## Abstract

We propose an amplified all-optical polarization phase modulator assisted by a local surface plasmon in Au-hybrid CdSe quantum dots. When the local surface plasmon of a spherical Au quantum dot is in resonance with the exciton energy level of a CdSe quantum dot, a significant enhancement of the linear and nonlinear refractive index is found in both the real and imaginary terms via the interaction with the dipole field of the local surface plasmon. Given a gating pulse intensity, an elliptical polarization induced by the phase retardation is described in terms of elliptical and rotational angles. In the case that a larger excitation than the bleaching intensity is applied, the signal light can be amplified due to the presence of gain in the CdSe quantum dot. This enables a longer propagation of the signal light relative to the metal loss, resulting in more feasible polarization modulation.

© 2012 OSA

## 1. Introduction

Currently, great interest has been focused on the advantages of nanocrystal quantum dots (NQDs) [1] such as fine size tunability, small size deviation, temperature insensitivity, and easy integration with nano-bio structures. In particular, application of the large optical nonlinearities of strongly confined NQDs is prospective [2, 3]. Nevertheless, the light-matter interaction in a nanoscale is limited to strong light excitation. Although the coupling is enhanced by using a high-finess cavity, metal-hybrid nanostructure can be a cavity-free alternative. The optical properties of NQDs can be dramatically enhanced when a metal surface is in close proximity via the coupling between excitons and surface plasmons (SPs) [4–12], where up to 23-fold enhanced photoluminescence intensity of the Purcell effect and a Rabi splitting (∼ 112meV) of the dressed state were measured in the weak [8] and strong [11] coupling regimes, respectively.

In the case of a planar hybrid structure, the propagation of the coupling, which is called SP-polariton (SPP), is well guided by the confined strong electric field along a metal-dielectric interface, where dielectric NQDs are layered on a metal surface. This enables a feasible all-optical modulation at low power densities (∼ 10^{2} Wcm^{−2}) in a micrometer-scale planar metal-hybrid structure [3], where the incident field is interfered with the propagating SPP between the CdSe NQDs and subwavelength-patterned Au. Furthermore, when the hybrid structure is composed of metal NQDs, the spectral resonance with the dielectric NQDs can be obtained easily by the size control, and the stronger local SP field of a spherical metal NQD induces remarkably large optical nonlinearities of the dielectric NQD. Recently, the plasmonic resonant coupling was manifested by optical Stark effect (OSE) in colloidal Au-CdSe core-shell hybrid structures under the sub-resonant excitation with respect to the ground exciton state, where the activated local SP of a Au NQD gives rise to a transient blueshift of the exciton absorption edge [2]. As a light-induced effective magnetic field (**H**_{Stark}) is defined along the laser direction in the OSE, the coherent control of a spin state in the Bloch sphere is achieved in combination with an external magnetic field. Although the tipping angle of the exciton spin was controlled over 90 degree with increasing the excitation intensity, the optical nonlinearities were not considered in terms of the precise refractive index or susceptibility.

In this work, we propose a feasible nonlinear photonic device specifically, the so-called amplified all-optical polarization phase modulator based on Au-hybrid CdSe NQDs, where the photo-induced phase retardation of an incident linear polarization can be predicted precisely in terms of the enhanced complex refractive index of an electron-hole pair in the Au-hybrid CdSe NQD, whereby the resultant elliptical polarization is controlled by the gating pulse intensity and propagation length. Additionally, we utilize the idea of SP-polariton amplification [13,14]; as the propagation loss is compensated by the gain in the dielectric medium, the polarization modulation becomes more feasible when the signal light is designed to propagate along the dielectric-metal interface.

