17 March 1997

ELSEWIER

PHYSlCS LETTERS A

Physics Letters A 227 (1997) 197-202

Chaotic light scattering from a system of osculating, conducting spheres Dat Ngo a, Gorden Videen b,c, Robert Dalling ’

a Edgewood Research and Development Engineering Center, E595I. Aberdeen Proving Grdund. MD 21010, USA h Army Research kzboratory, White Sands Missile Range, White Sands, NM 88002, USA

c Department of Physics, New Mexico State University, Las Cruces, NM 88003, USA

Received 3 January 1996; accepted for publication 31 December 1996 Communicated by C.R. Doering

Abstract

Feedback in the equations describing the scattering coefficients cause chaotic behavior in the scattering intensities from a system composed of a pair of osculating spheres. Unlike other systems which display chaos as a trajectory in phase space, this system exhibits chaotic behavior in a partial summation. As the region of osculation increases, tell-tale signs of chaos emerge, and the Lyapunov exponent, a measure of the system’s mean rate of exponential errOr growth, is shown to be greater than zero in these chaotic regions. We show bifurcation diagrams which depict regions of transformation from stability to chaos. including periodic windows and super-cycles.

PAC.!? 05.45.+b; 03.65.Nk; 03.80.+r; 24.6O.K~; 42.25.F~

Chaotic scattering of electromagnetic radiation has

recently been reported [ 1,2]. Doron et al. [ 1 ] demon- strated that reflection of microwaves from a contain-

ment vessel can become chaotic. Likewise, Jensen [ 21

was able to show that internally reflected light rays inside a “stadium-shaped” dielectric cylinder produce

chaotic trajectories. Chaotic manifestations from these

and other systems have always been associated with system trajectories or system evolution through time. In this paper we examine chaotic effects that mani- fest themselves in the scattering solution of a simple system composed of two osculating spheres. What is different about this system is that chaos is manifest in a partial summation:

(1)

where f$A) and $$,) are scattering coefficients for the

system and a,,,,, and b,, are system-dependent func- tions of the spherical harmonics [ 31. Sy, is a physical quantity, the back-scatter total intensity of the scat-

tered light. The system of interest is composed of two per-

fectly conducting spheres which overlap, or osculate,

as shown in Fig. 1. Sphere 1 of radius al is centered on the RI, jj ,z^l coordinate system. Sphere 2 of radius a:! is centered on the 22, F2, 22 coordinate system at a position 2, = 0, 91 = 0, .?I = d. When d < al + a2, the spheres osculate. The incident radiation is a plane wave traveling in the 2-2 plane, oriented at angle (Y

with respect to the i axis. The scattered field can be derived by performing a

multipole expansion of the fields about each individ- ual sphere and treating the scattered field from each

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D. Ngo et al./Physics Lxtters A 227 (1997) 197-202 198

Fig. 1. Geometry of the scattering system

sphere as part of the incident field on the other sphere [ 3-71. The scattering solution for this system is given by Videen et al. [ 31. We merely give the relevant equa-

tions in which chaos manifests itself. The chaotic be- havior is due to feedback in the scattering coefficients,

fc3’ = B;;j + nm c f(3)CU) + f(4Q)U) n’m nnr n’m nm ’ (2)

n’=l

where Bh$ is a function of the incident field and C,$ , DE are system dependent parameters defined by Videen et al. [ 31. It is important to note that the scattering coefficients f$,) and fi$ on the left hand side of Eqs. (2) and (3) are not linearly dependent on the incident field modes Bhz and B,$, but are also dependent on the scattering interaction that oc- curs between the spheres. This feedback is described mathematically in the last two terms of Eqs. (2) and (3). These two sets of equations can be expressed in a matrix form and solved using any one of a number

of methods. The method of choice, for non-osculating spheres, is the order-of-scattering approach [ 4-71.

This approach treats the interaction of the two spheres as a perturbation of the solution. For this method to

work, the interaction terms (off-diagonal elements)

must be small compared to the diagonal elements. This method is inadequate for solving the osculating

sphere system, since large values in the off-diagonal terms within the matrix cause divergence. We use di-

rect methods of solution: LU-decomposition, Gauss- Jordan and singular-value decomposition [ 81. These

results agree with the order-of-scattering techniques for non-osculating spheres [ 4,5] and agree with each other when the spheres are osculating.

In order to solve Eqs. (2) and (3), the number of terms used in the series expansion is truncated to

some number N which is generally found to be ap- proximately proportional to some size parameter of

the scatterer. The size parameter (X = 2rru/A) is a measure of the sphere’s circumference divided by the

wavelength of the incident radiation. For Mie theory a

truncation criterion was developed by Wiscombe [ 91 and Bohren and Huffman [ lo] who found the number

of terms, N, in the expansion needed for convergence to be dependent on the size parameter of the sphere:

N = x + 4x’f3 + 2. (4)

For ease and convenience, we will denote xi, x2, and xd to be the size parameters of sphere 1, sphere 2,

and the separation parameter, respectively. Although the series in Eqs. (2) and (3) show an infinite sum-

mation, convergence of the scattering coefficients for a single (Mie) sphere occurs quite rapidly (N is ap- proximately equal to 10 for spheres comparable to the wavelength). For most scattering systems consisting

of two non-osculating spheres (xd > XI + x2), con- vergence also occurs quite rapidly and this criterion may also be used [ 4-71. Mackowski [ 71 however, found that for some systems composed of Rayleigh

elements, convergence could be much slower and also depends on the system orientation.

