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Thakur Vishal, Kaur Prabhdeep, Kaur Amanpreet, Kant Niti. Pulse Slippage of Resonant Third Harmonic Generation in Electron-Hole Plasma. Communications in Theoretical Physics, 2018, 70(3): 311  
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Pulse Slippage of Resonant Third Harmonic Generation in Electron-Hole Plasma
Thakur Vishal1, Kaur Prabhdeep1, Kaur Amanpreet2, Kant Niti1, †
Department of Physics, Lovely Professional University, G. T. Road, Phagwara -144411, Punjab, India
Department of Applied Sciences, Chandigarh Engineering College Landran, Mohali 140307, Punjab, India

 

† Corresponding author. E-mail: nitikant@yahoo.com

Supported by a financial grant from CSIR, New Delhi, India, under Project No. 03(1438)/18/EMR-II

Abstract
Abstract

Effect of pulse slippage on resonant third harmonic generation of a short pulse laser in electron-hole plasma in the presence of wiggler magnetic field has been investigated. The group velocity mismatch of the third harmonic pulse and the fundamental pulse is significant in electron hole plasma. As the third harmonic pulse has higher group velocity than that of fundamental pulse, therefore, it moves faster than the fundamental pulse. It gets slipped out of the domain of fundamental pulse and its amplitude saturates. Phase matching condition is satisfied by applying wiggler magnetic field, which provides additional angular momentum to the third harmonic photon to make the process resonant. Enhancement in the efficiency of third harmonic generation of an intense short pulse laser in electron-hole plasma embedded with a magnetic wiggler is seen.

Keyword:Wiggler magnetic field;pulse slippage;electron-hole plasma;third harmonic generation;laser
1 Introduction

The interaction of short pulse laser with plasma and semiconductor has been of a great importance since the invention of high power laser and it plays a major role in the study of various non-linear phenomena.[1–5] One of which is harmonic generation, which is observed under different conditions.[6–7] Harmonic generation during laser plasma interaction is non-linear phenomenon as the electric field of incident laser pulse is comparable to the interatomic field of the medium, hence, nonlinearity arises. Harmonic generation in semiconductor plasma has been a fascinating field of research for last few decades. Third harmonic generation plays an important role in the coherent output of infrared lasers to shorter wavelengths in the visible and near ultraviolet, which makes intense laser filaments useful for remote sensing applications.[8] During the interaction of short pulse laser with semiconductor, the electron by absorbing photon of required energy get excited to the higher levels and leaving behind holes.

Singh et al.[9] studied the effect of magnetic field on the generation of third harmonic in plasma. When intense Gaussian laser beam is incident in the plasma, it exerts the ponderomotive force on electrons which results in self-focusing of the beam. Magnetic field affects the dynamics of the electrons which in turn modified the plasma waves. The ponderomotive force at second harmonic produced the electron density perturbation that beats with oscillatory velocity of electron to generate the current which drives the third harmonic generation. Kant et al.[10] studied the interaction of laser pulse with semiconductor plasma embedded with wiggler magnetic field which provides the phase matching for third harmonic generation in two different ways i.e., one by producing a transverse third harmonic current and the other by imparting sufficient momentum to third harmonic photons. Rajput et al.[11] gave description about the effect of external wiggler magnetic field on the pulse slippage in the third harmonic generation. As the third harmonic pulse has higher group velocity as compared to the fundamental pulse, so it moves forward and slips out from the domain of the fundamental pulse. The external magnetic field is required to satisfy the phase matching condition in order to enhance the efficiency of third harmonic. Rathore et al.[12] studied the phase matched third harmonic generation of laser pulses in the high density quantum plasma in the presence of wiggler magnetic field. They used the quantum hydrodynamic model and considered the quantum effect including the 1/2 spin effect. Thakur et al.[13] have seen the effect of density transition on the generation of third harmonic. They studied that the density transition improves the conversion efficiency of the generated harmonic in the presence of wiggler magnetic field.

