Fourphoton parametric amplification in semiconductorsM. Jain, J. Gersten, and N. Tzoar Citation: Journal of Applied Physics 46, 3969 (1975); doi: 10.1063/1.322147 View online: http://dx.doi.org/10.1063/1.322147 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/46/9?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Four-photon parametric mixing and interaction between filaments AIP Conf. Proc. 1629, 146 (2014); 10.1063/1.4902268 Twophoton resonant fourwave parametric amplification J. Appl. Phys. 51, 84 (1980); 10.1063/1.327306 Polarization effects in fourphoton conductivity in quartz Appl. Phys. Lett. 27, 48 (1975); 10.1063/1.88263 OBSERVATION OF TUNABLE FOURPHOTON PARAMETRIC NOISE IN CALCITE Appl. Phys. Lett. 14, 360 (1969); 10.1063/1.1652686 FOURPHOTON OPTICAL PARAMETRIC NOISE IN WATER Appl. Phys. Lett. 14, 32 (1969); 10.1063/1.1652647

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Four-photon parametric amplification in semiconductors *

M. Jain, J. Gersten, and N. Tzoar

Department of Physics. City College of the City University of New York. New York. New York 10031 (Received 17 March 1975; in final form 2 June 1975)

A theoretical study of four-photon parametric amplification in narrow-band-gap semiconductors is made. It is shown that phase matching is achievable in a linear geometry if a magnetic field is employed. Furthermore. a substantial cyclotron resonance enhancement occurs in the presence of a magnetic field. We calculate the growth rates and threshold fields associated with the parametric amplification and conclude that an efficient laser may be designed based on this process.

PACS numbers: 42.65.D. 42.60.J

I. INTRODUCTION

Much interest has been shown in recent years in devising new techniques for obtaining coherent tunable electromagnetic energy. In particular, Patel and Shawl have introduced the Raman spin-flip laser which is capable of generating tunability over a wide range of the infrared spectrum. The technique involves the Raman scattering of a primary laser beam from a sample of narrow-band-gap semiconductor embedded in a mag­netic field.

In this paper, we explore an alternate method for generating infrared radiation by employing a four­photon parametric process using the same type of semi­conductor. Interest in parametric phenomena stems from the initial investigation by Giordimaine,2 and in the four-photon process from the work of De Martini and Kelley. g The basic technique we propose is to allow two primary laser beams to interact in the narrow-gap semiconductor in the presence of a magnetic field. The nonlinear response of the medium, originating from the nonparabolic dispersion of the conduction electrons gen­erates the current denSity which acts as a parametric pump for the two secondary beams. By appropriately tuning the cavity it is possible to frequency select particular modes that have undergone parametric growth. The purpose of the magnetic field is to allow phase matching to be achieved in a collinear geometry, as well as to produce a resonant enhancement of the amplification. It is found that the parametric growth can be quite substantial and that a practical and efficient means for obtaining tunable infrared radiation may be obtained by employing the technique in the present paper.

In Sec. IT we outline the basic theoretical considera­tions relating to the parametric process and derive a formula for the growth rate of the secondary waves. In Sec. lIT we study the phase-matching criteria and the limitations set by them. Finally, Sec. IV deals with re­sults and a brief discussion.

II. PARAMETRIC AMPLIFICATION

Consider the case of a narrow-gap semiconductor embedded in a dc magnetic field B directed along the z axis. If the sample is irradiated with circularly polarized light along Z, the equation of motion of an electron with electronic charge - e and effective mass m* at the bottom of the conduction band can be written4

3969 Journal of Applied PhYSics. Vol. 46. No.9, September 1975

(1)

where v is the electronic velocity, c* = (E,/2m*)1 /2,

where Ef{ is the gap energy, and T is a phenomenological relaxation time: E is the ac electric field of the incident radiation, and we neglect the effect of the ac magnetic force since typical electronic speeds are much less than the speed of light.

