Efficiency of submillimeter-wave generation and amplification by coherent wave-packet oscillations in semiconductor structures

2470 J. Opt. Soc. Am. B/Vol. 11, No. 12/December 1994

Efficiency of submillimeter-wave generationand amplification by coherent

wave-packet oscillations in semiconductor structures

Kai Victor

Institutfdr Theoretische Physik B, Rheinisch-Westfdlische Technische Hochschule Aachen,Sommerfeldstrasse, 52056 Aachen, Germany

Hartmut G. Roskos and Christian Waschke

Institut fur Halbleitertechnik II, Rheinisch-Westfdlische Technische Hochschule Aachen, Sommerfeldstrasse,52056 Aachen, Germany

Received February 7, 1994; revised manuscript received May 23, 1994

We discuss the emission of submillimeter waves from optically excited coherent wave-packet oscillationsin quantum-well structures in terms of the emitted power. The material dynamics are treated in a two-band real-space density-matrix approach; the radiation process, by conventional antenna theory. Excitationgeometries that assist cooperative radiation from all points of the excited spot are proposed. We consideramplification by phase-locked feedback of the terahertz wave to the emitter in a resonator. Our results leadto concepts for optimized power extraction from coherent terahertz sources.

PACS numbers: 73.20.Dx, 42.50.Fx, 42.50.Md, 84.40.-x, 42.65.Re.

1. INTRODUCTIONRecently, terahertz (THz) radiation emitted from excitonicwave-packet oscillations in semiconductor heterostruc-tures has been observed in several experiments.1 6 Agreat deal of attention has been paid to this phenom-enon because of the fascinating insight it provides intothe quantum-mechanical dynamics of coherent excitation.Optically driven charge oscillations may also be of practi-cal interest. It is a tempting idea to imagine THz sourcesbased on the free-space emission of THz electromagneticwaves from optically excited quantum oscillations. At-tractive features are the high oscillation frequencies (atleast 4 THz) and their tunability by an applied dc elec-tric field (0.5-4 THz for superlattices). Still, until nowthe measured output power has been too small for mostapplications. It is our aim in this paper to discuss THzemission from coherent quantum oscillations, with a fo-cus on the emitted power.

In Section 2 we present a description of the mate-rial dynamics based on real-space density-matrix theory.The result is an oscillating classical polarization densityresponsible for the radiation observed. The principalreason for the low radiative output is that these coherentoscillations are interrupted after a few picoseconds by de-phasing long before the major part of the available energycan be radiatively emitted. Two concepts for enhance-ment of the power extraction from the polarizationcurrents during the coherence time are discussed inSection 3. These concepts are based on the fact thatthe radiation rate of coherent dipole oscillations is in-creased when a number of dipoles emit cooperatively.The dependence of this increase on the details of the exci-tation geometry can be described by conventional antenna

theory. Stressing the quantum-mechanical origin of thepolarization densities, we may consider the cooperativeemission as superradiance of initially coherent transitiondensities.7 We show that the emission efficiency can beenhanced by (i) an increased excitation density, (ii) properchoice of excitation geometries that support cooperativeradiation from large parts of the spot, and (iii) phase-locked feedback of the THz radiation to the repetitivelypumped sample in a resonator. Because attribute (iii)requires a high-amplitude THz field at the sample, wediscuss the backcoupling of THz radiation on the exci-tonic dynamics in Section 4 in the same density-matrixformalism as is used in previous sections. Section 5 isdevoted to the discussion of concepts for the realizationof attributes (ii) and (iii), in which we give estimates forthe achievable enhancement of radiation power.

2. IMPULSIVE GENERATION OFSUBMILLIMETER WAVESThe physical principle of THz-wave generation is illus-trated in Fig. 1 for the example of a coupled quantum-wellstructure consisting of two wells of different thicknessseparated by a tunneling barrier. The bias field acrossthe structure is chosen such that the lowest single-wellelectron eigenstates of the two wells are aligned. Theelectron wave functions of the composed structure aresplit and are delocalized. The hole states are not alignedat the-chosen bias. A femtosecond laser pulse resonantlyexcites the lowest interband transitions and populatesa coherent superposition of eigenstates of the quantumsystem. Both the generated electrons and the holes areinitially located in the wider well because of vertical tran-sitions in real space. When the laser pulse has passed,

0740-3224/94/122470-10$06.00 ©1994 Optical Society of America

Victor et al.

Vol. 11, No. 12/December 1994/J. Opt. Soc. Am. B 2471

2 + ~ D2 O) t3,,t=O

/CO 2 t7E

-- - co jt=2it

Fig. 1. Excitation of wave-packet oscillations leading to emis-sion of THz radiation for the example of a double quantum well.

the electronic part of the wave packet oscillates betweenthe wells with a frequency corresponding to the excitonicenergy splitting. The holes remain confined in the widewell. The wave-packet oscillation lasts until the phasecoherence between the quantum states is lost because ofscattering after a time T2. The oscillating dipole momentassociated with the spatial wave-packet dynamics leads tothe emission of electromagnetic radiation.8

A. Experimental BackgroundFigure 2 schematically shows the setup used to detectthe coherent electromagnetic radiation from the opticallyexcited quantum structures discussed in Refs. 1-6. Thedipole moment associated with the charge oscillation is di-rected perpendicular to the surface. The radiation fromeach of the individual dipoles is strongest in the direc-tion parallel to the surface of the sample. The rela-tive phase of the dipole densities at different points ofthe spot is determined by the arrival time of the opti-cal excitation pulse. According to a generalized Fresnellaw, constructive interference between the wavelets isfound roughly in the directions of the transmitted and thereflected optical beams.9 Radiation leaves the samplethrough the semitransparent large-area contacts for theelectrical bias. These contacts on top of the samples areformed by very thin (several tens of angstroms) Cr orCr-Au films. Emission also occurs into the substrate,with the radiation cone pointing in the direction of thetransmitted optical beam.

