Optical parametric amplification in the magnetoplasma in semiconductorsTakashi Iida and Yoshihiko Mizushima Citation: Journal of Applied Physics 77, 218 (1995); doi: 10.1063/1.359601 View online: http://dx.doi.org/10.1063/1.359601 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/77/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Optically pumped parametric amplification for micromechanical oscillators Appl. Phys. Lett. 78, 3142 (2001); 10.1063/1.1371248 A new magnetoplasma amplification effect in the optical frequency range J. Appl. Phys. 75, 2348 (1994); 10.1063/1.356253 Picosecond optical parametric amplification in lithium triborate Appl. Phys. Lett. 58, 213 (1991); 10.1063/1.104692 Parametric dispersion and amplification of acoustohelicon waves in piezoelectric semiconductors J. Appl. Phys. 69, 61 (1991); 10.1063/1.347657 Fourphoton parametric amplification in semiconductors J. Appl. Phys. 46, 3969 (1975); 10.1063/1.322147

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Optical parametric ampMication in the magnetoplasma in semiconductors Takashi lida and Yoshihiko Mizushima Hamamatsu Photon& 1126-l Ichino-cho, Hamamatsu-shi, Shizuoka 435, Japan

(Received 20 June 1994; accepted for publication 13 September 1994)

4 nonlinear parametric interaction is found in the magnetoplasma in a semiconductor. The spatial asymmetry is produced by the magnetic field in the Voigt configuration, where the Lorentz force acts on the drifting electrons. The nonpolar semiconductor can be used as the nonlinear optical element. The optical parametric conversion and the second-harmonic generation are discussed. The device operates in the infrared range. The efficiencies are found to be much larger than in conventional nonlinear optical crystals. 6 I995 American Institute of Physics.

I. INTRODUCTION

The authors have proposed a magnetoplasma instability in a semiconductor, wherein an external magnetic field pro- duces Lorentz motion of electrons, and a space-charge wave coupling occurs between the drifting electrons and the opti- cal waves.’ Also in a polar semiconductor with transverse and longitudinal phonons, the dielectric dispersion due to the polariton causes another type of coupling producing an instability.2

Since the mechanism is based on the Lorentz force term, the second-order effect is to be considered here.

The second order in the secular equation in the magne- toplasma at the Voigt configuration leads to nonlinear phe- nomena. The parametric and the second-harmonic generation (SHG) are both derived from the fundamental equations. Those phenomena are induced from- the Maxwell and the electron movement equations. The nonlinear optical coeffi- cient is found to be larger than in conventional nonlinear optical materials.

In this article the conversion of the optical wave was investigated based on the following two items: (1) SHG3,4 and (2) parametric conversion.5-7

II. MODE-COUPLING EQUATIONS

We treat an infrared electromagnetic wave propagating in an infinite semiconductor as a plane-wave approximation. The starting equations are the same as in the previous theory.“2 The relative orientation of the electric field, the magnetic field, and the electromagnetic wave is the Voigt configuration. The wave is assumed to propagate along the z direction. A static electric field E is applied parallel to the z direction, and a static magnetic field H is applied parallel to the y direction.

In this configuration, a Hall field appears and drives the current in the x direction. An auxiliary compensating bias is applied in the x direction to suppress the Hall field, and as a result the carrier drifts in the z direction. With a finite width device, a transverse bias appears automatically, and the gen- erated Hall field between the sidewall charges acts as the source for the auxiliary bias. In the following, the auxiliary bias voltage means is not discussed, and only the drift com- ponent in the z direction is a priori assumed.

Here only the electrons are considered. Holes in the semiconductor are neglected, since their effective mass is

heavy and their motion is slow. Since the frequency is larger than the dielectric relaxation frequency, only the space charge in the electron plasma wave is considered. Electrons are assumed to drift at a constant speed ue lying in the x-z plane by an external dc electric field, and 8 is the angle between the direction of the wave and the direction of the electron.

The equation of electron motion is

m” g+(u.V)u =p(E+vxB)-:, i i

and the Maxwell equations are

VxE=-$,

(1)

(3)

V.D=qn, (4)

V.B=O. (5)

where 4 is the electric quantum charge, n the electron den- sity, u the velocity of the electron, J the current density, E the electric field, D the electric flux density, H the magnetic field, and B the magnetic-flux density. The material param- eter is assumed to be intrinsically isotropic,

B=,xH, (6)

D=eE, (7)

where ,u is the permeability of the medium and E is the permittivity of the medium.

