Ab initio investigation of lasing thresholdsin photonic molecules

Denis Gagnon, Joey Dumont, Jean-Luc Déziel, and Louis J. Dubé*

Département de physique, de génie physique et d’optique, Faculté des Sciences et de Génie, Université Laval,Québec G1V 0A6, Canada

*Corresponding author: ljd@phy.ulaval.ca

Received April 23, 2014; accepted June 8, 2014;posted June 19, 2014 (Doc. ID 209915); published July 17, 2014

We investigate lasing thresholds in a representative photonic molecule composed of two coupled active cylindersof slightly different radii. Specifically, we use the recently formulated steady-state ab initio laser theory (SALT) toassess the effect of the underlying gain transition on lasing frequencies and thresholds. We find that the order inwhichmodes lase can bemodified by choosing suitable combinations of the gain center frequency and linewidth, aresult that cannot be obtained using the conventional approach of quasi-bound modes. The impact of the gaintransition center on the lasing frequencies, the frequency pulling effect, is also quantified. © 2014Optical Societyof America

OCIS codes: (140.3410) Laser resonators; (140.3945) Microcavities; (230.4555) Coupled resonators;(140.3430) Laser theory.http://dx.doi.org/10.1364/JOSAB.31.001867

1. INTRODUCTIONThe study of light–matter interactions in photonic molecules(PMs), which are formed by coupling several optically activemicrocavities (atoms), has been the object of much work inrecent years [1,2]. Applications of microresonators and PMsinclude optical communications [3], sensing [4–7], quantumcomputing [8], and metrology [9]. Photonic atoms [10] andmolecules [11] are also well suited for the fabrication ofmicrolasers owing to their high quality factor, or photonrecycling rate. These coupled systems also provide a test bedfor a panoply of fundamental phenomena including opticalbistability [12], coupled-resonator induced transparency [2],nonreciprocal light transmission [13], and exceptional points(EPs) [14,15]. EPs are ubiquitous in parameter-dependenteigenvalue problems and can lead to surprising physicaleffects [16]. For instance, coupled edge-emitting lasers mayturn off even as the pump power increases above threshold.This counter-intuitive behavior is achieved by pumping thecavities nonuniformly near an EP [15].

The lasing characteristics of microresonators are oftenobtained from the calculation of the cold-cavity (passive)modes. An alternative approach consists of introducing thethreshold material gain in the laser eigenvalue problem[17–19]. This approach allows for the assessment of the effectof the resonator geometry on lasing thresholds and emissiondirectionality. However, to take into account the spectralproperties of a given laser transition, for instance its positionand linewidth, formulations such as the Maxwell–Bloch orSchrödinger–Bloch (for 2D systems) theory must be used[20,21]. This is especially important in the case of nearly de-generate lasing frequencies, for instance near EPs. As weshow in this work, the conventional wisdom of cold-cavitymodes may not be sufficient in this case.

We use the steady-state ab initio laser theory (SALT) [22]to assess the effect of gain transition parameters on lasing

frequencies and thresholds of a simple two-dimensional PMcomposed of two coupled active cylinders in the vicinity ofa geometric EP. The term ab initio refers to the fact thatthis theory involves the solution of a set of self-consistentequations that explicitly take the gain medium parametersinto account. In other words, SALT allows one to determinethe steady-state solutions of the Schrödinger–Bloch equations[21,23]. The near-threshold behavior of the PM can be ob-tained by computing the threshold lasing modes (TLMs) asdescribed in [22]. This choice of basis states permits the studyof the effect of the Lorentzian gain transition on thresholdswhen there are several modes competing for efficient gainextraction.

This paper is organized as follows. In Section 2, we describethe theoretical background behind our computations, includ-ing the main equations of SALT and the method used to com-pute the lasing states for an arbitrary number of cylinders. InSection 3, we compute the lasing states of a diatomic photonicmolecule as a representative example. Our results show thatthe thresholds of closely spaced modes exhibit a nontrivialdependence on the parameters of the gain transition, specifi-cally the gain center frequency and its linewidth. We also in-vestigate how the lasing modes are subject to the frequencypulling effect, i.e., the effect of the gain center frequency onthe spectrum of the PM laser. We summarize our findings inSection 4 and mention a number of possible improvementsand extensions.