The radiation is detected with the help of a subpico-second photoconductive dipole antenna0 gated by atime-delayed replica of the excitation pulse. Both theamplitude and the phase of the electromagnetic tran-sients are measured at the same time. The duration ofthe observed THz pulses is determined by the scattering-dominated T2 of the excitons, although fluctuations ofthe oscillation period across the excited spot that aredue to variations of the quantum-well parameters mayplay a role, too. The experiments of Refs. 1-6 were per-formed on excitons at cryogenic temperatures to takeadvantage of the long dephasing times of several pico-seconds in high-quality quantum-well structures. Thefrequency of the radiation was bias tunable (i) from 1.4to 1.7 THz in the case of electron oscillations in coupledquantum wells,' (ii) from 1.4 to 2.6 THz in the case ofheavy-hole-light-hole quantum beats,3 and (iii) from 0.5to at least 4 THz (the detection limit) in the case of Blochoscillations in a GaAs-AlGaAs superlattice.46

In the superlattice experiment of Ref. 4 the time-averaged optical input power is 20 mW, from which 10%

is absorbed and excites pair densities of 109 cm-2 in eachof the 35 wells. We estimate' the THz output power tobe 0.1 nW in each cone of radiation. Until now, to ourknowledge this was the only available experimental es-timate of the emitted power. We compare our theoreti-cal results (given in Section 3) with this estimate. Therepetition rate of the laser pulses and hence of the THzpulses is 76 MHz (Ti:sapphire laser).

B. Density-Matrix Theory of Carrier DynamicsThe observed electromagnetic radiation is emitted frommacroscopic spatial oscillations of charge densities on alength scale that is large compared with atomic distances.We calculate the time-dependent expectation value of thecarrier densities across the heterostructure. All carriersinvolved in the quantum oscillations are optically excitedelectron-hole pairs; hence, to lowest order in the opti-cal field, all carrier densities can be derived from pairdensities' 2

(cl+cl) = (cj+dj+djcj), (1)

where c; (dj) denotes the annihilator of an electron (hole)at the atomic site j. If, for simplicity, we assume theexcitonic lifetime to be half the dephasing time T2 = /y,to the lowest order in the electric field the pair densitiesfactorize 3 and can be expressed as the absolute square ofexcitonic transition amplitudes Y(rh, re) :=(dhce). Thuswe obtain for the charge density p(r)

p(r) = e[f d3 reIY(r, re)12- f d3 rhlY(rh, r)I2]. (2)

If we take into account the layer-structured symmetry ofall samples used in Refs. 1-6 it is of advantage to use

Paraboloidal Mirror

THz Pulse

Dipoleip ya / Antenna

Excitation ample iPulse Optical

GatingPulse I

Sample

Fig. 2. Experimental setup used to detect the THz radiation.

Victor et al.

I


center-of-mass coordinates in the planes and the particlecoordinates perpendicular to the wells (z direction):

UAVV - e1 mh [mh( ) +me( X]

optical excitation to consist of a train of pulses, numberedby j, with vectors of polarization ej, complex amplitudesEj, times of arrival at the heterostructure t(U), and acommon center frequency wo:

E(U, V; t) = E ejEj A[t - t(U)]

( k) Yh = ( Ye (3)

In these coordinates the constitutive equation for Y readsas 1 V16

FL (~~~~Zh -Ze ) :

X Y(U, V; U, V; Zh, Ze; t)

= Mo8(u)8(v)(zh -ze)E(U, V; t),

withhf~y hwg 2 h 2 h 2 h 2 v 2

g 2 mh WZh 2m Ze - 2 (u)+ Vh(Zh) + Ve(Ze) + W(U, V, Zh - Ze),

(4)

(5)

where me (mh) is the electron (hole) effective mass and pis the reduced in-plane mass. V and Vh are particle-specific potentials that include the spatial variation ofthe band edges as well as a possible electrostatic bias.Ctg is the gap frequency; the homogeneous solution of Ywill oscillate with this frequency. E denotes the opticalfield, where the dependence on the z coordinate is ne-glected because of the long wavelength compared with thewell-localized particle eigenfunctions along z. The opti-cal field will resonantly drive Y by means of the optical in-terband transition dipole M08(re - rh). In contrast, theinfluence of the optical field on the exciton envelope isnegligible because E is off resonant on the left-hand sideof Eq. (4). Thus we neglect this term throughout thissection, but we will need it again in Section 4 when wediscuss the influence of the THz field. The differentialoperator [Eq. (5)] describes excitons formed by the stati-cally screened Coulomb potential W. We neglect in-planedispersion of the center of mass because of the long opticalwavelength compared with the exciton diameter.

