Influence of disorder on a Bragg microcavity
S. G. Tikhodeev, E. A. Muljarov, W. Langbein, N. A. Gippius, H. Giessen, T. Weiss
IInfluence of disorder on a Bragg microcavity
S. G. T
IKHODEEV , E. A. M
ULJAROV , W. L
ANGBEIN , N. A.G
IPPIUS , H. G
IESSEN , AND
T. W
EISS , M. V. Lomonosov Moscow State University, Leninskie Gory 1, Moscow 119991, Russia A. M. Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova Street 38, Moscow119991, Russia Cardiff University, School of Physics and Astronomy, The Parade, CF24 3AA, Cardiff, United Kingdom Skolkovo Institute of Science and Technology, Nobel Street 3, Moscow 143025, Russia Physics Institute and Research Center SCoPE, University of Stuttgart, Stuttgart D-70550, Germany * [email protected] Abstract:
Using the resonant-state expansion for leaky optical modes of a planar Braggmicrocavity, we investigate the influence of disorder on its fundamental cavity mode. We modelthe disorder by randomly varying the thickness of the Bragg-pair slabs (composing the mirrors)and the cavity, and calculate the resonant energy and linewidth of each disordered microcavityexactly, comparing the results with the resonant-state expansion for a large basis set and withinits 1 st and 2 nd orders of perturbation theory. We show that random shifts of interfaces causea growth of the inhomogeneous broadening of the fundamental mode that is proportional tothe magnitude of disorder. Simultaneously, the quality factor of the microcavity decreasesinversely proportional to the square of the magnitude of disorder. We also find that 1 st orderperturbation theory works very accurately up to a reasonably large disorder magnitude, especiallyfor calculating the resonance energy, which allows us to derive qualitatively the scaling of themicrocavity properties with disorder strength. © 2020 Optical Society of America
1. Introduction
Disorder plays an important role in photonics. For example, it drives the coloring and polarizationconversion of natural disordered light diffusers such as opals, birds feathers, or wings ofbutterflies [1–5]. Unavoidable technological imperfections can sometimes critically reduce thedesired performance of photonic crystal slab waveguides and nanocavities [6–9]. Differenttheoretical approaches have been proposed to describe the role of disorder, either numerically [10,11] or based on various versions of perturbation theory in electrodynamics [6, 12–15]. Theimportant prerequisite for any perturbation theory is a suitable basis, which, in the case of openelectrodynamical systems, is composed of resonant states (also known as quasi-normal or leakymodes) [16–23] that determine the resonant optical response, e.g., the Fano resonances in opencavities [24–26].Recently, the resonant-state expansion, a rigorous perturbation theory for calculating theresonant states of any open system in electrodynamics based on a finite number of resonantstates of some more elementary system, has been developed [19]. Originally proposed forpurely dielectric shapes (slabs, microspheres, microcavities [27]) with nondispersive dielectricpermittivity, the method was then generalized to dispersive open systems [28], photonic crystalslabs, and periodic arrays of nanoantennas at normal [29] and oblique incidence [30], and opensystems containing magnetic, chiral, or bi-anisotropic materials [31]. In addition, the method hasbeen extended to waveguide geometries such as dielectric slab waveguides [32, 33] and opticalfibers [34], with a possibility to account for nonuniformities [35] and nonlinearities [36].The perturbation in the resonant-state expansion can be of any shape within the basis volume.The difference from the basis reference can even be huge when using a sufficiently large number a r X i v : . [ c ond - m a t . d i s - nn ] S e p f resonant states as basis. In order to have a meaningful physical picture, it is, however, better todescribe the structure of interest using a minimum number of resonant states, see, e.g., examples ofcalculating the sensor performance with a single resonant state first order approximation [29, 37],and the interaction of spatially separated photonic crystal slabs with a pair of quasi-degeneratestates in Ref. [30].In this paper we concentrate on the impact of disorder on the resonant states of a Braggmicrocavity using full-wave calculations and the resonant-state expansion as well as its 1 st -and 2 nd -order perturbative formulations. In particular, we vary randomly the thickness of theBragg-pair slabs (acting as the mirrors) and the cavity itself, and derive how the resonantstates change with growing amplitude of random displacements. On the one hand, because ofthe simplicity of the system, its disorder-modified states (their energies, linewidths, and fielddistributions) can be calculated with any accuracy via linearization of the frequency dependenceof the inverse scattering matrix around the resonant state of interest [18, 38, 39] for each disorderrealization. On the other hand, we can calculate the same resonances using the resonant-stateexpansion for an increasing number of resonant states in the basis, and then compare them withthe exact values. Repeating the calculations many times and retrieving the statistically averagedresults yields relevant information about the influence of disorder on the optical properties of theBragg microcavities.The paper is organized as follows: The model of the disordered Bragg microcavity is describedin Sec. 2, the formulation of the resonant-state expansion is given in Sec. 3. Section 4 summarizesthe results of the comparison between the exact solutions and those obtained by the resonant-stateexpansion using different orders of perturbation theory. Special attention is paid to the disorder-induced inhomogeneous broadening in the ensemble of disordered cavities (Subsec. 4.1) and theanalysis of the influence of disorder magnitude on the statistically averaged resonance energiesand their homogeneous linewidths (Subsec. 4.2). Section 5 contains a discussion of the obtainedresults, which are summarized in Sec. 6. Details of the linearization scheme of calculating thepoles of the scattering matrix are given in Appendix 6. The accuracy of the different ordersof perturbation theory based on the resonant-state expansion, depending on the magnitude ofdisorder is discussed in Appendix 6.