2. THEORETICAL BACKGROUNDThe investigation of the lasing behavior of 2D cavities usuallyimplies the computation of the eigenstates of the passive cav-ity. These states are governed by the Helmholtz equation

�∇2 � ϵ�r�k2�φ�r� � 0; (1)

Gagnon et al. Vol. 31, No. 8 / August 2014 / J. Opt. Soc. Am. B 1867

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where an harmonic time dependence exp�−iωt� is assumedand ϵ�r� is the passive spatially varying refractive index. BothTM �φ ≡ Ez� and TE �φ ≡Hz� polarized waves can be consid-ered. By applying the usual Sommerfeld radiation condition,i.e., an outgoing wave component only, we obtain a set ofleaky, or quasi-bound (QB) states characterized by complexeigenfrequencies kQB � k0 � ik00 [24]. Since the radiation con-dition implies k00 < 0, QB states are nonorthogonal and exhibitexponential growth toward infinity [23]. Despite this unrealis-tic behavior, QB states provide a useful measure of the photonlifetime in the cavity by means of the quality factorQ � jk0∕2k00j. The Q-factor gives a qualitative indication ofwhich cavity modes will lase first. Considering a gain transi-tion with the center located at frequency ka, one will usuallyinfer that the modes with eigenfrequencies close to ka and asufficiently high Q-factor will lase first [20,25].

A. Steady-State Ab Initio Laser TheoryAlthough the QB states are widely used, they cannot describethe lasing behavior of microcavities or PMs in a completelyaccurate way since the gain medium parameters have noinfluence on the QB states described by Eq. (1). Dynamicaltheories such as the Schrödinger–Bloch model have beenused in the past decade to improve the treatment of lasingmodes [20,21]. In this respect, the recently developed SALToffers an alternative formulation as a stationary version of theSchrödinger–Bloch model. Since we are precisely interestedin the steady-state lasing behavior of PMs, we shall use SALTin the remainder of this work.

The most important feature of SALT is the introduction ofa new kind of eigenstate called a constant-flux (CF) state[21–23]. The CF states satisfy the following equation

�∇2 � ϵ�r�K2�k��φ�r� � 0; r ∈ C; (2a)

�∇2 � ϵ�r�k2�φ�r� � 0; r∉ C; (2b)

where C is the cavity region, which is defined as the union ofall optically active regions. The CF states are characterized bycomplex eigenfrequencies K inside C, but real frequencies koutside this region. Thus, they exhibit no exponential growthand are physically meaningful, contrary to the QB states [21].

The laser modes can be expanded on the basis of CF statesthat take into account effects such as nonlinear coupling be-tween modes (spatial hole burning) and nonuniform pumpprofiles [15,22]. For the purpose of this work, we restrictourselves to the case of near-threshold behavior and spatiallyuniform pumping inside all cylinders composing the PM.Under those conditions, the TLMs of the PM satisfy the follow-ing equation

�∇2 �

�ϵ�r� � γaD0F�r�

k − ka � iγa

�k2�φ�r� � 0; (3)

where ka is the gain center frequency, γa is the gain width, andD0 is associated with the “pump strength.” Since we supposeuniform pumping of all cylinders forming the PM, F�r� � 1 forr ∈ C and F�r� � 0 for r∉ C (the cavity region and the cylin-ders coincide). The TLMs must satisfy an additional realitycondition on D0 [22]. For a given choice of exterior real fre-quency k, the value of D0 is in a general complex, but when it

crosses the real axis at k � kμ, we obtain a pair of real num-bers �kμ; Dμ

0� defining the TLM lasing frequency and lasingthreshold, respectively. The first lasing mode is thereforethe TLM with the smallest threshold Dμ

0 [22].In the case of uniform pumping and a uniform dielectric

constant ϵ�r� � ϵc inside all cylinders, each TLM can be asso-ciated with a single CF state. By comparing Eqs. (2a) and (3),one obtains the relation

γaD0

k − ka � iγa� ϵc

�K2

k2− 1

�: (4)

Simply stated, for a given combination of gain parameters (ka,γa), each TLM of eigenvalue kμ corresponds to a CF state ofeigenvalueKμ for whichD0 is purely real, according to Eq. (4).

