accelerated convergence method for fast fourier transform

Accelerated convergence method for fast Fouriertransform simulation of coupled cavities

R. A. Day,1,* G. Vajente,2,3 and M. Pichot du Mezeray4

1European Gravitational Observatory (EGO), I-56021 Cascina (PI), Italy2INFN, Sezione di Pisa, I-56127 (PI), Italy

3Currently at LIGO Laboratory 100-36, California Institute of Technology, Pasadena, California 91125, USA4ARTEMIS/Observatoire de la Côte d’Azur, Nice, France

*Corresponding author: richard.day@ego‑gw.it

Received December 10, 2013; revised January 17, 2014; accepted January 23, 2014;posted January 29, 2014 (Doc. ID 202831); published February 27, 2014

Fast Fourier transform (FFT) simulation was used to calculate the power and spatial distribution of resonant fieldsin optical cavities. This is an important tool when characterizing the effect of imperfect geometry and mirror aber-rations. This method is, however, intrinsically slow when the cavities are of relatively high finesse. When this isthe case, an accelerated convergence scheme may be used to calculate the steady-state cavity field with a speedthat is orders of magnitude faster. The rate of convergence of this method, however, is unpredictable, as manydifferent factors may detrimentally affect its performance. In addition, its use in multiple cavity configurations isnot well understood. An in-depth study of the limitations and optimization of this method is presented, togetherwith a formulation of its use in multiple cavity configurations. This work has not only resulted in consistent im-provement in performance and stability of the accelerated convergence method but also allows the simulation ofoptical configurations, which would not previously have been possible. © 2014 Optical Society of America

OCIS codes: (070.7345) Wave propagation; (070.5753) Resonators.http://dx.doi.org/10.1364/JOSAA.31.000652

1. INTRODUCTIONOptical simulation is an important tool for understanding andbuilding optical cavities. The physics of optical cavities is wellunderstood [1], and so an analytical approach can give manyaccurate predictions about the design and resulting perfor-mance of such a cavity. However, when designing coupled op-tical cavities or even a single optical cavity to extremely highrequirements, it is necessary to take into account many subtle-ties such as cavity mirror roughness, alignment, and realisticlocking points. For such effects, an analytical approach be-comes less practical and optical simulations come into play.

One popular approach is to use modal simulations [2,3].Modal simulators expand the laser field into a base ofHermite–Gaussian or Laguerre–Gaussian modes for whichthe base is the eigenmode of the perfect cavity. These simu-lators have the advantage of being very fast when consideringlow spatial frequency aberrations in the cavity. However, thesimulation of higher spatial frequency aberrations requires anumber of higher order modes (HOMs) that increase as ρ4

(where ρ is the spatial frequency) [4], which in turn dramati-cally increases the simulation time.

An alternative approach is the so-called fast Fourier trans-form (FFT) simulation [5], or angular spectrum propagationmethod. The FFT simulations expand the laser field into planewave components with different transverse frequency compo-nents by means of a FFT. The plane waves are then propa-gated independently and the resulting field is computed byinverse Fourier transform where it can then be transmittedthrough or reflected off an optical component before beingpropagated again. In simple optical configurations such asa Michelson interferometer, a finite number of operations

are needed to propagate the input field to the output ports.However, in an optical cavity the fields propagate an infinitenumber of times inside the system, and the steady-state sol-ution is given by the interference of all the round trips. In thiscase, the propagations, transmissions, and reflections of theintra-cavity field may be iterated until it has converged toits steady state. The FFT simulation is much faster than modalsimulation when there are high spatial frequency aberrationsin the cavity. With such simulators, it is possible to simulatehighly degenerate cavities and even round-trip losses due tomirror clipping or scattered light.

The FFT simulator has proved to be an extremely importanttool for many applications such as the design of gravitationalwave (GW) interferometers [6]. These experiments, which arecomposed of high-finesse kilometer-scale coupled cavities,demand a perfect understanding of the mechanisms involvedin the use of imperfect cavity mirrors. It is the use of FFTsimulators that has allowed researchers to define theextremely high polishing requirements for the core opticsof GW interferometers.

