Optimal Filter Design for Annular Imaging

Andre Fedotowsky and Kurt Lehovec

Filters of the type exp(in O )sgn Jn(zr) with 0 the azimuthal angle and r the radial coordinate in the filterplane, are shown to maximize the energy content in a narrow annular image of radius z. with respect to in-cident energy. The simplest optimal filter, sgnJ0(zr), is well approximated by the binary circular phasegrating sgn cos(zr - r/4). The single lobe of the first order image of this filter contains 46% of the inci-dent energy within the half-width 0.4Xf/a, centered around the image radins NXf /(2a ), where N z/ - isthe number of filter sections.

1. Introduction

The purpose of this paper is the design of spatialfilters that maximize the ratio of the energy incidenton an annulus of specified radius and infinitely smallwidth in the image plane of a lens-filter combinationto the energy incident from a point source on the fil-ter. This optimal filter will be shown to be a phasefilter. An application for such a filter is the machin-ing of annuli or disks with an intense laser beam. Adisk of material is most efficiently removed from asuspended work piece by concentrating the incidentenergy of the laser cutting beam on the periphery ofthe disk. Recently, Engel et al.1 used the ring focusof a modulated zone plate to machine an annulus in achromium layer. Absorbing filters such as those ob-tained in many apodization problems2-5 would be un-suitable for this application.

The optimal filters considered in this paper are ofinterest also for the following application: certainSchottky barrier disk sensors, operating in the ava-lanche mode, are known6 to be most sensitive alongthe periphery of the disk-shaped Schottky barriercontact since the electric field peaks there. Focusingenergy on the sensor rim thus maximizes response.Other sensor types, such as lateral n-p-n or p-n-pphototransistors, junction- and insulated gate-fieldeffect transistors, have contour shaped sensitive re-gions (base layer of the phototransistor and channelof the field effect transistor), which are convenientlymade in the shape of an annulus separating emitterand collector, or source and drain, as the case may be.Although it is most efficient to focus the entire ener-gy on a point of the annular sensor, nonlinearity and

The authors are with the University of Southern California, LosAngeles, California 90007.

Received 14 November 1973.

saturation under intense radiation are reduced bydistributing the incident energy uniformly over theentire annular sensing region.

We shall show that the optimal phase filters forconcentrating coherent monochromatic radiation ho-mogeneously along a narrow annulus of radius z arethe Bessel phase filters exp(-in 0) sgnJn(zr). Thesimplest of these optimal filters, sgnJ0 (zr) is wellapproximated by the binary circular phase gratingsgn cos(zor - 7r/4), for which convenient analyticalexpressions for peak intensity and energy content ofthe annular image are available.7 These data areused for a comparison with the absorbing filter8

J0 (zr), which maximizes energy in the annulus of ra-dius z with respect to that transmitted through thefilter. 9

11. Optimal Filters of N-Fold Rotational Symmetry

The diffracted field V(z, ) in the focal plane of aperfect lens with circular aperture and using a filterof complex transmission F(r, ) is given by:

V(z, ) = Q Z V(z)exp(inp),

where

Vt,(z) = [F(r, 0)exp(ina)dti J(z~r)rdr,

and= ra2Vi/f .

The dimensionless coordinates

r = p/a and z = 2raR/xf

(1)

(2)

(3)

pertain to the filter and image planes, respectively(Fig. 1), with f the focal length, X the wavelength ofthe incident coherent radiation, and Vi the incidentfield. The flux incident on an annular sensor of radi-us Ro and narrow width AR is

December 1974 / Vol. 13, No. 12 / APPLIED OPTICS 2919

INCIDENT I AROPLANEMONOCHROMATIC |WAVE I

PHASEFILTER

LENS RING SENSORin Image Plane

Fig. 1. Physical arrangement of lens, phase filter, and ring sensorshowing the coordinates used.