B. 2D Generalized Lorenz–Mie TheoryWhile SALT allows for arbitrary pump profiles, we restrict ourdiscussion to uniformly pumped dielectric cylinders. Thispump uniformity permits the use of the 2D generalizedLorenz–Mie theory (2D-GLMT)—also called the multipolemethod—to compute the eigenstates of the PM [26–28]. Thiskind of theory can be used provided the cylinders composingthe PM possess enough symmetry to use the method ofseparation of variables [29]. It can also be used for the com-putation of the scattering of arbitrary beams by a complex ar-rangement of dielectric cylinders [30–32]. Multipole-basedmethods can also be used for modal analysis of quantum dots[33]. For the sake of completeness, we review briefly the mainequations of the application of 2D-GLMT for the computationof lasing states.

Consider an array of N cylindrical scatterers of radii un andrelative permittivity ϵn. Let also rn � �ρn; θn� be the cylindricalcoordinate system local to the nth scatterer. For modelingpurposes, we suppose that every cylinder is infinite alongthe axial z direction. The central hypothesis of 2D-GLMT isthat the total field outside the scatterers can be expandedin a basis of cylindrical functions centered on each individualscatterer, that is

φ�r� �XNn�1

X∞l0�−∞

bnl0H���l0 �k0ρn�eil0θn ; (5)

where H���l is a Hankel function of the first kind. Inside the

nth scatterer, the field can be written as

φ�r� �X∞l�−∞

cnlJl�knρn�eilθn ; (6)

where Jl is a Bessel function of the first kind.In order to apply electromagnetic boundary conditions at

the interface of the nth scatterer, one must find an expressionfor φ�r� outside the scatterers containing only cylindricalharmonics centered on the nth scatterer, that is

φ�r� �X∞l�−∞

�anlJl�k0ρn� � bnlH���l �k0ρn��eilθn : (7)

This can be achieved via the application of Graf’s additiontheorem for cylindrical functions, which allows a translationfrom the frame of reference of scatterer n0 to the frame ofreference of scatterer n [34]. The theorem states that

1868 J. Opt. Soc. Am. B / Vol. 31, No. 8 / August 2014 Gagnon et al.

H���l0 �k0ρn0 �eil0θn0 �

X∞l�−∞

ei�l0−l�ϕnn0H���

l−l0 �k0Rnn0 �Jl�k0ρn�eilθn ; (8)

where Rnn0 is the center-to-center distance between scatterersn and n0, and ϕnn0 is the angular position of scatterer n0 in theframe of reference of scatterer n. Substituting Eq. (8) inEq. (5) yields

φ�r� �X∞l�−∞

bnlH���l �k0ρn�eilθn

�X∞l�−∞

Xn0≠n

X∞l0�−∞

bn0 l0ei�l0−l�ϕnn0H���

l−l0 �k0Rnn0 �Jl�k0ρn�eilθn :

(9)

The comparison of Eq. (7) with Eq. (9) then yields the follow-ing relation between the fanlg and fbnlg coefficients

anl �Xn0≠n

X∞l0�−∞

ei�l0−l�ϕnn0H���

l−l0 �k0Rnn0 �bn0 l0 : (10)

A further relation between the fanlg and fbnlg coefficientsis obtained by applying electromagnetic boundary conditionsto Eqs. (6) and (7) at ρn � un. This finally leads to the homo-geneous equation for the coefficient vector b, T�kn; k0�b � 0,whose nontrivial solutions are given by the condition

det�T�kn; k0�� � 0; (11)

where kn is the frequency inside the nth cylinder and k0 is theexterior frequency (both can be complex). One immediatelyrecognizes the transfer matrix T as the inverse of the usualscattering matrix. T has a well defined structure; it is com-posed of blocks containing coupling coefficients betweencylindrical harmonics centered on each circular scatterer.Its elements are given by

Tnn0ll0 �kn; k0� � δnn0δll0

− �1 − δnn0 �ei�l0−l�ϕnn0H���l−l0 �k0Rnn0 �snl�kn; k0�:

(12)

The snl factor results from the application of electromagneticboundary conditions and is given by

snl�kn; k0� � −J 0l�k0un� − ΓnlJl�k0un�

H���0l �k0un� − ΓnlH

���l �k0un�

; (13)

where

Γnl � ξn0knJ 0

l�knun�k0Jl�knun�

(14)

and ξij � 1�ϵj∕ϵi� for TM (or TE) polarization. Prime symbolsindicate differentiation with respect to the whole argument. Ina typical implementation, T is composed of N × N blocks ofdimension 2lmax � 1, where lmax is chosen as a sufficientlylarge value to ensure convergence of the cylindrical functionexpansions. Its value is usually fixed by lmax ≥ 3 kmaxnfung.In the case of a diatomic photonic molecule (N � 2), thematrix is of the form

T ��T11 T12

T21 T22

��15�

with the diagonal blocks equal to identity matrices and denseoff-diagonal blocks, consistent with Eq. (12). More details onthe Lorenz–Mie method can be found in [26,27,29–31,35–37].

