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\begin{document} \title{The Role of Electric Polarization in Nonlinear optics}\author{ Sumith Doluweera\\  \\         Department of Physics\\          University of Cincinnati \\         Cincinnati, Ohio 45221\\[1em]} \date{ }\maketitle
\begin{abstract} Nonlinear optics became a very active field of research  in both experimental and theoretical physics after the invention of LASER. Nonlinear nature is very important for optical data processing applications and all physics of nonlinear optics can be extracted by studying the behavior of applied optical field-induced polarization of the medium. In this article very basic properties of nonlinear electric susceptibilities are discussed qualitatively.      \\[1em]
\end{abstract}
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\section{\bf Introduction}
In linear optics it is assumed that an optical disturbance propagating through an optical medium can be described by a linear wave equation. As a consequence of that assumption  regardless of the intensity of light, the principle of superposition is valid in the regime of linear optics. When the intensity of light becomes great enough, linear optics is no longer enough to describe the situation observed experimentally \cite{refB}. It seems that light waves interact with one another and optical medium. These nonlinear phenomena require an extension of the linear theory, that allows for a nonlinear response of optical materials to the electromagnetic radiation.\\ 
It was realized that the electric polarization induced in the medium is the key to study physics of nonlinear optics and so in general the polarization of the medium should be expressed as a power series of applied electric field {\bf E}.
\begin{equation}\label{E1}
{\bf P}=\epsilon_{0}[\chi^{(1)} {\bf E}+\chi^{(2)} {\bf E}{\bf E}+\chi^{(3)} {\bf E}{\bf E}{\bf E}+......].\end{equation}
 Where ${\bf P}$ is the polarization,$\epsilon_{0}$ is the free space permitivity and $\chi^{(i)}$ is the  $i$ th order susceptibility tensor(material coefficient) of a given medium. First approximation of this equation gives linear optics and higher order terms contribute for nonlinear optics.

\section{\bf Theoretical framework of Nonlinear optics}.
Two major theoretical approaches can be employed in nonlinear optics as well as laser optics. They are semi-classical theory and quantum electrodynamical theory. In the semi-classical theory  the media composed of molecules and atoms are described by quantum mechanics and light radiation is described by Maxwell's theory.\\
The key issue of semi-classical theory is to give the expressions of macroscopic nonlinear electric polarization for optical media. The density matrix method is used to derive the expressions for various orders of susceptibilities and the expressions for various orders of polarization components. Substituting the appropriate nonlinear polarization components into the generalized wave equation, in principle ,it is possible to predict many possible nonlinear optical responses of the medium for a given condition of the  input intense optical fields\cite{refA}. 

\section{\bf Fundamental understanding of nonlinear polarization.}
Under the interaction of an applied optical electric field, the atoms or molecules of the medium may respond in two ways: (i) a real transition of a certain amount of atoms or molecules from one quantum eigen state another, and (ii) the perturbation of normal distribution or motion of internal electric charges within an atoms or molecules. The first is predominant for resonant interactions and the second is predominant for non or near resonant interactions.
Optical field-induced electric dipole moment  which, in turn, will act as a new source to emit a secondary electromagnetic wave. This is the fundamental process describing the optical field-induced dipole moment of a molecular system and the re-emission of a secondary wave radiation. Assuming that the number of molecules in a unit volume is {\bf N} and induced dipole moment of $i$ th molecule is ${\bf p_{i}} $ then ${\bf P}$ is determined by
\begin{equation}\label{E2}
{\bf P(t)}=\sum_{i=1}^{N}{\bf p_{i}(t)}
\end{equation}.
From Eq. (2) it can be seen that the electric polarization of a medium is determined by two factors: one is the field induced dipole moment of each individual molecule of the medium, and the other is the statistically averaged property of a great number of molecules. That is the molecular dipole moment is determined by the microscopic structure or their eigen functions; whereas the results of the summation depends on the microscopic $symmetry$. Now taking the  $n$ th order Fourier component of polarization ${\bf P^{(n)}(t)} $and the Fourier component of $ {\bf E(t)}$ it can be shown that \cite{refA}
\begin{equation}\label{E3}
{\bf P}^{(n)}({\omega})=\epsilon_{0}\chi^{(n)}(\omega_{1},\omega_{2},...,\omega_{n}){\bf E}(\omega_{1}){\bf E}(\omega_{2})......{\bf E}(\omega_{n}).
\end{equation}.
 