2. Model
We consider a planar microcavity that is made of two Bragg mirrors with m pairs of layers of λ / ε and ε , surroundinga cavity layer of M × λ / ε . The cavityis surrounded by free space with permittivity ε =
1. In the numerical results presented weuse ε = ε = m =
4, and M =
2, the latter corresponding to a cavity layer of λ optical thickness. A schematic of the microcavity is displayed in Fig. 1. We have chosenthe parameters of the cavity such that the fundamental cavity mode at normal incidence is Ω = π (cid:126) c / λ = λ = . µ m). This corresponds to thicknesses of the Bragg λ / L = π c (cid:126) /( √ ε Ω ) ≈
98 nm and L = π c (cid:126) /( √ ε Ω ) ≈
155 nm, and the central cavitylayer is L C = L ≈
392 nm thick. Then the fundamental cavity mode linewidth appears to be Γ = . Q = Ω / Γ ≈ E and Im E of the resonant electric field of the fundamental cavity mode with eigenenergy E = Ω − i Γ are shown in Fig. 1 by blue and red curves, respectively.The optical scattering matrix of this simple microcavity has an infinite series of discreteFabry-Perot poles on the complex energy plane, which manifest themselves as peaks in thetransmission spectrum, as shown in Figs. 2a,b.The transmission spectrum in Fig. 2 has been calculated within a 2 ×
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Fig. 1. (Color online) Schematic of the unperturbed Bragg microcavity (gray back-ground) and spatial distributions of the real (blue solid line) and imaginary (red solidline) parts of the electric field E ( ) ( z ) of the fundamental cavity mode. Darker andbrighter gray shades indicate materials with dielectric susceptibilities ε =
10 and ε =
4, respectively. Yellow/bright gray shades illustrate a realization of a microcavitywith interfaces randomly displaced by shifts a j , with disorder strength a = . []00.51 T r a n s m itt a n ce (a) Energy (meV) -40-200 E n e r gy ( m e V ) (b)
996 998 1000 1002 1004Energy (meV)00.51 T r a n s m it a n ce (c) Fig. 2. (Color online) (a) Transmittance of the ideal microcavity as depicted in Fig. 1(without random displacements). (b) Map of resonant states of the microcavity on thecomplex energy plane (crosses); the vertical green dashed lines denote the positionsof the resonant states on the real energy axis. (c) Transmission spectra in the vicinityof the fundamental resonance at 1 eV (red curve). The dashed blue curve shows thesingle-pole resonant approximation given by Eq. (17). Green, red, and blue verticaldashed lines mark the energies Ω , Ω − Γ , and Ω + Γ , respectively. energy plane in Fig. 2b, as well as the the electric eigenfields in Fig. 1 can be calculated viathe scattering matrix energy dispersion linearization [18, 38, 39], as described in Appendix 6.The real part of the eigenenergy, Ω n = Re E n , corresponds to the resonance energy, while theimaginary part, 2 Γ n = − E n , gives the resonance linewidth. In what follows, we mark thevalues corresponding to the unperturbed (ideal) microcavity by the upper index ( ) , as shown inFigs. 1,2.We now investigate the behavior of the fundamental cavity mode denoted by eigenenergy under the influence of random displacements of the microcavity interfaces. We will leavethe external interfaces of the microcavity at their original positions, and assume that all other j = , , . . . J ( J = ) interfaces are shifted by a j = a β j L , (1)where β , β , . . . β J is a set of J uniformly distributed uncorrelated random numbers withinthe interval (-1,1). The disorder strength a is chosen between zero and 0.5 in order to keepall resulting layer thicknesses positive. Furthermore, we consider uncorrelated disorder withvanishing statistically averaged displacements (cid:104) β j (cid:105) = . (2)Note that random shifts a j of the interfaces have been measured experimentally before indisordered GaAs/AlAs cavities [40].