+I

a)

+1

sgn cos(Zor/ 4 )A

b)l b) IC,

-l

c)

C

, (zOr)

zor

, sgn J,(zr)

JO(zor) sgn JO(zor)

D "I \2F 4 6 8 10

zor

E = RAR 02?TS22 | V"(Z0 ) 12

is maximized by maximizing the field-

Vn(z0 ) = 2 f G(r)J(zr)rdr. (7)

Figure 2 illustrates the choice of G(r) for maximiz-ing V(z 0). Different annular zones, r, of the lenscontribute to the intensity at z0 with opposite signs inaccordance with the polarity reversals of the Besselfunction J(zr). To maximize the contributions ofall zones, the filter G(r) must introduce a phase re-versal between zones of opposite polarities of J0 (zr).Since the filter is subject to the constraint I G(r)l< 1,the integral is maximized by choosing I G(r)l = 1, i.e.,the optimized field is the binary Bessel phase filtersgnJ.(z~r). The same type of argument applies to allBessel functions J(rz 0 ), i.e., the filter with N-foldazimuthal symmetry that maximizes the fieldstrength at z is

F(r, 0) = [sgnJ(zr)exp(- in)exp(io), (8)

where is any real constant. These filters have acontinuous azimuthal phase variation and a discon-tinuous radial phase variation with the discontinui-ties located at

rns = in,sl/zo (9)

where in,s is the s th zero of Jn-.Formulas for the diffraction pattern associated

with filters consisting of annular zones of constanttransmission have been given by Boivin.5 The dif-fraction pattern associated with the binary Besselphase filter sgnJo(zor) can be expressed as the sum ofAiry patterns of apertures with different radii:

zor

Fig. 2. (a) Contributions J0 (zr) from the annulus of radius r inthe lens aperture to the field at radius z0 in the focal plane. (b)Binary phase Bessel filter sgnJ0 (z~r) to optimize intensity at z0 byphase reversal. Also shown is the circular binary grating sgncos(zr - r/4). (c) Contribution [sgnJ0(zr)] X J0 (z~r) to thefield at z0 by combination of (a) and (b); dotted line-same for sgncos(zr - r/4) X J,(zr). Dashed area is proportional to the errorarising by replacing the Bessel filter sgnJ0 (zr) by the binary grat-

ing sgn(cos zr - ir/4).

2V

E = RARo Is V(Z", 0 2dO,

6Z0/7r -

Fig. 3. Maximum energy flux I Vn(z0 )l2 along the perimeter ofradius z0 for filters of the type sgnJ(zr) exp(-inO). I Vn(z,)l isthe maximum field strength obtainable at a radius zo for dif-

fraction patterns of circularly symmetric intensity.

ati .6

-

o.4

(4)

where zo/R = AzO/ ARO = 27ra/ Xf.Hereafter we shall consider filters of n- fold rota-

tional symmetry having transmission

F(r, 0) = G(r)exp(- MO), (5)

which produce a diffraction pattern of circularlysymmetric intensity Q2J Vn(Z)J 2. The energy flux

2920 APPLIED OPTICS / Vol. 13, No. 12 / December 1974

with errors of 0.02/s2 according to McMahon's expan-sion.1" The reduction in peak intensity of the annu-lar image thereby caused is estimated in the Appen-dix and is found to be negligible for practical pur-poses. The physical dimensions of the filter zoneboundaries of Eq. (13) are

P = (s - 1/4)Xf/2R 0

(s = 1,2 .. .). All zones have the same width,

A p = Xf/2R 0

(14)

(15)

except for the innermost zone, which has the radius3Ap/4. Equation (15) is the familiar relation be-tween the period of the grating and the first orderdiffraction angle a = Rf. Intensity distributionand energy content for the first image of this filter iswell described by superposition of the beams of anaxicon pair of phase difference -ir/2 as is shown in anaccompanying paper. The first order image intensi-ty distribution is characterized by the function

Fig. 4. Intensity distribution C1 2(2) of the first order image 'ofthe binary circular phase filter sgn cos(rz,, - r/4), which closelyapproximates the optimal filter sgnJ 0(zr) and energy content

Hj(2) within the range z0 + 2.