The increasing complexity of applications such as GWinterferometers has highlighted limitations in FFT simulationtools. The FFT simulation, which uses standard convergence,replicates the light circulating inside the cavity. Therefore, thetime taken (or number of iterations) for the intra-cavity powerto buildup to steady state increases with the finesse of the cav-ity. This fundamental issue was addressed by Saha [7] with theproposition of an accelerated convergence scheme. Thismethod basically treats the convergence as a matrix inversionproblem and demonstrates how a traditional over-relaxationmethod may be used to arrive at steady state in as few

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http://dx.doi.org/10.1364/JOSAA.31.000652

iterations as possible. This work was an important contribu-tion to the field, and it is currently implemented in most FFTsimulators in the GW community.

The accelerated convergence scheme works extremelywell for near perfect stable cavities. However, it has beenfound that the performance of this method can be degradedby a large number of factors dependent on both the opticalconfiguration (e.g., cavity geometry, operating points, andbeam matching) and the numerical implementation (e.g., ali-asing and machine precision). The rate of convergence of thismethod is therefore relatively unpredictable in a general senseto the extent that, in some cases, it can take longer than thetraditional convergence method.

The second issue with this scheme is that its use withcoupled cavities is not straightforward. Attempts to acceler-ate two cavities that are coupled usually result in large in-stabilities that prevent convergence to the steady state.One solution to this problem is to accelerate the cavity con-taining high power and use traditional convergence on theone containing lower power. However, this solution mayonly be applied for a few specific optical configurations.A “global relaxation” scheme was proposed in [8,9] wherebysteady-state fields for each cavity are determined simultane-ously in order to minimize a specially weighted sum of theiteration errors for all relaxed fields. However, this system isalso reported to have instabilities that are heavily configura-tion dependent.

In this paper, we address these issues concerning the accel-erated convergence scheme. We annotate this discussion bygiving example results of simulations that have been devel-oped in MATLAB [10]. In Section 2, we will summarize thestate-of-the-art FFT simulation for a single cavity. We will alsodescribe our criteria for determining when steady state hasbeen reached, which will be used extensively throughout thisarticle. In Section 3, we will discuss how different optical con-figurations can degrade the performance of the acceleratedconvergence and what measures may be taken in order to op-timize the convergence using such configurations. Finally, inSection 4, we will propose an accelerated convergencescheme for multiple coupled cavities and give some examplesof where its use will be of great benefit. This work lays thefoundation for using accelerated convergence in any arbitraryoptical configuration.

2. ACCELERATED CONVERGENCE OF ASINGLE CAVITYIn this section, we will discuss the basic principles needed tosimulate a single optical cavity using FFT simulation with theaccelerated convergence scheme.

A. Simulation OperatorsAny FFT simulation consists of a series of propagations, trans-missions, and reflections. Figure 1 shows a schematic of afield in plane A propagated to plane B where it is then re-flected off and transmitted through a “thin” mirror having am-plitude reflectivity and transmission of r and t, respectively.Given the field in plane A, EA�x; y� as a function of the trans-verse coordinates, the field in plane B is given by:

EB�x; y� � P�EA�x; y��: (1)

Here, P is the propagation operator, which is the solution ofthe paraxial diffraction Eq. (1):

EB�x; y� � F−1�M�kx; ky� · F �EA�x; y��; (2)

where F �EA�x; y�� is the bi-dimensional Fourier transform ofthe field, which returns a function of the spatial frequency co-ordinates kx and ky; and F−1 is the inverse Fourier transform,which returns a function of the spatial coordinates. M�kx; ky�is the propagation kernel given by:

M�kx; ky� � exp�−ikL� i

k2x � k2y2k

L�; (3)

where k is the wavenumber and L is the propagation dis-tance. This expression is accurate within the paraxialapproximation.

