February 1, 1997 / Vol. 22, No. 3 / OPTICS LETTERS 145

Polyadic Cantor superlattices with variable lacunarity

Dwight L. Jaggard

Complex Media Laboratory, Moore School of Electrical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6390

Aaron D. Jaggard

Department of Mathematics, Wheaton College, Wheaton, Illinois 60187

Received October 23, 1996

Ref lection and transmission properties of polyadic fractal superlattices are formulated, solved analytically, andcharacterized for variations in stage of growth, fractal dimension, and lacunarity. This is the first time toour knowledge that the effect of lacunarity on wave interactions with such structures has been considered.The results are summarized by families of ref lection data that we denote twist plots. A new doubly recursivecomputational technique efficiently provides the ref lection and transmission coefficients for a large class ofCantor superlattices with numerous interfaces. 1997 Optical Society of America

Optical and electromagnetic wave interactions withfractal structures have been studied for severaldecades.1 – 3 We show here that polyadic (multigap)Cantor superlattices imprint scattered waves withinformation concerning their structure. This infor-mation depends on the fractal dimension, stage ofgrowth, and lacunarity of the superlattices. Waveinteractions with polyadic Cantor structures were pre-viously investigated for fractal aperture diffraction.4

Although research concerning wave interactionswith fractal superlattices started in 1990 and hascontinued to the present,5,6 this is apparently the f irsttime the effect of lacunarity, texture, or ‘‘gappiness’’,has been considered. We demonstrate here that thelacunarity of these fractal superlattices gives rise tocharacteristic patterns in the scattering data. Thisis the first major result of this research. We alsodevelop a doubly recursive method of computation thatexploits the self-similar geometry of these structures,providing an efficient method for solving the scatteringproblem for this geometry. This is the second majorresult of the research reported here. This researchhas applications to the characterization and design ofsuperlattices, the synthesis of coupled Fabry–Perotcavities, the remote description of finely divided strati-fied media, and the design of new classes of optical andmicrowave filters. In each case, one seeks the relationbetween the geometric properties of the structure andthe scattered waves.

In this problem an optical or electromagnetic wave isnormally incident upon a fractal superlattice composedof alternating layers of refractive indices n0 and n1.This variation in index is governed by a family ofgeneralized polyadic Cantor sets on the interval [0,L] at finite stages of growth. For the case governedby the set E, the refractive index at the coordinatez is n1 for z [ E and n0 otherwise. Generalizationsto oblique incidence are a straightforward extension ofthis research.

We construct these sets iteratively starting with [0,L] at each stage of growth S replacing the interval withN nonoverlapping copies of the interval at the previous

0146-9592/97/030145-03$10.00/0

stage, each scaled by r , 1. The construction of thisfamily of generalized Cantor sets is displayed in Fig. 1for three members of the family.

Fig. 1. Generation of a polyadic Cantor bar and its f irstthree stages of growth for three values of lacunarity. Thefractal dimension of these Cantor bars in the limiting caseis D ­ logs4dylogs7d.

1997 Optical Society of America

146 OPTICS LETTERS / Vol. 22, No. 3 / February 1, 1997

For the stylized fractal structures consideredhere, we use the similarity fractal dimension, D ­lnsN dylns1yrd of the limiting set sS ! `d.7 Fractalstructures can have identical fractal dimensions, evenwith disparate values of N . Although the values ofN s­4d and r s­1y7d, and hence of D, are the samefor each of the structures in Fig. 1, the three sets aredifferent. Thus the fractal dimension is only one ofseveral possible fractal descriptors.

Lacunarity is one measure of the texture of fractals,the distribution of gap sizes within a fractal set, or thetranslational invariance of a fractal set.7,8 Fractalsthat contain large gaps are usually more lacunar thanthose that contain only small gaps. For the polyadicCantor bars that we examine here, the values of Nand r determine the total gap length in the fractal aswell as the number of gaps sngaps ­ N 2 1d. Here wecharacterize the lacunarity of the set by e, which wedefine as the ratio of the length of the outermost gapto the total length L. Low lacunarity bars (Fig. 1, top)are constructed with all gaps the same size, whereashighly lacunar bars (Fig. 1, bottom) are the result ofallowing one gap to become very large while the othersbecome vanishingly small.

There is ongoing research to quantify lacunarity fora number of different fractal structures.8 This hasyielded several candidate expressions but no singledefinition for arbitrary geometry. For our purposes,the use of e will suffice.

Well-known matrix techniques do not recognize theinherent order of fractal superlattices. Here we ex-ploit this order to develop an efficient method of doublerecursion to generate ref lection (transmission) coeff i-cients for polyadic Cantor superlattices. In the f irstrecursion, generating functions are used to create gen-eral ref lection (transmission) functions through recur-sion on the number gaps. Noting that at stage S thebar contains N scaled copies of the interval at stageS 2 1, we use these functions to f ind these coeff i-cients through a recursion on the stage of growth.This forms the second recursion.