We assume that the excitonic eigenfunctions Spx thatare defined by

hfly5o.(u, v; Zh, z) =howxq,(u, v; Zh, Ze) (6)

and the corresponding eigenenergies hx are known.The solution of Eq. (6) is technically difficult (see, e.g.,Ref. 17 for the case of superlattices). Expanding Y withrespect to the orthonormalized Cpx,

Y(U, V; U, V; Zh, Ze; t) = Zy.(U, V; t)PGx(u, v; Zh, Ze) ,

(7)

we obtain uncoupled equations of motion for the time-dependent expansion coefficients yx:

(a, + ix + y)yx(U, V; t) = (i/h)Wx*MoE(U, V; t), (8)

with Wx := f dzKx(0, 0; z, z) being the spatial overlap ofelectron and hole for the exciton x. The spatial depen-dence of Yx is governed by the driving force in Eq. (8),namely, the optical field. To be specific, we assume the

x exp{-iwo[t - t(U)]}f(U, V), (9)

with

tj(U) = tjo + (U/c)sin(a), (10)

where a denotes the angle of incidence of the optical pulseand c is the speed of light. We choose the center of thespot, which has the envelope f(U, V), as the origin ofour coordinates and define tj0 := 0 for this pulse, whichimmediately precedes the time interval in which we areinterested. Equation (8) can easily be solved by convo-lution with its Green's function. For the case of pulseenvelopes A(t) that behave as 6 functions for time scalesthat are small compared with the inverse interexcitonicfrequencies ( - cwi)-l of all involved excitons, we find asimple oscillatory behavior:

y.(U, V; t) = /ni7 f(U, V)exp{-i[(&w. - iy)t' + 811},(11)

with t' = t - (U/c)sin(a) and

,/F; exp(-i ) := Wx* EjejMo exp[i(wox - iy)tj°]

(12)

for t' > 0. n is the density (per unit area) of excitonsexcited into the eigenstate x. The complex amplitudesEj of all the preceding pulses add in Eq. (12). Thus con-structive as well as destructive interference are possibleif the time delay between successive pulses is smallerthan the dephasing time 1/y, as experimentally proved inRefs. 18-20. Constructive superposition requires a de-fined optical phase relation between pulses that arriveduring one dephasing time. Luryi proposed2

1 that a con-tinuous quantum oscillation be maintained with the helpof synchronized optical excitation pulses of a THz repeti-tion rate. He suggested that the pulses be produced byamplitude modulation of an optical cw laser to achieve therequired optical phase locking between pulses with inter-ferometric time precision. In the other case, i.e., if thepulse separation is longer than 1/y, only the last pulsewill contribute to the sum in Eq. (12). In the sectionsbelow we focus on this situation.

When inserting Eq. (11) together with expansion (7)into Eq. (2) for the charge density, we may make use of thefact that the exciton eigenstates ,op are well localized on ascale of the optical wavelength. This permits evaluationof the integrals and easy calculation of the polarizationdensity P, that is due to excitons:

Pz(UW, V; t) := dz zp (U, V, z; t)

2 nn' eZ",f 2(U, V)LU"(xsxt )

X cos(coxitl - Oxxi + SX - 6xi)exp(-2yt'),

(13)

Victor et al.


where OXX, := cox - x denotes the interexcitonic fre-quency; w, the number of wells; and Zxx, the elec-tron-hole separation matrix element:

Zxx, exp(-iE)x.,) := f dzzf dz'dudv

X [Spx*(U, V; Z Z,) Dx,(U, V; Z. Z')

- x(U, V; Z', Z)pDx'(u, v; z', z)].

(14)

Because of Eq. (2) and because of the connection ofthe excitonic transition amplitude with the opticalpolarization,' 4 the intraband polarization [Eq. (13)] canbe written as resulting from a second-order susceptibility

(2)XTH' that is closely related to the linear interband sus-ceptibility X)t:

(2)XTHZ(t, t2 ) = ZeZxxw.[ez 0 Xx*(tl) 0 X42)]

xx'

X'1)= E Wx[Mo* ® Xx(t)],x

Xx(t) = (i/h)Wx*Mo exp[-i(&0x - iy)(t)]e(t).

Thus the process of generation of the THz polarization isequivalent to linear absorption within the phase-coherenttime regime. Within this regime this process appearsas an extreme case of difference-frequency generation,with both input frequencies originating from the opticalpulse. The extreme difference in the scale of frequencies&) - &02 << 01, 02 is related to well-separated scales ofreal-space dynamics in the material: whereas the opti-cal interband transitions involve the atomic part of theexcitonic wave functions, the THz intraband polarizationis build up by the envelope dynamics.

For the evaluation of the polarization density P, thesummation in Eq. (13) must be performed over thenonordered pairs of excitons (x, x'). For x = x' the con-tribution to the polarization is constant in time except forthe exponential decay. The amplitude of the THz signalis proportional to the second time derivative of P,. Thesteplike behavior during the optical excitation gives riseto an instantaneous signal in the THz-radiation pattern,22

but for t > 0 only the oscillating terms from pairs x x'will contribute to the emitted THz field. We restrict ourdiscussion to this more-or-less monofrequent radiation.

Equation (13) describes the sources of the observed ra-diation as classical polarization currents. For simplicitywe neglect all but two excitons. The emitted radiationhas the frequency of the excitonic energy splitting 012,

and its amplitude is determined by the coherent excita-tion densities n, and n2 and the matrix element Z12. TheTHz field is a linear superposition of the radiation fromall spot positions (U, V). According to the conventionaltheory of broadside antenna arrays (see, e.g., Ref. 23),this superposition leads to fully directed radiation onlyfor the case of a spot that is large compared with the THzwavelength.

