Tunable slow and fast light device based on a carbon nanotube resonator

Jin-Jin Li; Ka-Di Zhu

doi:10.1364/OE.20.005840

1. Introduction

Over the past several years, researchers have been intrigued by the possibility of using nonlinear optical methods to realize unprecedented control of light pulse propagation through different kinds of material systems [1,2]. The interaction of light with matter can lead to extreme changes of light velocity [3] i.e., fast light, slow light and even stored light [4,5], which have been investigated in several systems, mainly using electromagnetically induced transparency (EIT) [6], coherent population oscillation (CPO) [7], and stimulated Brillouin scattering (SBS) [8, 9]. These phenomena have been investigated in different kinds of media ranging from atomic vapors [10], solid materials [11] to cavity optomechanical system [4]. The slow and stopped light achieved in several solid state systems have thrust EIT to the forefront of detailed study during the past two decades. EIT has been proved to be a powerful technique that can be used to eliminate the effect of a medium on a propagating beam of electromagnetic radiation, while retaining the large and desirable nonlinear optical properties associated with the resonant response of a medium [12, 13]. Recently, at low temperature, Painter et al. [4] have reported an optically tunable delay of 50 nanoseconds with near-unity optical transparency, and superluminal light with a 1.4 microsecond signal advance in cavity optomechanical system, using electromagnetically induce transparency.

Here, we show for the first time that possibilities to obtain slow and fast light are offered by a suspended carbon nanotube (CNT) resonator, whose potentialities for this purpose remain unexplored to date. Carbon nanotube resonator are particularly attractive for their unique features of strong nonlinear response, long vibration lifetime, easy fabrication and high quality factor [14, 15]. Consequently, CNT resonators enable ultrasensitive photonic sensors such as photovoltaics, nanotherapeutics, bioimaging, and superconducting device [16–20]. Furthermore, carbon nanotube is not very sensitive to external temperature and perturbation, which will enhance the transmission spectrum and reduce the noise. In the present work, we design a tunable slow and fast light device based on a suspended carbon nanotube resonator, in the presence of a strong pump laser and a weak signal laser. The signal light group velocity can be accelerated and decreased significantly by turning the pump laser on- and off-resonance with the exciton frequency in CNT, which correspond to the negative and positive dispersion, respectively, associating with vanished absorption. Detailed analysis also shows that the linewidth of signal light spectrum is determined by the vibration decay rate of carbon nanotube. The smaller the decay rate of carbon nanotube is, the narrower the signal spectral bandwidth is.

2. Theory

As schematically represented in Fig. 1, the suspended carbon nanotube resonator is under the radiation of a strong pump laser (with frequency ω_p) and a weak signal laser (with frequency ω_s). Recently, such two-laser technique has been experimentally investigated by several groups to study the cavity optomechanical system and demonstrate the achievement of slow light and on-chip storage of light pulses [4, 5, 21, 22].

Fig. 1 Schematic of a suspended carbon nanotube resonator with a localized exciton. This system is driven by a strong pump laser and probed by a weak signal laser. The inset describes the energy level of exciton while dressing with the vibration modes of carbon nanotube resonator.

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The structure of a semiconducting carbon nanotube can be viewed as a graphene rolled into a cylinder, where the center of mass of the exciton is localized via the spatial modulation of the Stark-shift induced by a static inhomogeneous electric field. The quantum confinement is induced by the inhomogeneity in the field component along the CNT axis E_//. In turn the normal component E_⊥ can be used to induce a tunable parametric coupling between the exciton and the flexural motion of the CNT [23]. This localized exciton is formed in the segment of nanotube between the doubly clamped suspensions, leading to a quantized energy spectrum in the longitudinal direction. The inset of Fig. 1 shows the energy levels of localized exciton in suspended nanotube resonator when dressing with the vibration modes of CNT, which can be modeled as a two-level structure consisting of the ground state |g〉 and the first excited state (single exciton) |ex〉. Such an exciton can be characterized by the pseudospin −1/2 operators S^± and S^z. Then the Hamiltonian of this localized two-level exciton can be described as H_ex = h̄ω_exS^z, where ω_ex is the frequency of exciton. Besides, here we consider the flexural branch which is expected to present resonance with a free spectral range larger than the optical linewidth of the zero phonon line of the excitonic transition and focus on laser excitations near resonant with the lowest-frequency flexural phonon mode [23, 24]. In this case, this lowest-energy resonance corresponds to the fundamental in-plane flexural mode with the frequency ω_n and the resonator is assumed to be characterized by sufficiently high quality factors. The eigenmode of CNT can be described by a quantum harmonic oscillator with b and b⁺ (the bosonic annihilation and creation operators with a quantum energy h̄ω_n). The vibration Hamiltonian of nanotube resonator is given by H_n = h̄ω_nb⁺b, where the vibration modes of CNT can be treated as phonon modes.

