Electronic version of an article published as Huang, JG, Christian, JM & McDonald, GS 2017, 'Spontaneous spatial fractal pattern formation in dispersive systems' Journal of Nonlinear Optical Physics and Materials, vol 26, no. 1, 1750009. DOI: 10.1142/S0218863517500096 © copyright World Scientific Publishing Company https://www.worldscientific.com/worldscinet/jnopm

Spontaneous spatial fractal pattern formation in dispersive systems

J. G. Huang

Faculty of Computing, Engineering and Science,

University of South Wales, Pontypridd CF37 1DL, UK

J. M. Christian∗ and G. S. McDonald

Joule Physics Laboratory, School of Computing, Science and Engineering,

University of Salford, Greater Manchester M5 4WT, UK

∗ j.christian@salford.ac.uk

We report spontaneous spatial optical fractal patterns in a ring cavity containing a thin slice of diffusive Kerr-type material. The Turing threshold instability condition is derived through linear analysis, and static patterns are found to be described by spectra with multiple-minimum characteristics. These theoretical predictions are subsequently verified through numerical simulations with both one and two transverse dimensions. Our findings support that a proposed fractal-generating criterion for nonlinear wave-based systems with thin-slice host media can have independence with respect to both system geometry and nonlinearity.

Keywords: Spatial fractals; ring cavity; Turing instability.

1. Introduction

The classic patterns of Cantor, Sierpinski, Julia, von Koch, Lévy, and others, are well-known examples of fractals.1 A defining characteristic of such patterns is that more intricate substructure (representative of the pattern as a whole) is uncovered the closer one looks. For such exact fractals, the same proportion of extra detail is revealed upon magnification down to arbitrarily-fine scalelengths. Exact fractals thus possess an infinite number of scalelengths, or equivalently, no natural scalelength.

Fractal curves are often quantified by a dimension D, where 1 < D ≤ 2. The value D = 1 can be regarded as the topological dimension of a conventional Euclidean line; D = 2 then describes an area-filling pattern, or a line with an infinitely dense level of detail (these geometrical notions can be extended to the case of two Euclidean dimensions, where fractals may have 2 < D ≤ 3, with D = 3describing a volume-filling pattern). While D encapsulates a curve’s complexity (or degree of meandering), it has no universal definition. The various measures of dimension tend to be interrelated, but often yield different numerical values for the same object. Furthermore, not all definitions of D are meaningful in all circumstances.2,3

Scaling laws and fractality play fundamental roles in our general understanding of dynamical systems4–7 and critical phenomena.8 Since Mandelbrot’s seminal essays on their universality,1 fractals have continued to be discovered in many branches of physics, chemistry, engineering, biology, geology, ecology, meteorology, and economics. Such real-world patterns tend to involve a small-scale cut-off. In optics, for example, one might expect the diffraction limit to prevent light structure developing on sub-wavelength scales. Another property of Nature’s fractal patterns is that smaller-scale details tend to have statistical (rather than exact) similarity to larger-scale features. Such patterns are known as statistical fractals.

In this paper, the first prediction is made of spontaneous spatial fractal patterns in a ring cavity containing a thin slice of dispersive (Kerr-type) material. In Sec. 2, we review some key studies that have identified self-similarity and fractals in optical systems, highlighting some commonality and distinctiveness of the various contributions. Ring-cavity geometry beyond the mean-field approximation is discussed in Sec. 3, and linear analysis of the thin-slice system reveals multiple-minimum thresholds for Turing bifurcations. Full numerical simulations (with both one and two transverse dimensions) then test theoretical predictions in Sec. 4, and the fractal properties of emergent spontaneous light patterns are characterized in terms of system parameters. Conclusions are drawn in Sec. 5.

2. Fractals in Optics

We begin with a short review, by no means exhaustive, of some important contributions to the diverse optical fractals literature from the last three decades. Broadly speaking, there tend to be four main types of fractal pattern.