The field EB will then be reflected and transmitted at themirror to give the fields:

ER�x; y� � RB�EB�x; y��; (4)

ET �x; y� � T B�EB�x; y��: (5)

Here, RB and T B are the reflection and transmission opera-tors, respectively, and correspond to the operations:

ER�−x; y� � irD�x; y� exp�2ikH�x; y��EB�x; y�; (6)

ET �x; y� � tD�x; y� exp�−ikG�x; y��EB�x; y�; (7)

whereH�x; y� is the height map (in z) of the reflecting surfaceand G�x; y� is the optical path length map (in z) in transmis-sion. D�x; y� represents the finite aperture of the mirror,taking values of 1 for all points inside the mirror and 0 outside.Note that in the computation of the reflected field, it is neces-sary to invert the sign of one of the coordinates to conservethe right hand coordinate system. In this definition, we haveused the convention that all reflections are multiplied by i,although other literature may use a different convention.

In the actual numerical implementation of the above equa-tions, a discrete sampling of the fields over a window of fixedsize is used, with N samples per direction. Each field is thendescribed by a N × N matrix, as well as the functions G, H,and D. The continuous Fourier transform is finally substitutedwith the discrete FFT. The propagation kernel is also de-scribed by aN × N matrix that is sampled in the Fourier space.

Fig. 1. Definition of field propagated to, reflected off, and transmit-ted through a thin mirror.

Day et al. Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. A 653

B. Convergence SchemeThe accelerated convergence scheme for a single cavity wasfirst described in [7]. Figure 2 shows a schematic diagram ofthe single cavity setup. En, referred to as the guess field, is theestimation of the cavity field at a given iteration, while thepump field, after transmission through the input mirror, iscalled Et. The simple iterative approach to finding thesteady-state solution is to compute one round trip of the field:

E0n�1 � Et �A�En�; (8)

whereA � RAPRBP is the operator describing the round-trippropagation inside the cavity. The acceleration proposed in[7] is to construct a new guess field as a linear combination

En�1 � anEn � bnE0n�1 (9)

with coefficients an and bn to be determined at each iteration.The error we make in using this new guess field can be esti-mated as the amount of change in the field over an entireround-trip propagation:

Δn�1 � En�1 − �Et �A�En�1��: (10)

If we were able to find the exact solution, this change wouldbe zero. This error is a function of the an and bn coefficients,which can be found by substituting Eq. (9) into Eq. (10):

Δn�1 � an�En −A�En�� bn�E0n�1 −A�E0

n�1�� − Et: (11)

Introducing the following definitions

D �� En −A�En�E0n�1 −A�E0

n�1�

�T

XA �� anbn

�; (12)

where xT indicates the matrix transpose, the error can be writ-ten in a more compact form

Δn�1 � DXA − Et: (13)

The best choice of the coefficients is the one that minimizesthe squared value of the error:

e � Δ�n�1Δn�1 �

X�X�

AD�− E�

t ��DXA − Et�; (14)

where the summation is over all the samples of the windowused and x� indicates the Hermitian transpose. The minimum

is found through requiring the derivative of the above equationwith respect to X�

A to be zero:

�XD�D

�XA �

XD�Et; (15)

which corresponds to the system of equations of [7]. We maytherefore determine the coefficients an and bn by solvingnumerically

XA ��X

D�D�−1�X

D�Et

�: (16)

C. Convergence Figure-of-MeritAn important aspect of the convergence algorithm is to deter-mine when the guess field has reached steady state. It is there-fore necessary to define a figure-of-merit that gives thisinformation. This figure-of-merit will then be compared witha user defined threshold in order to decide if steady state hasbeen reached. The choice of threshold will be a compromisebetween the required simulation speed and required accuracy.A natural choice of figure-of-merit for this kind of simulationis the fractional error in power between the field after n roundtrips and n� 1 round trips yielding:

δ0 ��PE�

nEn −P

E�n�1En�1

��PE�n�1En�1

: (17)

However, by inspection, we can see that this type of figure-of-merit will not take into account errors in phase betweensuccessive iterations, nor will it take into account errors inthe transverse distribution of power. With this in mind, a pref-erable figure-of-merit would take the form of:

δ � 2

��P�E�n − E�

n�1��En − En�1�PE�n�1En�1

s: (18)

Fig. 2. Definition of field and operators in the simulation of a singleFabry–Perot cavity.

