Zone plates with cells apodized by Legendre profiles
Jorge Ojeda-Castaneda, Pedro Andres, and Manuel Martinez-Corral
By apodizing the cells of a zone plate and changing the opening ratio, it is possible to shape the relative powerspectrum of its foci. We describe a novel procedure that leads to an analytical formula for shaping the focuspower spectrum by using apodizers expressible as the Legendre series; these act on cells of arbitrary openingratio. Our general result is used to design zone plates that have missing foci and to discuss a synthesisprocedure using apodizers with various opening ratios. Our applications can also be used for shaping thepower spectrum of 1-D gratings.
1. IntroductionMultiple images of an input picture are used in
microelectronics for mask generation, producing newpatterns in the textile industry and automatic recogni-tion by pyramidal image processing.1 Multiple in-plane impulse responses are generated by gratings2 3;while multiple on-axis impulse responses are createdby a Fabry-Perot interferometer 4 or using zoneplates.5 6
For some applications, it is convenient that the mul-tiple impulse responses have prespecified characteris-tics. For example, in microelectronics it is useful forthe multiple responses to have high focal depth. Theimpulse response is usually tailored in instrumentalspectroscopy and in imaging systems by apodization.7
Recently, some efforts have been addressed to increas-ing the focal depth or reducing the influence of spheri-cal aberration by apodization.81 2 However, exceptfor a few examples,1314 it seems that apodization hasnot been applied to shape multiple impulse responsesalong the optical axis.
The aim of this paper is to report a novel procedurethat gives an analytical formula for evaluating therelative peak energy of the multiple responses of zoneplates with an arbitrary opening ratio. The elemen-tary cells are expressible as a series of Legendre poly-nomials. This approach is illustrated by designingzone plates that do not produce certain foci.
In Sec. II, we use Bauer's formula15 for discussing therelative irradiances of an in-plane multiple impulse
response. In Sec. III, we extend the discussion to on-axis multiple impulse responses, using the quasiperio-dic approach of Lohmann and Paris.16 In Sec. IV, wedefine novel profiles for achieving zone plates withmissing foci. In Sec. V, we describe a profile synthesisprocedure that uses apodizers with different openingratios.
II. In-Plane Multiple Impulse ResponsesWe start by considering a grating with period d. In
each period, the grating transmittance is assumed to bezero for all the points inside a band whose width is (1 -s)d. The parameter s, where 0 < s • 1, is called herethe opening ratio of the grating. The complex ampli-tude transmittance of a grating with opening ratio scan be written in terms of a Fourier series as
F(x,s) = E Cm(s) exp(i2-rxm/d),M=-
(1)
wheresd/2
Cm(S) = (lid) j F(x,s) exp(-i27rxm/d)dx. (2)
Next we recognize that the kernel in the integraltransform in Eq. (2) can be written using Bauer's for-mula. Since
In Eqs. (3) and (4), Pn denotes the n-order Legendrepolynomial, and jn represents the spherical Besselfunction of the n-order, also known as the Bessel func-tion of the fractional order.
Note that the change of variable in Eq. (6) makes theintegral in Eq. (7) independent of the opening ratio s.Furthermore, Eq. (6) indicates that from a given apo-dizing function G(t), it is possible to generate a wholefamily of apodized gratings, F(x,s), which have thesame transmittance profile with a different openingratio, as is shown in Fig. 1. We point out the interest-ing fact that any pair of functions belonging to thesame family is related by a scale transformation, namely,
F(x,sl) = F(S2X/s 1,S2 )-
'6 _ a L Nxis)-1 - ~ 1 -1 1
Ad. 2T 2
Fig. 1. Generation of apodizing functions with the same transmit-tance profile but different opening ratio.
(8)
Since we are interested in apodization profiles thatare expressible as a series of Legendre polynomials,
(9)G(t) = 7 aqPq(t).q=O
By substituting Eq. (9) into Eq. (7) and taking intoaccount the orthogonal property of the Legendre poly-nomials, we obtain
(10)Cm(S) = S 7 (i)aqjq(7rms).
q=O
Note that as a particular case Eq. (10) contains thesquare groove grating, which is characterized by ao = 1and aq = 0 for q 5$ 0. Then Eq. (10) becomes
Cm(s) = sjo(irms) = s sin(rms)/(rms). (11)
From Eq. (10) it is now possible to evaluate therelative power spectrum of in-plane multiple impulseresponses, I Cm(S)l 2, for variable opening ratio s and forany apodizing function which can be expressed as aseries of Legendre polynomials. This treatment isextended next to zone plates.
Ill. On-Axial Multiple Impulse ResponsesThe complex amplitude transmittance of any zone
plate, with opening ratio s, can be written as
H(r2,s) = E h.(s) exp[i2grm(r/R)2], (12)
where the radial coordinate is r, the period of the zoneplate in r2 is R2, and
The above formulation is illustrated next with someexamples.
IV. Zone Plates with Missing FociWe consider first the trivial case of rectangular cells.
Next, we discuss apodization by the first-order Legen-dre polynomial, and later we propose a compoundLegendre apodizer.
