0 A A I I

Stephen D. Fantone

Fractal fiber optics

Lee M. Cook

Analysis of the geometry of recursive tilings has led to the development of a new class of highly orderedoptical composites that exhibit fractal surface character. These objects are, we believe, the first engineeredfractal objects. The mathematics of tiling and examples of fractal fiber array devices are reviewed.

". . . It is a widespread source of irritation thathexagons put together do not quite make up a biggerhexagon."

B. Mandelbrot, The Fractal Geometry of Nature(1977)

Fiber-optic faceplates, microchannel plates, and im-agescopes may all be classified as fiber array devices(FAD's), i.e., they are coherent lattice arrays ofwaveguides. Such lattices are either close-packedhexagonal or cubic. Unfortunately, a homogenouslattice array envisioned by the optical engineer (Fig.1) is seldom encountered in commercially availableFAD's; various degrees of disorder are more common(Fig. 2). To improve the production of highly homoge-nous arrays we have applied fractal theory to thefabrication process.' The resulting products are, webelieve, the first engineered fractal objects.

The author is with Galileo Electro-Optics Corporation, P.O. Box550, Sturbridge, Massachusetts 01566.

Received 22 April 1991.0003-6935/91/365220-03$05.00/0.e 1991 Optical Society of Allerica.

A typical fabrication sequence for hexagonal latticeFAD's is as follows:

(1) A single fiber (with core and clad) is drawn.(2) A group of fibers is assembled into a hexagonal

array, fused, and drawn to any desired dimension toform a multifiber.

(3) A hexagonal array of multifibers is assembledand drawn down.

(4) Step (3) is repeated as many times as neces-sary to achieve the desired single-fiber dimensions.

(5) The final fibers are fused together to make thefinal device.

The simplest description of the surface of hexago-nal arays produced by such a recursive tiling processis the formation of a Gosper snowflake, which wasfirst described in 1976.2 Starting with a hexagon, onebreaks up each face into three segments of equallength so as to preserve the original area of thehexagon. When this operation is performed recur-sively the original hexagon is transformed into acomplex polygon, as illustrated in Fig. 3. The perime-ter length L of any member of the transformation

5220 APPLIED OPTICS / Vol. 30, No. 36 / 20 December 1991

A

Fig. 1. Electron micrograph of a Galileo CP Series fiber-opticfaceplate. The pitch of the waveguide array is 4.5 m, 2000xmagnification.

sequence may be described by

L = Kr1 -Dh, (1)

where K is the number of sides of the polygon, r is thelength of each side, and Dh is the characteristicdimension of the perimeter, commonly termed theHausdorf or fractal dimension. Such objects arecommonly termed fractals. For the example of Fig. 3the fractal dimension is 1.129 1. An infinite variety of

Fig. 2. Electron micrograph of a conventional fiber-optic face-plate. The pitch of the waveguide array is 3 jim, 2000x magnifica-tion.

)-7

Fig. 3. Fractal transform series for the Gosper snowflake. PartsA-E indicate the sequence of transformations.

fractals may be generated by changing the number ofsurface transformations per operation or by usingalternative transformations. 3

We have constructed FAD's that are precise ana-logs to the Gosper snowflake. Figure 4 is a photomicro-graph of a multifiber corresponding to the secondtransformation of Fig. 3, where the circular fiberelements are topologically equivalent to a hexagonconstructed from lines that are tangent to the circu-

Fig. 4. Fractal Fiberoptic array. The structure shown is theanalog of a second transformation Gosper snowflake (Fig. 3C), 50 xmagnification.

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Fig. 5. Fractal Fiberoptic array. The structure shown is theanalog of a fourth transformation Gosper snowflake (Fig. 3E), 20 xmagnification.

lar fibers. These fractal multifibers interlock perfectlyto form yet larger and more complex arrays. Anexample of a fourth-order Gosper snowflake made byrecursive tiling of subelements is illustrated in Fig. 5.The boundaries of the subelements are virtuallyinvisible. Alternatively, one may construct large-areaFAD's by tiling fractal fibers. A photomicrograph of afractal fiber-optic faceplate (trademarked Fiberoptic)with a 75-[um pitch is shown in Fig. 6. While approxi-mately 20 fractal multifibers are present in thephotograph, their boundaries cannot be discerned.

The examples shown above were produced prima-rily to illustrate the concepts of fractal tilings. We are

Fig. 6. Photomicrograph of a fractal Fiberoptic faceplate con-structed from a tiling of Gosper snowflakes (Fig. 4). The pitch ofthe waveguide array is 75 jim. There are approximately 20fractal multifibers in the field of view, 5 Ox magnification.

Fig. 7. Photomicrograph of a fractal Fiberoptic faceplate having a6-jim pitch, 200 x magnification.

Table 1. Comparative MTF Data for Conventional Versus FractalFiberoptic Faceplates,

Spatial Resolution MTF (%)(lines/mm) EZ (Conventional) FF0231 (Fractal)

14 50 8530 30 5550 20 35

aBoth samples have a 6-jim pitch and are 25 mm thick.

currently producing fiber-optic faceplates using avariety of similar fractal architectures (Fig. 7). Thehigh degree of internal order in these devices givessubstantial improvements in optical performance [themodulation transfer function (MTF)], which makethem particularly useful in electronic imaging applica-tions. (The MTF is a widely used measure of theimaging performance of optics. It describes the modu-lation, or change in contrast, of the image of a sinewave transmitted through the optic. This is typicallyexpressed as a plot of the ratio of input-to-outputimage contrast as a function of spatial frequency.)Table I gives comparative MTF data for analogoussamples of fiber optics, one made in the conventionalfashion (EZ) and the other using fractal architectures(FF0231). The improvement in the MTF is due solelyto the higher degree of order in the fractal sample; allother variables (glass type, fiber size, cladding thick-ness, etc.) were kept constant.

References1. L. Cook, D. Mancini, and S. Patterson, "New fiberoptic face-

plate architectures," in Electronic Image Tubes and ImageIntensifiers, I. P. Csorba, ed., Proc. Soc. Photo-Opt. Instrum.Eng. 1243, 196-204 (1990).

2. M. Gardner, "Mathematical games," Sci. Am. 235, 124-133(1976).

3. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, NewYork, 1977), Chap. 6.

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