Our modelling approach is described in several papers and books (see References). Using a “paraxial” or small-angle approximation throughout, we express the propagating electromagnetic fields as linear combinations of free-space modes (e.g. Hermite-Gaussian φpq) outside the guide, and guide modes (e.g. sinusoidal EHmn) inside the guide. At the boundaries (the exit/entrance apertures of the guide) we define “overlap integrals” or “coupling coefficients” which allow a change of basis. For example, the EH11 fundamental guide mode has the amplitude profile cos (πx/(2a)), that is a single lobe peaking at the guide centre, with zero amplitude beyond the guide limits [-a, a]. When leaving the guide aperture, this field can be represented in free space as a linear combination Σapq,11 ψpq(x,y,z) where
apq,11 = the amplitude coupling coefficient (possibly complex) from EH11 to ψpq(x,y,z)
φpq(x,y,z) = the scalar field for the Hermite-Gaussian beam at any point (x,y,z) in free space beyond the guide
z = the optical-axis distance relative to the coordinate origin; usually z = 0 is conveniently assigned to the guide aperture.
The waist plane can be chosen anywhere convenient, so that in general the free-space modes have curved wavefronts at the change-of-basis plane. Here we keep the usual choice z = 0 for both the aperture and the beam waist plane. Mistakes can happen with ψpq origins and phase shifts; my first independent scientific paper, in 1985, corrected a small bug which had sat in the literature for 15 years. This was a hint of the future time, effort and strife spent fixing mistakes; and colleagues could say the same of my work.
The Hermite-Gaussian beams have one extra parameter of scale, usually called the “beam waist radius” w0 (the radial distance over which the Gaussian intensity profile of the ψ11 mode decays by 1/e2). Their fields eventually tail off at radial distances of many w0, but they do not vanish; we make the particular sum of their fields (the above linear combination) as small as we wish outside [-a, a] by extending the limits of p and q as necessary.
These (p, q) and (m, n) are the pairs of integers used to notate the two dimensions of the mode geometry in the transverse plane. We may have considerable freedom to choose our sets of modes; the most common are perhaps the Hermite-Gaussians (with Cartesian x and y coordinates) and the Laguerre-Gaussians (with polar coordinates), but there are others. Sometimes there is an obvious fit with the geometry of the problem, as with rectangular-cross-section guides and Hermite-Gaussians. Sometimes an apparent symmetry in a resonator is annoyingly overcome in practice by small misalignments or imperfections or internal optical elements, so that for example the lasing modes in a resonator with circular-bore guides and well-aligned well-centred mirrors do not show circular symmetry.
Usually it is clear that there is a preferred local axis of optical propagation defined by the waveguide and/or the cavity mirrors. If there are tilts and bends, we may adjust the propagation directions of the sets of modes by including a phase factor such as exp(jkφx) in the overlap integral if the angle φ is small (<<1). Complications such as folded resonators with curved mirrors can be treated with more effort.
Computers do not represent infinite summations or infinitely fine meshes, so there will always be errors, likely to grow as the values of these integers or “mode orders” p, q, m and n increase. The number of modes (in one basis) required to account for a certain fraction of the power in a given mode (of the other basis) depends on the mode order of the latter and on the scale parameter w0/a. In a multimode resonator model it is not necessary (and may be unwise) to use the value of w0/a that maximises single-mode coupling efficiency. A value around 0.65-0.7 will give a fundamental mode ψ11 that closely approximates EH11, but we then have to work hard to calculate contributions from higher-order ψpq in order to account for the remaining 1 % or 2 % of the total. It may be better to use a lower value 0.2-0.5 in order to reduce the number of modes necessary for a reasonably accurate expansion of all the resonator modes of interest (not just the fundamental). Using ordinary software quadrature routines, we can usually obtain reliable results for at least the first few resonator modes of interest.
Once the coupling and propagation coefficients are found for all parts of the resonator round trip, including the exit/entrance apertures and any small tilts or other paraxial distortions, we can form them into matrices, multiply these together (in the correct sequence), and diagonalise the resulting “round-trip matrix” to find its “eigenvectors” (resonator modes, expressed as linear combinations of the basis modes) and their “eigenvalues” (round-trip phase shifts and losses). The answers will be approximate, but modern software routines for matrix manipulation are unlikely to introduce errors greater than those we have already implied with our first-order expressions and truncated mode expansions.
So, making various checks, we try to ensure that significant power is not (fictitiously) acquired or lost through numerical difficulties in the integration or matrix algorithms, or series truncation in the cascaded changes of basis.
The computed eigenvectors and eigenvalues refer to our particular initial plane. An obvious choice is the plane of the outcoupling mirror, so that the eigenvectors represent the field patterns expected at a detector there. But the starting point is arbitrary. We multiply the round-trip matrices, including all of them in the correct order, and the computed results will then refer to the chosen initial plane, chosen direction of propagation, and chosen basis set. My 1980s FORTRAN codes, and I expect similar codes from other researchers, were appropriate for finding resonator designs that (say) maximise EH11 content within the guide, or TEM00 content outside the guide. We can transform the output (for our given initial plane) to any other internal or external plane, by once more multiplying by the appropriate propagation and coupling matrices.
Below are some introductory and tutorial results, with remarks on the research background. When writing a note of appreciation about Rodney Loudon (https://www.malvernoptics.co.uk/rodney-loudon-1934-2022), I revisited this resonator subject with an eye on another short note for MRATHS, and prepared new illustrations with recent MATLAB versions of the 1980s-1990s code. I include a section on a possibly novel trigonometrical approach which avoids computer matrix calculations and lengthy algebra, and uses the standard fact that eigenvectors remain “parallel” after round-trip matrix “rotation” (in an appropriately defined space or plane). There is little here that I did not say as clearly as I could in the thesis, RSRE technical memos and published papers, but the historical context may be interesting for MRATHS and other readers. Perspectives differ sharply about civil service and MOD activities, and (though this reduces the gossip value) I have tried to be careful with classified/commercial/personal matters.
Chris Hill
Malvern Lidar Consultants
July-August 2025
