Photoinduced diffraction grating in hybrid artificial molecule

Zhi-Hong Xiao; Li Zheng; HongZhen Lin

doi:10.1364/OE.20.001219

1. Introduction

Modern nanotechnology opens a possibility to build nano-superstructures with various combinations. In particular, hybrid structure composed of semiconductor quantum dots (SQDs) and metal nanoparticles (MNPs) has recently received great attention [1–9]. SQDs have been extensively studied during the past decades owing to their unique optical properties for instance bright photoluminescence, tunable color, and high photostability. MNPs have also attracted a lot of research interests because of their light-scattering and surface plasmonic properties. Putting a SQD in the vicinity of a proper MNP, the SQD and the MNP can be electronically coupled and the coupling strength depends on the geometry of the hybrid. It has been demonstrated that such hybridization may bring out some new physical effects e.g. Fano-type asymmetric features in absorption spectra [1, 2], exciton/plasmonic induced transparency [5, 8], and enhancement of Rabi flopping [9]. Most of these studies are based on hybrid systems driven by traveling wave, and it is crucial to have at least one coupling laser to create necessary coherence.

In the present paper, we modeled the photoinduced formation of diffraction grating in a SQD-MNP hybrid system based on exciton/plasmonic induced transparency. When a proper traveling-wave coupling field is applied, the strong absorption of the MNP at a certain frequency range vanishes as a result of exciton-plasmon interaction, and meanwhile the line absorption of the SQD at the probe frequency spectrum is also eliminated so that the probe field can transmit trough the medium. While a coupling field with a standing-wave intensity pattern is applied, the absorption or refractive index for the probe will experience a periodic variation as a function of the probe detuning, or as a function of the phase of the probe light. The medium thus acts an absorption or phase grating and can effectively diffract the probe beam into the first and higher order components. Further analysis shows that the diffraction efficiency can be manipulated by adjusting the inter-particle distance between the SQD and MNP, i.e., tuning the interaction strength between the SQD and the MNP. Nearly pure phase grating on line modulation is created and an efficiency of approximately 33% is obtained. From the numerical results it is found that the efficiency of phase grating is dramatically improved due to the presence of the MNP, which plays a significant role in diffracting the probe beam. We note that a standing-wave coupling field is essential for the formation of this grating.

2. Hybrid molecule

We consider in this article a hybrid nanostructure composed of a spherical semiconductor quantum dot (SQD) with radius $b$ and a spherical metal nanoparticle (MNP) of radius $a$ , as depicted in Fig. 1a . The SQD and MNP are separated by a distance $R$ . $ε_{s}$ and $ε_{m}$ are the dielectric constants of SQD and MNP, respectively. Figure 1b shows the energy scheme of the system. The plasmonic excitations of the MNP are a continuous spectrum; the excitations of the SQD are excitons with discrete energy levels. The interband transition $| 2 〉 \leftrightarrow | 3 〉$ is excited by a strong coupling beam with frequency $ω_{c}$ and the Rabi frequency is given by $Ω_{c} = μ_{23} E_{c} / ℏ$ , and a weak probe beam with frequency $ω_{p}$ drives the interband transition $| 1 〉 \leftrightarrow | 2 〉$ , and the Rabi frequency $Ω_{p} = μ_{12} E_{p} / ℏ$ , where $μ_{12}$ and $μ_{23}$ are the transition dipole moments of the SQD.

Fig. 1 (a)Schematics of nanocrystal complex composed of spherical nanoparticle and quantum dot; (b) the energy structure of the system, the horizontal arrow depicts the Coulomb interaction.

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There is no direct electron tunneling between the SQD and the MNP. On the other hand, they are coupled in a Coulombic manner, leading to formation of hybrid excitons following optical excitation. Herein, we use the density-matrix approach to describe the SQD; while for description of the MNP, we employ classical electrodynamics and the quasi-static approach. The frequency of the MNP plasmonic resonance is set the same as that of the transition $| 1 〉 \leftrightarrow | 2 〉$ . The external applied fields are parallel to the major axis of the hybrid system ( $S_{a} = 2$ ). In the following, we assume that the MNP is made of Au, of which the plasmonic resonance happens around 2.38 eV [10]; and $R ≪ λ$ , where $λ$ is the wavelength of the probe beam. Moreover, we use the dipole approximation, assuming $a, b < R$ .

