Multiaperture Focusing Technique

Manoranjan De, John Wai Yu Lit, and Ral Tremblay

A method of focusing electromagnetic waves by diffraction alone is proposed. An incident beam is

diffracted by a series of circular apertures which have their diffracting edges falling on a surface of revolu-tion. A geometrical consideration of the principle involved is presented, followed by a series of experi-mental results.

I. Introduction

The study of free space propagation of an electro-magnetic wave in terms of Fresnel zone constructionhas led to the interesting development of phase reversedSoret zone plates. This has the property of focusing adivergent beam by diffraction alone. This latter con-sists of a system of concentric coplanar apertures sodesigned that the Fresnel zones related to S (source)and S' (detector) are alternately retarded by r, thusproducing a strong focusing effect at S'. The focusing,resolving, and diffraction properties of such zone plateshave already been analyzed by many authors. 1-6

Here we describe a device',8 which also possessesfocusing properties. The main difference between itand the Soret zone plates is that it makes use of iterateddiffraction, thus rendering it much more complicatedthan the other. To show the existence of focusingproperties, it has been found that, as a first approxima-tion, a simple geometrical consideration suffices. More-over, it is seen that as the lens is the limit of the Soretzone plate, so the ellipsoidal mirror is the limit of theellipsoidal diffracting system proposed.

11. Aperture Configuration

Let S and S', respectively, be the source and the de-tector (Fig. 1) and let the field due to S be limited byan aperture at any arbitrary position. It is knownthat if the size of the opening of the aperture is such thatit contains an odd number of Fresnel zones with re-spect to S and S', we should get some build up of energyat S'. The magnitude will evidently depend on theposition of the aperture relative to S and S'. It isobvious that the aperture may have a size and position

M. De is with the Applied Physics Department, UniversityCollege of Technology, Calcutta, India; the other authors arewith Laboratoire d'Optique et Hyperfrequences, D6partment dePhysique, Universit6 Laval, Qu6bec, P.Q.

Received 23 February 1967.

such that its rim coincides with a surface of revolutionabout the axis SS', with S and S' as the foci of thegenerating ellipse. We would call this ellipsoid ofrevolution an N ellipsoid, if any aperture of this con-tains N Fresnel zones relative to S and S'.

Let us consider two or more circular apertures withall their edges falling on the surface of an N ellipsoid[Fig. l(b)]. Since all the rays which proceed from thesource S, diffracted by the edges of the apertures andarrived at S', have equal path lengths, it is to be ex-pected that they will reinforce at S' to produce a strongsignal showing the focusing capability of the system.

In order to study the effects of the axial spacings be-tween the apertures, we have tried to space the aper-tures according to some physical considerations which,from a study of the Soret zone plates, seem to be mostnatural.

Let us first assume an aperture placed on the Nellipsoid as near to the source as feasible, depending onthe maximum usable divergence of the beam originatingfrom the source S. The divergent wavefront at thisaperture will then have N Fresnel zones pertaining tothe detector at S'. With a Soret zone plate, the(N - )th zone would have been phase retarded by theintroduction of a phase retarding plate. We couldvisualize a situation in which the ray passing throughthe margin of the (N - 1)th zone is allowed to proceedonwards and meet a second aperture on the N ellipsoidagain. We can continue to build up a system oflongitudinally spaced apertures in the above manner.For example, the ray passing through the margin of the(N - 1)th zone of the wavefront falling on the secondaperture may be allowed to proceed to meet a thirdaperture on the N ellipsoid, and so on till we reach thelast effective aperture. In fact, this may be generalizedto consider an M ellipsoid within the N ellipsoid [Fig.1 (b) ]. If there is only one aperture, it is true that, if Nis even, the signal at S' is minimum. For a series ofseveral apertures, since we expect that the most im-portant feature is that the direct rays, diffracted once

March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 483

.3 -2 41 0 ' 2 3 Ill. Nomenclature

N

- . -M

S

- L - -

(a)

5,ob.J I~e_

_ _ _ _ _ _ _ _ - C (0 )I Z r -

For easy reference, the symbols used in this com-munication are summarized as follows: S, S' = foci ofgenerating ellipse, N = Fresnel number of the outerellipsoid, = Fresnel number of the inner ellipsoid,ao = radius of the central aperture of the system, L =distance between the two foci of the generating ellipse,and n = number of diffracting apertures.

