We propose a technique to manufacture Fresnel lenses where the grooves are phase-synchronized at a givenwavelength. This means that the light from different grooves will superpose coherently at the focal point atthe specified wavelength. The focusing performance of such a "tuned" Fresnel lens will be better than that ofordinary Fresnel lenses by orders of magnitude and can, in principle, equal the performance of a conventionallens of the same diameter (diffraction limit). In optical communication systems, which use mostly narrow-band light sources (lasers, LEDs), tuned Fresnel lenses may offer a lower cost alternative to conventionallenses without sacrificing focusing performance.
1. Introduction
A Fresnel lens is a low-cost large-area flat focusingdevice. It finds application wherever a large collectorarea is required without the need for very accuratefocusing. Applications include concentration of sun-light onto a solar energy collector and open-air opticalcommunication systems. In the latter case use of theFresnel lens is motivated by the need to collect as muchlight as possible at the receiver end of the link.'
The main drawback of Fresnel lenses is their poorfocusing performance compared with conventionallenses. This limits their applicability to only the re-ceiver end of the link, where diffraction-limited perfor-mance is not required, and has the added consequencethat background light cannot be eliminated very effec-tively through spatial filtering in the receiver. In thispaper we describe a technique to improve substantiallythe focusing ability of a Fresnel lens at a specifiedwavelength. Such a tuned Fresnel lens can, in princi-ple, achieve diffraction-limited performance for its ap-erture size, while maintaining its characteristics of lowcost, low weight, and large collector area. Implemen-tation of the technique requires only a modification ofthe procedure for manufacturing the master mold forthe lens, so that the cost impact per lens is small whenthe lenses are manufactured in quantity.
The author is with AT&T Bell Laboratories, Holmdel, New Jersey07733.
Most optical communication systems use narrow-band light sources (lasers, LEDs, etc.). In such sys-tems tuned Fresnel lenses offer a lower cost alternativeto conventional lenses without sacrificing focusingperformance. Wherever a conventional lens or a fo-cusing mirror is used, a tuned Fresnel lens can poten-tially do the same job more economically.
The principle of operation of a tuned Fresnel lens issomewhat similar to that of a blazed diffraction grat-ing. It achieves focusing by combining dielectric re-fraction (as employed in conventional Fresnel lenses)with coherent wave superposition (as employed inFresnel zone plates).
II. Theory
A Fresnel lens is a flat optical device which focusesthe light by means of a series of concentric grooves.Figure 1 shows the principle behind the design of aFresnel lens. One can visualize a conventional plano-convex lens being sliced into thin cylindrical sections.In each section the unnecessary extra thickness of glassis then eliminated to obtain effectively a flat array ofthin annular lenses. In actuality the surface of thegrooves might not be exactly the same as the curvedsurface of the equivalent thick lens; for simplicity ofmanufacturing the surface of the individual grooves isusually flat. Thus a Fresnel lens can be more appro-priately described as an array of prisms, with the angleof each prism adjusted so that the rays of a collimatedbeam parallel to the lens axis are deflected toward thefocal point of the Fresnel lens.
The geometry of the light rays is shown in Fig. 2. Itis evident that if diffraction is neglected, the smallestspot size achievable by a Fresnel lens is equal to thegroove width, so that smaller grooves will achieve asmaller spot. However, when diffraction is taken intoaccount, we see that Fig. 2 is valid only in the limitwhere geometrical optics applies, or when d2 >> 2FX,
Fig. 2. Principle of operation of a Fresnel lens with flat grooves.
Fig. 1. Design of a Fresnel lens.
FRESNEL LENS
where d is the groove width, F is the focal length of thelens, and X is the wavelength of the light. In the otherlimit, when d2 << 2FX, the diameter of the spot can becalculated by considering each groove as a narrow slitand using Fraunhofer diffraction theory.2 The inten-sity profile of the spot formed by one groove can befound to be
I(P) - oFA sm FA 1r ~~~~(1)ir'd p3 0
where I is the total intensity of the spot, and p is theradial distance from the lens axis. Thus the diameterof the spot is approximately d = 2 F/d.
