Open Access Presentation + Paper
8 May 2017 Optical control of polaritons: from optoelectronic to spinoptronic device concepts
R. Binder, S. M. H. Luk, N. H. Kwong, P. Lewandowski, S. Schumacher, O. Lafont, E. Baudin, J. Tignon , A. Lemaitre, J. Bloch, Ch. K. P. Chan, P. T. Leung
Author Affiliations +
Abstract
Exciton-polaritons in semiconductor microcavities have been studied intensely, both with respect to their intriguing fundamental physical properties and with respect to their potential in novel device designs. The latter requires ways to control polaritonic systems, and all-optical control mechanisms are considered to be especially useful. In this talk, we discuss and review our efforts to control the polariton density, utilizing optical four-wave mixing instabilites, and the spin or polarization textures resulting from the optical spin Hall effect. Both effects are readily observable in the cavity’s far-field emission, and hence potentially useful for optoelectronic and spinoptronic device applications.
Conference Presentation

1.

INTRODUCTION

Exciton-polaritons in semiconductor microcavities have been studied intensely both with respect to their intriguing fundamental physical properties118 (for reviews see for example Refs. [13, 1719]), including polaritonic Bose condensation, and with respect to their potential in novel device designs.2023 The latter requires ways to control polaritonic systems, and all-optical control mechanisms are considered to be especially useful. In the following, we discuss and review our efforts to control various physical attributes. One of those attributes is the polariton density, which can be controlled by optical pump and control beams, utilizing optical instabilities that are based on four-wave mixing processes. Using a double-cavity design, which has two lower-polariton branches, we are able to pump the system in normal incidence and control the output direction with a combination of static system parameters and dynamical off-axis control beams. This allows for an efficient directional and low-energy switching of optical beams. A second example is the control of spin transport phenomena that leads to polarization patterns (or textures) in the far-field emission. We discuss the control of the so-called optical spin Hall effect. Utilizing again our double-cavity design, we are able to rotate the spin/polarization texture of the far field simply by changing the intensity of the pump beam. This effect is based on an optical manipulation of the polariton-spin orbit interaction, and may lead to novel applications in spinoptronic systems.

A semiconductor double-microcavity consists of two cavities,24 each of which has one or more quantum wells embedded, as sketched in Fig. 1(a). The cavity photons and the excitons in the quantum wells are coupled and form exciton-polaritons. For a double-cavity, two upper and two lower polariton branches are formed, which is in contrast to one upper and one lower polariton branch in the case of a single-cavity. The extra polariton branches allow resonant frequency windows for the creation of directly pumped polaritons. An optical pump is in normal incidence to the cavity’s surface. When the pump intensity is above a certain threshold, the nonlinear scattering of polaritons can lead to spontaneous symmetry breaking and pattern formation. Figure 1(b) shows a sketch of a possible hexagonal pattern observed in the far-field.

Figure 1.

(a)A sketch of the semiconductor double-microcavity with a pump beam at normal incidence. Two quantum wells are embedded inside three distributed Bragg reflectors (DBR), which form two cavities. The (b) A schematic of the hexagonal intensity pattern spontaneously created by a normally incident light source with intensity above instability threshold. A two-spot pattern can also be created using different parameters. (c) A schematic plot of the optical spin Hall effect (OSHE) pattern created by a “+”-circularly polarized source. “x” and “y”-polarized polaritons are spatially separated due to the polariton spin-orbit interaction.

00006_PSISDG10193_101930G_page_2_1.jpg

Another possible optical pattern that can be observed and controlled in a double-cavity is the optical spin Hall effect (OSHE) spin texture.2531 The OSHE is the formation of polarization/spin pattern of exciton-polaritons in an optically pumped semiconductor microcavity. The effect arises from the transverse-electric and transverse magnetic (TE-TM) splitting, which yields a polariton spin-orbit interaction. The spin-orbit interaction leads to an effective magnetic field that creates an anisotropic polarization texture of polaritons. The OSHE pattern with a linearly polarized (“x”-polarized) light source was recently shown in experiments,31, 32 and is sketched in Fig. 1(c).

In the following, we will rst discuss the theory of cavity photons and excitons in a semiconductor double-microcavity, and present numerical simulation results illustrating the formation of patterns and their optical control. In the next section, we will simplify the photon-exciton equations using the exciton-polariton picture and also formulate a pseudospin model. We will present results for the control of the OSHE pattern at different pump intensity using both the microscopic theory as well as the pseudospin formulation.

