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Self-induced transparency of excitons in semiconductors
Volume 57, number 3
OPTICS COMMUNICATIONS
1 March 1986
S E L F - I N D U C E D T R A N S P A R E N C Y O F E X C I T O N S IN S E M I C O N D U C T O R S W. H U H N lnstitut ft~r Theoretische PID'sik, R W T H Aachen, Fed. Rep. Germany
Received 17 July 1985: revised manuscript received 9 December 1985
Self-induced transparency of excitons is analysed in a model where intraband processes are described by density matrices and interhand processes by coherent pair amplitudes. We determine the dispersion law of the carrier wave and the exciton wave function. The theory predicts: (a) a forbidden energy gap centered at the exciton line which broadens with increasing intensity', (b) a critical dependence of the effect on intensity, (c) a strong influence of intensity and carrier frequency on the exciton wave function.
1. Introduction When noninteraction two level atoms are subjected to a short pulse of intense coherent light at resonance, the effect of self-induced transparency (SIT) can be observed [ 1 ]. It has been discussed controversially, whether this effect would also be possible in semiconductors [ 2 - 5 ] . A necessary condition for SIT is that the pulse duration is shorter than all incoherent relaxation processes in the system. If we assume that this condition can be met experimentally in the case o f excitonic transitions in a direct gap semiconductor, the question arises, how the mobility o f the electrons and holes in the fmal state will influence the coherent propagation of the pulse. We shall investigate this problem by using a real-space theory for the electrodynamics o f a two-band semiconductor which was developed in ref. [6]. This theory can be seen as an appropriate generalization of the M a x w e l l Bloch theory for two level atoms [7]. It describes electrons and holes b y density matrices and e l e c t r o n - h o l e pairs by coherent pair amplitudes. As a consequence of the real-space formulation b o t h quantities can be directly related to macroscopic electrodynamical quantities. It was shown in ref. [6] that therefore all interactions between excitons can be accounted for by self-consistent fields o f the transverse and longitudinal type. Thus incoherent interactions between excitons, which would lead to effective interaction terms in a theory which treats excitons as interacting bosons in reciprocal space [5,8], can be interpreted as contributions to a longitudinal scalar potential. The effect due to the compositeness o f the exciton, i.e. the Pauli repulsion of its constituent particles, appears naturally as a density dependent screening of the driving electric field. The method enables us to calculate the exciton wavefunction and a dispersion law for the carrier wave of the light pulse which is conspicuously different from the result obtained by GoU and Haken [4]. This disagreement is due to the fact that our theory considers coherence in the e l e c t r o n - h o l e configuration space and thus explicitly accounts for the mutual influence o f the light wave and the wave function in e l e c t r o n - h o l e relative space. The implied concept o f a non-rigid polariton whose wavefunction depends on the frequency o f the exciting electric field has shown to be successful especially in the context o f boundary value problems [9].
2. Basic equations Our constitutive equations can be regarded as the Bloch equations for two level systems including terms for the mobility and the mutual attraction o f electrons and holes [6]. For simplicity we consider a one-dimensional 0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Pubhshing Division)
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model [10], assume a contact potential for the e l e c t r o n - h o l e attraction and take the effective masses o f electrons and holes to be equal. Furthermore we suppose that the pulse duration is sufficiently shorter than the phenomenological relaxation times T 1 and T 2. ]g(z, X) + i~2eh Y(z,x) = (iMo/fi)[E(z)8(x) - E ( z + x/2)C(z,x) - E(z - x/2)D(z 1 - x ) ] ,
(la)
¢(z, x ) + ig2eeC(Z , x) = (iMo/l~)[E(z - x / 2 ) Y * ( z , - x ) - E ( z + x/2)Y(z, x ) ] ,
(lb)
Jg(z, x) + ig2hhD(Z , x) = (iMo/h) [E(z - x/2)Y*(z, x) - E(z + x/2)Y(z, - x ) ] ,
(lc)
z and x are the center o f mass and the relative coordinate o f electrons and holes. Y(z, x) is a coherent pair amplitude which describes the interband transitions. The density matrices C(z, x) and D(z, x) for electrons in the conduction band and holes in the valence band represent the intraband processes. In terms o f Fermi destruction operators Czl , dzl for electrons and holes at position z 1 respectively we may write:
Y(Zl , z 2 ) = (dzxcz2)= Yl - iY2,
(2a)
C(z 1 , z2) = (C+lCz2) = C 1 - iC 2 ,
(2b)
D(z 1, z2) = (d+zldz2 ) = D1 - iD2-
(2c)
The mobility is contained in the differential operators ~eh = COg - (fi/2M)32 - (h/2U)O 2 - U08(x),
(3a)
~ e e = g~hh = ( h / m ) a z a x ,
(35)
where M = m e + m h and 1/p = 1/m e + 1/m h. hcog denotes the energy gap between the bands, M 0 the dipole transition matrix element and U0 the strength o f the contact potential. Eqs. (1) are supplemented b y the wave equation for the transverse part of the electric field
O2E(z, t) - c2O2E(z, t) = --(2Mo/eoeb)a 2 Yl (z, x = O, t).