The QB states of a PM can be computed by substitutingkn → k

�����ϵn

pand k0 → k

�����ϵ0

pin Eq. (11) and looking for solu-

tions in the complex k-plane. As for the CF states, the appro-priate substitution is kn → K

�����ϵn

pif the nth cylinder is part of

the cavity region C and kn → k�����ϵn

potherwise, with real k. The

solutions are in this case located in the complex K -plane. Wenote that a countably infinite set of CF states can be computedfor each different value of the real exterior frequency k.

3. LASING STATES OF A SIMPLEPHOTONIC MOLECULEAs a proof of concept, we consider a diatomic PM composedof two coupled cylinders, and restrict the discussion to TM-polarized modes. More specifically, we consider the asymmet-ric PM as proposed in [14], which is shown in Fig. 1. Themotivation behind this choice is two-fold. First, it allows usto calibrate the combination of 2D-GLMT and SALT againsta previous calculation method, in this case the boundaryelement method [14]. Moreover, the proposed geometry ex-hibits an avoided crossing of QB states, which results fromthe proximity of an EP with a geometrical nature. EPs aregenerically defined as specific parametric combinations forwhich the eigenvalues of the non-Hermitian operator describ-ing a coupled system coalesce [15,16]. In the case of the asym-metric PM, the EP can be parametrically encircled by varyingthe inter-disk distance and the ratio between disk radii [14].Therefore, this choice of geometry is also motivated by theinteresting physics of EPs.

The PM is composed of two cylinders of radii u1 and u2,with u2 � 0.8908u1. The center-to-center distance is R12 �2.448u1, and the relative permittivity of the cylinders isϵc � 4 (see Fig. 1). The y axis is perpendicular to the lineconnecting the two cylinders, while the x axis is taken alongR12. The near-coalescent states located near k0u1 � 5.4 areshown in Fig. 2, and the evolution of the associated complexeigenfrequencies is shown in Fig. 3(a). The states are split intwo symmetry classes with respect to the x axis: odd modes(M1 and M4) and even modes (M2 and M3). They result fromthe coupling between whispering-gallery modes of slightlydifferent angular momenta of the uncoupled cylinders, whichcreates doublet states [2,38]. As a result of the proximity of thegeometrical EP, the four QB states located near k0u1 � 5.4have closely spaced resonance frequencies [14]. Moreover,the Q-factors are all similar (Q ∼ 200) and the four states com-pete for gain. The order in which the modes will lase is notobvious, especially if we consider a gain center with a fre-quency higher than that of M4. In other words, the conven-tional approach of considering only Q-factors does notallow a quantitative determination of the lowest threshold

u1 u2

R12

Fig. 1. Geometry of the diatomic photonic molecule used in thiswork. The cylinders radii are u1 and u2, with u2 � 0.8908u1. Thecenter-to-center distance is R12 � 2.448u1, and the relative permittiv-ity of the cylinders is ϵc � 4. This geometry is also used in [14].

Gagnon et al. Vol. 31, No. 8 / August 2014 / J. Opt. Soc. Am. B 1869

mode for nearly degenerate states. Despite this closespacing of modes, we assume that the stationary inversionapproximation—which is central to SALT—still holds for thisgeometry. Typical atomic relaxation rates for semiconductorlasers show that this is indeed the case [39].

A. Influence of Gain Medium ParametersAs stated previously, the computation of QB states cannotaccount for the influence of the gain center frequency andwidth on the lasing characteristics of a PM. However, if theexterior frequency k is chosen to be close to the frequencyRe�kQB� � k0 of a QB state, then this state can be associatedwith a single CF state and their intensity profiles look similarinside the active medium [23]. This correspondence can beseen by comparing the symmetries of the amplitude profilesbetween Figs. 2 and 4. For other values of the exterior fre-quency k, the intensity profiles may look somewhat different,but the one-to-one correspondence with the QB states stillholds (see for instance Figs. 3 and 4).

