Tuning single-walled aligned carbon nanotubes for optimal terahertz pulse generation through optical rectification of ultrashort laser pulses

Mohammad Hassani; Fazel Jahangiri

doi:10.1364/OE.442168

1. Introduction

Demand for efficient, stable, compact, high power, broadband, and tunable terahertz sources is still growing due to their fast-increasing applications. Terahertz radiation has become a useful tool in a wide span from fundamental scientific researches to advanced sensing, probing, time-resolved imaging and spectroscopy applications [1–4]. Among various methods for the generation of single-cycle sub-picosecond high field terahertz pulses [5–12], optical rectification of ultrashort laser pulses in nonlinear media and photoconductive switching of semiconductors have generally attracted more attention [13,14]. The conversion efficiency and the output power of the terahertz pulses are mainly determined by the properties of the interacting media. For terahertz generation using optical rectification, these factors include the length and the geometry, the effective nonlinear coefficient, phase- and velocity-matching conditions, and dispersion properties over the optical and terahertz frequencies. Besides the commonly used materials such as ZnTe, GaP, GaAs, LiNbO3 and DAST [15–18], research is being carried out for introducing alternatives with more suitable features for terahertz generation.

The potential of carbon nanotubes (CNTs) with exceptional and diverse optoelectronic properties has been the subject of many theoretical and experimental studies [19–23]. These materials have been employed as alternative nanoscale terahertz emitters, modulators, polarizers, and detectors [24–26]. It has been theoretically demonstrated that applying a dc bias voltage to CNTs could result in spontaneous terahertz emission [27,28]. Terahertz amplification in the analogy with the cascade lasers [29], transition radiation in the presence of the magnetic field [30], and Cherenkov type lasing [31] in CNTs have been also investigated. Moreover, CNTs have been proposed as terahertz nano-antenna with a geometric structure that could affect the radiation properties [32–34]. Single-walled carbon nanotubes (SWCNTs) have been also proposed as efficient terahertz photoconductive switches [35–38] based on their attractive properties including high mobility, high thermal conductance, and high absorption in the infrared region [19,33,35,39]. It has been shown that SWCNTs based photoconduction can enhance the optical to terahertz conversion efficiency with two orders of magnitudes compared to conventional materials such as LT-GaAs [35]. Titova et al. have experimentally observed the emission of broadband terahertz pulses from highly aligned SWCNTs irradiated by ultrashort laser pulses without applying voltage bias [40]. They suggested photocurrent surge induction in SWCNTs along the nanotube axis as the origin of terahertz radiation, which should be confirmed by more detailed studies involving the effects of the structural parameters of different types of SWCNTs.

On the other hand, huge second and third-order nonlinear susceptibilities, much larger than those of GaAs [20] and fused silica [41,42], respectively, have been predicted for SWCNTs. Despite that, the nonlinear optical excitation of SWCNTs as a promising scheme of terahertz radiation generation has been rarely investigated. Huang et al. have mentioned the second-order nonlinear optical effects as a probable contribution to the mechanism of terahertz emission in CNTs. They have experimentally studied terahertz wave emission from vertically aligned CNTs in a reflection configuration and observed that the amplitude of terahertz waves quadratically depends on that of the pump pulse [43]. Such a dependence could be attributed to a second order nonlinear mechanism. Moreover, the optical rectification hypothesis has been presumably tested by measuring a dc current in carbon nanotubes clusters and a nanoscale ultrafast nonlinear optical current created by the laser field has been proposed as the mechanism of terahertz enhancing [44]. Parashar et al. have also modeled the radiation mechanism by calculating the free electron current density induced by the ponderomotive force in a plasma-like medium [45,46]. They have hypothetically attributed the terahertz emission from the interaction of a modulated continuous laser field with CNTs to the optical rectification. However, it should be confirmed by taking into account a nonlinear optical model describing the induced polarization, nonlinear susceptibility, and conditions required for such interaction. Thus, the generation mechanism of terahertz radiation from CNTs has remained ambiguous [43]. Further elaborated results are required regarding the nonlinear optical excitation of CNTs with various geometry, structural alignment, and chirality as well as the dispersion and absorption to advance the understanding of the related THz emission mechanism.

In this paper, we present a comprehensive model to explain the terahertz generation via optical rectification in CNTs irradiated by ultrashort laser pulses. We pursue our investigation on (i) explaining the terahertz generation process, (ii) considering the propagation effects and providing coherent amplification conditions, and (iii) taking into account the dispersive and structural effects of SWCNTs. For such purpose, we initiate by extracting the optical rectification susceptibility tensor in chiral SWCNTs. Then, the nonlinear polarization responsible for optical rectification is calculated and introduced to the wave equation as a radiation source. Moreover, the effect of group velocity dispersion on the propagation of the optical pulse and the generated terahertz pulse is discussed and the velocity matching conditions required for coherent amplification are obtained. In addition, the effects of SWCNT’s structural parameters including the chirality, alignment, and filling factor on their dispersive behavior are taken into account.