Under the rotating-wave approximation, the system dynamics for the three-level SQD is described by density-matrix equations given as:

\begin{array}{l} \overset{\cdot}{σ_{21}} = - (γ_{21} + i Δ_{p}) σ_{21} + i Ω_{c} \cdot σ_{31} + i Ω^{ζ} \cdot (σ_{11} - σ_{22}) \\ \overset{\cdot}{σ_{31}} = - [γ_{31} + i (Δ_{p} + Δ_{c})] σ_{31} + i Ω_{c} \cdot σ_{21} - i Ω^{ξ} \cdot σ_{32} \\ σ_{21} = σ_{12}^{*} \end{array}

where

γ_{21}

and

γ_{31}

are the dephasing rates of SQDs. The dephasing rate in semiconductors realized so far appear to be comparable in systems [11];

Δ_{p}

and

Δ_{c}

denote the detuning given by

Δ_{p} = ω_{12} - ω_{p}

and

Δ_{c} = ω_{23} - ω_{c}

, respectively. Rabi frequency

Ω^{ξ} = μ_{12} E_{S Q D} / ℏ

, where

E_{S Q D}

denotes the total electric field felt by the SQD which consists of the external applied field

E_{p}

and the induced internal field from the MNP.

E_{S Q D}

is given by

E_{S Q D} = E_{p} + \frac{1}{4 π ε_{b}} \cdot \frac{S_{a} \cdot P_{M N P}}{ε_{e f f m} \cdot R^{3}}

where

ε_{e f f m} = (2 ε_{b} + ε_{m}) / 3 ε_{b}

,

ε_{b}

is the background dielectric constant;

ε_{m}

corresponding to 2.38 eV is taken from Ref. 10. The polarization

P_{M N P}

comes from the charge induced on the surface of MNP and depends on the total field

E_{M N P}

acting on the MNP.

P_{M N P}

and

E_{M N P}

can be respectively expressed as

P_{M N P} = (4 π ε_{b}) \cdot r \cdot a^{3} \cdot E_{M N P}

E_{M N P} = E_{p} + \frac{1}{4 π ε_{b}} \cdot \frac{S_{a} \cdot P_{S Q D}}{ε_{e f f s} \cdot R^{3}}

where

ε_{e f f s} = (2 ε_{b} + ε_{s}) / 3 ε_{b}

,

r = (ε_{m} - ε_{b}) / (2 ε_{b} + ε_{m})

,

P_{S Q D}

is the polarization of the SQD and has

P_{S Q D} = 2 μ_{12} \cdot N^{-} \cdot A

where

N^{-}

is the SQD density. Rabi frequency

Ω^{ξ}

is achieved by Eq. (2)-Eq. (4), as follows

Ω^{ξ} = Ω_{p} (1 + \frac{r \cdot S_{α} \cdot a^{3}}{ε_{e f f m} \cdot R^{3}}) + \frac{N^{-} \cdot μ_{12}^{2}}{2 π ℏ ε_{b}} \cdot \frac{r \cdot S_{α}^{2} \cdot a^{3}}{ε_{e f f m} \cdot ε_{e f f s} \cdot R^{6}} A

In the weak probe field limit and the steady state, the initial conditions are $σ_{22} = σ_{33} = 0,$ $σ_{11} = 1, σ_{i j} = 0 (i \neq j)$ , base on which we solve the density-matrix equations. And then from the polarization of medium $P_{p} = ε_{b} x_{p} E_{p} = N^{-} μ_{12} σ_{21}$ , we obtain the probe susceptibility

$x_{p} = (N^{-} μ_{12}^{2} / ℏ ε_{b}) \cdot x^{(1)}$ , where $x^{(1)}$ corresponds to the first-order linear part given by

Im (x^{(1)}) = \frac{- (1 + K^{'})}{[Δ_{p} - E + η K^{''}] \cdot (\frac{E - Δ_{p}}{F + γ_{21}}) - (F + γ_{21})}

Re (x^{(1)}) = \frac{- (E - Δ_{p}) (1 + K^{'})}{[Δ_{p} - E + η K^{''}] \cdot (E - Δ_{p}) - {(F + γ_{21})}^{2}}

with

E = \frac{{| Ω_{c} |}^{2} (Δ_{p} + Δ_{c})}{γ_{31}^{2} + {(Δ_{p} + Δ_{c})}^{2}}, F = \frac{γ_{31}^{2} \cdot {| Ω_{c} |}^{2}}{γ_{31}^{2} + {(Δ_{p} + Δ_{c})}^{2}}, K^{'} = \frac{r \cdot a^{3} \cdot S_{α}}{ε_{e f f m} \cdot R^{3}}, K^{''} = \frac{μ_{12}^{2}}{4 π ℏ ε_{b}}, η = \frac{r \cdot a^{3} \cdot S_{α}^{2}}{ε_{e f f m} \cdot ε_{e f f s} \cdot R^{6}}

Some parameters used in this paper are shown in Table 1 . In order to simplify the numerical simulation, we assume $ℏ = 1$ .