3. Resonant-state expansion
The resonant-state expansion [19, 27, 30, 41] relies on knowing the electric field distributions E ( ) n ( z ) of a set of resonant states with complex frequencies E ( ) n for a photonic structure witha spatial profile of the dielectric susceptibility ε ( ) ( z ) . These resonant states are used in theresonant-state expansion as a basis to expand the electric fields of the resonant state of a modifiedstructure with dielectric susceptibility ε ( z ) = ε ( ) ( z ) + ∆ ε ( z ) (3)as E( z ) = (cid:213) n b n E ( ) n ( z ) C n . (4)The normalization constants C n have the analytical form [19, 30] C n = ∫ L ε ( z )E n ( z ) dz + i k n (cid:2) E n ( ) + E n ( L ) (cid:3) , (5)where the range 0 to L covers exactly the microcavity structure, and the fields in the second termhave to be taken in the medium outside the cavity. However, the fields are continuous at theoutermost interfaces for the considered case of normal incidence, because this results in purelytransverse electric fields over the entire microcavity. The general orthonormality of resonantstates is given by [19, 30] δ n (cid:48) n = C n C (cid:48) n (cid:26)∫ L ε ( z )E n (cid:48) ( z )E n ( z ) dz (6) + ik n (cid:48) + k n [E n (cid:48) ( )E n ( ) + E n (cid:48) ( L )E n ( L )] (cid:27) , where (cid:126) k n = E n / c .The coefficients b n and new eigenenergies E can be calculated via the linear eigenproblem [19] (cid:213) n (cid:48) W nn (cid:48) b n (cid:48) = E b n , (7)where W nn (cid:48) = ( A − ) nn (cid:48) E ( ) n (cid:48) , (8) nn (cid:48) = δ nn (cid:48) + V nn (cid:48) , (9)and the matrix elements of the perturbation are V nn (cid:48) = C n C n (cid:48) ∫ L ∆ ε ( z )E ( ) n ( z )E ( ) n (cid:48) ( z ) dz . (10)Following from the resonant-state expansion, the resonance eigenenergy in 1 st order of perturbationtheory yields [30, 41] E ( ) n ≈ E ( ) n (cid:18) + V nn (cid:19) − , (11)whereas the resonant state eigenenergy up to 2 nd order of perturbation theory is given by [41] E ( ) n ≈ E ( ) n (cid:32) + V nn − (cid:213) n (cid:48) (cid:44) n E ( ) n V n (cid:48) n E ( ) n − E ( ) n (cid:48) (cid:33) − . (12)In the following, we keep explicitly the normalization constants C n in the resonant-stateexpansion formulas and use the eigenfields (e.g., the one shown in Fig. 1) satisfying the conditions E n ( ) = (− ) p n E n ( L ) = , (13)where p n = ,
4. Influence of disorder
While investigating the influence of disorder, we compare the scattering matrix result fromlinearization, which we call here “exact”, with the 1 st and 2 nd order approximations (11) and (12)as well as with the full resonant-state expansion obtained by solving the eigenvalue problem (7)with a truncation of an infinite matrix. The resonant-state expansion is asymptotically exact, andits only limitation is the basis size. The resonant states are calculated as described in Appendix 6.The basis size is taken as N =
419 in the present paper, symmetrically around the fundamentalcavity mode, see Appendix 6. The same resonant states are used in the 2 nd order perturbationtheory.Figure 3 illustrates changes of the real and imaginary parts of the fundamental cavity modeenergy and linewidth for 1000 different realizations of random shifts of interfaces with thedisorder parameter a = . ∼
15 nm.It can be seen that ( i ) introducing disorder causes an inhomogeneous broadening of theresonance energy position, with a standard deviation on the order of 10 meV; ( ii ) the linewidthof the resonance (homogeneous broadening) grows by approximately 10% (from ∼ . ∼ .
52 meV); ( iii ) the results for the resonance energy Ω (Fig. 3a), calculated exactly, in the1 st and 2 nd perturbation orders, and in the resonant-state expansion do visually coincide for alldisorder realizations, while for the linewidth Γ only the exact and the resonant-state expansionresults coincide.The difference, representing the calculation error between the exact results, the 1 st , 2 nd and“full” resonant-state expansion (with N =
419 resonant states in the basis) is analyzed versus thedisorder strength a in Appendix 6. Since the absolute error is similar for the real and imaginarypart, the relative error of calculating Ω is approximately Q times smaller than that of Γ .Figure 8 in Appendix 6 demonstrates that the calculation error and its standard deviationgrow with a and can be quite large, especially for the 1 st perturbation order. Interestingly,
200 400 600 800 1000
Realization ( m e V ) (a) Realization ( m e V ) (b) exact1st order2nd orderRSE Fig. 3. (Color online) Example of calculated resonance frequencies Ω (a) andlinewidths Γ (b) of the fundamental microcavity resonance, for 1000 interface shiftrealizations with disorder parameter a = . st and 2 nd order perturbationtheory [Eq. (11) and (12), respectively], and resonant-state expansion (RSE) [Eq. (7)]with 419 basis states. Cyan dotted and dashed horizontal lines in panel (a) denote themean values (cid:104) Ω (cid:105) and (cid:104) Ω (cid:105) ± σ Ω , where σ Ω is the standard deviation of Ω . Themagenta dashed horizontal line indicates the resonance energy Ω ( ) of the unperturbedmicrocavity. Lines in panel (b) give the equivalent values for the resonance linewidth,i.e., (cid:104) Γ (cid:105) , (cid:104) Γ (cid:105) ± σ Γ , and Γ ( ) , where σ Γ is the standard deviation of Γ . the calculation errors for the quantities, averaged over many (1000 in this work) realizations,remain relatively small over the investigated range of disorder parameter a ≤ .