We define generating taken from well-known opticalref lection (transmission) coeff icients for a single refrac-tive slab, functions genr s gentd for ref lection (transmis-sion), as

genrsngapsd

­x 1 s y2 2 x2dgenrsngaps 2 1dexps2in0kaiLd

1 2 x genrsngaps 2 1dexps2in0kaiLd, (1)

gentsngapsd

­y gentsngaps 2 1dexpsin0kaiLd

1 2 x genrsngaps 2 1dexps2in0kaiLd, (2)

subject to the initial conditions genrs0d ­ x andgents0d ­ y, k being the free-space wave number of theilluminating wave. Here the ai si ­ 1, 2, . . . , ngapsd de-note gap sizes normalized to the total length L. Theseare used in the first recursion to generate the ref lectionand transmission functions grsx, y, Ld ­ genrsngapsdand gtsx, y, Ld ­ gentsngapsd for an arbitrary number ofgaps in the Cantor bar. The ref lection and transmis-

sion coefficients for a Cantor bar of stage S and a totallength L are computed in the second recursion with

RsS, Ld ­ grsRsS 2 1, rLd, T sS 2 1, rLd, Ld , (3)

T sS, Ld ­ gtfRsS 2 1, rLd, T sS 2 1, rLd, Lg . (4)

The initial conditions for Eqs. (3) and (4) are givenby the ref lection and transmission coefficients for asingle slab of arbitrary length d, Rs0, d d ­ fr01 1

st10t01 2 r01r10dr10 exps2in1kd dg f1 2 r10 r10 3exps2in1kd dg21 and T s0, d d ­ ft01 t10 expsin1kd dg f1 2

r10 r10 exps2in1kd dg21, where rij f­ sni 2 nj dysni 1

nj dg and tij f­ 2niysni 1 nj dg are the Fresnel ref lectionand transmission coeff icients, respectively, for aninterface between media of refractive indices ni and nj .Equations (1)–(4) are valid for any stage of growth,refractive index, number of gaps, fractal dimension,and spacing of the scaled copies within [0, L] forbilaterally symmetric polyadic Cantor superlattices.

In Fig. 2, we examine the magnitude of the ref lection(jRj) as a function of normalized frequency kL for thefirst stage of growth sS ­ 1d and D ­ lns4dylns7d ø0.71241 with varying lacunarity. Here low, medium,

Fig. 2. Ref lection coeff icient amplitude jRj as a functionof normalized frequency kL for three fractal superlatticeswith S ­ 1, D ­ lns4dylns7d, and variable lacunarity e ­1y7, 1y14, 0 (top to bottom). The insets indicate thedistribution of refractive index, with black representing n1and white n0.

ebruary 1, 1997 / Vol. 22, No. 3 / OPTICS LETTERS 147

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Fig. 3. Amplitude of the ref lection coefficient jRj shownas a density plot (black is zero) as a function of thenormalized frequency kL and continuous variations e(vertical axis) for the first (top plot) and second (bottomplot) stages of growth. These twist plots clearly show thefeatures that are independent of lacunarity (vertical nulls)and those that are descriptors of lacunarity (diagonal andcurved nulls). The second-stage growth plot (bottom) issimilar to a scaled version of the first with an overlaidfine structure. This indicates the fractal nature of thesesuperlattices.

and high lacunarity are given by e ­ 1y7, 1y14, 0,respectively, and correspond to the bars displayed inFig. 1.

A clearer picture of the effect of lacunarity is givenin the twist plots of Fig. 3, in which the ref lectioncoeff icient amplitudes are displayed as a functionof normalized frequency kL and continuous varia-tions of e for two stages of growth. The nulls inFig. 3 are especially informative. There are threesets of these characteristic nulls in the top twistplot. The f irst group of nulls is the set of verti-cal dark lines at kL ­ 14pmy3 sm ­ 1, 2, 3, . . .d.We expect the nulls to be independent oflacunarity, as they are caused by destructiveinterference from the front and back surface re-

f lections of individual layers of index n1. The secondset of nulls are broad curves that are almost vertical forsmall normalized wave number but appear as arcs forlarger normalized wave numbers. These nulls are dueto the changed spacing between regions of index n1 inthe superlattice. Therefore these nulls are sensitiveto variations in both frequency and lacunarity. Thefinal set of nulls, which provide barber pole striations,are due to the gap width at the center of each super-lattice. These last two sets of nulls are indicatorsof the degree of lacunarity and demonstrate that, inaddition to information on stage of growth and fractaldimension,5 lacunarity is also distinctively imprintedon scattering data. The nulls in the bottom plot ofFig. 3 are explained in a similar way. As expected,there are additional f ine striations here that are due tothe finer structure of the second stage superlattice.

We have examined the interaction of waves with aclass of fractal superlattices characterized by fractaldimension, stage of growth, and lacunarity. For thefirst time to our knowledge, we find the effect of la-cunarity on scattering data. Twist plots of ref lectiondata can be used to extract lacunarity and other infor-mation. In addition, we have developed a novel doublyrecursive method of computation that greatly reducesthe computational burden compared with that of tradi-tional matrix methods.

We acknowledge the insightful comments of TerryPerciante (Wheaton College, Wheaton, Ill.) andseveral reviewers. Portions of this research werereported at the XXV General Assembly of the Interna-tional Union of Radio Science, August 28–September5, 1996 (Lille, France). This work is partially sup-ported through a NATO Collaborative Research GrantNo. 920 702 and the Complex Media Laboratory(University of Pennsylvania).

References

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Pietronero and E. Tosatti, eds. (Elsevier, New York,1986).

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4. See, e.g., T. Spielman and D. L. Jaggard, presentedat the 1992 IEEE Antennas and Propagation SocietyInternational Symposium and Union of Radio ScienceInternational Meeting, Chicago, Ill., July 20–25, 1992.

5. D. L. Jaggard and X. Sun, Opt. Lett. 15, 1428 (1990); X.Sun and D. L. Jaggard, J. Appl. Phys. 70, 2500 (1991).

6. V. V. Konotop, O. I. Yordanov, and I. V. Yurkevich,Europhys. Lett. 12, 481 (1990).

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8. See, e.g., B. B. Mandelbrot, in Fractals in Biology andMedicine, T. F. Nonnemacher, G. A. Losa, and E. R.Weibel, eds. (Birkhauser Verlag, Basel, Germany, 1994),pp. 8–21; see also B. B. Mandelbrot and D. Stauffer,J. Phys. A Math. Gen. 27, (1994).