Fig. 2a shows the back-scatter total intensity SF1 as a function of separation parameter for spheres having size parameters equal to xi = 5.0 and x2 = 2.0. These values were chosen arbitrarily. Plotted are the back- scatter Sf: solutions for N = 15 to N = 69. There are three regions of interest in this SF, bifurcationdiagram. The first region occurs where the separation parame-

D. Ngo et al. /Physics Letters A 227 (1997) 197-202 199

44.1....,....,..! 7 65 6

Xd

06

JO4

7 65 6

%

Fig. 2. (a) Back-scatter intensity Sf: as a function of separation

parameter &j for a pair of overlapping metallic spheres. The overlap region begins when x,j = XI + x:! = 7.0 and is shown

by the solid vertical line. The system is illuminated at end-on

incidence (u = 0). (b) Plot of the Lyapunov exponent for the

same parameters as in (a).

ter xd is greater than approximately 7.1. In this region, Sy, has only one solution for the entire range of N.

The second region occurs where the separation param- eter is comparable to the combined size parameter of the two spheres, xd N 7.0. In this region there is slow

convergence of the solutions, and the system under- goes a smooth transition into the chaotic region. This smooth transition implies a continuity in the boundary surface and overall good system behavior. The third region occurs where the separation parameter Xd is less than approximately 7.0. In this region, there can

be a multitude of possible values for SF,. This third

region displays chaotic behavior [ 161 as indicated by the multiple solutions in Fig. 2a. We note that the “drip-drops” near Xd - 7.0 is a transient effect and is normally seen in chaotic systems. We examine this last region in more detail later in this paper.

One important criterion for determining whether or not a system is chaotic is that small errors grow ex-

ponentially [ 111. These exponentially growing errors

are not due to round-off or numerical techniques, but are due to the intrinsic nature of the system. Calcula- tion or round-off errors grow as the square root of the number of arithmetic operations ( fi) [ 81. The error

in a chaotic system grows exponentially. This fact can be used to calculate an effective Lyapunov exponent

[ 12-151. For our system, if the back-scatter is cal- culated simultaneously using two different precisions, for example, double precision ( 16 significant digits)

and quadruple precision (32 significant digits), result-

ing in SF, and @,, then the exponential error growth

can be observed as a linear loss of agreement between

Sp, and @, through the number of terms in the partial summation [ 14,151. This produces an effective Lya- punov exponent of the form

&Y = L/N, (5)

where L is the number of digits lost in comparing SF, and @, , and N is the number of terms used in finding L lost digits. The units for &n is decimal digits per term

(multiply by ln( 10) to convert to binary digits/term).

The linear loss of digits occurs as more terms are calculated in the sum.

Fig. 2b shows the values of the effective Lyapunov exponent as a function of the separation parameter for the same system parameters used in Fig. 2a. The

effective Lyapunov exponents are tabulated using Eq. (5). We should point out that calculating just one value of the effective Lyapunov exponent requires over 500 CPU minutes on a dedicated SUN workstation running at 250 Mega-flops. For this reason, we only show a few points on this graph. But even so, we can see that the

effective Lyapunov exponent is zero when Xd is equal to or greater than x1 +x2. As the value of X,-J decreases, the value of the effective Lyapunov exponent grows. The positive effective Lyapunov exponent indicates that errors within that region grow exponentially and that the system is chaotic.

200 D. Ngo et aLlPhysics L&ten A 227 (1997) 197-202

Table 1 Back-scatter intensity Sy, for the system parameters of Fig. 2 calculated in double and quadruple precision. I is the number of digits

which agree between the two precisions

xd N Double precision Quadruple precision I

7.15 25 15.07236106081215 15.07236106081213804265.. 15

7.15 30 15.07243985154617 15.07243985154611092705.. 15 7.15 49 15.07245878543825 15.07245878543821057480.. 15

6.70 25 12.08525844303845 12.08525844303845583707.. 16 6.70 30 15.09979068818836 15.099790688 18834536944.. . 15 6.70 49 1 I .97230391640318 11.9723039 16403 17279956.. 15

5.90 25 6.201693205848710 6.201693205848697041114.. 13 5.90 30 8.107918087552068 8.107918087553062922308.. 12 5.90 49 6.354969965845736 6.3549699746 199049927 13.. 8

5.25 25 3.986238740304115 3.986238739960162459644.. 8

5.25 30 3.345115470636528 3.345115459865957756178.. 8 5.25 49 4.035169941824903 3.678072 102800096770306.. , 0