3, even in the 1 st perturbation order.In what follows we investigate the statistics of Ω and Γ as functions of the disorder parameter.However, we begin from the analysis of the most visible effect of the disorder, namely theinhomogeneous broadening of the fundamental cavity mode eigenenergy distribution due to thedisorder. The distribution of fundamental cavity mode energies Ω ( ν ) (where ν stands for the realizationnumber of the random disorder) broadens with increasing disorder strength a , as is clearlyseen in Fig. 3a and Fig. 8 in Appendix 6. Physically, this would result in an inhomogeneousroadening of the transmission spectrum of a hypothetical large-area microcavity with randomlydisplaced inner interfaces, where the displacement changes gradually on some large-distancescale, and assuming incoherent addition of the transmission of different parts of this largemicrocavity. Such inhomogeneous broadening was observed, e.g., in high-quality factor III-Vnitride microcavities [42] and attributed to homogeneous areas (at a local scale of ∼ µ m),separated by fluctuations occuring on a short distance scale. The realization of high Q-factor insuch microcavities is likely to be limited by the structural disorder [43, 44].From comparison with Fig. 3b we see that for a disorder parameter a = . a ≈ . β j (see Eq. 1), itturns out to be close to normal Gaussian. A discretized density of states can be defined on anenergy mesh with a small step δ < Γ as P δ ( E ) = δ (cid:213) ν ∫ E + δ / E − δ / δ ( E (cid:48) − Ω ( ν )) dE (cid:48) , (14)where the sum is evaluated over all random realizations. It can be smoothed on a larger energyscale by convolution with a normalized rectangular function of width ∆ ≈ Γ , resulting in thedistribution P δ, ∆ ( E ) = ∆ ∫ E + ∆ / E − ∆ / P δ ( E (cid:48) ) dE (cid:48) . (15)Typical densities of states Eqs.(14) and (15) for the distribution of poles in Fig. 3a with anamplitude of disorder of a = . δ = . ∆ = . P Gauss ( E ) = √ πσ Ω exp (cid:32) − ( E − Ω ( ) ) σ Ω (cid:33) , (16)plotted as a blue dashed line in Fig. 4a, in accordance with the Central Limit Theorem. Thistheorem states that if you sum up a large number of random variables, the distribution of the sumwill be approximately normal (i.e., Gaussian) under certain conditions, see, e.g., Ref. [45].The transmission spectrum of the ideal microcavity in the vicinity of the fundamental cavitymode is approximated quite well by a Lorenzian T ( E , Ω ) = Γ ( E − Ω ) + Γ , (17)see the solid and dashed lines in Fig. 2c. Thus, the inhomogeneously broadened spectrumof a large microcavity with the distribution of resonances is expected to exhibit the Voigtfunction [46, 47] shape, (cid:104) T ( E )(cid:105) = ∫ T ( E , Ω ) P Gauss ( Ω ) d Ω . (18)In the limit Γ (cid:28) σ Ω the averaged transmission is approximately Gaussian, (cid:104) T ( E )(cid:105) ≈ Γ σ Ω (cid:114) π (cid:32) − ( E − Ω ( ) ) σ Ω (cid:33) , (19) (cid:25)(cid:19) (cid:28)(cid:27)(cid:19) (cid:20)(cid:19)(cid:19)(cid:19) (cid:20)(cid:19)(cid:21)(cid:19) (cid:20)(cid:19)(cid:23)(cid:19) (cid:40)(cid:81)(cid:72)(cid:85)(cid:74)(cid:92)(cid:3)(cid:11)(cid:80)(cid:72)(cid:57)(cid:12) (cid:19)(cid:19)(cid:17)(cid:19)(cid:20)(cid:19)(cid:17)(cid:19)(cid:21)(cid:19)(cid:17)(cid:19)(cid:22)(cid:19)(cid:17)(cid:19)(cid:23)(cid:19)(cid:17)(cid:19)(cid:24)(cid:19)(cid:17)(cid:19)(cid:25)(cid:19)(cid:17)(cid:19)(cid:26)(cid:19)(cid:17)(cid:19)(cid:27)(cid:19)(cid:17)(cid:19)(cid:28) (cid:39) (cid:72) (cid:81) (cid:86) (cid:76)(cid:87) (cid:92) (cid:3) (cid:82) (cid:73) (cid:3) (cid:86) (cid:87) (cid:68) (cid:87) (cid:72) (cid:86) (cid:3) (cid:11) (cid:80) (cid:72) (cid:57) (cid:16) (cid:20) (cid:12) (cid:11)(cid:68)(cid:12)(cid:68)(cid:32)(cid:19)(cid:17)(cid:20)(cid:24)(cid:19)(cid:24)(cid:15)(cid:49)(cid:32)(cid:23)(cid:20)(cid:28) (cid:51)(cid:51) , (cid:42)(cid:68)(cid:88)(cid:86)(cid:86)(cid:76)(cid:68)(cid:81)
960 980 1000 1020 1040
Energy (meV) T r a n s m i ss i on (b)
960 980 1000 1020 1040
Energy (meV) -3 -2 -1 T r a n s m i ss i on (c) Fig. 4. (Color online) (a) Densities of states [Eqs. (14) and (15)], calculated with δ = . ∆ = . Ω ± σ Ω and the solid line Ω ; (b) Averaged inhomogeneously broadenedtransmission spectrum of a large cavity with all realizations of disorder (red solid curve).The green solid line is the homogeneously broadened transmission spectra of the idealcavity calculated by Eq. (17). Blue dashed curves display Gaussian distributions [Eqs.