The diffracting planes are numbered as 0, +1, 42,etc..., with the one midway between the two focibeing 0, those towards the source being -, and thosetowards the detector being +. Whenever a pure num-ber may easily lead to misunderstanding, the symbol ofa subscripted C is used. Thus C-1 means the firstcircular diffracting aperture next to the central one and

X placed on the side close to the source.

(b)

Fig. 1. Curves N and I1! represent, respectively, the N and Mellipsoids with foci S and S': (a) geometrical picture of the raysproceeding from S, diffracted by the circular apertures (C-3, C-2,

etc.) and converging at S'; (b) Ao0 o represents central plane ofsystem. Coordinates of various points have CS' as X axis and

CA0 as Y axis.

iLIILi.L.

Fig. 2. Experimental setup: P = power supply, D = detector,A = amplifier, and R = register.

by the apertures, will have traveled through equal pathlengths before arriving at S', we shall not lay any re-striction on the nature of N and ll. Thus we shallconsider the cases where N and 11 are fractional orintegral, even or odd.

Since the ellipsoid is symmetrical about a plane per-pendicular to axis SS' and midway between S and S',we may construct a symmetrical system by first startingwith an aperture placed on this central plane, and thenproceeding backward and forward. This would needcalculation for only one half of the system.

It should be noted that the above consideration dis-regards all diffractional effects produced by one apertureon the others. Nevertheless, it does provide a veryconvenient way to describe the system which can bespecified simply by the two numbers N and 11/. Thezone effects, which are very important in the Soret zoneplates, can be studied by allowing the zone betweenthe two ellipsoids (namely, the value N - l) to takeon various values.

The present analysis is preliminary in nature, and ex-cept for a case where theoretical results had been ob-tained by iterated diffraction method, the results aremainly experimental.

IV. Layout of Apertures

Referring to Fig. 1(b) and consideringtriangles, we have:

a bi

(L/2) + z + z (L/2) + 1

Equations of ellipses are:

x2 qv2

N: + =1a 02 + (L/2)2 ao2

x2 b21'bo2 + (L/2)2 b02 =1

1.0

0.8

0.6

0.4

0.2,

0.00 5

the similar

(1)

(2)

(3)

AXIAL INTENSITY

n THEORETICAL

1 4.0 3.5

2 9.7 8. 5

3 16 6 I4.5

} - - I T.EOR ET ICA L...................

@ .l . @@ EX-.RIMIZNTAL

n 1

Io 15 2.0 X()

Fig. 3. Transverse intensity distribution in principal focal planeof system with N = 5, T = 2, a = 2, and L = 318.75X. Thetable shows a comparison of the theoretical and experimental

values of the intensities at the principal focus.

484 APPLIED OPTICS / Vol. 7, No. 3 / March 1968

l (x)

1.0*

0.8

0.6

0.4-

0-2~

0

I

III

IIII: IIII

I

Ii �I A

i B.r ,I It � : 1�

% I .1 11I-

0 5 lo IS 20 X(A)

Fig. 4. N = 5, M = 4, a = 20X, L = 318.75X. Experimentalresults of the principal focal plane transverse intensity distribu-tions. Curve A is that produced by diffracting apertures C-1,C0, and C+1. B is that of C-,5, C- 4 , C-3, C-2, and C-1. C is that

of C-2, C-1, Co, C+1 , and C+2.

Having chosen the values of N and M, it can be seenthat if z is also fixed the coordinates of the points Al,B1, and A can be determined by substituting them intothe appropriate equations above. Then, by treatingthe point A as A1, the positions and radii of all otherdiffracting apertures can be found.

The above method had been used to program anIBM (Fortran) 360 computer to give us all the dimen-sions of the systems which we have studied and whichwe describe further on.

V. Theory

Consider two coaxial circular apertures with theircenters being each the origins of the circular coordinatesystems for points in their planes. If Ul(ri, 01) repre-sents the wave function in the first aperture, thewave function in the second aperture can be given by

i f 2 e-ikRU2(r2,,,) JO ± ( r2 R Arldkldrl .. , (4)

where a is the radius of the first aperture; R is thedistance between two points considered, one in each ofthe apertures; A is the inclination factor; r, 01 andr2, 02 are the coordinates of points in the apertures; X isthe wavelength; and k is the wave propagation constant27r/X. The value of A can be taken2 as + b/R, whereb is the perpendicular distance between the two aper-tures.

