Difference frequency sideband generation in semiconductors

Houquan Liu; Chuanxin Teng; Hongyan Yang; Hongchang Deng; Ronghui Xu; Shijie Deng; Li Zhang; Ming Chen; Libo Yuan

doi:10.1364/OSAC.2.000244

1. Introduction

Recently, the extensive study of the interaction of intense electromagnetic fields with semiconductors results in the reveal of many new effects, including dynamical Franz-Keldysh effect [1,2], non-linear excitonic effects [3,4], high-order harmonic generation [5–7], etc. The high-order THz sideband generation (HSG) is one of the most prominent effects [8–14], in which semiconductors are simultaneously illuminated by a weak near-infrared (NIR) laser with frequency ω_NIR and an intense THz field with frequency ω_THz. The NIR laser excites electron–hole pairs, which will be accelerated by the intense THz field. The recombination of the generated electron–hole pairs will then produce sidebands with frequency evenly spaced by twice the THz frequency, i.e. ω = ω_NIR + 2Nω_THz. HSG has received strong attentions thanks to its potential usefulness in many applications [9,15–19]. For example, since the sideband spectrum is a frequency comb, the HSG is expected to be useful in wide-band optical multiplexer. Moreover, considering the intense THz field as a controlling electric field, the HSG is a high speed electro-optical modulation process, which can be utilized to realize Tbit/s light modulation in optical communication.

In this work, we propose another driving scheme to obtain a cousin phenomenon of HSG, named difference frequency sideband generation (DSG), in which an intense mid-infrared (MIR) bichromatic laser with frequencies ω₁ and ω₂ (the frequency difference, i.e. Ω = ω₁-ω₂, is in dozens of GHz frequency range) is employed to replace the intense THz field. In this driving scheme, the electron–hole pairs created by the NIR laser are accelerated by the intense MIR laser, and the recombination of the electron–hole pairs emits sidebands with frequency evenly spaced by Ω, i.e. ω = ω_NIR + NΩ. Figure 1

Fig. 1 The schematic diagram of the frequency spectrum of the DSG.

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gives the schematic diagram of the frequency spectrum of the DSG. Similar to HSG, the generated sideband spectrum in DSG is also a frequency comb. However, the frequency spacing between the teeth of this comb isdifferent with that of the HSG. It enables the DSG’s capability to generate a new kind of frequency comb. Actually, we find that the DSG is more suitable for generating frequency comb with frequency spacing of dozens of GHz rather than THz, while the HSG is the opposite (more detail about this will be given in the following). This makes us think that the DSG is a new effect worthy of detailed study.

Similar to the HSG, the emitted sideband spectrum in DSG exhibits a nonperturbative plateau, where the intensity of the sidebands remains approximately constant up to a cutoff frequency. The location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser are discussed via semi-classical saddle-point method. We find that a positive detuning will induce equal frequency shift of the upper and lower cutoffs relative to the NIR laser frequency, thus the bandwidth of the sideband spectrum of the DSG is independent of the positive detuning. In the following, we begin our discussion.

2. Implement method of the driving scheme

In our driving scheme, the MIR bichromatic laser is one of the most important elements. Generally, a coherent MIR bichromatic laser can be written as

F (t) = F_{1} e^{i θ} \cos (ω_{1} t) + F_{2} \cos (ω_{2} t),

where F₁ and F₂ are the amplitudes of the two frequency components, and

θ

is their initial phase difference. If the laser is a continuous laser or a pulse with the pulse duration much larger than

T = 2 π / (ω_{1} - ω_{2})

, the relative phase between the two frequency components will appear equivalently from 0 to 2π during the duration. In this case, there should be no influence of

θ

on the DSG. If the laser is a short pulse whose duration is about several T or shorter than T, the relative phase no longer appears equivalently from 0 to 2π, there must be influence of

θ

on the DSG, which is the so-called carrier envelope phase effect. In this paper, we consider only the former case, i.e. we do not pay our attention to the carrier envelope phase effect. Thus, without loss of generality we can assume

θ = 0

.