In terms of the vector potential A, which is a circu­larly polarized vector, the equation becomes

(d 1) u dA . dt + T (1- u2)1 /2 = J.J. dt +lWeU,

where J.J. =e/m*c*c, we=eB/m*c and u=v/c is also a circularly polarized vector.

(2)

While conSidering the parametric conversion of fre­quencies WI and w2 into Wg and w4, the total vector po­tential A is taken to consist of the sum of the vector potentials due to the four frequencies:

4

A = 6 eA. exp[- i(k.z - w.t - x.»), .=1

where e = (x +iy), x. are the phase factors, and k. are the propagation vectors:

e . e = e* . e* = 0

and

k. are taken to be complex to allow for growth, deple­tion, or absorption due to collisions. Since we are in­terested in the parametric growth of Wg and w4, we write

and

(3)

(4)

k 4 =P+iy. (5b)

We solve the equation of motion given by Eq. (2) in terms of the power series

u = J.J. UI + J.J. gU2 + . . . . (6)

The first-order solution is obtained as 4

Uj =6 e UI • exp[ - i(k.z - w.t - x .. ») .. =1

(7)

where

UI .. = (w..A..I ~)(w.- we +i/T) (Sa)

Copyright © 1975 American Institute of Physics 3969

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and

(8b)

The second-order solution is obtained by solving

(:t + ~) [U2 + tU1 (U1 . U1)] = iweU2• (9)

The presence of the tu1(U. U1) terms shows that U2 will contain many mixed-frequency components, each of which will act as a source term for the growth of that frequency. We are only interested in the parametric growth of W3 and W4 components and instead of blindly solving Eq. (9) for all components, we only look for w3 and W4 components of U2, which are written as U23e exp[ - i(k3z - W3f - X3)] and U24e exp[ - i(k4z - w4t - X4)]' U23 and U24 are complex to allow for the phase difference.

The equation for the W3 component of U2 is obtained:

(:t + ~ ){u23 + tl U1312U13 + U11 U12 Uj4 exp[i(X1 + X2 - X3 - X4»)}

Xe exp[- i(k3z - W3t - X3)]

(10)

where we have used the fact that the four-photon parametric process, in going from from w1 and w2 into W3 and w4 obeys the energy and momentum conserva­tion relations given, respectively, by

(lla)

(llb)

Equation (10) is easily solved to obtain

U. - (W3- i/ T)(wa- We+ i!r){l.lu 12u U U U 23 - - ~~ 2 13 13 + 11 12 14

X exp[i(X1 + X2 - X3 - X4)])'

The phase factors Xv are the initial phase factors and are determined essentially by the experimental setup. If the experimental conditions are chosen such that

the expression for U23 becomes

U. - (W3- i!r)(W3- We+ i!r)(l.lu 12u ·u U T"") 23-- 2 2 13 13+ Z 11 12 u 14' ~3

(12)

The w3 component of the electron velocity is, therefore, given by

~ *{~( i) fl2(W3-i!r)(W3-We+i!r) eflc A2 w3- we+- - A2

~3 T ~3

[1 w~~ ( i ) iWjW2W4A1AzA4 ( i ) xL"2 ~! w3-we+"T + ~I~~~i w1 -we+"T

X (W2 - we + ~)(W4 - we+ ~)]}exp[- i(k3z - W3 t - X3)]

where we have substituted for U1:s from Eq. (8). Using the velocity as the source of current in the wave equation,

(13)

3970 J. Appl. Phys., Vol. 46, No.9, September 1975

where EL is the dielectric function, we obtain the equa­tion for the W3 frequency as

( - k~ + ~ W~ As = W!;L [W~3 ( W3 - we + ~ ) fl 2 (W3 - ijr)(W3 - We + i/ T)2 (waA3)3

- 2 ~~

- i~r2:~1:~2:t (W1-We+~)(W2-We+~)

X (ws - We +~) 04 - We + ~)A1A2A4} (14)

where w; = 47Tne 2/m*E L is the plasma frequency, with n the concentration of electrons in the conduction band.