Our model could anticipate the optimal conditions for terahertz radiation generation in all types of chiral SWCNTs, based on their structural parameters along with the dispersion and absorption over both pump and terahertz pulse frequency regions. To address the optimum coherence amplification of terahertz pulse, we explore within the available ranges of the pulse duration and wavelengths according to the specifications of practically available laser systems. This set of variables provide the feasibility of tunable terahertz emission from SWCNTs. Our results reveal that SWCNTs with higher alignment, and lower filling factor at chirality (6,4) irradiated by ultrashort laser pulse with the wavelength of 1550 nm could provide the conditions for maximum terahertz radiation generation.

2. Theory and design

2.1 Terahertz generation process

Optical rectification is expected to contribute to terahertz generation only in noncentrosymmetric nonlinear materials that exhibit second-order nonlinear optical behavior. Among different configurations of SWCNTs that are schematically compared in Fig. 1, we focus on those with chiral geometry [47]. To calculate the features of terahertz radiation generated by optical rectification in SWCNTs, we first extract the time-dependent nonlinear second-order polarization induced by the laser fields. In general, nonlinear second-order polarization induced at frequency $\omega$ by laser fields with amplitudes of $E_{j,k}$ and frequencies of $\omega _{n,m}$ is obtained as

(1)$$P_{i}^{(2)}(\Omega)=\epsilon_{0} \sum_{j k}\sum_{(nm)} \chi_{i j k}^{(2)}\left(\Omega; \omega_{n}, \omega_{m}\right) E_{j}\left(\omega_{n}\right)^{*} E_{k}\left(\omega_{m}\right),$$

where, $\epsilon _{0}$ is the electric permittivity of the vacuum, $\chi _{ijk}^{(2)}$ denotes the second-order nonlinear susceptibility tensor, $ijk$ indices represent the Cartesian components of the fields and $(nm)$ denotes the varying frequency components involved in the interaction so that $\Omega = \omega _{n}+\omega _{m}$ is kept fixed at the desired frequency [48]. For optical rectification ($\Omega = 0$) we have $\omega _{n}=-\omega _{m}=\omega$ and thus Eq. (1) can be written as

(2)$$\begin{gathered} P_{i}^{(2)}(0)=P_{i}^{O R} \\ =\epsilon_{0} \sum_{j k}\left(\chi_{i j k}^{(2)}(0 ; \omega,-\omega) E_{j}(\omega) E_{k}(\omega)^{*}+\chi_{i j k}^{(2)}(0 ;-\omega, \omega) E_{j}(\omega)^{*} E_{k}(\omega)\right). \end{gathered}$$

Fig. 1. Schematic comparison of different configurations of SWCNTs and their corresponding chiral vectors. This vector is defined as $\vec {C}=p \vec {a}_{1}+q \vec {a}_{2}$ using the vectors $\vec {a}_{1}$ and $\vec {a}_{2}$ for the hexagonal graphene lattice. $p$ and $q$ are integer positive numbers, which determine the chirality angle as $\theta =\cos ^{-1}((2 p+q) / 2 \sqrt {p^{2}+ pq+q^{2}})$.

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Considering the SWCNTs with intrinsic permutation symmetry, Eq. (2) can be simplified as

(3)$$P_{i}^{O R}(0)=2\epsilon_{0} \sum_{j k}\left(\chi_{i j k}^{(2)}(0 ;-\omega, \omega) E_{j}(\omega)^{*} E_{k}(\omega)\right),$$

in which $\chi _{i j k}^{(2)}(0 ;-\omega, \omega )$ denotes the optical rectification susceptibility tensor of SWCNT that needs to be determined.

One may consider the Lorentz model as a straightforward approach for obtaining the optical rectification susceptibility [48] tensor as

(4)$$\chi^{(2)}(0 ;-\omega, \omega)=\frac{\chi^{(1)}(0) \chi^{(1)}(\omega)^{*}}{\chi^{(1)}(2 \omega) \chi^{(1)}(\omega)} \chi^{(2)}(2 \omega ; \omega, \omega),$$

where $\chi ^{(1)}$ and $\chi ^{(2)}(2 \omega ; \omega, \omega )$ are associated with the available values of the linear refractive index [49] and the second harmonic susceptibility [20,50,51] of SWCNTs, respectively. However, this model yields only the scalar values for the susceptibility, instead of Cartesian components, which are not also valid at near-resonant frequencies [48].

Here we propose another approach for extracting all 27 elements of the optical rectification susceptibility tensor in materials with full symmetry. The susceptibility tensor of optical rectification can be written as

(5)$$\chi^{(2)}(0 ;-\omega, \omega)= \chi^{(2)}(\omega ; \omega, 0),$$

where $\chi ^{(2)}(\omega ;\omega,0)$ denotes the susceptibility tensor for Pockels effect, which has been recently obtained for chiral carbon nanotubes [39]. Therefore, the elements of optical rectification tensor could be extracted as schematically shown in Fig. 2.

