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\begin{document}
\title{Raman Spectroscopy}
\author{ SUN, TSENG-BAO\\
         \\
         Department of Physics\\
         University of Cincinnati\\
         Cincinnati, Ohio 45221\\[1em]}

\date{\today}
\maketitle

\begin{abstract}

The Raman effect is a most useful tool for the study of elementary excitation in solids, such as phonons and plasmons, as well as molecular vibrations and molecular structure. This review deals with basic investigations of the Raman scattering of light by materials. A variety of techniques exist for performing coherent Raman spectroscopy. Coherent Anti-Stokes Raman Scattering (CARS) and a semiconductor example will be given later.


\end{abstract}

\pagebreak

\def\baselinestretch{1.1}

That the laser has revolutionized since 1960 the field of molecular spectroscopy is a well known commonplace. This is indeed so emphatically true in the case of Raman spectroscopy, although the Raman effect was experimentally discovered in 1928. This review concludes an introduction of Raman theory and techniques. 

\section{Introduction}

There are two kinds of processes involved for light waves or visible photons scattered by phonons. One is known as Brillouin scattering when the phonon is acoustic. The other is Raman scattering when the phonon is optical.
Now consider that energy conserves in the scattering process, that is,

\begin{eqnarray}
\omega = \omega_{0} \pm \omega(\bf{q})
\end{eqnarray}

Here, $\omega_{0}$ and $\omega$ are the frequencies of the incident and scattered phonon, respectively, and $\omega(\bf{q})$ the frequency of the phonon involved. The positive sign refers to the phonon-absorption case, while the minus sign refers to the phonon emission case. Then the spectrum of the scattered beam should reveal two lines displaced relative to the incident frequency, shown in Fig.~1, by an amount equal to the phonon frequency, and the centural unshifted line is not produced by phonon scattering. This is so called Rayleigh scattering caused by static impurities in the sample and called the Rayleigh line. There are the same structure of the spectra, called Brillouin wings, for both Brillouin and Raman scatterings. The lower Brillouin wing, arising from phonon emission, is called the Stokes line, and the upper wing, which arises from phonon absorption, is known as the anti-Stokes line. 

\begin{figure}[htb]
\centerline{\psfig{figure=fig2_1.ps,height=2.5in}}

\caption[]{\em{Brillouin Wings}}
\end{figure}

The long-wavelength approximation $\omega(\bf{q}) = {\em{v}}_{s}q$ holds true near the center of the Brillouin zone. The frequency shift is

\begin{eqnarray}
\Delta\omega = {\pm}2n{\omega_{0}}\left(\frac{{\em v}_{s}}{\em c}\right)\sin {\theta}
\end{eqnarray}

where the shift increases with the scattering angle $\theta$.

\section{Non\-li\-near Op\-tics and Co\-her\-ent Ra\-man Spec\-tro\-sco\-py}

The field of third-order nonlinear spectroscopy is a rich and diverse collection of techniques for studying the excitation of matter. Generally, this nonlinear response of matter to the incident light fields is written in the form of a polarization density power-series expansion

\begin{eqnarray}
\vec P = {\vec P}^{(1)} + {\vec P}^{(2)} + {\vec P}^{(3)} + \cdots
\end{eqnarray}

Here, ${\vec P}^{(n)}$ represents the {\em{n}}th-order term whose amplitude responds to incident fields according to the relation

\begin{eqnarray}
\vec P = {\chi} : \vec E + {\chi}^{(2)} {\vdots} {\vec E}{\vec E} + {\chi}^{(3)} {_{:}^{:}} {\vec E}{\vec E}{\vec E} + \cdots
\end{eqnarray}

Here, ${\chi}^{(n)}$ is an $n+1$ rank tensor which relates the input fields $\vec E$ to the induced polarization density ${\vec P}^{(n)}$.
This study is concerned with the third-order term in the series expansion

\begin{eqnarray}
{\vec P}^{(3)} = {\chi}^{(3)} {_{:}^{:}} {\vec E}{\vec E}{\vec E}
\end{eqnarray}

