Spectral resolution of a zoneplate with circular sensor

Andre Fedotowsky and Kurt Lehovec

The energy intercepted by a circular sensor exposed to panchromatic light from a point source transmittedby a zoneplate lens is expressed as the convolution of the spectral source distribution with a wavenumberspread function. This wavenumber spread function, which represents the monochromatic energy encircledby the sensor as a function of wavenumber, depends on the product of the number of zones and the distancebetween sensor and focal plane and is invariant for a constant ratio of sensor radius to Airy radius. Resultsobtained by the diffraction integral for sensor radii larger than or comparable with the Airy radius are com-pared with approximate expressions based on geometrical optics and with the analytical expression for thelimiting case of sensor small compared with Airy radius. The dependence of the energy intercepted by thesensor on the halfwidth of a Lorentzian source distribution is described. Applications of the sensor-zone-plate lens combination to a scanning spectrometer and monochromator are discussed.

1. Introduction

The spectral distribution of a light source can bedetermined by using a sensor of radius smaller than theAiry disk radius, or a pinhole, and varying its positionalong the optical axis of a Fresnel zoneplate (FZP) or ofa point hologram." 2

In this paper we consider circular sensors having ar-bitrary diameter in conjunction with a zoneplate opticsof moderate f number, for which the vectorial nature ofthe light beam3 can be ignored. The energy interceptedby the sensor as a function of sensor radius and sensorposition is obtained as the convolution of the spectralsource distribution with a wavenumber spread function,defined in this paper. This function not only charac-terizes the change of sensor response with wavelengthat a fixed sensor position but also expresses the changeof sensor response with displacement of the sensor fromthe image plane in the case of monochromatic light.The wavenumber spread function depends on the sen-sor radius and zoneplate properties, such as diameterand number of zones. In the case of sensor radius smallvs Airy disk radius, the sensor responds substantiallyonly to the axial intensity, and an analytical expressionis available. For the case of sensor radius comparablewith the Airy disk radius, the exact diffraction opticalwavenumber spread function is computed and com-pared with an approximate expression based on geo-metrical optics. The case of a Lorentzian source dis-tribution is treated in detail. The spectral resolution

The authors are with University of Southern California, Los An-geles, California 90007.

Received 1 April 1976.

of the combination of a zoneplate lens with a sensor offinite radius is analyzed for use in a scanning spec-trometer and monochromator.

II. General Considerations

The focusing elements to be considered are FZPs andpoint holograms, or, in general, optical elements withcomplex filter function T periodic with the square of thenormalized radial coordinate r = p/a in the aperture ofradius a. Such a filter function can be expanded intothe sum

T(r) = E bn exp(-i27rnNr 2), (1)

with N the number of zones.4 The coefficients bn forFZPs are listed in Table I. The term exp(-i27rnNr 2)is the filter function of a perfect lens of focal length

Fn = a 2 k/(47rNn) (2)

for light of wavenumber k = 27r/X. The sensor locationwill be assumed to be in the vicinity of the primary focallength for which n = +1, i.e., at or near

F = a2k/(47rN). (3)

A circular sensor of radius a* positioned at the distanceR from the zoneplate intercepts the energy flux

ssR) = I biJ2S(k)E(,,R,k)dk, (4)

where S(k)dk is the energy flux of the source within therange dk incident on the zoneplate and E(o,R,k) is thefraction of monochromatic energy of wavenumber kwithin radius a* at distance R from the zoneplate lens.

