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Rabi splitting in quantum well (QW) embedded in microcavities under strong coupling condition is modeled by a time-dependent transfer matrix model. The spectral response of QW under the influence of excitonic effects is simulated by infinite impulse digital filters. It is shown that the splitting energy obtained from the proposed model match well with that deduced from the reflection spectrum analysis. The lasing spectra observed from different transmission angles of the QW microcavity can also be calculated. Hence, it is proved that the proposed model can be used to design and analyze the lasing characteristics of QW microcavities under strong coupling condition.
©2008 Optical Society of America
1. Introduction
For quantum wells (QWs) embedded in microcavities, strong coupling between cavity modes and excitons leads to the formation of half-light half-matter quasi-particles (i.e., polaritons) [1-11]. Under some conditions, high-density polaritons can be generated and condensed inside the microcavities to emit coherent photons (i.e., polariton lasers). As the generation of coherent photons does not require population inversion as if the conventional semiconductor lasers, thresholdless lasing can be obtained from the polariton lasers [1, 2, 7].
In order to design and realize polariton lasing, the recognition of strong coupling between cavity modes and excitons in QW microcavities is crucial. The strong coupling results in two light-matter entangled eigenstates spectrally separated, namely Rabi splitting [1-10]. The appearance of Rabi splitting, which is usually identified as two dips at the bandgap of the reflection spectrum from the QW microcavities, is viewed as a signature of cavity polariton. Therefore, reflection spectrum analysis was used to design and investigate QW microcavities for the realization of cavity polariton [7, 8]. However, a more direct proof of Rabi splitting should be obtained from the emission spectrum of the QW microcavity [3-5]. This is because the dips of the reflection spectrum can be related to the weak-coupling of cavity modes inside the QW microcavity if the exciton inhomogeneous broadening exceeds the width of cavity modes [1].
In this paper, a time-dependent transfer matrix model (TMM) is proposed to study the lasing characteristics of QW microcavities with strong coupling of cavity modes and excitons taken into consideration. The spectral response of QW under excitonic effects is modeled by two digital filters (DFs) to filtrate the spectral response of the propagating waves inside the cavity. Using this model, the Rabi splitting behavior and angle-dependence of lasing spectrum of QW microcavities are studied and compared with those obtained from the reflection spectrum analysis.
2. Theory
Fig. 1. (a) Schematic of field transmission in a microcavity, where n is the refractive index of dielectric layer and z is the interface position; (b) digital filter treatment of reflection and transmission of active region, where RDF and TDF (i.e., reflection and transmission digital filters) are reflection and transmission digital filters, respectively.
The GaAs QW microcavity similar to that given in Ref. [12] is considered in our analysis. In order to model the lasing characteristics from the QW microcavity with strong light-exciton coupling, the light generation mechanisms inside the microcavity are assumed as follows: In the QW region, a large quantity of excitons is generated either under optical or electrical excitation. The strongly coupled cavity modes and excitons form polaritons, which distribute within the reciprocal space [1]. After some radiative scattering processes (e.g., polariton-polariton scattering), the high-energy polaritons scattered into the ground state to emit photons with perfect coherence. Due to the strong coupling effects, the coherent photons have spectral response dependent on the amount of Rabi splitting energy. On the other hand, some portions of the excitons can be recombined radiatively within the QW (i.e., due to exciton-exciton scattering) and leads to spontaneous emission. As a result, the coherent photons and spontaneous emission can both be coupled into the laser cavity to form the propagating electric waves. They can be re-absorbed and amplified when re-entering the QW active layer. This process will lead to the formation of stable lasing spectrum and the corresponding lasing modes should have energy separation equals to the Rabi splitting energy.
Hence, a time-dependent TMM is developed to model the propagating electric fields consisting of coherent photons and spontaneous emission. The spectral response of these electric fields will be determined by the structure of the microcavity as well as the light-exciton interaction inside the QW.
2.1 Time-dependent transfer matrix model
The time-dependent electric field (TE waves) in the jth layer can be represented as
where E + j and E - j are the complex amplitudes of the forward and backward-propagating waves [Fig. 1(a)]. After travelling a distance of dj=zj−zj−1 (i.e., layer j thickness), next relation can be established for E + and E -
where Δt=λ0/4/C is the system sampling time, C is the light speed in vacuum, kj is defined as kj=k0njcos(θj)−iαj, k0=λ0/2/π, nj, θj, and αj are respectively the refractive index, light transmission angle, and material’s absorption coefficient in layer j.
At the interface between layers j and j+1, the waves in opposite sides can be related with a scattering matrix as
where from Fresnel formula Mj can be expressed as
The boundary conditions in the top and bottom interfaces used in this study are [13]
where r1 (=0.55) and rN (=0.55) are the field reflectivities of these air-dielectric interfaces (with positions z 1 and zN), respectively.