If the two frequency components of the MIR bichromatic laser in Eq. (1) come from two independent laser sources, it is difficult to ensure the coherence of the two frequency components. To overcome this, we propose to generate the MIR bichromatic laser through a cosine amplitude modulation of a monochromatic MIR laser. Considering the monochromatic MIR laser is $F_{0} (t) = 2 F \cos (ω_{a} t)$ with ω_a = (ω₁ + ω₂)/2 being the average frequency and the modulation frequency of the modulator is $ω_{\mod} = Ω / 2$ , the amplitude modulation and the obtained MIR bichromatic laser can be expressed as

F (t) = F_{0} (t) \cos (ω_{\mod} t) = 2 F \cos (ω_{a} t) \cos (ω_{\mod} t) = F \cos (ω_{1} t) + F \cos (ω_{2} t) .

In Eq. (2), the amplitudes of the two frequency components are the same, therefore in the following discussion we consider only the situation of F₁ = F₂ = F.

3. Comparison between DSG and HSG

As we propose to obtain the MIR bichromatic laser by modulating the amplitude of a monochromatic MIR laser beam, the frequency spacing of the comb spectrum of the DSG is determined by the modulation frequency of the modulator. Since the modulation frequency of current commercial electro-optical modulator is about 1~100GHz, the DSG is suitable for generating frequency comb with dozens of GHz frequency spacing. While in contrast, the HSG is not suitable for producing such kind of frequency comb when considering device integration. To illustrate this, we assume to generate a frequency comb with 100GHz frequency spacing using the HSG. Since the frequency spacing of the comb spectrum of the HSG is twice of the driving field (intense THz field) frequency, the frequency of the driving field should be chosen as 50GHz. The corresponding wavelength of this field is about 6 mm. Due to the limitation of diffraction limit, the focal spot obtained by focusing this field is about several cm in size, which is obviously not suitable for fabricating integrated optical devices. Therefore, to meet the current demand of high integration devices, the DSG is more suitable than the HSG for generating frequency comb with dozens of GHz frequency spacing.

4. Theories, numerical results, and discussions of DSG

In the above sections, some qualitative properties about the DSG have been discussed. In this section, we investigate the DSG quantitatively. To achieve this, we consider an effective mass model for a parabolic band material driven by the intense MIR bichromatic laser Eq. (2). The parabolic band approximation is reasonable under near resonance excitation and could significantly simplify our following calculation (leading to the integration over k in Eq. (6) can be performed analytically). The exciting NIR laser used is a continuous wave with the form of F_NIR(t) = F_NIRexp(−iω_NIRt). In the single active electron approximation, the optical response of the semiconductor is determined by two-band (conduction band and valence band) Bloch equation

i ℏ \frac{\partial ρ_{c v} (k, t)}{\partial t} = {\frac{1}{2 m_{c v}} {[ℏ k - e A (t)]}^{2} + E_{g} - i ℏ γ_{2}} ρ_{c v} (k, t) + d_{c v} \cdot F_{NIR} (t),

where ρ_cv = <c|ρ|ν> is the nondiagonal element of the density matrix with |c> and |v> respectively denoting the wave function of conduction band and valence band electrons, A(t) = -Fsin(ω₁t)/ω₁-Fsin(ω₂t)/ω₂ is the vector-potential of F(t), m_cv is the reduced mass of electron-hole pair, E_g is the band gap of the semiconductor, γ₂ is interband dephasing rate, and d_cv is the interband transition dipole moment.

It should be noted that the excitation of electron–hole pair by the intense MIR laser is ignored in Eq. (3). This is reasonable since the recombination of the MIR laser excited electron–hole pair will emit high harmonics of the MIR laser frequency (high order harmonic generation) rather than the difference frequency sideband. In the meantime, there are of course influences of the MIR laser excitation on the DSG. It plays its role via exciting high carrier densities, increasing the carrier-carrier scattering rate, then leading to a high dephasing rate, which can drastically suppress the sideband generation. Thus, in the DSG, the intensity of the MIR laser should be appropriate. More details about this will be discussed in the following.