Fig. 3. Comparison of figure-of-merit δ and δ0 for standard and accel-erated convergence. δ and δ0 for standard convergence are super-posed. Concave–concave cavity of length 3000 m was used for thiscase, with a radius of curvature for the input and end mirror being1420 and 1683 m, respectively. Transmission values of input andend mirrors were 1.4% and 1 ppm, respectively. Mirror diameters were350 mm. Input beam was mode-matched to the cavity.


In Fig. 3, we compare the evolution of these two figures-of-merit as the cavity field converges. In the x axis, we prefer tocount the number of FFT propagations rather than the num-ber of iterations used in other published works. The reason forthis choice is the introduction of “smoothed” and “averaged”acceleration in Section 3.B, where the time to calculate oneiteration greatly depends on the smoothing and averagingnumber chosen. Counting the number of FFT propagationsis therefore a better indicator of convergence time when com-paring different convergence methods and parameters. Wesee that for standard convergence the two figures-of-meritare equivalent. However, for accelerated convergence δ0,the fractional error in power has a relatively erratic descentand prematurely indicates that steady state has beenachieved. This generates some doubt as to the validity of thestate of convergence. The preferred figure-of-merit δ, how-ever, exhibits a smooth descent. The shape and rate ofdescent of this figure-of-merit will be a useful visual guide forthe optimization of the convergence throughout this paper.

D. Resonance LengthFinding the resonance length or “locking point” for the cavityis another important aspect to consider in order to faithfullyreproduce the physical system. The simulator should normallyuse the same type of error signals that are used experimen-tally. For example, for a cavity that is locked using thePound–Drever–Hall technique [11], it would be necessaryto generate an error signal using also the upper and lowersidebands of the phase modulated laser. The simulation of ad-ditional fields poses no fundamental difference with respect tothe carrier field. It is only necessary to modify the propagatorof Eq. (3) to include the additional phase shift due to the fre-quency difference of the sideband with respect to the carrier.

However, the simulation of a Pound–Drever–Hall error sig-nal is outside the scope of this paper and we will restrict ourdiscussion to a “coarse” lock of the cavity. We base this lockon the principle that the resonant cavity field should undergoa round-trip phase change that is an integer multiple of 2π. Wetherefore compared the phase of the cavity field and the samefield after one round trip. We then modified the phase of thepropagation matrix accordingly, such as to modify the micro-scopic length of the cavity. For a linear Fabry–Perot cavitythis yields

ϕ � Arg�X

�A�En��En

�; (19)

M 0�kx; ky� � e−ϕ∕2M�kx; ky�: (20)

Here, Arg is a function that returns the phase of a complexnumber. The factor 1∕2 in the phase corresponds to the factthat the matrix propagator, M , is used twice to achieve theround trip of the field. In different optical configurations thisfactor might vary; for example, for a triangular cavity wheneach space propagator is used once, one can add the phaseto only one propagator or divide it among all of them.

It would be inefficient to wait for convergence of the fieldbefore applying this length change. We prefer to apply thelength change after each iteration such as to progressivelyapproach the locking point as the cavity field converges tosteady state. In Fig. 4, we compare the convergence of the

cavity field for two cases. Specifically, these cases included(1) starting with an arbitrary microscopic cavity length andconverging the field while modifying the phase ofM after eachiteration, and (2) starting with the resonant microscopiclength and converging without modifying the phase of M .We found that very little time was lost determining the coarselock during convergence of the field. An extension of thiswork would be to then use “real” locking signals, once thecoarse lock has been found, in order to fine tune the lockingpoint.

3. UNDERSTANDING AND IMPROVINGPERFORMANCEAs we saw in the previous section, the accelerated conver-gence scheme is capable of dramatically increasing the con-vergence speed of a high finesse optical cavity. However,when using this scheme in an arbitrary configuration withrealistic mirror aberrations, a number of technical difficultiesare encountered. In this section, we will discuss some of theseissues and, where possible, we propose how to best deal withthem in order to optimize the convergence speed. These sug-gestions not only apply for a single cavity but also for the caseof multiple coupled cavities that will be presented in the nextsection.