By Ref. 1, the scattering is shown to be unimportant when w+l, which is valid within the optical frequency range, and r is neglected.

In the following, suffix 0 implies a dc component. Suftk 1 indicates an ac component. In the small-signal treatment, with assumptions as below,

n0+nl, 03)

dul,+uo. ~1+ % I, dt the equation of electron motion and the Maxwell equations become

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111% ~+(uo.v)u,+(u,.v)ul ! =I VXH,=qnluo+qnoul+qnlvl+Edt,

VXE,=-p, z, ,uoV.H1=O.

(11) By substituting dilt-tjw in Eq. (ll), we obtain

(12)

(13)

(14)

[i l+“;; ‘;)]Elx

a d d +

I! z+uo cos 8 z

1 u. sin’9 z--WC

! g+uo cos ti-$]Elz+s (k%)($+uo ms eg)El,=o, (15)

I! ;+uo cos e -5 dr)i~-~~)+U:j~+uo~~e~)]~~y+~(~~)(~+~o~o~~~)E,~=O, (16)

[~~(~-~~)--:~~]~~~+[(~+uo cos og)2+wCuo sin t3~+w~]Elz

where suffix x, y, or z implies polarization of the electric field in X, y, or z direction and 8 is an angle between the wave and the electron-beam directions,

q”n If2 WE- P ( i em* ’

where in* is the effective mass of the electron,

@ @b=m*.

(18)

(19)

We assume that the nonlinear property is derived from u 1 XB 1 . Other components such as (u 1 .V)u 1 or nlu 1 are neglected, since they are much smaller.

When a TEM wave with E ly mexp[j(w+-k2z)] and a TM wave with Elx, EI,~exp[j(o,t-kk,z)] are assumed, the above equation leads to the one below, (in the following, suffixes 1 and 2 refer to modes 1 and 2, respectively):

I! kzC2 2

(w,-kluo cos 0) ol- k-- 2 El,+j(w,+jkluo sin B?El, ]+j f$ (2 Elx)~(o,-kluO cos B)El,l=O, (20)

(02-k2uo cos e) [(ml--k,uo cos @El,]=O, Gw

[ -juc( ~l-!$i)-~S kluLF ‘1 E1,+[(ol-k,uo cos e)2-W~+jW,kluo sin B]El,

Expressed as a matrix,

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k?c” w; 0 ---I_ 1

ml 01

2 kluo sin 6’ P

Wl

+j 5 (k,q, sin BEI,) %

0

k2c2 2 2 *P a?- -- - I w2 6J2

.I I k;c2\

j(w,+ jkluo sin 8)

X w,-kluo cos 6, r+--k2uo cos 8

(al--kluo cos 0)2 -o~+jo,kluo sin 6

+j--$($Elx)( q--z) 7. 02--

’ O2 ’

We assume El is composed of two waves,

Eli=eli+ezi,

0 (i 0 . 03) 0

where eri is a first-order term and cai a second-order term, and the first-order components eli satisfy the equation

w1---- 01 *I

-jo,

0

\ 0 +j(w,+jk,u, sin 0)

k;c2 0; WJ- -- -

02 6J2

kluo sin 0 0

-0:

0

(q-k,u,, cos ej2 -co~fjw,kluo sin 0

I/ ml /

el, 0 ely = 0 . fb 1 i) 0

(24)

!W>

The wave elY, which has a polarization in the y direc- tion, propagates independent of the static field. The total re- flection appears at a frequency less than wP , while the propa- gation appears at a frequency exceeding wp . Therefore, the wave that is polarized parallel to the static magnetic flux does not interact with the electrons.

The other wave with er, and el, components, which has a polarization perpendicular to the static magnetic field, propagates and is amplified. That amplifying frequency band

is reduced from the determinant of the combination matrix of Eq. (25). Electrons that circulate by the static magnetic flux interact with the TM mode electromagnetic wave and the wave becomes amplifying. Discussions of the mode and the meaning of Eq. (25) are given in Ref. 1.