Table 1. Parameter Values Used in the Numerical Simulations

View Table

From the Eq. (6), the absorption and refraction spectrum of $x^{(1)}$ is plotted in Fig. 2 . It finds that the dip on the absorption spectrum becomes deep until it forms a transparency window and the abrupt dispersion at the transparency region when the intensity of the coupling field increases. The width of the transparency window becomes broad in comparison to conventional EIT investigated in atomic, quantum well or QD systems. The reason is that the absorption of the MNP can be controlled by adjusting the intensity of the coupling field when the MNP is at the vicinity of the SQD, and the strong absorption of the MNP is eliminated at near resonance [7]. The plasmonic induced transparency is formed due to the coherent interaction of the external coupling field with the SQD. The hybrid absorption for the probe disappears in the hybrid SQD-MNP nanostructure.

Fig. 2 (a) Imaginary part (absorption) of probe susceptibility $Im [x^{(1)}]$ versus probe detuning $Δ_{p}$ for different $Ω_{c}$ ; (b) Real part (dispersion) of probe susceptibility $Re [x^{(1)}]$ versus probe detuning $Δ_{p}$ for different $Ω_{c}$ .Other parameters: $Δ_{c} = 0, R = 33 n m .$

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3. Photoinduced diffraction grating

A coupling field with an intensity pattern can produce absorption and phase pattern of the probe field because of the coupling-intensity-dependent absorption and refraction index. While the coupling field is a standing-wave field, the absorption or refractive index of the probe field will experience a periodic variation. Hence, the probe, which propagates perpendicularly to standing wave, is diffracted into high-order components by photoinduced diffraction grating (PIDG). Figure 3 shows schematic diagram of the PIDG. Two counter-propagating components of the coupling fields form a standing wave along the x-direction. Therefore, the dynamic response of the probe field exhibits periodical modulation, and leads to form the absorption or phase grating.

Fig. 3 Sketch of the spatial configuration of the probe and coupling beam.

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To obtain the dynamic response of the probe field, we use Maxwell’s equation to describe the probe field. Under the slowly varying envelope approximation and in the steady state regime, the equation is reduced to $\partial E_{p} / \partial z = i (π / λ ε_{b}) \cdot p_{p}$ or $\partial E_{p} / \partial z^{'} = {- Im [x^{(1)}] + i Re [x^{(1)}]} E_{p}$ , where $z^{'} = (N^{-} μ_{12}^{2} / 2 ℏ ε_{b}) k_{p} z_{0}$ with $k_{p} = {2 π / λ}_{p},$ and $z_{0} = (N^{-} μ_{12}^{2} / 2 ℏ ε_{b}) k_{p}$ is treated as the unit for z. Thus we may derive the transmission function for an interaction length L of medium

T = \exp (- Im [x^{(1)}] \cdot L + i Re [x^{(1)}] \cdot L)

where the first and second terms correspond to the absorption and phase modulation, respectively. For a standing-wave coupling field with Rabi frequency

Ω_{c} = Ω \sin (π x / Λ)

,

Ω

being the intensity of standing-wave field.

Λ

is the spatial frequency of standing-wave, the transmission function is modulated in the x-direction as gratings. By performing the Fourier transformation on the transmission function, we obtain the Fraunhofer distribution (i.e. far-field distribution) of a single space period in the form

I_{p} (θ) = {| \int_{0}^{1} T \exp (- i 2 π x \cdot \sin (θ) M) d x |}^{2} \cdot \frac{\sin^{2} (N π \sin (θ) M)}{M^{2} \sin^{2} (π \sin (θ) M)}

with

θ

denoting the diffraction angle of the probe field with respect to z-direction, and

M = Λ / λ_{p}

being a constant factor designating the spatial width of the probe beam. The n-order diffraction intensity is determined by Eq. (8), with

\sin (θ) = n \cdot M

. The first-order diffraction intensity of grating is given by

I_{p} (θ_{1}) = {| Ω_{p}^{I} (θ_{1}) |}^{2} = {| \int_{0}^{1} T \exp (- i 2 π x) d x |}^{2}