(16) and (19)]; (c) same as panel (b) in logarithmic scale. except for the Lorenzian tails for | E − Ω | > σ Ω . An example of averaged spectra correspondingto the distribution of poles at disorder parameter a = . (cid:104) T ( E )(cid:105) (redcurves in panels b,c) coincide quite well with the Gaussian spectra, Eq. (19) (blue dashed curves)in the central part of the broadened resonance. In contrast, the Lorenzian tails approaching thehomogeneously broadened spectrum Eq. (17) (solid green curves) are clearly visible in panel cdue to its semi-log scale. ( m e V ) exact1st order2nd orderRSE ( m e V ) (b) ( m e V ) (c) ( m e V ) (d) Fig. 5. (Color online) Resonance energy (cid:104) Ω (cid:105) (a) and linewidth (cid:104) Γ (cid:105) (b) as functions ofdisorder parameter a , averaged over 1000 realizations of random interface displacements.Panel (c) shows the standard deviation of resonance energy σ Ω as a function of a ,and panel (d) depicts (cid:104) Γ (cid:105) as a function of a . The values of Ω ( ) [ Γ ( ) ] for the idealmicrocavity without disorder are shown as horizontal dashed lines in panel (a) [panels(b-d)]. The dependence of the averaged parameters of the fundamental cavity mode on the disorderparameter a are illustrated in Fig. 5. Panels a and b display the averaged fundamental cavitymode energy (cid:104) Ω (cid:105) and linewidth (cid:104) Γ (cid:105) , respectively, as functions of the disorder parameter a .The averaging is carried out over 1000 random realizations, different for each value of a . Panel cdepicts the inhomogeneous broadening. It displays the fundamental cavity mode energy standarddeviation σ Ω = (cid:104)( Ω − (cid:104) Ω (cid:105)) (cid:105) / as a function of disorder parameter a . Panel d contains thesame dependence as in panel b, but plotted instead as a function of a .The averaged position of the resonance does not shift significantly with the growth of thedisorder parameter a . Fluctuations are due to the finite number of realizations used. Themagnitude of the inhomogeneous broadening, which is given by σ Ω , grows linearly with a , andthe averaged linewidth (cid:104) Γ (cid:105) grows quadratically with a (the latter is clearly visible in paneld). The inhomogeneous broadening matches the homogeneous linewidth of the resonance at a ≈ . a , as illustrated in Fig. 6.
5. Discussion
The reasons for the power scaling a α of (cid:104) Ω (cid:105) , σ Ω and (cid:104) ∆Γ (cid:105) with α = , st order approximation of the resonant-state expansion.The characteristic feature of the fundamental cavity mode electric field distribution for an a Q N = 419exact1st order2nd orderRSE
Fig. 6. (Color online) Dependence of the microcavity quality factor on the disorderparameter a , calculated exactly and in the 1 st and 2 nd orders of perturbation theory, aswell as the resonant-state expansion with 419 states. The averaging is carried out over1000 realizations of random displacements of inner interfaces in the microcavity. unperturbed microcavity with exactly λ / λ cavity layer, and with theboundary conditions of Eq. (13) can be clearly seen in Fig. 1. Namely, the values of real andimaginary parts of the electric eigenfield are subsequently zeroed exactly at successive interfaces.As a result, in the vicinity of each interface, either the real or the imaginary part of the field iseither a constant or a linear function of the distance to this interface z − z , j , i.e.,Re E ( ) ( z ) ≈ D j + O( z − z , j ) , Im E ( ) ( z ) ≈ ( z − z , j ) F j + O (cid:16) ( z − z , j ) (cid:17) , (20)or Re E ( ) ( z ) ≈ ( z − z , j ) D j + O (cid:16) ( z − z , j ) (cid:17) , Im E ( ) ( z ) ≈ F j + O( z − z , j ) , (21)where D j , F j are constants. The signs of D j , F j are identical (negative or positive) on theright-hand sides of the layers with larger dielectric susceptibility (i.e., for odd j = m + j = m ). Note that ∆ ε = | ∆ ε | sign ( z − z , j ) on such right-hand side interfaces, and ∆ ε ( z ) = −| ∆ ε | sign ( z − z , j ) on the left-hand side ones.Additionally, the normalization constant of the fundamental cavity mode is real, as discussedin Appendix 6. All this results in the following equation for the fundamental cavity modeeigenenergy, averaged over random realizations: (cid:104) (cid:126) ω (cid:105) = (cid:126) ω (cid:18) − (cid:104) V (cid:105) (cid:19) ≡ (cid:126) ω + (cid:126) ∆ ω , (22)with (cid:126) ∆ ω = − Ω (cid:213) j (cid:104) V , j (cid:105) , (23)where the sum is over all inner interfaces and V , j = C − ∫ z , j + a j z , j ∆ ε ( z )E ( z ) dz ≈ | ∆ ε | C − ± D j a j / ∓ F j a j + i | D j F j | a j , j = m + ∓ D j a j ± F j a j / + i | D j F j | a j , j = m (24)fter averaging the odd powers of a j vanish, and, as a result, we obtain in the 1 st resonant-stateexpansion order and up to the 2 nd order in a (cid:104) ∆Ω (cid:105) = , (cid:104) ∆Γ (cid:105) ∝ a , (25)in agreement with the numerical results in Fig. 5a,b. As to the inhomogeneous broadening of Ω ,due to the terms linear in a j , σ Ω is proportional to a and thus σ Ω ∝ a , in agreement with thenumerical results in Fig. 5c.This shows in particular the well known fact that the unperturbed planar Bragg microcavity isan optimized structure from the point of view of the maximum quality factor Q (or minimum ofhomogeneous linewidth Γ ): Any change of its structure causes a decrease of Q and increase of Γ . In fact, in the case the unperturbed structure would not correspond to a mimumum of Γ versus layer thicknesses, a linear dependence of Γ with a would be present.