## 2. Results and discussion

Provided that the local dipole field of a spherical Au NQD is excited resonantly by the electric field of the excitation light (*E*_{0}(*t*)), the total electric field applied to the exciton energy levels of a CdSe NQD (*E*_{CdSe}(*t*)) is significantly strengthened (Fig. 1(a)), where the three bright exciton states (±1^{L}, ±1^{U}, and 0^{U}) are considered for the CdSe NQD [15–17] and the multipole fields of the local SP in the Au are ignored. As a result, large optical nonlinearities are expected in the hybrid structure. The local SP dipole can be brought into resonance with the exciton energy level by size control of the NQDs. For example, a 3 nm-radius CdSe NQD is roughly in resonance with a 100 nm-radius (*R*) Au NQD. Although the coupling with the local SP dipole is enhanced for decreasing separation between the Au and CdSe NQD, the dissipative energy transfer to the Au becomes significant, resulting in non-radiative energy transfer [18]. In this model, the non-radiative energy transfer is assumed to be negligible at the optimum separation (∼ 10nm) between the Au and CdSe NQDs [9], which can be realized by using the overall thickness *d* of the shell (ZnS) and a spacer molecular linker such as biomolecules and 1,6-hexanedithiol (HDT) [4, 19, 20].

The Hamiltonian of the hybrid structure is described as [6]

*ε*=

_{i}*h*̄

*ω*is the energy level of the ground (

_{i}*i*= 0) and bright exciton states (

*i*= 1, 2, 3) with the density matrix elements ${d}_{ij}\left(={d}_{ji}^{*}\right)$, and creation ( ${\alpha}_{i}^{\u2020}$) and annihilation (

*α*) operators of the excitons are used. The total electric field applied inside the CdSe (

_{i}*E*

_{CdSe}(

*t*)) and the induced dipole on the spherical Au NQD (

*p*

_{m}) are given by

*ε*

_{0},

*ε*

_{s}, and

*ε*

_{m}are the dielectric constant of the background, CdSe, and Au, respectively [5, 16, 21, 22]. The induced dipole

*p*

_{m}is also affected by the dipole electric field of a spherical CdSe NQD,

*i.e.*an interaction with the image dipole. It is noticeable that the enhancement of

*E*

_{CdSe}(

*t*) and

*p*

_{m}depends on the direction between the two dipoles

*p*

_{m}and

*p*

_{s}, leading to the different geometric factor

*ξ*= 2 and

*ξ*= −1 at position A and B in Fig. 1(c), respectively. Additionally, since the real part of dielectric constant in Au is negative in Eq. (3) [21, 22], the direction of the induced dipole in Au (

*p*

_{m}) is opposite to the applied electric field of the excitation light (

*E*

_{0}(

*t*)),

*i.e.*$\eta =\frac{{\epsilon}_{\text{m}}-{\epsilon}_{0}}{{\epsilon}_{\text{m}}+2{\epsilon}_{0}}<0$. Therefore,

*E*

_{CdSe}(

*t*) is strengthened at position B, but suppressed at position A (here all results are for the position B). The dipole (

*p*

_{s}) of a CdSe NQD is defined in terms of the off-diagonal elements of the density matrix as ${p}_{\text{s}}={\sum}_{i}\left({d}_{i0}^{*}{p}_{i}+{d}_{i0}{p}_{i}^{\u2020}\right)$, where the microscopic polarization is given by ${p}_{ij}=<{\alpha}_{i}^{\u2020}{\alpha}_{j}>$. As the intraband transition between the bright exciton states is neglected, both

*p*and

_{ij}*d*are 0 for

_{ij}*i*≠

*j*≠ 0 or

*i*=

*j*. Therefore, only the interband polarizations ${p}_{i}\equiv <{\alpha}_{i}^{\u2020}{\alpha}_{0}>$ are considered.