(4)
In order to obtain equations for slowly varying envelopes we set:
E(z, t) = E(z, t) cos (kz - cot), (Y1)(zx't')=2Y2
()
C~ (z,x,t) 1)]
I { ( Q )(z'x't)exp[i(kz-cot)]
:'{(') ~
S]
.(z,x,t) exp(-ikx/2) +
(5)
+(P*i(z,x,t)[exp[-i(kz-cot)]}\Q,]
\S?]
(z,x,t)exp(ikx/2)
,
} '
/ = 1 2. '
(6)
(7)
ff~, P, Q, RI, S/are slowly varying quantities on the length and time scales o f the carrier wave. Note that we retain the full dependence on the relative coordinate x except for a phase factor exp(ikx/2) in the intraband variables which is a direct consequence o f the high density saturation not being localized (cf. rhs of eq. (1)). Using (5) and (6) in (4) we obtain two field equations:
~t ~ + c2 (k/co)~z ~ = -(Moco/eOeb ) Im P(z, x = O, t), (1
222
--
c2k2/co2)ff,
=
--(2M0/e0eb) Re P(z, x = 0, t),
(8) (9)
Volume 57, number 3
OPTICS COMMUNICATIONS
1 March 1986
and by substituting the ansatz ( 5 ) - ( 7 ) into (1) we are left with 12 coupled equations for 12 unknowns. Choosing appropriate linear combinations fr and gr of the interband quantities P and Q, and n r of the intraband variables R i and S i the system of equations decouples into two sets: a resonant and an antiresonant set. The resonant one reads:
(a t + Vexaz)fr + i(h/2/a)(t~ 2 - a 2 - ( 2 / a B ) 6 ( x ) ) g r = 0,
(10a)
(a t + Vexaz)g r + i(h/2/a)(• 2 -- a 2 -- ( 2 / a B ) f ( x ) ) f r = i(2Mo/h)ff~(5(x ) - ½nr) ,
(10b)
(a t + Vexaz)n r = -i(Mo/h)ff?gr,
(lOc)
with a B = h/(PUo) (exciton Bohr radius); Vex = l i k / M (exciton velocity) and t~2/a = (2/.t/hXcog -T-6o + h k 2 / 2 M ) .
(11)
Note that we only take the resonant terms into account (rotating wave approximation) and therefore neglect terms exp [+ 2i(kz - cot)].
3. Steady-state pulse
solution
In the steady state one can assume if?,fr, gr and n r to be functions of the variable w = t - z/v alone, where v is the pulse velocity. After Fourier transformation with respect to the relative coordinate x, eqs. (10) become: (1 - Vex/V)a w fr(W, q) + i(h/2/a)(r 2 + q2)~r(W, q) - i(h/PaB)gr(W, 0) = 0,
(12a)
(1 -- Vex/V)a w ~r(W, q) + i(h/2tz)(K 2 + q 2 ) f r ( W , q) -- i(h/PaB)fr(W , O) = -i(Mo/h)ff~(w)lfr(W, q),
(12b)
(1 - Vex/V) awir(W , q) = - i ( M o / h )ff(W)~r(W , q),
(12c)
where we have introduced afr = h r - 2. We observe that the terms arising from the additional degree of freedom, i.e. the relative motion of electrons and holes, seem to play a similar role in (12) as the inhomogeneous broadening in the theory o f two level systems [ 1,11 ], leading to a detuning of the system. We therefore first try to solve (12) at "exact resonance". If we drop the corresponding terms we obtain: /r(W) = - 2 cos p(w)
~r(W) = 2i sin p(w),
(13,14)
with M0
w
p(w) - h(1 -- Vex/V) f
E(w')dw'.