6. Conclusion
To conclude, we have demonstrated that introducing random shifts of interfaces in a standardplanar Bragg microcavity causes a growth of the inhomogeneous broadening of the fundamentalcavity mode, linear in the disorder strength a , which quantifies the relative change of the layerthicknesses. In contrast, the linewidth increases proportionally to a , with an according decreaseof the quality factor. The inhomogeneous broadening starts to exceed the homogeneous one at acertain value of disorder parameter, which is a ≈ .
02 for the considered microcavity. The 1 st order perturbation theory within the resonant-state expansion works accurately up to a disorderstrength of a ≈ .
1, especially for calculating the resonance energy. Furthermore, it allows tofind a quantitative scaling of the microcavity parameters with disorder strength.
Appendix A: Poles of the scattering matrix via linearlization
For normal incidence, the solutions of Maxwell equations for electric E and magnetic H fields ineach layer of the microcavity E = ( E x , , ) , H = (cid:0) , H y , (cid:1) , (A1)with E x = A + exp (− i ω t + ik l z ) + A − exp (− i ω t − ik l z ) , (A2) H y = n l A + exp (− i ω t + ik l z ) − n l A − exp (− i ω t − ik l z ) , where n l = √ ε l , l = , , ε = k l = n l ω / c . Note that we are using the SGS units. Defining the amplitude vector as | A (cid:105) = (cid:169)(cid:173)(cid:171) A + A − (cid:170)(cid:174)(cid:172) , (A3)the transfer matrix over a distance d inside a homogeneous and isotropic material is˜ T l , d = (cid:169)(cid:173)(cid:171) exp ( ik l d )
00 exp (− ik l d ) (cid:170)(cid:174)(cid:172) , (A4)with | A ( z + d )(cid:105) = ˜ T l , d | A ( z )(cid:105) ). The transfer matrix over the interface from material l to l (cid:48) is T l (cid:48) , l = (cid:169)(cid:173)(cid:171) + K − K − K + K (cid:170)(cid:174)(cid:172) , K = n l n l (cid:48) . (A5) Fig. 7. (Color online) Real (blue solid lines) and imaginary (red solid lines) parts ofthe electric field distributions E ( ) n ( z ) of the resonances of the ideal microcavity with n = − , − , . . .