A. Limits of Working PrecisionThe numerical calculation of the guess field requires solvingEq. (16). This necessitates finding the inverse or pseudo-inverse of the term

PD�D. However, as the convergence

approaches steady state this matrix becomes more and moresingular. The calculated guess fields, and hence the conver-gence, can therefore become unstable before the requiredaccuracy has been achieved. This problem is accentuatedwhen accelerated convergence is considered for multiplecoupled cavities where there are potentially large differencesin power. The nature of this instability will depend on how theinverse is calculated. In Fig. 5, we compare the use of differentnumerical methods for solving Eq. (16).

Fig. 4. Comparison of convergence with two different algorithms.Solid lines: when starting with an arbitrary microscopic cavity lengthand converging the field while modifying the phase of M after eachiteration. Dashed lines: when starting with the resonant microscopiclength and converging without modifying the phase of M .


The default working precision for the simulations is doubleprecision. A direct inverse of the matrix results in an uncon-trollable instability. This problem is addressed using a singularvalue decomposition (SVD) method. The SVD method is astandard computation that decomposes the matrix into threematrices such that

XD�D � USV�: (21)

Here, S is a diagonal matrix with nonnegative diagonal ele-ments in decreasing order and U and V are unitary matrices.In order to filter out singular values, elements on the diagonalof S whose ratio with the maximum value is inferior to a cer-tain threshold are set to zero. An element-wise reciprocal isthen made of all remaining nonzero values. The resulting ma-trix Srec is used to determine the inverse matrix

�XD�D

�−1 � VS0recU�: (22)

We can see in Fig. 5 that by choosing an appropriate thresholdthe instability is avoided. However, the system continues toconverge at a much slower rate. There are many other meth-ods for finding the pseudo-inverse of this matrix, however, tothe best of the authors’ knowledge, the working precision willcontinue to hinder the convergence speed or stability below acertain value for the figure-of-merit.

The easiest and most effective solution to this bottleneck isto double the working precision for this part of the calcula-tion. We therefore calculate Eq. (16) and only Eq. (16) in quad-ruple precision. We used a third party toolbox [12] to providethis functionality in MATLAB. Because the FFT and inverseFFT calculations were the most time consuming operationsin the simulation, the implementation of quadruple precisionfor Eq. (16) had a negligible effect on the time taken for oneiteration. In the third plot of Fig. 5, we can see that the dou-bling of precision (using quadruple precision in this case) forthis part of the calculation results in the rate of convergenceremaining stable down to the required value for the figure-of-merit.

B. Cavities with Large HOM ContentIn order to measure the usefulness of accelerated conver-gence in a general case we must consider its use in differentoptical configurations with various kinds of imperfections.The empirical conclusion of this study is that a high propor-tion of HOMs in the cavity slows down convergence. This canbe caused by various factors such as high degeneracy of thecavity, resonance condition, beam mismatching, mirror aber-rations, and large apertures. Figure 6 shows how the conver-gence is modified by the presence of HOMs in the cavity. Thesolid red curve shows the accelerated convergence, as before,of an ideal cavity. The remaining red, blue, and green curvesshow the convergence of the same cavity for which the mir-rors have a random roughness with varying rms. These mapswill scatter light into HOMs that could be on or near reso-nance. We can see that, at first, the cavity converges in thesame way. However, at a certain point (depending on the mir-ror rms) the rate of convergence suddenly reduces. This effectis interpreted as the error in the HOM convergence becomingdominant. As the steady-state solution for a large number ofmodes has many more degrees of freedom, it takes longer toconverge. It should be noted that the standard convergence(black curves) was totally unaffected by the presence ofHOMs. The authors believe that this reduction in convergencerate due to HOMs is an intrinsic limitation of the presentedaccelerated convergence scheme.

However, we identified other possibilities that we mayconsider. To further improve the convergence rate of the ac-celerated convergence method described above, we foundempirically that it is useful to extend the number of fields usedin the guesses. For example, in the case of a simple cavity wecan then write

En�1 � anEn � bnE0n�1 � cnE0

n�2 � � � � � dnE0n�N: (23)

This change is easily inserted into the equations described inthe previous sections by simply allowing the vectors definedin Eqs. (12) and (35) to accommodate more elements. Werefer to this method as “smoothed acceleration.”

