Excitation of terahertz surface magnetoplasmons by nonlinear mixing of laser and its second harmonic on a rippled surface of n type semiconductor

We investigated the excitation of terahertz (THz) surface magneto plasmons (SMPs) by nonlinear mixing of laser and its second harmonic on a rippled surface of n-type semiconductor-free space interface. Obliquely incident p-polarized lasers exert a nonlinear ponderomotive force on free electrons of n-type semiconductor. Nonlinear ponderomotive force induces the oscillatory velocities at frequencies 2ω1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\omega _{1}$$\end{document} and (ω1-ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{1}-\omega _{2}$$\end{document}). These oscillatory velocities beat the modulated electron density nq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{q}$$\end{document} to get the charge density perturbation at (2ω1,2k1z+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2\omega _{1}, 2k_{1z}+q$$\end{document}) and (ω1-ω2,k1z-k2z+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{1}-\omega _{2}, k_{1z}-k_{2z}+q$$\end{document}). Perturbed charge density couples with the linear oscillatory velocities to produce a nonlinear current density, which resonantly derives THz SMPs at frequency ω=2ω1-ω2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega =2\omega _{1}-\omega _{2}$$\end{document} and propagation constant kz=2k1z-k2z+q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k_{z}=2k_{1z}-k_{2z}+q$$\end{document}. Here, q provides the extra wave number for the phase matching condition. The efficiency of THz SMPs wave amplitude was attained up to 9%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$9\%$$\end{document} and THz SMPs wave amplitude controlled by the electron cyclotron frequency ωce\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _{ce}$$\end{document} and incident angle θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta$$\end{document}.


Introduction
Generation and use of terahertz (THz) waves for diverse applications has quickly become a hot topic in science and technalogy in the last few decades. These waves have potential applications in medical imaging (Mittleman 2003), material diagnostics (Bergé et al. 2019), explosive detection (Shen et al. 2005) and spectroscopy (Jepsen et al. 2011). THz radiation can be produced by using a several methods such as optical rectification with crystals (Yeh et al. 2007;Fülöp et al. 2012), photoconductive antenna (Yardimci et al. 2015;Ropagnol et al. 2016), semiconductors (Gupta 2021), plasma-based techniques, and air or gas plasma interaction with lasers (Kumar et al. 2015), relativistic free electron laser interaction with ripple density plasma (Malik et al. 2017), interaction of high intensity short pulse with nonlinear dielectrics materials and plasma Sharma 2013, 2014;Bakhtiari et al. 2017). Wang et al. (2010) investigated excitation of THz emission by two-color femtosecond laser-induced filaments in the air in the presence of an external dc field. Sharma and Vijay (2018) studied the THz generation by two collinear laser beams over anharmonic carbon nanotube with power conversion efficiency up to the order of 10 −6 . Chauhan and Parashar (2015) reported the excitation of THz SPWs over thin films by nonlinear mixing of two laser beams. Jatav and Parashar (2019) investigated analytically the excitation of second harmonic generation by nonlinear mixing of two crossed surface plasma waves which propagate to each other at angle on the metal-air interface. Bhasin and Tripathi (2010) analytically examined the THz generation by optical rectification on the rippled density of the magnetized plasma. Kumar et al. (2016) studied the excitation of THz SMPs over the n-InSb by linear mode conversion of THz radiation. Srivastav and Panwar (2022) analytically derived generation of THz SMPs on rippled surface of n-InSb in the presence of external magnetic field by nonlinear mixing of two laser beams. Xie et al. (2006) demonstrated experimentally THz radiation generation by fundamental wave and its second harmonic in ionized plasma using four-wave mixing. Kumar (2013) examined theoretically excitation of THz radiation using the beat frequencies of laser and its second harmonic over a rippled density plasma and reported that the maximum THz radiation amplitude for the laser beams are in the same polarization state. In the present article, we investigated the excitation of THz SMPs by beating two lasers and frequency shifted second harmonic over a rippled surface of n-type semiconductor. Two lasers incident at an angle on the rippled surface of n-type semiconductor (cf Fig. 1). These lasers impart oscillatory velocities ⃗ v 1 and ⃗ v 2 and exert a nonlinear ponderomotive force on free electrons of n-type semiconductor and induce the perturbed charge densities n 2 1 and n 1 − 2 with frequencies of 2 1 and ( 1 − 2 ) and propagation constants of 2k 1z + q and k 1z − k 2z + q . Perturbed charge densities couple with the oscillatory velocities of electrons to get a nonlinear current density, which resonantly generates THz SMPs wave with frequency = 2 1 − 2 and propagation constant k z = 2k 1z − k 2z + q . The rippled wave number q gives the appropriate wave number for the excitation of THz SMPs wave. We derived the nonlinear ponderomotive force in sec. II. Sec.III represents the excitation process of THz SMPs and conclusion presented in sec.IV.