Equation (14) describes the parametric growth of frequency W3. As expected for a four-photon parametric process, the equation has a term proportioned to A 4•

Similarly, the equation for A4 would contain a term proportional to A 3• Using Eq. (5a), equating the real parts on both sides of Eq. (14), and keeping only the lowest- order terms in 1/ T, one gets

+ _1 __ + __ 1_)_ 1J (W1 - We)(W2 - We)(W3 - we) Ws - We W4 - We

X(W4-We)(~1~2~3~4)"2, (15)

where -? has been neglected compared to q2. Equation (15) is essentially the dispersion relation for w3 with small corrections due to absorption, nonlinearity, and parametric effects. In fact in the limit T-oo, it re­duces to the nonlinear dispersion relation obtained before. 4

Similarly, equating the imaginary terms on both sides of Eq. (14) and keeping only the lowest-order terms in l/T, one obtains,

_ 2 A - WpEL ~ _.l!:... WS- we (w A )3_ fl W1W2W3W4 2 ( . A 2 (2 2) 2

qy S - C2 T~~ 2T ~~ 3 3 (~j ~2~3~4)2

X (W1 - We)(W2 - We)(W3 - We)(W4 - We)A1A2A4)'

The A~ term can be dropped compared to the A1A2A 4 term since A1 and A2 are large, being pumped, while As and A 4, the parametrically produced outputs would be small. Also, the first term on the right-hand side is interpreted as arising due to the linear absorption of frequency Ws. Introducing the linear absorption coeffi­cient 0'3 = W;ELWa!2c2 ~~Tq, the equation for y becomes

2 2 ( )A _ WpEL!J. WtW2W3W4 y + 0'3 3 - 2qc2 (~1 ~2~3~4)2

X (Wj - We)(W2 - we)(ws - We)(W4 - we)AjA2A4'

(16a)

Jain. Gersten. and Tzoar 3970

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Proceeding similarly for the w4 frequency, one would obtain

2 2 _ WpELjJ. W1W2W3W4 (Y+0'4)A 4- 2c2p (.!l1.!l2.!l3.!l4)2

X (W1 - Wc)(W2 - Wcl(W3 - We)(W4 -: We)A1A2A S'

(16b)

where 0'4 is the linear absorption coefficient for w4'

Combining Eqs. (16a) and (16b) to eliminate A3A 4 results in

1 1 r 2 4 (WlELt2 W1W2WSW, Y= - 2«'(3 + 0'4) +"2 00's - ('(4) + pq 2c . (.!l1.!l2.!lS.!l,)2

2J1/2 X (w1 - We)(W2 - wc)(wa - wc)(w, - Wc)A1A 2) ,

which gives the parametric growth rate. For T- 00,

(17)

i. e., no absorption, the growth rate will always be positive. But for finite T, one needs a threshold value for the power product A 1A 2, only above which the parametric growth will take place. Also, the growth depends only on the product A1A2 and not on the individ­ual vector potentials for the primary beams. It should be noted that the product A1A2 appearing in Eq. (17) refers to the vector potentials inside the semiconductor. We relate it to the incident field in the usual way and in terms of the mean incident intensity I j we can write

(18)

where II = (/11121)112, 111 and Iu being the incident inten­sities for w1 and W2 beams, respectively.

Letting y - 0 in Eq. (17) and using Eq. (18), the threshold value of the mean incident intensity is given by

(I) _ 1 + [E(W1) ]1/2 1 + (E(W2)]1/2 .!l~.!l~.!la.!l4

I th- 2 2 81TCjJ.2(wSW4)1/2T

X[(W1 - Wc)(W2 - we)(ws - we)(wc we)]-1. (19)

Only when the mean incident intensity exceeds the above value, will the parametric production of wa and w4 waves take place.