V1 (z) = 2(- 1)N + J 1(Z)/Z

12 (- 1)thOeNnuJb(eOrNoz/zf)(zO/z)ec t (1)

where the number N of filter sections is

N(z0) = 1 + max s, (11)

such that o's zo.Figure 3 shows z 0 1 Vo(Zo)l 2, which is proportional to

the energy flux E incident on a narrow circular regionof radius z in the image plane. Correspondingvalues for the filters of Eq. (8) with n = 1 and 2 arealso shown. At z = r, the filter n = 1 of angular de-pendence expiO produces a flux that is about twicethat of the circular filter n = 0 (Ref. 10). However,as z increases, the detected flux becomes indepen-dent of filter index n and of sensor radius z, ap-proaching the value 0.46 asymptotically. The rota-tionally symmetric filter n = 0 is easiest to construct,since it consists only of annular regions of alternatingphases and is fully satisfactory for any z larger thana few times the Airy disk radius.

Ill. Approximation of the Optimal Filter sgnJ0(zr) bya Circular Grating

The binary Bessel phase filter sgnJ0('zr) is wellapproximated by the binary circular grating

sinceF = sgn cos(zr - 7r),

jos 7T(s - 1/4),

(12)

VQ I = 16r-2 z,-0 C,2 (z)

shown in Fig. 4, where

z = -

and

Cl = /-J cos(zr)r' 12 dr

= -- [I sinz - (27Y)-/2. sin(z-r)r.4/2dr.

(16)

(17)

(18)

The second term on the right-hand side of Eq. (18) isrelated to the tabulated" Fresnel integral S(2).The peak intensity of Eq. (16) is

I V(o) 2 = 128(973z_)-1 = 0.46z,-0 . (19)

The fraction of the total energy H = E(a 2 ,IrVd 2

contained within the range z + is7

H(z-) = 78 _ C (-) 2dZ = 82 H(z (20)

and is plotted in Fig. 4. The total intensity in thefirst order image is 8/r 2. Forty-six percent of the in-cident flux is located within 2 = 0.47r (half-width ofmain lobe) -and 56% within +0.87r (full lobe).

IV. Comparison of the Optimal Filter sgnJ0(zr) withthe Absorbing Filter J(zr) I

Boivin8 has analyzed the absorbing filter J(zr),and Papoulis9 has shown that it maximizes the ratioof the flux incident on the annular sensor to the totalenergy transmitted through the filter. The peak in-tensity of the annular image generated. with the aidof the absorbing filter J(zr) (Ref. 9) is

I V(z0)12 = [J 0

2 (z 0) ± + J 12 (z0)]2 4(7WZ0)-

2 . (21)

(13) Comparison with the corresponding value, Eq. (19),

December 1974 / Vol. 13, No. 12 / APPLIED OPTICS 2921

U

0:Dco

rI-Cl)

U)Cn

zLUI-z1

1.0

.8

I-z.4 8

(5

LUr

Phase Filter

e P t L sI I I I I

lPa e l l l l l

Zone Plate Lens

il I11 1 l ll

I ntegrated_Phase Filter)Zone Plate Lens|

Fig. 5. Binary circular phas grating combined with a phased zoneplate into an integrated structure. Each surface step intro-

duces a phase shift 7r, i.e., a path difference of X/2.

for our nonabsorbing filter sgnJ0 (rz1,) shows that theabsorbing filter reduces the peak intensity by the fac-tor 97r(32zo)-l. The same is true for the annularflux, since the half-widths of the images generated byboth filters are comparable, Az _ r. The flux 2?rzi-Azj V(z1 )12 decreases as zo- in the case of the ab-sorbing filter but remains constant for our phase fil-ter. The loss in energy associated with the absorbingfilter is very significant for images of radius largecompared with the Airy disk radius; e.g., in the caseof an eight-section filter the peak intensity would bedecreased by the ratio of Eqs. (21) and (19), i.e., thefactor 4/(0.467r3N) = 1/28. Moreover, construction ofthe absorbing filter J0 (zor) is more difficult than thatof the binary phase filter sgnJ0 (zr).