By making the usual assumptions and the approxima-tion:

R = b + (rl' + r)/2b - r1r2 COS(l - 02)/b2 *-- (5)

and noting that, with the source placed on the axis ofthe system, the wave functions are independent of 0,it can easily be shown that

P=i7r fa U ( exp -ik[b + (r2l + r)/2b] }o R

X (1 + b/R) Jo[k(rjr2/b)]rjdr ... , (6)

where Jo is the Bessel function of zero order.When U2(r2) has been calculated, it can be used to

replace Ul(r2 ) in Eq. 6 to give U3(r3) in a third aperture,and so on. Thus, by this iteration method, the wave-front in the final plane can be found.

This method had been used to program a computerwhich gave us the theoretical results presented here.

For the cases where the apertures are close together,the Huygens-Fresnel-Kirchhoff formulation of diffrac-tion, as used in the above iteration method,9 becomesinapplicable. Further analysis using the AIaggi-Som-merfeld-Rubinowicz boundary wave theory, which ismore suitable for edge diffraction, is pursued. M\'ore-over, multidiffraction phenomena are considered also.

VI. Experiment

To verify the feasibility of the principle sketchedabove and to study the properties of such systems, anumber of microwave systems were constructed forX = 1.25 cm. The experimental setup is shown inFig. 2.

Two attenuators were used: one giving the initialattenuation needed in order not to have too great asignal on the detector, and a second one of high pre-cision for measuring the differences in intensities.

The two horns, which were chosen to serve as emitterand receiver, had a gain of 10 dB, and a half-powerwidth in both the E plane and H plane equal to ap-proximately 580.

VII. Results

A. Intensity Distribution in Focal PlaneFigures 3, 4, and 5 show a series of transverse intensity

distributions in the focal planes. It can be noticedthat, as the number of diffraction (n) increases, boththe widths of the central lobes and the intensities ofthe secondary maxima decrease indicating a focusingeffect. This effect is again strongly borne out by thetables in Figs. 5 and 6 which show that the focal in-tensity at S' increases very rapidly with the number ofdiffracting apertures used. It is well known that thecontributions from each of a pair of two consecutiveFresnel zones practically cancel each other. That iswhy the focal intensity for N = 6 has a very low valuefor one diffracting aperture. This effect makes itnecessary to have two more diffracting apertures for asystem with an even N to produce approximately thesame signal at S' as a system with an odd N.

March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 485

. .

1.0

0

In (0)

0, Ii1 007

2 1 0.363 1 2.5

4 6.85 12.6

6 224

7 I 34

8 47I 9 60

0' 78

[ I!198

nl

W._ I - > X(i)10 15 200

Fig. 5. N = 6, 11 = 5, a = 20X, L = 265.17X. Transverseintensity distribution obtained experimentally in the principalfocal plane. Table gives the intensities at focus due to various

number of diffracting apertures.

Figures 3 and 7 show four of the results of the in-tensity distributions in the focal plane of a system withN = 5, I = 2, a = 2, and L = 318.S X. It wasthought that, with the same number of diffractingapertures, the central lobe would be narrower if theapertures were closer to the detector, because the anglesubtended at the detector by the aperture nearest toit would be bigger. This is shown to be true, as inFig. 7.

Figure 4 shows the experimental results of anothersystem with all the parameters being the same as thelast one, except that M was equal to 4 instead of 2,which means that there was only one instead of threeFresnel zones between the two ellipsoids. For thesame L, a greater number of diffracting apertures couldtherefore be used, which means that a more effectiveuse of the energy emitted by the source could beachieved, resulting in a much larger gain in intensity atthe focus S'.

In Fig. 6 the intensity distribution produced byeleven apertures is compared with that of the patternthat would be produced by a perfect system with adiameter equivalent to the one of an eleven-zone phase-reversed Soret zone plate placed midway between thesource and the detector.

The result of a system with an even value of N isshown in Fig. 5. The parameters of the system wereN = 6, 1M1 = 5, a = 20 X, and L = 265.2 X. Here thegeneral focusing effect was the same as the other sys-tems, but the intensities at focus, as noted above, wereless than those of the corresponding systems with anodd value of N.

B. Focal Intensity

In order to see the effect of the axial spacings on theintensity at the focus, the system N = 5 had beenexamined with values of M = 4.5, 4, 3.5, 3, and 2.The results are given in Table I. It can be seen thatfor the same number of diffracting apertures (e.g.,n = 3) the focal intensity is a function of the axialspacings. It changes rather quickly when the aper-tures are very close together, as expected. It seems toattain a maximum in a region where N - M is ap-proximately equal to 2, namely, M = 3 in Table I.