(15)
__oo
Following the method used by McCall and Hahn [1] we make the following ansatz: gr(W, q) : 2i~,r(q) sin p(w).
(16)
Inserting (16) into (12) we obtain a condition for the pulse profile h3
sin p(w) -
-
- o r = const.
(17)
4/a2M0(1 - Vex/V ) aw'ff(w ) and an integral equation for "Yr(q) 223
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1 March 1986
OPTICS COMMUNICATIONS
ar _ (2/aB)ar f [.i,r(q)/(K2 + q2)] dq/Zrr + 2a 4r (Kr2 + q2)fZ/r(q)dq/2rr 5'r(q) = - -
4 2 q2)2 1 +Or(K r +
,
(18)
with a r = KrqB/(gra B -- 1). Eq. (18) can be readily solved by noting that it has the structure 2 2 4 2 ~'r(q) = JAr + ar(Kr + q 2 ) B r ] / [ 1 + Or (Kr +q2)2].
(19)
A little algebra yields A r = [(k-taB) 2 -- GaB cos S0r]/[1 - 2 g r a B cos ~or + ( g r a B ) 2 ] ,
(20)
B r = (graB)2/[1 -- 2k-ra B cos Cr + (~raB) 2],
(21)
where k-r = (K4 + Or 4 )1/4; ~r = 1 arctan (Kr 2 Or2). After Fourier transformation the result can be written as
()
PQ = (Fr(x)ff(w) + ½i sin p(w)3,r(X))
-
1
i + ( f a ( x ) E ( w ) - il sin p(X)Ta(X))
(1)
-i
,
(22)
=tS2(_x))=3'r(X)sin2½P(w)(1)+Ta(x) where we have defined Fr(X) = ~ [(%/Kra B) exp(-K r Ix [) 6r(0) + ~r(X)], 4/1/140 r e x p ( - K r l x ~r(X) = T k 2K r
[)
__1
(1 - A t ) + 2k-r (A r cos ~r - Br sin ~3r) exp( •r cos S0rlX[)l
(24)
(25)
Vr(x) = (1/2Go 2) exp(- G cos ~rlx I)(Ar sin ~r + Br cos/3r),
(26)
~r(x) = Tr + k-r sin ~rlX 1.
(27)
The antiresonant quantities (index " a " ) can be obtained by replacing Kr -+ Ka; o r -+ %. Insertion of (22) into the field equation (9) gives a dispersion law for the carrier wave:
c2k2/co 2 = 1 + (Mo/2eOeb)(arthr(O) + aa~a(0)) .
(28)
Eq. (8) yields a differential equation for the pulse profile:
r2 ( d/ d w ) E ( w ) = (lt/Mo )(1 - Uex /V ) sin p(w) ,
(29)
r2 _ 2/ie0eb c2 k/cou - 1 M 0 2 ~ - ( 1 - Vex/U) ~r--~---- 7a(i3)
(30)
with (pulse width).
The solution of eq. (29) is the well known stable hyperbolic sech pulse of area 2rr. Note that therefore the pulse width is a measure of the intensity of the pulses.
224
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1 March 1986
~10 -2
~._..._.. W: CI< 257,0
10 1
wi > >,
WT
o~ 10-2 CD 2 3...
c 25 z,,0 (11 (2) (3) (4)
4
o >
T= 0,1ps T= 0,5p$ "r= 1,0 pS ~= 2,0 ps
10~3
£ 10 ~
251,0
I 8,00
,
. ] ,
g,00 WQvenumber
10,00
11,00._10 2
,
I
252,0
I
254,0
J] I %WL256,0
, 4~10-2
k QBohr
Energy [eV] Fig. 1. Dispersion curves calculated by using material constants
of CdS. The parameter is the pulse duration (as a measure o f the intensity): (1) 0.1 ps, (2) 0.5 ps, (3) 1.0 ps, (4) 2 . 0 ps. T h e dotted line is the dispersion curve o f the exciton polariton. Spatial dispersion is accounted for throughout.