V. Integration of Circular Gratings with Zone Plates

The ring images described in the previous sectionsare produced in the Fraunhofer plane, i.e., very farfrom the grating or else in the focal plane of the lens.A focusing means that is ideal for integration withthe circular phase grating is the zone plate (Zp).12

An extensive bibliography of zone plates can befound in Ref. 13. Several authors' 4"15 have con-structed phase ZP's using photolithographic meth-ods. The transmissions of a phase ZP, Tf, and of acircular grating, Tg, have binary character +1. TheZP and the circular grating can be integrated12 into astructure of transmission T = TfTg, which is again abinary phase function (Fig. 5). Illuminating thisstructure with a plane wave creates a ring image inthe principal focal plane at the radius

R = NXf/2a, (22)

where N is the number of filter sections. Engel andHerziger13 have described a modulated zone platethat is equivalent to a zone plate and filter simulatingan axicon transmission. The method of constructingthe zones was not discussed in detail, but from the re-

sults given it appears that the zone plate-filter com-binations are similar to those described here. Themodulated ZP of Ref. 13 consists of opaque andtransparent zones, which would reduce the intensityin the ring images by one fourth compared with thatof a phase zone plate of TfT. = +1.

Figure 6 compares the ring images of a ten-zone ZPand of a lens, respectively, each integrated with a six-zone phase grating. There is a close agreement aftertaking into account the ratio 4/7r2 between intensitiesproduced by phase zone plate and lens. The agree-ment improves with the number of zones in the ZP.

The radius of the first order annular image of agiven binary phase filter with zone boundaries suchas are described by Eq. (14) is proportional to thewavelength, if a lens with focal length independent ofwavelength is used. However, if a zone plate lens isused, the focal length is inversely proportional towavelength; and the ring images for different wave-lengths, now lying in different focal planes, have thesame radius. Images in the same plane and havingdifferent radii for different wavelengths can be gen-erated by a zone plate lens if higher order focalplanes are used and if the wavelengths in questionhave the appropriate ratiog.12 For instance, for X1/X2= 5/3, the third order (k = 3) focal plane for wave-

.03 ZONE PLATE (xr 2 /4)

LENS

.02.. '

.01 _ '

0 5 10 15 20Z -

Fig. 6. Comparison of the focal annular image of a six-zone bina-ry circular grating of Eq. (12) combined with a phased ten-zone plate (solid line) and an ideal lens (dotted line), respec-tively. Intensity pertaining to the zone plate is multiplied by72/4. To obtain the energy content in the annulus z I Az/2,

multiply I V(z) 2 by 2rzAz.

2922 APPLIED OPTICS / Vol. 13, No. 12 / December 1974

length X, and the fifth order (k = 5) focal plane forwavelength X2 generated by the same zone plate coin-cide. Radius and width of the kth order focal imageare reduced by k-1 compared with the image in thefirst order focal plane, but the peak intensities re-main the same. Therefore the energies in the lobesdecrease in proportion to k 2.

VI. Discussion

The general problem of maximizing the flux on asensor of prescribed spatial sensitivity distribution iscomplex: optimization is achieved by a phase filterobtained as a solution of nonlinear eigenvalue prob-lem.16 Circular symmetry in sensor and aperturedoes not necessarily imply rotational symmetry forthe optimum phase filter; e.g., a simple prism thatplaces the center of the Airy diffraction pattern onthe rim of the sensor is. more efficient by about 20%than the best rotationally symmetric filter if the radi-us R of the annular sensor is larger than several Airyradii. The same effect can, of course, be achievedwithout a prism by having the optical axis intersectthe off-axis ring-shaped sensor. However, the circu-'larly symmetric filter decreases the sensitivity to ra-dial misalignment compared with direct focusing on apoint of the annular sensor. The detected intensitydecreases as (Az)-' in the case of the circular filter,but as (z)- 2 in the case of direct focusing, where themisalignment Az is assumed to be larger than theAiry disk radius but smaller than the sensor radius.