The Bloch Eq. (3) can be solved exactly. The expression of ρ_cv can be found to be

ρ_{c v} (k, t) = - \frac{i}{ℏ} \int_{0}^{\infty} e^{- i S (k, t, τ) / ℏ - i ω t - γ_{2} τ} d_{c v} \cdot F_{NIR} d τ,

with the semiclassical action

S (k, t, τ) = \int_{t - τ}^{t} \frac{{[ℏ k - e A (t^{″})]}^{2}}{2 m_{c v}} d t^{″} - (ℏ ω_{NIR} - E_{g}) τ - ℏ (ω - ω_{NIR}) t,

where

ℏ (ω - ω_{NIR})

is the sideband energy relative to the NIR laser frequency and, τ denotes the delay between the recombination and the creation of the electron-hole pair. The interband polarization is given by the expectation value of the dipole moment operator. It reads

P (t) = \frac{1}{{(2 π)}^{3}} \int d_{c v}^{*} \cdot ρ_{c v} (k, t) d^{3} k

Here we consider that the working medium is bulk material and the integral over k is 3D integral. If the working medium is 2D material, such as semiconductor quantum well, the integral of k needs to be converted into 2D integral. Having Eq. (6), the sideband spectrum can be calculated via the Fourier transform of the polarization. Substituting Eqs. (4) and (5) into Eq. (6), then performing the integral over k, the polarization becomes

\begin{array}{l} P (t) = - \frac{i d_{c v}^{*} d_{c v} \cdot F_{NIR}}{{(2 π)}^{3} ℏ} \int_{0}^{\infty} {\sqrt{\frac{2 π m_{c v}}{i ℏ τ}}}^{3} e^{i (ω_{NIR} - E_{g} / ℏ) τ - i ω_{NIR} t - γ_{2} τ} e^{i κ (τ)}, \\ \times e^{i ξ_{1} (τ) \cos (2 ω_{1} t - ω_{1} τ)} e^{i ξ_{2} (τ) \cos (2 ω_{2} t - ω_{2} τ)} e^{i ξ_{a} (τ) \cos (2 ω_{a} t - ω_{a} τ)} e^{i ξ_{Ω} (τ) \cos (Ω t - Ω τ / 2)} d τ \end{array}

where

κ (τ),

ξ_{1} (τ),

ξ_{2} (τ),

ξ_{a} (τ)

and

ξ_{Ω} (τ)

are introduced auxiliary functions, which are

κ (τ) = \frac{e^{2} F^{2}}{2 m_{c v} ℏ τ} [\frac{1 - \cos (ω_{1} τ)}{ω_{1}^{4}} + \frac{1 - \cos (ω_{2} τ)}{ω_{2}^{4}}] - \frac{e^{2} F^{2} τ}{4 ℏ m_{c v}} {\frac{1}{ω_{1}^{2}} + \frac{1}{ω_{2}^{2}}},

ξ_{a} (τ) = \frac{e^{2} F^{2}}{m_{c v} ℏ ω_{1} ω_{2}} {\frac{1}{2 ω_{a}} \sin (ω_{a} τ) - \frac{2}{ω_{1} ω_{2} τ} \sin (\frac{ω_{2} τ}{2}) \sin (\frac{ω_{1} τ}{2})},

ξ_{1} (τ) = \frac{e^{2} F^{2}}{2 m_{c v} ℏ ω_{1}^{3}} {\frac{1}{2} \sin (ω_{1} τ) - \frac{1}{ω_{1} τ} [1 - \cos (ω_{1} τ)]},

ξ_{2} (τ) = \frac{e^{2} F^{2}}{2 m_{c v} ℏ ω_{2}^{3}} {\frac{1}{2} \sin (ω_{2} τ) - \frac{1}{ω_{2} τ} [1 - \cos (ω_{2} τ)]},

ξ_{Ω} (τ) = \frac{e^{2} F^{2}}{m_{c v} ℏ ω_{1} ω_{2}} {\frac{2}{ω_{1} ω_{2} τ} \sin (\frac{ω_{2} τ}{2}) \sin (\frac{ω_{1} τ}{2}) - \frac{1}{Ω} \sin (\frac{Ω τ}{2})} .