The values and properties of $\chi^{(n)}$ can be determined from the semi-classical theory mentioned above and it is very complex and difficult. So By considering special features of interaction between the incident light beam and the given medium and then simplifying the above model it is possible to extract some useful conclusions.
\section{\bf Various mechanisms causing Nonlinear polarization in a medium.}
Depending on the applied field frequency and the phase of the medium one or more of the mechanisms discuss below become the major contributor to the nonlinear behavior of the medium.
\subsection{\em Distortion of electronic cloud}
The change of the outer-shell electronic cloud of an atom, ions or molecule compare to the undisturbed status without  applying an external optical electric field. This mechanism has very fast response time and approximately less than ($10^{-16}-10^{-15}$) seconded: most optical frequency mixing effects such as second harmonic and third harmonic generation, sum-frequency generation, optical parametric oscillation and four-photon parametric interaction.

\subsection{\em Intramolecular motion}
It denotes the polarization contribution from an optical-field induced relative motion (vibration, rotation, etc .) between the nuclei(or ions) within a molecule. The response time of this mechanism is around $10^{-12}$ second. Ex: some Raman resonance-enhanced four-wave-mixing effects and Raman enhanced refractive index change.

\subsection{\em Molecular reorientation}
It denotes the additional electric polarization contribution from an optical field induced reorientation of anisotropic molecules in a liquid. The response time of this process is dependent on the rotational viscosity of molecules in the liquid and is approximately $(10^{-12}-10^{-13})$ second. Ex: stimulated Kerr scattering and Kerr-effect related refractive index change.

\subsection{\em Induced acoustic motion}
It is the polarization contribution from an optical induced macroscopic acoustic motion related to the so called electrostriction interaction. the response time of this mechanism is ($10^{-9}-10^{-10}$) second depending on the phase state of the medium. Ex ; Brillouin scattering, self focusing and optical breakdown damage.

\subsection{\em Induced population change}
The distribution of molecules in their different eigen states may be changed owing to one-photon or two-photon absorption or Raman transition for resonant interaction. the response time of this  additional polarization  of the medium is strongly depend on the dynamic properties of molecular transition and relaxation of the medium. and in general slower than that of first three mechanisms.

\section{Manipulation of nonlinear susceptibilities}
The main issue of nonlinear optics is to study the interaction between the laser field and media. In many cases, the applied laser field is a coherent monochromatic electromagnetic  wave with narrow spectral line width and a small divergent angle. Since the laser field has only one or more monochromatic Fourier components, the field-induced nonlinear polarization can also be recognized as a combination of limited number of Fourier components. 
According to Eq.(3), the linear component is given by
\begin{equation}\label{E4}
{\bf P}^{(1)}(\omega)=\epsilon_{0}\chi^{(1)}(\omega){\bf E}(\omega) 
\end{equation}
Second-order nonlinear polarization is given by 
\begin{equation}\label{E5}
{\bf P}^{(2)}(\omega=\omega_{1}+\omega_{2})=\epsilon_{0}\chi^{(2)}(\omega_{1},\omega_{2})
{\bf E}(\omega_{1}){\bf E}(\omega_{2})\end{equation}
Physical meaning of this equation is that the radiation at new frequency $\omega=\omega_{1}+\omega_{2}$ can be generated by two incident monochromatic waves with frequencies  of $\omega_{1}$ and $\omega_{2}$.The $ \chi^{(2)} $ is a third order tensor having $3\times9=27$ elements. The third order nonlinear polarization is given by
 \begin{equation}\label{E6}
{\bf P}^{(3)}(\omega=\omega_{1}+\omega_{2}+\omega_{3})=\epsilon_{0}\chi^{(2)}
(\omega_{1},\omega_{2},\omega_{3}){\bf E}(\omega_{1}){\bf E}(\omega_{2}){\bf E}(\omega_{3})
\end{equation} .
 
This equation tells that in the third order approximation, the radiation at a new frequency $\omega=\omega_{1}+\omega_{2}+\omega_{3}$  can be generated by an intense  incident field containing $\omega_{1}$, $\omega_{2}$ and $ \omega_{3}$.The $\chi^{(3)}$ is a fourth order tensor having $3\times27=81$ elements.
The above equations  (4),(5) and(6) tell that various-order of susceptibilities are the key parameters to describe the nonlinear coupling between different incident waves, as well as the generation of new frequency radiation through the induced nonlinear electric polarization of the medium.  