Fig. 5. Comparison of three numerical configurations for calculatingEq. (16).

Fig. 6. Comparison of convergence when realistic mirror maps, withvarying rms, are added to cavity mirrors. Plots for standard conver-gence case are superposed.


A second improvement in the convergence rate can beobtained if the simple fields E0

j used in the estimates aresubstituted with averages over multiple round trips:

hE0niM � 1

M

XM−1

j�0

E0n�j : (24)

Clearly, Eq. (23) must be modified accordingly to avoid over-lapping averages:

En�1 � anhE0niM � bnhE0

n�M iM � � � � � cnhE0n�NM iM: (25)

We refer to this technique as “averaged acceleration.” Whenboth smoothed and averaged acceleration is used, the numberof FFT propagations per iteration is simply given by NM .

In Fig. 7, we show how the smoothed and averaged accel-eration can improve the convergence rate. We can see that, inthis example, the convergence rate may be improved by up toa factor of five both for ideal cavities and cavities with realisticmirrors. The smoothed acceleration method was more effec-tive than the averaged acceleration method. However, for theformer all the fields must be kept in the computer memorywhereas as for the latter it is sufficient to progressivelysum the fields. Keeping the computer memory limitation inmind, it is often convenient to use a combination of bothmethods in one simulation.

C. Problems of AliasingThe angular spectrum beam propagation method that is usedin this study has the intrinsic problem of aliasing. Light thatreaches the side of the FFT window should normally disap-pear. However, the aliasing effect results in it reappearingon the opposite side of the window with the same directionof propagation. In normal FFT simulations this effect is unde-sirable, but it may be tolerated to a certain extent. However, inaccelerated convergence, this aliasing can have catastrophiceffects. This unwanted light resulting from aliasing is equiva-lent to HOMs and therefore greatly reduces the convergence

speed. As this effect is totally nonphysical, we wish to com-pletely eliminate this problem. We will use an approach sim-ilar to that described in [9,13]. In Fig. 8, we show a schematicdiagram of light scattered at large angles off mirror A andpropagating to mirror B. Ray 1 shows the limiting case ofthe aliased scattered light hitting the edge of mirror B. Wetherefore wish to eliminate light scattering at angles greaterthan θ1. Ray 2 shows the limiting case of light scattered offthe bottom of mirror A and hitting the top of mirror B. Wetherefore wish to keep light scattering at angles smaller thanθ2. If we choose the FFT window, W , to be exactly two timeslarger than the mirrors, then θc � θ1 � θ2. Using the smallangle approximation we may write

θc ≃W2L

: (26)

The angle at which light is scattered is related to the spatialfrequency, ρ, of the mirror structure by

θ ≃ λρ: (27)

This again uses the small angle approximation. Therefore, thespatial frequency cutoff to avoid aliasing is given by

ρc �W2Lλ

: (28)

This filtering is implemented after building the propagationkernel, M , in Eq. (3). All points for which �k2x � k2y� > ρ2care set to zero. Figure 9 gives an example of convergence withand without the filtering. We can see that with no filteringeven a cavity with ideal mirrors can result in the characteristicsudden reduction in rate of convergence; for nonideal mirrors,the effect was worse. After implementation of the filtering,however, we found that we were no longer susceptible tothe increased convergence time due to aliasing.

4. DOUBLE AND MULTIPLE CAVITIESIn this section, we will extend the accelerated convergencescheme for a single cavity to that of any arbitrary combinationof coupled cavities.

The extension to an arbitrary number of cavity and pumpfields is more easily understood starting from a double cavity

Fig. 7. Improvement in accelerated convergence by using smoothedand averaged acceleration in the case of perfect mirrors (red) andrealistic mirrors with rms 5 × 10−1 nm. In the case of N � 1, M � 1corresponds to the standard accelerated convergence. CasesN � 9, M � 1 and N � 1, M � 5 carry out the same number of prop-agations per iteration.

Fig. 8. Schematic diagram showing limiting condition for onset ofFFT aliasing.


example, like the one shown in Fig. 10. For the sake of gen-erality, we considered also the possibility of a pump field en-tering directly into the second cavity. This will allow us togeneralize to an arbitrary system.