The contributions of the secondary foci for which n5 1 have been neglected in Eq. (4) since they are small

582 APPLIED OPTICS / Vol. 16, No. 3 / March 1977

Table I. Coefficients b, of Eq. (1) for Transparent/Opaque and BinaryPhased Zoneplates

Type of Zone Plate n = 0 even n ( 0) odd n

Transparent/Opaque 0 O i/irn

Phased Zone Plate 0 0 2i/77n

in the vicinity of the primary focal plane n = 1.Moreover, they can be estimated and added to Eq. (4)by using the geometrical approximation discussed in theAppendix. The contributions of the secondary foci canbe suppressed by a centrally masked FZP."5

The intensity distribution generated by an opticalsystem is conveniently expressed in terms of the coor-dinates

y = a 2 k(R-1 -F-1)/2 (5)

and

z = akx/R, (6)

where x is the distance from the optical axis. From thisdistribution the normalized encircled energy flux E(z,y)within the radius x = o- shown in Fig. 1 can be obtained. 6

These data are listed also in Table II since they are notgiven in standard reference books7 for the planes y 0 outside the image plane.

The factor E (o-,R,k) in the integrand of Eq. (4) is theprofile of E(z6,y) along the trace obtained by eliminatingk from Eqs. (5) and (6). The focal length F of an ordi-nary lens is independent of k, and this trace is a straight

line through the origin of the y-z coordinate system.However, for a zoneplate lens the focal length is inproportion to k, Eq. (3); and the trace is

z, = 4rNa/a + 2y/z. (7)

The focusing condition y = 0 can be satisfied only forone wavenumber, k for which z = 47rNc/a. Thesensor is out of the focal plane for light of the otherwavenumbers.

Let

R0 = F(k 0 ) = a2 k0 /(47rN) (8)

be a reference position given by the focal plane of lighthaving a reference wavenumber ko. Introducing therelative deviation of the sensor position R from thatreference position

f = (R-Ro)/Ro (9)

and the relative deviation of the wavenumber from thereference wavenumber

= ( -k)/ko (10)

into Eqs. (5) and (6) with x = a, and using Eqs. (3) and(8), we obtain

y = 2rN( -R)1(1 + )= 2rN[(l + k)/(1 + ) - 11, (11)

andz = (47rNa/a)( + )/(1 + ). (12)

Fig. 1. Monochromatic energy flux within radius z at a distance yfrom focal plane normalized to full plane energy flux equal unity.Dotted lines obtained by the geometrical optics approximation dis-

cussed in the Appendix.

We shall now assume the reference position F(ko) ischosen at or close to the sensor position so f? I << 1. Itwill be shown shortly that significant values of E (za,y)arise only if JI -f 1 << 1, so we may assume also that Ik I<< 1. We may thus approximate z by the constantvalue

zf- 4rNa/a (13)

and use

(14)

The energy flux incident on the sensor becomes, thus,the convolution

Esct a )= dis r b S'G( )E'(a,m d- ) y h

of the spectral distribution S'(ki), modified by the

March 1977 / Vol. 16, No. 3 / APPLIED OPTICS

(15)

ens

583

y _ 2rN ( - ).

Table II. Normalized Encircled Energy Flux (z,y) in %; Stairway Marks Boundaries of the GeometricalOptics Approximation'

(z/TT) 1/2

(y / Tr)

1 3/2 2 5/2 3 7/2 4 9/2 5 11/2

4 5. 6 82.6 85.0 90.6

37.0

91.4 93.6 93.9 95.1 95.3 96.1 96.2

69. 6 79. 0 88. 1 89. 9 92.8 93.4 94.8 95. 1 95. 9 96. 1

18.7 40.6 62.3 78.0 84. 1 89.7 91.4 93.6

4.35 15.7 39.6 58.1 71.8 81.9

0.157 5.04 19.4 34.3 53.0 67.1

86. 5 90. 7

76.8 84.2

1. 53 4.19 8.28 16.9 32.9 47.1 61.6 72. 5

2.07 4.29 5,83 10.1 18.2 28.5 43.4 55.8

0.793 2.59 6.11 9.22 11.8 17.5 27.6 38.2

94.2 95.3 95.6

92. 2 94. 1 94. 7

88.0 91.4 92.8

8 0. 0 8 5. 9 8 9. 1

67.4 76.2 82.3

51.3 62.1 71. 5

0.039 1.27 4,87 7.92 10.6 14.0 18.3 25.1 35.7 45.9 57.4

0.474 1. 36 2. 83 5. 33 9. 56 12. 8 1 5. 1 1 9. 0 24. 7 32.2 42. 6

0.747 1.54 2.07 3.76 7.12 10.4 13.9 17.1 19.6 24.1 30.8

0.320 1.02 2.35 3.67 5.00 7.50 11.7 15.1 17.8 20.9 24.10.017 0.56 2.17 3.46 4.45 6.17 8.8 11.9 15.9 19.1 21.5

