Marius P. Schamschula, H. John Caulfield, and Avery Brown
Department of Physics, Alabama A&M University, Normal, Alabama 35762
Received December 20, 1993
Repeated folding of the optical axis can be used to design space- and volume-efficient optical systems. We suggestthat space-filling curves, such as the Peano and Hilbert curves, offer a useful way of realizing compact modularoptics.
Some complex optical systems require long opticalaxes. Examples of such systems are optical corre-lators, optical symbolic substitution systems, and op-tical shuffle networks. If such systems are built upalong a straight line, they become cumbersome as aresult of to their length. Efficient use of space isdesirable for such optical systems. By folding theoptical axis it is possible to reduce the physical di-mensions of a complicated optical system.
The goal of this Letter is to introduce a new wayof folding the optical axis to reduce the volume andlinear dimensions of an optical system. To achievea practical solution we further require that such asystem can be assembled from a minimal number ofmodular components. This provides greater ease ofmanufacture if such a system is to reach the produc-tion stage. Furthermore, unlike in previous studies,except for Refs. 1 and 2, we restrict our search to ar-chitectures that permit normal incidence to all lensesand active elements. This reduces aberrations thatare due to powered surfaces and ensures the effec-tiveness of spatial light modulators.
There have been numerous approaches to foldingthe optical axis to reduce the physical size of suchsystems. Most of these architectures are ad hoc so-lutions to the specific requirements of a given systembut can be generalized to other problems. In anotherpaper3 we develop several regular methods of foldingsuch a system into three dimensions.
In this Letter we develop more general methods offolding. We use some of the same figures of meritintroduced in Ref. 3. These figures of merit includethe volume, the airline dimensionality (AD), a com-bined geometrical figure of merit (GFM), the anglesof incidence to optical surfaces, and minimum f-numbers.
To introduce the concept of space filling curves, weneed to understand how a series of straight linescan be folded to cover an area. In mathematicalterms a line is an infinitely thin one-dimensional ob-ject. So how can such an object completely cover atwo-dimensional area? Since a line is infinitesimallythin, a finite area can be covered only by an infin-itely long line. Methods of folding a line to cover anarea completely were proposed by Peano4 in 1890 andby Hilbert5 in 1891. Both introduced self-similarcurves that are iteratively lengthened to cover thewhole area; that is, their fractal dimension is two.
Figure 1 shows the first step of the construction ofboth curves.
In the limit of infinitely repeating the substitutionin Fig. 1(a), we obtain a Peano curve. We take a lineof length three and generate a curve of length nine.This construction gives a fractal dimensionality
D log9 = 2log 3
(1)
as required. To generate other Peano curves, wecould divide the initial line into any odd number N ofline segments. Each step in the construction processlengthens the curve by N2. Each prime number hasa unique Peano curve.
The Hilbert curve [Fig. 1(b)] substitutes a U-shaped line segment into each of four quadrants.The line segment's orientation depends on its posi-tion in the curve. The ends of the line segmentsare then joined to form the next stage of the Hilbertcurve. The area is now divided into four times as
(a)
(b)Fig. 1. (a) First iteration in the creation of a Peanocurve. In each step of construction a line segment isreplaced by the nine line segments on the right. (b)The first and second stages of the Hilbert curve. Eachsquare is successively replaced by the curve shown inthe left-hand figure, oriented as shown in the right-handfigure.
Fig. 2. Comparison of two curves: the area within theprism is covered twice, once by the incoming beamand once by the exiting beam. This occurs only inself-touching curves.
many squares as before, and the procedure is re-peated.
On closer examination of Fig. 1 we can see a dif-ference between the two curves. The Peano curvetouches itself, while the Hilbert curve does not. TheHilbert curve is classified as a self-avoiding curve.Accordingly, we hereafter generalize such curves asbeing either self-touching or self-avoiding.
Because real optical systems have a finite beamdiameter, we can construct a two-dimensional space-filling curve with lines of finite width. This meansthat a finite-length optical axis can be used to fillan area. By comparison with the two mathematicalcurves, we find the finite width curve has an inter-esting property: near a folding mirror the area iscovered twice by a collimated beam. Consider, forexample, Fig. 2. The incoming and outgoing linescover the same area.
We can build an optical system by adding depthto the two-dimensional finite-width curves. For thePeano scan-based system, the only structural compo-nent required is a right-angle prism with a reflection-coated hypotenuse. If the sides of the prisms are thesame length as the width of the optical beam, thenthe whole volume is used twice.
If we base a system on the Hilbert curve, we alsoneed clear cubes as a second building block. Lightpasses straight through these cubes only once. Thusthe cubes represent single usage of volume. A sec-ond volume inefficiency arises because the Hilbertcurve is self-avoiding. The volume is only singly oc-cupied, since there are no back-to-back corner prisms.
The design of an optical system strongly depends onthe system's intended use. We can, however, makesome general observations.
For a modular optical system to be useful the num-ber of optical components needs to be minimized.For example, if we base an optical system on thePeano curve, we need one solid optic module, a right-angle prism. To make a meaningful system we needother components, such as lenses, diffractive optics,spatial light modulators, and detectors.
In Ref. 3 we introduce three figures of merit forthe physical size of an optical system. These are thevolume (V), the AD, and the GFM.