One of the earliest reports of optical fractals in a linear-waves context was Berry’s “diffractals” (plane-waves scattered by fractal objects).9 Although it is intuitive that light diffracted by complex gratings might acquire fractal character,10–12 Berry also showed that even the simplest diffraction grating imaginable —a Ronchi (i.e., square wave) grating—can produce fractal light patterns. In this latter case, repeated self-imaging of the grating itself (the Talbot effect) can generate fractals.13,14

Self-imaging is, however, a property of many linear optical systems. The transverse modes of unstable-cavity lasers can have fractal character.15–23 There, the cavity eigenmode (i.e., the self-reproducing profile) condition involves small-scale diffraction effects and successive round-trip magnifications. Mode fractality was discovered in the so-called “kaleidoscope lasers”, for which the transverse symmetry of the aperturing element (e.g., a polygonal coupling mirror) was shown to have a profound effect on the complexity and character of the laser patterns.19–23 Alternative schemes for optical self-imaging, such as multiple-reduction copiers24,25 and pixellated video feedback setups,26–31 have provided further (potentially linear) contexts for spatial fractal formation.

A range of self-similar and fractal patterns in soliton-supporting systems has been identified during the last two decades. These patterns are nonlinear in origin as their existence is directly related to self-action effects within the light-material coupling. In the extreme-transient limit, narrowband32 and broadband33,34 stimulated- Raman-scattering geometries can sustain self-similar soliton pulse trains. In optical fibres, self-similarity has also been predicted during the amplification of parabolic pulses.35,36 Moreover, Cantor-type distributions of soliton profiles were predicted in systems where series of abrupt material discontinuities induce splitting phenomena.37,38 Each splitting can introduce a new scale, so that many scales may arise from a long sequence of distinct material changes.

A third category of fractal pattern can appear in the parameter characterization and bifurcation structure of nonlinear optical phenomena (while their real space and time representations remain nonfractal). Particular examples of systems where self-similarity appears in phase space include chaotic pixel–pixel mappings in analyses of optical memory devices,39 and the properties of interacting vector solitons.40

Finally, reaction–diffusion systems may become spontaneously patterned due to intrinsic nonlinear dynamical processes. Turing demonstrated in 1952 that simple spatial structures may emerge spontaneously from near-uniform states41; well-known examples include hexagons, honeycombs, squares, stripes, spirals, and vortices. Self-patterning was proposed as arising from arbitrarily-small background fluctuations to states that are otherwise homogeneous in space and stationary in time. The mathematical origin of simple-pattern emergence was described in terms of the existence of a single instability minimum whose position (in Fourier space) dictates the dominant scalelength of patterning. Universality was anticipated as many nonlinear systems, regardless of their physical nature (optical, fluidic, chemical, etc.), possess a single-minimum instability spectrum.

More recently, it was proposed that any system whose instability characteristic is composed of multiple Turing-type minima with comparable depth may be susceptible to multi-scale self-patterning, and truly spontaneous fractal formation.42 The first predictions of such nonlinear spatial structures were made for a simple dispersive model: the classic Kerr slice with a single feedback mirror43,44 (SFM). The multiple-minimum signature was expected to have independence with respect to both the geometry of the physical system and the nature of the nonlinearity present. Having previously identified fractal patterns in a ring cavity containing a purely-absorptive thin slice,45 here we further test the notion of universality by considering the purely dispersive case.

3. Nonlinear Ring Cavity

We consider a diffusive Kerr-type material in the form of a thin slice with near-negligible width L << d, as introduced by Firth.43 While a detailed (nonparaxial) treatment of small-scale optical structure is unnecessary in the slice configuration, one expects such complications to arise for finite light-medium interaction lengths. In such cases, one must typically reformulate the governing equations to accommodate higher-order contributions from the nonlinear polarization.

3.1. Boundary condition & spatial filtering

Optical ring cavities have been studied extensively in terms of spontaneous pattern formation. Many models involve mean-field approximations,46 where light propagation effects are averaged away and the spatiotemporal complexity is subsequently reduced.47–58 Most mean-field models to date have Turing threshold spectra that possess a single instability minimum. Spontaneous symmetry-breaking is thus predicted to drive the emergence of simple (or “single-K”) patterns, i.e., those associated with a single dominant scalelength. On the other hand, the relaxation of mean-field constraints permits the spontaneous development of multiple transverse spatial frequencies in the circulating field.59–68

Here, we consider a ring cavity consisting of a four-mirror loop (see Fig. 1). The coupling mirror has intensity reflectivity and transmissivity coefficients r12 and t12 ≡ 1−r12, respectively, while and other mirrors are assumed lossless. During each transit, light traverses a free-space path d in round-trip time tR. The longitudinal coordinate z measures the distance along the cavity axis, with z = 0 (L) located at the input (output) face of the slice. Denoting the Fourier transform of the electric field envelope E with a tilde, and where K is the spatial frequency, the general ring-cavity boundary condition is

