Parametric dispersion and amplification of

Parametric dispersion and amplification in piezoelectric semiconductors

of acoustohelicon

waves

A. Neogi and S. Ghosh Schwl of Studies in Physics, Vikram University, Ujain 456010, India

(Received 29 August 1989; accepted for publication 13 August 1990) Assuming that the origin of the nonlinear interaction lies in the second-order optical susceptibility arising from the nonlinear induced current density and using the coupled-mode theory, the parametric dispersion and amplification of acoustohelicon waves is analytically investigated in a longitudinally magnetized piezoelectric semiconductor of noncentrosymmetric nature. The relevant experiments have not been reported. The threshold value of the pump electric field E,,,, and its corresponding excitation intensity is obtained. The longitudinal magnetic field decreases the required magnitude of E,,,, for the excitation of parametric amplification. The phenomenon of self-defocusing of the signal in the prevailing case is found to be a consequence of the negative dispersive characteristics exhibited by the acoustohelicon waves. Numerical analyses are performed for an InSb crystal at 77 K, duly irradiated by frequency-doubled pulsed 10.6~pm CO, lasers. The parametric gain constant is observed to be maximum when the cyclotron frequency w, attains the magnitude equal to that of wO, the incident laser frequency ( = 1.78~ lOI s ‘I.

I. INTRODUCTION Nonlinear optics has developed into a significant subfield of physics. It was opened up by the advent of lasers with high peak powers. The availability of tunable dye lasers has made detailed nonlinear spectroscopic studies possible throughout the visible region of spectrum, from 0.35 to 0.9 nm. Conversely, nonlinear techniques have extended the range of tunable coherent radiation. The study of nonlinear phenomena such as harmonic generation, parametric up and down conversions, and stimulated scattering processes of different orders has been made extending over the entire range of the spectrum, i.e., from extreme ultraviolet’ to the far infrared.’ Nonlinear optical processes are essential for the development of modulators and demodulators which are extensively used in optical communication systems.” The phenomena of parametric interaction exhibits a distinctive role in nonlinear optics. Parametric processes have been widely used to generate tunable laserlike radiation in a nonlinear crystal at a frequency that is not directly available from a laser source.4’s Parametric oscillators, amplifiers, optical phase conjugators, etc.,6-8 are the outcome of parametric interactions in a nonlinear medium. Besides these technological uses there are several other applications of parametric interactions in which basic scientists are interested. It is a fact that the origin of parametric interaction lies in the second-order optical susceptibility xc2’ of the medium. This lowest-order nonlinear susceptibility is a third rank tensor which is nonvanishing in a media which lacks inversion symmetry. The polarization which is quadratic in terms of field amplitude leads to the optical phenomena of second harmonic generation, sum and difference frequency mixing, as well as rectification of light. In general, the terms in xc2’ provide a coupling between sets of three electromagnetic waves; each of these waves is characterized by frequency wi, wave vector ki, state of polarization e,, as well as a complex amplitude E, = A, exp( hit). The theoretical study of sec-

ond-order nonlinear susceptibility x(” which gives rise to various nonlinear optical phenomena like parametric amplification, harmonic generation, etc., was made in different regimes by Flytzanis.’ The sum rules for nonlinear susceptibilities in solids has been studied by Peiponen.” The parametric amplification of acoustic waves in a piezoelectric semiconductor has been studied by Ghosh and Khan.” Economou and SpectorI have reported the parametric interaction of acoustic phonons with microwave electric fields. Ghosh and Agarwal’3,‘4 have reported the excitation of acoustohelicon wave by parametric and modulational interactions in magnetoactive piezoelectric semiconducting media. Recently the second-order nonlinear susceptibility originating from the finite induced current density in both unmagnetized and transversely magnetized semiconductors are reported by Aghamkar and Sen15 and Neogi and Ghosh,” respectively. They have studied the dispersive as well as absorptive characteristics of the interaction. The investigation carried out by Sen and Sen” further reveals that the parametric dispersion for both scattered electromagnetic wave and generated acoustic waves is possible only in case of noncentrosymmetric (NCS) crystals. The propagation of helicon waves is of substantial importance in the investigation of the fundamental properties of solids. It is also applicable in super-high-frequency (SHF) technology and for transmission purposes at microwave frequencies. The study of transverse phonon-helicon interactions in piezoelectric semiconductors has been one of the most active fields due to its vast potentialities in semiconductor diagnostics and devices.” It appears from the available literature that no attempt has been made to study second-order nonlinear susceptibility x(*) arising due to parametric interaction of acoustohelicon waves in longitudinally magnetized piezoelectric semiconductors. Motivated by the intense interest in the field of parametric interactions in semiconductors, we have taken up the investigation of second-order susceptibility x(*’ due to parametric interac-

