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\title{
Nonlinear optical effects and carbon nanotubes
} 
\author{
Chiyat Ben Yau
}
\address{
Department of Physics, University of Cincinnati, OH 45221, USA
}
\date{\today}
\maketitle

\begin{abstract}
Materials with large second or third order optical nonlinearities are of great 
interest for data processing in photonic computing. Since the invention of laser 
in 1960 by Maiman and the first discovery of nonlinear optical (NLO) effect of quartz 
crystals, there has been a great deal of research directed at finding new materials with 
higher order NLO properties. Only noncentrosymmetric or polar materials could possess second 
order NLO susceptibility $\chi^{(2)}$. For most third order NLO materials, 
their $\chi^{(3)}$'s are generally 
too low for practical applications. Since the discovery of carbon nanotubes, which possess 
optimal properties (like highest thermal conductivity, Young modulus 100 times higher than steel, 
zero bandgap metal, ballistic conductance, simple structure, nanometer-size and ultra-light weight) 
by a NEC scientist Sumio Iijima in 1991, a heat of studying and engineering the properties of carbon 
nanotubes has been raised worldwide. Experimentalists recently discovered that carbon nanotube is also 
a NLO material with amazingly high third order NLO susceptibilities. And, up to now, there is no 
publication about the experimental proof of the second order NLO properties of carbon nanotubes. 
In this paper, we will review the basic physics of NLO properties of materials, and state the reasons 
of high $\chi^{(3)}$ and practically zero $\chi^{(2)}$ of carbon nanotubes.
\end{abstract}
%\pacs{PACS numbers: 75.30.Vn, 71.45.Lr, 75.25.+z, 71.70.Ej}
%]
\newpage
\narrowtext

Before the invention of laser, the intensity of light is approximately 1 V/cm and nearly all 
materials exhibit linear relationship between the polarization of the 
materials and the incoming optical electric field {\bf E}. When the high intensity 
(10$^6$-10$^9$ V/cm) coherence light was first provided by the first laser, which was 
invented by Maiman in 1960 \cite{maiman}, the nonlinear relationship between the averaged electric 
field inside material {\bf D} and the incoming optical {\bf E} field, as shown in Fig. 1, implying 
nonlinear polarization was discovered. When a red laser light enters a quartz crystal, 
as shown in Fig. 2, the outcoming light is blue, and this indicates that the frequency 
of outcoming light is doubled.  This is a nonlinear optical (NLO) effect called Second 
Harmonic Generation (SHG). Since then, there has been a great deal of research directed 
at finding new materials with higher order nonlinear optical properties.



     







NLO effects are defined on the tensor of optical susceptibility $\chi$, 
which 
is the transformation matrix of the incoming {\bf E} vector to the resulting 
polarization {\bf P} vector inside the material. For linear materials or incoming 
light with intensity $\sim 1$ V/cm, the relation ${\bf P}=\chi {\bf E}$ is 
essentially linear. 
However, when the light intensity reaches $\sim$ 10$^6$-10$^9$ V/cm, the 
relationship between {\bf P} 
and {\bf E} is no longer linear. For second order NLO behavior, {\bf P} will be related with {\bf E} as follows:
\begin{equation}
P =\epsilon_{0}(\chi^{(1)}E_0\cos\omega t+\chi^{(2)}E_0^2\cos^2\omega t) 
\end{equation}

For third order NLO behavior, {\bf P} will be related with {\bf E} as follows:

\begin{equation}
P = \epsilon_{0}(\chi^{(1)}E_0\cos\omega t+
\chi^{(2)}E_0^2\cos^2\omega t +\chi^{(3)}E_0^3\cos^3\omega t) 
\end{equation}



The tensors $\chi^{(2)}$ and $\chi^{(3)}$ in equations (1) and (2) are second 
and third 
order NLO susceptibilities respectively. 

In order to see the properties 
of the outcoming light, we expand equations (1) and (2) by simple 
trigonometry, yield the following equations (3) and (4) respectively.
\begin{equation}
P = \frac{1}{2}\epsilon_0\chi^{(2)}E_0^2+
\epsilon_0\chi^{(1)}E_0\cos\omega t+\frac{1}{2}\epsilon_0E_0^2\cos2\omega t
\end{equation}
\begin{equation}
P = \frac{1}{2}\epsilon_0\chi^{(2)}E_0^2+
(\epsilon_0\chi^{(1)}E_0 + \frac{3}{4}\epsilon_0\chi^{(3)}E_0^3)\cos\omega t+\frac{1}{2}\epsilon_0E_0^2\cos2\omega t +
\frac{1}{2}\epsilon_0\chi^{(3)}E_0^3\cos3\omega t
\end{equation}
\begin{equation}
\chi_{ijk}(2)(0;-\omega,\omega)=\chi_{ijk}(2)(\omega;0,-\omega)
\end{equation}

