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Theoretical and experimental aspects of the nonlinear spectroscopy of biexcitons in CuCl
Phymca II7B& II8B (1983)301-302 North-HoUandPubhslungCompany
301
THEORETICAL AND EXPERIMENTAL ASPECTS OF THE NONLINEAR SPECTROSCOPY OF BIEXCITONS IN CuCI I. Abram, A. Maruanl, D.S. Chemla R, F. Bonnouvrier and E. Batifol CNET ~m~ 196 Rue de Paris 92220 BAGNEUX - FRANCE
Forward degenerate fourwave mixing in CuCl in the vicinity of the two-photon absorption on the blexciton can be analyzed from two complementary theoretical points of view :(a) In terms of the traditional perturbational formulation of nonlinear optics, the low intensity observations can be accounted for in terms of interferences and re-normalizations among different-order elementary mixing processes (b) In terms of a perturbational scheme which provides directly an analytic expresslon for the intensitydependent dielectric function of the material. This last approach polnt~to a potentially bistable optical response when local field effects are included in the calculation.
Recent experimental and theoretical work has shown very large optical nonlinearities associated wlth the ground state-exclton-blexciton system in semiconductors, in the vicinity of the two photon hxexciton resonance. This is due to the giant oscillator strengths involved in the corresponding transitions, the quasi resonant nature of the one photon intermediate ahsorptlon and the low effective damping constants. ] The high efficiency of two photon resonant nonlinear processes was dramatized in our laboratory through the observation by naked eye of twelvephoton mixing processes in CuCI. That observation raises at once two intimately connected ques~ons: a) If the observed processes are to be described through the conventional nonlinear susceptibility, what is the structure of the relevant X (2n+]) and what are the elementary mixing processes involved ? b) wlth such a huge nonlinear response is the perturbational formulation of the susceptibility still valid ~ Indeed one can think that a non perturbational approach is necessary under those circumstances. Some aspects of question (a) were tackled with In a previous series of papers 2 ~ so we shall content ourselves with a very brief sketch of the method and its results : an experiment of the degenerate four wave mixing type, in the forward configuration, was carried out. Through the mixing of a ~ump wave with UV frequency around 25707 cm -* and wave vector kp and a test beam (~,k t) we could observe several generated beams at the same frequency ~, in the directions k n = (,I+I) kp - nk t and k m = (m+])k t - ~ p .
2 Bell Laboratorie% Holmdel, NJ, USA • Laboratoire associ~ au CNRS (LA 250)
0 378-4363/83/0000-0000/$03.00 © 1983 North-HoUand
Beams up to nffi5 could be easily seen through their fluorescence of on a white index card. The number of distinct mixing processes contributing to each beam increases very rapidly with n : we may distinguish direct processes, in which the nonixnear polarization PNL is proportional to x(2n+l) E~ +I (E~) n and lowest order cascades, involving the electric field of the previous order beam : PNL ~ X (3) F~-I EpE~. As could be expected, the experimental result could be described well in terms of interferences among those elementary processe~. The treatment invoked a model for X (~n+|) and a propagation analysis accounting for the degeneracy of the elementary processes. The results for x(2n+]) ~s ~h@t basically, it is proportional to (X~3)/X~I)) n where X~ l) is the linear susceptibility a~soclated to t~e exclton system. A tricky point In the propagation problem is that the signals undergo violent nonlinear absorption and phase shifts, and both those effects distort strongly the gain profiles. A strictly numerical analysis of the coupled wave problem obscurs somewhat the physics. Now, in the hyperparametric approximation, which considers only pump induced phenomena and holds when Ep>>Et, the equation for the (complex) ratio rn ffiEn+|/E . is linear and depends only on the galn factors for the n th and (n+l) th - order beams. This enables us to solve exactly for r 2 and dlscuss each of the different processes. With no adjustable parameter, we obtain a satisfactory agreement, at low intenslties,2between experimental and theoretical results . Let us turn now to the second problem. Marz et al. 3 have derived a non perturbatlonal dielectric function involving the exciton polariton dressed by virtual transitions to biexcltonic states, through Green'a function techniques. The result was expressed as a function of the polariton density np which in turn could be related to the laser intensity I through the