3. EFFICIENT POWER EXTRACTIONFROM EXCITONIC OSCILLATIONTo maintain the energy balance, emission of THz radia-tion implies transitions of carrier densities from the upper

to the lower exciton level. Therefore the energy that canbe converted into THz radiation after an optical excitationis limited by the interexcitonic energy

tEmax = Nih a) 2 , (15)

where the total number N, of carriers in the upper exci-tonic level (labeled 1) is given by the sheet density nwntimes the spot area. Unfortunately the process of ra-diation is slow compared with the excitonic dephasing,which interrupts the oscillations after a few picoseconds.22

Thus only a small part of the interexcitonic energy Emaxis converted into radiation; the larger part is wasted andheats the sample. It is our main goal in this section todescribe a way to enhance the THz emission by super-radiant cooperative emission. Here we assume that thematerial dynamics are completely described by Eq. (13),although backcoupling of the THz radiation was neglectedin Subsection 2.B. This assumption is correct only if ra-diation damping, THz absorption or lasing, etc., is small.Nevertheless, the basic phenomenon of cooperative emis-sion is a matter of conventional antenna theory, startingfrom the dipole density [Eq. (13)]. We discuss correctionsin Section 4. Throughout this paper we further neglectthe dielectric properties of the sample. These propertiesmight require adjustments of our results to include inter-nal reflections and possible waveguiding effects.

In the case of no dephasing (y = 0) the material oscil-lation would require a radiative lifetime 1 /Yrad, with

Prado

r max(16)

to emit the total interexcitonic energy. Prad' is the emit-ted power. Thus the ratio of Yrad and y determines theefficiency of a possible THz source.

There are several ways to increase Yrad. First, notethat the polarization density as well as TEmax is propor-tional to the excitation densities n. but that the radiationpower is quadratic in the polarization and hence in theexcitation density. Thus Yrad is proportional to the exci-tation density. The THz output increases quadraticallywith the optical intensity. This result is not surprising,as it is a direct consequence of the fact that conversionof light into THz waves is a X(2)-type process.24 Unfortu-nately, in practice the increase of THz efficiency by higherexcitation densities is limited by even faster dephasingthat results from large carrier densities.2 5

It is known to be a more general phenomenon that radi-ation rates can be enhanced by a factor of N if N oscil-lators emit collectively into the same electromagneticmode.2 2 26 For quantum systems this superradiance wasfirst pointed out by Dicke,7 who discussed two distinctsystems: (i) N inverted atoms (with incoherent popula-tion of the upper level) show spontaneous-emission ratesenhanced by a factor of N (superfluorescence). Most dis-cussions of superradiance are restricted to this situation(see Ref. 27 for an overview). (ii) N thermalized two-level atoms are excited by an electromagnetic pulse intoa coherent transition state, which is identical to a super-radiant state into which the initially incoherent systemof situation (i) evolves by cooperative spontaneous emis-sion. Thus, also in this case, without inverted incoherent

Victor et al.


populations the atoms will emit collectively.2 8 Becausethe radiation of coherent transitions in most cases is lim-ited by dephasing and not by the radiative decay, thisphenomenon has been called limited superradiance.2 7 Asuperradiant state in an extended medium can be visual-ized as one in which a macroscopic polarization is estab-lished over a region of space. In a quantum-mechanicaltreatment the polarizations are represented as transitionelements of the density matrix of stationary states; forN >> 1 this polarization is equivalent to a phased arrayof dipoles29 and can be treated classically. It is exactlythis situation that matches our problem, except that onegenerates the macroscopic polarization by pumping froma third level not involved in the radiation process.

Dicke7 especially discussed the radiation into full spaceof N atoms positioned in a volume of dimensions that issmall compared with the emitted wavelength. For thecase of initially coherent polarizations the power of the su-perradiant emission is that of a classical Hertzian dipole.The power is proportional to the square of the total dipolemoment and is thus proportional to N2 . This result en-ables us to increase THz efficiency not only by raisingthe optical intensity but also by enlarging the illuminatedspot while keeping the excitation density n, constant.The total excited area, however, must remain small com-pared with (A12).2

The situation becomes more complicated for spot di-mensions larger than A12 in any direction: for this casethere will be only one central direction7 in the cone ofTHz radiation in which the emitted intensity is quadraticin N. We are interested in the overall power emitted co-herently from the sample. To calculate the efficiency ofa THz source we integrate the power over the hole ra-diation pattern of the antenna instead of concentratingon the central direction. As the emitted lobe becomesnarrower by enlargement of the spot, we may expect themore intuitive result, i.e., that the integrated power isproportional to the emitting area. However, if we defineN as the number of cooperatively radiating dipoles thatdetermines Yrad, we find N to be only a fraction of the to-tal number N of excited excitons throughout the spot. Ndepends on the geometrical details of the excitation spot.We calculate it immediately below for the case of a rect-angular spot of dimensions a X b, where a is measured inthe direction of incidence of the optical excitation.