~--
Fig. 2. Ratio o f pulse velocity to the velocity o f light in the m e d i u m for a 0 . 5 ps pulse. The model substance is C d S . T h e dotted line gives the exciton velocity Uex = hk/M.
4. Results Eqs. (28) and (30) provide two coupled transcendental equations for wavenumber k and pulse velocity o for given frequency co and pulse width r. The solution is given in figs. 1 and 2. For small pulse widths there is a section with anomalous dispersion. The forbidden energy gap centered at the exciton line increases with growing intensity. The gap is readily explained by the observation that near resonance the pulse velocity (fig. 2) approaches the exciton velocity. In this case excitons cannot be excited coherently so that the coherent propagation process is destroyed and the pulse is absorbed. Off resonance our solution (fig. 1) asymptotically approaches the dispersion curve of the linear polariton. The polariton is the limiting case for long pulses, but in contrast to the results obtained by Goll and Haken [4] there is no polaritonlike solution below a certain pulse duration. The qualitative change in the shape of the dispersion curve at a certain pulse duration resembles a phase transition. Above a critical pulse duration there are no intersection points with the mode of the free light field and below there are two such points. As we have devised our theory in electron-hole configuration space we can calculate the exciton wave function, which we identify as the real part of P in eq. (22). While the wave function has the simple exponential shape o f the 1s exciton at some distance to the intersection points, some conspicuous modifications occur in a small interval containing these points (fig. 3). The wavefunction resembles that of an excited state and it is there where it has its strongest dependence on intensity (fig. 4). Thus our prediction is: If it is possible to observe SIT in semiconductors at all, it will occur below and above a certain energy gap but there will be no effect on resonance (exciton-line). Below a critical pulse duration the carrier wave has a dispersion law which differs both from the dispersion of the free light field and the dispersion of polaritons. The inset of the effect will resemble a phase transition.
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I
OPTICS COMMUNICATIONS
0,2
4,0 "g
D
2
1 March 1986
0,0 -0,2
>,,,
2,0
-0,4
(3
-0,6
~ 1,o "s 0-
E o,o
"5
-0.8
4
1,0
~" -10
-6
-2,0
-1,2 -1,4 I
0,0
1,2 2,4 Relative coordinate
3,6
/.,8
/ClBohr
Fig. 3. Exciton wave function for a small energy interval containing an intersection point of the dispersion curve with the mode of the free light field (1) 1.5166 eV ~
0,0
1.2
2/.
Relative c o o r d i n o i e
3,6 X/Olqoh
I
L
L,8 r
Fig. 4. Exciton wave function for a small intensity interval containing an intersection point of the dispersion curve with the mode of the free light field. (1) 1.02 ps < r ~< (4) 1.08 ps. The energy was chosen to be h~o = 1.5125 eV. The model substance is GaAs.
Acknowledgement The a u t h o r wishes t o t h a n k Prof. A. Stahl for criticism and m a n y h e l p f u l discussions.
References [1 ] S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1969) 457. [2] D.H. Auston, Picosecond nonlinear optics, in: Ultrashort light pulses, ed. S.L. Shapiro (Springer Verlag, 1977) p. 123. [3 ] B. Bosacchi, Picosecond spectroscopy and solid state physics, in: Coherence in spectroscopy and modern physics, eds. t:.T. Arecchi, R. Bonifacio, M.O. Scully (Plenum Press, 1978). [4] J. Goll and H. Haken, Phys. Rev. A 18 (1978) 2241. [51 E. Hanamura, J. Phys. Soc. Japan 37 (1974) 1553. [6] W. Huhn and A. Stahl, Phys. Stat. Sol. (b) 124 (.1984) 167. [7] A. Stahl, Coherence and saturation of exciton polaritons in semiconductors, in: Festk6rperprobleme, Vol. 25, ed. P. Grosse, (Vieweg Verlag, 1985) p. 287. [8] M. Inoue, J. Phys. Soc. Japan 37 (1974) 1560. [91 L. Gotthard, A. Stahl and G. Czajkowski, J. Phys. C 17 (1984) 4865. [10] I. Balslev and A. Stahl, Phys. Stat. Sol. (b) 111 (1982) 531. [11 ] G.L. Lamb, Jr., Rev. Mod. Phys. 43 (1971) 99.