The iterative relations to be used without any accelerationare:

E0A;n�1 � EA;t �D�EB;n� �A�EA;n� (29)

E0B;n�1 � EB;t � B�EB;n� � C�EA;n�: (30)

Following the same approach used for the single cavity, wewrite the best guess for the fields at the next iteration as linearcombinations:

EA;n�1 � aEA;n � bE0A;n�1 (31)

EB;n�1 � cEB;n � dE0B;n�1: (32)

The error we make using these estimations is given, as before,by the change of this field when undergoing a full round-trippropagation:

ΔA;n�1 � EA;n�1 − �EA;t �D�EB;n�1� �A�EA;n�1�� (33)

ΔB;n�1 � EB;n�1 − �EB;t � B�EB;n�1� � C�EA;n�1��: (34)

To simplify to some extent the notation, we introduce thefollowing definitions:

XA �� a

b

�

XB �� c

d

�

FA �

EA;n −A�EA;n�E0A;n�1 −A�E0

A;n�1�

!

FB �

EB;n − B�EB;n�E0B;n�1 − B�E0

B;n�1�

!

GA �

C�EA;n�C�E0

A;n�1�

!

GB �

D�EB;n�D�E0

B;n�1�

!; (35)

which allows us to write the errors in the following form

ΔA;n�1 � FTAXA −GT

BXB − Et;A (36)

ΔB;n�1 � FTBXB −GT

AXA − Et;B: (37)

This time we require both errors to be minimum when aver-aged over the transverse coordinates. In other words, we haveto search for the set of coefficients a, b, c, and d that simulta-neously minimize the following merit functions

eA �X

Δ�A;n�1ΔA;n�1 (38)

eB �X

Δ�B;n�1ΔB;n�1 (39)

where the summation is, as before, over all the samples of thewindow used. The optimal solution is found when the deriva-tive of the above equations with respect to the coefficients ofthe guess field is zero. To be more precise, we need to set thederivative of eA with respect to X�

A to zero, assuming XB isalready known. In the same way we must set the derivativeof eB with respect to X�

B to zero, assuming XA to be known.The simultaneous solution of both the resulting equations willyield the desired optimal coefficients. After some algebra, thefollowing equations are found:

�XF�AFA

�XA −

�XF�AGB

�XB �

XF�A Et;A (40)

�XF�BFB

�XB −

�XF�BGA

�XA �

XF�B Et;B: (41)

From this result, it is straightforward to extend to the moregeneral case of an arbitrary number of cavities. To the ith cav-ity, there is a corresponding equation contained on the lefthand side the term �PF�

i Fi�Xi and all the terms−�PF�

i Gj�Xj with j ≠ i. The right hand side will containthe term

PF�i Et;i, which corresponds to the input field that

enters directly into the cavity, if any.

Fig. 9. Comparison of convergence with (dashed lines) and without(solid lines) aliasing filter using perfect cavity mirrors (red curves)and mirror maps having rms of 5 × 10−1 (blue curves).

Fig. 10. Definition of field and operators in the simulation of a dou-ble Fabry–Perot cavity.


We will give, as an example, the simulation of a double cav-ity similar to those found in GW interferometers. In Fig. 11, wepresent the cavity parameters and results of convergence.

Using realistic mirror maps with rms of 0.5 nm, the power inthe cavities for 1 W at input was 50 W and 14 kW for the firstand second cavity, respectively. Because of the much higherpower in the second cavity, previous simulations have usedthe approach of accelerating the second cavity while contin-uing to use standard convergence on the first cavity. We referto this technique as single cavity acceleration. This results inthe convergence given by the dashed red line in Fig. 11. Whenusing the coupled cavity acceleration described in thissection, we obtained the blue dashed line. We can see, in thisexample, that there was no advantage to using coupled cavityacceleration. The reason for this is the unfavorable combina-tion of mirror aberrations and low power in the first cavity.However, we gain considerably by implementing thesmoothed acceleration. We can see that the use of smoothedacceleration for the single cavity acceleration (solid redcurve) is totally ineffective. This is because the convergenceof the second cavity, which is accelerated, is limited by thefirst cavity that uses standard convergence. The coupled cav-ity acceleration was, however, able to profit from smoothedacceleration (solid blue curve), and, in this example, it con-verged more than seven times faster than the other methods.