III. PHASE MATCHING

In the process of parametric amplification through four-photon interaction, two incident photons combine in such a way as to parametrically pump two other photons. Phase matching is the requirement that energy and mo­mentum conservation conditions be simultaneously satisfied. In general, the pumped photons will not al­ways travel along the direction of the beam, resulting in a noncollinear process requiring complicated theoreti­cal and experimental analysiS. However, by the ap­propriate use of a dc magnetic field, it is possible to alter the dielectriC- properties of the semiconductor in such a way as to achieve phase matching along the beam. This leads to a coherent loss mechanism in which the pumped waves grow along the length of the primary beams, as was considered in Sec. II. It was for this reason that the momentum consl!rvation equation, 9(b), waS a scalar equation, since aU the waves prop-

3971 J. Appl. Phys., Vol. 46, No.9, September 1975

agate along the z direction. We now proceed to obtain the requirement on the magnetic field for collinear phase matching to be possible.

Since phase matching is essentially a correction to the parametriC process, it suffices to take the disper­sion relation without absorption and nonlinearity and write it in the usual form in the presence of a magnetic field,

(20)

Using Eq. (20), we can approximate the dispersion relations for the four frequencies by the relations

where we have made use of the energy conservation relation given by Eq. (l1a).

(21)

Using Eqs. (21) in the momentum conservation rela­tion given by Eq. (l1b) results in the phase-matching condition,

(22)

Another requirement for successful parametric amplification is that all the waves be able to propagate through the semiconductor, i. e., E(Wv ) should be posi­tive for all W v'

An inspection of the expression for E(W v ),

(23)

shows that for w;/ w~ « 1, E(W v) is negative only in the range we < Wv < We + w!/ We' Therefore, if we choose the frequencies such that w1 < we < W2 and Ws < w" use of the phase-matching condition given by Eq. (22) results in the following requirements for E(Wv ) to be positive:

(24a)

and

(24b)

It is to be noted that the requirement for E(W4) to be positive has already been included. w4 does not appear explicitly in the above inequalities, since chOOSing Ws immediately fixes w4 as required by Eq. (l1a). Equa­tions (22), (24a), and (24b) together with (l1a) repre­sent all the restrictions on the frequencies for com­plete phase match and possibility of propagation through the semiconductor.

Jain, Gersten, and Tzoar 3971

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Using Eqs. (22) and (11a), the expression for the growth parameter 'Y can be rewritten in the limit T- 00,

_ 1. WM/2 I __ e_) 2 A1A2W1 (2we - W1)W! /2 (2We - W3)1 /2 'Y- 2 c \m*c*c (w1- we)2(W3 - we)2

[( w2 w~ ) ( w2 (2we _ W3)2 )~.1 /2

X 1- 2(1- W w3) 1- 2[1- W (2we - W3)] ~ ,

(25)

where we have substituted for p and q from Eq. (21). Equation (23) gives the growth rate for parametric amplification in the absence of absotption due to collisions.

IV. RESULTS AND DISCUSSION

In the previous sections we have derived a theory for four· photon parametric amplification in narrow-band­gap semiconductors. The nonlinearity associated with the nonparabolicity of the conduction band was re· sponsible for the four-photon process. Two incident lasers at frequereies w1 and w2 were allowed to interact to parametrically drive two output beams at frequen­cies w3 and w4' A magnetic field was utilized for two reasons. First it permitted one to obtain phase match­ing in a linear geometry, i. e., all the four beams could be collinear. In addition it lead to a cyclotron en­hancement of the nonlinear effect itself. The latter effect has recently been employed in other discus­sions4• 5 relating to nonlinear propagation in narrow-gap semiconductors.

10'

'0'

",'".,/,')/

, , "

, , , ,

, , , , " . , ,

, '.