And specifically with the various forms of coherent Raman spectroscopy based on the Raman effect contribution to the third-order nonlinear susceptibility, ${\chi}^{(3)}$.
Several nonlinear optical techniques exist which generate a coherent, well-defined Raman-induced field,, we can consider a four-photon process illustrated in Fig.~2.

\begin{figure}[htb]
\centerline{\psfig{figure=fig2_2.ps,height=2.5in}}

\caption[]{\em{Diagram for Coherent Raman Scattering}}
\end{figure}

The electronic states $|a\rangle$, $|b\rangle$ and $|c\rangle$ are three independent fields with angular frequencies $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$. A fourth field, which is phase-coherent relative to the input fields, is generated at frequency $\omega_{4} = \omega_{1} - \omega_{2} + \omega_{3}$. When the frequency difference $\omega_{1} - \omega_{2}$ equals the Raman excitation frequency $\omega_{R}$, the signal wave at $\omega_{4}$ is enhanced, indicating the Raman resonance. The susceptibility must be corrected to include factors which express the local fields in terms of the macroscopic fields, and the polarization density amplitude expressed as

\begin{eqnarray}
P^{(3)}_{i}\left(\omega_{4}, t\right) = \chi^{(3)}_{ijkl}\left(-\omega_{4}, \omega_{1}, -\omega_{2}, \omega_{3}\right)E_{j}\left(\omega_{1}, t\right)E^{\star}_{k}\left(\omega_{2}, t\right)E_{1}\left(\omega_{3}, t\right)
\end{eqnarray}

$\chi^{(3)}_{ijkl}$ is the third-order nonlinear susceptibility tensor, the $E_{\alpha}\left(\omega_{\beta}, t\right)$ are real incident fields of frequency $\omega_{\beta}$ and polarization axis $\alpha$ and the susceptibility is expressed as

\begin{eqnarray}
\chi^{(3)}_{ijkl} = \chi^{NR}_{ijkl} + \chi^{R}_{ijkl} = \chi^{NR}_{ijkl} + \frac{N\alpha_{ij}\alpha_{kl}}{\omega_{R}-\left(\omega_{1}-\omega_{2}\right)-i\Gamma_{R}}
\end{eqnarray}

Here, $\omega_{R}$ and $\Gamma_{R}$ are espectively the resonance frequency and linewidth.

\section{Coherent Anti-Stokes Raman Spectroscopy}

One of the most widely used coherent Raman techniques today is coherent anti-Stokes Raman spectroscopy which has been given the acronym CARS. To begin the signal analysis of CARS, we may write the nonlinear polarization density induced in the material, when $\omega_{1} > \omega_{2}$, as

\begin{eqnarray}
P_{x}\left(\omega_{3}\right) = \frac{3}{8}\chi_{1111}\left(-\omega_{3}, \omega_{1}, -\omega_{2}, \omega_{1}\right)E_{x}\left(\omega_{1}\right)E^{\star}_{x}\left(\omega_{2}\right)E_{x}\left(\omega_{1}\right)
\end{eqnarray}

\begin{figure}[htb]
\centerline{\psfig{figure=fig2_3.ps,height=2.5in}}

\caption[]{\em{Raman Contribution to $\chi_{(3)}\left(\omega_{1}-\omega_{2}\right)$}}
\end{figure}

The technique of CARS is usually defined by the use of two lasers at frequencies $\omega_{1}$ and $\omega_{2}$, with parallel polarizations. when the difference frequency $\omega_{1} - \omega_{2}$ is equivalent to a Raman resonance frequency $\omega_{R}$, the signal field generated by the polarization density at $\omega_{3}$ is enhenced. In Fig.~3, we have

\begin{eqnarray}
\chi_{1111} = \chi^{NR}_{1111} + \chi'_{1111} + i\chi''_{1111}
\end{eqnarray}

