Vol. 8, No. 1/January 1991/J. Opt. Soc. Am. A 33

Archaic focusing properties of Fresnel zone plates

Jing-Ao Sun and Anni Cai

Beijing University of Posts and Telecommunications, Internal Box 113, Beijing, China

Received November 17, 1989; accepted July 11, 1990

Results published previously [J. Opt. Soc. Am. 57, 610 (1967)] concerning the focusing properties of Fresnel zoneplates were obtained by implicitly invoking the Fraunhofer approximation. A comparison of these results withthose derived from Fresnel diffraction theory shows that the focusing properties of Fresnel zone plates predicted byFraunhofer diffraction theory are incorrect except for resolution defined by the Rayleigh criterion.

INTRODUCTION

A quantitative focusing analysis of Fresnel zone plates isgiven in another paper.' Waldman2 has pointed out thatthe Fraunhofer diffraction theory is not appropriate for pre-dicting the focusing properties of Fresnel zone plates. How-ever, we have found that the analysis in Ref. 1 was obtained,in fact, by invoking the Fraunhofer approximation. In thispaper we will first show how their analysis leads to an equiva-lent of the Fraunhofer approximation. Then we will com-pare the results in Ref. 1 with those obtained by using theFresnel diffraction theory.

COMMENTS ON REFERENCE 1

In Ref. 1 the authors claimed that the Kirchhoff diffractiontheory had been applied to the calculation of the complexamplitude of the diffracted light from a zone plate. Theircalculation is started by using the following integration [i.e.,the equation before Eq. (2) in Ref. 1]:

uc, = 2 Jr| uo exp(-i -AL )rdrdO, (1)f'r -7r/2A

where X is the wavelength of the incident light, P and P2

represent the multiple integral limits, which means that theintegration is carried out over the entire visible area of thezone plate. The coordinates are illustrated in Fig. 1.

According to Kirchhoff diffraction theory,3 the pre-cise form of the phase term in the above integrand shouldbe exp(j27rAC'/X). Replacing AC' with AL is a keystep in Ref. 1. We can write exp(j27rAC'/X) asexp(j27rAC/A)exp(-j2-7rAL/A) (see Fig. 1). Clearly, omittingexp(j27rAC/X) from the above integrand implies that thisterm is regarded as unity or constant over the entire aper-ture.

Unfortunately, this is too restrictive an approximation forthe Fresnel region. Since AC and AC' are both functions of rin the same order, the quadratic term r2 involved in AC',which represents the nature of the Fresnel diffraction, mustbe canceled out during the subtraction of AC' from AC.This can be immediately seen by following the procedure ofRef. 1: Expanding AC = (r2 + F2)/ 2 and AC' = (r2 + F2 + r0

2

- 2ror sin 0)1/2 in binomial series and retaining only the firsttwo terms, we find that

-AL = - _ r sin 0.2F F (2)

Note that, to be explicit, ,B in Ref. 1 is represented by ro/Fhere, where r is the off-axis distance in the focal plane and Fis the focal distance.

Now let us compare Eq. (1) with what was obtained by theFraunhofer approximation. In the Fraunhofer region, thediffracted light at a point r on the observation plane, whichis parallel to and a distance F apart from the zone plate, canbe calculated by4

° exp(j27rF/X) ( )jXF Jo fo

X exp 2r r - sin 0]rdrdO,Xe PL X 2Fj~ F (3)

where u(r), in our case, is equal to the complex amplitude Uoof the incident light over the transparent area of the zoneplate and zero elsewhere. In Eq. (3) we have chosen theobservation point along the x axis, which loses no general-ity, owing to the circular symmetry of the zone-plate diffrac-tion pattern.

The phase term in the integrand of Eq. (3) is the same asthat of Eq. (1). The physical dimension of uo defined in Ref.1 is unclear (there should be something that represents thewave unit on the right-hand side of the definition equation).For comparison we may assume that uo = Uo/XF. Thus Eqs.(1) and (3) are equal, except for the insignificant termexp(j27rF/X)/j, which is ignored in Eq. (1).

It is clear that replacing AC' with AL in the phase term ofEq. (1) leads to an equivalent of the Fraunhofer approxima-tion. Consequently, the result that they obtained [Eq. (3) ofRef. 1] consists of Bessel functions J,, which correspond tothe Fraunhofer diffraction patterns of circular rings.

A COMPARISON

The focal point of a zone plate always falls in the Fresnelregion. We now present an analysis that uses the Fresnelapproximation. The light diffracted by a zone plate can becalculated in the Fresnel region by4

0740-3232/91/010033-03$05.00 © 1991 Optical Society of America

J.-A. Sun and A. Cai

J.-A. Sun and A. Cai34 J. Opt. Soc. Am. A/Vol. 8, No. 1/January 1991

VAX-11/780 computer by using the Runge-Kutta tech-nique. Since the phase change of the integrand in terms ofeach transparent zone is within 7r, it does not cause problemsin the numerical integration. The calculation results areplotted with solid curves in Figs. 2-4 as functions of thenumber of zones. For easy comparison, the corresponding

y yo

Fig. 1. Coordinate system involved in Eq. (1).

