kärtner, and A. Rubio, “Impact of electronic band structure in high-harmonic generation spectra of solids,” Phys. Rev. Lett. 118, 087403 (2017).
[Crossref]

A. A. Lanin, E. A. Stepanov, A. B. Fedotov, and A. M. Zheltykov, “Mapping the electron band structure by intraband high-harmonic generation in solids,” Optica 4, 516–519 (2017).
[Crossref]

S. Gholam-Mirzaei, J. Beetar, and M. Chini, “High harmonic generation in ZnO with a high-power mid-IR OPA,” Appl. Phys. Lett. 110, 061101 (2017).
[Crossref]

K. K. Hansen, T. Deffge, and D. Bauer, “High-order harmonic generation in solid slabs beyond the single-active-electron approximation,” Phys. Rev. A 96, 053418 (2017).
[Crossref]

T.-Y. Du and X.-B. Bian, “Quasi-classical analysis of the dynamics of the high-order harmonic generation from solids,” Opt. Express 25, 151–158 (2017).
[Crossref]

J.-B. Li, X. Zhang, S.-J. Yue, H.-M. Wu, B.-T. Hu, and H.-C. Du, “Enhancement of the second plateau in solid high-order harmonic spectra by the two-color fields,” Opt. Express 25, 18603–18613 (2017).
[Crossref]

X. Liu, X. Zhu, X. Zhang, D. Wang, P. Lan, and P. Lu, “Wavelength scaling of the cutoff energy in the solid high harmonic generation,” Opt. Express 25, 29216–29224 (2017).
[Crossref]

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order harmonic generation of a doped semiconductor,” Phys. Rev. A 96, 043425 (2017).
[Crossref]

G. Vampa and T. Brabec, “Merge of high harmonic generation from gases and solids and its implications for attosecond science,” J. Phys. B 50, 083001 (2017).
[Crossref]

2016 (2)

G. Ndabashimiye, S. Ghimire, M. Wu, D. A. Browne, K. J. Schafer, M. B. Gaarde, and D. A. Reis, “Solid-state harmonics beyond the atomic limit,” Nature 534, 520–524 (2016).
[Crossref]

T. Otobe, “High-harmonic generation in α-quartz by electron-hole recombination,” Phys. Rev. B 94, 235512 (2016).
[Crossref]

2015 (8)

M. Hohenleutner, F. Langer, O. Schubert, M. Knorr, U. Huttner, S. W. Koch, and R. Huber, “Real-time observation of interfering crystal electrons in high-harmonic generation,” Nature 523, 572–575 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, D. D. Klug, and P. B. Corkum, “All-optical reconstruction of crystal band structure,” Phys. Rev. Lett. 115, 193603 (2015).
[Crossref]

T. T. Luu, M. Garg, S. Y. Kruchinin, A. Moulet, M. T. Hassan, and E. Goulielmakis, “Extreme ultraviolet high-harmonic spectroscopy of solids,” Nature 521, 498–502 (2015).
[Crossref]

G. Vampa, T. J. Hammond, N. Thiré, B. E. Schmidt, F. Légaré, C. R. McDonald, T. Brabec, and P. B. Corkum, “Linking high harmonics from gases and solids,” Nature 522, 462–464 (2015).
[Crossref]

M. Wu, S. Ghimire, D. A. Reis, K. J. Schafer, and M. B. Gaarde, “High-harmonic generation from Bloch electrons in solids,” Phys. Rev. A 91, 043839 (2015).
[Crossref]

N. Moiseyev, “Selection rules for harmonic generation in solids,” Phys. Rev. A 91, 053811 (2015).
[Crossref]

P. G. Hawkins, M. Y. Ivanov, and V. S. Jakovlev, “ Effect of multiple conduction bands on high-harmonic emission from dielectrics,” Phys. Rev. A 91, 013405 (2015).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, P. B. Corkum, and T. Brabec, “Semiclassical analysis of high harmonic generation in bulk crystals,” Phys. Rev. B 91, 064302 (2015).
[Crossref]

2014 (3)

T. Higuchi, M. I. Stockman, and P. Hommelhoff, “Strong-field perspective on high-harmonic radiation from bulk solids,” Phys. Rev. Lett. 113, 213901 (2014).
[Crossref]