302
1 Abram et al
/ The nonhnear spectroscopy
signal velocity function. We have adopted here a different point of view, based on operator techniques, which yields an analytic expression for the dielectri~ function directly dependant on the intensity-. We conslder in our model a system composed of exciton and biexclton particles described quantum mechanlcally by Bose operators b~/b k and B~/B K respectively ; the semlconductor Hamlltonlan is thus
• '
N0 = kZ ~(k) b E b k ÷ ~ ~ (k) B +k Bk where ~(k) and ~(k) are the dlspersion relations for the two types of elementary excitations. The electric field is described classically as the real plane wave Ecos (qr-~t). Considering only transitions from the ground state to excitons and excltons to biexcltons, the radiative interactlon hamiltonian is written in the electric dipole approximatlon as :
where ~ is the excltonzc transition dipole element per unlt cell, N is the number of unlt cel~ in the c r y s t a l , ~ is the exciton-bxexcxton matrix element and ~=qr-~0t. The polarization induced xn the system is given by the expectation value of the total dipole operator taken over the state obtained when the radiative interaction H l ~s swltched-on adlabatically 5. Neglecting antlresonant terms and assuming that the exclton biexciton system is ~nltlally in its ground state, the polarlzatxon due to the exciton transition dipole xs : Px=(N/V)(A~=E)/ {A~-($E/2) 2} while the exciton-biexc~ton dipole gives P B = ( N / 4 V ) ( A ~ E 3 ) / { A ~ - ( ~ ) - } 2 where A=~(2q)-2~, 6ffi~(q)-~ are the detunln~s of the two photon biexciton and one photon exclton transltlons respectively, while V xs the crystal volume. Damping can be phenomenologxcally introduced at thls stage by letting A and 6 be complex. We may now obtain the overall nonlinear optical susceptibility from x(E)={Px(E)+PB(E}/E and the intensity dependant dielectric function ~LT A ~ = ] + (A~_($_~F,)2)2 2 where ~LT is the longitudinal transverse splitring. Figure la is a graphic representation of eq(1) for small det~nings wlth respect to the two-photon transition. Numerical values concern CuCI, with no damping. Series expansion of ~ xt powers of the intensity I = ! E 2 gives at zeroth .4~ order the polariton dispersion relation at first order the classic X (3) and so on. The results of ref (3) are obtained if PB is neglected and a linear relationship is assumed between I and the polarlton density. (])
E = I+4~x(E)
o f btexcttons m CuCI
3
' ....
I ' '~'
ill 50
{b)
100
Figure | : Calculated refractive index of CuCI as a function of incident intensity in the vicinity of the two-photon blexcltnn resonance (3.186 eV). (a) Without local field considerations (b) including local field effects. If we assume, as is reasonable in mary physical cases , that E~ .zoc = -E+2 = ~, L~ Enen the true intensity dependant daelectr~c~function is : ~LT AJ~ C+2 (2) e = ! + --3-Figure lb presents a numerical solution of eq.(2) with the same parameters as in fig la. We note that the inclusion of the local field effect provides a feedback in the system, and that causes the refractive index to be a multivalued function of the incident intensity, implying a potentially blstable optical behaviour. As ms evident from that figure, the swltching intensities, if any, are within current experimental posslblllties. Thls point, however, deserves further theoretical study in terms of stability analysls.
! - C. Kllngshlrn and H. Haug, Phys. Reports 70, 316 (1981) 2 - A. Maruanl and D.S. Chemla, Phys. Rev. B23, 841 (1981) 3 - R. Marz, S. S c h m i t t - R x n k and H. Haug, Z. P h y s . B4__O0, 9 (1980) 4 - I. Abram and A. Maruani to be published
However, the field actually felt by the elementary excitation is not the external field E but the local field E loc"
5 - J.F. Ward, Rev. Mod. Phys. 37, I (1965) 6 - A. Maruani and D.S. Chemla, J. PHys. Soc. Japan 49 Suppl. A, p. 583 (1980).