0

*11.4

q

1.2

1.0

0.81 10 100

Number of zones (N)

Fig. 2. D6/(f) versus the number of zones for positive and negativeFresnel zone plates (P-FZP and N-FZP, respectively).

.8

-

00

:Z

a

0.03

0.02

0.01

0.001 10 100

Number of zones (N)

Fig. 3. Maximum value of the first sidelobe of the normalizedintensity distribution versus the number of zones for positive Fres-nel zone plates (P-FZP).

u(ro) = 2ir ex(j 2r F)exp( 7r ro2)

X u(r)rJ0( t;<F r expj AZ; r 2)dr. (4)

The resolution 6 (with the same definition made in Ref. 1)

as well as the maximum of the first sidolobe of the normal-ized intensity distribution are computed based on Eq. (4) forvarious zone plates. The calculation was performed on a

0.20

0.15

0.10

0.05

0.001 10 100

Number of zones (N)

Fig. 4. Maximum value of the first sidelobe of the normalizedintensity distribution versus the number of zones for negative Fres-nel zone plates (N-FZP).

Dr. /(XF)

(a)

z

¢

._

'2

E9

0 1 2 3 4

Dr. /(XF)

(b)

Fig. 5. Normalized intensity distributions at the focal plane: (a)positive Fresnel zone plate with N = 3, (b) negative Fresnel zoneplate with N = 2. The solid curves show the results computed byusing Eq. (6) of this paper, the dashed curves by using Eqs. (6) and(7) of Ref. 1.

x I xoA

rI -

F

I

Vol. 8, No. 1/January 1991/J. Opt. Soc. Am. A 35

results in Figs. 2, 3, and 4 of Ref. 1 are, respectively, plottedin Figs. 2, 3, and 4 here with dashed curves.

Comparing the solid curves with the dashed curves in Fig.2 shows that the zone-plate resolution, when it is defined as

case more significant discrepancies between the two analy-ses can be observed. For example, the complex light ampli-tudes at the focal points of a positive and a negative zoneplate are

[=[rUO(N + 1)/(2j)]exp(j27rF/X) for positive zones

[irU0 N/(2j)jexp(j27rF/X) for negative zones

in the Fraunhofer case [Eq. (3)] and

[ (N + 1) U0 exp(j2,xF/X) for positive zones

l-NUO exp(j27rF/X) for negative zones

(5)

(6)

the distance from the maximum to the first minimum of theimage response, predicted by the Fraunhofer and Fresnelapproximations are nearly the same. For example, the nor-malized resolution D3/(XF), where D is the diameter of thezone plate, equals 1.355 for N = 3 and 0.888 for N = 2 shownby the solid curves, and the corresponding results shown bythe dashed curves are 1.39 and 0.89, respectively. For largeN the discrepancy is even smaller.

The maximum values of the first sidelobe shown by solidcurves in Figs. 3 and 4 are higher than those shown bydashed curves, most notably for a small N. This is so, inpart, because the analysis based on Fraunhofer diffractionfails to predict the effects of the unfocused background light.A zone plate is capable of bringing only a certain amount ofincident light into the principal focal point.5 The unfocusedlight makes the minima of the intensity distribution non-zero.6 To examine this point, we computed the normalizedintensity distributions of a positive zone plate with N = 3and a negative zone plate with N = 2 based on Eq. (4) andplotted them as solid curves in Figs. 5(a) and 5(b), respec-tively. The abscissa in the figure is Dro/(XF). As expected,all minima are nonzero. For comparison, the same exam-ples are computed by using Eqs. (6) and (7) of Ref. 1, and theresults are plotted with dashed curves in Figs. 5(a) and 5(b).In this case all minima are zero.

These comparisons are made with the intensity distribu-tions normalized to one at the center. In the unnormalized

in the Fresnel case [Eq. (4)]. With normal monochromaticincidence, the diffracted light at the focal point of a positivezone plate has r phase difference with that of a negative zoneplate.7 Equation (5) shows that the analysis based on theFraunhofer approximation gives the wrong phase and over-estimates the amplitude by a factor of 7r/2.

ACKNOWLEDGMENT

This research was performed under the support of the Na-tional Natural Science Foundation of China, grant68972033. The authors thank the first unknown reviewerfor his or her valuable comments and suggestions.

REFERENCES

1. D. J. Stigliani, R. Mittra, and R. G. Semonin, "Resolving power ofa zone plate," J. Opt. Soc. Am. 57, 610-613 (1967).

2. G. S. Waldman, "Variations on the Fresnel zone plate," J. Opt.Soc. Am. 56, 215-218 (1966).

3. M. Born and E. Wolf, Principles of Optics (Pergamon, New York,1964), Chap. 8.

4. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, NewYork, 1968), pp. 57-62.

5. M. H. Horman, "Efficiencies of zone plates and phase zonesplates," Appl. Opt. 6, 2011-2013 (1967).

6. M. Bottema, "Fresnel zone-plate diffraction patterns," J. Opt.Soc. Am. 59, 1632-1638 (1969).

7. A. Sommerfeld, Optics (Academic, New York, 1954), Chap. 5.

J.-A. Sun and A. Cai