A new tool: TIFOSI (Transfer Matrix Fourier Domain Simulation); by J Harms

The goal of TIFOSI is to provide a simulation tool of quantum-scattering effects so that for example the impact of higher-order modes on quantum noise in squeezed fields can be studied. This work is well underway, but requires further understanding of the fundamentals of multi-mode squeezing. The main open challenge is to understand the operation of an OPA so that numerical problems can be clearly discriminated from the physical effects. Quantum fields need to be represented as matrices (classical fields as vectors), therefore the building blocks of TIFOSI are transfer matrices, and solutions of quantum fields cannot be found directly as stationary solution as is done in other Fourier simulations. However, as outlined in the following, TIFOSI has already been used to (re)produce results for classical problems.

One example is to calculate the eigenmodes of a cavity. A straight-forward method to simulate cavity eigenmodes is to first calculate the round-trip transfer matrix of fields inside the cavity, and then to evaluate the eigenvectors and eigenvalues of this matrix. Not all optical simulations provide direct access to transfer matrices. Especially Fourier domain simulations such as Oscar, SIS, and FOG evaluate fields as a stationary solution of a recursive field propagation. TIFOSI takes the approach known from various modal codes (MIST, Finesse), to calculate transfer matrices, and to use these matrices to calculate fields.

The main disadvantage of this approach, and probably the reason why it has not been done in the past, is that Fourier codes need a very large number of modes to obtain accurate results. For example, simulations with N^2 modes require transfer matrices with N^2 x N^2 components. It should be noted though that these transfer matrices are calculated by TIFOSI using almost entirely N x N representations of elementary transfer matrices. So whereas the final results for cavity reflection and transmission into and through the cavity are all N^2 x N^2 matrices, there is not a single multiplication of such large matrices in TIFOSI, and it is also not necessary to invert any matrix!

The classical version is fully functional already, and here we give a first idea of what this code can provide. The main advantage of a Fourier matrix code over modal matrix codes is that it can investigate systems where fields experience significant clipping loss (e.g. when scatter loss are of interest, or when aperture size affects the eigenmode spectrum of a cavity).

The attached zip file (TIFOSI; first eigenmode results) contains the first 100 eigenmodes of a linear cavity sorted in ascending order by their clipping loss. The cavity parameters were taken from Arm cavity loss with maps, however without applying the mirror maps, and also reducing the pixel number from 512^2 to 80^2. The title of each plot shows the phase accumulated by each eigenmode during one roundtrip, and the clipping (i.e. scatter) loss.

TIFOSI; numerical considerations

After comparison with Oscar, it turned out that the first TIFOSI version had a couple of numerical problems. The two reasons for this were that all transfer matrices were represented as N^2 x N^2 matrices, and that in certain simulations aliasing caused artificial contributions to the spatial modes. Both problems have been solved now. The most important step was to substitute a single-step 2D Fourier transform calculated by matrix multiplication with the field vector by an FFT. Since FFT can only be applied to the field itself and not directly transform a transfer matrix, the corresponding transfer matrix must be obtained by mapping all unit vectors of the field's basis. In Fourier domain, these unit vectors can either be plane waves, or beam pixels. The pixel basis is used in TIFOSI. Now the advantage is that most elementary transfer matrices such as propagation, transmission through and reflection from a mirror do not have to be represented as N^2 x N^2 matrices, but only as N x N matrices. There is only one step where it cannot be avoided to work with a large N^2 x N^2 matrix, which is the inverse of (1-R), where R is the cavity round-trip matrix. However, it turns out that R and (1-R)^-1 are used in two ways only. First, the eigenvectors and values of R define the cavity eigensystem, and the only expression with (1-R)^-1 is in multiplication with the input transmission of the cavity T1, which in Matlab language can be calculated as (1-R)\T1, i.e. without explicitly calculating the inverse of (1-R). Another advantage of an FFT based propagation of fields is that one can also apply a trick to completely avoid aliasing problems (as described in Applied Optics 45, p1102 (2006)). With these two changes, it was possible to obtain a very high level of numerical precision with TIFOSI. As a side effect, it has also decreased computing time significantly.