301
THEORETICAL AND EXPERIMENTAL ASPECTS OF THE NONLINEAR SPECTROSCOPY OF BIEXCITONS IN CuCI I. Abram, A. Maruanl, D.S. Chemla R, F. Bonnouvrier and E. Batifol CNET ~m~ 196 Rue de Paris 92220 BAGNEUX - FRANCE
Forward degenerate fourwave mixing in CuCl in the vicinity of the two-photon absorption on the blexciton can be analyzed from two complementary theoretical points of view :(a) In terms of the traditional perturbational formulation of nonlinear optics, the low intensity observations can be accounted for in terms of interferences and re-normalizations among different-order elementary mixing processes (b) In terms of a perturbational scheme which provides directly an analytic expresslon for the intensitydependent dielectric function of the material. This last approach polnt~to a potentially bistable optical response when local field effects are included in the calculation.
Recent experimental and theoretical work has shown very large optical nonlinearities associated wlth the ground state-exclton-blexciton system in semiconductors, in the vicinity of the two photon hxexciton resonance. This is due to the giant oscillator strengths involved in the corresponding transitions, the quasi resonant nature of the one photon intermediate ahsorptlon and the low effective damping constants. ] The high efficiency of two photon resonant nonlinear processes was dramatized in our laboratory through the observation by naked eye of twelvephoton mixing processes in CuCI. That observation raises at once two intimately connected ques~ons: a) If the observed processes are to be described through the conventional nonlinear susceptibility, what is the structure of the relevant X (2n+]) and what are the elementary mixing processes involved ? b) wlth such a huge nonlinear response is the perturbational formulation of the susceptibility still valid ~ Indeed one can think that a non perturbational approach is necessary under those circumstances. Some aspects of question (a) were tackled with In a previous series of papers 2 ~ so we shall content ourselves with a very brief sketch of the method and its results : an experiment of the degenerate four wave mixing type, in the forward configuration, was carried out. Through the mixing of a ~ump wave with UV frequency around 25707 cm -* and wave vector kp and a test beam (~,k t) we could observe several generated beams at the same frequency ~, in the directions k n = (,I+I) kp - nk t and k m = (m+])k t - ~ p .
2 Bell Laboratorie% Holmdel, NJ, USA • Laboratoire associ~ au CNRS (LA 250)
0 378-4363/83/0000-0000/$03.00 © 1983 North-HoUand
Beams up to nffi5 could be easily seen through their fluorescence of on a white index card. The number of distinct mixing processes contributing to each beam increases very rapidly with n : we may distinguish direct processes, in which the nonixnear polarization PNL is proportional to x(2n+l) E~ +I (E~) n and lowest order cascades, involving the electric field of the previous order beam : PNL ~ X (3) F~-I EpE~. As could be expected, the experimental result could be described well in terms of interferences among those elementary processe~. The treatment invoked a model for X (~n+|) and a propagation analysis accounting for the degeneracy of the elementary processes. The results for x(2n+]) ~s ~h@t basically, it is proportional to (X~3)/X~I)) n where X~ l) is the linear susceptibility a~soclated to t~e exclton system. A tricky point In the propagation problem is that the signals undergo violent nonlinear absorption and phase shifts, and both those effects distort strongly the gain profiles. A strictly numerical analysis of the coupled wave problem obscurs somewhat the physics. Now, in the hyperparametric approximation, which considers only pump induced phenomena and holds when Ep>>Et, the equation for the (complex) ratio rn ffiEn+|/E . is linear and depends only on the galn factors for the n th and (n+l) th - order beams. This enables us to solve exactly for r 2 and dlscuss each of the different processes. With no adjustable parameter, we obtain a satisfactory agreement, at low intenslties,2between experimental and theoretical results . Let us turn now to the second problem. Marz et al. 3 have derived a non perturbatlonal dielectric function involving the exciton polariton dressed by virtual transitions to biexcltonic states, through Green'a function techniques. The result was expressed as a function of the polariton density np which in turn could be related to the laser intensity I through the
