In this paper, a near-infrared (NIR) laser (with frequency ωNIR) exciting and an intense mid-infrared (MIR) bichromatic laser (with frequencies ω1 and ω2) driving scheme is proposed in semiconductors. It is predicted that if the frequency difference between the two components of the MIR laser, i.e. Ω = ω1-ω2, is in dozens of GHz frequency range, the emitting light contains spectra with frequencies ω = ωNIR + NΩ, where N is integer. In analogy with high-order THz sideband generation (HSG) in semiconductors, this phenomenon is named difference frequency sideband generation (DSG). Similar to the HSG case, the emitted sideband spectrum in DSG exhibits a nonperturbative plateau, where the intensity of the sideband 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 the semi-classical saddle-point method.
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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 ωNIR and an intense THz field with frequency ωTHz. 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 = ωNIR + 2NωTHz. 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 = ωNIR + NΩ. Figure 1 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
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 with ωa = (ω1 + ω2)/2 being the average frequency and the modulation frequency of the modulator is , the amplitude modulation and the obtained MIR bichromatic laser can be expressed asEq. (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 FNIR(t) = FNIRexp(−iωNIRt). In the single active electron approximation, the optical response of the semiconductor is determined by two-band (conduction band and valence band) Bloch equation
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 ρcv can be found to beEq. (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 becomesEq. (9) is inspirational, in which NΩ determines the order of the difference frequency sidebands, while Na, N1 and N2 determine the order of high harmonic of frequencies ωa, ω1 and ω2, respectively. Since the cutoff frequency of the high harmonic generation is about 3.17Up with Up = e2F2/4mcvωa2 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 Na = 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 (in the following, we will use these parameters for calculating the DSG sideband intensity), which gives for GaAs crystal with mcv = 0.058me, where me is free electron mass. In this case, the polarization becomesEq. (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 .). 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 , where ω0 is the resonate frequency. Since in semiconductors, the linear optical absorption in semiconductors is . We then consider that GaAs is the working material (whose band gap is Eg = 1.519eV) and choose that the NIR laser is near-resonant, the frequencies of the MIR laser are , and the NIR laser intensity is ~105W·cm−2 (it is same as the NIR laser intensity of the HSG experiment in Ref , under which is about several meV and the sideband generation can occur). Under these conditions, we have . 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. . 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 , where = 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 , , , , , , (in this case) and . The results are shown in Fig. 2, where we have introducedto 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 equation9]. It means that the electron must return the position of the hole to recombine. Substituting Eq. (11) into Eq. (5), the action becomes to beEquation (12) determines the phase of the sideband and, if we consider only the difference frequency sideband of the fundamental frequency ωNIR, the high frequency parts of Eq. (12) should be zero, i.e. Na = N1 = N2 = 0. Thus, the action becomesFig. 2 (). Thus can be simplified asEqs. (14) and (15) into Eq. (13), the action becomesEqs. (17a) and (17b) we have
According to Eq. (18), the dimensionless difference frequency sideband frequency shift as a function of the delay time could be calculated numerically. The results are shown in Fig. 3 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 4Up 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 △ = 0.5Up, Up and 1.5Up with same calculation conditions as Fig. 2. The numerical results are shown in Fig. 4(a). 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 △ = Up has been shifted up by 1 and the sideband spectrum of △ = 1.5Up 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 Eg to Eg + 4Up. 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.
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.
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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