In the intermediate case, when d2 - 2FX, the deter-mination of the exact spot size requires the more com-plex calculations of Fresnel diffraction theory; howev-er, it is a good approximation to simply assume
fd when d2 > 2FX
d., -2FX/d when d2 2FX. (2)
To optimize lens performance at a given wavelength,we can pick the groove width that achieves the smallestspot size. From Eq. (2) we see that this optimum valueis d = 2F5;, which corresponds to a spot diameter d VSX. It is interesting to observe that, if the lens is tobe used for imaging or for spatial filtering, the corre-sponding best angular resolution is dIF = OAF,which improves proportionally to the square root ofthe focal length. This is in contrast to what happenswith conventional lenses, where the angular resolutionis the diffraction limit for the given aperture size andthus is determined only by the lens diameter regard-less of the focal length.
Although not explicitly stated, it was assumed in theforegoing discussion that the light rays from the indi-vidual grooves are superposed incoherently at the focalpoint. This is implicit in the assumption that the spotdiameter due to a single groove is the same as the spot
.dj.acet gr... .remthd
Fig. 3. Light deflection by phase-matched grooves.
diameter achieved by the entire lens. However, if thedepth of the grooves is adjusted so that the wave frontsfrom adjacent grooves superpose coherently at the fo-cal point, the performance of the Fresnel lens can besubstantially improved.
Coherent superposition can be achieved at a givenwavelength for wave fronts coming from a point sourcelocated in the lens axis at a given distance from thelens. We shall consider the situation where the pointsource is located at infinity, corresponding to incomingplane wave fronts perpendicular to the lens axis. Sucha "tuned" Fresnel lens combines the principle of oper-ation of the conventional Fresnel lens with the princi-ple of operation of the Fresnel zone -plate.
Figure 3 shows the situation where the groove depthis such that the wave fronts emerging from..two adja-cent grooves are exactly in phase. As the wave frontsenter the material of the Fresnel lens, they are de-flected according to Snell's law:
Fig. 4. Setup for laser interferometry during groove cutting.
where the angles a and f are the angle of incidence andangle of refraction, respectively, as defined in Fig. 3,and n is the index of refraction of the material (typical-ly n 1.5). The distance between wave fronts in air is,of course, X, while the distance between wave frontsinside the material, as measured along the lens axis,can be calculated using standard geometry:
sinl X 1(4sina cos(a - ) n cos(a-3
For a groove of depth h, the phase difference betweenthe waves going through adjacent grooves will thus be
A =5 2rh
= 2r h [n cos(a--1]- (5)
For coherent superposition Aq5 must be an integer mul-tiple of 27r, so h must be an integer multiple of
X/[n costa-o3)-1]. (6)
To get an idea of the typical values of the variousparameters involved, we observe that a typical Fresnellens might have a 300-mm diameter, a focal length alsoof 300 mm, and a groove width of 0.15 mm. 3 With atypical index of refraction of '1.5 the groove angle avaries from a minimum of 0° in the center to a maxi-mum of -45° at the edge of the lens. The angle a -avaries between 0 and 17°, so that we can use theapproximation cos(a - f) 1 in Eq. (6), which yields
) - X(n -1) = 2 (7)
regardless of the groove position. The groove depthwill vary from zero at the center to d tana 150 ,um atthe end. If we assume X = 0.8 Atm, we see that thedepth of the typical groove is of the order of -70 A.This means that the requirement that the groovedepth be an integer multiple of X can be met withoutsubstantially altering the basic structure of the lens, asthe nominal depth of the individual grooves must bemodified only by a small percentage to achieve thedesired result.
The performance of a tuned Fresnel lens can, inprinciple, be as good as that of a normal lens of thesame diameter. In other words, it can achieve thediffraction limit for its aperture size. This can be seenby observing that the parameters given above for a
Fig. 5. Geometry of light paths during groove cutting.
typical Fresnel lens meet the condition d2 << 2FX.When this condition is met, the plane wave frontemerging from the groove is indistinguishable from aspherical wave front, i.e., the two differ only by a smallfraction of a wavelength. Thus the overall wave frontemerging from the Fresnel lens will be, rigorouslyspeaking, piecewise plane, but it will differ from theoptimum spherical wave front only by a negligibleamount. With the lens parameters given above, thiscorresponds to a spot size which is smaller than thatgiven by Eq. (2) by several orders of magnitude.
Ill. Manufacturing Considerations
It is important to address the question of how thephase coherence among grooves can be achieved in themanufacturing process. Fresnel lenses are pressuremolded from a brass or aluminum master; the mastermold is machined on a lathe with the edge of thecutting tool adjusted at the proper angle for eachgroove.