factor lb 1 12, with the wavenumber spread function,E(a,k - R) = E(zy). The wavenumber spread func-tion, which provides the encircled energy flux for apanchromatic point source, is analogous to the pointspread function, which provides the image intensity fora monochromatic spatially extended source.

The energy flux intercepted by the sensor does notdepend on the arbitrary choice of reference wavenumberk 0 and reference plane F(ko ). The implicit dependenceof Es'(af) on k 0 through fR is compensated by an ex-plicit dependence on k [not indicated in Eq. (15)],which can be proven as follows: replace E(a,R,k) in Eq.(4) by E(z,,y). According to Eqs. (11) and (12), thecoordinates z and y depend on k , k, and R onlythrough the combination u = (1 + M)/(1 + ), whichvaries between 0 and for 0 S k < a. By changing theintegration variable in Eq. (4) from k to u, we haveS(k)dk = S[u(1 + f?)k 0] (1 + fl)kodu with (1 + R)k 0 =2NR/a 2 independent of k0, as seen from Eqs. (8) and(9).

Figure 2 shows as full curves the wavenumber spreadfunction calculated from Fig. 1 with the approximationsof Eqs. (13) and (14). The dependence of the wave-number spread function on N( - h) is invariant tochanges of N, a, or a, provided that Za, i.e., the productNa/a, remains constant. Since the Airy radius is A 0.3a/N, these systems are characterized by the sameratio aA' = a/aA. The broken lines in Fig. 2 representa geometrical approximation discussed in the Appendix.The dotted line for A' = 0.3 is the analytical expres-sion

E'(a,k) = 3.67 (A )2 sin2(1rNk)/(zrNk)2 (16)

I< Y?

b

-2 0

aCA 2

2

Fig. 2. Wavenumber spread function, i.e., the fraction of mono-chromatic energy flux intercepted by a circular sensor of radius a fora zoneplate of N zones as function of relative deviations of wave-number I and of sensor position k, from some reference values.

I I << 1 is assumed.

584 APPLIED OPTICS / Vol. 16, No. 3 / March 1977

0

1/2

1

3/2

2

5/2

3

7/2

4

9/2

5

11/2

6

IEIII

I

c'J

b,

Lu

0.3 -

I ---. 0 .04 -.06 .. 100 .02 .04 .06 .08 .10

b --

Fig. 3. Encircled energy for a sensor of radius A' expressed inmultiples of the Airy disk radius as function of halfwidth 2b for aLorentzian spectral distribution of a point source focused by a zo-neplate of N = 40 zones. Broken line represents the geometricaloptics approximation, and dotted line is the analytical expression for

A' <<1.

for the limit aA' << 1, in which case the sensor interceptssubstantially only the radiation along the optical axisz =0.

Figure 2 shows that significant values of the wave-number spread function are obtained only for N I - I<< 1; i.e., the encircled energy flux for a (small) sensordecreases rapidly with increasing displacement of sensorposition from the focal plane. This justifies the as-sumption I << 1 used to derive Eq. (13) from Eq. (12),since N >> 1, and provided that IR I << 1.