Here, a TEM wave with e,wexp[j(o,t-kaz)], a TM wave with el,, er,~exp[j(,rt-krz)], a TEM wave with e2yccexp[.i(w-Wl, and a TM wave with e>, e,mexp[ j( @at-- k3z)] are assumed. Substituting Eq. (24) into E$ (23), and taking the second-order term,

(q-kg0 cos 6) w3- k;c” 0; -- -

w3 03 ezx+ (- k3u0 sin 0+ jti3w,)eZr ]+.i [(WI--kluo cos 8)el,]=0, (26)

(W3-k3~0 cos e) 03- i

g- z)e2y+j 5 (2 e,,,)[(q-k,u,, cos O)el,]=O, (27) .

e2x+[(ti3-k3u0 cos 0)2-o$tjco,k3u0 sin B]ezl

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Then, the dispersion of the wave Eli is controlled by elL and the second-order quantity e2i acts as a perturbation. As- suming three hinds of waves, as E Ix, ~~,~w[j(~~~-k~~)l, E l,~exp[j(w2t-k2zjl, and El,, El,,, EIZKexp[j(03t -k3zj], the power transformation co- efficient is displayed below:

are the nonlinearity coefficients. In the Voigt configuration, there are two modes. One is the TBM mode elY, and the other is TM mode er, and el,. The modes are essentially independent from each other, but in considering the second- order interaction, eaY pp a ears from the product of the first- order terms, elY and e,,, while eti appears from the square of elY. Then, EI,~exp[j(o,t--kazj] and El,, E,, ~expLi(q~-klzjl,

e2y -z-* q w3 ol-klu,, cos 8 k2 J-T- -

elgelr m w2 w,-k3uo cos 8 w$-k:c2-wz

xexp[j(kk-k&l, (29)

w,-kluo cos 6, w3-k3u,, cos e w,-kluo cos e w3-k3uo cos 8

while ea,Kexp[j(2o,t-2krz)] or ezzKexp[j(20at -2k,z)].

+kluo sin 8(o~-k~c2--w~) exp[j(k3-2kl)z],

(30)

Ill. INDEX ELLlPSOlD IN THE VOIGT CONFIGURATION

The dispersion relation of the wave propagating along the t axis and the x axis is derived from the above consid- eration by replacing 0 by 0 or rr/2, respectively, in Eq. (25). By substituting 8=0, the dispersion relation of the wave propagating along the z axis becomes

e2z 1 q kl z=j pm*G (w;-k&‘-w;)

X expUUG-- 2W1, (31)

i

W2-,&2-w2 P 0 + jww,

0 w2-k2C2- 2 wP 0 e2,- -3 ca k2

LpP”sG \ -jwc(w-F) 0 (w-kuoj”- w; X expW3 - 2W3, (32)

where

P=(w;-k’&2-w;)[(wg-k3u0 cos 8)2-w;]

- w:(w~--~$z~)- o.$k$i sin a 8. (33) The terms e2,,lelyelZ, e2zlelxelr, ed&, and e2,1efy

el, 0 x ely = 0 ) i 10 el, 0

(34)

and for e=?rJ2, the one along the x axis becomes,

W2-&2- w;

0

0 + jww,- wkuo W2-,&2- w2

P 0

0 w2- wz+ jo,kuo (35)

I

Similarly, the dispersion relation of the wave propagating The secular relation along each axis is obtained from the along the y axis is above dispersion relations; on the z axis, for El, and EIZ,

2 2 c,J"-k2C2- df!-%&

2

w2- w; 0 wP

+jww, w2- ,z c

0 W;- W; 2 kuo

-wp - W

- wkuo w2-k2C2- W2W2 P

W2-W,

(w2-k2c2-w;)[(w-ku0)2-a;]= ;;z (w2-k2c2), (37)

and for El,,,

w2-k2C2- w2,0 P * (38)

On the x axis, for El, and El=,

T lida and Y. Mizushima 221

el, 0 x ely = 0 .

i io el, 0

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I f5 a, 23 ‘.+-... : F--‘T=-

0 WV

liicI.l kfP----

2-v g -I

I.11 1. I2

ANGUL~l$RR~UENCY

Y (a) id I 4

fj “8 ;I Ek -,

1.11 1. 12

ANGULAR FREQUENCY (10%

(b)

.#-J* . ..-.-

i -? --- 1.12

ANGiAR FREQUENCY

Y (1O'pJ

id 22

1 -

g! z

F

8 -1' .H

11 1. 12

ANGULAR FREQUENCY uo%

Cd)

FIG. 1. The dispersion relation k-o and the optical index n-o. (a) The dispersion of the wave propagating along z axis, and (b) the optical index of the same. (c) Th.e dispersion of the wave propagating along x axis, and (d) the optical index of the same.