In the simultaneous presence of a strong pump field and a weak signal field, the Hamiltonian of the total system can be written as [23, 25, 26]

\begin{array}{l} H = H_{ex} + H_{n} + H_{ex - n} + H_{ex - o} \\ = \bar{h} ω_{ex} S^{z} + \bar{h} ω_{n} b^{+} b + \bar{h} ω_{n} η S^{z} (b^{+} + b) \\ - μ (S^{+} E_{p} e^{- i ω_{p} t} + S^{-} E_{p}^{*} e^{i ω_{p} t}) - μ (S^{+} E_{s} e^{- i ω_{s} t} + S^{-} E_{s}^{*} e^{i ω_{s} t}), \end{array}

where H_ex_–_n = h̄ω_nηS^z(b⁺ + b) represents the interaction between the nanotube resonator and the exciton [23, 27], and η is the coupling strength.

H_{ex - o} = - μ (S^{+} E_{p} e^{- i ω_{p} t} + S^{-} E_{p}^{*} e^{i ω_{p} t}) - μ (S^{+} E_{s} e^{- i ω_{s} t} + S^{-} E_{s}^{*} e^{i ω_{s} t})

describes the exciton coupling to the two optical fields, where E_p and E_s are slowly varying envelopes of the pump field and signal field, respectively, and μ is the electric dipole moment of the exciton. In a frame rotating at the pump field frequency ω_p, the total Hamiltonian of the coupled system reads as follows

H = \bar{h} Δ_{p} S^{z} + \bar{h} ω_{n} b^{+} b + \bar{h} ω_{n} η S^{z} (b^{+} + b) - \bar{h} (Ω S^{+} + Ω * S^{-}) - μ (S^{+} E_{s} e^{- i δ t} + S^{-} E_{s}^{*} e^{i δ t}),

where Δ_p = ω_ex – ω_p is the pump field-exciton detuning, δ = ω_s – ω_p is the signal-pump field detuning, and Ω = μE_p/h̄ is the Rabi frequency of the pump laser.

Applying the Heisenberg equations of motion for operators S^z, S⁻ and Q and introducing the corresponding damping and noise terms, we derive the quantum Langevin equations as follows [28, 29]:

\frac{d}{d t} S^{z} = - Γ_{1} (S^{z} + \frac{1}{2}) + i Ω S^{+} - i Ω * S^{-} + \frac{i μ E_{s} e^{- i δ t}}{\bar{h}} S^{+} - \frac{i μ E_{s}^{*} e^{i δ t}}{\bar{h}} S^{-},

\frac{d}{d t} S^{-} = - (i Δ_{p} + Γ_{2}) S^{-} - i ω_{n} η Q S^{-} - 2 i Ω S^{z} - \frac{2 i μ E_{s} e^{- i δ t}}{\bar{h}} S^{z} + {\hat{F}}_{e},

\frac{d^{2}}{d t^{2}} Q + \frac{1}{τ_{n}} \frac{d}{d t} Q + ω_{n}^{2} Q = - 2 ω_{n}^{2} η S^{z} + \hat{ξ},

where Q = b⁺ + b, Γ₁ and Γ₂ are the exciton spontaneous emission rate and dephasing rate, respectively. τ_n is the vibration lifetime of carbon nanotube [23, 30], F̂_e is the δ-correlated Langevin noise operator, which has zero mean 〈F̂_e〉 = 0 and obeys the correlation function

〈 {\hat{F}}_{e} (t) {\hat{F}}_{e}^{+} (t^{'}) 〉 \sim δ (t - t^{'})

.

