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\begin{document}
\title{Yttrium Aluminum Garnet}
\author{ Scott Freeman\\
         \\
         Department of Physics\\
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

\date{December 8, 1995}
\maketitle
\begin{abstract}
Yttrium Aluminum garnet (YAG) is commonly employed as a laser host for a 
variety of rare earth elements, especially neodymium.  This paper examines
the crystal structure of Nd:YAG and the physical and thermal properties which
make it such an excellent laser host. Finally, some of the features of the 
Nd:YAG luminescence and emission spectra are considered.    
\end{abstract}

\newpage

\section{Introduction}
\hspace{1cm} 
YAG was first synthesized in 1951 and was first employed as a laser host for
a research group at Bell Laboratories in 1964[1].  Today Nd:YAG is the most 
widely used solid state laser medium, with a wide variety of applications in 
medicine, materials processing, and the military.  It is a clear, hard, 
optically isotropic crystalline substance, possessing essentially all of the 
characteristics materials used in solid state lasers should exhibit, including 
good mechanical strength, a high degree of transparency in the portion of the 
spectrum where the laser operates, good thermal properties, and the ability 
to sustain a precise optical polish[2].

\section{The Nd:YAG Crystal}
\hspace{1cm}
Chemically, YAG is $Y_3$$Al_5$$O_{12}$. Like other garnet systems, it 
crystallizes in the cubic group, space group $O_h$(10) Ia3d.The unit cell of a 
compound with the garnet structure $A_3$$B_5$$O_{12}$ is shown in Figure 1.  
The unit cell of such a compound contains eight formula units. The B atoms 
occupy two types of sites with different oxygen coordination: 16 octahedral a 
positions and 24 tetrahedral d positions.  The a sites form a body centered 
cubic lattice while the 24 A ions lie in dodecaheral c positions.  These ions 
are surrounded by eight oxygen ions which form the corners of a distorted cube.
The replacement ions which are responsible for the laser properties of the 
crystal mainly occupy the c positions.  The site symmetry at the place of the 
yttrium ions is orthorhombic and the substituted neodymium ions experience a 
low symmetry crystal field in the YAG host lattice[3].

\begin{figure}[htb]

\centerline{\psfig{figure=figure1.ps,height=2.5in}}

\caption[]{\em{Motif of Garnet Crystal Structure[3]}}
\end{figure}  

\hspace{1cm}
In Nd:YAG about 1 percent of the $Y^{3+}$ ions are substituted by $Nd^{3+}$ 
ions.  The atomic radii of the two ions differ by about 3 percent, so as more 
Nd ions are added the crystal becomes strained and YAG lattice can become 
distorted. Consequently, the concentration of Nd ions in Nd:YAG can have a 
considerable effect on the quality of the crystal's laser properties[4]. 
Specifically, the cubic structure of YAG favors a narrow fluorescent 
linewidth, which results in high gain and low threshold for laser operation, 
so deviations from the YAG crystal structure can impair the quality of the 
laser[4]. When the concentration of Nd ions becomes too great, the laser 
begins to suffer from flourescent line broadening and reduction in 
fluorescent lifetimes[5].   


\section{Physical and Thermal Properties}
\hspace{1cm}
The YAG structure is highly stable from the lowest temperatures up to its
melting point.  No transformations have ever been reported in its solid phase.
The complex cubic lattice does not have any apparent easy slip directions.  It 
has low dislocation densities and very small strain and inhomogeneity as long 
as the crystals are grown sufficiently slowly. Nd:YAG also has a high 
resistance to thermal induced stress fracture[3]. A table of physical,
thermal, and and optical properties of Nd:YAG are givien in Table 1[4].

\begin{table}
\begin{center}
\begin{tabular}{|c|c|}
\hline
Melting point&1970 C\\
Mohs Hardness&8.5\\
Young's Modulus&3*$10^3$ kg-$cm^{-2}$\\
dislocation denities& approx.$10^{-2}$$cm^{-2}$\\
Thermal Expansion coefficient&\\
\hspace{.25cm}[100]orientation&8.2*$10^-6$ $C^{-1}$, 0-250 C\\
\hspace{.25cm}[110]orientation&7.7*$10^-6$ $C^{-1}$, 0-250 C\\
\hspace{.25cm}[111]orientation&7.8*$10^-6$ $C^{-1}$, 0-250 C\\
Index of refraction&1.82 (at 1.0$\mu$m)\\
Scatter Losses&$\alpha_{sc}$=1.82\\
Thermal Conductivity&0.13 W-$cm^{-1}$-$K^{-1}$ (at 300K)\\
Thermal Expansion&7.5 $K^{-1}$ (at 300K) \\
Specific Heat&0.59 W-s$(g-K)^-{1}$ (at 300K)\\
\hline
\end{tabular}
\end{center}
\caption[]{\em{Some Physical, Optical, and Thermal Prpoerties of Nd:YAG]}}
\end{table}  

\hspace{1cm}
One of the more significant thermal properties of Nd:YAG is its high thermal 
conductivity relative to other neodymium hosts.  This allows for the rapid 
removal of wasted heat due to inefficient excitation in the medium and 
enables to Nd:YAG laser to produce a continuous beam of high quality[6]. 
Such continous operation is virtually impossible to achieve with other solid 
state lasers, whose lower thermal conductivity restricts them to operating 
for only short pulses.