According to the property of Bessel function

e^{i β \cos α} = \sum i^{n} J_{n} (β) e^{i n α}

, it’s easy to obtain the Fourier transform of the interband polarization, which reads

\begin{array}{l} P (ω_{NIR} + N_{a} 2 ω_{a} + N_{1} 2 ω_{1} + N_{2} 2 ω_{2} + N_{Ω} Ω) = \frac{- i d_{c v}^{*} d_{c v} \cdot F_{NIR}}{{(2 π)}^{3} ℏ} \int_{0}^{\infty} {\sqrt{\frac{2 π m_{c v}}{i ℏ τ}}}^{3} e^{i (ω_{NIR} - E_{g} / ℏ) τ - γ_{2} τ} e^{i κ (τ)} \\ \times i^{- N_{a} - N_{1} - N_{2} - N_{Ω}} J_{- N_{a}} {ξ_{a} (τ)} J_{- N_{1}} {ξ_{1} (τ)} J_{- N_{2}} {ξ_{2} (τ)} J_{- N_{Ω}} {ξ_{Ω} (τ)} e^{i (N_{a} ω_{a} + N_{1} ω_{1} + N_{2} ω_{2} + N_{Ω} Ω / 2) τ} d τ \end{array}

The Eq. (9) is inspirational, in which N_Ω determines the order of the difference frequency sidebands, while N_a, N₁ and N₂ determine the order of high harmonic of frequencies ω_a, ω₁ and ω₂, respectively. Since the cutoff frequency of the high harmonic generation is about 3.17U_p with U_p = e²F²/4m_cvω_a² being the ponderomotive energy, appropriate electric field intensity F and frequencies of the MIR laser could ensure that the vast majority of the polarization of Eq. (9) comes from N_a = N₁ = N₂ = 0, i.e. the relative intense of the harmonic is very low. This can be satisfied by choosing F = 8x10²kV.cm⁻¹ and

ℏ ω_{1} = 301 meV,

ℏ ω_{2} = 300 meV

(in the following, we will use these parameters for calculating the DSG sideband intensity), which gives

3.17 U_{p} / ℏ ω_{a} \approx 0.25

for GaAs crystal with m_cv = 0.058m_e, where m_e is free electron mass. In this case, the polarization becomes

\begin{array}{l} P (ω_{NIR} + N_{Ω} Ω) = - \frac{i d_{c v}^{*} d_{c v} \cdot F_{NIR}}{{(2 π)}^{3} ℏ} \int_{0}^{\infty} {\sqrt{\frac{2 π m_{c v}}{i ℏ τ}}}^{3} e^{i (ω_{NIR} - E_{g} / ℏ) τ - γ_{2} τ} e^{i κ (τ)} \\ \times i^{- N_{Ω}} J_{0} {ξ_{a} (τ)} J_{0} {ξ_{1} (τ)} J_{0} {ξ_{2} (τ)} J_{- N_{Ω}} {ξ_{Ω} (τ)} e^{i N_{Ω} Ω τ / 2} d τ \end{array}

Until now, according to Eq. (10), a numerical calculation can be performed to explore the sideband intensity and spectrum shape. Since in our driving configuration, the MIR laser is intense, it may excite high carrier densities, leading to a high dephasing rate, which can drastically suppress the sideband generation (see the supplementary material of Ref [15].). So, to avoid the high dephasing, the electric field F used in our calculation should satisfy that the concentration of exciton created by the MIR laser is low. An appropriate F could be obtained through a simple estimate as follows. Firstly, for the case of exciting a two-level system by light with frequency ω, the optical susceptibility is