(3.1)

where Ein is the amplitude of the plane-wave pump field, δ(K) is the Dirac delta function, and 0 is a linear (interferomic) cavity mistuning. The phase factor θHelm(K) = Θ(K, k0), obtained from

(3.2)

allows for Helmholtz propagation in the free-space path69 and k0 = 2π/λ, where λ is the optical wavelength. Diffraction becomes paraxial in the limit that |K|/k0 << O(1), where θHelm(K) ~ K2d/2k0 ≡ θ(K). The function F(K,KC) in Eq. (3.1) is defined so that its effect, in combination with diffraction, is to provide a low-pass

spatial frequency filter with a (continuously variable) cut-off wavenumber KC:

(3.3)

where 0 < K ≤ KC. Spectral components with K >KC are thus attenuated on each round trip. In practical terms, F could represent a physical element placed inside the cavity to allow the control of pattern formation. Alternatively, it may mimic spatial filtering that is inherent to a cavity design. The filter can be removed by setting KC = k0, in which case all transverse wavenumbers with K ≤ k0 may propagate freely around the cavity. In the paraxial limit, and with no spatial filtering, Eq. (3.1) simplifies somewhat to

(3.4)

This form of boundary condition, which lumps together the free-space diffraction and round-trip time delay, is appropriate for thin-slice cavity configurations.66,70

3.2. Model equations

In normalized units,43,44 the envelope E and the medium photoexcitation density n are coupled through a system of first-order equations

(3.5a)

(3.5b)

where ∇⊥2 is the transverse Laplacian, τ and lD are the relaxation time and diffusion length of n, respectively, and χ parameterizes the nonlinearity (being positive in self-focusing and negative in self-defocusing media, respectively).

By recognizing that n must be independent of z (because the medium is thin and hence diffusion cannot occur in the longitudinal direction), one may immediately write down a formal solution to Eq. (3.5a) that connects transverse spatial profiles of the electric fields at z = 0 and z = L according to E(x, L, t) = E(x, 0, t) exp[iχLn(x, t)]. The field is assumed to travel the distance L effectively instantaneously so that diffraction in the medium may be ignored. Material effects thus result solely in a phase modulation of the circulating light. Combining the formal solution for E with Eq. (3.4), the boundary condition (in real space) becomes

(3.5c)

The uniform states of cavity, characterized by constants E0 and n0, are identified by solving the plane-wave problem. Diffraction and diffusion are neglected by discarding transverse effects, and we seek time-independent solutions that are denoted by E(x, z, t) → E(0)(z) = E0 exp(iβz) and n(x, t) → n0.

Substitution of the uniform-state E and n into Eqs. (3.5a) and (3.5b) yields the propagation constant β = χn0 and the photoexcitation density n0 = |E0|2. Boundary condition (3.5c) subsequently determines the complex amplitude E0 = t1Ein/[1−r1 exp(iΦ0)], where Φ0 ≡ φ0 +χL|E0|2 includes both linear and nonlinear mistunings. The connection between pump intensity |Ein|2 and intracavity intensity |E0|2 is then given by the well-known result

(3.6)

demonstrating that the multistable character of the dispersive ring resonator operating in the plane-wave regime is independent of the thin-slice approximation. Lachinova and Lu70 have analyzed a similar system to illustrate simple pattern competition.

3.3. Linear stability analysis

Linear analysis can be used to establish the threshold condition for predicting spontaneous pattern emergence by examining the susceptibility of the uniform states to small background perturbations.43,44 We set

(3.7a)

(3.7b)

and where the complex function f describes a deviation from E(0)(z) while δn is a (real) fluctuation around the stationary value n0. The perturbations are assumed to be small, satisfying inequalities |f| << O(1) and |δn|/n0 << O(1).

By substituting the decompositions for E and n into Eq. (3.5a) and linearizing in small quantities around the stationary state, it follows that f must satisfy

(3.8a)

Direct integration then gives a simple form of solution, namely f(x, z, t) = f(x, 0, t) + iχδn(x, t)z. From Eq. (3.5b), it also follows that δn is governed by a linear equation that is parametrized by the intracavity plane-wave intensity |E0|2:

(3.8b)

Note that the right-hand side of Eq. (3.8b) always remains real, as should be the case. Finally, substituting for E and n from Eqs. (3.7a) and (3.7b) into boundary condition (3.5c) and cancelling the plane-wave solution leaves a delay relation capturing the effect of ring-cavity feedback on the perturbations:

(3.8c)

It now proves convenient to consider a single spatial frequency K which is associated with a pattern scalelength 2π / K. Transforming to the Fourier domain, the linearized delay system of Eqs. (3.8b) and (3.8c) is given by

(3.9a)

and

(3.9b)

which may be solved analytically.