First TIFOSI results with mirror profiles

The plots in TIFOSI; eigenmodes with profile were calculated for the benchmark cavity in Arm cavity loss with maps using set A of mirror profiles, however with pixel number reduced from 512^2 to 96^2. Comparing with the first eigenmode results, one finds that the spectrum of the first 100 eigenmodes does not change significantly by adding a random mirror profile. The eigenvalues are practically the same, there are only small changes of the shape of some eigenmodes, and in the ordered sequence of eigenmodes, only a few with very similar eigenvalues exchanged places. Note that the round-trip loss calculated with TIFOSI is only 31ppm. This should be consistent with the higher estimates (around 43ppm) obtained by the other simulations since round-trip loss increases with increasing pixel number.

NOTE THAT THE LOSS FUNCTION RESULTS ARE PRONE TO BEING AFFECTED BY SPATIAL ALIASING. MORE AGGRESSIVE METHODS TO SUPPRESS ALIASING STILL NEED TO BE TESTED TO VALIDATE THE LOSS FUNCTION RESULTS. NONETHELESS, THE RESULTS DO NOT SHOW CLEAR SIGNS OF ALIASING PROBLEMS.

TIFOSI loss function

In the study of scatter loss in cavities, a very useful result is the loss function. It is calculated as follows. A symmetric aberration a(r) = a0*cos(2*pi*r/s) is added to the unperturbed mirror profile (typically applied to "ETM" only). Here r is the radial coordinate of the mirror with origin at the center of the mirror, a0 is the amplitude of the aberration, and s is the perturbation scale. There are two regimes that we believe to understand intuitively, if s is very small, then light scattered from the mirror will be almost completely lost since it is scattered into large angles alpha~lambda/s. If s is larger, then we would expect that it generates higher-order mode content in the cavity. Most of the energy in these modes may stay inside the cavity, but increased clipping loss could still occur. For very large perturbation scales though, negligible scattering of the fundamental mode into HOMs is generated and therefore the round-trip loss should be similar to the loss of the fundamental mode, which can typically be neglected.

The total loss from small-scale (large-angle) scattering should obey the "golden rule": loss_gold = (4*pi*(a0/sqrt(2))/lambda)^2. Since this loss corresponds to all of the scattered light being lost, it would be somewhat surprising if one ever observed scatter loss larger than loss_gold. This however is exactly what was found in some scattering simulations. The excess loss always occurs around very specific aberration scales. This suggests that a careful balance between certain parameters is necessary for this effect to occur. People have no good explanation for this yet, but it is possible that it is related to some kind of impedance matching condition between fields that may include the input field, as well as various higher-order modes (that couple in the basis of the input beam; naturally, eigenmodes of the cavity don't couple).

The first loss function calculated with TIFOSI is shown here: TIFOSI; first loss function It was calculated with a0 = 1e-9m, and a near confocal cavity of length 1m, aperture 3mm, simulation window of length 6mm, and 80^2 pixels. The power transmission of the input mirror is 1ppm, whereas the output mirror is fully reflective. The beam waist is 0.4mm. The dashed horizontal line in the plot represents the golden rule, which nicely approximate the round-trip loss at shorter aberration wavelengths. The loss falls to a value close to zero at longest aberration wavelengths. In between, it can be seen that round-trip loss can exceed the golden rule prediction. The phenomenon is known as resonant scatter loss, anticipating the, yet to be confirmed, explanation that this has to do with impedance matching conditions, or other kinds of coherent effects related to higher-order modes.

TIFOSI eigenmode study of resonant scatter loss

In the following, we will present the first 100 eigenmodes for the loss-function simulation with 1e-9m ripple amplitude evaluated on peak at perturbation scale 0.6mm (TIFOSI; eigenmodes, 0.6mm ripple), and near minimum at scale 1mm (TIFOSI; eigenmodes, 1mm ripple). First of all, one can see that the eigenmode spectrum is strongly perturbed at these scales. This is not surprising since a perturbation amplitude of 1nm is very high given an input transmission of 1ppm, and consequently high cavity finesse (F=6300). Note that in both cases, 0.6mm and 1mm perturbation, none of the eigenmodes looks similar to a TEM00. The Gouy phases are therefore given relative to the phase of the lowest-loss mode. Since the loss peak at 0.6mm reaches a value of about 1400ppm, the TEM00 input beam must couple strongly to higher-loss eigenmodes (mode 11 is the first to exceed 1400ppm round-trip loss). The low round-trip loss at perturbation scale 1mm of about 1ppm is a bit paradoxical given the fact that the lowest loss of a cavity eigenmode is 99ppm. Here one needs to keep in mind though that this is not any less "mysterious" than the loss peak.