2.2 Digital filter modeling of quantum-well active region under strong coupling condition
In strong coupling case, due to the formation of vast one-dimensionally confined excitons, the optical properties of the QW can be significantly modified. It has been shown that reflection and transmission of electric fields form the QW exhibit an unique frequency-selectivity due to the strong light-exciton interaction. Using the so-called nonlocal dielectric response theory, analytic expressions of the reflection, rQW, and transmission, tQW, coefficients of TE-polarized waves can be obtained [1, 7]
where Γo (ħΓo=0.6 meV) is the radiative width of the exciton line, γ (ħγ=1 meV) is the homogeneous broadening of the exciton resonance, and ω is the light frequency. In this paper, only TE waves are taken into consideration.
Using the results from Eq. (6), the spectral response of the electric fields under the influence of strong light-exciton coupling can be modeled as [see Fig. 1(a)]
where the delay of 2Δt is due to the light transmission in the active region (d QW=λ0/2/n QW). k QW=k0nQWcos(θQW)+i(g-α QW), where both the absorption loss αQW and the optical gain (g) provided by the QW region are also considered. In this model, we have introduced a broadband noise source [12], i.e., S + and S - in Eqs. (7) and (8), to represent the spontaneous emission as described previously.
In order to implement the response spectra shown in Eq. (6), we designed two types of infinite impulse DFs used to filtrate the calculated time series of propagating waves [see Fig. 1(b)]. According to our calculation, the following frequency response functions are found to be appropriate to reproduce r QW and t QW
where the subscripts r and t denote the RDF and TDF, respectively. The coefficients A and B mainly influence the response amplitudes of the filters, while ΔT is the filter sampling time, which determines the bandwidth of the response spectrum. From Eq. (9), following differential equations for the time series can be established [13]
where x(t) and y(t) are respectively the input and output signals of the filters at time t. It should be noted that before using the TDF to filtrate the light across the active region, the experienced phase shift as well as the optical loss and gain, included in the term of exp(-ikQWdQW) in Eqs. (7) and (8), should be computed in advance. Alternatively, one can also treat this term as a temporally varied coefficient to be included as the amplitude of TDF.
3. Simulation results
In the following calculation, the excitonic wavelength λ0=850 nm is chosen to be at the central wavelength of the distributed Bragg reflectors (DBRs). The refractive indices of low-index DBR film, high-index DBR film, spacer layer, and active region are set to n L=2.95, n H=3.41, n Spa=3.39, and n QW=3.524, respectively.
In order to reproduce the special frequency responses of r QW and t QW shown in Eq. (6), the parameters of DFs should be designed properly. It involves 1) to seek the proper values for A and B to obtain the peak response amplitudes calculated from Eq. (6) [i.e., fulfill the condition of A/(1-B)=Γo/(Γo+γ)] and 2) to optimize ΔT to obtain the needed response bandwidth. According to our calculation, A=0.0370, B=0.9014 and ΔT=60Δt is an appropriate set of filter parameters. Results are shown in Fig. 2, where the frequency responses of both the designed DFs and those calculated directly from Eq. (6) are displayed for comparison. It is found that the designed response spectra are very similar to the theoretical forms. Hence, it is expected that the strong-coupling-induced special frequency-filtering effects can be considered in our time-domain model by including these DFs.
Figure 3(a) shows the calculated lasing spectrum obtained by employing fast Fourier transformation (FFT) to the electric fields over a selected ‘time window’ when stable lasing spectrum was formed. From this figure, two pronounced peaks (i.e., lasing modes with wavelengths λ1 and λ2) adjacent to the central wavelength λ0 (~1.46 eV) are observed. The energy difference of these two peaks is found to be ~14.5 meV. Apparently, they are not the longitudinal modes supported by the cavity, but two eigenstates due to the strong coupling of the cavity modes and excitons. This is so-called Rabi splitting and can be verified by using the reflection spectrum analysis for the entire microcavity. Figure 3(b) plots the reflection spectrum of the QW microcavity obtained by the method given in the literature [1, 7]. Two reflection dips are observed from the reflection spectrum which corresponding to two exciton resonance frequencies. Comparing Figs. 3(a) with 3(b), we found the dips and peaks occur almost at the same light energies (the same frequencies), i.e., Rabi splitting is certainly observed in the lasing spectrum using our time-dependent TMM.
Fig. 3. Normalized lasing spectra [(a) & (c)] and reflection [(b)] of the QW-embedded microcavity. The function of splitting energy with digital filter sampling time is given in (d).
The validity of the DFs in the description of excitonic effects can easily be checked by removing the filters from the model. As shown in Fig. 3(c), where the DFs are not included into the model, the splitting disappears with only the central cavity mode left. That is because without the excitonic effects represented by the DFs, the strong coupling condition is not satisfied anymore and the laser is now operating in weak coupling regime. Moreover, it is found that the splitting energy can be effectively controlled by adjusting the ΔT parameter of the digital filter. As displayed in Fig. 3(d), the Rabi splitting energy will be greatly decreased with increasing ΔT. In fact, the change of ΔT reveals the variation of the response bandwidth modified by varying the excitonic parameters. Therefore, the designed DFs can be used to simulate the polariton lasing processes under various excitonic conditions.