As we have mentioned previously, the coupled cavityscheme is of particular interest when there is a large amountof power in all cavities that are coupled. An example configu-ration that is therefore well adapted to this technique is adispersively coupled optomechanical system [14]. This con-figuration consists of placing a membrane inside a cavity witha finesse of the order of one million. The membrane typicallyhas a reflectivity of the order of a few percent. The ensemblemay therefore be treated as two coupled cavities. In Fig. 12,we present the cavity parameters and results of convergence.

Using realistic mirror maps with rms of 0.05 nm, the powerin the cavities for 1 W at input was 287 and 430 kW for the firstand second cavity, respectively. Both the single and coupledcavity acceleration were unable to converge within a reason-able time; the extremely high finesse of the cavity appears togenerate an unstable state that, at the time of writing, couldnot be explained. However, the dashed and dotted curves inFig. 12 show that, by using smoothed acceleration with a suf-ficiently large value for N, the stability was regained resultingin an extremely fast convergence.

5. CONCLUSIONIn this paper, we have presented a complete method for ac-celerated convergence of FFT simulations. Our approach isa direct extension of the work done by Saha [7]. We have com-plemented his work by studying in more detail the intrinsiclimitations of such a scheme. We found that there were a num-ber of important factors that could hinder the performance ofthe accelerated convergence such as the working precision,the proportion of HOMs in the cavity, and FFT aliasing. Wepresented methods for tackling these issues effectively. Nota-bly, we presented the concept of smoothed and averaged ac-celeration, which provides a consistent speed improvementover classic accelerated convergence. The product of thiswork was a convergence scheme that was much more reliableand predictable.

We went on to present a method that extends Saha’s singlecavity accelerated convergence to the case of double andmultiple cavities. We demonstrated that there was consider-able benefit in using such a scheme for a GW-interferometertype optical configuration. However, we also demonstratedthat this scheme is absolutely essential in the case of high fi-nesse coupled cavities whereby there is a similar powerstored in each cavity. Our numerical simulations showed thatfor such a challenging simulation, the smoothed accelerationmethod not only allowed faster convergence but also acted to

Fig. 11. Comparison of convergence for a double cavity using singlecavity acceleration (red lines) and coupled cavity acceleration (bluelines). For clarity, the figure-of-merit has been shown only for the lowpower cavity. First cavity is concave–convex of length 11.953 m andsecond cavity is concave–concave of length 3000 m. Radius of curva-ture values for input, middle, and end mirrors were 1431 m, 1420 m,and 1683 m, respectively. Transmission values of input, middle, andend mirrors were 5%, 1.4%, and 1 ppm, respectively. Mirror diameterswere 350 mm. Input beam was mode-matched to the first cavity.

Fig. 12. Comparison of convergence for a double cavity using singlecavity acceleration (red line) and coupled cavity acceleration (bluelines). For clarity, the figure-of-merit has been shown only for thelow power cavity. First cavity is concave–flat of length 59.5 mmand second cavity is flat–concave of length 59.5 mm. Radius of cur-vature for the input and endmirrors was 200 mm. Transmission valuesof input, middle, and end mirrors were 1 ppm, 96%, and 1 ppm, respec-tively. Mirror diameters were 1.2 mm. Input beam was mode-matchedto first cavity.


reduce numerical instabilities resulting in a more robustmethod.

The increased speed, versatility, and robustness of theaccelerated convergence scheme presented in this papermakes this an important tool for FFT simulation in many fieldsof laser physics.

ACKNOWLEDGMENTSThe research activity of one of the authors (GV) was partiallysupported by Regione Toscana (Italy) through the programPOR CreO FSE 2007-2013 of the European Community, withinthe project no. 18113 (ISAV). LIGO was constructed by theCalifornia Institute of Technology and Massachusetts Instituteof Technology with funding from the United States NationalScience Foundation under grant PHY-0757058.

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accelerated convergence method for fast fourier transform

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