J. Appt. Phys. 69 (I), 1 January 1991 0021-8979/91 /01061 61 @ 1991 American Institute of Physics Downloaded 02 Feb 2004 to 152.3.232.17. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

61

tion of acoustohelicon waves in a longitudinally magnetized piezoelectric crystal like n-InSb of NCS nature and reported the resultant dispersive and absorptive characteristics of the decay products. The nonlinearities taken into account in the present study are the electronic nonlinearity due to nonlinear current density and nonlinear polarization resulting from the effective nonlinear susceptibility of the medium, which is also the cause ofthe nonlinear coupling between the produced acoustic wave and the scattered transverse electromagnetic waves.

mode, and crystal elastic constant, respectively. The other basic equations used in this analysis are

%+w,, +e[Eo + %+w,

azu,

(vo XB,

),

au,

(4)

-uo

(5)

susceptibility

au +zy,p*+P*=c*,

aE

a*U

(la)

with p, y,, and C being the mass density of the crystal, phenomenological damping parameters of the acoustohelicon J. Appl. Phys., Vol. 69, No. 1,l January 1991

JE, -+aazU=

-nl

dX

E

e -,

(6)

E

ax2

(7) D = EE + P.

(8) Equations (2) and (3) are the zeroth- and first-order momentum transfer equations in which v0 and vl are the zeroth- and first-order oscillatory fluid velocities of the electron having effective mass m and charge - e. Y is the phenomenological electron collision frequency. Equation (4) represents the continuity equation in which n, and n, are the equilibrium and perturbed carrier densities, respectively. The space-charge field E is determined by the Poisson equation (6) in which ~.andPare the scalar dielectric and piezoelectric constants of the semiconductor, respectively. Equations (5) and (7) are the Maxwell equations in which J represents the sum of the linear and nonlinear induced conduction current densities. Equations (2)-(5) are soarranged that the terms on the right-hand side of equations represent the nonlinear contribution. These terms on the right-hand side of Eqs. (2)-(5) give the basic nonlinearities built into them because they involve the product of fields due to the laser and scattered waves. It is also a well-established fact that the optical nonlinearities of a plasma arise from the (V-V) v terms and the Lorentz force on the electrons in Eq. (3) .2’ In a piezoelectric semiconductor, the low-frequency phonon mode (w) as well as the pump electromagnetic mode (wg ) produce density perturbations (n, ) at the respective frequencies in the medium which can be obtained by using the standard approach.** We assume that the lowperturbations proportional to are frequency exp[ i( kx - wt) ] while high-frequency perturbations vary as exp( - hot), and using Eqs. (l)-( 8) in the collisiondominated regime ( v$-u, k*u, ), we get

a2n,

an,

-

(lb)

62

m

-+nez= at

We consider the well-known hydrodynamical model of a homogeneous n-type piezoelectric semiconductor plasma of infinite extent with electrons as carriers. This model restricts the validity of the analysis to the limit kZ< 1, where k is the wave vector and I is the mean free path of the electrons. In order to study the parametric excitation processes arising due to the effective nonlinear optical susceptibility (XEN ), the medium is subjected to a magnetic field B, parallel to the propagation vector k which is along the direction of a spauniform high-frequency pump electric field tially E, exp(iwt) (i.e., pump vector Ik, 1z-0). All these fields are applied along thex axis. We could neglect the nonuniformity of the high-frequency electric field under the dipole approximation when the excited sound and helicon waves have wavelengths which are very small as compared to the scale length of the electromagnetic field variation.lY We apply the coupled-mode theory*’ to obtain a simplified expression for the effective nonlinear susceptibility. The origin of this nonlinear optical susceptibility lies in the coupling of pump, signal, and helicon waves via density perturbation. According to the coupled-mode theory,20 if there are a couple of normal modes or waves Xand Y with respective frequencies wl and w2 in the linear system, then considering the situation in which these two normal modes interact with each other through the action of another wave Z with frequency wo, then X (or Y) are forced to oscillate at the beat frequency of Y (or X) and Z. If w0 is close to (wl + w2 ), then this forced oscillation resonates with the natural oscillation, leading to a resonant energy conversion between Z and the set of X and Y. Due to this exchange of energy X and Ymay get amplified. The applied pump wave E, gives rise to a time-varying electrostrictive strain and is thus capable of driving the acoustic waves in the medium. Assuming the acoustic wave generated internally to be a pure shear wave propagating along the cubic axis [OOl ] of the crystal, the lattice displacement u is then along the cubic axes [ 1 lo] and [ 1701 of the crystal. The equation of motion of the lattice in the piezoelectric crystal is then given by ’ at2

