Mode Matching

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In order to maximize coupling between a free space Gaussian beam and a specific mode (typically TEM00) of an optical device (single mode fiber, Fabry-Perot Resonator etc) it is necessary to match the parameters of the incoming Gaussian beam to the parameters of the target mode. We assume here that the parameters ( $q 1, q 2$ ) of the two modes are known. The parameters can be found using two razor blade measurements. We present two methods of mode matching, either using a single lens or using a pair of lenses in a telescope configuration. The latter has the advantage that it can be considerably shorter than the single lens configuration and can use shorter lenses. The two lens technique will piggyback on the one lens technique, however, and so we present that technique first.

1 Single Lens Mode Matching
- 1.1 Existence
- 1.2 Matlab script
2 Two Lens Mode Matching
- 2.1 Matlab Script

Single Lens Mode Matching

The simplest method of matching is to use a single converging lens placed between the two waists. The ABCD matrix for this system is

$ABCD = \begin{pmatrix} 1-\frac{L_2}{f} & L_2+L_1 \left(1-\frac{L_2}{f}\right) \\ -\frac{1}{f} & 1-\frac{L_1}{f} \end{pmatrix}$

where $L 1$ and $L 2$ are the distance to the lens from the first waist and from the lens to the second waist. $f$ is the focal length of the lens. This system will then transform the Gaussian beam parameters by

$q_2=\frac{A q_1+b}{C q_1+D}.$

Substituting in from the ABCD matrix and solving (easy with Mathematica) gives

$L_1=\frac{f q_2+\sqrt{|q_1 q_2| f^2+q_1^2 q_2^2}}{q_2}$

and

$L_2=f-q_2-\frac{f^2 q_2}{f q_2+q_1 q_2-f q_2-\sqrt{q_1 q_2 f^2+q_1^2 q_2^2}}.$ Here we have chosen the solution that corresponds to the physical problem that we are interested in. The trick to solving this system is to note realize that the equation is complex and can be thought of as two separate equations.

Existence

Solutions exist to these equations only when the radical is purely imaginary. Thus the focal length must satisfy the relation

$f>\sqrt{q_1 q_2}.$

Matlab script

getModeMatchLength.m

Two Lens Mode Matching

We can make a considerably shorter mode matching network by employing two lenses. I will present the following variant of the technique. We use a network where the beam waist due to the first lens is between the first and second lenses. The matching network is then broken down as follows. The first lens focuses the incoming beam at a small, short waist. The second lens then mode-matches this spot to the desired output mode. The second part is just the single lens matching that we had above.

We will assume that the experimenter wants to choose $L 1, f 1,$ and $f 2$ . Our task is to find $L 2$ and $L 3$ . We solve the transformational ABCD equation from above for the parameters $q I N T$ and $L I N T 1$ giving:

$L_{INT1}=-\frac{f_1(f_1L_1-L_1^2+q_1^2)}{f_1^2-2L_1f_1 + L_1^2 - q_1^2},\qquad q_{INT}=\frac{f1^2q_1}{f_1^2-2L_1f_1 + L_1^2 - q_1^2}.$

The single lens matching technique used above is now used to find $L I N T 2$ ( $L 2 = L I N T 1 + L I N T 2$ ) and $L 3$ using $q I N T$ as the input beam parameter.