While the optical Bloch equations are used for the light-matter interaction of a two-level atomic system, the semiconductor Bloch equations (SBEs) were developed for the optical response of electron-hole pair in semiconductors by including Coulomb interaction and many-body effects. Although the validity range depends on the time-dependent Hartree-Fock approximation, the numerous optical nonlinearities of semiconductors have been described successfully in bulk, quantum well, and quantum dots [23]. As the optically-generated electrons and holes likely exist only as pairs (excitons) for resonant excitation, the bosonic fine excitons can be represented using quasiboson operators in the new basis [24]. The optical nonlinearities of the bright excitons (*i* = 1, 2, 3) in the CdSe NQD are described in terms of the interband polarization (*p _{i}*) and the occupancy (

*f*≡

_{i}*p*) by the semiconductor Bloch equations (SBEs) [23] as

_{ij}δ_{ij}*=*

_{i}*d*

_{i0}

*E*

_{CdSe}(

*t*)/

*h*̄, and a phenomenological transverse relaxation rate

*γ*

_{0}= 1ps was used. The equations have been linearized with respect to strong pump (

*E*(

_{p}*t*)

*e*

^{ikp·r}) and weak test probe (

*E*(

_{t}*t*)

*e*

^{ikt·r}) fields in the rotating-wave approximation, the SBEs for arbitrary order

*n*in the excitation electric field (

*E*

_{0}(

*r⃗*,

*t*) =

*E*(

_{t}*t*)

*e*

^{ikt·r}+

*E*(

_{p}*t*)

*e*

^{ikp·r}) are given by

*= Ω*

_{i}*+ Ω*

_{it}*, and*

_{ip}*k*=

_{t}*K*+

*k*and

*k*=

_{p}*K*−

*k*are introduced for spatial Fourier series expansions in the form ${p}_{i}={\sum}_{n=-\infty}^{\infty}{p}_{i}^{n}{e}^{iK\cdot r+ink\cdot r}$ and ${f}_{i}={\sum}_{n=-\infty}^{\infty}{f}_{i}^{n}{e}^{ink\cdot r}$.

If the system initially has no excitation, the odd harmonics of *f _{i}* and the even harmonics of

*p*remain zero. In the first order of the field, only ${p}_{i}^{1}$ is excited with no occupancy (

_{i}*f*= 0) leading to a linear response. As shown in Fig. 2(a) and 2(b), both the real and imaginary terms of the linear refractive index are significantly enhanced in the presence of Au compared to those in a CdSe NQD [5,16], where the 1

^{L}state is in resonance with the local SP dipole with 10 nm separation between CdSe and Au NQD. The detuned states of ±1

^{U}and 0

^{U}with respect to the local SP dipole are also enhanced. In particular, the intrinsic absorption enhancement (Fig. 2(b)) is very useful from the viewpoint of efficient light collection in solar cells.

Regarding only the SP-assisted enhancement, the real and imaginary parts of the linear refractive index for the Au-separation distance *d* were also calculated at 2.1502 eV (Fig. 2(c)). Although the enhancement is mainly attributed to the induced dipole field of a Au NQD compared to the dipole-dipole interaction between an exciton in a CdSe NQD and the image exciton in a Au NQD (Eq. (3)), it can be compensated by non-radiative energy transfer to the Au NQD at short distances via the dipole-dipole interaction [18]. Therefore, the optimum distance is determined as a consequence of the competing two effects. This precise engineering of the linear refractive index by size and distance control is a quite challenging technique in nano-photonic applications, and recent work reported that the transition from fluorescence enhancement to quenching occurs below ∼ 10nm separation [18].

To determine the optical nonlinearities with finite order, the set of equations is truncated up to third order for
${p}_{i}^{\pm 3}$ and
${f}_{i}^{\pm 2}$. In the third order of the field, the occupancy is increased with a second order in the field and polarization. As shown in Fig. 1(b), the induced 1^{L} occupancy under weak excitation intensity is obtained for the detuning factor *δ*, which is the difference between the central wavelength and that of the pumping laser. As mentioned earlier, the 1^{L} occupancy is enhanced in the presence of Au due to the strengthening of the total electric field applied to a CdSe NQD (*E*_{CdSe}(*t*)). The occupancy enhancement is significant up to *δ* ∼ ±100meV, and the asymmetry originates from the ±1^{U} and 0^{U} states.