We seek Fourier mode solutions to Eqs. (3.9a) and (3.9b) whose time dependence is of the exponential form

(3.10a)

(3.10b)

where , μ, and η are complex constants while λ ≡ Λ − iΩ is a complex Lyapunov exponent with growth rate Λ and Hopf frequency Ω. Substitution into Eq. (3.9b) provides the relationships = Dη/[exp(λtR) − C] and μ = D∗η/[exp(λtR) − C∗], where C ≡ r1 exp[i(Φ0 − θ)] and D ≡ iχLC. Substituting into Eq. (3.9a) then yields

(3.11)

Our interest lies with Turing (i.e., static) patterns, which are nonoscillatory in time and thus one may set Ω = 0. Henceforth, we look for the threshold intracavity intensity |E0|2 → Ith at which point Λ = 0 (corresponding to the boundary between temporal growth and decay of the perturbation). Equation (3.11) leads to the threshold instability spectrum for such Turing patterns:

(3.12)

where Φth ≡ Φ0(|E0|2 → Ith) represents the combination of linear and nonlinear mistunings at threshold.

Condition (3.12) divides the (K, χLIth) plane into a number of discrete, coexisting bands (see Fig. 2). The width and separation of these bands tends to decrease smoothly with increasing K, while the minima increase within an envelope ∼ lD2K2. Within certain parameter regimes, such as for a sufficiently small diffusion length, one can find many comparable instability minima. The Turing threshold spectra of the diffusive-Kerr ring cavity are thus qualitatively similar to those of the SFM system.42–44 This key result provides the analytical basis for the expectation that thin-slice dispersive cavities are potentially capable of generating spontaneous spatial fractal patterns.

4. Spontaneous Spatial Fractals

The computational models employed throughout this research include the filter function F(K,KC) [c.f., Eq. (3.3)]. To illustrate conventional pattern formation (i.e., the emergence of a single dominant scalelength), KC is selected so that only those spatial frequencies within the first (i.e., low-K) instability band propagate

freely. In the following simulations, we have chosen normalized units such that d/2k0 = 1 and χL = 1, with results presented for the parameter set r12 = 0.9 and 0 = π/4.

4.1. Simple & fractal pattern formation

For an instantaneous (lD = 0 = τ) Kerr nonlinearity, the static Turing instability threshold bands share a global minimum [see Fig. 2(b)]. When the intracavity intensity exceeds that threshold, spatial frequencies defined by these minima will all have the same growth rate. One then expects that the resultant intensity distribution in the one- (two-) dimensional transverse plane will evolve into an extremely complicated area- (volume-) filling pattern with fractal dimension 2 (3).

For a given plane-wave pump field, n is initialized at the corresponding stationary-state value n0, plus a 0.1% level of noise (the addition of which merely accelerates the pattern formation process). After a time of about 400tR, the transverse intensity distribution has transformed into a finite-amplitude static stripe pattern. The filter is then removed and the subsequent evolution is monitored. Results are first presented for a single transverse dimension (these patterns have topological dimension 1). To understand the transition to a fractal, it is instructive to compare the real-space intensity pattern I(x) with the power spectrum P(K) [defined here to be the modulus-squared of the Fourier transform of I(x)]. The power spectrum of the stripe pattern comprises a dominant K, plus (generally weaker) harmonics [see Fig. 3(a)].

When the filter is removed, I(x) develops an increasing amount of detail [see Figs. 3(b) and 3(c)]. In K-space, sets of spatial-frequency harmonics (associated with each instability minimum) develop very rapidly due to intrinsic nonlinear dynamics. This nonlinear cascade leads to sustained growth at the high-K edge of P(K) whereby, after a sufficient time, the majority of frequency components with K ≤ k0 attain roughly the same intensity [see Fig. 3(d)]. This property of P(K) verifies that comparable levels of detail persist in I(x) down to a smallest scalelength ∼2π/k0. The resulting pattern is thus a statistical fractal that has an area-filling character (i.e., dimension 2).