The motion of carbon nanotube resonator is affected by thermal bath of Brownian and non-Morkovian process [28,31]. The quantum effects on the resonator are only observed in the limit of high quality factor, that obeys Q = ω_n/γ_n ≫ 1. The Brownian noise operator can be modeled as Markovian with the decay rate γ_n (γ_n = 1/τ_n) of the resonator mode. Therefore, the Brownian stochastic force has zero mean value 〈ξ̂〉 = 0 that can be characterized as [31]

〈 {\hat{ξ}}^{+} (t) \hat{ξ} (t^{'}) 〉 = \frac{γ_{n}}{ω_{n}} \int \frac{d ω}{2 π} ω e^{- i ω (t - t^{'})} [1 + coth (\frac{\bar{h} ω}{2 k_{B} T})] .

Following standard methods from quantum optics, we derive the steady-state solution to Eqs. (3)–(5) by setting all the time derivatives to zero. They are given by

S_{0}^{-} = \frac{- 2 Ω S_{0}^{z}}{(Δ_{p} + ω_{n} η Q_{0}) - i Γ_{2}}, Q_{0} = - 2 η S_{0}^{z},

where

S_{0}^{z}

is determined by Eq. (16) (see below). To go beyond weak coupling, we can always rewrite each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value as follows:

S^{-} = S_{0}^{-} + δ S^{-}, S^{z} = S_{0}^{z} + δ S^{z}, Q = Q_{0} + δ Q .

Inserting these equations into the Langevin equations Eqs. (3)–(5), one can safely neglect the nonlinear term δQδS⁻. Since the optical drives are weak, but classical coherent fields, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms [21]. Then the linearized Langevin equations can be written as:

〈 δ {\dot{S}}^{z} 〉 = - Γ_{1} 〈 δ S^{z} 〉 + i Ω 〈 δ (S^{-}) * 〉 - i Ω * 〈 δ S^{-} 〉 + \frac{i μ E_{s} e^{- i δ t}}{\bar{h}} 〈 δ (S^{-}) * 〉 - \frac{i μ E_{s}^{*} e^{i δ t}}{\bar{h}} 〈 δ S^{-} 〉,

〈 δ {\dot{S}}^{-} 〉 = - (i Δ_{p} + Γ_{2}) 〈 δ S^{-} 〉 - i ω_{n} η (〈 δ S^{-} 〉 Q_{0} + S_{0}^{-} 〈 δ Q 〉) - 2 i Ω 〈 δ S^{z} 〉 - \frac{2 i μ E_{s} e^{- i δ t}}{\bar{h}} 〈 δ S^{z} 〉,

〈 \ddot{δ} Q 〉 + \frac{1}{τ_{n}} 〈 \dot{δ} Q 〉 + ω_{n}^{2} 〈 δ Q 〉 = - 2 ω_{n}^{2} η 〈 δ S^{z} 〉 .

In order to solve Eqs. (9)–(11), we make the ansatz [21, 25] 〈δS⁻〉 = S₊e^−iδt + S₋e^iδt, $〈 δ S^{z} 〉 = S_{+}^{z} e^{- i δ t} + S_{_}^{z} e^{i δ t}$ and 〈δQ〉 = Q₊e^−iδt + Q₋e^iδt. Upon substituting the approximation to Eqs. (9)–(11) and working to the lowest order in E_s, but to all orders in E_p, we finally obtain the linear optical susceptibility S⁺ in the steady state as the following solution

χ^{(1)} {(ω_{s})}_{eff} = ρ μ \frac{S_{+}}{ɛ_{0} E_{s}} = \frac{ρ μ^{2}}{ɛ_{0} \bar{h} Γ_{2}} χ^{(1)} (ω_{s}) = Σ_{1} χ^{(1)} (ω_{s}),

where Σ₁ = ρμ²/ε₀h̄Γ₂, ρ is the number density of CNT, which during the measurement the identical resonator arrays are needed because of the weak measurable signal and weak interaction with light produced by a single nanotube resonator. For a better experimental observation, ρ should be about 10⁹m⁻³ [32]. ε₀ is the dielectric constant of vacuum. In Eq. (12), the imaginary part and real part of χ⁽¹⁾ (ω_s) correspond to the absorption and dispersion of the signal field, respectively [25]. In this case, all the interacting elements are considered during the physical treatment, except for the light-phonons coupling which is induced indirectly through the light-exciton interaction via deformation potential coupling. The dimensionless linear optical susceptibility is given by

