# Compact Aberration-Free Relay-Imaging Multi-Pass Layouts for High-Energy Laser Amplifiers
## 2 The ABCD-Matrix Model  
Using the paraxial approximation, any classical optical system containing only translations of lengths \(d\) and imaging elements with focal length \(f\) can be described by the matrices \(T\) for translation and \(L\) for the imaging element:  
\[T= \begin{bmatrix}1&d\\ 0&1\end{bmatrix} \qquad L= \begin{bmatrix}1&0\\ -1/f&1\end{bmatrix}  \tag{1}\]  
In the following, we assume that a round-trip in an imaging multi-pass amplifier is described by alternating free space propagation and ideal imaging elements. Hence, the matrix representing one round-trip in a system of \(n\) elements is calculated by:  
\[G_{n} = T_{n+1}\cdot L_{n}\cdot T_{n}\cdot L_{n-1}\cdot\ldots\cdot L_{1} \cdot T_{1} \tag{2}\]  
An imaging system suitable for allowing several round-trips has to restore both the spatial and the angular distribution after each round-trip. Hence, it is required that the optical system matrix equals either the unity matrix \(U\) or in the case of a symmetric profile, the negative unity matrix \(-U\). If we furthermore demand a constant small angle add-on per pass of the system to separate the different beam passes geometrically, e.g., by introducing a little tilt of the flat optics in the image plane, the latter solution has to be omitted, as the alternating direction of the angular add on will prevent a summation. Under these constraints, the prescriptions for building such an imaging system can be derived by solving the matrix equation \(G_{n}=U\). In the following, we will give the results for \(n=1\ldots 3\). Solutions for systems of more than three elements are no longer distinct, as there are too many degrees of freedom. Nevertheless, the used algorithm could be applied for these cases accordingly, especially as most of such systems can be described by a concatenation of systems with less than four imaging elements. In this case, the results presented in the following can be directly applied to the sub-systems. Starting with a system containing only one imaging element per round-trip, there is no solution, as the focusing or de-focusing effect of the single optic cannot be mitigated and will always alter the collimation state of the input beam. In the case of \(n=2\), there is again no solution that allows one to generate a positive unity matrix. Nevertheless, a solution is found for the negative unity matrix. Hence, double passing such a system back and forth will result in a unity matrix again. This is in a narrower sense a special case for four imaging elements. The solution in this case is:  
\[d_{1} = \frac{f_{1}f_{2}\cdot(f_{1}+f_{2})-d_{3}f_{1}^{2}}{f_{2}^{2}} \tag{3}\] \[d_{2} = f_{1}+f_{2}\] (4) \[f_{1} = f_{2} \tag{5}\]  
Here, \(d_{1}\) to \(d_{3}\) are the corresponding free space propagation distances, and \(f_{1}\) and \(f_{2}\) are the focal lengths of the imaging elements. This corresponds to the widely-used version of a double-passed 4f-telescope for a multiple imaging layout as, e.g., in [4; 6; 7]. We will refer to such a kind of system in the following as a Type I system. Schematic setups for optical systems of this type accomplishing three round-trips are exemplarily given in Figure 1.  
As this system needs to be double passed to fulfill the constraints, it is not really necessary to have a unity magnification, as was originally requested by the constraints used for deriving this solution. This renders Equation (5) unnecessary, but since all distances have to be \(\geq\)0, it follows from Equations (3) and (4) that \(f_{1}\) and \(f_{2}\) must be at least positive. For a system with three imaging elements, there are already too many degrees of freedom with respect to the four different distances to get a  
Figure 1: Schematic drawings of Type I systems with three round-trips based on lenses (**a**) and on mirrors (**b**) as imaging elements. The numbers in the arrows denote the sequence of passing beams. closed solution for each distance using our matrix model.