Coupling of higher-order-modes

Note that much of this content is based on this review [12]

Solving the overlap integral

The coupling of a mode refers to how a spatial mode in one basis is represented in another; e.g. which sum of modes in the cavity basis \(q_2\) produces the \(\mathrm{HG}_{00}\) mode in the \(q_1\) basis. Hermite–Gauss modes are coupled whenever a beam is not matched or aligned to a cavity or beam segment. This coupling is sometimes referred to as scattering into higher-order modes because in most cases the laser beam is a considered as a pure \(\mathrm{HG}_{00}\) mode and any mode coupling would transfer power from the fundamental into higher-order modes. However, in general every mode with non-zero power will transfer energy into other modes whenever mismatch or misalignment occur, and this effect also includes the transfer from higher orders into a low order.

To compute the amount of coupling the beam must be projected into the base system of the cavity or beam segment it is being injected into. This is always possible, provided that the paraxial approximation holds, because each set of Hermite–Gauss modes, defined by the beam parameter at a position \(z\), forms a complete set. Such a change of the basis system results in a different distribution of light power in the new Hermite–Gauss modes and can be expressed by coupling coefficients that yield the change in the light amplitude and phase with respect to mode number.

Let us assume that a beam described by the beam parameter \(q_1\) is injected into a segment described by the parameter \(q_2\). Let the optical axis of the beam be misaligned: the coordinate system of the beam is given by (\(x, y, z\)) and the beam travels along the \(z\)-axis. The beam segment is parallel to the \(z'\)-axis and the coordinate system (\(x', y', z'\)) is given by rotating the (\(x, y, z\)) system around the \(y\)-axis by the misalignment angle \(\gamma\). The amplitude of a particular mode \(\mathrm{TEM}_{nm}\) in the beam segment is then defined as:

\[u_{n m}(x,y;\,q_2)\exp{\left(i(\omega t -k z)\right)}=\sum_{n',m'}k_{n,m,n',m'}u_{n' m'}(x,y;\,q_1)\exp{\left(i(\omega t -k z')\right)}, \]

where \(u_{n' m'}(x,y;\,q_1)\) are the HG modes used to describe the injected beam, \(u_{n m}(x,y;\,q_2)\) are the “new” modes that are used to describe the light in the beam segment and \(k_{n,m,n',m'}\) is the coupling coefficient from each \(\mathrm{TEM}_{n'm'}\) into \(\mathrm{TEM}_{nm}\).

Note that including the plane wave phase propagation within the definition of coupling coefficients is important because it results in coupling coefficients that are independent of the position on the optical axis for which the coupling coefficients are computed.

Using the fact that the HG modes \(u_{n m}\) are orthonormal, we can compute the coupling coefficients by the overlap integral [24]:

\[k_{n,m,n',m'}=\exp{\left(i 2 k z' \sin^2\left(\frac{\gamma}{2}\right)\right)}\int\!\!\!\int\!dx'dy'~ u_{n' m'}\exp{\left(i k x' \sin{\gamma}\right)}~u^*_{n m}. \]

Since the Hermite–Gauss modes can be separated with respect to \(x\) and \(y\), the coupling coefficients can also be split into \(k_{n m n' m'}=k_{n n'}k_{m m'}\). These equations are very useful in the paraxial approximation as the coupling coefficients decrease with large mode numbers. In order to be described as paraxial, the angle \(\gamma\) must not be larger than the diffraction angle. In addition, to obtain correct results with a finite number of modes the beam parameters \(q_1\) and \(q_2\) must not differ too much.