χ \propto 1 / (ℏ ω - ℏ ω_{0} + i ℏ γ_{2})

[20], where ω₀ is the resonate frequency. Since

ℏ ω_{0} = E_{g}

in semiconductors, the linear optical absorption in semiconductors is

α \propto Im χ \propto ℏ γ_{2} / [{(ℏ ω - E_{g})}^{2} + {(ℏ γ_{2})}^{2}]

. We then consider that GaAs is the working material (whose band gap is E_g = 1.519eV) and choose that the NIR laser is near-resonant, the frequencies of the MIR laser are

ℏ ω_{1} ~ ℏ ω_{2} ~ 300 meV

, and the NIR laser intensity is ~10⁵W·cm⁻² (it is same as the NIR laser intensity of the HSG experiment in Ref [11], under which

ℏ γ_{2}

is about several meV and the sideband generation can occur). Under these conditions, we have

ℏ ω_{NIR} - E_{g} ~ ℏ γ_{2} ~ meV

. Therefore, it can be calculated that the absorption to the NIR laser is about six orders of magnitude larger than that to the MIR laser, i.e.

α_{NIR} / α_{MIR} ~ 10^{6}

. So, as long as the MIR laser intensity is no larger than six orders of magnitude of the NIR laser intensity, i.e. the MIR laser intensity is lower than 10¹¹W·cm⁻², the excitation of the MIR laser should be low and the majority of the excitons should be created by the NIR laser, which leads to the change of the dephasing rate induced by the MIR laser being ignorable. This can be satisfied by considering F = 8x10²kV.cm⁻¹, which results in the MIR laser intensity

I_{0} = 4 n c ε_{0} F^{2} \approx 2.24 \times 10^{10} W \cdot {cm}^{- 2}

, where

n

= 3.3 is refractive index, ε₀ is the vacuum permittivity and c is the speed of light in vacuum. With these knowledge, we calculate the sideband intensity by using

m_{c v} = 0.058 m_{e}

,

d_{c v} = 6.7 e \overset{\circ}{A}

,

ℏ γ_{2} = 1 meV

,

E_{g} = 1.519 eV

,

F = 8 \times 10^{2} kV \cdot {cm}^{- 1}

,

ℏ ω_{1} = 301 meV

,

ℏ ω_{2} = 300 meV

(

ω_{\mod} = Ω / 2 \approx 120 GHz

in this case) and

ℏ ω_{NIR} - E_{g} = 0

. The results are shown in Fig. 2

Fig. 2 The intensity of the sideband spectrum. The black arrow gives the position of the cutoff frequency of the spectrum plateau, which is $N_{\max} ℏ Ω = 4 U_{p}$ .

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, where we have introduced

χ (ω_{NIR} + N_{Ω} Ω) = P (ω_{NIR} + N_{Ω} Ω) / ε_{0} F_{NIR}

to represent the sideband intensity.

The numerical results in Fig. 2 clearly show that the DSG can occur in our MIR bichromatic laser driving scheme. We can see that the emitted sideband spectrum exhibits a plateau, where the intensity of the sidebands remains approximately constant up to a cutoff frequency. Similar to the HSG, the location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser can be discussed via semi-classical saddle-point method. Firstly, by minimizing the action, we have saddle point equation

\nabla_{k} S (k, t, τ) = 0 \Rightarrow k = \frac{e}{ℏ τ} \int_{t - τ}^{t} A (t^{″}) d t^{″} .