C. Intensity Distribution Along Axis

In the intensity distribution along the axis of the sys-tem N = 5, M = 2, which has been described above, hasbeen computed with the help of an IBM (Fortran) 360computer (Fig. 8). As does a Soret phase-reversedzone plate, this system exhibits the multifoci property.

D. Optimum ConfigurationWith N = 5 and M varying by steps of 0.1 from 4.5

to 1.9, focal intensities have been measured by usingonly two diffracting apertures. Figure 9 is a graph ofthe results obtained. It seems that, for this particularcase when the zone width is about 2.2, the maximumfocusing effect is attained.

From the above results it appears that in order toobtain the optimum configuration for maximum focalintensity it may be necessary to have spacings betweenthe apertures that correspond to zone widths greater

l(x)1.0

O9

0.8

0.1

0.6-

0.5

04

03

02

0.1

n 10)l1(0)

0 1.0I 3.1Z . 53 13.04 ZZ.4S 34.76 s0.1a la S

- LSLCLP50iO&L CQMi6L01iQM IEYT)

---- EQiVLEI PEqFECT 5'ISTM

0 i 2 ! A 5 6 1 a 9 10 11

Fig. 6. Comparison of transverse intensity distributions in theprincipal focal plane of the ellipsoidal system (full line) and of anequivalent perfect lens (dotted line). The ellipsoidal system had

N = 5, 1 = 4, and n = 11.

486 APPLIED OPTICS / Vol. 7, No. 3 / March 1968

I/

l(,o)

I;

11i.I -l

i -_ __-- THEORETICAL

lI S@Y°X0 EXPERIMENTAL

I I

I I EI l llIl

I P

than that of a system with larger zone width and thesame n. Thus, in building a system, much more workhas to be done in order to attain the same intensity.So, in designing any specific system, a compromisemust be made between the focal intensity and theamount of apertures to be used.

VIII. Application and Further Development

From the above discussion, it can be concluded thatthe proposed system has definite focusing properties.Future applications may include the following.

(1) The characteristics of an iris beam waveguidemay be improved by having the equally spaced aper-tures9 replaced by those constructed according to theabove principle.

(2) Work being carried on in this laboratory withan ellipsoidal system built into a Fabry-Perot resonatorindicates that the diffraction loss is decreased, thus in-creasing the Q factor of the resonator.' 0

,(z)

0 5 10 15 20 X(A

Fig. 7. N = 5, M = 2, n = 2. A comparison of the effects pro-duced by diffracting apertures placed in different planes. Fullline is the transverse intensity distribution in principal focal plane

due to C-1 and Co; dotted line, that due to C0 and C+i.

Table . Experimental Results of Focal Intensitiesa of Sys-tems with N = 5 and Different Values of M; n s the Number

of Diffracting Apertures.

N/M 4.5 4 3.5 3 2

1 3.6 3.1 4.0 3.5 3.52 6.2 7.5 9.0 9.1 8.53 9.6 13.0 18.9 19.0 14.54 12.3 22.4 32.1 - -5 16.2 34.7 47.4 - -

6 - 50.1 - - -7 - 70.8 - - _8 - 89.1 - - -9 - 112 - - -

10 - 135 - - -11 -- 166 ---

a The intensities have been normalized with respect to free space propa-gation.

than 1. Nevertheless, it must be noted that for a givendimension of the system, namely, L and a are fixed, themaximum number of apertures that can be insertedis limited by the choice of zone width. With zone widthequal to 2, the axial spacings are more than twice thatof the corresponding spacings of zone width equal to 1.Thus, referring to Table I, in the same given space ofthe system M = 3, n = 3 (intensity = 19.0) five aper-tures could have been inserted for the system M = 4giving an intensity of 34.7. On the other hand, if theapertures are too closely spaced, the situation cor-responds to that with zone width less than one. It isprobably true that focal intensity still increases with thenumber of apertures, but it has a value much lower

Fig. 8. N = 5, M = 2, n = 3. Theoretical axial intensity dis-tribution, normalized with respect to that at the principal focus.