Fig. 2. Schematic illustration for extraction of 27 elements of optical rectification susceptibility tensor according to that of Pockels effect for SWCNTs oriented along x-axis.

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By ensuring that the frequencies of the interacting fields are kept well away from the resonant frequencies of the medium, the full permutation symmetry and consequently the simplified contracted matrix notation [48] can be applied to reduce the susceptibility tensor to a matrix with a sole independent element as

(6)$$[d]_{CNT}^{OR}=\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \beta & 0 \\ 0 & 0 & 0 & 0 & 0 & -\beta \end{array}\right].$$

Therefore, the nonlinear polarization of optical rectification in SWCNTs could be calculated as

(7)$$\left[\begin{array}{l} P_{x}^{(2)} \\ P_{y}^{(2)} \\ P_{z}^{(2)} \end{array}\right]_{C N T}^{OR}=4 \epsilon_{0}[d]_{C N T}^{O R}\left[\begin{array}{c} \left|E_{x}(\omega)\right|^{2} \\ \left|E_{y}(\omega)\right|^{2} \\ \left|E_{z}(\omega)\right|^{2} \\ E_{y}(\omega) E_{z}^{*}(\omega)+E_{z}(\omega) E_{y}^{*}(\omega) \\ E_{x}(\omega) E_{z}^{*}(\omega)+E_{z}(\omega) E_{x}^{*}(\omega) \\ E_{x}(\omega) E_{y}^{*}(\omega)+E_{y}(\omega) E_{x}^{*}(\omega) \end{array}\right].$$

By considering the electric field components of the incident light in polar coordinates (Fig. 3(a)) as

(8)$$E=\vec{E}^{o p t}=\left[\begin{array}{l} E_{x}^{o p t} \\ E_{y}^{o p t} \\ E_{z}^{o p t} \end{array}\right]=E^{o p t}\left[\begin{array}{c} \sin \theta \cos \emptyset \\ \sin \theta \sin \emptyset \\ \cos \theta \end{array}\right]$$

and then substituting in Eq. (6), Eq. (7) can be written as

(9)$$\begin{gathered} {\left[\begin{array}{c} P_{x}^{(2)} \\ P_{y}^{(2)} \\ P_{z}^{(2)} \end{array}\right]_{C N T}^{O R}=4 \epsilon_{0} \beta\left|E^{opt}(\omega)\right|^{2}\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & -1 \end{array}\right]\left[\begin{array}{c} \sin ^{2} \theta \cos ^{2} \varphi \\ \sin ^{2} \theta \sin ^{2} \varphi \\ \cos ^{2} \theta \\ 2 \sin \theta \cos \theta \sin \varphi \\ 2 \sin \theta \cos \theta \cos \varphi \\ 2 \sin ^{2} \theta \sin \varphi \cos \varphi \end{array}\right]} \\ \qquad=8 \epsilon_{0} \beta\left|E^{opt}(\omega)\right|^{2} \sin \theta\left[\begin{array}{cc} 0 \\ -\cos \theta \cos \varphi \\ \sin \theta \sin \varphi \cos \varphi \end{array}\right]. \end{gathered}$$

This equation reveals that the nonlinear polarization and consequently terahertz generation via optical rectification would be zero when the optical field polarization is oriented in a direction parallel ($\theta =90^\circ,\varphi =0^\circ$) or transverse ($\theta =0^\circ,\varphi =90^\circ$) to the SWCNT’s axis. It is also clear from this equation that the direction of the induced nonlinear polarization and consequently terahertz field’s polarization would be transverse to the x-axis in the y-z plane without x-component. Therefore, we choose for simplicity the conditions at which $\theta =45^\circ$ and $\varphi =0^\circ$, as shown in Fig. 3(b). In this case, Eq. (9) could be written as

(10)$$P_{C N T}^{O R}=\left[P_{y}^{(2)}\right]_{C N T}^{O R}=4 \epsilon_{0} \beta\left(E^{o p t}\right)^{2}$$

Fig. 3. The electric field vector of the incident optical pulse in polar coordinates (a) and in respect with the nonlinear polarization (b) induced through the interaction with SWCNTs.

