Even-order high-harmonic generation from solids in velocity gauge

Jiayuan Cao; Na Li; Ya Bai; Peng Liu; Ruxin Li

doi:10.3788/COL202119.043201

Journals >Chinese Optics Letters >Volume 19 >Issue 4 >Page 043201 > Article

Chinese Optics Letters
Vol. 19, Issue 4, 043201 (2021)

Even-order high-harmonic generation from solids in velocity gauge | Editors' Pick

Jiayuan Cao^1,2, Na Li¹, Ya Bai^1,*, Peng Liu^1,2,**, and Ruxin Li¹

Author Affiliations

¹State Key Laboratory of High Field Laser Physics and CAS Center for Excellence in Ultra-Intense Laser Science, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China

²Center of Materials Science and Optoelectronics Engineering, University of Chinese Academy of Sciences, Beijing 100049, China

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DOI: 10.3788/COL202119.043201 Cite this Article Set citation alerts

Jiayuan Cao, Na Li, Ya Bai, Peng Liu, Ruxin Li, "Even-order high-harmonic generation from solids in velocity gauge," Chin. Opt. Lett. 19, 043201 (2021) Copy Citation Text

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Abstract

We present a velocity-gauge model for the generation of even-order high harmonics, and reveal that the even-order harmonics originate from the multiple-step transitions among the energy bands in momentum space, while the odd-order harmonics are mainly from direct transitions. The lower valence band is found vital for the generation of even harmonics. Relative intensity of even-order harmonics versus the odd orders is calculated and shows a growing trend as the laser field amplitude increases.

Keywords

even-order harmonics inversion asymmetry solid high-harmonic generation

V (x) = \sum_{R} \sum_{i = A, B, C} V_{i} [1 + \tanh (x - x_{i, R})] [1 - \tanh (x - x_{i, R})],

(1)

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\frac{\partial}{\partial t} P_{n, n'} (k, t) = - i ({\tilde{ϵ}}_{n, k} - {\tilde{ϵ}}_{n', k} - i / T_{2}) P_{n, n'} (k, t) + i A (t) [ρ_{n} (k, t) - ρ_{n'} (k, t)] p_{n, n'} (k) - i A (t) \sum_{m} [P_{m, n'} (k, t) p_{n, m} (k) - P_{n, m} (k, t) p_{m, n'} (k)],

(2)

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\frac{\partial}{\partial t} ρ_{n} (k, t) = - ρ_{n} (k, t) / T_{1} - 2 A (t) Im [\sum_{m} P_{n, m} (k, t) p_{m, n} (k)],

(3)

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j (t) \propto \sum_{k, n, n'} P_{n, n'}^{*} (k, t) p_{n, n'} (k) + \sum_{k, n} ρ_{n} (k, t) [p_{n, n} (k) + A (t)] .

(4)

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S (ω) \propto ω^{2} | j (ω) |^{2} .

(5)

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\frac{\partial}{\partial t} P_{c, v} (k, t) = - i ({\tilde{ε}}_{c, k} - {\tilde{ε}}_{v, k} - i / T_{2}) P_{c, v} (k, t) + i A (t) [1 - ρ (k, t)] p_{c, v} (k) .

(6)

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P_{c, v} (k, t) = i \int_{- \infty}^{t} [1 - ρ (k, t)] A (τ) p_{c, v} (k) d τ e^{i S} e^{- \frac{t - τ}{T_{2}}},

(7)

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j (t) \propto i \sum_{k} p_{c, v}^{*} (k) \int_{- \infty}^{t} [1 - ρ (k, t)] A (τ) p_{c, v} (k) d τ e^{i S} e^{- \frac{t - τ}{T_{2}}} + c.c .

(8)

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Jiayuan Cao, Na Li, Ya Bai, Peng Liu, Ruxin Li, "Even-order high-harmonic generation from solids in velocity gauge," Chin. Opt. Lett. 19, 043201 (2021)

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