Liu et al.[14] studied the harmonic generation due to femto second laser pulses from Ti-Sapphire in BBO crystal. They accounted that due to high non linearity of BBO crystal laser light up to ultra violet region can be produced. They calculated the results by considering the effect of group velocity mismatch along with group velocity dispersion of second harmonic. Aggarwal et al.[15] studied the harmonic generation of laser pulses in atomic cluster embedded with wiggler magnetic field. They studied the consequence of change in cluster size on the efficiency of second harmonic. Rax et al.[16] demonstrated that the interaction of laser pulses with plasma induces the non-linearity at the third harmonic. But due to nonlinear dispersion the pump and harmonic are unlocked and results in saturation. To make the phase matching conditions satisfy they have used the concept of resonant density modulation. Gibbon[17] studied the harmonic generation of femtosecond laser interaction with overdense plasma. They used the concept of particle-in-box to study the harmonic generation by reflection of intense light from overdense plasma. They also concluded that there is no cut of frequency corresponding to lower intensity laser reflected from the upper density. By this method they studied the generation of water window x-rays with higher efficiency.

Thakur et al.[18] studied the propagation of high power laser pulse, through a semiconductor under chirped pulse effect and observed the very interesting non-linear phenomenon of second harmonic generation. Further, it was noticed that when a high intense laser beam propagates through a semiconductor under the combined effect of wiggler magnetic field, the transverse current density starts oscillating with a frequency twice of the laser beam. In case of semiconductor, as free charge carriers are created in pairs consisting of an electron and a hole. Absorption process is contributed by the mobility of the holes which, behaves like electrons. The total absorption coefficient comprises of the lattice and the carrier absorption coefficients. Further on heating more carriers are produced, which in turn reinforces the absorption. This is how electron-hole plasma helps to generate third harmonic generation.

In the present work we consider the propagation of short pulse laser in the electron-hole plasma with dielectric constant εl = 14 in the presence of wiggler magnetic field and observe the effect of pulse slippage on the generation of third harmonic in electron-hole plasma. We solve the corresponding equations for the generation of third harmonic. It has been noticed that presence of the wiggler magnetic field plays a very crucial role in the enhancement of the efficiency of third harmonic generation.

In Sec. 2, we describe the variation of normalized amplitude of the third-harmonic wave A3/A0 with normalized propagation distance ξ. The numerical results are discussed in Sec. 3, and the conclusions are presented in Sec. 4.

2 Theoretical Considerations

Non Linear Current Density and Third Harmonic Field

Consider the propagation of a strong short pulse laser in plasma in the existence of a wiggler magnetic field

where A1 (z,t) = F(z − vg1t), , ω1 is the laser frequency, k1 is the wave number in an n-type semiconductor, B0 is the wiggler magnetic field, c is the velocity of light in vacuum, ωp is the plasma frequency and k0 is the wiggler wave number. The nonlinear interaction of laser with the electrons in the company of wiggler magnetic field gives rise to the third harmonic, whose electric vector can be presumed as, where ω3 = 3ω1. The fundamental and third harmonic waves follow the linear dispersion relation, , where, ωl is the permittivity, ωp = (4πn0 e2/m)1/2 is the frequency of plasma, n0 is the electron density and e, m are the charge and mass of the electron respectively. The wave vector rises more than linearly with frequency ω; hence, k3 > 3k1. The modification in the variation of momentum can be provided to third harmonic photon on the condition of the wiggler magnetic field; then phase-matching condition becomes, k3 = 3k1 + k0, where, k0 satisfies the phase matching condition. Laser imparts oscillatory velocity at . From Eq. (1a), we get , where, beats with and employ ponderomotive force , at to generate an oscillatory velocity at given by, . It will induce the density perturbation at in the agreement of equation of continuity. We get, . To exert a ponderomotive force on electrons at , and beats together as, , which gives an oscillatory velocity to the electrons at , . Further and beat with and , to produce a transverse second harmonic ponderomotive force at , provides the oscillatory velocity to electron at