226
OPTICS COMMUNICATIONS
1 March 1986
S E L F - I N D U C E D T R A N S P A R E N C Y O F E X C I T O N S IN S E M I C O N D U C T O R S W. H U H N lnstitut ft~r Theoretische PID'sik, R W T H Aachen, Fed. Rep. Germany
Received 17 July 1985: revised manuscript received 9 December 1985
Self-induced transparency of excitons is analysed in a model where intraband processes are described by density matrices and interhand processes by coherent pair amplitudes. We determine the dispersion law of the carrier wave and the exciton wave function. The theory predicts: (a) a forbidden energy gap centered at the exciton line which broadens with increasing intensity', (b) a critical dependence of the effect on intensity, (c) a strong influence of intensity and carrier frequency on the exciton wave function.
1. Introduction When noninteraction two level atoms are subjected to a short pulse of intense coherent light at resonance, the effect of self-induced transparency (SIT) can be observed [ 1 ]. It has been discussed controversially, whether this effect would also be possible in semiconductors [ 2 - 5 ] . A necessary condition for SIT is that the pulse duration is shorter than all incoherent relaxation processes in the system. If we assume that this condition can be met experimentally in the case o f excitonic transitions in a direct gap semiconductor, the question arises, how the mobility o f the electrons and holes in the fmal state will influence the coherent propagation of the pulse. We shall investigate this problem by using a real-space theory for the electrodynamics o f a two-band semiconductor which was developed in ref. [6]. This theory can be seen as an appropriate generalization of the M a x w e l l Bloch theory for two level atoms [7]. It describes electrons and holes b y density matrices and e l e c t r o n - h o l e pairs by coherent pair amplitudes. As a consequence of the real-space formulation b o t h quantities can be directly related to macroscopic electrodynamical quantities. It was shown in ref. [6] that therefore all interactions between excitons can be accounted for by self-consistent fields o f the transverse and longitudinal type. Thus incoherent interactions between excitons, which would lead to effective interaction terms in a theory which treats excitons as interacting bosons in reciprocal space [5,8], can be interpreted as contributions to a longitudinal scalar potential. The effect due to the compositeness o f the exciton, i.e. the Pauli repulsion of its constituent particles, appears naturally as a density dependent screening of the driving electric field. The method enables us to calculate the exciton wavefunction and a dispersion law for the carrier wave of the light pulse which is conspicuously different from the result obtained by GoU and Haken [4]. This disagreement is due to the fact that our theory considers coherence in the e l e c t r o n - h o l e configuration space and thus explicitly accounts for the mutual influence o f the light wave and the wave function in e l e c t r o n - h o l e relative space. The implied concept o f a non-rigid polariton whose wavefunction depends on the frequency o f the exciting electric field has shown to be successful especially in the context o f boundary value problems [9].
2. Basic equations Our constitutive equations can be regarded as the Bloch equations for two level systems including terms for the mobility and the mutual attraction o f electrons and holes [6]. For simplicity we consider a one-dimensional 0 030-4018/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Pubhshing Division)
221
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model [10], assume a contact potential for the e l e c t r o n - h o l e attraction and take the effective masses o f electrons and holes to be equal. Furthermore we suppose that the pulse duration is sufficiently shorter than the phenomenological relaxation times T 1 and T 2. ]g(z, X) + i~2eh Y(z,x) = (iMo/fi)[E(z)8(x) - E ( z + x/2)C(z,x) - E(z - x/2)D(z 1 - x ) ] ,
(la)
¢(z, x ) + ig2eeC(Z , x) = (iMo/l~)[E(z - x / 2 ) Y * ( z , - x ) - E ( z + x/2)Y(z, x ) ] ,
(lb)
Jg(z, x) + ig2hhD(Z , x) = (iMo/h) [E(z - x/2)Y*(z, x) - E(z + x/2)Y(z, - x ) ] ,
(lc)
z and x are the center o f mass and the relative coordinate o f electrons and holes. Y(z, x) is a coherent pair amplitude which describes the interband transitions. The density matrices C(z, x) and D(z, x) for electrons in the conduction band and holes in the valence band represent the intraband processes. In terms o f Fermi destruction operators Czl , dzl for electrons and holes at position z 1 respectively we may write:
Y(Zl , z 2 ) = (dzxcz2)= Yl - iY2,
(2a)
C(z 1 , z2) = (C+lCz2) = C 1 - iC 2 ,
(2b)
D(z 1, z2) = (d+zldz2 ) = D1 - iD2-
(2c)
The mobility is contained in the differential operators ~eh = COg - (fi/2M)32 - (h/2U)O 2 - U08(x),
(3a)
~ e e = g~hh = ( h / m ) a z a x ,
(35)
where M = m e + m h and 1/p = 1/m e + 1/m h. hcog denotes the energy gap between the bands, M 0 the dipole transition matrix element and U0 the strength o f the contact potential. Eqs. (1) are supplemented b y the wave equation for the transverse part of the electric field
O2E(z, t) - c2O2E(z, t) = --(2Mo/eoeb)a 2 Yl (z, x = O, t).