The electrical radiation field can be calculated fromEq. (13) by convolution with the retarded Green's functionof the wave equation. We focus on the oscillatory timedependence when differentiating P, twice, assuming thaty << (012. Additionally, we assume that fields from allparts of the spot can interfere, i.e., that light can travelalong the spot dimensions in a fraction of the dephasingtime. This assumption is specified by c/2y >> a, b. Weobtain the radiation field

E(r, t) = 12 ° i() F(6, 0; a, a, b)wnn7T810 r

X COS(w12T - 012 + 61 - 62)OXP(-2yr), (17)

where 0, 0 are the angles of polar coordinates; r := t -r/c is the time relative to the arrival of the THz pulse at

a distance r from the sample; and F is the spatial profileof the emitted pulse:

sini 2A [sin(6)cos() - sin(a)]j

F(6, 0; a, a, b) = sin(0)cos(0) - sin(a)

sin[ 2A sin(6)sin(k)]X in

sin(o)sin(o)(18)

Here F is a product of two slit-diffraction functions and iswell localized near 0 = 0, 0 = a V 0 = i - a for a, b >>A := c/C012; thus THz emission is directed roughly in thedirections of the transmitted ( = - a) and the reflected(6 = a) optical beams.9 We calculate the THz poweremitted into full space (both radiation lobes, time aver-aged over one THz oscillation) as the energy flux (givenby the Poynting vector) through a closed surface far fromthe sample and obtain

1 t+-/2

Prad(t) = - dt' do-n(soc 2 E X B)7 / 2

= §Prado exp(-4yt),

a= ce2Z122 a2 b

6iso A(19)

with

C(a, a, b) = 12 doJ dO A2 Sin3(0)xr f-, °ab abx F 2 (, 0; a, a, b). (20)

Inserting Proad into Eq. (16), we obtain a radiation rate:

(012 3e2Z12 2

Yra = ~6irh C30 N ,

N = 2n2WnA 2C(a, a, b).

(21)

(22)

Thus the radiation rate is enhanced by a number N ofcooperatively radiating dipoles. The radiation rate foran individual dipole (N = 1) would be the same as forspontaneous emission. This enhancement was pointedout in Ref. 22 with respect to THz-wave emission. Thenumber N was, however, assumed to be equal to N. Infact, N must be used. This number is proportional to theintegral C(a, a, b), which accounts for all the geometrydependencies. Figure 3 shows C for various angles ofincidence a in the case of a quadratic spot (a = b). Forsmall spots with a << A, C is proportional to the spotarea, whereas for large spots C becomes constant. Thedependence on a is stronger than expected from a sin2(a)law for a single Hertzian dipole. For a ' 900, when theTHz radiation leaves the sample under a grazing angle, Cincreases up to very high values of a. For a = 90°, C isproportional to the square of the spot length for a >> A andis not limited at all, apart from the condition that c/2y >>a, b, which was an assumption of our calculation. These

Victor et al.


1000

100

C.)

10

1

0.1 /0s

0.01 , , , , . .0.01 1 100 10000

spot area a * a ()

Fig. 3. Integral C(a, a, b), defined in Eq. (20), as a Ithe area of a quadratic spot (a = b) for the angles ofa = 45o, a = 80O, and a = 90°. The area is measureof (A)2.

results can also be obtained by asymptotic evalintegral (20), which leads to

N = 2n 2wnab if a, b << A,

A2N = 6grn 2wn sin2(a)

cos(a)

N = 6irn2wnAf6aA0.376

if b >> A-, a >

if a, b >> A, a

Typical experimental parameters (a, b >> A;Wn = 35; n2 = 0.5 X 109 cm72; Z12 = 50 A; W12 =

lead to

N 5 x 106,

1 - 1Yrad N4 ms 1 ns

For an absorbed optical input power of 2 mW aRef. 4, a time-averaged THz output of 2 x 10-9 of the two cones is expected, taking into account:losses at the semiconductor interfaces. This isof 20 larger than the estimated power of theradiation (see Subsection 2.A). This discrepanbe due to considerable absorption and reflectionthe semiconductor material30 and to the semitraSchottky contact.

The calculation shows that radiation is slowerphasing by a factor of several hundred. This ables us to increase the THz output within tiwithout raising the excitation densities if weto speed up radiation. Equations (23) disclosstep toward this goal. We can profit from co,radiation by flattening the excitation geometry.diation efficiency is optimized in the arrangen90°, called continuous end-fire array in antennaIn Section 5 we propose a technique for the realthis geometry.

A better visual understanding of the cooperatition mechanism can be obtained if the radiation

the antenna is calculated with the induced electromotiveforce (emf) method.23 Consider for a moment that theexcited spot is divided into two parts. For each part i wedefine an electromagnetic field (Es, Bj) that is the solu-tion of Maxwell's equations, neglecting the other part ofthe polarization currents. Because of the superpositionprinciple, the sum of both fields is the solution for the to-tal polarization. If we are interested in energy-flux den-sities that are quadratic in the fields there will be crossterms between the two fields, which add to the Poyntingvectors Sij oc E, X Bi. The source densities of these ad-ditional contributions to the energy flux, time averagedover one oscillation period, are

(VS+) oc2 (V(Ei X B2 + E2 X B))= -(E 1 j2 + E2 j1 ) - o(at(c2BjB2 + E1 E2 )), (24)I

where we have used the Faraday-Maxwell and the"unction of Ampere-Maxwell laws. The last time average in'incidence Eq. (24) vanishes for oscillatory fields because it con-ed in units cerns a time derivative. The first term is the classical

counterpart of superradiance between dipoles prepared

uation of in coherent transitions: it is precisely the emf on onepolarization current in the external field that was gener-ated by dipoles in the other region. For correct relativephase this term will be a positive contribution to the field

A energy. Of course, this contribution will vanish if thecos() radiation of one part of the spot does not pass through

the other part. This explains why, for emission perpen-a 0 90o, dicular to the surface, only small subensembles of the

- 90g. excited carriers can radiate collectively. If, in contrast,we are able to realize emission parallel to the surface,

(23) the radiation will pass through the whole spot length aa = 45°; and is amplified if we manage to keep the phase relation2ng THz) constant.