\begin{array}{l} χ^{(1)} (ω_{s}) = \frac{w_{0}}{f (δ_{0})} \times \\ {2 e_{1} Ω_{R}^{2} (e_{1} + δ_{0}) (e_{2} + ω_{n 0} η^{2} w_{0} ζ) - e_{1} e_{2} [2 Ω_{R}^{2} (e_{1} + ω_{n 0} η^{2} w_{0} ζ) - e_{1} (2 i + δ_{0}) (e_{1} + δ_{0})]}, \end{array}

where e₁ = Δ_p₀ – ω_n₀η²w₀ + i, e₂ = Δ_p₀ – ω_n₀η²w₀ – i, and

w_{0} = 2 S_{0}^{z}

, Γ₁ = 2Γ₂, ω_n₀ = ω_n/Γ₂, γ_n₀ = γ_n/Γ₂, Ω_R = Ω/Γ₂, δ₀ = δ/Γ₂, and Δ_p₀ = Δ_p/Γ₂.

The auxiliary function ζ(δ₀) and function f (δ₀) are given by

ζ (δ_{0}) = \frac{ω_{n_{0}}^{2}}{ω_{n_{0}}^{2} - i δ_{0} γ_{n 0} - δ_{0}^{2}},

\begin{array}{l} f (δ_{0}) = e_{1} e_{2} (e_{1} - δ_{0}) [2 Ω_{R}^{2} (e_{1} + ω_{n 0} η^{2} w_{0} ζ) - e_{1} (2 i + δ_{0}) (e_{1} + δ_{0})] \\ - 2 e_{1}^{2} Ω_{R}^{2} (e_{1} + δ_{0}) (e_{2} + ω_{n 0} η^{2} w_{0} ζ) . \end{array}

The population inversion w₀ of the exciton is determined by the following equation

(w_{0} + 1) [{(Δ_{p 0} - ω_{n 0} η^{2} w_{0})}^{2} + 1] + 2 Ω_{R}^{2} w_{0} = 0.

In terms of this model, we can determine the light group velocity as [33, 34]

v_{g} = \frac{c}{n + ω_{s} (d n / d ω_{s})},

where

n \approx 1 + 2 π χ_{eff}^{(1)}

, and then

\frac{c}{v_{g}} = 1 + 2 π R e χ_{eff}^{(1)} {(ω_{s})}_{ω_{s} = ω_{ex}} + 2 π ω_{s} R e {(\frac{d χ_{eff}^{(1)}}{d ω_{s}})}_{ω_{s} = ω_{ex} .}

It is clear from this expression for v_g that when Reχ (ω_s)_{ω_s=ω_ex} is zero and the dispersion is steeply positive or negative, the group velocity is significantly reduced or increased, and then we define the group velocity index n_g as

n_{g} = \frac{c}{v_{g}} - 1 = \frac{c - v_{g}}{v_{g}} = \frac{2 π ω_{ex} ρ μ^{2}}{\bar{h} Γ_{2}} R e {(\frac{d χ^{(1)} (ω_{s})}{d ω_{s}})}_{ω_{s} = ω_{e x}} = Γ_{2} Σ R e {(\frac{d χ^{(1)} (ω_{s})}{d ω_{s}})}_{ω_{s} = ω_{ex}},

where

Σ = 2 π ω_{e x} ρ μ^{2} / ɛ_{0} \bar{h} Γ_{2}^{2}

. One can observe the slow light when n_g > 0, and the superluminal light when n_g < 0 [35].

3. Numerical results and discussion

In order to show the tunable slow and fast light clearly, we choose the realistic suspended carbon nanotube resonator with an ambient temperature 4.2K, for (ω_n/2π,Γ₂,γ_n,η ) = (725MHz,310MHz,0.8MHz,0.17) [23, 26]. The quality factor of carbon nanotube is Q = 900.