(w2- a$- o$k2c2=(02- w;)2- &d2, and for Ely,

a2-k2c2- o;+).

On the y axis, for El, and ErZ,

(39)

(40)

d (,2-k2c2-a#Z=z (u2-k2c2)2. (41)

From the above three sets of secular equations, the elec- tron drift plays an important part only for the wave propa- gating along the z axis. For a wave propagating in the x axis, there is an extraordinary mode and an ordinary mode within the Voigt configuration. For the wave propagating in the y axis, there are normal clockwise and counterclockwise polar- ized modes within the propagating wave in the Faraday con- figuration.

Based on the above theory, the index ellipsoid is ob- tained as described in the following section.

IV. PHASE MATCHING AND NUMERICAL ESTIMATION

In the configuration with the static magnetic field, the second ordered term, u, XB, , appears as important. It results in the optical SHG or optical parametric conversion.

From the above consideration, four hinds of nonlinear interaction are derived. Two cases in Eqs. (29) and (30), are an optical parametric conversion and the others in Eqs. (31) and (32) are a SHG.

Phase matching is required. There are four kinds of phase-matching conditions as stated below.

When 03=w1-l-w2,

(42)

(43)

when 0,=2or,

k3[e,,,=(W3)l=2kl[elx,z(wl)l, (44)

and when 03=202,

(45)

Utilizing the actual parameters, feasibility of the phase matching is estimated from the relation among the index ellipsoid of the two or three frequencies.

For a numerical estimation of the nonlinear characteris- tics, we consider InSb and assume the following constants: ~=17.7% (Q: permittivity of vacuum); m” =O.O145mu (m. : free electron mass); II = 101' cme3, the electron den- sity; ~=vr,/c~O.OO1, the ratio between the electron drift velocity and the light velocity in vacuum; B = 0.1 T; w =l 1135X1014s-*, ~,=1.2129XlO’~s-~. P *

A. SHG

For the case of Eq. (44), o1 is expressed by w, and 0, is 2~. ‘&en, the condition of the phase matching is expressed by %(k&) =2%(kQ, that is R(n$ =!R(n&). ‘3(k) is the real part of k.

The optical index of refraction is calculated by n = kc/w. The dispersion relation of the wave propagating along the z axis is depicted in Fig. l(a), and the one along the x axis is in Fig. l(c). The index of the wave propagating along the z axis is depicted in Fig. l(b), and the one along the x axis is in Fig. l(d).

When we assume 01=02=wp, a”=1.004, ni =0.2095, nz*=O.2886, and r~;~=0.86661 Then, the two ellipsoids of o and 20 cross was seen in Fig. 2(a). At phase

0-3 FIG. 2. The cut plane of optical index ellipsoid by the X-Z plane. At the crosspoints, the condition of the phase matching is satisfied. (a) Self-mode SHG; in this case the condition of the phase matching is satisfied by select- ing the polarized plane to the crosspoints. (b) Different-mode SHG; in this case phase matching is not achievable since there are no crosspoints.

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s -3 -

ANG;LAR FREQUENCY 2

(1 O14/s) . .

-1

ANGULAR FREQUENCY (IO'%)

b)

FIG. 3. The dispersion relation k-w and the optical index n-o: (a) The optical index of the wave propagating along y axis, and (b) the dispersion of the same situation.

matching, 0 is 27.5”. An o, wave leads to’s double frequency, as ti3=202,

From Eq. (32),

%=j4.1931 x 10-l’. eiy

The output wave has an internal gain. As the denomina- tor becomes zero, the output becomes very large. Limited by other factors hereby neglected, the SHG constant is beyond the validity of the present simple theory.

In the other case in Eq. (45), w2 is expressed by w, and 0, by 20. Here, the condition of the phase matching is ex- pressed by R(kf&) =2iR(kyy), that is, %(n:,“,,) =%(nt).