This insight leads us to a second possibility for increas-ing the power extraction from the quantum oscillation,namely, feedback in a resonator. The idea that the am-plification scheme for emitted waves should be applicableto any external coherent field with correct phase relationto the material oscillation seems quite natural. We are

s used in especially concerned with THz waves that are generatedr in each during earlier pulses and that are fed back to the samplereflection by means of a resonatorlike arrangement of mirrors (see

a factor Section 5). We now calculate the additional power ex-observed traction by a wave reflected back to an independently pre-Cy might pared polarization current in a resonator (see Fig. 4). Welosses in shall assume that the material dynamics are not affected.nsparent by the THz field and defer a discussion of corrections to

Section 4. The additional energy flux through a closedsthan de- surface far from the sample is determined by the cross

result en- terms between the backcoupled and the emitted fields.is rangemanage

3e a firstoperative

The ra-nent a =theory.2 3

ization of

Ive radia-power of

opt.T=0 T>0

Fig. 4. Possible setup of a cavity-based oscillator.

Victor et al.


This flux is equal to the emf power because of energyconservation. As the external field has been generatedby earlier pulses, it has the same geometry as the radia-tion field [Eq. (17)]. We assume this field to have an am-plitude E+/r, vectors of polarization e+, e_, and phases

E+(r, t) E+ F12(0, 0)e+ cos 1 2(t- -J + °+]r I ci

+ e_ cos[W12 (t + c ] (25)

Only the field traveling in the same direction as the radi-ated field contributes to the additional energy flux. In-tegration leads to the additional radiation power

P+(t) fdrnsoc2(Erad X B+ + E+ X Brad)

ab sin(a)= rceZ 2 wni A2 cos(a) E+ exp(-2yt)

X cos L (e+, eo)cos(E)1 2 - al + 32 - 0) (26)

The maximum gain is obtained if the external field is inphase with the radiated one and has the same polarizationbecause then both cosine terms will become unity. Thegain is proportional to the amplitude of the external field.We give an estimate in Section 5.

Equation (26) describes cooperative radiation from dif-ferent pulses that is possible because of coherence in time,whereas the dependence on N in Eq. (22) indicates coop-erative radiation that is due to spatial coherence amonga number of dipoles. To profit from Eq. (26), it is neces-sary to preserve the correct phase relation between thematerial oscillations and the THz field in the resonator.This requires a stable repetition rate of the optical pulses.

4. INFLUENCE OF THE TERAHERTZFIELD ON MATERIAL DYNAMICSIn the cavity setup discussed in Section 3 the samplewas assumed to be penetrated by a strong cw electro-magnetic field. We should worry about whetherbackcoupling of the THz field modifies the materialdynamics significantly. It is especially necessary toestimate whether the inevitable microwave absorption inthe sample is small compared with the gain [Eq. (26)].Whereas in Section 3 it was sufficient to specify theexternal field far from the sample, we now must considerthis field at the position of the active region. We assumethat the electrical-field amplitude, which in a steady-stateoperation is stored in the resonator, is large comparedwith the field emitted by a single pulse. Consistent withEq. (25), we write the standing-wave ansatz:

E+ = E°+eoi cos(col 2t + E+)cos(kU + (D+). (27)

The parameters of Eq. (27) can be related to those ofEq. (25) if we again calculate ?+, this time by integratingthe emf that is due to P, given by Eq. (13), and equatethe result to Eq. (26). We obtain

k = 12 sin(a),c

0+ - (D+ - (/2) = O' ,

L(e°, e.) = L (e+, eo),

+= tan(a). (28)

To discuss the influence of the external THz field on ex-citon dynamics, we must add the THz field to the op-tical one in Eq. (4). Although the THz and the opticalfields enter the equation through the same terms, theymust be treated differently in the subsequent calculation.Whereas we neglected the influence of the optical fieldon the exciton envelopes because it is highly oscillating,the same argument does not hold for the THz field. Incontrast, this almost static field (on optical time scales)is nonresonant on the right-hand side of Eq. (4) and willnot lead to interband transitions. Thus the influence ofthe THz field on the exciton dynamics is restricted to theenvelope effects that are due to its parametric appearancein the drift operator. Like any coherent nonlinear-opticalphenomenon, THz emission can be interpreted as a para-metric process. In this case it is the parametric transferof energy from an excitonic polarization wave (pump) toan electromagnetic THz wave (signal) and a second polar-ization wave (idler wave).