(4)
In order to obtain equations for slowly varying envelopes we set:
E(z, t) = E(z, t) cos (kz - cot), (Y1)(zx't')=2Y2
()
C~ (z,x,t) 1)]
I { ( Q )(z'x't)exp[i(kz-cot)]
:'{(') ~
S]
.(z,x,t) exp(-ikx/2) +
(5)
+(P*i(z,x,t)[exp[-i(kz-cot)]}\Q,]
\S?]
(z,x,t)exp(ikx/2)
,
} '
/ = 1 2. '
(6)
(7)
ff~, P, Q, RI, S/are slowly varying quantities on the length and time scales o f the carrier wave. Note that we retain the full dependence on the relative coordinate x except for a phase factor exp(ikx/2) in the intraband variables which is a direct consequence o f the high density saturation not being localized (cf. rhs of eq. (1)). Using (5) and (6) in (4) we obtain two field equations:
~t ~ + c2 (k/co)~z ~ = -(Moco/eOeb ) Im P(z, x = O, t), (1
222
--
c2k2/co2)ff,
=
--(2M0/e0eb) Re P(z, x = 0, t),
(8) (9)
Volume 57, number 3
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1 March 1986
and by substituting the ansatz ( 5 ) - ( 7 ) into (1) we are left with 12 coupled equations for 12 unknowns. Choosing appropriate linear combinations fr and gr of the interband quantities P and Q, and n r of the intraband variables R i and S i the system of equations decouples into two sets: a resonant and an antiresonant set. The resonant one reads:
(a t + Vexaz)fr + i(h/2/a)(t~ 2 - a 2 - ( 2 / a B ) 6 ( x ) ) g r = 0,
(10a)
(a t + Vexaz)g r + i(h/2/a)(• 2 -- a 2 -- ( 2 / a B ) f ( x ) ) f r = i(2Mo/h)ff~(5(x ) - ½nr) ,
(10b)
(a t + Vexaz)n r = -i(Mo/h)ff?gr,
(lOc)
with a B = h/(PUo) (exciton Bohr radius); Vex = l i k / M (exciton velocity) and t~2/a = (2/.t/hXcog -T-6o + h k 2 / 2 M ) .
(11)
Note that we only take the resonant terms into account (rotating wave approximation) and therefore neglect terms exp [+ 2i(kz - cot)].
3. Steady-state pulse
solution
In the steady state one can assume if?,fr, gr and n r to be functions of the variable w = t - z/v alone, where v is the pulse velocity. After Fourier transformation with respect to the relative coordinate x, eqs. (10) become: (1 - Vex/V)a w fr(W, q) + i(h/2/a)(r 2 + q2)~r(W, q) - i(h/PaB)gr(W, 0) = 0,
(12a)
(1 -- Vex/V)a w ~r(W, q) + i(h/2tz)(K 2 + q 2 ) f r ( W , q) -- i(h/PaB)fr(W , O) = -i(Mo/h)ff~(w)lfr(W, q),
(12b)
(1 - Vex/V) awir(W , q) = - i ( M o / h )ff(W)~r(W , q),
(12c)
where we have introduced afr = h r - 2. We observe that the terms arising from the additional degree of freedom, i.e. the relative motion of electrons and holes, seem to play a similar role in (12) as the inhomogeneous broadening in the theory o f two level systems [ 1,11 ], leading to a detuning of the system. We therefore first try to solve (12) at "exact resonance". If we drop the corresponding terms we obtain: /r(W) = - 2 cos p(w)
~r(W) = 2i sin p(w),
(13,14)
with M0
w
p(w) - h(1 -- Vex/V) f
E(w')dw'.