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2.2 Propagation and coherent amplification

By obtaining the nonlinear polarization responsible for optical rectification in SWCNTs, now we introduce it as a radiation source to the general wave equation to extract the electric field vector of terahertz radiation ${E}_{THz}$. By assuming that the medium is non-magnetized, free of the electric charge and current densities, and with a length small enough (submillimeter) to be able to ignore the nonlinear propagation effects for the low-intensity generated terahertz field, the wave equation can be written as

(11)$$\vec{\nabla}\left(\vec{\nabla} . \vec{E}_{T H z}(\vec{r}, t)\right)-\nabla^{2} \vec{E}_{T H z}(\boldsymbol{r}, t)+\frac{1}{c^{2}} \frac{\delta^{2}}{\delta t^{2}} \vec{E}_{T H z}(\vec{r}, t)={-}\frac{1}{\epsilon_{0} c^{2}} \frac{\delta^{2}}{\delta t^{2}} \vec{P}(\vec{r},t),$$

where, $c$ is the speed of light in the vacuum and $\vec {P}(\vec {r},t)$ denotes the time-dependent induced polarization. Here we also consider the laser beam to be collimated with a definite spatial profile and plane phase front to ensure that the direction of optical electric field is approximately fixed inside the interaction region. Under such conditions, which could be practically provided in the Rayleigh range of the beam, the spatial distribution of the generated terahertz field would be a replica of that of the laser field and the results for a specific transverse position could be extended to all points of the cross-section. Therefore, Eq. (10) could be more simplified without transverse spatial dependence as

(12)$$\frac{\delta^{2}}{\delta z^{2}} E_{T H z}(z, t)-\frac{1}{\epsilon_{0} c^{2}} \frac{\delta^{2}}{\delta t^{2}} D_{T H z}(z, t)=\frac{1}{\epsilon_{0} c^{2}} \frac{\delta^{2}}{\delta t^{2}} P_{C N T}^{O R}(z, t),$$

where

(13)$$D_{THz}(z,t)=\epsilon_{0} E_{T H z}(z, t)+\epsilon_{0} \int_{-\infty}^{+\infty} \chi^{(1)}\left(z, t-t^{\prime}\right) \cdot E_{T H z}\left(z, t^{\prime}\right) d t^{\prime}$$

describes the electric displacement field of the interaction medium. The wave equation could be solved in the frequency domain as

(14)$$\frac{\delta^{2}}{\delta z^{2}} E_{T H z}(z, \omega)+\frac{\omega^{2}}{\epsilon_{0} c^{2}} D_{T H z}(z, \omega)={-}\frac{\omega^{2}}{\epsilon_{0} c^{2}} P_{C N T}^{O R}(z, \omega),$$

where, $D_{T H z}(z, \omega )$ is Fourier transform of the electric displacement field in an isotropic homogeneous medium. Although SWCNTs are not isotropic, their electric permittivity matrix is diagonal with two identical elements, and because the nonlinear polarization is orthogonal to the SWCNT’s axis, $D_{T H z}(z, \omega )=\epsilon _{\omega } E_{T H z}(z, \omega )$ can be used in a scalar from.

A general solution of the Eq. (13) in the frequency domain would be as

(15)$$E_{T H z}(z, \omega)=A\left(z, \omega-\omega_{0}\right) e^{i k_{0} z}+A^{*}\left(z, \omega+\omega_{0}\right) e^{{-}i k_{0} z},$$

which could be expressed in the time domain as

(16)$$E_{T H z}(z, t)=A(z, t) e^{{-}i\left(\omega_{0} t-k_{0} z\right)}+c.c.,$$

where $\omega _{0}$ and $k_{0}$ denote the central frequency and wavenumber of the generated terahertz wave, respectively. By inserting Eq. (14) in Eq. (13) and considering the homogeneity of this equation, the wave equation could be written as

(17)$$\left(\frac{\delta^{2}}{\delta z^{2}} \pm 2 i k_{0} \frac{\partial}{\partial z}+k(\omega)^{2}-k_{0}^{2}\right)\left\{\begin{array}{c} A\left(z, \omega-\omega_{0}\right) \\ A^{*}\left(z, \omega+\omega_{0}\right) \end{array}\right\}={-}\frac{\omega^{2}}{\epsilon_{0} c^{2}} P_{C N T}^{O R}(z, \omega) e^{{\mp} i k_{0} z}.$$

By solving the above equation we have:

(18)$$E_{T H z}(z, \omega)=c_{1}(z, \omega) e^{i k(\omega) z}+c_{2}(z, \omega) e^{{-}i k(\omega) z} \\ +c_{1}^{\prime}(z, \omega) e^{{-}i k(\omega) z}+c_{2}^{\prime}(z, \omega) e^{i k(\omega) z},$$

where

(19)$$\begin{gathered} c_{1}(z, \omega)=\frac{i \omega^{2}}{2 \epsilon_{0} c^{2} k(\omega)} \int_{0}^{z} e^{{-}i k(\omega) z} P_{C N T}^{O R}(z, \omega) d z, \\ c_{2}(z, \omega)=\frac{-i \omega^{2}}{2 \epsilon_{0} c^{2} k(\omega)} \int_{0}^{z} e^{i k(\omega) z} P_{C N T}^{O R}(z, \omega) d z, \\ c_{1}^{\prime}(z, \omega)=c_{2}(z, \omega), \\ c_{2}^{\prime}(z, \omega)=c_{1}(z, \omega). \end{gathered}$$

Now we introduce an optical pulse to the equations with a Gaussian temporal profile as

