Second Harmonic Generation Induced by Surface Plasma Wave on Metallic Surface in the Presence of Wiggler Magnetic Field

Under the influence of wiggler magnetic field, the phenomenon of second harmonic generation at the metal-semiconductor interface, induced by surface plasma wave (SPW) has been investigated. Metals like Cu, Ag and Al, each with a thin layer of n-InSb over it, are considered for our study. Laser light is incident on metal layered on glass prism in attenuated total reflection Kretschmann configuration (ATR) which generates SPW. The SPW further interacts nonlinearly with the electrons of n-type semiconductor layered over the metal, leading to second harmonic generation (SHG). The presence of an external wiggler magnetic field makes the process resonant and helps in phase matching. Relatively more enhancement in the amplitude of the second harmonic is observed for Cu-InSb as compared to Ag-InSb and Al-InSb. Numerical analysis shows that the enhancement in the amplitude of SHG increases with the wiggler magnetic field.


Introduction
Surface plasma waves (SPW) are electromagnetic waves that are formed by the interaction of laser with metals. These waves propagate along the interface of two different media and have amplitude larger than the amplitude of the laser [1][2]. The nonlinear interaction of laser with the matter has gained much attention in the past few decades because of its unique applications like high harmonic generation, laser driven plasma based accelerators, sensors, photoelectron spectroscopy, etc. [3][4][5][6][7]. Harmonic generation on different materials and in the presence of external agents like wiggler field has been studied by many researchers. Jha et. al. [8] studied SHG with an intense laser beam in magnetized plasma and found high conversion efficiency in the presence of the transverse magnetic field. Rajput et.al. [9] investigated the generation of third harmonic by a short pulse laser in plasma and observed that the presence of the wiggler magnetic field makes the generation process resonant accompanied by the increase in the efficiency of harmonic generation.
Vinay et.al. [10] investigated propagation of laser pulse through plasma and found that laser interacts nonlinearly with electrons exerting a ponderomotive force which in turn oscillates them.
It results in the harmonic generation. Abedi-Varaki [11] investigated the acceleration of electrons of metal by SPW in the presence of a helical magnetostatic wiggler and observed that under the effect of both SPW and wiggler field the electrons can travel more distance in the direction of propagation of laser. Vij et.al. [12] studied the production of Terahertz (THz) radiation by carbon nanotubes in the presence of a wiggler magnetic field and observed that the presence of the wiggler field enhances the efficiency of the generation of THz radiation. Abedi-Varaki and Jafari [13] studied the generation of THz radiations by mixing two Cosh-Gaussian laser beams in the presence of a wiggler magnetic field and observed the increase in efficiency with the increase in wiggler frequency. Ghimire et.al. [14] had investigated harmonic generation in solids and reported that terahertz or infrared fields are suitable for metals and semiconductors due to their small band gap.
The presence of external factors like wiggler magnetic field helps to make the process resonant.
Singh et. al. [15] studied the acceleration of electrons by a laser pulse in the presence of magnetic wiggler, in vacuum and plasma. Their work showed that the suitably tapered magnetic wiggler can maintain electron-laser resonance conditions for a longer time and electrons can gain sufficient energy.
In the present manuscript, second harmonic generation induced by surface plasma wave at metalsemiconductor interface in the presence of wiggler magnetic field has been studied. The phase matching condition is fulfilled by the application of the wiggler magnetic field. Here, in section 2, we have obtained the expression for normalized amplitude for second harmonic generation.
Results are discussed and presented in graphical form in section 3 and conclusion is presented in section 4.

Theoretical considerations
A thin metal layer over a glass prism with a thin layer of semiconductor over it has been considered in Kretschmann ATR configuration as shown in Fig.1, where, laser light is incident at an acute angle on metal glass interface leading to SPW in metal [16]. The em fields of laser interact with electrons of metal and produce SPW of nearly the same frequency as laser but with higher amplitude. The component of the wave vector of the laser along the surface matches with the surface plasma wave number, leading to efficient coupling [17]. A thin layer of n-InSb with the thickness of a few nanometers is considered above the metal surface. The electromagnetic fields of SPW at the metal-semiconductor interface interact with electrons of n-InSb. The semiconductor can accumulate energetic electrons which can easily tunnel through plasmonic entities [18]. Electric and magnetic fields of laser are considered as where, A is the amplitude of field of laser,  is the frequency of laser and 1 k is the propagation constant. These fields interact with electrons of metal and produces surface plasma waves where, s k is the propagation constant such that,  and s  as the permittivity of metal and semiconductor respectively and s  is the frequency of surface plasma wave. The frequency of SPW is nearly same as the frequency of the laser, so    s [16]. Amplitude of surface plasma wave s A depends on amplitude of incident laser A with n is a positive real number. Fields of surface plasma waves interacts with electrons of semiconductor and accelerates them. Acceleration so produced is calculated using equation of motion, The velocity gained by electrons is, As the wave vector increases more than linearly with incident frequency, the momentum is not conserved, i.e. the momentum of second harmonic is more than two times the momentum of incident photon. Here, which is a positive real number. Wiggler magnetic field is applied in transverse direction given by, where, 0 B is the amplitude of wiggler field and 0 k is the wiggler wave number. The presence of wiggler field helps to achieve the phase matching condition by providing additional momentum to the photons of second harmonic. The velocity 1 v beats with wiggler magnetic field of laser and generates ponderomotive force F , where, due to this force velocity of electrons changes and it is taken as ' To exert ponderomotive force on electrons at   and imparts velocity, NL v 2  to electrons given as, This give rise to second harmonic nonlinear current density at metal semiconductor interface, The linear current density due to self consistent second harmonic field The amplitude ' 2 A for second harmonic is calculated by using equation, By taking the laser profile to be Gaussian as and  is laser pulse width. Using a new set of variables , we get normalized second harmonic amplitude as, respectively. The values of other parameters for Fig.2(b) and Fig.2(c) are same as for Fig.2(a) As observed graphically under given conditions, increase in amplitude of SHG has been found more for Cu-InSb than for Ag-InSb and Al-InSb. The reason for this observation can be understood as the value of the conduction electron density for Cu is more as compared to Ag and Al so the fields of SPW of Cu are expected to be comparatively stronger and can efficiently interact with electrons of n-type semiconductor. As per the study by Takagi et al. [21] surface plasma waves in Kretschmann geometry shows good agreement with bulk dielectric function for Cu than Ag. Similarly in our work, we have seen better enhancement in the amplitude of second harmonic for Cu. Sharma et al. [22] studied SHG of cosh-Gaussian laser beam in magnetized plasma and found enhancement in efficiency in the presence of wiggler field.
Similarly, in the present work good enhancement in amplitude of second harmonic has been observed in the presence of wiggler field.

Conclusion
The electromagnetic fields of laser interact with electrons of metal and generate SPW, which are electromagnetic waves with higher amplitude. The SPW of different metals like Cu, Ag and Al interact nonlinearly with electrons of n-InSb, leading to second harmonic generation. The interaction has been studied in the presence of wiggler field with different amplitudes. It has been observed that amplitude increases sharply at the Cu-InSb interface as compared to Ag-InSb and Al-InSb. The enhancement in second harmonic amplitude has been observed with the increase in wiggler field strength. Presence of wiggler field helps in phase matching and resonates the process of second harmonic generation. Figure 1 Kretschmann con guration with thin layer of a metal and a semiconductor  Variation of normalized second harmonic amplitude with normalized propagation distance for Cu-InSb interface for different values of wiggler eld at time