To exert a ponderomotive force on electrons at , and also beat together as, , giving oscillatory velocity at (, ), . and will beat together to exert a ponderomotive force , on electrons at , and will induce an oscillatory velocity . and beat with and , to enhance a transverse third harmonic ponderomotive force at , and expression is given by which generates non-linear oscillatory velocity at ,

The non-linear current density at the third harmonic comes out to be , where, , is administered by the equation of motion , and comes out as . Therefore, third harmonic current density can be deduced as,

For the third harmonic field , wave equation can be written as,

To solve Eq. (5), consider , we get

where, , and Now by considering the temporal profile of laser pulse to be Gaussian, where τ is the laser pulse length. By Presenting a new set of variables z − vg1t = ξ, and z = η, we can write

Equation (6) can be written as

where β = (1 − vg1/vg3), ξ0 = τ vg1. The complimentary solution of the above equation is and the particular integral for this is given by where . Hence, the complete solution of Eq. (7) can be written as

where erf(ψ) is the error function of argument ψ. The fraction of amplitudes of third harmonic and the fundamental is

where z′ = z/ξ0 and t′ = vg1t/ξ0 are dimensionless quantities.

Normalized amplitude of third harmonic comes out as,

3 Results and Discussion

In case of semiconductor, as free charge carriers are created in pairs consisting of an electron and a hole. Absorption process is contributed by the mobility of the holes which, behaves like electrons. The total absorption coefficient comprises of the lattice and the carrier absorption coefficients. Further, on heating, more carriers are produced which in turn reinforces the absorption. This is how electron-hole plasma helps to generate third harmonic generation. By considering the propagation of short pulse laser in the electron-hole plasma with dielectric constant εl = 14 in the presence of wiggler magnetic field, we have observed the effect of pulse slippage on the generation of third harmonic in electron-hole plasma. A combination of numerical parameters can be chosen depending upon experimental conveniences and requirements. For typical case, electron hole plasma (n-type Ge semiconductor) with a doping level n0 ≈ 1017 cm−3 is irradiated by a CO2 laser having angular frequency ω1 = 1.8 × 1014 rad/sec, with collisional frequency ν = 1012 s−1. Figure 1 shows the variation of the third harmonic pulse |A3/A0| and the fundamental pulse |A1/A0| with the normalized propagation distance z′ for t′ = 0, 4, 8, 12 with (ωc/ω1) = 0.6, , ωp/ω1 = 0.8. For t′ = 0, generation of third harmonic with very small amplitude takes place in the domain of fundamental pulse. At t′ = 4 there is increase in the amplitude of the third harmonic pulse and the generated third harmonic starts slipping from the domain of fundamental pulse. At t′ = 8, third harmonic pulse slips out of the domain of the fundamental pulse with appreciable increase in its amplitude. Whereas at t′ = 12 amplitude of the third harmonic pulse gets saturated as shown in Fig. 1(d). The presence of wiggler magnetic field also affects the efficiency of the third harmonic pulse in the electron hole plasma significantly. It has been found that as time passes the amplitude of the third harmonic pulse increases and pulse starts slipping out of the domain of fundamental laser pulse. It is because of the phase mismatch between the fundamental and third harmonic pulse as the group velocity of the third harmonic wave is greater than that of fundamental wave. Effect of pulse slippage is seen and found that the amplitude of third harmonic increases with time as depicted in Fig. 1. Similar results were found by Thakur et al.[13] Vij et al.[19] examined the effect of the pulse slippage on the resonant third-harmonic generation in clusters in the presence of a density ripple. They observed that the third-harmonic amplitude increased with time.