Since we restrict ourselves to the case of uniform pumping,each QB state shown in Fig. 2 is also associated with a TLM,which makes it possible to keep the same labels and computethe associated TLMs to assess the influence of the gainmedium parameters. To achieve this goal, one can devise thefollowing procedure for computing the TLM associated with asingle QB state:

1. Compute the complex eigenfrequency kQB � k0 � ik00

of the QB state;2. Compute the complex eigenfrequency K�k0� of the

corresponding CF state using the fact that K�k0� and kQB areusually close [23], as seen in Figs. 3(a) and 3(b) for instance;

3. Compute the values of K�k� in the real neighborhoodof k0;

4. Using Eq. (4), map the values of K�k� to values ofD0�k�. The TLM is characterized by the pair of values�kμ; Dμ

0� for which D0 becomes purely real. An example of thisbehavior is shown in Fig. 5.

Once the values of K�k� are computed using Eq. (11), it isnot necessary to repeat steps 1–3 when varying the values ofka and γa in step 4 as long as the cavity geometry and pumpprofile are unchanged.

Using this straightforward approach, the dependence ofthe lasing thresholds and lasing frequencies on the gainmedium parameters can be readily investigated for each ofthe four modes depicted in Fig. 2. We find that the lowestthresholds modes are always M1 and M4 owing to their higherquality factor. Therefore, we restrict our discussion to thesetwo modes. The dependence of the lasing thresholds D1

0 andD4

0 on ka and γa is shown in Figs. 5–7. As expected, M1 is thefirst lasing mode when the gain center frequency ka is smallerthan the value of k0 for that mode. However, as the value of

(a) M1 (b) M2

(c) M3 (d) M4

Fig. 2. Profile of four TM-polarized QB states of a diatomic photonicmolecule composed of two cylinders of different diameters, as shownin Fig. 1. The z-coordinate is proportional to the intensity (arbitraryunits).

Fig. 3. (a) Map of log jdet�T�j in the complex k plane for each of the four QB states of the photonic molecule. Eigenvalues correspond to the zerosof the function (dark spots) and are located at: M1, kQBu1 � 5.3830 − i0.0122; M2, kQBu1 � 5.3958 − i0.01756; M3, kQBu1 � 5.3993 − i0.0154; M4,kQBu1 � 5.4078 − i0.0133. (b)–(c) Map of log jdet�T�j in the complex K plane for two different values of the exterior frequency k (purely real). EachQB state can be associated to a unique CF state, which allows the use of the same labels for QB and CF states.

Fig. 4. Profiles of two TM-polarized CF states of a diatomic photonicmolecule, which are counterparts to the QB states M1 and M4 shownin Fig. 2. These profiles were computed for the same values of theexterior frequency used in Figs. 3(b) and 3(c). The z-coordinate is pro-portional to the intensity (arbitrary units).

1870 J. Opt. Soc. Am. B / Vol. 31, No. 8 / August 2014 Gagnon et al.

ka is increased, M1 can still lase first even if ka is greater thanthe position of mode M4 (see Fig. 7). This is especially true fora large gain width γa as mode M1 is able to extract energy

more efficiently from the gain transition in that case, whilefor a narrow gain transition M4 is favored.

Interestingly, at the intersection of the two surfacesshown in Fig. 7, both modes have exactly the same thresholdand lase concurrently. Although this lasing behavior is qualita-tively consistent with the fact that M1 corresponds to thehighest Q cold-cavity mode, the use of SALT is needed toobtain quantitative predictions of its dependence on ka and γa.

It is also instructive to examine the evolution of lasingthresholds when the gain transition center frequency is farfrom the QB eigenfrequencies. The dependence of Dμ

0 on thevalue of γa for a gain transition for large values of jk − kaj isshown in Fig. 6. Our numerical results show that the thresh-olds of modes approximately quadruple when the value of γais reduced by half. Accordingly, one can derive the followingexpression for Re�D0� from Eq. (4), under the conditionsIm�D0� � 0 and jk − kaj ≫ γa

D0 ≃ −2εc Re�K �Im�K � �k − ka�2γ2ak2

: (16)

This behavior (D0 ∼ γ−2a ) is consistent with the observationthat modes extract energy more efficiently from a broad lasingtransition.