Given the polarization for the occupancy, the imaginary nonlinear refractive index spectrum (*κ* or absorption coefficient
$\alpha =2\frac{\omega}{c}\kappa \propto \text{Im}\left[\frac{P(\omega )}{{E}_{t}(\omega )}\right]$) for the injected pulse area (
$\mathrm{\Theta}=\left|\frac{d}{\overline{h}}\int {E}_{p}(t)dt\right|$) was obtained as shown in Fig. 2(e) [25]. The real part of the refractive index spectrum *n* was also deduced by using the Kramers-Kronig relations (Fig. 2(d)). All bright exciton states (±1^{L}, ±1^{U} and 0^{U}) were included in the broad laser spectrum (dotted line in Fig. 2(e)), and the photo-induced refractive index change was calculated at zero delay time. The central laser wavelength is tuned to ±1^{L}, and a relatively weaker intensity is present at ±1^{U} and 0^{U}. Also, the oscillator strength at ±1^{L} is higher than that at ±1^{U} and 0^{U}. As a result, the nonlinearities are dominated by ±1^{L}. In the case of a single two-energy level system, the upper state is fully occupied for a *π* pulse area. For a Au-hybrid CdSe NQD, full absorption bleaching of ±1^{L} was observed to appear when the injected pulse area was nearly a half of *π* (Θ ∼ 0.538*π*). Thus a quarter of the intensity is necessary in the presence of Au to obtain the equivalent nonlinearity in a CdSe NQD to that in the absence of Au, as *I* ∝ Θ^{2}. This benefit supports the possibility of using Au-hybrid CdSe NQDs as a prospective medium for nonlinear photonic devices. It is interesting that the required pulse area for gain onset is slightly larger than that for full occupancy (Θ ∼ 0.423*π*) (shown in Fig. 3(a)) because the gain for ±1^{L} is compensated partly by the absorption in the case of ±1^{U} and 0^{U} for a spectrally-integrated broad pulse.

The maximum refractive index change of the real (Δ*n*) and imaginary (Δ*κ*) coefficient for the bright exciton states (±1^{L}, ±1^{U}, and 0^{U}) and the occupancy of ±1^{L} in a Au-hybrid CdSe NQD are compared those in a CdSe NQD in the absence of Au with increasing the intensity, which is defined in terms of Θ^{2}/4*π*^{2} in Fig. 3(a), 3(b), and 3(c). The enhancement is such dramatic as the refractive index changes of ±1^{U} and 0^{U} in the presence of Au become comparable to those of ±1^{L} in a CdSe NQD in the absence of Au. For small excitation intensity (*I* ∝ *f _{i}* ≪ 1), the complex refractive index change is often assumed to be linearly dependent on the excitation intensity (Δ

*ñ*∝

*I*), where the optical Kerr scheme is valid. However, as the intensity becomes large (Θ

^{2}/4

*π*

^{2}> 0.025), both the state occupancy and the refractive index change (Δ

*ñ*= Δ

*n*+

*i*Δ

*κ*) eventually become nonlinearly dependent on

*I*. Consequently, the intensity upper limit of the optical Kerr scheme is also significantly reduced in the presence of Au compared to that (Θ

^{2}/4

*π*

^{2}∼ 0.2) in a CdSe NQD in the absence of Au [16, 17].

Suppose the electric field of a gating (excitation) pulse is applied along the *x*–axis (Fig. 3(d)), a phase retardation is induced relative to linearly polarized incident light (+45°-rotated with respect to *x*–axis). As long as a CdSe NQD has cylindrical symmetry, *i.e.*, the cross section of a NQD ellipsoid is circular, the circular polarization-based spin-degenerate states (|+ 1^{L}〉 and | − 1^{L}〉) can be transformed equivalently into the linear polarization-based states (
$|x\u3009=\left(|+{1}^{\text{L}}\u3009+|-{1}^{\text{L}}\u3009\right)/\sqrt{2}$ and
$|y\u3009=\left(|+{1}^{\text{L}}\u3009-|-{1}^{\text{L}}\u3009\right)/\sqrt{2}$). If the cylindrical symmetry is broken, a splitting between |*x*〉 and |*y*〉 occurs (2 ∼ 3meV) [17]. Nevertheless, this scheme is applicable if the homogeneous linewidth in an inhomogeneous ensemble is comparable to the *x*–*y* splitting [26]. Assuming that the broadened spectrum of the refractive index is comparable to the experimental results in an NQD ensemble [5] and that the refractive index spectrum is dominated by ±1^{L}, the Jones vector of an induced elliptical polarization for an injected pulse intensity can be described as