(20)$$E_{opt}(z, t)=E_{0} e^{{-}a(z)\left(t-\frac{z}{v_{g r}\left(\omega_{0}\right)}\right)^{2}} e^{{-}i \omega_{0}\left(t-\frac{z}{v_{p h}\left(\omega_{0}\right)}\right)},$$

where, $E_{0}$ is the amplitude, $v_{gr}$ and $v_{ph}$ denote the group and phase velocities of the optical pulse which are defined as $v_{ph}(\omega _{0})=\frac {\omega _{0}}{k_{0}}$ and $v_{gr}(\omega _{0})=\frac {\partial \omega }{\partial k}|_{\omega =\omega _{0}}$, respectively. Moreover, $a(z)=a_{R}(z)+ia_{I}(z)$ in which $a_{R}(z)$ and $a_{I}(z)$ are respectively defined as [52]

(21)$$a_{R}(z)=\frac{a_{0}}{1+4 a_{0}^{2}\left(\left.\frac{d^{2} k}{d \omega^{2}}\right|_{\omega=\omega_{0}}\right)^{2} z^{2}}$$

and

(22)$$a_{I}(z)=\frac{\left.2 a_{0}^{2} \frac{d^{2} k}{d \omega^{2}}\right|_{\omega=\omega_{0}} z}{1+4 a_{0}^{2}\left(\frac{d^{2} k}{d \omega^{2}} \mid \omega=\omega_{0}\right)^{2} z^{2}},$$

where, $a_{0}=\frac {4ln(2)}{\tau _{p}^{2}}$, in which $\tau _{p}$ is the temporal duration. Thus the nonlinear polarization responsible for optical rectification is obtained by using Eq. (9) as

(23)$$P_{C N T}^{O R}(z, t)=4 \epsilon_{0} \beta\left|E_{0}\right|^{2} e^{{-}2 a_{R}(z)\left(t-\frac{z}{v_{g r}\left(\omega_{0}\right)}\right)^{2}}$$

in the time domain and

(24)$$P_{C N T}^{O R}(z, \omega)=4 \epsilon_{0} \beta\left|E_{0}\right|^{2} e^{i \frac{z}{v g r} \omega} \frac{1}{\sqrt{4 a_{R}(z)}} e^{-\frac{\omega^{2}}{8 a_{R}(z)}}$$

in the frequency domain. By substituting Eq. (23) in Eq. (18) and supplying the parameters responsible for various initial conditions, the amplitude of the generated terahertz radiation could be numerically calculated. Since this process would yield the terahertz emission at each specific slab of SWCNTs, the total emission from the medium should be calculated by taking into account the propagation effects along the interaction length including the dispersion and absorption. By considering these effects, the conditions of coherent amplification and velocity matching are extracted and optimum parameters of the SWCNTs structures, as well as the laser pulses are achieved. Thus, we discuss the dispersion behavior of SWCNTs over terahertz and visible frequency ranges in the following.

2.3 Dispersion and structural effects

As mentioned above, optical rectification would only occur in the material without centrosymmetric inversion and thus we concentrate on chiral SWCNTs in the present work. Table 1 presents the structural parameters of four different types of chiral SWCNTs which are used in our calculations [20].

We consider SWCNTs to be vertically aligned along the z-direction as bulk, nonlinear, homogeneous, and nonisotropic media. The linear optical properties could be described by using the complex electric permittivity of $\epsilon =\epsilon ^{\prime }+i \epsilon ^{\prime \prime }$ that is presented in a matrix form of

(25)$$\epsilon=\left[\begin{array}{ccc} \epsilon_{11} & 0 & 0 \\ 0 & \epsilon_{22} & 0 \\ 0 & 0 & \epsilon_{33} \end{array}\right],$$

where $\epsilon _{11}=\epsilon _{22}$ [20]. Thus, depending on the polarization direction of the incident electric field, the real and imaginary parts of the refractive indices can be calculated for SWCNTs with different chiralities. The results presented in Fig. 4 reveal how the chirality could affect the behavior of SWCNTs over the visible frequencies and predict that the optimum optical wavelength should be adjusted by considering the chirality. Moreover, the absorption of SWCNTs would be relatively lower at optical frequencies in the case that the incident field is polarized in a transverse direction with respect to the nanotube axis.

Fig. 4. Real and imaginary parts of the refractive indices calculated for SWCNTs with different chiralities of (a):(6,2), (b):(6,4), (c):(8,4) and (d):(10,5) over optical frequency region.

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Table 1. Structural parameters of four different chiral SWCNTs.

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On the other hand, the dispersion of SWCNTs over the terahertz frequency region can be extracted from those of graphene multilayers graphite [49]. In such materials, the complex electric permittivity matrix is similar to that in Eq. (24), when the graphene layers are perpendicular to the z-axis. Therefore, we similarly calculate the real and imaginary parts of the refractive indices of SWCNTs with different chiralities for parallel and normally polarized fields at the terahertz frequency range as presented in Fig. 5.