Next, we assess the influence of the gain center frequencyka on the lasing frequencies of individual modes. The exactlasing frequency of a mode is always shifted by a small amountfrom the cold-cavity resonance frequency toward the gaincenter frequency, an effect known as frequency pulling[22,25]. Since the studied PM geometry exhibits closelyspaced lasing modes, this small shift may be of the orderof magnitude of the mode spacing. As shown in Fig. 8(a),the order of the lasing frequencies kμ can be altered by chang-ing the gain center frequency. For instance, for kau1 ≃ 5.4, thelasing frequencies are in the order of the QB eigenfrequencies(1, 2, 3, 4), whereas for kau1 ≃ 6.0, the order is (4, 3, 1, 2). Thiscan be explained by the fact that lower Q modes are pulledmore strongly. As seen in Fig. 8(b), mode M2 is generallythe most strongly pulled mode. However, for large valuesof ka, mode M1 is also subject to strong pulling since it isthe mode located further away from the gain center fre-quency. This result shows that in the case where there are

Fig. 5. Evolution of complexD0 values of the four CF states shown inFig. 3 for different values of the gain transition central frequency ka.The threshold for each mode is given by Dμ

0 when Im�Dμ0 � � 0; this is

indicated by circles on the curves. Note the reversal of M1 and M4 asthe first lasing mode when kau1 is increased.

Fig. 6. Evolution of complexD0 values of the four CF states shown inFig. 3 for different values of the gain transition width γa. The thresholdfor each mode is given by Dμ

0 when Im�Dμ0 � � 0; this is indicated by

circles on the curves.

Fig. 7. Evolution of lasing thresholds of modes M1 and M4 as a func-tion of the gain center frequency ka and gain width γa. The thresholdsof modes M2 and M3 are higher for this range of parameters (data notshown).

Gagnon et al. Vol. 31, No. 8 / August 2014 / J. Opt. Soc. Am. B 1871

closely spaced cold-cavity modes with a similar Q-factor, forinstance in the vicinity of an EP, the spectral characteristics ofthe PM laser may be strongly affected by the gain transitionparameters.

4. CONCLUSIONIn summary, we have used the TLMs of SALT to obtain accu-rate quantitative predictions of the lasing thresholds andfrequencies of a simple diatomic PM composed of twocoupled cylinders. These predictions were obtained fromthe computation of TLMs using 2D-GLMT. This combinationof SALT and 2D-GLMT is general and not limited to diatomicPMs. For instance, it can readily be applied to the computationof modes of random lasers for an arbitrary number of activescattering centers [27].

We found that the lasing thresholds of closely spacedmodesof the diatomic PM are strongly influenced by the gain centerfrequency ka and its linewidth γa.More specifically, the order inwhichmodes lase can be changed by a suitable combination ofthose gain medium parameters. We also highlighted thefrequency pulling effect and found that lower Q modes areusually subject to stronger pulling. These results show theimportance of using ab initio theories to take the gainmediumcharacteristics into account in microcavities research. Futurework will include an extension to nonuniformly pumpedPMs, which precludes, however, the use of the generalizedLorenz–Mie approach. Although these kind of computationsmay be achieved via a finite-element method [15], we haverecently developed a versatile scattering approach [40] thatalso allows for inhomogeneous pumping and arbitrary 2Dgeometries. The generalization to 3D geometries is more chal-lenging, although approaches based on solving the underlyingdifferential equations directly have recently been proposed[41]. By combining these various numerical schemes withSALT, the path is laid out to investigate, engineer, and harness

the lasing properties of PMs for ultra-low threshold and direc-tional single-mode emission.

ACKNOWLEDGMENTSThe authors acknowledge financial support from the NaturalSciences and Engineering Research Council of Canada(NSERC). D.G. was supported by a NSERC PostgraduateScholarship. J.D. and J.L.D. are grateful for a research fellow-ship from the Canada Excellence Research Chair in PhotonicInnovations of Y. Messaddeq. We also acknowledge the freesoftware projects Armadillo [42] and Mayavi [43]. Colorblindcompliant colormaps were taken from [44]. Finally, wethank an anonymous referee for directing our attention toreferences [17,36].

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