Negative axial distance means towards the system.

g10

12.0-

10.0-

8.0-

6.0-

4.0-

OS 1.0 I.s 2.0 25 3.0 ZONE-WITM

(M-M )

Fig. 9. N = 5. Effect of the axial spacing on the focal intensity.Focal intensities were measured always with two aperturesplaced in position: the C0 = 20X fixed in the central planeof the system, and the C+1: at various axial distances from C0 butalways of such a radius as to have its edge falling on the ellipsoidN = 5. Intensities were normalized with respect to free space

propagation.

March 1968 / Vol. 7, No. 3 / APPLIED OPTICS 487

1.0.

0.8

0.6

0.

0.2

on ] . . . . . bo.u

I.

(3) In a laser system, bored disks have been usedto help cool the system and confine the plasmas."If the diameters and spacings of the apertures are sodesigned as to have an appropriate ellipsoidal configura-tion, then, not only can they fulfill the above functions,but they can help increase the efficiency of the systemas well.

The authors are very grateful to Marcel Rouillard ofLaboratoire d'Optique et Hyperfr6quences for his helpin realizing the experimental work. Thanks are alsodue to Defence Research Board of Canada for support-ing this project financially under a grant.

References

1. A. Boivin, Torie et calcul des figures de diffractions derevolution (Les Presses de 'Universite Laval, Quebec, 1964).

2. F. Sobel, Electronic Design (March, 1962) (Sic-Ed.)3. A. Boivin, A. Dion, and H. P. Koenig, Can. J. Phys. 34,

166 (1956).

4. G. Toraldo Di Francia, Atti Fond. Giorgio Ronchi Contrib.Inst. Natl. Ottica 6, 3 (1951).

5. R. W. Wood, Phil. Mag. 45, 511 (1898).

6. J. L. Soret, Arch. Sci. Phys. Nature 52, 320 (1875).

7. R. Tremblay and AI. De, Appl. Phys. Letters 9, 136 (1966).8. M\. De, R. Tremblay, and J. W. Y. Lit, J. Opt. Soc. Am. 54,

1437 (1966).

9. A. G. Fox and T. Li, Bell System Tech. J. 40, 453 (1961).10. P. Gosselin, "Ensemble d'ouvertures circulaires dans un

r6sonateur Fabry-P6rot." Th6se de maitrise, Departementde Physique, Universit6 Laval, aoit 1967.

11. H. G. Hernqvist and J. R. Fendley, Jr., IEEE J. QuantumElectron. QE-3, 66 (1967).

Summer Course on Fundamentals of OpticsInstitute of Optics, University of RochesterJuly 1968

The program is an intensive two-week course designed for physicists and engineers inindustry and government, and for college teachers of physics and electrical engineering.About 53 hours of lectures, drawn primarilyfrom the Institute's regular academic program,have been selected to provide a useful foundation in certain optical areas of practicalinterest. The lecturers are drawn from the Institute of Optics'faculty: P. W. Baumeister,M. P. Givens, M. Hercher, R. E. Hopkins, W. L. Hyde, J. C. Heurtley, and P. J. Sands;

and from elsewhere: A. F. Turner, Bausch & Lomb, J. W. Goodman, Stanford, and J. F.Hamilton, Eastman Kodak.Lectures will treat coherence (1 h), polarized light (4 h), Gaussian optics, instruments,and system layout (8 h), lens design (5 h), image structure and transfer (5 h), lasers (4 h),interferometry and Fourier methods in spectroscopy (4 h), modulators and deflectors (2 h),multilayer filters (4 h), holograms (3 h), Hamiltonian optics (3 h), modern spectrographicmethods (2 h), Fabry-Perot interferometers (2 h), and photometry, radiometry, and detectors(6h).Academic level is that of an advanced undergraduate course, but with a reduced empha-sis on formal proofs and an increased emphasis on applications. Participants are as-sumed to have the equivalent of a bachelor's degree in physics or engineering. Tuitionfor the course, including an extensive set of notes and two books,will be $400 ($425 forapplications received after 1 June). Living and dining facilities will be available on cam-pus. Tuition remission, stipends and travel allowances will be provided for a maximumof 20 college teachers of physics and electrical engineering; stipends and travel funds comefrom a National Science Foundation grant in support of the program. The deadline forapplications for financial aid is March 1968. The course carries no academic credit.Requests for further information and for enrolment applications should be directed toFundamentals of Optics, Institute of Optics, University of Rochester, Rochester, New York14267.

488 APPLIED OPTICS / Vol. 7, No. 3 / March 1968