Our further calculation follows that given in Sub-section 2.B. Expanding Y in exciton-mode functions (p,we obtain coupled equations of motion that can be solvedin a rotating-frame approximation. Using Eq. (2), weobtain

P (U, V; t) = ef 2(U, V){i'1(t')exp(-i E)u)ZI

+ V2 2(t)exp(-iE)2 2)Z22

+ [ 2 (t)exp(-iE 12 )Zl 2 + C.C.]}, (29)

Pv i(t, U) = exp(-2yt)[ni cos2(WRt) + n2 sin2(CORt)

+ /fiuin sin(2 C0Rt)cos(l - 82 + D)], (30)

z-12(t, U) = exp(-2yt)exp[i(ol - cW2 )t]exp(-iI)

X (V/-nl2 {coS2 (Rt)exp[i(6l - 82 + (t)]

+ sin2 ((wRt)exp[-i(8l - 82 + CF)]}

+ 12(nl - n2)sin(2WRt)}),

WR(U) = (e+ -e.) eZ 12E+ cos(kU + cF+),2h

(D(U)= (D T - - 12 U sin(a)2 c

(31)

(32)

(33)

instead of Eq. (13). Below we restrict our discussion tolow repetition rates, which allow the material excitationto relax between successive pulses. For this case thedensity of carriers n. excited into one exciton state isthe same as is given in Subsection 2.B; in particular,it is not dependent on the position in the spot. Thesecoherent population densities perform Rabi oscillationsbetween the two excitonic levels, as is described by thetime-dependent population densities v, As the exter-nal field is a standing wave, the Rabi frequency (OR is afunction of the position on the spot. It is remarkable thatthe above results are identical to those for a simple two-level atom initially excited into a coherent superposition

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of the stationary states,3 ' although both exciton densitiesare built up by coherent transitions from a third level(the common ground level). In fact, because of Eq. (2),these coherent populations are still involved in interbandprocesses with optical frequencies while they performRabi oscillations connected with the THz interexcitonictransition.

For OR = 0, Eq. (29) is, of course, the same as Eq. (13).For a moderate external field (R < y) it makes sense toexpand the results [Eqs. (29)-(31)] with respect to toRtbecause of the exponential time decay. The first-ordercorrections to the results given in Subsection 2.B are pro-portional to the inversion (n - n2) if we again assumethat to12 >> > OR in performing time derivatives. Cal-culating the corresponding correction to the radiated en-ergy by integrating the emf that is due to p(l) and E+over the hole spot, we obtain

P,')=h(w1 - .o2)(nl - n2)( 2 i )

X abwnt' exp(-2yt'). (34)

This result indicates that the system is lasing for n1 > n2and acts as an absorber otherwise (with subscript 1 de-noting the upper level). If the interexcitonic two-levelsystem leaves the optical excitation in an inverted situa-tion (e.g., because of different spatial overlap of electronsand holes in the states 1, 2), X>(1) is an additional gain toEq. (26). If §PWi) is a loss term the total gain P+ + P()might nonetheless be positive for a moderate amplitude ofthe external field. For this reason amplification based onthe total gain was called lasing without inversion in a cal-culation concerning two-level atoms.31 This terminologymight be misleading3 2 because it effaces the differencesbetween the antennalike radiation of initially preparedpolarization currents and the two-step process of lasing,which requires inversion and an external field to effect atransition density that in a second step emits radiation.

Whether a possible absorption for n, < n2 would sub-stantially decrease the total gain depends on the am-plitude of the external field. We give an estimate inSection 5.

5. CONFIGURATIONS WITHOPTIMIZED POWER EXTRACTIONThe possibility of obtaining cooperative emission of anelectromagnetic wave in a coherent quantum system sug-gests that optical concepts could be adapted to raise thepower output of possible ultrahigh-frequency emitters.Below we discuss two main approaches, which may also beused in combination. We call the first a traveling-waveapproach. The second approach is based on a cavity inwhich the coherent quantum system functions as a gainmedium.

A. Traveling-Wave OscillatorThe most efficient radiation geometry according toEqs. (23) is realized if the THz radiation is emitted par-allel to the surface. For this case the first spot region,which is excited by the optical pulse, will emit a THz wavethat extracts energy from the other oscillators along its

path through the sample. For the operation of sucha phase-coherent process it is essential that the quan-tum beat be prepared at any place in phase with thepropagating wave. According to the calculation givenin Section 3, this phase adjustment would require thatthe optical excitation pulse propagate parallel to the sur-face, too.

Nevertheless, emission parallel to the surface (a = 90°)can be realized with other angles of incidence for theoptical beam. Remember that the only requirement forphase synchronization is the correct time of arrival of theoptical transients at a specific place on the sample accord-ing to Eq. (10). We can profit from a setup analogous tothe traveling-wave dye laser used for the generation of in-frared pulses from optically pumped dyes with a picosec-ond excited-state lifetime.3 3 There the synchronizationis accomplished by tilting of the wavefront of the pumppulse, achieved with the help of a grating, which reflectsthe pump beam before it hits the dye cell.34

A possible setup is sketched in Fig. 5. For an estimateof the gain to be expected for a traveling-wave scheme, wecompare N, obtained from Eqs. (23), for a = 90° and theusual setup, where a = 45°:

N(90') = a= 0.376 _

N(45o) A(35)

For a spot length of 2 mm the emitted power of 1-THzradiation would be enhanced by a factor of 3.5 by atraveling-wave setup. Equation (35) predicts better ef-ficiencies for radiation with a shorter wavelength or forlarger spot dimensions. Unfortunately, Eq. (35) is re-stricted to spots with a < c/2y. For large a one endof the spot has already stopped emission, whereas theother end has not even begun to radiate. This com-plicates interference and might lead to less-collimatedemission. Nevertheless, the basic idea that the pulseamplitude will rise while the pulse is traveling throughthe sample is expected to hold. The emf of the field

Grating

Front

Fig. 5. Schematic representation of a traveling-wave setup forefficient generation of electromagnetic waves.