The physical meaning of this equation is same as that interpreted in Ref [9]. It means that the electron must return the position of the hole to recombine. Substituting Eq. (11) into Eq. (5), the action becomes to be

\begin{array}{l} S (t, τ) = - ℏ κ (τ) - ℏ ξ_{Ω} (τ) \cos (Ω t - Ω τ / 2) - ℏ ξ_{1} (τ) \cos (2 ω_{1} t - ω_{1} τ) \\ - ℏ ξ_{2} (τ) \cos (2 ω_{2} t - ω_{2} τ) - ℏ ξ_{0} (τ) \cos (2 ω_{a} t - ω_{a} τ) - Δ τ \\ - N_{a} 2 ℏ ω_{a} t - N_{1} 2 ℏ ω_{1} t - N_{2} 2 ℏ ω_{2} t - N_{Ω} ℏ Ω t \end{array}

where

ℏ (ω - ω_{NIR}) = N_{a} 2 ℏ ω_{a} + N_{1} 2 ℏ ω_{1} + N_{2} 2 ℏ ω_{2} + N_{Ω} ℏ Ω

and

Δ = ℏ ω_{NIR} - E_{g}

are introduced. Equation (12) determines the phase of the sideband and, if we consider only the difference frequency sideband of the fundamental frequency ω_NIR, the high frequency parts of Eq. (12) should be zero, i.e. N_a = N₁ = N₂ = 0. Thus, the action becomes

S (t, τ) = - ℏ κ (τ) - ℏ ξ_{Ω} (τ) \cos (Ω t - Ω τ / 2) - Δ τ - N_{Ω} ℏ Ω t

In addition, to emit difference frequency sideband, the delay τ should be at about time scale of 1/Ω for the electron-hole pair to gain energy. With this knowledge, it can be estimated that

e^{2} F^{2} [1 - \cos (ω_{i} τ)] / 2 m_{c v} ℏ τ ω_{i}^{4} \approx 2 (U_{p} / ℏ ω_{i}) [1 - \cos (ω_{i} τ)] / ω_{i} τ \approx 1 \times 10^{- 3} ≪ 1

under the same calculation conditions of Fig. 2 (

U_{p} / ℏ ω_{i} \approx 0.08; ω_{i} τ \approx ω_{i} / Ω \approx 300

). Thus

κ (τ)

can be simplified as

κ (τ) \approx - \frac{e^{2} F^{2} τ}{4 ℏ m_{c v}} {\frac{1}{ω_{1}^{2}} + \frac{1}{ω_{2}^{2}}} \approx - \frac{2 U_{p} τ}{ℏ} .

Similarly,

ξ_{Ω} (τ)

can be simplified as

ξ_{Ω} (τ) \approx - \frac{e^{2} F^{2}}{m_{c v} ℏ ω_{1} ω_{2} Ω} \sin (\frac{Ω τ}{2}) \approx - \frac{4 U_{p}}{ℏ Ω} \sin (\frac{Ω τ}{2}) .

Substituting Eqs. (14) and (15) into Eq. (13), the action becomes

S (t, τ) = 2 U_{p} τ + \frac{4 U_{p}}{Ω} \sin (\frac{Ω τ}{2}) \cos (Ω t - Ω τ / 2) - Δ τ - N_{Ω} ℏ Ω t .

Taking derivative of action (16) with respect to t and τ, respectively, and then minimizing the action again, we can obtain other two saddle point equations

\partial_{τ} S (t, τ) = 0 \Rightarrow \cos (\frac{Ω τ}{2}) \cos (Ω t - \frac{Ω τ}{2}) + \sin (\frac{Ω τ}{2}) \sin (Ω t - \frac{Ω τ}{2}) = \frac{Δ - 2 U_{p}}{2 U_{p}},

\partial_{t} S (t, τ) = 0 \Rightarrow N_{Ω} ℏ Ω = - 4 U_{p} \sin (\frac{Ω τ}{2}) \sin (Ω t - Ω τ / 2)

Solving Eqs. (17a) and (17b) we have

N_{Ω} ℏ Ω = [1 - \cos (Ω τ)] (2 U_{p} - Δ) \pm \sin (Ω τ) \sqrt{4 U_{p} Δ - Δ^{2}}