2.

PATTERN FORMATION IN SEMICONDUCTOR DOUBLE-MICROCAVITIES

In this section, we describe one of the optical control mechanisms on the polariton patterns in a semiconductor double-microcavity as demonstrated in Ref. [24]. We start with considering the dynamics of the excitons and cavity photons in the two cavities. The equations describing the dynamics of the exciton and cavity photon fields in the two cavities and quantum wells are

00006_PSISDG10193_101930G_page_3_1.jpg
00006_PSISDG10193_101930G_page_3_2.jpg
00006_PSISDG10193_101930G_page_3_3.jpg
00006_PSISDG10193_101930G_page_3_4.jpg

Here 00006_PSISDG10193_101930G_page_3_5.jpg is the cavity photon field, and 00006_PSISDG10193_101930G_page_3_6.jpg is the exciton field in the ith cavity (i = 1, 2 in a double-cavity) of plus (+) and minus (-) circular polarization. For cavity photons, 00006_PSISDG10193_101930G_page_3_7.jpg and 00006_PSISDG10193_101930G_page_3_8.jpg. For excitons, 00006_PSISDG10193_101930G_page_3_9.jpg. 00006_PSISDG10193_101930G_page_3_12.jpg and εx are the k = 0 energies of the cavity mode and the 1s exciton, respectively. The two cavities have the same 00006_PSISDG10193_101930G_page_3_13.jpg and εx as they are assumed to be identical. mTE and mTM are the cavity photon masses for the TE and TM modes, respectively. The spin-orbit coupling H± turns two units of spin into two units of orbital angular momentum and vice versa. It is non-zero when there is splitting in the TE and TM dispersion (mTEmTM). mX is the 1s heavy-hole exciton mass, which is assumed to be infinite in the range of interest. γC is the cavity loss rate and γX the exciton dephasing. ΩX is the coupling strength between excitons and cavity photons in each cavity, and the photon fields in the two cavities are coupled with strength ΩC. 00006_PSISDG10193_101930G_page_3_10.jpg and 00006_PSISDG10193_101930G_page_3_11.jpg are the effective pump sources in cavity 1 and 2, respectively.

The nonlinear couplings contain two parts: the phase space filling effect among the Fermionic constituents of the excitons (electrons and holes), denoted by APSF, and the effective interactions T++ and T+−, in the co-circularly polarized and counter-circularly polarized exciton channels, respectively. Retardation (quantum memory) effects are neglected due to the quasi-monochromatic excitation conditions.

Steady state far-field patterns are obtained by numerically solving Eqs. (1)-(4) and Fourier transforming the cavity photon and exciton fields into transverse momentum (k) space. A continuous normally incident pump source, linearly polarized at 45°, with intensity above the instability threshold leads to a spontaneous hexagon formation in k-space, as shown in Fig. 2(b). The hexagon consists of two strong spots and four weaker spots, with the two strong spots orientated perpendicularly to the polarization of the pump. When the pump is tilted to the direction perpendicular to the pump polarization, as illustrated in Fig. 2(a), an anisotropy is introduced to the system. With a sufficiently strong anisotropy, the hexagon is destabilized and a two-spot pattern emerges, as shown in Fig. 2(c). Note that all detections are in the cross-polarized channel, which is the linear polarization at 135° in this case. Further examples of switching, including using a control beam, can be found in Ref. [24].

Figure 2.

(a)A schematic illustration of the tilted pump (the tilt angle in the sketch is exaggerated for clarity). (b) A hexagonal intensity pattern is created by a normally incident polarized pump source, linearly polarized at 45°, with intensity above instability threshold. The polarization direction is indicated by the red arrow. The orientation of the two dominant spots in the hexagon is perpendicular to this polarization angle. The detection is cross-polarized with the pump, which is linearly polarized at 135°, in this case. (c) A two-spot pattern is observed when the pump is slightly tilted perpendicularly to the pump polarization direction, which is indicated by the small yellow arrow.

00006_PSISDG10193_101930G_page_4_1.jpg

3.