Fig. 5. Calculated real and imaginary parts of the refractive indices of SWCNTs for ordinary (a) and extraordinary (b) field polarized configurations at the terahertz frequency range

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In addition, the scalar values of effective dielectric constants for SWCNTs could be extracted using effective medium approximation and based on the parameters available in Fig. 5. For the cases in which the electric field is polarized parallel and normal to the nanotube axis, these values are obtained respectively as

(26)$$\varepsilon_{e f f}^{{\perp}}=\frac{\sqrt{\varepsilon_{e}/ \varepsilon_{o}}(1+f)+(1-f)}{\sqrt{\varepsilon_{e}/ \varepsilon_{0}}(1-f)+(1+f)}$$

and

(27)$$\varepsilon_{e f f}=f \varepsilon_{o}+(1-f),$$

where, $\varepsilon _{e}$ and $\varepsilon _{o}$ denote the dielectric constants for extraordinary and ordinary cases, respectively. Moreover, $f$ represents the filling factor of SWCNTs, which is defined as $\pi d^{2}/4a^{2}$, in which $d$ and $a$ are the diameter and the adjacent distance, respectively.

We further take into account the alignment factor ($\psi$) of SWCNTs in our simulation for a more realistic estimation of the radiation generation process. This factor will modify the effective dielectric constants for extraordinary and ordinary cases as [49]

(28)$$\varepsilon_{V A C N T}^{{\perp}}=\psi \varepsilon_{e f f}^{{\perp}}+(1-\psi) \varepsilon_{e f f}^{\|},$$

and

(29)$$\varepsilon_{V A C N T}^{\|}=\psi \varepsilon_{eff}^{\|}+(1-\psi) \varepsilon_{eff}^{{\perp}}.$$

Here we utilize the typical values for $f$ and $\psi$ in the ranges of 0.01-0.1 and 0.9-0.99, respectively, which are practically more available. According to the above-mentioned equations, we conclude that dispersion of SWCNTs at the terahertz frequency range does not pronouncedly depend on the chirality, however, it would effectively depend on structural factors of $f$ and $\psi$. The real and imaginary parts of the refractive index of SWCNTs with different filling and alignments factors are compared in Fig. 6 at the terahertz frequency region. The field polarization is taken to be normal to the nanotubes axis. According to this figure, one could prefer highly aligned SWCNTs with fewer filling factors for efficient terahertz generation, because of their less linear absorption and lower dispersion effects. A similar influence of the alignment factor on terahertz power has been also observed in photoconductive SWCNTs switches [35]. Moreover, it can be seen that the influence of the structural parameters of SWCNTs on terahertz generation mechanism would be less considerable at higher THz frequencies.

Fig. 6. Calculated real and imaginary parts of the refractive index of SWCNTs with different alignments factors of (0.9,0.95 and 0.99) and different filling factors of (a) 0.01, (b) 0.05 and (c) 0.1, over terahertz frequency range

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3. Results and discussion

According to Eqs (23),(24), we first obtain the temporal variations of the nonlinear polarization induced by optical rectification in SWCNTs under various initial conditions. Then, by substituting the resulted polarization into the Eq. (18), the spectral distribution and consequently the temporal electric waveform of the generated terahertz radiation is numerically calculated. The values of $c_{1}^{\prime }(z, \omega )$ and $c_{2}(z, \omega )$ in those equations are also obtained by numerically solving the integral. For a more simple comparison of the results, $\beta$ and $E_{0}$ are normalized to unity. The chirality and the geometry of SWCNTs as well as the laser pulse duration and wavelength are the main parameters we explore their influences on the radiation generation process. We present the results for four different chirality cases of (6,2), (6,4), (8,4), and (10,5), which are chosen among those are practically available.

For a quantitative comparison of propagation loss of the interacting waves, we utilize the attenuation length as a benchmark defined as $l_{atn}=c/k(\omega )\omega$, which is inversely proportional to the extinction ratio $k(\omega )$. Figure 7 presents the wavelength dependence of attenuation length for SWCNTs with different chiralities, which are obtained based on linear absorption coefficients shown Fig. 4. It can be seen that the attenuation length is relatively larger at longer wavelengths so that for wavelengths above 1100 nm, which are sufficiently away from optical resonance regions of SWCNTs, more efficient terahertz generation could be expected. Figure 7 implies that SWCNTs with chirality of (6,4) yield a relatively large attenuation length of 0.8 $\mu$m at the practically available ultrashort laser wavelength of 1550 nm.

Fig. 7. Attenuation length versus the laser wavelength in SWCNTs with different chiralities.

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Moreover, it is expected that filling and alignments factors would affect the coupling conditions for the electric fields in the interacting region and consequently affect the dispersion behaviors. To investigate this dependency, we compare the attenuation length versus the laser wavelength in SWCNTs with chirality of (6,4) for various filling and alignments factors. As shown in Fig. 8, SWCNTs with higher filling and alignment provide larger attenuation lengths and consequently lower propagation loss. In the following, we present our simulation results for SWCNTs with filling and alignments factors of $f=0.1$ and $\psi =0.99$, according to the typical practically available values [49,53,54]. These parameters provide a maximum attenuation length of 48 $\mu$m at the frequency of 2.8 THz, which could be equivalent to a laser pulse with a duration of 190 femtosecond.