According to the experimentally found relation (5) as above \cite{Bass}, the effects of the first and second terms in equation (3) are cancelled, leaving the last term describing the outcoming light with doubled incoming frequency. Hence this second order NLO effect causes frequency doubling. For equation (4), the effects of the first two terms are also cancelled, leaving the last two terms. If the last term is predominant, the outcoming light will possess frequency three times that of the incoming light. 
     In order to understand the relation between centrosymmetry and NLO properties, Prasad and William have given an argument as follows \cite{Prasad}. Consider a third order NLO material that can exhibit polarization liked the following equations (6) or (7).
\begin{equation}
P_x = \alpha E + \gamma E^3 
\end{equation}

\begin{equation}
P_x = \alpha E + \beta E^2 + \gamma E^3 
\end{equation}


For materials exhibit polarization as equation (6), the relation between potential V(E) and the {\bf E} field will be symmetrical like equation (8).
 
\begin{equation}
V_x(E)= - [(1/2) \alpha E^2 + (1/4) \gamma E^4] 
\end{equation}


The symmetrical potential of materials under the influence of an incoming optical {\bf E} field implies that the material is centrosymmetrical that the electron polarizations in the positive and negative x direction are the same. The relation can be plotted as Fig. 3. The materials possessing centrosymmetry must have structures lower than cubic symmetry. For materials exhibit polarization as in equation (7), the potential {\bf V(E)} is asymmetrical as the form of equation (9) below:
\begin{equation}
V_x(E) = - [(1/2) \alpha E^2 + (1/3) \gamma E^3 + (1/4) \gamma E^4] 
\end{equation}



Then for second order NLO materials possessing the second term 
of equation (7) must have an asymmetrical potential that is a 
symbol of noncentrosymmetry. The polarization versus {\bf E} curve 
is shown in Fig. 4. Thus the basic requirement for second order NLO 
behavior is noncentrosymmetry. Materials possessing noncentrosymmetry 
are anisotropic crystals having structures from cubic to triclinic. 
     
As the anisotropic materials generally possess second order NLO 
properties while the materials of third order NLO behavior are having too 
low $\chi^{(3)}$ for any practical applications, the search for materials 
having large $\chi^{(3)}$ is of first priority in NLO research. Since the 
discovery of carbon nanotubes (CNT) by a NEC scientist Sumio Iijima in 
1991\cite{Iijima}, optimal properties like highest thermal conductivity 
\cite{Ruoff}, Young's modulus higher than steel\cite{Wang}, ballistic 
conductance\cite{Thess}, nanometer-size\cite{Ajayan} and ultra-light 
weight\cite{Colbert} of CNT have been revealed by experiments. Amazingly, 
experimentalists recently discovered that CNT is also a NLO material with 
very high $\chi^{(3)}$\cite{Liu}.  
     Carbon nanotubes are rolled-up graphene sheets with hemisphere caps.  
All carbon atoms in a molecule of CNT are sp$^2$ hybridized that each 
carbon atom has a $\pi$ electron and 3 sharing $\sigma$ orbital electrons. 
The 
$\pi$ electrons are very delocalized that they can delocalized over the 
whole tube, while the $\sigma$ electrons are very localized. The structure 
of carbon nanotube is composed of hexagonal benzene rings except the 
hemisphere caps, which have pentagons too. A benzene molecule has 
centrosymmetry and possesses zero $\chi^{(2)}$. When benzene is bonded with 
an electron donor or acceptor group, it is essentially a 
noncentrosymmetric molecule, and it has non-zero $\chi^{(2)}$. For 
symmetrical benzene has a $\pi$ delocalized system, the $\pi$ electron 
distribution is thus highly deformable, and the $\chi^{(3)}$ becomes 
significant. CNT has significant amount of benzene rings and forms a very 
efficient $\pi$ delocalization system and behave the same as benzene. This is 
consistent with the experiments that the CNT was found to have large 
$\chi^{(3)\prime}$s($\sim10^{-11}$ esu) compared to the $\chi^{(3)}$ of 
C$_{60}$ 
($\sim 3\times 10^{-34}$ esu), as shown in Table 1, but no experiments 
showed that CNT has non-zero $\chi^{(2)}$'s \cite{Liu}.