Firstly, we interpret the basic principle of the slow and fast light in a carbon nanotube resonator, under the radiation of a strong pump laser and a weak signal laser. According to Eq. (12), we depict the resonance absorption spectrum of carbon nanotube as a function of signal-pump beam detuning δ with Δ_p₀ = 0 (as shown in Fig. 2). Since the original energy levels of the localized exciton have been dressed by the vibration modes of the nanotube (here the vibration modes is the same as the phonon modes), the uncoupled energy levels (|ex〉 and |g〉) split into dressed states |ex,n〉 and |g,n〉, which is shown in Fig. 2(b) (|n〉 denotes the number state of the resonance mode). The curve shown in the Fig. 2(a) displays three prominent features - the middle part consisting of three weak peaks and two sharp peaks at the both sides. The middle part is analogy with conventional atomic two-level systems [25], which shows the origin of nanotube vibration induced stimulated Rayleigh resonance. Here the electrons make a transition from the lowest dressed level |g,n〉 to the dressed level |ex,n〉, which is signified by the transition 2. Besides, it’s worth notice that the two sharp peaks at the both sides in absorption spectrum are totally different from those in atomic system [25]. These two sharp peaks represent the resonance amplification and absorption of the carbon nanotube, which can be interpreted by the transition 1 and 3. The left peak corresponds to the amplification of the signal laser, where electrons making a transition from the lowest dressed level |g,n〉 to the highest dressed level |ex,n + 1〉 by the simultaneous absorption of two pump laser photons and emission of a photon at ω_p – ω_n. This process can amplify a wave at δ= −ω_n. Otherwise, the right peak is an absorption process, which corresponds to the usual absorption resonance as modified by the ac-Stark effect. The underlying physical mechanism for this phenomenon can also be understood as follows. The simultaneous presence of the pump and signal fields generates a beat wave oscillating at the beat frequency δ= ω_s – ω_p to drive the CNT via the localized exciton. If the beat frequency δ is close to the resonance frequency ω_n, the CNT starts to oscillate coherently, which will result in Stokes (ω_s = ω_p – ω_n) and anti-Stokes (ω_as = ω_p + ω_n) scattering of light from the pump field via the localized exciton. For the near-resonant signal laser, the signal field will interfere with the Stokes field and the anti-Stokes field, respectively. As a result the signal spectrum can be modified significantly. In the picture of the dressed states, the system is similar to the conventional three-level system in EIT. Here coupling to phonons seems to provide the exciton with additional energy levels to realize EIT phenomena. Therefore in our structure one can obtain the slow output light without absorption only by simply adjusting the pump-exciton detuning to the vibrational frequency of the nanotube resonator. In this case, we conclude that the coupling between exciton and the vibration of carbon nanotube plays an important role in the implement of these specific features.

Fig. 2 (a) The absorption spectrum of a signal field as a function of signal-pump detuning for the case $Ω_{R}^{2} = 2$ ,ω_n₀ = 6, Δ_p₀ = 0, γ_n₀ = 0.003, and η = 0.17. (b) The energy levels of localized exciton while dressing with the vibrational modes of nanotube resonator. The transition 1,2,3 correspond to the characteristic parts shown in (a), respectively.

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Figure 3 displays a tunable slow and fast light device by fixing the pump laser off- and on-resonant with exciton frequency of CNT resonator. The left part shows the fast light spectrum with Δ_p = 0, while the right part exhibits the slow light situation with Δ_p = ω_n. Figure 3(a) shows the imaginary part and real part of linear optical susceptibility while fixing Δ_p = 0, which correspond to the absorption and dispersion of the signal light, respectively. From this figure, we find that the the imaginary part has a zero absorption and the real part has a negative steep slope at Δ_s = 0, which signifies the potential of superluminal light achievement. We next plot the group velocity index of signal laser n_g (in the unit of Σ) as a function of the Rabi frequency Ω², as shown in Fig. 3(b). Figure 3(b) indicates that the output signal pulse can be about 10 times faster than input signal pulse in vacuum simply via tuning the pump laser on the resonant with exciton frequency in CNT resonator(Δ_p = 0). Furthermore, in the case of pump-off resonant (Δ_p = ω_n), the imaginary part and real part of linear optical susceptibility exhibit zero absorption and positive steep slope at Δ_s = 0 in Fig. 3(c), which denotes the possibility of ultraslow light realization. Figure 3(d) exhibits the slow light curve, where the most slow-light index can be produced in CNT resonator device as 180 as Ω² = 0.02(GHz)². That is, the output signal pulse will be 180 times slower than the input light with a single CNT resonator. The total magnitude of slow light and fast light is determined by the number density of CNT resonator. The physical origin of this result is the coupling between exciton and CNT vibration, which makes quantum interference between the CNT and the two optical fields via the exciton as δ = ω_n. Such nonlinear process is analogy with electromagnetically induced transparency (EIT), which has been discussed in Fig. 2(b).