302
1 Abram et al
/ The nonhnear spectroscopy
signal velocity function. We have adopted here a different point of view, based on operator techniques, which yields an analytic expression for the dielectri~ function directly dependant on the intensity-. We conslder in our model a system composed of exciton and biexclton particles described quantum mechanlcally by Bose operators b~/b k and B~/B K respectively ; the semlconductor Hamlltonlan is thus
• '
N0 = kZ ~(k) b E b k ÷ ~ ~ (k) B +k Bk where ~(k) and ~(k) are the dlspersion relations for the two types of elementary excitations. The electric field is described classically as the real plane wave Ecos (qr-~t). Considering only transitions from the ground state to excitons and excltons to biexcltons, the radiative interactlon hamiltonian is written in the electric dipole approximatlon as :
where ~ is the excltonzc transition dipole element per unlt cell, N is the number of unlt cel~ in the c r y s t a l , ~ is the exciton-bxexcxton matrix element and ~=qr-~0t. The polarization induced xn the system is given by the expectation value of the total dipole operator taken over the state obtained when the radiative interaction H l ~s swltched-on adlabatically 5. Neglecting antlresonant terms and assuming that the exclton biexciton system is ~nltlally in its ground state, the polarlzatxon due to the exciton transition dipole xs : Px=(N/V)(A~=E)/ {A~-($E/2) 2} while the exciton-biexc~ton dipole gives P B = ( N / 4 V ) ( A ~ E 3 ) / { A ~ - ( ~ ) - } 2 where A=~(2q)-2~, 6ffi~(q)-~ are the detunln~s of the two photon biexciton and one photon exclton transltlons respectively, while V xs the crystal volume. Damping can be phenomenologxcally introduced at thls stage by letting A and 6 be complex. We may now obtain the overall nonlinear optical susceptibility from x(E)={Px(E)+PB(E}/E and the intensity dependant dielectric function ~LT A ~ = ] + (A~_($_~F,)2)2 2 where ~LT is the longitudinal transverse splitring. Figure la is a graphic representation of eq(1) for small det~nings wlth respect to the two-photon transition. Numerical values concern CuCI, with no damping. Series expansion of ~ xt powers of the intensity I = ! E 2 gives at zeroth .4~ order the polariton dispersion relation at first order the classic X (3) and so on. The results of ref (3) are obtained if PB is neglected and a linear relationship is assumed between I and the polarlton density. (])
E = I+4~x(E)
o f btexcttons m CuCI
3
' ....
I ' '~'
ill 50
{b)
100
Figure | : Calculated refractive index of CuCI as a function of incident intensity in the vicinity of the two-photon blexcltnn resonance (3.186 eV). (a) Without local field considerations (b) including local field effects. If we assume, as is reasonable in mary physical cases , that E~ .zoc = -E+2 = ~, L~ Enen the true intensity dependant daelectr~c~function is : ~LT AJ~ C+2 (2) e = ! + --3-Figure lb presents a numerical solution of eq.(2) with the same parameters as in fig la. We note that the inclusion of the local field effect provides a feedback in the system, and that causes the refractive index to be a multivalued function of the incident intensity, implying a potentially blstable optical behaviour. As ms evident from that figure, the swltching intensities, if any, are within current experimental posslblllties. Thls point, however, deserves further theoretical study in terms of stability analysls.
! - C. Kllngshlrn and H. Haug, Phys. Reports 70, 316 (1981) 2 - A. Maruanl and D.S. Chemla, Phys. Rev. B23, 841 (1981) 3 - R. Marz, S. S c h m i t t - R x n k and H. Haug, Z. P h y s . B4__O0, 9 (1980) 4 - I. Abram and A. Maruani to be published
However, the field actually felt by the elementary excitation is not the external field E but the local field E loc"
5 - J.F. Ward, Rev. Mod. Phys. 37, I (1965) 6 - A. Maruani and D.S. Chemla, J. PHys. Soc. Japan 49 Suppl. A, p. 583 (1980).