## 2 Review of Cryogenic-Cooling Benefits
### Spectroscopic, Lasing, Linear and Nonlinear Optical Parameters
#### 2.7.2 Linear and Nonlinear Optical Parameters
If we consider an input pulse if field \(E_{\text{in}}(t)\) with spectrum \(A_{\text{in}}(\omega)\) propagating through a dispersive dielectric crystal, the output pulse can be written as the Fourier relationship  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]  
\(\omega\) is the angular frequency, and \(\psi\left(\omega\right)\) is the spectral phase at frequency \(\omega\). Upon passage through a medium of length L, we can relate the spectral phase to the propagation constant of the medium \(\beta(\omega)\), through  
\[\psi\left(\omega\right)=-\beta\left(\omega\right)\cdot L \tag{82}\]  
It is commonplace [70] to expand both \(\psi\left(\omega\right)\) and \(\beta\left(\omega\right)\) about a central frequency \(\omega_{0}\) using a Taylor series approach, resulting in  
\[\psi\left(\omega\right)=\psi\left(\omega_{0}\right)+\frac{\partial\psi}{ \partial\omega}\cdot\left(\omega-\omega_{0}\right)+\frac{1}{2}\frac{\partial^ {2}\psi}{\partial\omega^{2}}\cdot\left(\omega-\omega_{0}\right)^{2}+\frac{1} {6}\frac{\partial^{3}\psi}{\partial\omega^{3}}\cdot\left(\omega-\omega_{0} \right)^{3}+\ldots \tag{83}\]  
or,  
\[\psi\left(\omega\right)=\psi\left(\omega_{0}\right)+\psi_{1}\cdot\left(\omega -\omega_{0}\right)+\frac{1}{2}\psi_{2}\cdot\left(\omega-\omega_{0}\right)^{2} +\frac{1}{6}\psi_{3}\cdot\left(\omega-\omega_{0}\right)^{3}+\ldots \tag{84}\]  
\(\beta(\omega)\) is then given by  
\[\beta\left(\omega\right)=\beta\left(\omega_{0}\right)+\beta_{1}\cdot\left( \omega-\omega_{0}\right)+\frac{1}{2}\beta_{2}\cdot\left(\omega-\omega_{0} \right)^{2}+\frac{1}{6}\beta_{3}\cdot\left(\omega-\omega_{0}\right)^{3}+\ldots \tag{85}\]  
and we write in general  
\[\psi_{n}=\frac{\partial^{n}\psi}{\partial\omega^{n}}\cdot\beta_{n}=\frac{ \partial^{n}\beta}{\partial\omega^{n}} \tag{86}\]  
The derivatives are evaluated at \(\omega=\omega_{0}\), and we also have  
\[\psi_{n}=\ -\ \beta_{n}\cdot L \tag{87}\]  
\(\beta(\omega)\) can also be expressed as  
\[\beta\left(\omega\right)=\frac{\omega\ n\left(\omega\right)}{c} \tag{88}\]  
where \(c\) is the speed of light. Using Equation (88) we can then calculate the dispersion parameters \(\beta_{i}\) as  
\[\beta_{1}=\frac{d\beta}{d\omega}=n\left(\omega\right)+\omega\frac{dn}{d\omega} \tag{89}\]  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]0  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]1  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]2  
Most index of refraction dispersion equations are expressed in terms of wavelength, rather than angular frequency. The most common dispersion equation used is the Sellmeier equation, often written in its' most general form as  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]3Here n is the wavelength dependent index of refraction, and \(A_{0}\), \(A_{i}\) and \(\lambda_{i}\) are fitting constants. In many cases \(A_{0}=~{}1\), and i is usually \(\leq 4\). Another equation often found in the literature is the Laurent series formula,  
\[E_{out}\left(t\right)=\left(\frac{1}{2\pi}\right)\cdot\int A_{in}\left(\omega \right)e^{i\omega t}e^{i\psi\left(\omega\right)}d\omega \tag{81}\]4  
where \(A\), \(B\), and \(C_{i}\) are constants, and the index \(i\) takes the values 1, 2, 3, and 4. The Cauchy equation is also often used, which may be obtained by setting \(B=0\) in Equation (94).