It has been indicated that the splitting of polariton modes is extremely sensitive to the transmission angle, θ, in the system with strong light-exciton coupling [7]. Therefore, the θ dependence of the lasing spectrum is studied and compared with the reflection spectrum analysis. Figures 4(a) and 4(b) plot respectively the reflection and lasing spectra of the QW microcavity for some values of θ. It is found that with the increase of θ, both upper-polariton branch (UPB) and low-polariton branch (LPB) modes shift toward the higher energy side. However, the UPB mode is blue-shifted more rapidly than that of the LPB modes [1, 2-4, 7]. Furthermore, we found that the intensity of the LPB mode is suppressed with the increasing of θ (i.e., this unique phenomenon cannot be observed from reflection spectrum analysis). This is because the UPB mode has a relatively lower reflectivity than that of the LPB at large values of θ so that photons can be easily escaped from the microcavity at the energy of the UPB mode. On the other hand, it was observed experimentally that the LPB mode can be sustained from the emission spectra of a light-emitting diode even at a large value of θ (i.e., >10°) [3]. This is because the light-emitting diode was biased at a low injection level so that the corresponding modal discrimination is weak. The laser considered in our study, however, is operating under high injection level, leading to strong modal competition as well as weak Rabi splitting. Hence, the LPB mode is suppressed at a small value of θ.
Fig. 4. Reflection [(a)] and, lasing [(b)] spectra with different transmission angles θ. In (b), arrows indicate the positions of the suppressed LPB modes.
4. Conclusion
Lasing behavior of a QW microcavity is studied using a time-dependent TMM that incorporated DFs to represent the influence of excitonic effects. We have shown that the phenomenon of Rabi splitting due to strong light-exciton coupling inside the microcavity can be reproduced by the model. This is because the calculated energy of Rabi splitting is found to be well matched with that obtained from the reflection spectrum analysis. The influence of transmission angle on the shift of lasing spectra from the QW microcavity is also examined. Furthermore, the proposed method can be modified to simulate emission characteristics of microcavities embedded with quantum wire or quantum dot active medium by using nonlocal dielectric response theory to model 2D or 3D quantum confinement systems [1].
Acknowledgement
This work is supported by Singapore MoE grant ARC 02/06. X. F. Li is also supported by DARPA under agreement HR0011-07-2-0002.
References and links
1. A. V. Kovakin and G. Malpuech, Cavity Polaritons (Elsevier, Amsterdam, 2003).
2. G. Khitrova, H. Gibbs, F. Jahnke, M. Kira, and S. W. Koch, “Nonlinear optics of normal-mode-coupling semiconductor microcavities,” Rev. Mod. Phys. 71, 1591–1639 (1999). [CrossRef]
3. S. I. Tsintzos, N. T. Pelekanos, G. Konstantinidis, Z. Hatzopoulos, and P. G. Savvidis, “A GaAs polariton light-emitting diode operating near room temperature,” Nature 453, 372–375 (2008). [CrossRef] [PubMed]
4. D. Bajoni, E. Semenova, A. Lemaitre, S. Bouchoule, E. Wertz, P. Senellart, and J. Bloch, “Polariton light-emitting diode in a GaAs-based microcavity,” Phys. Rev. B 77, 113303 (2008). [CrossRef]
5. S. Christopoulos, G. Baldassarri Höger von Högersthal, A. J. D. Grundy, P. G. Lagoudakis, A. V. Kavokin, and J. J. Baumberg, “Room-temperature polariton lasing in semiconductor microcavities,” Phys. Rev. Lett. 98, 126405 (2007). [CrossRef] [PubMed]
6. F. Tassone and Y. Yamamoto, “Lasing and squeezing of composite bosons in a semiconductor microcavity,” Phys. Rev. A 62, 063809 (2000). [CrossRef]
7. A. V. Kavokin and M. A. Kaliteevski, “Excitonic light reflection and absorption in semiconductor microvcavities at oblique incidence,” Solid State Commun. 95, 859–862 (1995). [CrossRef]
8. E. L. Ivchenko, M. A. Kaliteevski, A. V. Kavokin, and A. I. Nesvizhskii, “Reflection and absorption spectra from microcavities with resonant Bragg quantum wells,” J. Opt. Soc. Am. B 13, 1061–1068 (1996). [CrossRef]
9. E. L. Ivchenko, “Excitonic polaritons in periodic quantum well structures,” Sov. Phys. Solid State 33, 1344–1346 (1991).
10. L. C. Andreani, F. Tassone, and F. Bassani, “Radiative lifetime of free excitons in quantum wells,” Solid State Commun. 77, 641–645 (1991). [CrossRef]
11. L. V. Butov, “A polariton laser,” Nature 447, 540–541 (2007). [CrossRef] [PubMed]
12. S. F. Yu, “Dynamic behavior of vertical-cavity surface-emitting lasers,” IEEE J. Quantum Electron. 32, 1168–1179 (1996). [CrossRef]
13. C. Nyack, “Infinite impulse response filters,” http://dspcan.homestead.com/files/IIRFilt/zfiltiiri.htm.