\

\ \

\ , \ ,

, ,

"

A

, , ,

" ~·1.5~' \ ,/ WI Wz. " , ~

~"~~~--~~~--~~~~~~--~~~~' ~ ~ ~ ~~ ~

11), (IOI4 I1AD/SiC)

FIG. 1. Growth rate 'Y vs Ws for different intensities 1j and plasma frequencies w~. Solid curves w~ =3.0 x1012 rad/sec; broken curves w, = 1. 742 x1ot2 rad/sec; curves A and B, 1j =107 W/cm2 ; curves C and D, 1j =106 W/cm2 •

3972 J. Appl. Phys., Vol. 46, No.9, September 1975

1~.I~L-~J--L~I~.O~~~I.~~~~5-7~-2~.o~~-L~~~~3P (0)5 (10 14 RAD /Sf')

FIG. 2. Growth rate 'Y vs w3 for different WI' w2, and we' Other parameters are the same as in Fig. 1.

In Fig. 1 we present the growth rate of Eq. (17) as a function of the frequency of one of the parametrically pumped waves at frequency w3' The input frequencies w1 and w2 are denoted by arrows on the abscissa, as is their mean, which equals the cyclotron frequency we'

Values of the growth rate are given for two plasma fre­quencies (w, =3. OX1012 and 1. 742 x1012 rad/sec) and two mean incident laser intensities (Ij = 107 and 106 W/cm2). The calculations were made for InSb which was chosen for its low energy gap and low effective electron mass. Employing parameters for InSb (E, = 0.234 eV, m* = mj60, EL = 16), the doping densities for curves A and C were n = 7. 56 X 1014 cm·3 while for curves Band D, n=2. 55x1014 cm·3

• The we corresponded to a magnetic field of 152 kG. One notes that the growth rate is quite substantial, particularly in the region near we' One main limitation on the growth rate is the short photon mean free path, especially at frequencies close to we because of resonant absorption. At these frequencies, the finite electron lifetime T results in a large linear absorption thus reducing significantly the optical ab­sorption length or the photon mean free path. For this reason, the plasma frequencies were so chosen that the mean free path A for frequencies w1 and w2 was of the order of a reasonable sample size (A = 1. 24 cm for w, = 3. Ox 1012 rad/sec and A =4.83 ·cm for w, = 1. 742 X 1012

rad/sec) and w3 and w4 were not allowed to get any closer to we than Wi and w2' Thus, for these sample sizes, for all four frequencies, the photon mean free path is at least the sample length.

In Fig. 2(a) a similar plot of the growth rate is given for different values of w1 and W2. The strong rise of the

Jain. Gersten, and Tzoar 3972

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3.0

1.0

FIG. 3. Threshold intensity (fIlth vs w3 for the same parame­ters as in Fig. 1.

growth rate in the vicinity of we can be understood simply in terms of the cyclotron enhancement of the nonlinear coupling parameter. Compared to the values in Fig. I, w1 and W2 here are much closer to each other and hence to we' resulting in much larger values of the growth rate than in Fig. 1. The value of we here corre­sponded to a magnetic field of 144 kG, other parameters being the same as in Fig. 1. The lowest (out of the four frequencies) mean free paths were A == 0.21 cm for wp == 3. 0 X 1012 rad/ sec and A = O. 82 cm for wp = 1. 742 X 1012

rad/sec.

In Figs. 3 and 4 the threshold value of the mean in­cident intensity (fl)th given by Eq. (19) is plotted as a function of w3 for the frequencies and other parameters of Fig. 1 and 2, respectively. A high growth rate, especially near we' results in relatively low values of (fl)th' As expected, thresholds increase with in­creasing wp and are much lower for the parameters of Fig. 4 where the growth rate is larger. At these in­tenSities, the parametric growth will just be balanced by linear absorption. Much higher intensities are re­quired for the parametrically produced beams W3 and W4 to be observable.