111. Lorentzian Source Distribution

The energy flux of the Lorentzian spectral distribu-tion

S'(ki) = 7r-lb/(b2 + k2), (17)

of half-power width 2b focused by a zoneplate of N =40 on sensors of CA' = 1 and 2, is plotted in Fig. 3. Thespectral distribution has been normalized so S+ S'( )di = 1. The integral limits become unimportant ifb << 1. We may thus replace =-1 by = -, not-withstanding the fact k = - has no physical signifi-cance. The coefficient b1 in Eq. (15) varies only littlewith k and can be considered as being constant forsources of narrow spectral range b << 1. For sensorssmall compared with the Airy radius, the interceptedenergy flux of Eq. (15) with Eqs. (16) and (17) becomesthe function

Ed/lb1 I 2 = 7.32 CA 12 [exp(--y) - 1 + y]/,y2 (18)

of y = 2i7rNb, which is shown in Fig. 3 assuming aA' =0.3. For b << 1 the significant ki values are small vsunity, and 1 + I 1. In the geometrical approximation(dashed lines in Fig. 3) the energy flux intercepted bythe sensor depends only on the combination a/ab (Ap-pendix).

IV. Spectral Resolution

A small sensor moving along the optical axis of a zo-neplate lens registers primarily the intensity of thewavelength corresponding to that image plane, whichcoincides with the momentary position of the sensor andmay thus serve as a spectrometer. A monochromatoris obtained when an aperture (pinhole) replaces thesensor.

The wavenumber spread function for the combina-tion of a zoneplate optics with a sensor of radius equalto or larger than the Airy radius has no null, and theRayleigh criterion is therefore not directly applicable.The definition of resolving power as the inverse of thehalf-intensity width a shown in Fig. 2 gives N/0.88 forthe sinc2 function, which is close to the resolving powerN obtained by the Rayleigh criterion. Moreover, thepeak-to-saddle intensity ratio 7r2/8 = 1.234 obtained bythe Rayleigh criterion for the sinc2 function is similarto that obtained from our definition of resolving power,which provides 1 + E'(a,I = a) for the peak-to-saddleratio, e.g., 1.20 for the Lorentzian wavenumber spreadfunction of the point hologram of Ref. 1 and 1.125 forthe geometrical optics approximation.

The resolving power of the wave spread function ofthe geometrical approximation discussed in the Ap-pendix is

a-1 = (a/o)/2,/2 = 1.16 NAIA'. (19)

As Fig. 2 shows, this approximation underestimates theactual half-intensity width and thus overstates the re-solving power. For instance, the exact curve of Fig. 2shows that a-' = 0.83 N for a ' = 1. Resolution im-proves with decreasing size of the sensor or pinhole; and,when the sensor radius becomes small compared withthe Airy radius CA' << 1, the resolving power reaches thelimiting values 1.11N pertaining to the sinc2 function.Keating et al.' and Childers and Stone2 have discussedthe resolving power for this special case.

Equation (19) shows that the resolving power be-comes independent of the number of zones for sensorradii large compared with the Airy radius, being afunction only of the ratio of the aperture and sensorradii. With decreasing sensor size, the resolving powerincreases; but the energy flux on the sensor decreases.For k = = 0, the energy flux on the sensor is the curve(z,0) shown along the axis y = 0 of Fig. 1. For CA' =

1, the intensity halfwidth of E'(aj) in Fig. 2 is 1.2/N,and E(zjTO) = 0.83. Decreasing CA' to 0.75 reducese(z,0) only by about 2%, since we are at the flat portionof the curve. Thus aA' values of the 0.5-0.75 rangeappear to be a desirable compromise for good resolutionat reasonable flux levels.

In the limiting case of a/aA << 1, the sensor respondssubstantially to the axial field, which is proportionalto

T(r) exp(iyr2)dr2 = fo T(v'r') exp(iyr')dr', (20)

with T(r) the filter function of the aperture. Using therelation T'(r) T(Vr), it is seen that E'(a,k) is pro-

March 1977 / Vol. 16, No. 3 / APPLIED OPTICS 585

portional to the squared modulus of the Fourier transform of the modified filter function T', which is the sinfunction for a clear aperture. The Fourier transformrelation between the wavenumber spread function anthe filter function suggests the use of spatial filterintechniques such as those used in conventional imaginapplications. In particular, the side lobes could be reduced by apodization to enhance detection of a fainspectral line adjacent to a strong one. Apodizatio:should be particularly useful for suppressing the atpreciable side lobes produced by apertures with a central stop. A filter can be designed for a given spectrumto produce a strong response at a prescribed axial pcsition.