\section{Luminescence and Spectral Emission Properties of Nd:YAG Crystals}
\hspace{1cm}
       

A figure of the most significant energy levels associated with the Nd:YAG 
laser are shown in Figure 2.  The most prominent of these transitions is the 
1.064$\mu$m line from the $R_2$ component of the $^4F_{3/2}$ energy level to 
the $Y_3$ component of the $^4I_{11/2}$ level. The upper level is a 
metastble state with a lifetime of .203ms.  The levels shown in this 
figure are all associated with the inner 4f shell of Nd which is partially 
screened from external fields by the 5d and 6s outer shells[7].     

\begin{figure}[htb]

\centerline{\psfig{figure=figure2.ps,height=2.5in}}

\caption[]{\em{Partial Energy Level Diagram of Nd:Yag[4]}}
\end{figure}

Although it is the Nd ions in Nd:Yag are responsible for the the laser 
transitions in the medium,  the YAG host does have an effect on the Nd ion's 
energy levels since the Nd ions interact with the ctrystal field.  These 
interactions with the crystal field are responsible for Stark splittings of 
the Nd energy levels with $J > 3/2$ [3]. The principle contributor to the 
crystal field is its strong cubic componenent while axial and orthorhombic 
components are responsible for the fine structure of the Nd ion's energy 
spectrum. The cubic component of the Crystal field arises from exchange and 
Coulomb interactions between the $Nd^{3+}$ ion`s 4f electrons and its eight 
nearest neighbor $O^{2-}$ ions.  

\begin{figure}[htb]

\centerline{\psfig{figure=figure3.ps,height=2.5in}}

\caption[]{\em{Concentration dependence of luminescence of 1.06415 $\mu$m 
line intensity (1) and excitation threshold (2) of the A emission line for 
a Nd:YAG crystal $^4F_{3/2}$ to $^4I^{11/2}$ transition at 300K.[3]}}
\end{figure}

Not only are the energy level splittings produced by the crystal field 
significant in their own right, but they can be employed to answer some 
critical questions about the operation of the Nd:Yag laser. For example, the 
optimum concentration of $Nd^{3+}$ ions in a Nd:YAG crystal can be determined
by studying the luminescence intensity of the 9382 and 9458 Angstrom lines due
to transitions to the highest stark component of the $^4I_{9/2}$ state. This 
component is spaced far from the ground state and is essentially unpopulated 
at room temperature. A plot of the luminescence intensity dependence for the 
most intense line of the $^4F_{3/2}$ to $^4I_{11/2}$ transition.  As is 
evident from the graph, the maximum luminescence intensity is at a neodymium 
concentration of about 1.1 percent.  Similarly, The laser threshold energy can
be plotted in the same figure as a function neodymium concentration.  As the 
figure indicates, the minimum excitation threshold  is at nearly the same 
concentration as the maximum luminescence intensity, indicating that a 
crystal doped with about 1.1 percent neodymium is optimal laser medium[3].    
       
\section{Conclusions}

While it is not the ideal laser medium, Nd:YAG has a number of desirable 
properties, particularly, thermal properties which which make it one of the 
best laser media used today.  No other laser crystal has achieved the 
widespread commercial success of Nd:YAG, and over thiry years after it was 
first discovered to be a laser medium, it continues to find new applications 
in science and technology.
                      
\begin{thebibliography}{99}
\bibitem{refA} J.E. Geusic, H.M. Marcos, L.G. Van Uitert, App. Phys. Lett. 10,
182 (1964).
\bibitem{refB} CRC Handbook of Laser Science and Technology (CRC PRess, Boca 
Raton).
\bibitem{refC} Laser Crystals, Alexander A. Kaminskii (Springer-Verlag, New 
York).
\bibitem{refD} Solid State Laser Engineering, Walter Koechner (Springer-
Verlag, New York).
\bibitem{refE} J.B. Willis, M. Dixon, Journal of Crystal Growth 3 (1968).
\bibitem{refF} Handbook of Optics, Optical Society of America (Mcgraw-Hill,
New York).  
\bibitem{refG} Lasers and Nonlinear Optics, B.B. Laud (John Wiley and Sons, 
York). 
\end{thebibliography}

\end{document}