Fig. 8. Attenuation length versus terahertz frequency in SWCNTs with different filling and alignments factors.

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The conditions for optimum terahertz generation should also include the coherent amplification requirements, which are adjusted by relative velocities of optical and terahertz pulses inside SWCNTs. For terahertz generation at a definite frequency (e.g. 2.8 THz as mentioned above), the group velocity of the laser pulse should be adjusted using the laser wavelength for the minimum velocity mismatch. However, the laser wavelength is limited to the more available ultrashort laser systems e.g. central wavelength of 800 nm, 1050 nm, and 1550 nm, which might be far away from the ideal wavelength with negligible mismatch.

As a measure of the mismatch between group velocity of the optical pulse $v_{gr}(\omega )$ and phase velocity of terahertz pulse $v_{ph}(\omega )$ we compare the refractive index of SWCNTs in the terahertz frequency range $n_{THz} (\omega )=c/v_{ph}(\omega )$ with the group refractive index in the optical region $n_{gr}^{opt}=cv_{gr}(\omega )$. Figure 9 presents the dispersion of SWCNTs with filling and alignments factors of $f=0.1$ and $\psi =0.99$ and different chiralities. It can be seen that the dispersion of SWCNTs at the terahertz frequency range is independent of the chirality. The regions with minimum velocity mismatch and consequently efficient terahertz generation could be addressed in these figures. For instance, in SWCNTs with chirality (10,5) irradiated by laser pulses at a wavelength of 1035 nm, one would obtain optimal terahertz generation at the frequency of 4.2 THz at which both refractive indices approach a value of 1.195.

Fig. 9. Refractive index of SWCNTs over the terahertz and optical frequency ranges for chiralities of (a):(6,2),(b):(6,4),(c):(8,4), and (d):(10,5).

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To quantitatively emphasize how the velocity mismatch could affect the terahertz generation process in SWCNTs, here we utilize a parameter of saturation length as

(30)$$l_{w}=\frac{c \tau_{p}}{\left|n_{g r}^{o p t}-n_{T H z}\right|},$$

which is inversely proportional to the velocity mismatch. This parameter implies the length at which the generated terahertz field reaches its maximum value. Thus, coherent amplification conditions are satisfied when a lower velocity mismatch, and consequently a larger saturation length, is provided. Figure 10 depicts the variations of the saturation length versus the laser pulse duration and central wavelength for SWCNTs with filling and alignments factors of $f=0.1$ and $\psi =0.99$. Figures 10(a)-(c) are calculated over three typical central wavelengths of 800 nm, 1040 nm, and 1550 nm, respectively. The regions with red colors in this figure correspond to higher saturation length and, consequently, more efficient terahertz generation. However, one should regard the practical availability of the chosen laser wavelengths. Moreover, around these maximum regions, small variations of the laser pulse duration and wavelength would drastically reduce the terahertz output power. Therefore, the slower variation of terahertz power around 800 nm and 1550 nm could promise higher stability and less sensitivity to the fluctuations of the laser pulse duration and wavelength.

Fig. 10. The variations of the saturation length versus the laser pulse duration and central wavelength for SWCNTs with filling and alignments factors of $f=0.1$ and $\psi =0.99$ over three typical central wavelengths of (a) 800 nm, (b) 1050 nm, and (c) 1550 nm.

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In the following, we simulate terahertz generation in SWCNTs under the conditions designed based on the above-discussed considerations and the obtained parameters listed in Table 2. To confirm the optimum conditions, we explore the variation of terahertz intensity versus each effective parameter by keeping fixed the others. It should be noted that terahertz pulses would suffer from the linear absorption of the medium at both optical and terahertz frequencies. We take into account these absorption coefficients in the simulation in terms of the dispersion parameters to find the net intensity obtained at the end of the interaction length. Figure 11 presents the intensity evolution of terahertz radiation versus the interaction length at conditions listed in Table 2. It can be seen that the terahertz output intensity gradually increases inside SWCNTs, reaches its maximum at the middle of interaction length, and then slightly decreases at longer lengths of the medium. This implies that there exists an optimal length for SWCNTs that could be adjusted according to the effective parameters. Here we adjust the length of the interaction medium at 4 $\mu$m to unify the comparison and also diminish the concern over the velocity mismatch.

Fig. 11. Terahertz intensity evolution inside SWCNTs. The other effective parameters are adjusted according to Table 2.