3.1 Absorption grating

We know from the above discussion that under the strong coupling field the absorption of the hybrid system vanishes accompanied with eliminated refraction at near resonant frequency range. Therefore, under the action of a coupling field with standing wave pattern, periodic absorption modulation across the profile of the probe beam takes place, while phase modulation is absent. Figure 4 shows the effect of the inter-particle distance R between the SQD and the MNP on the amplitude and phase of the transmission function, respectively. At large R, the amplitude modulation is small [blue dotted line in Fig. 4a] which limits the diffraction efficiency of grating [blue solid line in Fig. 5a ]; while the amplitude modulation becomes large at small R [red dashed (green solid) line in Fig. 4a] leading to the increment of the diffraction efficiency [red (green) solid line in Fig. 5a]. According to Eq. (7), it is seen that both amplitude and phase of the transmission function depend on the inter-particle distance R, i.e., are influenced by the interaction strength between the SQD and the MNP. At the resonant case, the amplitude modulation is very sensitive to the value of R. This effect can be understood as that the dipole-dipole interaction between the SQD and the MNP becomes weak when increasing the value of R. The R-dependent destructive or constructive interference causes the change of the probe absorption at the nodes or antinodes of standing wave.

Fig. 4 (a) Amplitude of transmission function with various inter-particle distances $R$ (nm); (b) Phase of transmission function for different inter-particle distances $R$ (nm). Other parameters: $Δ_{c} = Δ_{p} = 0, L = 100 z_{0}, Ω = 13.$

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Fig. 5 Diffraction pattern as a function of $\sin (θ)$ . (a) for different inter-particle distances $R$ ; (b)with the MNP and without the MNP. Other parameters: $N = 4, M = 5, Δ_{c} = Δ_{p} = 0, Ω = 13,$ $L = 100 z_{0} .$

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The MNP plays an important role to enhance the diffraction efficiency of grating [see blue line in Fig. 5b]. On one aspect, the SQD and the MNP is coupled due to the plasmonic resonance of the MNP, and transition $| 1 〉 \leftrightarrow | 3 〉$ to generate the transparency is not dipole forbidden; on the other aspect, the absorption modulation depth is increased due to energy transfer between the SQD and the MNP, which results in more probe light being diffracted into first- and higher- order components.

The diffraction efficiency of grating, which is modulated by absorption, depends on the interaction length L and the coupling intensity of standing-wave field $Ω$ as well as on the inter-particle distance R. The first-order diffraction efficiency at different inter-particle distance R is plotted as a function of the interaction length L and the coupling intensity of standing-wave field $Ω$ in Fig. 6 and Fig. 7 , respectively. As one can clearly see from Fig. 6, the first-order diffraction efficiency is relatively high at small R and large L or at large R but small L. Figure 7 indicates that the first-order diffraction efficiency for a fixed interaction length L can be also slightly improved by properly increasing both the coupling intensity of standing-wave field and the inter-particle distance. Figure 6 and Fig. 7 illustrate that the absorption modulation depth and correspondingly the efficiency of the grating can be optimized by choosing the above physical parameters properly.

Fig. 6 First-order diffraction intensity as a function of $L$ for different inter-particle distances $R$ (nm). Other parameters are the same as in Fig. 5.

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Fig. 7 First-order diffraction intensity as a function of $Ω$ for different R (nm), with $L = 100 z_{0}$ . Other parameters are the same as in Fig. 5.

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3.2 Phase grating

We know from above simulation results that the first-order diffraction efficiency by pure absorption modulation is very limited (only up to 4% in our results). This is because that the probe light tends to be collected at the center maximal and the available light for the first-order and high-order is of very minor fraction under absorption modulation. In order to obtain a grating with high diffraction efficiency, phase grating is concerned. Ideally, for phase grating with high diffraction efficiency, the medium should be completely transparent to the probe beam, but has a $π$ phase modulation across the probe beam. Although the ideal grating is hard to achieve, phase grating with high transmissivity can be obtained. In Fig. 8a , a large phase modulation (black dashed-dot line) and an absorption modulation (red dashed line) of the transmission function $T (x)$ are displayed for the SQD in the presence of the MNP. It is seen from Fig. 8b that a phase grating with high transmissivity is formed. Due to phase modulation, the center light intensity becomes weak, and the first-order diffraction is increased dramatically. On the contrary, the probe beam is hardly diffracted via absorption modulation. The phase grating in the absence of the MNP is also investigated (see Fig. 8c). We find that in this case the phase modulation is ineffective. The role of the MNP lies in the following aspects. Firstly, in the presence of the MNP, the exciton-plasmon coupling makes the internal field of the MNP to be involved, and transparency is formed in a relatively broad range around the plasmon frequency. Secondly, the degree of transparency may be controlled by tuning the coupling intensity (see Fig. 2a). Finally, a high average transmissivity is maintained as the phase modulation depth being enhanced