Nonlinear current density
Let us assume x = 0 is a interface of rippled n-type semiconductor ( x ≤ 0 ) and free space ( x > 0 ), and n-type semiconductor rippled surface with modulated perturbed electron density n = (n 0 ∕2)(1 + cos qz) (Kumar et al. 2016), qa ≥ 1 , q and a are the rippled wave number and amplitude, respectively in the existence of external magnetic B 0ŷ (cf. Fig. 1). Obliquely incident p-polarized laser beams electric field with frequencies j and propagation constant k j at incidence angle may be written as, where A j = A j,0 cos , k jx = k j cos , k jz = k j sin , k j = j ∕c and, j = 1, 2 belong to the first laser and second laser, respectively. Transmitted electric field of incidence lasers are given as, 68 , the electron cyclotron frequency is ce , is the collision frequency of electrons, p = √ (n 0 e 2 )∕(m * e 0 r ) is the electron plasma frequency, m * e = 0.014m e is the effective mass of the electron, n 0 = 2.59 × 10 24 m −3 is electron density for the n-InSb and the respective electron plasma frequency The transmission coefficients ( T j ) (Srivastav and Panwar 2022) are given by, The transmitted electric fields impart linear oscillatory velocity ⃗ v ,j to electrons in the rippled regime, ] by incidence lasers electric field at their frequency difference, ( 1 − 2 ) and the second harmonic frequency 2 1 , . and where F px couples with the ripple density n q to produce the perturbed electron density Similarly using the same process, the perturbed density n 2 1 , where In the rippled surface area the nonlinear current density ⃗ J nl = −1∕2(n 2 1 e⃗ v * 2 − n ( 1 − 2 ) e⃗ v 1 ) develops by the perturbed density from the Eqs. 10 and 11 at frequency = 2 1 − 2 . Nonlinear current density components are, where J x = n 2 1 (ṽ x ,2 ) * + n ( 1 − 2 )ṽ x ,1 and J z = n 2 1 (ṽ z ,2 ) * + n ( 1 − 2 )ṽ z ,1