A. Zero-Order Legendre RulingAs we indicate in Eq. (11), for this example we find
that
aq = q, G(t) = Po(t) = 1, (18)
and consequently
hm(s) = sjo(rms) = sin(rms)/rm. (19)
In Eq. (18) 60q denotes Kronecker's delta. The resultin Eq. (19) is the well-known formula for rectangularprofiles. This formula predicts that for an openingratio of one half, s = 0.5, the even orders vanish. Thistype of zone plate (or grating) is known in the optics
literature as a Fresnel-Soret plate (or Ronchi ruling).The above results are shown graphically in Fig. 2.
B. First-Order Legendre RulingIn this case the only coefficient different from zero is
a,, that is, aq = 1q or equivalently G(t) = Pi(t) = t;hence
hm(s) = (-is)j(lrms)
= (-is) [sin(rms)/(7rms)2- cos(rms)/(irms)]. (20)
As can be seen in Fig. 3, there are some values of s forwhich the coefficients hm, in Eq. (20), are eliminated.For example, the value of s can be chosen to satisfy thefollowing roots of jl:
irms = 7r(1.43) or rms = 7r(2.47). (21)
Note from Fig. 3 that the first on-axis diffraction ordercannot be canceled by using an apodizer proportionalto the first-order Legendre polynomial. However, byusing the first root in Eq. (21), the second focus can becanceled by setting s = 0.71. The third focus vanishesfor two different values of s. The fourth focus candisappear for s = 0.36 and so on. Any interestedreader can use the above procedure for eliminatingcertain foci with suitable values of s.
C. Compound Legendre RulingThe same procedure holds for other Legendre rul-
ings. In Fig. 4 we show the amplitude transmittanceprofile obtained combining the zero-order Legendrepolynomial and the second-order Legendre polynomi-al:
G(t) = (2/3)P(t) - (2/3)P2 (t) = 1 - t2. (22)
The focal power spectrum vs the opening ratio of thiskind of apodizer is
as shown in Fig. 5.Instead of discussing other Legendre apodizers, we
next outline a synthesis procedure which considers thepossibility of adding functions with the same profilebut with a different opening ratio.
V. Synthesis Procedure: Various Opening RatiosWe show now that it is possible to synthesize coeffi-
cients hm(s) by the weighted sum, and/or difference, ofcoefficients hm(SkS), where k = 1,2,3,. .. These latercoefficients are obtained from individual functionsJ(x,sk), which are members of the set of apodizersgenerated with the same apodizing function G(t).
In other words, we can start with a certain generat-ing apodizer G(t). Then one can obtain any memberof the family of apodizers J(x,s) having the same pro-file but different opening ratio, as expressed in Eq. (6),namely,
x = (sd/2)t, J(x,s) = G(t). (24)
It is valid to consider an apodizing profile that is the
OPENING RATIOtS
Fig. 2. Traditional method of focus elimination by changing theopening ratio of rectangular cells.
0.12
E
i
I0enw
a-U)
30a-
0.08
0.04
0.00 0.0 0.5 1.0
OPENING RATIOSs
Fig. 3. Focus elimination by changing the opening ratio of cellsapodized with the first-order Legendre polynomial, as in Fig. 1.
1.0
wC.
I-
-J
a-
0.5
0.0
-0.5-1.0 0.0 1.0
DIMENSIONLESS COORDINATE't
Fig. 4. Amplitude transmittance: (a) dotted line, the zero-orderLegendre polynomial; (b) dashed line, the second-order Legendrepolynomial; (c) solid line, the combination of (a) and (b) as in Eq.
Fig. 5. Focal power spectrum of the apodized zone plate in Fig. 4.
1.0
0.9
wz
I-I.-
z
0
0-
-J
0.3
0.1
0.0
-0.2
-1.0 0.0
DIMENSIONLESS COORDINATEst
Fig. 6. Pyramidlike apodizer (solid line) generated by adding withvarious weights three rectangular functions (discontinuous lines) of
different opening ratio.
weighted sum of several apodizers of equal profile butdifferent opening ratio, that is,
K
f(X) = E ekJ(XSk)- (25)k=1
We consider now that the resultant profile, f(x) inEq. (25), can be thought of as a new generating apo-dizer G'(t), namely,
K
G'(t) = f[(d/2)t] = E ekJ[(d12)tskI- (26)k=1
As in Eq. (24), we can generate a new family ofapodizers, J'(x,s), that have the same profile as thegenerating function G'(t) but with a variable openingratio. In this case the formula equivalent to that inEq. (24) is
K
x = (sdI2)t, J(x,s) = > ekJ(X/sSk)k=1
K
a ekJ(X,8k8), (27)k=1
-C
0
0-
0
0.08
0.04
0.00 'I _s .0.0 0.5 1.0
OPENING RATIOs
Fig. 7. Focal power spectrum of pyramidlike apodized zone plate inFig. 6.
As the formation of multiple impulse responses is alinear process in complex amplitude, coefficients h'm(s)for the new apodizers J'(x,s) are
K
h'm(s) = > ekhm(sks).
k=l
(28)
The result in Eq. (28) is remarkable, since it allows oneto calculate the coefficients h'm for the variable openingratio of a synthesized apodizer composed of a series ofapodizers with the same profile and variable openingratio. Next we consider some examples.