OPTICAL SPIN HALL EFFECT

To observe the OSHE pattern numerically with a “x”-polarized source, instead of computing the cavity photon and exciton fields as in Eqs. (1) - (4), we simplify the problem using the exciton-polariton picture taking into account only the two lower polariton branches, LP1 and LP2, as in:33, 34

00006_PSISDG10193_101930G_page_3_14.jpg
00006_PSISDG10193_101930G_page_3_15.jpg

Here 00006_PSISDG10193_101930G_page_3_16.jpg and 00006_PSISDG10193_101930G_page_3_17.jpg are the polariton fields at the lowest polariton branch, LP1, and the second lowest polariton branch, LP2, in real space, respectively. The k = 0 energies of LP1 and LP2 polaritons are 00006_PSISDG10193_101930G_page_3_18.jpg and 00006_PSISDG10193_101930G_page_3_19.jpg, respectively. We also make an assumption that 00006_PSISDG10193_101930G_page_3_20.jpg due to the (+) polarized pump at energy equal to 00006_PSISDG10193_101930G_page_3_21.jpg. 00006_PSISDG10193_101930G_page_3_22.jpg are the effective source terms on the two lower polariton branches and the assumption leads to 00006_PSISDG10193_101930G_page_3_23.jpg. Moreover, 00006_PSISDG10193_101930G_page_3_24.jpg and 00006_PSISDG10193_101930G_page_3_25.jpg. The spin-orbit coupling is 00006_PSISDG10193_101930G_page_4_2.jpg. Here 00006_PSISDG10193_101930G_page_4_3.jpg and 00006_PSISDG10193_101930G_page_4_4.jpg are the inverse sum of the effective TE and TM masses of LP1 and LP2, respectively. The nonlinear terms are

00006_PSISDG10193_101930G_page_4_5.jpg
00006_PSISDG10193_101930G_page_4_6.jpg

The coupling between polaritons are modified from the photon-exciton picture with 00006_PSISDG10193_101930G_page_4_7.jpg 00006_PSISDG10193_101930G_page_4_8.jpg 00006_PSISDG10193_101930G_page_4_9.jpg and 00006_PSISDG10193_101930G_page_4_10.jpg. The coefficients are the combinations of the Hopfield coefficients defined as

00006_PSISDG10193_101930G_page_5_1.jpg00006_PSISDG10193_101930G_page_5_2.jpg00006_PSISDG10193_101930G_page_5_3.jpg00006_PSISDG10193_101930G_page_5_4.jpg00006_PSISDG10193_101930G_page_5_5.jpg00006_PSISDG10193_101930G_page_5_6.jpg

where 00006_PSISDG10193_101930G_page_5_7.jpg. For detailed derivation please refer to Ref. [34].

Before discussing the numerical simulation results of Eqs. (5) and (6), we introduce the pseudospin model.31, 35 The components of the pseudospin vector in k-space Sk are described by the Stokes parameters formed by the polariton wavefunctions:

00006_PSISDG10193_101930G_page_5_9.jpg

where 00006_PSISDG10193_101930G_page_6_1.jpg is the spatial Fourier transform of 00006_PSISDG10193_101930G_page_6_2.jpg. The evolution of the pseudospin vector can either be obtained from the equations of motion for the polariton wave functions as detailed above, or from a phenomenological pseudospin model, given by the equations31, 35

00006_PSISDG10193_101930G_page_6_3.jpg

with the effective magnetic field Bk, the decay term γS and a source term R(k). The pseudospin vector can be represented in the Poincare sphere as shown in Fig. 3(a). The north and south pole correspond to “+” and “-”-polarized polaritons, respectively, while the positive and negative parts of the S1 axis correspond to mostly “x” and “y”-polarized polaritons, respectively. In other words, for a positive S1 (k) the polarization state of the field is mostly “x”, while for negative S1 (k) it is “y”.

Figure 3.

(a)The Poincare sphere representation of the pseudospin vector S. The north and south pole correspond to “+” and “-”-polarized polaritons, respectively, while the positive and negative parts on the “S1”-axis correspond to “x” and “y”-polarized polaritons, respectively. (b) Numerical results for S1(k) with a “+”-polarized normally incident source. The “x” (red) and “y” (blue) polarized polaritons are spatially separated into four quadrants in k-space. A further increase of the pump intensity will lead to a further rotation of the S1(k) pattern.31 The data are normalized such that the maximum and minimum values are ±1.