VI - A-

an,

=O,

+(v,xBo)]

+;[E,

=-

II. THEORETICAL FORMULATION A. Second-order

CvoxBo)]

m

at2

-

+y +

at

+

@pc* ‘o(Zro,+iv) #;I3 -2

a(k2CZ, -w~)(w+o,

X

n, --

+iv)

>

noeD

me

k 2wT5 UT = -B--, anI ax (k2c; - w2) A. Neogi and S. Ghosh

(9) 62

Downloaded 02 Feb 2004 to 152.3.232.17. Redistribution subject to AIP license or copyright, see http://jap.aip.org/jap/copyright.jsp

5 = w - kv,, and vg is the carwhereE= [ - (e/m)&], rier velocity under the influence of the pump field. tip [ = ( n,,e2/mq,,eL ) “’ 1 is the plasma frequency. w, ( = eB,/m) is the electron cyclotron frequency and CL [ = (lw”EL) - “‘1 the electromagnetic wave velocity in the crystal with ~~ as the relative dielectric constant of the lattice and E, is the absolute dielectric permittivity. In obtaining Eq. (9) we have taken u k = u, f iu,, and E, =E,fiE,, “t = vv f. iv,, B, =B,,+iB,, where plus and minus signs correspond to the right- and lefthand circular polarization, respectively, and used the following equations:

v, = - i(e/m) (Z/w) E (Z+iv+w,)

(10)

+

velocity of light in vaccum. In this present investigation, in order to study the effect of nonlinear conduction current density on the induced polarization in a longitudinally magnetized doped semiconductor, the effect of the transition dipole moment is neglected while analyzing parametric interaction in the crystal. The Stokes component of the induced conduction current density is given by J, = - n’ev,. Thus under the quasi-static approximation (14) yield the component of J, as J,

(14) Eqs. (13) and

w;w

= -

k2cZ,(WTtic

-iv)

and i($-or)E,

=n,ev,

+wkfiui.

The above equations are derived using Eqs. (3)-( 8). Here both vY and v, represent the sum of the slow and the fast components of v along the y and z axes, respectively. The perturbed electron concentration n, also has components known as slow and fast, the slow component (n, ) being associated with the phonon mode and the fast components (n/) associated with the high-frequency scattered electromagnatic wave, arising due to the three-wave parametric interaction. These waves will propagate at generated frequencies w and w0 rtw,, respectively. For these modes, the phasematching conditions w0 = w, +wandk, = k, + k,known as the energy and momentum conservation relations, respectively, should be fulfilled. From Eq. (9) under spatially uniform laser irradiation (Ik, 1~0 such that Jk, 1 = lkl), we obtain nj =

ik%T

mq(k

in,, eflk “Z%E +‘ci - w2) (a2 - k ‘vf + 2iy,,w,) 4-2 *PmP

w(k’ci

-w’)(Wf~, -I

P + =

J _t dt.

s

(16)

Now the second-order optical susceptibility can be obtained by defining the induced polarization at frequency (w, ) as p

+

(a,

) =

%XENEOE:.

The effective nonlinear optical susceptibility in the coupledmode scheme obtained by using Eqs. (15)-( 17) on neglecting the induced polarization due to transition dipoles comes out to be as follows: ee 1 Aw2Z P

(2) -

x EN

-

2mkc~y,w,w,Z~ -

c&z3 1+

k2@[’ -’ ZT~((s2- iw, y) > ’

k ‘ci (6’ - iv)

(18)

whereS’=Z-tiw,. The above formulation reveals that the total crystal SUSceptibility is influenced by the equilibrium carrier concentration (n, ) through w,, #0 and by the longitudinal magnetostatic field through w, #O, which is one of the preconditions to evoke the helicon mode in the crystal.

+iv)

k ‘IE 1’ (13) (75; - 13: + iw, y) > ’ where vi = C/p; v,, is the velocity of acoustic wave in the lattice. It is evident from Eq. (13) that n, depends on the pump intensity (I) which is given by I= +~,v~cI&, I’, in which 77is the background refractive index of the crystal, E, the absolute permittivity of the medium, and c being the -

where A = W’k ‘vf, K* = fl ‘/EC, and S’ = 5; - w:. Henceforth treating the induced polarization P t as the time integral of current density J + , we have

(12)

(Tj:, - cd: + iw, Y) ’

where CL),= w0 - o and w; ’ = wi’ij/ti (on assuming CL)