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Table 2. Laser and SWCNTs parameters utilized in the simulation

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Figure 12 depicts the spectrum of the terahertz radiation generated in SWCNTs with different chiralities of (6,2), (6,4), (8,4), and (10,5). The other initial conditions are adjusted according to Table 2. It can be seen that the intensity of terahertz radiation is maximized in SWCNTs with a chirality of (6,4). Moreover, terahertz radiation is generated at a central frequency of 2.44 THz that seems to be independent of the chirality. We further study the dependence of terahertz radiation intensity on the filling factor and the alignment of SWCNTs as shown in Fig. 13. One may conclude from Fig. 13(a) that SWCNTs with less filling factor produce stronger terahertz radiation at central frequencies that are slightly shifted to lower values, provided that such filling factor could be practically feasible. Moreover, Fig. 13(b) implies that the alignment of SWCNTs would significantly improve the terahertz intensity by slightly changing the central frequency.

Fig. 12. The spectrum of terahertz emission generated in SWCNTs with different chiralities. The other effective parameters are adjusted according to Table 2.

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Fig. 13. The spectrum of terahertz emission generated in SWCNTs with different filling factors (a) and alignments (b). The other effective parameters are adjusted according to Table 1.

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Now we examine how the laser parameters including the wavelength and pulse duration could affect the mechanism of terahertz emission from SWCNTs. Figure 14(a) compares the terahertz spectrum obtained by three commercially available laser wavelengths of 800 nm, 1045, and 1550 nm. It is clear that the dispersion and absorption effects establish coherent amplification conditions much well at 1550 nm, comparing to those at wavelengths of 1045 nm and 800 nm. Our further investigation reveals the laser pulse duration determines not only the central frequency but also the spectrum of the terahertz radiation. As compared in Fig. 14(b), one may adjust the laser pulse duration for optimizing the output intensity, whereas terahertz pulses with a larger frequency bandwidth could be achieved using shorter laser pulses.

Fig. 14. The spectrum of terahertz emission generated in SWCNTs irradiated by laser pulses with different wavelengths (a) and different time duration (b). The spectra obtained at 800 nm and 1045 nm are magnified for visual comparison. The initial conditions are adjusted according to Table 2.

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Finally, we simulate the typical results which could be obtained in time resolved sampling of the sub-picosecond terahertz pulse produced by optical rectification in SWCNTs under conditions described in Table 2. Figure 15 presents the temporal electric field waveform along with the spectrum of terahertz pulse generated at optimal conditions. The temporal waveform of the electric field of terahertz pulse generated by optical rectification in SWCNTs under optimal conditions. The inset shows the corresponding spectrum according to that in Fig. 14.

Fig. 15. The temporal waveform of electric field and the spectrum of terahertz pulse generated by optical rectification in SWCNTs under optimal conditions.

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4. Conclusion

We have studied the terahertz radiation via optical rectification in chiral SWCNTs irradiated by ultrashort laser pulses, in which the noncentrosymmetric optical properties provide nonzero second-order optical nonlinearity. The three-dimensional susceptibility tensor responsible for optical rectification has been obtained based on that of the Pockels electro-optic effect. The effective dielectric constants of SWCNTs responsible for group velocity dispersion and absorption at both pump and terahertz pulse frequency regions have been extracted. The structural parameters of SWCNTs including the filling factor, alignment, and chirality along with the pulse duration and wavelengths have been adjusted according to those practically available to minimize the mismatch between laser group velocity and terahertz phase velocity along SWCNTs and, consequently, ensure the optimum coherence amplification of terahertz pulse. By comparing the variation of terahertz intensity and spectrum in various conditions of SWCNTs and laser pulse we concluded that SWCNTs with higher alignment and lower filling factor at chirality (6,4) could provide the conditions for maximum terahertz radiation generation when irradiated by ultrashort laser pulse with the wavelength of 1550 nm. It has been confirmed that the electric field waveform and the spectrum of the generated terahertz pulse, which are typically expected in terahertz time-resolved sampling, could be obtained for SWCNTs under various structural conditions. Finally, we have confirmed that the presented model could be employed for simulation of terahertz emission process in any kind of SWCNTs as well as other second order nonlinear materials.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Chirality type	Diameter( $A^{\circ}$ )	Chirality angle( $^{\circ}$ )
(6,2)	5.64	13.89
(6,4)	6.82	23.41
(8,4)	8.28	19.10
(10,5)	10.35	19.10

Chirality type	Diameter( $A^{\circ}$ )	Chirality angle( $^{\circ}$ )
(6,2)	5.64	13.89
(6,4)	6.82	23.41
(8,4)	8.28	19.10
(10,5)	10.35	19.10

Tuning single-walled aligned carbon nanotubes for optimal terahertz pulse generation through optical rectification of ultrashort laser pulses

Abstract

1. Introduction

2. Theory and design

2.1 Terahertz generation process

2.2 Propagation and coherent amplification

2.3 Dispersion and structural effects

3. Results and discussion

4. Conclusion

Disclosures

Data availability

References

Data availability

Cited By

Figures (15)

Tables (2)

Equations (30)

Optics Express