00006_PSISDG10193_101930G_page_5_8.jpg

We perform numerical simulations of Eqs. (5) and (6) with a “+”-circularly polarized light and observe the far-field polarization texture using S1(k). An example is shown in Fig. 3(b), which consists of four quadrants of mostly linearly polarized polaritons on the elastic circle. The polarization texture was shown to be rotated with an increase in pump intensity,31 and in Fig. 4(a) we show the change in polarization of polaritons on the elastic circles at 0-degree under different pump intensity. Here we dene the radial average of the Stokes parameters as

00006_PSISDG10193_101930G_page_6_4.jpg

Figure 4.

(a)The polarization ellipse of the far-field emission from polaritons at zero degrees on the elastic circle for low and high pump intensity. The azimuth angle of the polarization is rotated with a higher pump intensity. At the same time, the polarization is closer to a circle, i.e. the ellipticity is closer to π/4, with a higher pump intensity. (b) The three radially integrated Stokes parameters 00006_PSISDG10193_101930G_page_6_7.jpg, 00006_PSISDG10193_101930G_page_6_8.jpg and 00006_PSISDG10193_101930G_page_6_9.jpg on the elastic circle at zero degrees for low and high pump intensities. The increase in 00006_PSISDG10193_101930G_page_6_10.jpg and decrease in 00006_PSISDG10193_101930G_page_6_11.jpg effect a change in the azimuth angle.

00006_PSISDG10193_101930G_page_6_6.jpg

where

00006_PSISDG10193_101930G_page_6_5.jpg

is the radially integrated polariton density measured at angle ϕk. The range between k1 and k2 is determined by the thickness of the elastic circle so that further increase in the integration range does not change the Iϕk, η. Here, corrections to the Stokes parameters due to the small deviation of the emission from the surface normal has been neglected.

The ellipticity ϵ and the azimuth angle θaz of the polarization of polaritons plotted in Fig. 4(a) is then given by

00006_PSISDG10193_101930G_page_7_1.jpg
00006_PSISDG10193_101930G_page_7_2.jpg

The results in Fig. 4(a) show that the azimuth angle of the polarization is rotated and the polarization is closer to a circle, i.e. ellipticity is closer to π/4, with a higher pump intensity. These changes in the azimuth angle and ellipticity can be understood by studying the changes of 00006_PSISDG10193_101930G_page_7_3.jpg with low and high pump intensity. The values of 00006_PSISDG10193_101930G_page_7_4.jpg are shown in Fig. 4(b) for the two cases, and the change of ellipticity is coming from the increase of 00006_PSISDG10193_101930G_page_7_5.jpg in the high intensity case. From Eq. (14), a higher 00006_PSISDG10193_101930G_page_7_6.jpg weighting leads to a ellipticity ϵ closer to π/4. The change in azimuth angle comes from the combined effect of the increase in 00006_PSISDG10193_101930G_page_7_7.jpg and decrease in 00006_PSISDG10193_101930G_page_7_8.jpg.

4.

CONCLUSION

In conclusion, we have discussed two kinds of all-optical pattern control in a semiconductor double-microcavity, namely the hexagonal/two-spot intensity pattern and the OSHE spin texture, and showed that these patterns could be optically manipulated. In the intensity pattern case, a small tilt of pump can destabilize the hexagonal pattern and lead to the formation of a two-spot pattern. In the OSHE, increasing the pump intensity can lead to a rotation of the spin texture. These control mechanisms could be the fundamentals of future all-optical computation and communication utilizing polaritons in semiconductor microcavities.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support from the US NSF under grant ECCS-1406673, TRIF SEOS, and the German DFG (research centre TRR142 Tailored nonlinear photonics and SCHU 1980/5). S.Sch. further acknowledges support through the DFG Heisenberg programme.

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© (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
R. Binder, S. M. H. Luk, N. H. Kwong, P. Lewandowski, S. Schumacher, O. Lafont, E. Baudin, J. Tignon , A. Lemaitre, J. Bloch, Ch. K. P. Chan, and P. T. Leung "Optical control of polaritons: from optoelectronic to spinoptronic device concepts", Proc. SPIE 10193, Ultrafast Bandgap Photonics II, 101930G (8 May 2017); https://doi.org/10.1117/12.2263383
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KEYWORDS
Polaritons

Polarization

Control systems

Excitons

Semiconductors

Photons

Double patterning technology

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