11, normalized satisfying the conditions (13). The Q-factors andeigenenergies E n = Ω n − i Γ n (in meV) are shown in the title of each panel. We can calculate the transfer matrix over the entire microcavity as T ( ω ) = T , (cid:16) T − BP (cid:17) ˜ T , L C ( T BP ) T , , where T BP = T , ˜ T , L T , ˜ T , L , so that the amplitude vectors from the left and right sides of the microcavity are connected as | A L (cid:105) = (cid:169)(cid:173)(cid:171) A + L A − L (cid:170)(cid:174)(cid:172) , | A R (cid:105) = (cid:169)(cid:173)(cid:171) A + R A − R (cid:170)(cid:174)(cid:172) , | A L (cid:105) = T ( ω )| A R (cid:105) . (A6) able 1. The eigenenergies E n = Ω n − i Γ n and normalization constants C n of the first − ≤ n ≤
10 resonances of the original ideal microcavity. The parameters for thefundamental cavity mode with n = n Ω n (meV) Γ n (meV) Re ( C n ) (nm) Im ( C n )/ Re ( C n ) -1 0 24.8 7.12 · · − -9 99.2 26.5 6.69 · · − -8 186.3 25.0 7.08 · · − -7 295.6 25.7 6.88 · · − -6 375.3 25.0 7.07 · · − -5 485.3 23.7 7.45 · · − -4 565.4 23.6 7.44 · · − -3 659.5 19.1 9.20 · · − -2 746.6 17.3 1.01 · · − -1 797.9 9.18 1.90 · · − · · − · -4.28 · − · -3.72 · − · -2.98 · − · -2.39 · − · -1.84 · − · -1.39 · − · -9.99 · − · -6.54 · − · -3.17 · −
10 2000.0 24.8 7.12 · · − Using the vectors of incoming and outgoing amplitudes | in (cid:105) = (cid:169)(cid:173)(cid:171) A + L A − R (cid:170)(cid:174)(cid:172) , | out (cid:105) = (cid:169)(cid:173)(cid:171) A − L A + R (cid:170)(cid:174)(cid:172) (A7)the optical scattering matrix is defined as | out (cid:105) = S ( ω )| in (cid:105) . (A8)From this definition, it is seen that the physical meaning of the scattering matrix components is S = (cid:169)(cid:173)(cid:171) r LL t RL t LR r RR (cid:170)(cid:174)(cid:172) , (A9) -3 -2 -1 -12 -10 -8 -6 -4 -2 | | / -3 -2 -1 -10 -8 -6 -4 -2 (b) Fig. 8. (Color online) Relative accuracy of the 1 st and 2 nd order perturbation theory,as well as the resonant-state expansion with 419 states for real (a) and imaginary (b)parts of the resonance energy as functions of disorder parameter a for the fundamentalcavity mode, averaged over 1000 realizations of random interface displacements. Blackdashed, solid, and dashed-dotted lines show a , a , and a , respectively, dependencies.The relative accuracy of the quality factor is same as shown in panel (b). The relativeaccuracy is calculated as the relative difference between the exact and the approximatemethods. a -2-1.5-1-0.500.511.5 C a l c u l a ti on E rr o r -3 (a) | |/ / / a -0.15-0.1-0.0500.050.1 C a l c u l a ti on E rr o r (b) | |/ / / Fig. 9. (Color online) Relative calculation error of the 1 st order perturbation theoryfor real (a) and imaginary (b) parts of the resonance energy, as functions of a forthe fundamental cavity mode, averaged over 1000 realizations of random interfacedisplacements. The dashed lines with open triangles are the same as shown in Fig. 8.The dashed red lines are averaged calculation errors (cid:104) ∆Ω / Ω (cid:105) and (cid:104) ∆Γ / Γ (cid:105) . Yellowregions show the width of the error distribution, e. g., (cid:104) ∆Γ / Γ (cid:105) ± σ ∆ in panel b. where, e.g., r LL is the amplitude reflection coefficient from the left side of microcavity to left,and t RL is the amplitude transmission coefficient from left to right. The connection with thecomponents of transfer matrix is S = (cid:169)(cid:173)(cid:171) − T − T T − T − T T − T T T − (cid:170)(cid:174)(cid:172) , T = (cid:169)(cid:173)(cid:171) T T T T (cid:170)(cid:174)(cid:172) . (A10)Eigensolutions (resonances) are found as nonvanishing outgoing solutions | out (cid:105) = | o n (cid:105) (cid:44) | in (cid:105) =
0, which results in the homogeneous equation for the resonant outgoingeigenvectors | o n (cid:105) and eigenfrequencies ω n : S − ( ω n )| o n (cid:105) ≡ R ( ω n )| o n (cid:105) = . (A11)Equation (A11) can be solved iteratively via a frequency-dependent linearization, described, e.g.,n Ref. [18]. Assuming that ω n = ω + ∆ ω, and linearizing Eq. (A11) over ∆ ω , we obtain0 = R ( ω n ) | o n (cid:105) = R ( ω ) | o n (cid:105) + ∆ ω dR ( ω ) d ω | o n (cid:105) , which requires R ( ω ) | o n (cid:105) = − ∆ ω dR ( ω ) d ω | o n (cid:105) . Thus, we arrive at a linear 2 × ∆ ω : W | o n (cid:105) = ∆ ω | o n (cid:105) , (A12)where the matrix W ( ω ) ≡ − (cid:20) dR ( ω ) d ω (cid:21) − R ( ω ) = S ( ω ) (cid:20) dS ( ω ) d ω (cid:21) − (A13)can be easily calculated and diagonalized. The latter equation follows from d (cid:0) SS − (cid:1) / d ω = ∆ ω of W generates the corrected frequency ω (cid:48) = ω + ∆ ω , which ispresumably closer to the solution of Eq. (A8). The procedure can be iteratively continued untilfinding the solution with the desired accuracy. As a starting point for iterations, it makes sense touse the real values of frequency, that correspond to the transmission maxima (see in Fig. 2).As for the resonance eigenvector, it is known in the case of mirror-symmetric structure inadvance due to symmetry constraints: | o n (cid:105) = (cid:169)(cid:173)(cid:171) (− ) p n (cid:170)(cid:174)(cid:172) . (A14)The parity is p n = E n ( z ) = A + n ( z ) exp ( ik n z ) + A − n ( z ) exp (− ik n z ) , (A15)with (cid:169)(cid:173)(cid:171) A + n ( z ) A − n ( z ) (cid:170)(cid:174)(cid:172) = T z ( ω n ) (cid:169)(cid:173)(cid:171) (cid:170)(cid:174)(cid:172) , (A16)where T z ( ω n ) is the transfer matrix from the left side of the microcavity to point z inside.The calculated eigenenergies and normalization constants for the 21 resonances around theresonance with Ω = Ω = − ≤ n ≤