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on the polarization currents at the wave front and theefficiency of power extraction will rise during the propa-gation of the THz pulse, even for a > c/2y. Thepossibility of fulfilling both phase matching and syn-chronization with the traveling THz pulse is limited bythe difference between group and phase velocities in thespectral regime of the THz emission. For narrow-bandradiation well below absorption resonance this differencedoes not impose severe restrictions. For example, inGaAs the dispersion dn/dco has a value O.001 THz-'below 2 THz.30 Even for a 500-GHz-bandwidth pulsethis dispersion does not limit amplification for spotlengths to as large as 1 cm.

A traveling-wave geometry may be especially interest-'ing for large-area excitation, which requires high-energyamplified optical pulses for maintenance of the optimalexcitation densities.

B. Cavity-Based OscillatorIn Section 3 we pointed out the general possibility of en-hancing the power extraction from the wave-packet os-cillations by phase-locked feedback of the THz radiation.Phase locking requires that the time delay between suc-cessive pulses be an integral multiple of the THz oscilla-tion period. Only the power envelope, and not the phaseof the optical pulse, is involved. This property leads torather relaxed experimental conditions for establishingphase locking in the generation of submillimeter-waveradiation.

The required cavity (see Fig. 4) may consist of twospherical mirrors, with the sample being positioned in thecommon center. The cavity length must be an integralmultiple of the THz wavelength to permit standing waves.A cavity permits the possibility of creating a continuouselectromagnetic wave, even by pulsed excitation and emis-sion. To build up a cw, the time delay between succes-sive pump pulses, measured in oscillation periods, andthe cavity length, measured in wavelengths, should haveno common nontrivial divisor. With this arrangementthe patches of radiation emitted after excitation with in-dividual pump pulses will become stitched together toform a cw.

To estimate the efficiency of the cavity feedback we re-strict ourselves to this case of a cw setup. We assumethat all loss terms of the cavity are proportional to theinternal power that is reflected by the mirrors. We as-sume a transmittivity T for the output coupler and an in-ternal loss coefficient A. Equating the energy loss duringthe repetition time A of the optical pulse with the time-integrated single-pulse gain that is due to Eqs. (19) and(26) leads to the steady-state useful cw output power:

° Prad T4yA (A + T)2 yA {1 + [1 + (A +

. ?rad 4T4yA (A+ T)2 yA

max Prad 1

4yA AyA

for A + T <<-YA

for T = A; A + T «I 1.YA

Here dtd/(4A) is the time-averaged power that isemitted without the resonator setup. Thus the power

extraction can be increased by a factor of (AyA)-1 if thetransmittivity of the output coupler is optimized with re-spect to the internal loss. One possible loss mechanismthat is inherently related to the emission process is theabsorption of the THz wave by the interexcitonic tran-sition. Relating the time-integrated result [Eq. (34)] tothe internal cavity power, we obtain the correspondingloss coefficient Amin that is due to absorption:

(37)AmjnyA = rad n2 -n.V n2

If by chance or by arrangement the excitation of the up-per level is larger than that of the lower one, Amin will benegative. In this case Amin describes a lasing gain thatraises the overall efficiency of the cavity. In the case ofpositive Amin the corresponding absorption will be small.For a typical coherent excitation with the aim of opti-mizing the interexcitonic amplitude proportional to /-ii[see Eq. (13)], we will have n1 - n2. Thus the secondfraction in Eq. (37) is less than unity, and the absolutevalue of Amin, is determined by the relation of Yrad [theradiation rate without resonator according to Eq. (21)] tothe dephasing rate. As discussed in Section 3, this termwill be small in the normal case. Clearly, additional lossterms, e.g., absorption losses in the semiconductor ma-terial (by phonons,3 0 free carriers, etc.) and the contactmetallization, must be taken into account.

A cavity-based oscillator might substantially enhancethe efficiency of coherent oscillation if it is possible to re-alize a small parameter AyA, that is, a small loss coeffi-cient A and a high repetition rate A-1. Ideally, A ' 1/yshould be realized to achieve optimal power input with-out destructive interference of the interband transitionsgenerated by successive pulses. As it is difficult to ob-tain such high repetition rates where A 1/y, a slightlymodified setup might be useful. If A is chosen to be anintegral multiple of the round-trip time of a THz pulse,the THz field will build up not a cw but a highly peakedpulse. For this case the principal mechanism of extract-ing the power of the quantum oscillation by use of the emfin the THz pulse should work even better.

6. CONCLUSIONThe power efficiency of radiation emitted from coherentcharge oscillations in semiconductor heterostructures isdetermined by both quantum dynamics and electromag-netic propagation. Although the quantum-mechanicaldynamics can in most respects be understood to be inde-pendent of the irradiated fields, optimized setup geome-tries that include optical elements such as mirrors maysubstantially increase the THz output power. Thus THzradiation generated by excitonic oscillations is an inter-disciplinary challenge for quantum electronics as well asfor conventional electromagnetic theory.

ACKNOWLEDGMENTSHelpful discussions with V. M. Axt, S. L. Chuang,H. Kurz, M. Nuss, and A. Stahl are gratefully ac-knowledged. H. G. Roskos is indebted to J. Shah forsuggesting coherent charge carrier phenomena as a fas-cinating topic of semiconductor physics. This research

Victor et al.


was supported by the Deutsche Forschungsgemeinschaftand the Volkswagen Stiftung.

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Efficiency of submillimeter-wave generation and amplification by coherent wave-packet oscillations in semiconductor structures