Fig. 3 A tunable device of slow and fast light using a carbon nanotube resonator. (a) The dimensionless imaginary part and real part of the linear optical susceptibility (in units of Σ₁) as a function of the signal-exciton detuning for Δ_p = 0; (b) The group velocity index n_g of superluminal light (in units of Σ) as a function of pump Rabi frequency Ω²; (c) The same plots with (a) except Δ_p = ω_n; (d) The group velocity index n_g(= c/v_g) of slow light (in units of Σ) as a function of the pump Rabi frequency Ω². The other parameters used in plot(a)–(d) are Ω² = 0.1(GHz)², ω_n = 0.725GHz, γ_n = 0.8MHz, and η = 0.17.

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In the picture of the dressed states shown in the Fig. 2(b), the condition Δ_p = ω_n just corresponds to that the pump field couples to the optical transition via the Stokes process and the system becomes fully transparent to the signal beam. In this case, the system is similar to the conventional three-level systems in EIT [36]. Here coupling to a CNT resonator seems to provide the exciton with additional energy levels to realize EIT phenomena. Therefore in our structure one can obtain the fast or slow output light without absorption only by simply adjusting the pump laser on- or off-resonant with exciton frequency.

Figure 4 displays the the imaginary part of χ⁽¹⁾ as a function of Δ_s for three different decay rates of γ_n. The inset of Fig. 4 shows the amplification of the most remarkable region of transparency. From this figure, we can demonstrate that the width of the signal spectrum increases as the decay rate γ_n increases. Therefore the shorter the CNT resonator decay is, the narrower of the signal spectrum width is. When the decay rate of the resonator is 0.1GHz, the hole width in the spectrum becomes flat as shown in the inset. As a result, the CNT resonator with small decay rate is beneficial to the transparency window. Due to the hight quality factor and short decay rate of CNT resonator, the slow light and fast light effect performed in CNT is obviously better than that in other quantum systems such as quantum wells and quantum dots.

Fig. 4 The absorption spectrum of a signal field as a function of the detuning Δ_s with three different decay rates of CNT resonator. The other parameters used are Ω² = 0.1(GHz)², ω_n = 0.725GHz, Δ_p = 0.725GHz, and η = 0.17. The inset shows the amplification of the most remarkable region of transparency.

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Furthermore, there is an interesting feature we should notice, that the abscissa of the two sharp peak shown in Fig. 3(a) just corresponds to the vibrational frequency of CNT resonator (Δ_s = ω_n = 725MHz). That is, fixing the pump laser on the exciton frequency Δ_p = 0 and scanning the exciton frequency using another signal laser, one can obtain the vibrational frequency of CNT resonator at signal absorption or dispersion spectrum efficiently. This is a precise and easy method to get the vibrational frequency of CNT resonator in all optical domain.

4. Conclusion

In conclusion, we have proposed a theoretical model for a tunable slow and fast light device based on a suspended carbon nanotube resonator, in the presence of a strong pump laser and a weak signal laser. Pump laser on- or off-resonant with exciton frequency in CNT resonator results in the fast or slow output signal light simultaneously, together with zero absorption. Absorption spectrum displays that the width of such slow and fast light is decided by the decay rate of CNT resonator. Furthermore, our model can also be used to measure the vibrational frequency of CNT resonator either in absorption spectrum or dispersion spectrum. We believe that such a tunable slow and fast light device may be have the potential applications in optical networks and engineering in nanometer scale.

Acknowledgments

The authors thanks for National Natural Science Foundation of China (No. 10774101 and No. 10974133), the National Ministry of Education Program for Ph.D.

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Tunable slow and fast light device based on a carbon nanotube resonator

Abstract

1. Introduction

2. Theory

3. Numerical results and discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (19)

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