This paper is concerned with maximizing the ener-gy flux homogeneously distributed along an annularsensor with respect to the energy incident on a lens-filter combination. It is believed that only rotation-ally symmetric filters of type G(r) exp(inO) providecircularly symmetric energy distribution. The opti-mal filter for maximizing the energy in a narrow an-nulus with respect to incident energy is a phase filterI G(r) = 1. Computation shows that for a sensor ra-dius R slightly larger than 3/ of the Airy radius, asuitable filter having this type of transmission func-tion with n #= 0 gives higher detection efficiency thanany circularly symmetric (i.e., n = 0) filter. How-ever, if R exceeds several Airy radii, the performanceof the optimal filter n id~ 0 is not appreciably differ-ent from the optimal circularly symmetric filter n =0, which is the circular binary Bessel phase filtersgnJ0 (zr). For practical purposes, this filter can bereplaced by the circular binary phase grating sgncos(zr - r/4), which concentrates 56% of the inci-dent energy within an annulus of width Az = 1.67r.

This work was supported by the Joint ServicesElectronics Program through the Air Force Office ofScientific Research under Contract F44620-71-C-0067.

Appendix: Error Due to Replacement of the Filter'sgnJ0(z~r) by sgn[cos(z,r) - 7rl4]

The peak field of the first image V(o) of Eq. (19) isdecreased by AV due to phase reversal of the contri-bution from the annuli

j ,S17 - s '7 i S/)where

- (s - 1/4)]zo -

[see shaded area in Fig. 2(c)]. Approximating

we obtainJ0,(U) C~,J1 0 ,3) (U - , )

AV - 2 E 2j' r~ 0 0,,,)drI

= - 2s e2j O's ,J 1(j0,S)I-

The absolute value has been used since any displace-ment of a zone boundary of the optimum filter sgnJo-(zr) decreases the peak intensity. Using McMa-hon's expansion,"

Es = 0. 02s 2Z01 ,

we obtain= - 8. 10-zo-2 (1.25 + 0.11 + 0.03 +. .

= - 1.1 x 10X3z o-,so

(AEI/E] = (2Av)/ V(o = - 3. lo-3 z0-3 2 ,

where Eq. (19) has been used.

References

1. A. Engel, J. Steffen, and G. Herziger, Appl. Opt. 13, 269(1974).

2. R. K. Luneburg, Mathematical Theory of Optics, Brown Uni-versity, Providence, R.I., lecture notes, 1964 (Univ. of Califor-nia Press, Berkeley, 1964).

3. G. Boivin and G. Lansra'ux, Can. J. Phys. 39, 158 (1961).4. 0. Slepian, J. Opt. Soc. Am. 55, 1110 (1965).5. A. Fedotowsky and G. Boivin, Q. Appl. Math. 30, 235 (1972).6. D. Shepherd, Jr., Air Force Cambridge Research Laboratories;

private communication.7. A. Fedotowsky and K. Lehovec, Appl. Opt. 13, 2638 (1974).8. A. Boivin, Theorie et Calcul des Figures de Diffraction de

Revolution (Gauthier Villars, Les Presses de 'Universit6Laval, Quebec, 1964).

9. A. Papoulis, J. Opt. Soc. Am. 57, 362 (1967).10. The simple binary filter sgnJ,(rzo)sgn(pr - 0) produces at z

or a flux that is about 80% of that produced by the filtersgnJi (rz0 ) expiO.

11. M. Abramovitz and L. A. Stegun, Eds., Handbook of Mathe-matical Functions, (National Bureau of Standards, Appl.Math. Series, Washington, D.C., (1964), p. 321.

12. K. Lehovec and A. Fedotowsky, Theoretical Studies of ZonePlates Monolithically Integrated with Sensors, Final Report,Air Force Cambridge Research Laboratory, Office of Aero-space Research, U.S. Air Force, Bedford, Mass. 01730, Con-tract F19628-71-C-0241 (1972).

13. A. Engel and G. Herzinger, Appl. Opt. 12,471 (1973).14. A. H. Firester, Appl. Opt. 12, 1698 (1973).15. L. D'Auria, J. A. Huignard, A. M. Roy, and E. Spitz, Opt.

Commun. 5, 232 (1973).16. A. Fedotowsky and K. Lehovec, "Filter Design for Maximizing

Sensor Response", Optik, in print.

December 1974 / Vol. 13, No. 12 / APPLIED OPTICS 2923