Based on the theoretical analyses \cite{refA} it is understood that various orders of susceptibilities of a medium possess the following basic properties, which help researchers to considerably simplify theoretical work and have a better understanding the related nonlinear optical processes.
\subsection{\em Relative magnitude of various orders of susceptibilities}
If the major contribution for polarization is the distortion of electronic cloud of a molecular system, the approximation
\begin{equation}  
|\chi^{(n)}|/|\chi^{(n-1)}| \simeq{1/|E_{0}|} \end{equation}
is valid and where $|E_{0}|$ is the magnitude of average electric field strength inside an atom ( it is about $10^{11}$ V/m for a hydrogen atom.). according to Eq.(3) the ratio between successive polarization can be roughly estimated by
\begin{equation}
|p^{(n)}|/|p^{(n-1)}| \simeq{|\chi^{(n-1)}|/\chi^{(n)}|}\simeq{|E|/|E_{0}|}.\end{equation}
Here $|E|$ is the magnitude of an applied optical field. For ordinary light sources the ratio of $ |E|/|E_{0}|$ is so small that all nonlinear terms can be neglected. If the applied incident light is of high spectral intensity the ratio $|E|/|E_{0}|$ is not small and second and third order contributions may play vital roles.

\subsection{\em Spatial symmetry restrictions on susceptibilities}
The susceptibility tensors must remain unchanged upon the symmetric operation allowed for the medium. This reduces the number of the independent and nonzero elements of the susceptibility tensors. The most important conclusion from this property is that for all centrosymmetric crystals and isotropic media (gases, liquids  and amorphous solids), all tensor elements of even-order susceptibilities $(\chi^{(2)},\chi^{(4)}...)$ must be zero. Therefore no second order nonlinear effects can be observed for such media. But odd-odder susceptibility tensors will not be zero and they provide nonlinear effects.

\subsection{\em Resonance enhancement of susceptibility}
When one or  a combination of several frequency components of the applied electric field approaches a resonant frequency of the medium , the magnitudes of the tensor elements for appropriates orders of susceptibilities can be significantly increased.

\subsection{\em Permutation symmetry of susceptibilities}
The following permutation symmetry relations of tensor elements (in Cartesian coordinates) for second order and third order susceptibilities hold \cite{refA};
\begin{equation}
\chi_{ijk}^{(2)}(\omega_{1},\omega_{2})=\chi_{ijk}^{(2)}(\omega_{2},\omega_{1}) \end{equation}
and
\begin{equation}
\chi_{ijkl}^{(3)}(\omega_{1},\omega_{2},\omega_{3})=\chi_{ikjl}^{(3)}(\omega_{2},\omega_{1},\omega_{3}) =\chi_{iljk}^{(3)}(\omega_{3},\omega_{1},\omega_{2}) =....\end{equation}

\subsection{\em complex conjugation and time reversal symmetry}
It can be proved that  various orders of susceptibilities are complex quantities in general. Since the polarization response must be a real quantity, the manipulation of complex conjugation for $\chi^{(n)}$ leads to
\begin{equation}
[\chi^{(n)}(\omega_{1},\omega_{2},..,\omega_{n})]^{\ast}=\chi^{(n)}(-\omega_{1},-\omega_{2},..,-\omega_{n})\end(equation}
Furthermore if we consider the non resonant case, $\chi^{(n)}$  can be approximately viewed as real quantities, and then we obtain \cite{refA}
\begin{equation}
[\chi^{(n)}(\omega_{1},\omega_{2},..,\omega_{n})]^{\ast}=\chi^{(n)}(\omega_{1},\omega_{2},..,\omega_{n})\end{equation}
And finally considering both of above
\begin{equation}
\chi^{(n)}(\omega_{1},\omega_{2},..,\omega_{n})=\chi^{(n)}(-\omega_{1},-\omega_{2},..,-\omega_{n})\end{equation}
This implies that susceptibility tensors are invariant for simultaneous change of the sign for all frequency arguments. This is the time reversal symmetry and holds only for non-resonant interaction.
\\By using  these symmetries the number of independent and nonzero elements of $\chi^{(n)} $ can be further reduced and greatly simplify the analyses.


\section{Conclusion}Nonlinear phenomena are ultimately due to inability of dipoles in the optical medium to respond in a linear fashion to the alternating electric field associated with a light beam. Understanding the various orders(mainly second and third) of susceptibility tensors and the behavior of their elements of the optical medium subject to given condition is the basic problem of nonlinear optics.

\begin{thebibliography}{99}
\bibitem{refA} Guang S. He and Song H. Liu, Physics of Nonlinear Optics.\bibitem{refB} Frank L.Pedrotti, S.J.and Leno S. Pedrotti., Introduction to Optics.\end{thebibliography}
\end{document}