$\chi'_{1111}$ and $\chi''_{1111}$ represent the real and imaginary Raman contributions and $\chi^{NR}_{1111}$ the nonresonant contribution to $\chi_{1111}$.
The detected intensity is proportional to $\chi_{1111}$; thus the CARS signal power has four contributions: the nonresonant background intensity, the dispersive and resonantintensities from $\left(\chi'\right)^{2}$ and $\left(\chi''\right)^{2}$ and a cross-term $2\chi^{NR}\chi'$ which will have both positive and negative values as $\left(\omega_{1} - \omega_{2}\right)$ is tuned through the Raman resonance. Consider that the wave vector is $k_{i} = n_{i}\omega_{i} / c$ and ${\Delta}k = |\left(2{\vec k}_{1} - {\vec k}_{2} - {\vec k}_{3}\right) \cdot$ \^z$| = 0$, where \^z is a unit vector. It is called the phase-matching condition. If the incident lasers are overlapped at an angle $\theta$ between the vectors $\vec k_{1}$ and $\vec k_{2}$ is

\begin{eqnarray}
\cos \theta = \frac{4\omega_{1}\omega_{2}n^{2}_{3} - 4\omega^{2}_{1}\left(n^{2}_{3} - n^{2}_{1}\right) - \omega^{2}_{2}\left(n^{2}_{3} - n^{2}_{2}\right)}{4\omega_{1}\omega_{2}n_{1}n_{2}}
\end{eqnarray}

\begin{figure}[htb]
\centerline{\psfig{figure=fig2_4.ps,height=2.5in}}

\caption[]{\em{Schematic Diagram of CARS}}
\end{figure}

A schematic illustration of the apparatus required to perform CARS is shown in Fig.~4 where the mirrors are labelled M, and the beams at $\omega_{1}$ and $\omega_{2}$ are overlapped at the phase-matching angle $\theta$. The lens L prior to the sample S focuses both the $\omega_{1}$ and $\omega_{2}$ beams, while the lens at the exit face of the sample collimates $\omega_{3}$ for detection. The iris I provides the initial spatial discrimination of $\omega_{3}$ from $\omega_{1}$ and $\omega_{2}$. The detector D is generally a broadband diode or photomultiplier tube, followed by waveform-shaping electronics and an amplifier, a signal-averager such as a boxcar integrator, and a chart recorder for data display. Usually the two laser sources are high-peak-power dye lasers of $\sim$10 nanosecond pulse duration. One of the two lasers maybe fixed in frequency $\omega_{1}$ while the other, $\omega_{2}$, is tuned. As a result the output frequency $\omega_{3}$ is constantly changing. Although the degree of spatial filtering of the phase-matching field at $\omega_{3}$ from $\omega_{1}$ and $\omega_{2}$ is high, small portions of the $\omega_{1}$ and $\omega_{2}$ fields may tend to scatter from lenses and sample windows into the $\vec k_{3}$ direction and coping with this problem requires the use of frequency-dispersive elements. Over large scanning ranges of several wavenumber change in $\omega_{2}$, the detection of $\omega_{3}$ may require a synchronous tuning of any frequency dispersive elements such as these elements is the use of two tunable lasers which are scanned at the proper relative rates to maintain a constant difference $2\omega_{1} - \omega_{2}$.

\section{Raman Shift in Doped {\em GaAs-AlAs} Mul\-ti\-ple Quan\-tum Wells}

This example is performed on ({\em 001}) - oriented {\em GaAs-AlAs} MQWs, grown by molecular beam epitaxy, in which only {\em GaAs} layers were uniformly doped with {\em Si} donors. The thicknesses $d_{1}$ and $d_{2}$ of {\em GaAs} and {\em AlAs} layers measured by X-ray diffraction and the 2D electron concentrations determined by Hall-measurements. For $d_{1}$, sample A is 270 \AA, B is 455 \AA and C is 341 \AA. For $d_{2}$, sample A is 290 \AA, B is 589 \AA and C is 543 \AA. For $n_{s}$ at 10 $K$, sample A is $4.7 \times 10^{11} cm^{-2}$, B is $2.7 \times 10^{11} cm^{-2}$ and C is $2.4 \times 10^{11} cm^{-2}$. Each sample contains 10 periods with a cover layer of ${\em{Al}}_{0.2}{\em{Ga}}_{0.85}{\em{As}}$ of thickness $\sim$50 \AA.