*ñ*(0) is the linear complex refractive index. For a 1

*μ*m propagation distance

*z*, the elliptical polarization with increasing gating intensity is characterized in terms of the elliptical (

*ε*= 0.5sin

^{−1}(

*S*

_{3}/

*S*

_{0})) and rotational (

*θ*= 0.5tan

^{−1}(

*S*

_{2}/

*S*

_{1})) angles (Fig. 1(a)) by using Stokes parameters (

*S*

_{0}= |

*E*|

_{x}^{2}+ |

*E*|

_{y}^{2},

*S*

_{1}= |

*E*|

_{x}^{2}− |

*E*|

_{y}^{2}, ${S}_{2}=2\text{Re}\left\{{E}_{x}{E}_{y}^{*}\right\}$, and ${S}_{3}=-2\text{Im}\left\{{E}_{x}{E}_{y}^{*}\right\}$. As

*ε*and

*θ*are mainly governed by

*κ*and

*n*, respectively, the phase retardation also depends on the dispersion.

As an example, the Stokes parameters (Fig. 3(e)), *ε*, and *θ* (Fig. 3(g)) for the gating pulse intensity are calculated at 2.1502 eV, where the real part of refractive index is a maximum (Fig. 2(a) and 2(b)). As shown schematically in Fig. 3(f), the signal light of initially +45°-linear polarization evolves into an elliptical polarization as the excitation intensity is increased. Regarding the position B (Fig. 1(a)) for the field strengthening, the signal light is supposed to propagate along the interface between the CdSe and Au NQD rather than penetrating through the hybrid structure (Fig. 3(d)). This hybrid structure (CdSe/ZnS-spacer-Au) can be easily prepared by using a layer-by-layer method. Despite of the difficulty in achieving spectral resonance and large inhomogeneity, a rough Au film can be used as an alternative to Au NQDs for convenience. The propagation distance of a SP-polariton at the interface between a dielectric medium and a metal is often limited by losses in the metal [3]. However, this loss can be compensated if gain is supported in the dielectric medium. Currently, this idea is of great interest in terms of SP-polariton amplification by stimulated emission of radiation (SPASER) [13, 14]. Likewise, the signal light in Fig. 3(d) can be amplified if the pumping intensity is larger than the bleaching intensity, where gain begins to appear at ±1^{L} in a Au-hybrid CdSe NQD (Fig.2(e)). This enables a longer propagation of the signal light relative to the metal loss, resulting in more feasible polarization. Recently, it was found that a large optical gain can be obtained due to multiexcitons in strongly confined CdSe NQDs for resonant or state-resolved optical pumping [27, 28], otherwise the gain is suppressed due to the Auger recombination for the far off-resonant excitation (for example, 400nm-excitation). Therefore, the multiexciton effect can be enhanced in Au-hybrid NQDs [29] as the local surface plasmon dipole is in resonance with the exciton level.

## 3. Conclusion

We have calculated an enhancement of the linear and nonlinear refractive index of a CdSe NQD when a Au NQD is in resonance by using the semiconductor Bloch equations due to the local surface plasmon of a spherical Au NQD. The enhancement results from a strengthening of the electric field applied to a CdSe NQD, which requires the particular location of a CdSe NQD where the induced dipole of a local surface plasmon is parallel to the electric field of the incident light. We also propose an amplified all-optical polarization modulator, where the polarization phase retardation can be controlled precisely by microscopic (the size and separation of CdSe and Au NQDs) and macroscopic (excitation intensity and propagation length) parameters, and the propagation loss can be compensated by the gain in a CdSe NQD.

## Acknowledgments

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by Ministry of Education, Science and Technology ( 2010-0008942, NRF-2011-013-C00025, and 2010-0021173).

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