The volume is a useful measure, since in a solidoptic system it is a linear function of the systemmass. The AD (sum of length, height, and widthof an enclosing box) is a measure of bulkiness. Thesmallest AD for a given volume is given by a cube.The GFM is a weighted combination of the volumeand the AD:
GFM- (AD + 3 V)6
(2)
These weights are chosen that, for a cube, each factorcontributes half of a total GMF of 1.
To make these numbers meaningful we have to de-fine the units of measurement. The volume of anoptical system such as a correlator or symbolic substi-tution processor is a function of the focal length andf-number of the system. We assume that all lenseshave the same focal length and f-number. In a realsystem this may not be appropriate, as the resolu-tion of the spatial light modulators (SLM's) and CCDarrays may be different.
Optical components such as lenses and SLM's workbest if the optical axis is normal to the component. Ifthe angle of incidence is beyond the paraxial regime,then lens wave-front and polarization aberrations be-come significant.
Many SLM's operate by changing the polarizationstate of the incoming light. Some of the materialsused are birefringent and consequently have differ-ent phase retardations depending on the angle ofincidence of the illuminating light. The resultingoutput light state therefore will be elliptical, ratherthan linear or circular, given linear or circular inputlight.
Space utilization or multiplicity is yet another mea-sure of geometrical efficiency. The multiplicity canbe defined as the number of independent beams inany region of space if all beams are collimated. Themultiplicity for a self-touching curve, such as thePeano curve shown in Fig. 1(a), is two. The multi-plicity of a self-avoiding curve, such as the Hilbertcurve in Fig. 1(b), is one. For a collimated beam allvolume is doubly occupied in the self-touching curve.The self-avoiding curve includes straight-line seg-ments, and the folding prisms are not back to back.In both instances the volume is effectively usedonly once.
Finally, we must not allow a general optical moduleto lock the user into a particular f -number. Both lowand high f-numbers must be readily achievable.
We have developed an architecture based on afolded solid optic Peano curve. This variant usesonly right-angle prisms, lenses, and active elements.We show the layout of this architecture in Fig. 3.This example is a f/2 system that uses a collimatedinput beam with an axis length of 4f.
If we disregard the volume and linear dimensionsof the lenses and active optical elements, we can cal-culate the figures of merit for this system. We com-pute two volumes: the volume enclosed by the ac-tual system and the bounding volume. The productof the lengths is then used to calculate the AD.
Fig. 3. Correlator design based on the self-touchingtwo-dimensional Peano curve. This example is an f/2system with a collimated input.
Input SLM
Waveguide M-Illuminator
Fourier Plane SLM
Fig. 4. 4f, f/3 correlator based onHilbert curve.
-4-. CCDDetector
the self-avoiding
The system volume is given as
VOlsystem (f#)3
where f is the focal length and f/# is the f-number.This expression is true for integer f-numbers greaterthan or equal to two. For noninteger f-numbers, the
volume is given by the integer value. To generatesystems with f-numbers higher than f/2 we havetwo choices. First we could chain together multiplecopies of the system shown in Fig. 3. This can bedone in the plane and by vertical stacking. The sec-ond option is to use a Peano curve with a differentbase, such as 5 or 7. A system with a base of 5 (7)is useful for f-numbers greater than f/4 (f/8) for a6f correlator system with a collimation stage. If acollimation stage is not required, then a base 5 (7)system is appropriate for f/6 (f/12) and up.
The bounding volume for an f/2 system is 3f3/4.The corresponding AD is 3f . The GFM combines thesystem volume with the AD. For this system theGFM is 0.897f. For comparison, an unfolded f/2system has a bounding volume of f3, an AD of 5f, anda GMF of 1.333f. This geometry offers substantialreductions in all three figures of merit.
We may also construct a solid optic system basedon the Hilbert curve. In this case we have chosenthe f/3 system shown in Fig. 4. Disregarding thevolume of lenses, etc., the system volume is 7f3/27.The bounding volume is 4f3 /9, and the AD is 8f/3.For the Hilbert system the GFM is 0.763f. An un-folded f/3 system has a bounding volume of 4f3/9,an AD of 14f/3, and a GMF of 1.159f. The Hilbertcurve-based system has an equal bounding volumeto the unfolded system, but the AD is 0.571 of theunfolded system.
The first conclusion supported by this researchis that the mathematical field of integer fractal di-mensionality (space-filling) self-similar curves can beused to inspire highly useful modular optical systemdesigns. The second conclusion is that the first de-sign has outstanding behavior in terms of volume,AD, GFM, normal incidence, f-number flexibility,and number of modular components. We are explor-ing many variants of this simple system.
This research was sponsored by Teledyne BrownEngineering.
References
1. R. P. Bocker, H. J. Caulfield, and K. Bromley, Appl.Opt. 22, 804 (1983).
2. Y. Ichioka, Department of Applied Physics, Faculty ofEngineering, Osaka University, 2-1 Yamadaoka, Suita565, Japan (personal communication, 1993).
3. M. P. Schamschula, P. Reardon, H. J. Caulfield, andC. F. Hester, "Regular geometries for folded opticalmodules," Appl. Opt. (to be published).
4. G. Peano, Math. Ann. 36, 157 (1890).5. D. Hilbert, Math. Ann 38, 459 (1891).