O. Schubert, M. Hohenleutner, F. Langer, B. Urbanek, C. Lange, U. Huttner, D. Golde, T. Meier, M. Kira, S. W. Koch, and R. Huber, “Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations,” Nat. Photonics 8, 119–123 (2014).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generations in solids,” Phys. Rev. Lett. 113, 073901 (2014).
[Crossref]

2011 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

2008 (2)

V. Turkowski and C. A. Ullrich, “Time-dependent density-functional theory for ultrafast interband excitations,” Phys. Rev. B 77, 075204 (2008).
[Crossref]

J. Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B 78, 075204 (2008).
[Crossref]

1995 (1)

M. Protopapas, D. G. Lappas, C. H. Keitel, and P. Knight, “Recollisions, bremsstrahlung, and attosecond pulses from intense laser fields,” Phys. Rev. A 53, R2933–R2936 (1995).
[Crossref]

1994 (1)

M. Lewenstein, P. Balcou, M. Y. Ivanov, A. L’Huillier, and P. B. Corkum, “Theory of high-harmonic generation by low-frequency laser fields,” Phys. Rev. A 49, 2117–2132 (1994).
[Crossref]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71, 1994–1997 (1993).
[Crossref]

1991 (2)

N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207–1211 (1991).
[Crossref]

D. Bennhardt, P. Thomas, A. Weller, M. Lindberg, and S. W. Koch, “Influence of Coulomb interactions on the photon echo in disordered semiconductors,” Phys. Rev. B 43, 8934–8945 (1991).
[Crossref]

1986 (2)

D. C. Khandekar, V. A. Singh, K. V. Bhagwat, and S. V. Lawande, “Functional integral approach to positionally disordered systems,” Phys. Rev. B 33, 5482–5486 (1986).
[Crossref]

D. C. Khandekar and S. V. Lawande, “Feynman path integrals: some exact results and applications,” Phys. Rep. 137, 115–229 (1986).
[Crossref]

1977 (1)

V. Sayakanit, “Mobility of an electron in a Gaussian random potential,” Phys. Lett. 59, 461–463 (1977).
[Crossref]

1973 (1)

V. Samathiakanit, “An average propagator of a disordered system,” J. Phys. A 6, 632–639 (1973).
[Crossref]

1970 (1)

V. Bezak, “Path integral theory of an electron gas in a random potential,” Proc. R. Soc. London A 315, 339–354 (1970).
[Crossref]

1968 (1)

A. V. Chaplik, “The Green’s function and mobility of an electron in a random potential,” Sov. Phys. JETP 26, 797–800 (1968).

1965 (2)

L. V. Keldysh, “Ionization in the field of a strong electromagnetic wave,” Sov. Phys. JETP 20, 1307–1314 (1965).

T. Lukes, “On the electronic structure of disordered systems,” Philos. Mag. 12(118), 719–724 (1965).
[Crossref]

1964 (1)

S. F. Edwards and V. B. Gulyaev, “The density of states of an highly impure semiconductor,” Proc. Phys. Soc. 83, 495–496 (1964).
[Crossref]

1962 (1)

E. I. Blount, “Formalisms of band theory,” Solid State Phys. 13, 305–373 (1962).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7, 138–141 (2011).
[Crossref]

Akiyama, H.

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Figures (4)

Fig. 1.
Fig. 1. (a) Schematic pictorial description of the generic Feynman trajectories contributing to the dipole moment. (b) Generic Feynman trajectories satisfying the stationary-phase conditions Eq. (15). The curves are not solutions of the Newton equations of motion but only a pictorial description of the Green functions ${\bar G_c}$ (blue asterisks), ${G_v}$ (red circles), $G_v^*$ (solid black line). The vertical dotted line represents the time instant $t^\prime $ when the conduction-band electron is created.
Fig. 2.
Fig. 2. HHG spectrum generated by a disordered semiconductor with ${\Gamma _1} = 0.25\hbar {\omega _L}$, interacting with a laser field with $I = 2.0\;{\rm TW}/{{\rm cm}^2}$ and $\lambda = 2.5 \;{\unicode{x00B5}{\rm m}}$. Blue line is the total spectrum; red line is the intraband contribution.
Fig. 3.
Fig. 3. As in Fig. 2, but with ${\Gamma _1} = 0.7\hbar {\omega _L}$.
Fig. 4.
Fig. 4. Detailed view of the HHG spectrum as a function of the disorder level in the solids. The topmost curve (black solid line with crosses) is the HHG spectrum generated by a crystal without disorder ${\Gamma _1} = 0.0$, the middle curve (dashed red) is the spectrum emitted by a solid with disorder strength ${\Gamma _1} = 0.25\hbar {\omega _L}$, and the bottom curve (blue line) is the spectrum radiated by the solid when ${\Gamma _1} = 0.7\hbar {\omega _L}$. Intensity and frequency of the laser are as in Fig. 2. The three curves have been shifted by arbitrary factors to improve visibility.