Generation of THz surface magnetoplasmon
THz SMPs wave develops by nonlinear current density ⃗ J nl at ( , k z ) in rippled area and let ⃗ E is the self-consistent THz electric field inside the rippled region. By using the Faraday's law, , the wave equation may be written as, Here, is n-type semiconductor effective permittivity at frequency and it has following components as, and xy = yx = yz = zy = 0 . By solving x and z components of Eq. 14, one can obtain, Here (x) gives the nonlinear current density in the confined ripple area and h is ripple height. In absence of nonlinear current in Eq. 15 then (15) r.h.s term is zero and the solution of THz SMPs may be taken as, z − ( 2 ∕c 2 ) and decaying constants satisfy the SMPs dispersion relation (Brion et al. 1972;Srivastav and Panwar 2022) Figure 2 represents the dispersion curve of SMPs (eq. 18), dispersion curve of electromagnetic wave in free space ( = k z c ) and cut off frequency ( 2 = 0 ) regime for the different values of normalized magnetic field ce ∕ p = 0.00 (black line), ce ∕ p = 0.04 (red dotted line), ce ∕ p = 0.08 (green dashed line) and ce ∕ p = 0.12 (blue dot dashed line). Orange line represents the dispersion curve of electromagnetic waves in free space. Normalized propagation constant k z c∕ p varies linearly with normalized THz frequency ∕ p and shifts away from the dispersion curve of electromagnetic waves in free space for higher normalized THz frequency ∕ p . Cut off frequency of SMPs appears with 2 = 0 i.e. k 2 z − ( 2 ∕c 2 )(( 2 xz + 2 xx )∕ xx ) . In this Fig. 2, the nearly straight lines represent the cut off frequency of THz SMPs for the different values of the normalized magnetic field. We observed that the cut off frequency of THz SMPs decreases with increase in normalized magnetic field and due to this reason normalized THz frequency ∕ p drops with increase in normalized magnetic field ce ∕ p . For the excitation of THz SMPs, phase matching condition requires, Figure 3 shows the variation ratio of THz frequency ∕ p with ripple wave vector qc∕ p at angle = 60 0 for the normalized cyclotron frequency ce ∕ p = 0.00, 0.04, 0.08 , and 0.12. Eq. 19 clearly shows that the ripple wave vector q directly proportional to the propagation constant k z due to this reason in Fig. 3 ∕ p drops with increase in ce ∕ p . Figure 4 shows the variation ratio of incidence THz frequency ∕ p with ripple wave vector qc∕ p at particular normalized cyclotron frequency ce ∕ p = 0.1 for a various values of incidence angle = 30 0 , 45 0 and 60 0 . A close-up of the graph is shown in the inset and we observe that the ripple wave number q varies linearly with frequency ∕ p and ∕ p increases slightly with increase in incidence angle .
The THz SMPs of ripple wave so the THz SMPs mode structure is stable when r.h.s of Eq. 15 is finite , Solving Eq. 15 with Eq. 20 and letting k z → (k z − i ∕ z) , we get Integrating Eq. 21 from −∞ to ∞ and multiplying by ⃗ * (x)dx , we obtain, ,zẑ dx and I 3 = ∫ ∞ −∞ ⃗ * (x) ⋅ J nl ,xx dx Eq. 22 integrats from 0 to d, the THz SMPs wave amplitude becomes as, where d is illumination length.
We plotted the normalized amplitude of THz SMPs wave |A∕A 1 | with respect to normalized electron cyclotron frequency ce ∕ p and angle of incidence for the following laser parameters, 1 = 10.64 m , 2 = 9.2 m corresponding to the CO 2 laser, I = 2 × 10 15 W∕cm 2 in Fig. 5 and 6. The variations of normalized THz SMPs wave amplitude |A∕A 1 | with repect to the normalized electron cyclotron frequency ce ∕ p for the THz frequencies = 3THz, 4THz, 5THz and 6THz at angle of incidence = 60 • as plotted in Fig. 5. Normalized amplitude increases with the increase in value of ce ∕ p and attains the maximum value at ce ∕ p ≈ 0.11 ( B 0 ≈ 0.2696T ) and then start decreasing with further increase in value Fig. 6, we plotted the variation of normalized amplitude of THz SMPs wave |A∕A 1 | with respect to the angle of incidence for THz frequencies = 3THz, 4THz, 5THz and 6THz at normalized electron cyclotron frequency ce ∕ p = 0.1 . In the Fig. 6, normalized THz SMPs wave amplitude swiftly grows with incident angle and reaches to a maximum value and then abruptly decreases to zero because of its proportionality to the transmission coefficient.

Conclusion
THz waves of sub THz and 1.5 THz can be produced experimentally as well as theoretically by using the photonics and electronics techniques namely photoconductive antenna (Isgandarov et al. 2021;Chizhov et al. 2022), optical rectification (Avetisyan et al. 2017;Guiramand et al. 2022). SPWs in THz frequency regime can also be excited by laser-plasma interaction (Varshney et al. 2022) and electron beam-plasma interaction (Shu et al. 2016). In this paper, we analytically formulated the excitation of THz SMPs by the beating of a laser and its second harmonic over a magnetized n-InSb. Dispersion curve follows the electromagnetic wave in low frequency limit and further starts drop with increase in normalized applied magnetic field at higher normalized propagation constant as shown in Fig. 2. In Fig. 3, normalized ripple wave number for the resonant excitation increases with increase in normalized THz frequency and further decreases with normalized magnetic field for large values of normalized propagation constant. Normalized ripple wave number gets slightly increase with angle of incidence as shown in Fig. 4. Figure 5 shows the normalized THz SMPs wave amplitude increases with increase in normalized magnetic field upto a maximum value and falls off further with increase in normalized magnetic field. Normalized THz SMPs wave amplitude also increases with increase in angle of incidence and after a maximum value it starts fall off further with increase in angle of incidence as shown in Fig. 6. THz SMPs wave amplitude of higher frequency enhanced by the applied external magnetic field with efficiency ≈ |A∕A 1 | 2 upto the 9% for the magnetic field B 0 = 0.2942 T. In the presence of an external magnetic field, the THz SMPs amplitude is 10 3 times greater than the THz amplitude reported by Kumar (2013). THz SMPs amplitude strongly depends upon the external magnetic field and angle of incidence of lasers. This study could be useful in the application of THz detectors, sensors, THz plasmonic devices, and THz communication (Pitchappa et al. 2021;Gu et al. 2012;Bhattacharya et al. 2021Bhattacharya et al. , 2022.