Figure 4 shows the real-space pattern evolution when two transverse dimensions are considered. Once the filter is removed, the static stripe pattern I(x, y) (which has topological dimension 2) develops an increasing amount of detail as it tends toward a pattern with volume-filling character (i.e., fractal dimension 3).

4.2. Power-spectrum dimension

We now consider the dynamical evolution and final pattern states when finite diffusion is present. For 0 < lD << O(1), one expects to find a broad spectralplatform consisting of a space-filling fractal that spans a range of large scales (i.e., at low-K).42 Such a scenario would provide continuity with the lD = 0 results [c.f., Fig. 3(d)] in spectral regions where diffusion is negligible compared with the scalelengths under examination. We can confirm that this is indeed the case. It is thus instructive, instead, to focus on moderate-to-high levels of diffusion, and to examine precisely how lD modifies fractal characteristics at scalelengths where it has greatest effect. In so doing, we have been able to quantify (semi-analytically) the diffusion-induced quenching of fractality, the corresponding fractal-dimension variation, and the extent of space-filling fractality as a function of lD.

Figure 5 shows the dynamical evolution of the power spectrum when lD = 0.4 and in the absence of any strong spatial filtering. For highest accuracy, a single transverse dimension is considered. The rate of bandwidth growth depends upon system parameters such as |Ein|2 and r1. The growth is generally fast: even when the system has a finite response time τ, a dynamic equilibrium state is reached after ∼100 cavity transits. After that point, I(x) continues to change in time but the average distribution of power in P(K) remains invariant. For example, Figs. 5(c) and 5(d) are very similar but they are separated in time by 2,400tR.

In common with the diffusion-limited dispersive system of Ref. 42, we find that the average power spectrum of the dynamic equilibrium state has a significant portion that follows a remarkably simple linear relationship

(4.1)

where a and b < 0 are constants. The negative slope means that high spatial frequencies contribute less to the fractal pattern than low frequencies (in agreement with physical intuition, since the effect of diffusion is stronger at smaller spatial scales).

The effect on the dynamic equilibrium power spectrum of increasing lD is shown in Fig. 6. The abrupt change in slope that appears in Figs. 6(a) and 6(b) is a manifestation of the computational noise floor, where the data becomes susceptible to finite-precision effects. Detailed series of simulations have shown that the slope parameter b is sensitive to the level of diffusion present. In particular, we have uncovered a simple proportionality between b and lD:

(4.2)

where b0 < 0 is a constant. Figure 7(a) shows excellent correlation between this proposed proportionality and results derived from individual simulations.

For patterns with topological dimension 1, the power-spectrum fractal dimension DPS(K) is defined as71,72

(4.3)

Using P(K), as given by Eq. (4.1), it is possible to quantify fully the three scale regimes of the equilibrium patterns that emerge. These regimes are: (i) a uniform space-filling regime (pure fractal), (ii) a transitional band (scale-dependent fractal), and (iii) nonfractal (at the very smallest scales). In common with the SFM

configuration, one finds that

(4.4)

In the absence of diffusion (where lD → 0), Eq. (4.2) gives b → 0 and, hence, DPS → 2 (i.e., an area-filling pattern) across the entire spatial frequency range. For 0 < lD << O(1), a significant plateau of scales at low-K retain DPS = 2 [see Fig. 7(b)]. For moderate and strong diffusion, one finds that the curve I(x) is predominately a scale-dependent fractal whose dimension changes linearly from 2 to 1 within a mid-frequency band of width ∝ 1/lD. It is not surprising that diffusion can quench uniform space-filling structure, but it is perhaps surprising that scale-dependent fractal characteristics are so uniform, and potentially broad, in K.

5. Conclusions

We have predicted, to the best of our knowledge, the first truly spontaneous formation of spatial optical fractals in dispersive ring cavities. Both initial pattern formation and the subsequent development of many pattern scales arise spontaneously in these homogeneous systems. Predictions were made for a purely-dispersive model (complementing earlier analyses of purely-absorptive systems45), with linear analysis uncovering multi-Turing instability bands with comparable minima — a signature proposed, in Ref. 42, as indicative of potential fractal pattern formation. We have thus confirmed that this signature may indeed be universal for nonlinear wave-based systems in thin-slice regimes (more detailed studies are obviously required, extending similar considerations to bulk-medium geometries).