Also, the $\chi^{(3)}$ is more significant when the {\bf E} field is along 
the tube axis \cite{Colbert}.  And this indicates that the $\pi$ 
delocalization has a significant contribution to the third order NLO properties of CNT.
     
\begin{center}
Table 1. $\chi_{1111}^{(3)}$ measured by laser Nd:YAG for the carbon 
nanotube solution.\cite{Liu}
\end{center}
\begin{tabular}{|c|c|c|c|}\hline
Wavelength & $\chi_{1111}^{(3)}$(solution) & 
$\chi_{1111}^{(3)}$(solvent) & $\chi_{1111}^{(3)}$(CNT)\\
(nm) & ($\times 10^{-11}$ esu) & ($\times 10^{-11}$ esu) &($\times 
10^{-11}$ esu)\\ \hline
1064 & 2.429 & 1.250 & 1.179\\
332 & 0.586 & 0.277 & 0.309 \\ \hline
\end{tabular}

\vspace{1cm}
To characterize the NLO properties of CNT, we need to dissolve CNT in 
a polar solvent. Sometimes a non-zero $\chi^{(2)}$ could be yielded because (1) the caps of CNT are very chemically active and will bond with donor or acceptor groups to make the whole tube becomes polarized (2) molecular reorientation can be possible in a non-ultrasonically dispersed sample (3) doped CNT is essentially noncentrosymmetrical as the doped sites are effective electron traps.
     
Due to the existence of centrosymmetry and $\pi$-electrons that can 
be delocalized throughout the whole molecule, carbon nanotube is 
experimentally proved to be a third order NLO material with high enough 
$\chi^{(3)}$ for photonic applications like third harmonic 
generation (THG).  However, there is not much experimental 
research on NLO properties of carbon nanotubes and it is still 
required to search for higher order NLO properties of carbon nanotubes.  
Also the unique tube-filling property of carbon nanotubes provides an 
insight to change the photonic properties by filling the carbon nanotubes 
with dielectrics and metals.  As defected carbon nanotubes have obviously 
deformed $\pi$-delocalization and the photonic properties are still 
unknown, more efforts are required to study in this area.   


\begin{references}
\bibitem{maiman}  T.H. Maiman, Nature 187, 493 (1960)
\bibitem{heavens} O.S. Heavens, R.W. Ditchburn, Insight into Optics, Wiley, England, 1991,p.142
\bibitem{Bass} M. Bass, P.A. Franken, J.F. Ward, G. Weinreich, Phys. Rev. Lett. 9, 446 (1962)
\bibitem{Prasad} P. Prasad, D. Williams, Introduction to nonlinear optical effects in molecules and polymers, Wiley, New 
York, p. 17
\bibitem{Iijima} S. Iijima, Nature 354, 56 (1991)
\bibitem{Ruoff}
R.S. Ruoff, D.C. Lorents, Carbon 33, 925 (1995)
\bibitem{Wang} Z.L. Wang, R.P. Gao, P. Poncharal, W.A. de Heer, Z.R. Dai, Z.W. Pan, Materials Science and Engineering C 
16, 3 (2001)
\bibitem{Thess}
A. Thess, R. Lee, P. Nikolaev, H. Dai, P. Petit, J. Robert, C. Xu, Young H. Lee, S. Gon Kim, A. G. Rinzler, D. T. Colbert, G. Scuseria, D. Tom�nek, J. E. Fischer, R. E. Smalley, Science 273,  483 (1996).
\bibitem{Ajayan}
 P.M. Ajayan, T.W. Ebbesen, Rep. Prog. Phys. 60, 1025 (1997)
\bibitem{Colbert}
D.T. Colbert, R.E. Smalley, Nanotechnology 17, 46 (1999)
\bibitem{Liu} X. Liu, J. Si, B. Chang, G. Xu, Q. Yang, Z. Pan, S. Xie, P. Ye, J. Fan, M. Wan, Appl. Phys. Lett. 74, 
164 (1999)
\end{references}

%\begin{figure}[h]
%\centerline{epsfxsize=2.0in \epsfbox{fig1.ps}}
%\caption[*]{fig1's caption}
%\end{figure}


\begin{figure}
\caption{
Nonlinear relationship between ${\bf D}$ and ${\bf E}$ at 
high optical ${\bf E}$ field.
}
\end{figure}

\begin{figure}
\caption{Laser beam enters quartz crystal as red light 
and emerges as blue light (a second order NLO effect: 
second harmonic generation).$^2$}
\end{figure}

\begin{figure}
\caption{A symmetric P-E curve of centrosymmetrical materials. The figure in lower right corner indicates centrosymmetry.$^4$}
\end{figure}

\begin{figure}
\caption{An asymmetric P-E curve of noncentrosymmetrical materials. The figure in lower right corner indicates noncentrosymmetry.$^4$}
\end{figure}

\begin{figure}
\caption{TEM picture of carbon nanotubes.$^8$}
\end{figure}

\begin{figure}
\caption{Pictorial structure of single walled carbon nanotube}
\end{figure}




\end{document}