Fig. 8 (a) Amplitude (red dashed line) and Phase (black dashed-dot line) of transmission function with the MNP; Blue solid line is phase modulation of transmission function without the MNP; (b) with $N = 4, M = 5$ , Diffraction patterns with the MNP, transmission function shown in (a); (c) with $N = 4,$ $M = 5,$ Diffraction pattern by phase modulation in the absence of the MNP, transmission function shown in (a); other parameters: $Δ_{c} = 0, Δ_{p} = 28, Ω = 13,$ $R = 33 n m, L = 190 z_{0} .$

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In Fig. 9 and 10 , the first-order diffraction patterns of phase grating are displayed with the MNP. These two figures illustrate that the inter-particle distance R between the SQD and the MNP can affect the diffraction efficiency of phase grating. The interaction strength between the SQD and the MNP varies as R changes. The diffraction efficiency is low at small R while increases at larger R. For instance, the diffraction efficiency is approximate 33% at $R = 33 n m,$ which is much closed to that of an ideal sinusoidal phase grating. The increment of R not only enhances the refractive index for the probe field, but also restrains the Columbic coupling strength between the SQD and the MNP. Therefore, in order to obtain a phase grating with the best performance in such a hybrid artificial molecule, R has to be properly chosen.

Fig. 9 First-order intensity as a function of detuning $Δ_{p}$ for different $R$ in nm. Other parameters are the same as in Fig. 8.

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Fig. 10 First-order intensity as a function of $L$ for different inter-particle distances $R$ (unit of nm). Other parameters are the same as in Fig. 8.

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To further comprehend the role of the MNP, the first-order diffraction patterns of phase grating without the MNP are plotted in Fig. 11 . We find that, in the absence of the MNP, the diffraction efficiency that can be achieved by phase modulation is only $\approx$ 23%. The simulation results prove that excitons and plasmons interact via the Coulomb forces in the hybrid system, which leads the diffraction efficiency of phase grating to be improved in comparison to the SQD barely.

Fig. 11 First-order intensity as a function of the detuning $Δ_{p}$ without MNP. Other used parameters are the same as in Fig. 8.

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4. Conclusions

In conclusion, photoinduced diffraction grating is investigated in a SQD-MNP hybrid nanostructure. The results show that the strong absorption of the MNP can be eliminated by modulating the strength of the coupling field, and the medium becomes transparent for the probe. Photoinduced diffraction grating is formed based on the excitons induced transparency when a standing-wave coupling field is applied. We find that the existence of the MNP effectively improves the diffraction efficiency of the grating in comparison to that without the MNP. Pure absorption and almost pure phase grating are realized and the diffraction efficiency of gratings can be manipulated by tuning the interaction strength between the SQD and the MNP. The diffraction efficiency of pure absorption grating is rather limited (only $\approx$ 4%), while a high diffraction efficiency can be achieved for the phase grating. The presence of the MNP causes the enhancement of phase modulation depth, and a relatively large fraction of light is diffracted into first-order component. Nearly pure phase grating accompanied with a high transmissivity is demonstrated, and the first-order diffraction efficiency reaches $\approx$ 33%. The MNP plays an important role in such a hybrid artificial molecule, which lies in two aspects: one is that the plasmon effect of the MNP exerts an internal field on the SQD; on the other is that the existence of the MNP can enhance the absorption and phase modulation depths, so that the diffraction efficiency of the gratings is improved. Photoinduced diffraction grating has potential application in probing the optical property of materials, light switching and routing, diffracting and switching a quantized probe field, etc.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant No. 211732 53), China Postdoctoral Science Foundation (Grant No. 2011M500951).

References and links

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Symbol	Value	Unit	Symbol	Value	Unit
μ₁₂	0.67	enm	α	6	nm
γ₂₁	1	ns^–1		1
γ₃₁	3	ns^–1		6
N^–	3 × 10²¹	m^–3		2

Photoinduced diffraction grating in hybrid artificial molecule

Abstract

1. Introduction

2. Hybrid molecule

3. Photoinduced diffraction grating

3.1 Absorption grating

3.2 Phase grating

4. Conclusions

Acknowledgments

References and links

Cited By

Figures (11)

Tables (1)

Equations (11)

Optics Express