The multi-Turing criterion appears to be robust in the sense that it is independent of the details of the configuration (e.g., bi-directional light interactions due to a SFM versus uni-directional ring cavity geometries), and that it is independent of the particular form of the nonlinearity driving pattern formation (e.g., dispersive versus absorptive). Consequently, there may be many other (optical and nonoptical) contexts for application of the multi-Turing signature within the wider arena of nonlinear science.73,74

Results from an extensive series of numerical simulations have been summarized, with further analysis permitting characterization of optical fractal patterns in terms of system parameters. Surprisingly common characteristics have emerged from the study of distinct systems. For the dispersive cavity, and in common with the SFM,42 we were able to obtain semi-analytical descriptions of diffusion-induced quenching of fractality, the corresponding fractal dimension variation, and the extent of pure fractality — all as a function of the medium diffusion length. In discussion of results for the absorptive cavity, we noted that (in each system studied to date, and for each type of small-scale limiting effect considered), the K-space boundary layer between fractality and nonfractality depends on a subtle balance of linear and nonlinear processes. This is suggestive of potential solitonic character in which, instead of a fractal composed of many soliton profiles,37,38 one might have a single boundary soliton composed of a scale-dependent fractal.

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FIGURE 1

Huang, Christian, and McDonald

Fig. 1. Geometry of a four-mirror ring cavity of length d containing a thin slice of nonlinear material with near-negligible thickness L << d. The coupling mirror has intensity reflectivity r12, and a spatial filter controls the maximum transverse wavenumber KC that can circulate freely within the system.

FIGURE 2

Huang, Christian, and McDonald

Fig. 2. Multi-Turing threshold instability spectra for the dispersive nonlinear ring cavity with (a) finite diffusion (lD = 0.4) and (b) no diffusion (lD = 0). Other parameters are: d/2k0 = 1.0, χL = 1.0, r12 = 0.9, and 0 = π/4.

FIGURE 3

Huang, Christian, and McDonald

Fig. 3. Intensity patterns I(x) in real space (top row) and the corresponding power spectra P(K) (bottom row) with no diffusion (lD = 0 = τ) and t1Ein = 0.40729. (a) Static Turing stripe pattern, emerging from the homogeneous steady-state solution plus 0.1% white noise, after t = 400tR (KC = 1.2). The transition to a fractal, after the spatial filter is removed (through setting KC = k0 = 172), is shown in frames (b) t = 420tR, (c) t = 496tR, and (d) t = 590tR.

FIGURE 4

Huang, Christian, and McDonald

Fig. 4. Spontaneous intensity patterns I(x, y) in the two-dimensional transverse plane of the purely dispersive system with no diffusion (lD = 0 = τ). (a) Static Turing stripe pattern emerging from the homogeneous steady-state solution plus 0.1% white noise (KC = 0.8 and t = 400tR). The transition towards a fractal pattern (after the spatial filter is removed by setting KC = k0 = 45 at t = 400tR) is shown in frames (b) t = 454tR, (c) t = 487tR, and (d) t = 496tR. Other parameters are the same as in Fig. 2(b).

FIGURE 5

Huang, Christian, and McDonald

Fig. 5. Dynamic evolution of the (natural logarithm of the) power spectrum P(K) for a dispersive cavity, with a single transverse dimension and a moderate level of diffusion (lD = 0.4, τ = 1.0, and t1Ein = 2.0). Spectra shown at times (a) t = tR, (b) t = 25tR, (c) t = 100tR, and (d) t = 2500tR.

FIGURE 6

Huang, Christian, and McDonald

Fig. 6. Variation of the equilibrium power spectrum P(K) with diffusion length lD for (a) lD = 0.4, (b) lD = 1.0, (c) lD = 2.0, and (d) lD = 5.0. Results are for t1Ein = 2.0, τ = 1.0, and a time 1500tR (after system initialization with the steady-state plane-wave solution plus 0.1% white noise).

FIGURE 7

Huang, Christian, and McDonald

Fig. 7. (a) Variation of the slope b of the average power spectrum with diffusion length lD for t1Ein = 2.0. (b) Variation of the power-spectrum fractal dimension versus K, as given by Eq. (4.4). For systems with relatively-weak medium diffusion effects, there is a broad plateau at low K where DPS = 2. This is the regime of a space-filling optical fractal. Three levels of diffusion are shown: lD = 0.050, lD = 0.025, and lD = 0.005.

Page 1 of 21

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