Fig. 1 (Color online) Variation of the normalized third harmonic amplitude |A3/A0| and the normalized fundamental laser amplitude |A1/A0| with the normalized propagation distance z′ for different values of normalized time t′ = 4, 8, and 12 with ωc/ω1 = 0.6, and ωp/ω1 = 0.8.

Figures 2 shows the variation of the normalized third harmonic amplitude |A3/A0| and the normalized fundamental amplitude |A1/A0| with the normalized propagation distance z′ for different values of plasma frequency ωp/ω1 = 0.4, 0.6 and 0.8 with ωc/ω1 = 0.8 and a sharp increase in the normalized third harmonic amplitude is seen for ωp/ω1 = 0.8 at z′ = 15 for different values of normalized time t′ = 4, 8 and 12. Therefore, third harmonic generation process is more pronounced in high density plasma. Similar results were obtained by Singh et al.[9] Figure 3 shows the variation of the normalized third harmonic amplitude |A3/A0| and the normalized fundamental laser amplitude |A1/A0| with the normalized propagation distance z′ for different values of applied wiggler magnetic field ωc/ω1 = 0.4, 0.6, and 0.8 with ωp/ω1 = 0.8 for different values of normalized time t′ = 4, 8, and 12. With the increase in wiggler magnetic field, there is also increase in normalized amplitude of third harmonic pulse |A3/A0|. Wiggler magnetic field plays a crucial role in the enhancement of the intensity of third harmonic generation. Therefore, efficiency of third harmonic generation increases greatly with the increase in wiggler magnetic field on account of the self-focusing of fundamental laser beam.[20] The dynamics of oscillating electrons is changed due to Lorentz force, which modifies the plasma wave which affects the third harmonic significantly. The efficiency of third harmonic generation increases with the strength of the wiggler magnetic field. Similar results were found by Thakur et al.[13] Also Jyoti et al.[11] successfully reported that enhancement in the efficiency of the third harmonic generation strongly depends on the strength of the wiggler magnetic field. Similar results were obtained by Aggarwal et al.[15] They have seen the enhancement in second harmonic generation with the effect of Wiggler magnetic field in clusters. It is because of the group velocity of second harmonic is greater than that of fundamental pulse, hence results in pulse slippage of the second harmonic wave. Wiggler strength ≥ 10 kG is required to achieve high efficiency of third harmonic generation.

Fig. 2 (Color online) Variation of the normalized third harmonic amplitude |A3/A0| and the normalized fundamental laser amplitude |A1/A0| with the normalized propagation distance z′ for different values of normalized time t′ = 4, 8, and 12 with ωc/ω1 = 0.8 and for different values of normalized plasma frequency ωp/ω1 = 0.4, 0.6, and 0.8.
Fig. 3 (Color online) Variation of the normalized third harmonic amplitude |A3/A0| and the normalized fundamental laser amplitude |A1/A0| with the normalized propagation distance z′ for different values of normalized time t′ = 4, 8, and 12 with ωp / ω1 = 0.8 and for different values of normalized wiggler magnetic field ωc / ω1 = 0.4, 0.6, and 0.8.
4 Conclusion

The mismatch of group velocities of the third harmonic pulse and the fundamental pulse is significant in the interaction of laser pulse with electron hole plasma. The group velocity of the third harmonic pulse is found to be greater than that of the fundamental pulse hence, the third harmonic pulse starts slipping out from the domain of fundamental pulse. Pulse slippage effect becomes more significant with the normalized distance of propagation. Third harmonic pulse slips out of the fundamental domain and gets saturated. This happens because of high intensity of fundamental pulse having Gaussian beam profile on the axis, it repels the electrons away from the axis, which causes to decrease in density which in turns decrease the plasma frequency hence, less energy is converted into third harmonic pulse. The presence of external magnetic field plays a very crucial role to enhance the efficiency of the third harmonic pulse. It is calculated that as the strength of the wiggler magnetic field is increased, the normalized amplitude of the third harmonic pulse also increases significantly.

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