Raman spectra are taken at $\sim$10 $K$ using continuous flow {\em He}-cryostat in a backscattering configuration. The result shown in Fig.~5 are obtained in near resonance by using 6764 \AA (1.833 eV) line of krypton ion-laser which is close to the ($E_{0} + \Delta_{0}$) gap of {\em GaAs} ($\sim$1.88 eV).

\begin{figure}[htb]
\centerline{\psfig{figure=fig2_5.ps,height=2.5in}}

\caption[]{\em{Raman Shift}}
\end{figure}

Fig.~5 shows Raman spectra of three MQWs in polarized Z($X'X'$)\=Z scattering configuration. A sharp peak occurs at 295 $cm^{-1}$, corresponding to the longitudinal optical phonon of the {\em GaAs} layers. Broad bands at lower and higher frequencies of the {\em GaAs} longitudinal optical phonon are seen. There are centred at around 230 $cm^{-1}$ and 360 $cm^{-1}$ for sample A, 250 $cm^{-1}$ and 360 $cm^{-1}$ for sample B and 230 $cm^{-1}$ and 330 $cm^{-1}$ for sample C. These bands are assigned to the coupled collective intersubband plasmon longitudinal optical phonon modes.

\section{Conclusion}

Because of the lasers developed, the coherent Raman spectroscopy has made a very good prograss. Today's techniques include a variety of designs, for example, CARS, CARE (Coherent Anti-Stokes Raman Ellipsometry), ASTERISK, RIKES (Raman-induced Kerr-Effect Spectroscopy), SRS (Stimulated Raman gain Spectroscopy), and PARS (Photo-Acoustic Raman Spectroscopy), achieve the various goals in the different fields. It is a very useful tool to study a number of researches, such as solid state, semiconductors, superconductors, phase transitions, surface applications, molecular structures, inorganic and organic materials and medical applications.


\begin{thebibliography}{99}

\bibitem{1} A.\ Sood, J.\ Menendez, M.\ Cardona and K.\ Ploog, Phys. Rev. Lett.\ {\bf 54}, 2111(1985).
\bibitem{2} N.\ Bloembergen, {\em Nonlinear Optics}, Benjamin Press(1965).
\bibitem{3} G.\ Eesley, {\em Coherent Raman Spectroscopy}, Pergamon Press(1981).
\bibitem{4} M.\ Omar, {\em Elementary Solid State Physics}, John Wiley \& Sons(1975).
\bibitem{5} H.\ Ibach and H.\ L\"uth, {\em Solid-State Physics}, Springer-Verlag(1991).
\bibitem{6} S.\ Mukamel and Y.\ Yan, {\em Raman Spectroscopy and Four Wave Mixing; A Probe for Solvation Dynamics}, in Recent Trends in Raman Spectroscopy (Edited by S.\ Banerjee and S.\ Jha), p.160, World Scientific Publishing(1989).
\bibitem{7} A.\ Sood, M.\ Cardona, A.\ Fischer and K.\ Ploog, {\em Raman Scattering from Plasmons in Doped GaAs-AlAs Multiple Quantum Wells}, in Recent Trends in Raman Spectroscopy (Edited by S.\ Banerjee and S.\ Jha), p.289, World Scientific Publishing(1989).
\bibitem{8} J.\ Taran, {\em CARS, Non-Linear Raman Spectroscopy, and the Frontiers of the Day}, in International Conference on Raman Spectroscopy (Edited by R.\ Clark and D.\ Long), sec. PL8, p.31, John Wiley \& Sons(1988).

\end{thebibliography}

\end{document}