According to Eq. (18), the dimensionless difference frequency sideband frequency shift $N_{Ω} ℏ Ω / U_{p}$ as a function of the delay time could be calculated numerically. The results are shown in Fig. 3

Fig. 3 The dimensionless sideband frequency shift $N_{Ω} ℏ Ω / U_{p}$ as a function of the delay time for various frequency detuning △. Where (a) △ = 0; (b) △ = 0.5U_p; (c) △ = U_p and (d) △ = 1.5U_p. The red and blue curves are the results of $N_{Ω} ℏ Ω = [1 - \cos (Ω τ)] (2 U_{p} - Δ) + \sin (Ω τ) \sqrt{4 U_{p} Δ - Δ^{2}}$ and $N_{Ω} ℏ Ω = [1 - \cos (Ω τ)] (2 U_{p} - Δ) - \sin (Ω τ) \sqrt{4 U_{p} Δ - Δ^{2}}$ respectively, which coincide with each other in (a).

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for various frequency detuning △. The results present that the cutoffs of the DSG plateau depend on the frequency detuning. When △ = 0, the upper cutoff frequency of the spectrum plateau is 4U_p and, the lower cutoff frequency is 0. This agrees with the result of Fig. 2. When △≠0, a positive detuning will induce equal shift of the upper and lower cutoffs, and the shift equals to △. To demonstrate this, we calculate the intensity of the difference frequency sideband spectrum under △ = 0.5U_p, U_p and 1.5U_p with same calculation conditions as Fig. 2. The numerical results are shown in Fig. 4(a)

Fig. 4 (a) The intensity of the sideband spectrum under △ = 0.5U_p, U_p and 1.5U_p. (b) The intensity of the sideband spectrum as a function of △ and the sideband frequency. The black and red dotted lines in (b) are respective the upper and lower cutoffs extracted through Eq. (15). The calculation conditions are same as that of Fig. 2 except △.

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. Here we declare that in order to keep the sideband spectrum of the three cases from overlapping each other so that the spectrum cutoffs can be displayed clearly, the sideband spectrum of △ = U_p has been shifted up by 1 and the sideband spectrum of △ = 1.5U_p has been shifted down by 1 in Fig. 4(a). We can see that the cutoffs of the calculated sideband spectrum in Fig. 4(a) agree well with the results of Figs. 3(b)-3(d). Moreover, it can be found that the bandwidth of the sideband spectrum is independent of the positive detuning and, the frequency range of the plateau is always from E_g to E_g + 4U_p. This can be seen clearly from Fig. 4(b), in which the color indicates the sideband intensity and, the red and black dotted line give the upper and lower cutoffs extracted through Eq. (18), respectively.

5. Conclusion

In conclusion, DSG is predicted when a semiconductor is excited by a NIR laser and driven by an intense MIR bichromatic laser at same time. The sideband spectrum of the DSG is a frequency comb which exhibits a nonperturbative plateau. The location of the cutoff frequency of the sideband plateau is discussed via semi-classical saddle-point method. This work provides a deeper insight in the aspect of the interaction of intense electromagnetic fields with semiconductors.

Funding

National Natural Science Foundations of China (NSFC) (61535004, 61735009, 61675052, 61705050, 61765004, 61640409, 11604050), Guangxi project (AD17195074), National Defense Foundation of China (6140414030102), National Key R&D Program of China (2017YFB0405501), Guangxi Natural Science Foundation of China (2017GXNSFAA198048, 2016GXNSFAA380006), Guangxi Project for ability enhancement of young and middle-aged university teacher (2018KY0200).

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Difference frequency sideband generation in semiconductors

Abstract

1. Introduction

2. Implement method of the driving scheme

3. Comparison between DSG and HSG

4. Theories, numerical results, and discussions of DSG

5. Conclusion

Funding

References

Cited By

Figures (4)

Equations (23)

OSA Continuum