A sample of the data used to calculate these effec- tive Lyapunov exponents is shown in Table 1. This

table shows the values of SF, tabulated using double and quadruple precisions for x1 = 5.0, x2 = 2.0 at

four different values of the separation parameter xd. Calculation of the effective Lyapunov exponent for

&, = 7.15 results in a value of zero, indicative of a convergent solution. By using Eq. (5), and comparing

double precision with quadruple precision over a wide

range of N values, it can be seen that the double pre- cision values agree with those of quadruple precision

out to the 13th or 14th decimal places for all values of N. Thus, there is no loss of information between double and quadruple precision solutions (L = 0). In the same manner, calculation of the effective Lya-

punov exponent when the spheres become slightly os- culating (&j = 6.70), yields a value greater than zero (j&~ = 0.0677 bits/term). In this case there is a slow loss of information between the double and quadru-

ple precision solutions (L = 1 over a large range of N values). Likewise, when the spheres become even more overlapped (xd = 5.90)) the effective Lyapunov exponent is found to be j& = 0.4797 bits/term. And finally, when the spheres are very much overlapped (xd = 5.25), the effective Lyapunov exponent is peg = 0.7326 bits/term. The general trend is that as the sep- aration parameter decreases, the larger the effective Lyapunov exponent. We note that if these results were due to bad numerical procedure then L would never

be equal to zero. In addition, these results are repro-

ducible using different numerical techniques. Within Fig. 2a, we also see periodic windows and

supercycles [ 17,181 enveloping the chaotic regions. Fig. 3a shows a magnification of the periodic win-

dows shown in Fig. 2a for the region 5.7 < Xd < 6.2. The biggest difference between this region and

the nonchaotic region (6.99 < xd < 7.04) is that Syl does not slowly converge for larger N. In fact, even

though the solutions for SF, are bounded, there is no systematic prediction of the values of SF1 at a particu-

lar xd. In other words, the value of Syl depends solely on the number of terms N in the summation. In our

numerical studies, we found that as N increases, the

scattering coefficients do not converge to a single Sy, value within the bounded vertical region. We find in- stead that there are windows where the coefficients

seem to split into M quasi-distinct solutions (similar to the three arms in the period 3 window of the logis-

tic map) [ 161, which then split into M’ solutions as _&J decreases, and so on. In these almost-perio regions, the order of the solutions is retained as N increases. This is illustrated in Table 2 which shows the values

of SF1 for xd = 5.98 and N = 25 to 66. At first glance it looks as though the system is displaying a period- icity of M = 6. To that end, Fig. 3b shows an even more expanded region from 5.96 < &J < 6.0. Upon closer examination, what appears to be a periodic win- dow turns out to be almost-periodic because there is

D. Ngo et al./Physics Letters A 227 (1997) 197-202

Table 2

Back-scatter intensity Sy, for the system parameters of Fig. 2 with xd = 5.98. In this table, N = 25 + 6i + j

201

j

i 0 I 2 3 4 5

0 12.01446 7.86127 6.46610 6.70105 7.69470 10.06243

I I 1.98547 7.73546 6.49787 6.75988 7.783 14 10.18034

2 I 1.95665 7.63658 6.51850 6.80254 7.85333 10.279 19

3 1 I .92620 7.55402 6.53305 6.83604 7.91282 10.36598

4 I 1.89342 7.48248 6.54398 6.86392 7.96561 10.44462

5 I 1.85800 7.41898 6.55263 6.88817 8.01401 10.51747

6 II.81974 7.36167 6.55980 6.90998 8.05946 I058599

12

10

m,'

8

6

1 7 41...,...,...,...~...

62 6.1 6 5.9 5.6 5.7

Id

14 ...,'.,..',.,.',..,' (b)

12

10

%

a

6

45 6 5.99 598 5.97 5.96

Xd

Fig. 3. (a) Expanded view of Fig. 2a for the regions between

5.7 < xd 6 6.2 showing transitions from chaotic regions to

periodic windows and back to chaos. (b) An expanded view of

(a) for 5.96 < rd < 6.00.

a thickness associated with each arm. This thickness allows the solution to move along the arms so that the

periodicity is not exactly M = 6. We also find regions where all the Sy, values are different for all N. These

are regions where the effective Lyapunov exponents are greater than zero.

In this paper we have presented a novel system where chaotic light scattering can be encountered. Tra- ditionally, chaos has always been associated with the

trajectories of the system, or with the evolution of

the system through time. Our system displays chaotic

behavior which depends on a partial summation. Al- though there are a number of tests one can use to

determine whether or not a system is chaotic, we have shown two definitive results which validates our

chaotic system: the bifurcation diagram and the posi- tive Lyapunov exponent. This system not only shows signs of stability (Xd > XI + x2), but also signs of almost-periodicity and chaos when the parameters at- tain certain conditions.

Dat Ngo and Gorden Videen hold National Research Council fellowships with Edgewood Research and De-

velopment Engineering Center and the Army Research

Laboratory. We also would like to thank Ivy Sindoni. Brian Crenna and John Videen for their computer as-

sistance, and Matt Hart, Ron Pinnick, Bob Armstrong, and Hiroshi Brunette for their enthusiastic support.

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