Equations (42)

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ϵ c ( k ) = E g 2 Δ cos ( k a ) , ϵ v ( k ) = E g + 2 Δ cos ( k a ) ,
H ^ I = l = N N ( E g | w lc w lc | E g | w lv w lv | Δ [ | w l , c w l + 1 , c | + | w l + 1 , c w lc | ] ) + Δ l = N N ( | w lv w l + 1 , v | + | w l + 1 , v w l , v | ) ,
| w lc = k e ikal | k c 2 N + 1 , | w lv = k e ikal | k v 2 N + 1 .
w nv | H ^ D | w mv = w nc | H ^ D | w mv = 0 , w nc | H ^ D | w mc = α n δ nm .
H ^ ( t ) = H ^ 0 + eF ( t ) l = N N ( d cv [ | w lc w lv | + | w lv w lc | ] + la [ | w lc w lc | + | w lv w lv | ] ) .
i d | m ( t ) d t = ( H ^ 0 + exF ( t ) ) | m ( t ) ,
b ˙ c m ( n , t ) = η nc ( t ) i b c m ( n , t ) Δ i [ b c m ( n + 1 , t ) + b v m ( n 1 , t ) ] + e d cv F ( t ) i b v m ( n , t ) ,
b ˙ v m ( n , t ) = η nv ( t ) i b v m ( n , t ) + Δ i [ b v m ( n + 1 , t ) + b v m ( n 1 , t ) ] + e d cv F ( t ) i b c m ( n , t ) ,
η nc ( t ) = E g + α n + enaF ( t ) , η nv ( t ) = E g + enaF ( t ) ,
J ( t ) = t r [ ρ p ^ ] = m = N N c 0 t r [ | m ( t ) m ( t ) | p ^ ] = m = N N c 0 m ( t ) | p ^ | m ( t ) ,
i b ˙ c ( p , t ) = a α ( p p ) b c ( p , t ) d p ( i b c p d cv b v ) × eF ( t ) 2 Δ cos ( a p ) b c ( p , t ) ,
i b ˙ v ( p , t ) = ( 2 E g + 2 Δ cos ( a p ) ) b v i b v p eF ( t ) ,
i ϕ ˙ c ( x , t ) = ( exF ( t ) + D ( x ) ) ϕ c ( x , t ) 2 2 m c x 2 ϕ c ( x , t ) + ed cv ϕ v ( x , t ) F ( t ) ,
i ϕ ˙ v ( x , t ) = ( ϵ g + e x F ( t ) ) ϕ v ( x , t ) 2 2 m v x 2 ϕ v ,
ϕ c ( x , t ) = e d cv G c ( x , x , t , t ) ϕ v ( x , t ) F ( t ) d x d t ,
i G ˙ c ( e x F ( t ) + D ( x ) ) G c + 2 2 m c x 2 G c = δ ( x x ) δ ( t t ) .
G ¯ c ( x , x , t , t ) = x ( t ) = x x ( t ) = x [ d x ] e i S c ( [ x ] t t x x ) e ρ W 0 2 2 2 t t d s t t d s γ ( x ( s ) x ( s ) ) ,
G ¯ c ( x , x , t , t ) = G c ( x , x , t , t ) e Φ ( x , x , t , t ) ,
e Φ ( x , x , t , t ) = x ( t ) = x x ( t ) = x [ d x ] e i h S c ( [ x ] t t x x ) G c ( x , x , t , t ) e ρ W 0 2 2 2 t t d s t t d s γ ( x ( s ) x ( s ) ) .
Φ ¯ ( x , x , t , t ) 2 1 2 ρ W 0 2 = t t d s t t d s x x [ d x ( τ ) ] e i S c ( [ x ] t t x x ) γ ( x ( s ) x ( s ) ) G c ( x , x , t , t ) .
Φ ¯ ( x , x , t , t ) = π 2 ρ W 0 2 L 2 2 t t d s t t d s e A ( x , x , s , s , t , t ) B 1 2 ( s , s , t , t ) ,