11 are shown in Fig. 7. The resonance with n = −
10 is ‘static’, Ω − =
0. All other resonances are mirror-symmetric on the complex energy plane around it: theresonances with ˜ n = n + < Ω ˜ n − = − Ω − ˜ n − < Γ ˜ n − = Γ − ˜ n − , and the eigenfieldsare complex conjugate, i.e., Re E ˜ n − ( z ) = Re E − ˜ n − ( z ) , Im E ˜ n − ( z ) = − Im E − ˜ n − ( z ) . Theparity of the resonance with odd (even) n is odd (even). These simple symmetry properties of thereal and imaginary parts of the resonant fields are the consequences of the mirror symmetry ofthe microcavity and the definition of normalized resonant states using the boundary conditionsEq. A14. Note that the normalization constants C n are, generally, complex (except those ofthe fundamental cavity mode and other high-Q states, see below). We use in the main text upo N =
419 states in the resonant-state expansion basis, positioned symmetrically around thefundamental cavity mode, i.e., with Ω n for −( N − )/ (cid:54) n (cid:54) ( N − )/ C , = ∫ L ε ( x )E ( x ) dx ≈ . · − . · i and C , = i k (cid:2) E ( ) + E ( L ) (cid:3) ≈ − . · − + . · i for the unnormalized eigenfield E , shown in Fig. 1. This field is normalized according to E ( ) ( ) = E ( ) ( L ) = , which follows from Eq. (A16). It appears that for the fundamental cavity mode with n = C = C , + C , ≈ . · + . · − i , so that C is real with the accuracy of our numerical procedure. Appendix B: Accuracy of different approximations of the resonant-state expan-sion
The averaged absolute values of relative errors for calculating Ω and Γ by the 1 st and 2 nd orderapproximations and the resonant-state expansion with 419 nearest poles are illustrated in Fig. 8(panels a and b, respectively) as functions of the disorder parameter a . It can be seen that the 1 st order perturbation theory becomes, as expected, less accurate with increasing disorder parameter,but it gives in most cases quite accurate results, especially for the calculation of Ω , and for smallamplitude of disorder, a < . a , and Fig. 8bcontains black dashed and dashed-dotted ones, proportional to a and a , respectively. Themagnitude of the calculation errors grows as a and a for the 1 st perturbation order over theinvestigated range of a for Ω and Γ , respectively. For Ω , the second order only provides afactor of 2 improvement and is limited by the basis size used in the resonant-state expansion.The calculation error in the 2 nd perturbation order scales instead as a for Γ , but is limitedfor small a by the finite size of the resonant-state expansion basis used and merges with theerror of the corresponding resonant-state expansion. As to the calculation error of the fullresonant-state expansion, in the case of Γ it saturates around 2 × − for a > .
1. For a (cid:46) . nd order perturbation theorywhich means that the full resonant-state expansion becomes redundant. However, the value of a where the second order matches the full resonant-state expansion depends on the basis size. Thesaturated accuracy of the full resonant-state expansion for larger a depends on the chosen basissize. With decrease of the size of the resonant state basis this saturated accuracy worsens, e.g., to ∼ × − for N = st order perturbation theory of resonant-state expansion works well for a < . a corresponds to the amplitude of interface displacement of up to 30%of the thinner Bragg layer thickness, or in the present case as large as ∼
30 nm. The averagedcalculation error of the 1 st order perturbation is still smaller than 10% for a = .
3. Of course, ascan be understood from Fig. 2 and Fig. 3, there occur relatively rare displacement realizations witha very large calculation error. However, the majority of disorder realizations is still reasonablyell described by 1 st order perturbation theory. Figure 9 illustrates the width of the range wheremore than half of the disorder realizations are confined (filled by yellow color). With growingdisorder parameter systematic errors arise (cid:104) ∆Ω / Ω (cid:105) < (cid:104) ∆Γ / Γ (cid:105) <
0. However, for weakdisorder these systematic errors are small, and (cid:104)| ∆Ω |/ Ω (cid:105) ≈ σ ∆Ω / Ω , (cid:104)| ∆Γ |/ Γ (cid:105) ≈ σ ∆Γ / Γ . FundingAcknowledgments
The authors acknowledge support from DFG (SPP 1839, Mercator-Professorship), and RussianAcademy of Sciences. S.G.T. and N.A.G. thank the Russian Science Foundation (Grant No.16-12-10538 Π ) for support in part of the calculations of the exact microcavity resonant states. Disclosures
The authors declare no conflicts of interest.
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