(15)
__oo
Following the method used by McCall and Hahn [1] we make the following ansatz: gr(W, q) : 2i~,r(q) sin p(w).
(16)
Inserting (16) into (12) we obtain a condition for the pulse profile h3
sin p(w) -
-
- o r = const.
(17)
4/a2M0(1 - Vex/V ) aw'ff(w ) and an integral equation for "Yr(q) 223
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ar _ (2/aB)ar f [.i,r(q)/(K2 + q2)] dq/Zrr + 2a 4r (Kr2 + q2)fZ/r(q)dq/2rr 5'r(q) = - -
4 2 q2)2 1 +Or(K r +
,
(18)
with a r = KrqB/(gra B -- 1). Eq. (18) can be readily solved by noting that it has the structure 2 2 4 2 ~'r(q) = JAr + ar(Kr + q 2 ) B r ] / [ 1 + Or (Kr +q2)2].
(19)
A little algebra yields A r = [(k-taB) 2 -- GaB cos S0r]/[1 - 2 g r a B cos ~or + ( g r a B ) 2 ] ,
(20)
B r = (graB)2/[1 -- 2k-ra B cos Cr + (~raB) 2],
(21)
where k-r = (K4 + Or 4 )1/4; ~r = 1 arctan (Kr 2 Or2). After Fourier transformation the result can be written as
()
PQ = (Fr(x)ff(w) + ½i sin p(w)3,r(X))
-
1
i + ( f a ( x ) E ( w ) - il sin p(X)Ta(X))
(1)
-i
,
(22)
=tS2(_x))=3'r(X)sin2½P(w)(1)+Ta(x) where we have defined Fr(X) = ~ [(%/Kra B) exp(-K r Ix [) 6r(0) + ~r(X)], 4/1/140 r e x p ( - K r l x ~r(X) = T k 2K r
[)
__1
(1 - A t ) + 2k-r (A r cos ~r - Br sin ~3r) exp( •r cos S0rlX[)l
(24)
(25)
Vr(x) = (1/2Go 2) exp(- G cos ~rlx I)(Ar sin ~r + Br cos/3r),
(26)
~r(x) = Tr + k-r sin ~rlX 1.
(27)
The antiresonant quantities (index " a " ) can be obtained by replacing Kr -+ Ka; o r -+ %. Insertion of (22) into the field equation (9) gives a dispersion law for the carrier wave:
c2k2/co 2 = 1 + (Mo/2eOeb)(arthr(O) + aa~a(0)) .
(28)
Eq. (8) yields a differential equation for the pulse profile:
r2 ( d/ d w ) E ( w ) = (lt/Mo )(1 - Uex /V ) sin p(w) ,
(29)
r2 _ 2/ie0eb c2 k/cou - 1 M 0 2 ~ - ( 1 - Vex/U) ~r--~---- 7a(i3)
(30)
with (pulse width).
The solution of eq. (29) is the well known stable hyperbolic sech pulse of area 2rr. Note that therefore the pulse width is a measure of the intensity of the pulses.
224
V o l u m e 57, number 3
OPTICS COMMUNICATIONS
1 March 1986
~10 -2
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10 1
wi > >,
WT
o~ 10-2 CD 2 3...
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4
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T= 0,1ps T= 0,5p$ "r= 1,0 pS ~= 2,0 ps
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251,0
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,
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254,0
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Energy [eV] Fig. 1. Dispersion curves calculated by using material constants
of CdS. The parameter is the pulse duration (as a measure o f the intensity): (1) 0.1 ps, (2) 0.5 ps, (3) 1.0 ps, (4) 2 . 0 ps. T h e dotted line is the dispersion curve o f the exciton polariton. Spatial dispersion is accounted for throughout.
~--
Fig. 2. Ratio o f pulse velocity to the velocity o f light in the m e d i u m for a 0 . 5 ps pulse. The model substance is C d S . T h e dotted line gives the exciton velocity Uex = hk/M.