A ( x , x , s , s , t , t ) = ( s s ) 2 [ x x t t + λ ( s ) λ ( s ) s s λ ( t ) λ ( t ) t t ] 2 4 B ( s , s , t , t ) , B ( s , s , t , t ) = 1 2 [ L 2 + i m c ( ( s s ) Θ ( s s ) + ( s s ) Θ ( s s ) ( s s ) 2 t t ) ] ,
Φ ¯ ( x = x , t , t ) = ρ W 0 2 L 2 ( π / 2 ) 1 / 2 2 ( t t ) 2 i ( t t ) 2 m c log ( 1 + i ( t t ) 4 m c L 2 1 i ( t t ) 4 m c L 2 ) .
Φ ¯ ( x = x , t , t ) = 2 ρ W 0 2 L 5 ( π ) 3 / 2 2 4 L 2 ( t t ) 2 i ( / m c ) ( t t ) 3 ( 2 ( t t ) 2 m c 2 + 16 L 4 ) .
d ¯ ( t ) e d cv = d x ϕ ¯ c ( x , t ) ϕ v ( x , t ) ed cv + c . c . = d x d x 1 0 t d t F ( t ) G ¯ c ( x , x 1 , t , t ) G v ( x 1 , 0 , t , 0 ) G v ( x , 0 , t , 0 ) + c . c . ,
S c cl t ( [ x ] t t x x ) + σ I t ϵ G ω = S v cl t ( [ x ] t 0 x 0 ) ,
S c cl t ( [ x ] t t x x ) + σ I t + ϵ G = S v cl t ( [ x ] t 0 x 0 ) ,
S c cl x ( [ x ] t t x x ) + σ I x = S v cl x ( [ x ] t 0 x 0 ) ,
S c cl x ( [ x ] t t x x ) + σ I x = S v cl x ( [ x ] t 0 x 0 ) ,
S v cl x ( [ x ] t 0 x 0 ) S c cl x ( [ x ] t t x x ) S c cl x ( [ x ] t t x x ) = S v cl x ( [ x ] t 0 x 0 ) .
x e c ( τ ) = x + π + P m c ( τ t ) + λ ( τ ) λ ( t ) m c ,
x e ¯ v ( τ ) = x π P m c ( τ t ) λ ( τ ) λ ( t ) m c ,
p c ( [ x ] t t x x , τ = t ) p v ( [ x ] t 0 x 0 , τ = t ) = σ I x ( [ x ] t t x x ) ,
4 m c ( x x ) 2 π ρ W 0 2 L 2 = t t d s t t d s ( s s ) 2 ( x x t t + Λ ) Im ( e A B 3 / 2 ) ,
( x x ) 2 π ρ W 0 2 L 2 4 m c I 1 ( 1 + 2 π ρ W 0 2 L 2 4 m c I 2 ) 1 ,
B 3 2 1 2 3 / 2 L 3 + 3 i 2 m c L 2 ( ( s s ) Θ ( s s ) + ( s s ) Θ ( s s ) + ( s s ) 2 t t ) 2 3 / 2 L 3 ,
F ( t ) = F 0 sin 2 ( t / T t o t ) ( Θ ( t ) Θ ( t T t o t ) ) sin ( ω L t ) ,
Φ ¯ ( x , x , t , t ) ρ L 2 W 0 2 2 3 2 = t t d s t t d s d k γ F ( k ) x ( t ) = x x ( t ) = x [ d x ] e i S k ( [ x ] t t x x ) G c ( x , x , t , t ) ,
S k ( [ x ] t t x x ) = S c + k ( x ( s ) x ( s ) ) = t t [ x ˙ 2 2 m c x ( τ ) ( e F ( τ ) + F k ( τ , s , s ) ) ] d τ .
Φ ¯ ( x , x , t , t ) = ρ L 2 W 0 2 2 3 2 d s d s d k γ F ( k ) G k ( x , x , t , t ) G c ( x , x , t , t ) ,
G k ( x , x , t , t ) = A ( t , t ) e i S k cl ( [ x ] t t x x ) , G c ( x , x , t , t ) = A ( t , t ) e i S c cl ( [ x ] t t x x ) ,
Φ ¯ ( x , x , t , t ) = ρ L 2 W 0 2 2 3 2 t t d s t t d s d k γ F ( k ) e i ( S k cl S c cl ) .

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