It is not feasible to determine the depth of eachgroove with sufficient accuracy (a fraction of a micron)in an open-loop fashion. Therefore, we have devised alaser interferometric measurement technique whichmonitors the groove depth as it is being cut. Figure 4shows the setup for the interferometric measurement.The mold is illuminated with light from a He-Ne laserwhile it is being machined. The spot of laser light fallson the groove being cut as well as on a few adjacentgrooves that have already been cut. The light reflect-ed by the grooves is collected by a light detector whichdisplays the received light level. The geometry of thelight path is shown in Fig. 5. As the new groove be-comes deeper and deeper the output of the light detec-tor will oscillate up and down. The local maximaoccur when the path length difference for the lightreflected by the new groove with respect to the oldgrooves is an integer multiple of the wavelength of thelaser, X0 = 0.6328 Am. From the geometry of Fig. 5, wesee that the path length difference is
so that the condition for constructive interference isthat h be an integer multiple of
X,XP__ 2cos4!' Cosa (9)
We can now pick 4/-the angle of incidence of the laserbeam-so that Xp = X, or
cosl = n cos(a - X-lo (10)2cosoa A
With this choice of ,, the maxima in the reflected laserlight correspond to groove depths that achieve thedesired phase coherence.
It is important to observe that even with this tech-nique, it may not be possible to achieve exact phasecoherence over the entire surface of the Fresnel lens;however, the lens will still provide substantially im-proved performance over the equivalent untuned lens.For example, if the technique can achieve reasonablygood coherence over only -100 adjacent grooves, thespot diameter achieved by this lens will be -100 timessmaller than that achieved by the untuned lens.
IV. Conclusion
Using the manufacturing technique described in theprevious section, it is possible to produce tuned Fres-nel lenses with performance approaching that of con-ventional lenses (diffraction limit) at a given wave-length. It should be noted that the technique requiresextra effort only in the manufacturing of the mold; theproduction of the individual lenses from the mold willuse the same technique presently used to produce con-ventional untuned Fresnel lenses. As a result, withproduction in quantity, the extra cost associated withthe more complex manufacturing procedure for themold will not significantly affect the cost of the indi-vidual lenses. This retains the advantages of low cost,low weight, and large aperture size of ordinary Fresnellenses while achieving focusing capabilities which arebetter by several orders of magnitude.
It should be noted that the same technique could beused to make a Fresnel mirror consisting of a series ofconcentric reflecting grooves, phase matched toachieve diffraction-limited performance at a specificwavelength.
References1. B. G. King, P. J. Fitzgerald, and H. A. Stein, "An Experimental
Study of Atmospheric Optical Transmission," Bell Syst. Tech. J.62, 607 (1983).
2. M. Klein, Optics (Wiley, New York, 1970), Chap. 7.3. Ealing Optics Catalog (The Ealing Corp., South Natick, MA.).
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With manganese substituting for 45 percent of thecadmium atoms (the resulting crystal looks much like aruby), polarized helium-neon laser light passing througha centimeter of the crystal is rotated 0.17 degree pergauss of magnetic field. (The earth's magnetic field isabout half a gauss; a toy magnet produces about 100gauss.)
Butler and Martin have demonstrated that this largerotation effect allows variations in magnetic fieldstrength to be readily monitored at frequencies up to5 GHz. Combining this ability with optical fiber tech-nology will allow extremely rapid fluctuations in mag-netic fields to be measured.
In the magnetic field sensor, light from a diode laseror a light-emitting diode is passed through a three-portcoupler and an optical fiber link. The light emergesfrom the fiber and is collimated by a lens to pass inparallel rays through the magnetic semiconductor,which is placed within the magnetic field to be mea-sured. The magnetic field causes the light's plane ofpolarization to be rotated.
Reflected back through same path
This light is then reflected by mirrors back throughthe same path (doubling the rotation effect when itpasses through the magnetic semiconductor the secondtime). The coupler this time routes it to a polarizer that,like a second polarized sunglass lens, filters out polar-ized light according to the relative angles of polariza-tion. A photodetector measures the modulated intensity(the modulation being due to the varying polarizationdirection) of this returned light.
Experimental work at Sandia is continuing. The mag-netic semiconductor Sandia used for the first experi-ments was obtained from outside the labs, but recentlySandia has developed its own capacity to grow magneticsemiconductor crystals. This will make it easier to ex-plore effects of various crystal compositions.
"This sensor is particularly appropriate for measure-ment of rapidly varying magnetic fields in electricallynoisy environments," say Butler and Martin. "It mayalso find application in the direct detection of intenseradio frequency fields."