In Figs. 1-4, the entire domain of values of W3 has been considered. But in practice, the growth of the values beyond 10.6 jJ. will be restricted because of two­photon absorption cutoff at high frequencies. Since the same absorption takes place for W4 also, there is also a cutoff at low frequencies because of Eq. (l1a). For these reasons, the parametriC output is really limited to a frequency band whose width depends upon the parameters of the sample. The output frequency band can be expanded by the use of wider-gap crystals. The particular output frequency required is obtained by ap-

3973 J. Appl. Phys., Vol. 46, No.9, September 1975

propriate tuning of the output cavity to select w3' Be­cause of the high growth rate achieved here, the output lasers with extended frequency range would be valuable spectroscopic tools. The input frequencies used here are currently available through various tunable laser devices like the spin-flip laser. 1

It should be noted that the choice of wavelength around 10 jJ. was only taken as an example to illustrate the parametric process and the theory is not limited to these wavelengths. The parametric process described here can be used to extend the range of tunable lasers to other domains like the far-infrared region.

Here one must work in a pulse situation where typi­cally intensities of 1-5x106 W/cm2 can be achieved. For the far-infrared region we need much smaller magnetic fields, thus we have no problem there. The three important parameters to consider are the incident intensity f, the laser frequency w, and the electron density n. For an order-of-magnitude estimate, let us omit the magnetic field to simplify. Then the growth parameter Y is essentially given by Y = Yl - Y2, where Yt> the growth in the absence of linear attenuation, is proportional to (w/c)(w;Jw2)(eE/m*wc*)2, while n, the linear attenuation, varies as (w/C)(W!/W 3T). Thus, for the purpose of simple scaling, we can write

y=CnI/w 3 -Dn2/w 2, (26)

where w; varies linearly with n. For illustrative purposes we have taken 1/ T to be proportional to n as for the case of Coulomb electron-ion scattering situa­tion. (The estimate can be extended to cases where other scattering mechanisms are involved. )

4.0

(ld ..

(lol~.)

3.0

Z.O !AI, "1.142.10" RAO/SIC

1.0 L..L.-"----'---L--:~-'---L++lL_---'-~':--'--'--L.--'-~ .1 1,0 Us 1.57 Z.O 3.0

Ws (10 ,. RADisfC)

FIG. 4. Threshold intensity (fl)th vs w3 for the same parame­ters as in Fig. 2.

Jain, Gersten, and Tzoar 3973

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Now take the case of a 10- Jl versus a 100- Jl laser at the incident frequencies. From Eq. (26) one simply ob­tains that on taking nl00 = 10 1110 and /100 = rto /10, the growth rates obey

'Yl0 = YtOO'

(Here the subscripts are self-explanatory. ) Both the density nl00 and the intensity /100 are within the availa­ble range. Thus the high growth rate would make it possible to design tunable far-infrared lasers.

In conclusion, in th is paper we have discussed a four­photon parametric process with collinear phase match­ing and cyclotron resonance enhancement through a magnetic field and have shown that the resulting device

3974 J. Appl. Phys., Vol. 46, No.9, September 1975

would be a new type of laser source in a frequency do­main not currently available.

*Research sponsored by the V.S. Air Force Office of Scientif­ic Research, Air Force System Command, under AFOSR Grant No. 71-1978.

lC.K.N. Patel and E.D. Shaw, Phys. Rev. Lett. 24, 451 (1970); C.K.N. Patel, in Laser Spectroscopy, edited by R.G. Brewer and A. Mooradian (Plenum Press, New York, 1973), p. 471; R.E. Slusher, C.K.N. Patel, and P.A. Fleury, Phys. Rev. Lett. 18, 77 (1967).

2See J.A. Giordimaine, in Proceedings of the International School of Physics, "Enrico Fermi" Quantum Optics, edited by R. J. Glauber (Academic Press, New York and London, 1969), p. 493, where reference to earlier work is cited.

3F. De Martini and P.L. Kelley, in Ref. 2, p. 574. 4M. Jain, J.I. Gersten, and N. Tzoar, Phys. Rev. B 8, 2710 (1973).

5M. Jain and N. Tzoar, Phys. Rev. B 10, 5159 (1974).

Jain, Gersten, and Tzoar 3974

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