4. Results Eqs. (28) and (30) provide two coupled transcendental equations for wavenumber k and pulse velocity o for given frequency co and pulse width r. The solution is given in figs. 1 and 2. For small pulse widths there is a section with anomalous dispersion. The forbidden energy gap centered at the exciton line increases with growing intensity. The gap is readily explained by the observation that near resonance the pulse velocity (fig. 2) approaches the exciton velocity. In this case excitons cannot be excited coherently so that the coherent propagation process is destroyed and the pulse is absorbed. Off resonance our solution (fig. 1) asymptotically approaches the dispersion curve of the linear polariton. The polariton is the limiting case for long pulses, but in contrast to the results obtained by Goll and Haken [4] there is no polaritonlike solution below a certain pulse duration. The qualitative change in the shape of the dispersion curve at a certain pulse duration resembles a phase transition. Above a critical pulse duration there are no intersection points with the mode of the free light field and below there are two such points. As we have devised our theory in electron-hole configuration space we can calculate the exciton wave function, which we identify as the real part of P in eq. (22). While the wave function has the simple exponential shape o f the 1s exciton at some distance to the intersection points, some conspicuous modifications occur in a small interval containing these points (fig. 3). The wavefunction resembles that of an excited state and it is there where it has its strongest dependence on intensity (fig. 4). Thus our prediction is: If it is possible to observe SIT in semiconductors at all, it will occur below and above a certain energy gap but there will be no effect on resonance (exciton-line). Below a critical pulse duration the carrier wave has a dispersion law which differs both from the dispersion of the free light field and the dispersion of polaritons. The inset of the effect will resemble a phase transition.
225
Volume 57, number 3
I
OPTICS COMMUNICATIONS
0,2
4,0 "g
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2
1 March 1986
0,0 -0,2
>,,,
2,0
-0,4
(3
-0,6
~ 1,o "s 0-
E o,o
"5
-0.8
4
1,0
~" -10
-6
-2,0
-1,2 -1,4 I
0,0
1,2 2,4 Relative coordinate
3,6
/.,8
/ClBohr
Fig. 3. Exciton wave function for a small energy interval containing an intersection point of the dispersion curve with the mode of the free light field (1) 1.5166 eV ~
0,0
1.2
2/.
Relative c o o r d i n o i e
3,6 X/Olqoh
I
L
L,8 r
Fig. 4. Exciton wave function for a small intensity interval containing an intersection point of the dispersion curve with the mode of the free light field. (1) 1.02 ps < r ~< (4) 1.08 ps. The energy was chosen to be h~o = 1.5125 eV. The model substance is GaAs.
Acknowledgement The a u t h o r wishes t o t h a n k Prof. A. Stahl for criticism and m a n y h e l p f u l discussions.
References [1 ] S.L. McCall and E.L. Hahn, Phys. Rev. 183 (1969) 457. [2] D.H. Auston, Picosecond nonlinear optics, in: Ultrashort light pulses, ed. S.L. Shapiro (Springer Verlag, 1977) p. 123. [3 ] B. Bosacchi, Picosecond spectroscopy and solid state physics, in: Coherence in spectroscopy and modern physics, eds. t:.T. Arecchi, R. Bonifacio, M.O. Scully (Plenum Press, 1978). [4] J. Goll and H. Haken, Phys. Rev. A 18 (1978) 2241. [51 E. Hanamura, J. Phys. Soc. Japan 37 (1974) 1553. [6] W. Huhn and A. Stahl, Phys. Stat. Sol. (b) 124 (.1984) 167. [7] A. Stahl, Coherence and saturation of exciton polaritons in semiconductors, in: Festk6rperprobleme, Vol. 25, ed. P. Grosse, (Vieweg Verlag, 1985) p. 287. [8] M. Inoue, J. Phys. Soc. Japan 37 (1974) 1560. [91 L. Gotthard, A. Stahl and G. Czajkowski, J. Phys. C 17 (1984) 4865. [10] I. Balslev and A. Stahl, Phys. Stat. Sol. (b) 111 (1982) 531. [11 ] G.L. Lamb, Jr., Rev. Mod. Phys. 43 (1971) 99.
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