In Fig. 6 we show the synthesis of a discontinuousfunction, obtained by properly adding and subtractingrectangular functions (zero-order Legendre polynomi-als) with a different opening ratio. From the generat-ing apodizer, we have apodizers with a different open-ing ratio and the same profile; consequently, one is ableto shape the coefficients h',(s) as shown in Fig. 7.
The same procedure applies for synthesizing contin-uous profiles, as shown in Fig. 8, where we display apiecewise continuous apodizer that results from add-ing and subtracting the apodizer of Fig. 4 with a differ-ent opening ratio. The coefficients I h'm 2, for m = 1, 2,3, and 4, vs the opening ratio are displayed in Fig. 9.
The two examples indicate how to shape the focuspower spectrum by using novel apodizing profiles ob-tained from the same Legendre ruling with a differentopening ratio.
VI. ConclusionsWe describe a novel approach for evaluating analyti-
cally the relative power spectrum of the multiple im-pulse responses, which are generated by gratings orzone plates, if the cells of these diffraction elements areapodized, by functions expressible as a Legendre se-ries. Our formula considers explicitly the openingratio of the coll, and it allows us to synthesize theapodizing function by using Legendre polynomials of
Fig. 8. Continuous apodizing function (solid line) generated byadding with different weights three functions (discontinuous lines)
like that of Fig. 4 but with a different opening ratio.
0.024
m-l
. _ m:3
To ..b ....... As .0.016
a-
C3 0-0 -
0
0.0 0.5 ho
OPEN ING RAT I *s
Fig. 9. Focal power spectrum of the continuous apodizer in Fig. 8.
any degree and any opening ratio. We illustrate ourformula by designing apodizers, called Legendre rul-ings, that eliminate certain on-axis diffraction orders,and in this way we obtain zone plates that exhibitmissing foci. Finally, we propose a synthesis proce-dure for designing apodizers by adding the same trans-mittance profile with a different opening ratio.
One of us (J. O.-C.) gratefully acknowledges thefinancial support of the Direccion General de Investi-gacion Cientifica y Tecnica (Ministerio de Educacion yCiencia), Spain. The work was partially supported bythe Direccion General de Investigacion Cientifica yTecnica (grant PB87-0617), Ministerio de Educacion yCiencia, Spain.
Jorge Ojeda-Castaneda is on leave from the NationalInstitute of Astrophysics, Optics, and Electronics,Mexico.
References1. P. J. Burt, Multiresolution Image Processing and Analysis, A.
Rosenfeld, Ed. (Springer-Verlag, New York, 1984), Chap. 2.2. 0. Bryngdahl, "Image Formation Using Self-Imaging Tech-
niques," J. Opt. Soc. Am. 63, 416-419 (1973).3. B. J. Thompson, "Multiple Imaging by Diffraction Tech-
niques," Appl. Opt. 15, 312 (1976).4. G. Indebetouw, "Self-Imaging Through a Fabry-Perot Interfer-
ometer," Opt. Acta 30, 1463-1471 (1983).5. J. Ojeda-Castaneda, P. Andres, and E. Tepichin, "Spatial Filters
for Replicating Images," Opt. Lett. 11, 551-553 (1986).6. A. Davila and J. E. A. Landgrave, "Simultaneous Imaging of
Periodic Object Planes," Appl. Opt. 27, 174-180 (1988).7. P. Jaquinot and B. Roizen-Dossier, "Apodisation," Prog. Opt. 3,
31-184 (1964).8. J. Ojeda-Castaneda, P. Andres, and A. Diaz, "Objects that Ex-
hibit High Focal Depth," Opt. Lett. 11, 267-269 (1986).9. G. Indebetouw and H. X. Bai, "Imaging with Fresnel Zone Pupil
Masks: Extended Depth of Field," Appl. Opt. 23, 4299-4302(1984).
10. J. P. Mills and B. J. Thompson, "Effect of Aberrations andApodization on the Performance of Coherent Optical Systems.I. The Amplitude Impulse Response," J. Opt. Soc. Am. A 3,694-703 (1986).
11. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, "Bes-sel Annular Apodizers: Imaging Characteristics," Appl. Opt.26, 2770-2772 (1987).
12. J. Ojeda-Castaneda and L. R. Berriel-Valdos, "Arbitrarily HighFocal Depth with Finite Apertures," Opt. Lett. 13, 183-185(1988).
13. J. Ojeda-Castaneda, P. Andres, and A. Diaz, "Annular Apodizersfor Low Sensitivity to Defocus and to Spherical Aberration,"Opt. Lett. 11, 487-489 (1986).
14. J. Ojeda-Castaneda, E. Tepichin, and A. Pons, "Apodization ofAnnular Apertures: Strehl Ratio," Appl. Opt. 27, 5140-5145(1988).
15. M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathemat-ical Functions (Dover, New York, 1970), p. 440.
16. A. W. Lohmann and D. P. Paris, "Variable Fresnel Zone Plate,"Appl. Opt. 6, 1567-1570 (1967).