This work was supported by the Joint ServiceElectronics Program through the Air Force Office cScientific Research.

Appendix: Sensor Response According toGeometrical Optics

According to geometrical optics the light from a poixsource is confined to a cone with the focal point as apeand the lens aperture as basis. Uniform light intensitis assumed within the cross-sectional circle of the conThe boundaries of the light cone expressed in tkcoordinates of Eqs. (5) and (6) are

z= ±2y, (Aand the normalized encircled energy is thus

EG(z,y) = (z/2y) 2 forz 5 2lyI (A

and

Or, usingnotation,

eG(z,y) = 1 forz > 21y1. (A

Eqs. (11) and (12) with Ri = 0 to simplil

EG'(a,k) = (a)2[(1 + k)/fi]2 (A

for < il = -a/(a + a) and > = a(a - a), and

EC'/l0) = 1 (A

fork <Ein Fig. 2.

< 2. This function is shown as broken lineFor a << a the limits become

/1,2 = ta/a, (A

lC

nId

gg

itn

nI-

esof

and the half-intensity width of Eq. (A4) is 2/2a/a =1.22 aA'/A/2N. The intercepted energy flux of Eq. (15)becomes for the Lorentzian spectral source distributionof Eq. (17), using Eqs. (A4) and (A5),

E.(G)/lbI2 = (la) 2 7- 1 b j 5 dk(b2 + /12)/12

+ d/1 + ir ' +0,/a d+ a r(b2 +k2)k 2 J + a b2+ k2

= (2/7r) [1 + (/ab) 21 tan-((a/ab)

+ (2/7r)(a/ab) - (/ab) 2 . (A7)

According to geometrical optics the contribution bythe lens of focal length Fn = F/n to the field at the pri-mary focal point of a FZP is

(A8)

where the upper (lower) sign corresponds to a positive(negative) n. A negative n pertains to a divergent lens,while a positive n pertains to a convergent lens. Thefield intensity at the focal point F of the lens n = 1 isinfinity in the geometrical approximation.

The intensity resulting from the two lenses charac-terized by indices n -/ 1 and n = 1 is not only the su-

it perposition of the intensities V,- V,, VI VI of the separateex lenses but contains also the interference term (Vn VI +ty Vn V,), which oscillates in polarity and, therefore, av-e. erages to zero over a sufficiently large sensor area. Itshe contribution to the radiation flux received by the sensor

placed near F1 is less than VnVn, therefore, notwith-standing that I Vn << I V1 1. Moreover, Vn = 0 at the

*) optical axis in the focal plane where V1 has its maxi-mum, so the interference term Vn V1 + V VI vanishesthere.

,2)References1. P. N. Keating, R. K. Mueller, and T. Sawari, J. Opt. Soc. Am. 62,

3) 945 (1972).fy 2. H. M. Childers and D. E. Stone, Am. J. Phys. 37, 721 (1969).y 3. A. Boivin and E. Wolf, Phys. Rev. B 138, 1561 (1965).

4. One zone of an FZP corresponds to one opaque and one transparent4) subzone.

5. G. Elwert and J. V. Feitzinger, Optik 31, 600 (1970).6. G. Lansraux and G. Boivin, Can. J. Phys. 36, 1696 (1958).

5) 7. L. Levi, Ed. Handbook of Tables of Functions for Applied Optics(CRC Press, Cleveland, Ohio, 1974), p. 496.

Is 8. K. Lehovec and A. Fedotowsky, "Theoretical Studies of ZonePlates Monolithically Integrated with Sensors," Final Report, AirForce Cambridge Research Laboratory, Office of Aerospace Re-

6) search, Contract F19628-71-C-0241 (1972).

586 APPLIED OPTICS / Vol. 16, No. 3 / March 1977

Vn = bVAI(n 1F)