The integral can be computed directly using numerical integration methods. However, this can potentially be computationally very expensive depending on how difficult the integrand is to evaluate and complex it is. The following part of this section is based on the work of Bayer-Helms [24] and provides an analytic solution to the integral. In [24] the above integral is partly solved and the coupling coefficients are given by multiple sums as functions of \(\gamma\) and the mode mismatch parameter \(K\), which is defined by

\[K=\frac{1}{2} (K_0+iK_2), \]

where \(K_0=(z_R-z_R')/z_R'\) and \(K_2=((z-z_0)-(z'-z_0'))/z_R'\). This can also be written using \(q=i\zr +z-z_0\), as

\[K=\frac{i (q-q')^*}{2 \Im{q'}}. \]

The coupling coefficients for misalignment and mismatch (but no lateral displacement) can then be written as

\[k_{n n'}=(-1)^{n'} E^{(x)} (n!n'!)^{1/2} (1+K_0)^{n/2+1/4} (1+K^*)^{-(n+n'+1)/2}\left\{S_g-S_u\right\}, \]

where

\[\begin{array}{l} S_g=\sum\limits_{\mu=0}^{[n/2]}\sum\limits_{\mu'=0}^{[n'/2]} \frac{(-1)^\mu \bar{X}^{n-2\mu}X^{n'-2\mu'}}{(n-2\mu)!(n'-2\mu')!} \sum\limits_{\sigma=0}^{\min(\mu,\mu')}\frac{(-1)^\sigma \bar{F}^{\mu-\sigma} F^{\mu'-\sigma}} {(2\sigma)! (\mu-\sigma)! (\mu'-\sigma)!},\\ S_u=\sum\limits_{\mu=0}^{[(n-1)/2]}\sum\limits_{\mu'=0}^{[(n'-1)/2]} \frac{(-1)^\mu \bar{X}^{n-2\mu-1}X^{n'-2\mu'-1}}{(n-2\mu-1)!(n'-2\mu'-1)!} \sum\limits_{\sigma=0}^{\min(\mu,\mu')}\frac{(-1)^\sigma \bar{F}^{\mu-\sigma} F^{\mu'-\sigma}} {(2\sigma+1)! (\mu-\sigma)! (\mu'-\sigma)!}. \end{array} \]

The corresponding formula for \(k_{m m'}\) can be obtained by replacing the following parameters: \(n\rightarrow m\), \(n'\rightarrow m'\), \(X,\bar{X}\rightarrow 0\) and \(E^{(x)}\rightarrow 1\) (see below). The notation \([n/2]\) means

\[\left[\frac{m}{2}\right]=\left\{ \begin{array}{ll} m/2 & \text{if}\ m\ \text{is even,}\\ (m-1)/2 & \text{if}\ m\ \text{is odd.} \end{array}\right. \]

The other abbreviations used in the above definition are

\[\begin{array}{l} \bar{X}={(i \zr'-z')\sin{(\gamma)}}/({\sqrt{1+K^*}w_0}),\\ X={(i \zr+z')\sin{(\gamma)}}/({\sqrt{1+K^*}w_0}),\\ F={K}/({2(1+K_0)}),\\ \bar{F}={K^*}/{2},\\ E^{(x)}=\exp{\left(-\frac{X\bar{X}}{2}\right)}. \end{array} \]

Scattering matrices

Finesse computes and stores scattering matrices to describe how spatial modes couple with respect to each other between beam parameter bases. A scattering matrix is simply a square matrix of coupling coefficients where each coefficient represents how one mode couples into another - i.e. the scaling between the modes in terms of both the amplitude and phase of the field.

Fig. 10 shows the four scattering matrices present at a mirror M. For a mode-matched and aligned system, each of these matrices are just identity matrices.

Fig. 10 Scattering matrices for a mirror. K11, K22 are the reflection coupling matrices whilst K12 and K21 are the transmission coupling matrices.

To compute the overall field couplings at the mirror M, Finesse multiplies the corresponding element of the mirrors’ local coupling matrix by the associated scattering matrix. For example, to compute the coupling of the reflected field from the first surface, \(m_{11}\), the equation,

\[m_{11} = r K_{11} \exp{\left(i\varphi\right)}, \]

is applied, where \(r = \sqrt{R}\) with \(R\) as the refectivity of the mirror and \(\varphi\) is the phase.

API for coupling coefficient calculations

Functions for computing and modifying scattering matrices can be found in this submodule:

finesse.knm

Library of coupling coefficient data structures and calculations.