The dispersion of the wave propagating along the y axis is depicted in Fig. 3(a). The optical index of the wave propa- gating along the y axis is depicted in Fig. 3(b). In this form, the two ellipses of o and 20 do not cross as seen in Fig. 2(b); therefore, the condition of phase matching is not realizable.

B. Optical parametric conversion

In the case in Eq. (42), the condition of the phase match- ing is displayed by %(k;“y3) = ~!.R(k~&) + %(ky;). This equation is also expressed by nyw303 = .Ezwt + IzF?W~. Here, nWl =

XJ (ny” w3 - PZ~~W~)/W~ means that the phase matching is

achieved when the ellipse cut by the z-x plane of the index ellipsoid crosses the circle in n$ As described later, this condition is realizable.

Two waves of different frequencies wr, w2 yield the third frequency wave w,. Considering that w,=~,,+‘o,, by chang- ing the suffixes in Eqs. (29),

“2Y q o3 ol- kluo cos 13 =-j----

k2 elyelr m* w2 t,+-k3q, cos 8 &-k~c2--$

Xexp[j(k3-kl-k2)z]. -’ (47)

Since the denominator (oi- k$z”-(I$) is approxi- mately zero, the w3 wave increases and fmally leads to a negative resistance (parametric oscillator).

In the case shown in Eq. (43), this case is almost the same as in Eq. (44). If the basic waves have different fie- quencies, the method for the phase matching is similar to the cases in Eqs. (42) and (44).

C. Discussion about numerical estimation

The nonlinear optical constant is estimated and com- pared with that of conventional nonlinear optical crystals. For SHG d=657X 1O-23 CV2 at a 16.9 m wavelength. The corresponding value in optical crystals is d,,,=4.76 X1O-23 CV-2inLiNb03,andd311=18X10-23 CV2in BaTiO, but at the 1.06 pm wavelength, In the infrared region there have been no utilizable optical crystals, however this theory supplies such new materials. For Eq. (46) the constant is very much larger, and can not be estimated by the present simple theory.

V. DISCUSSION

Although the above illustrates a simple theory, the second-order term is derived from the cross term u XB which leads to the new nonlinear interaction. In our device, the rotating electron clays a significant part in the coupling of the x and z components, the energy exchange between the light wave and the magnetoplasma wave. The rotation causes an interaction between polarization modes. As shown, the second-order treatment yields a parametric effect which has an enormous gain.

A wide variety of frequencies becomes possible. The power conservation law between the different frequency waves will be generalized by considering the Manley-Rowe theorem. This theorem was derived on the nonlinear reac- tance circuit without any loss. If the medium is without loss, we can expect that the Manley-Rowe equation applies, and the up-conversion has a corresponding gain. Also negative resistance will appear in the parametric amplifier, accompa- nied with low noise property, since there is no dc component in that polarization mode.

The parametric amplifier is essentially of distributed na- ture, so that an arbitrarily long device can be selected for a large gain.

This device utilizes the spatial asymmetry produced by the magnetic field, but an asymmetrical crystal structure is not required. Any semiconductor material, even with inver-

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sion symmetry, can be applied to the optical nonlinear mate- rial. This implies a robustness by avoiding the structural weakness problems associated with conventional nonlinear crystals, such as optical damage which is caused by the bonding weakness in those inherently asymmetrical struc- tures. For limited application, only the band-gap absorption should be considered.

Vi. CONCLUSION

The nonlinear interaction is derived by application of the magnetic field. The SHG and optical parametric conversion

224 J. Appt. Phys., Vol. 77, No. 1, 1 January 1995

are discussed. Optical frequency conversion is found to be possible with a larger coefficient then in conventional optical crystals. The device will operate in the infrared region.

‘T. Iida and Y. Mizushima, J. Appl. Phys. 75, 2348 (1994). ‘T. Iida and Y. Mizushima, J. Appl. Phys. 75, 2479 (1994). 3R. C. Miller, Appl. Phys. L&t. 5, 17 (1964). 4C. G. B. Garrett and F. N. H. Robinson, IEEE J. Quantum Electron. QE-2,

328 (1966). 5C. C. Wang and G. W. Racette, Appl. Phys. Lett. 6,169 (1965). ‘R. H. Kingston, Proc. IRE! 50, 472 (1962). 7A. Yariv and W. H. LouiselI, IEEE J. Quantum Electron. QE-2, 418

(1966).

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