Polychromatic axial behavior of axial apodizing andhyperresolving filters
Maria J. Yzuel, Juan C. Escalera, and Juan Campos
The polychromatic behavior in defocused planes produced by different types of axial apodizing and hyperre-solving filter is studied. The image quality is determined by comparing the illuminance and chromaticitydistributions of the aberration free system with different filters and without a filter. The comparison is donein equivalent planes according to the normalized illuminance. Key words: Polychromatic image, apodiza-tion, hyperresolution, depth of focus.
1. Introduction
The use of nonuniform transmission filters to pro-duce apodization" 2 or hyperresolution2 3 on the pointspread function either with monochromatic or poly-chromatic4 illumination is well known.
More recently, the effects of nonuniform transmis-sion filters in the axial response have been studied inthe monochromatic case.5-1' In a recent paper weshowed results in the polychromatic case of filters thatproduce axial hyperresolution or apodization.12
In this paper we study the image quality in defo-cused planes of the aberration free system with differ-ent types of transmission filter. The distributions ofilluminance and chromaticity in different planes arestudied to evaluate the image quality. The distribu-tions of chromaticity and illuminance of the aberrationfree system without a filter in different planes aretaken into account to establish an image quality criter-ium.
11. Theory
The variation of color in the polychromatic pointspread function (PSF) with apodizing filters may beexpressed in terms of the tristimuli values1314:
X = f S(X)Gx(pP@xdX,
X
y = |" S(X)Gx(p,4,Yxd~XJI
(1)
Z= fr S(X)Gx(p,/)2dX.
where S(G) is the spectral distribution of the source (inour case illuminant C), G(p,4^) is the monochromaticirradiance for wavelength X at the point (p,iP) on theimage plane, jx, yx, and 2x are the three sensitivityfunctions of the human eye (CIE 1931) and X1,X1repre-sent the spectrum interval in which the receiver issensitive. The coordinates (p,') of the point in theimage plane are calculated from the change to polarcoordinates of the Hopkins reduced ones.15 For anaxial object point, it is enough to know the color distri-bution on the image along a radius owing to the revolu-tion symmetry. So At = 0 will be considered, and thepoint on the image will be given only by its radialcoordinate. The tristimulus value Y defined by Eq.(1) represents the illuminance, and the chromaticitycoordinates can be obtained from the tristimulus val-ues by
xx+ Y+Z
Yx+ Y+Z (2)
The monochromatic irradiance is obtained from thecomplex amplitude by
J. Campos is with University of Barcelona, Department of AppliedPhysics & Electronics, Diagonal 647, 08028 Barcelona, Spain; andthe other authors are with Autonomous University of Barcelona,Physics Department, 08193 Bellaterra, Barcelona, Spain.
Fig. 1. Aberration free system without afilter: (a) illuminance and (b) chromaticity
along the axis.
0.8
10.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
l E I | | | ~~~~(a)~
0 0.2 0.4 0.6 0.8 1
p
/tl||()
0 0.2 0.4 0.6 0.8
Fig. 2. Aberration free system without afilter: (a) illuminance and (b) chromaticity
in the W2 0 = 0 plane.
1.0
0.8
1 0.6
0.4
0.2
U.U0
1.0
0.8
0.6
0.4
0.2
0.0
0.2 0.4 0.6 0.8
0 0.2 0.4 0.6 0.8
Fig. 3. Aberration free system without afilter: (a) illuminance and (b) chromaticity
in the W20 = 0.14 plane.
f(r,,o) = {r(r,(o) exp[i2rW(r,p)] within Aoutside,
where (reo) are the coordinates of a point on the exitpupil, A is the pupil area, W(r,p) is the wave aberrationpolynomial, and r(r,so) is the amplitude function due tothe filter. With the axial object point and filter withrevolution symmetry they do not depend on so, and wemay write
W(r) =>E Wr m, (r) = A00 + A20r2 + A40r
4 + .... (6)
To evaluate integral (4), we use the Hopkins-Yzuelnumerical method modified by Calvo.15 The tristimu-lus values in Eq. (1) are determined by the Simpsonrule from monochromatic irradiances. The integra-tion interval is taken from 380 to 780 nm, and forty-onewavelengths equally distributed in this interval areconsidered to obtain good accuracy for the chromatic-ity coordinates on the whole image. Each diffractionpattern will be determined by the illuminance and thechromaticity coordinates (x,y) distribution. Two dif-ferent normalizations are used for the illuminance:
J S(X)Gx(p)YAdX
XN S(X)1 XdXA"~X 2
J S(X)G\(p)Y\dX
YNF =JA,
The first is normalized to the maximum illuminancefor the aberration free system without a filter, and thesecond is normalized to the maximum illuminance forthe aberration free system with the same filter. YN isuseful to compare different filters (all will have thesame normalization) and it shows the loss of illumi-nance due to the filter. YNF is useful to appreciate thecurves better.
Ill. Aberration Free System Without a Filter
First, we study the illuminance and chromaticitydistributions of the aberration free system (AFS) with-out a filter along the axis and in some transverse planesthat will be useful for making comparisons. In all thefigures p (in the transverse illuminance distributions)is the distance from the optical axis normalized to thesystem aperture. W20 (in the axial illuminance distri-butions) is the defocus coefficient. For further detailssee Ref. 13.
In Fig. 1 we show the illuminance and chromaticitydistributions along the axis. We have marked somepoints in the illuminance distribution and the corre-sponding points in the chromaticity curve. We ob-serve that there are rapid chromaticity variations nearthe minimum. For example, from bW20 = 0.48 to bW20= 0.56 there is a great change in the chromaticitycoordinates (x goes from 0.5 to 0.19). Moreover, itmay be observed that there are slow chromaticitychanges where illuminance is high. So, from bW20 = 0(YN = 1) to W20 = 0.24 (YN = 0.5) there is a slightchange in the chromaticity coordinates (x goes from0.26 to 0.3). The illuminance and chromaticity distri-
Fig. 4. Aberration free system without afilter: (a) illuminance and (b) chromaticity
in the 6W 2 0 = 0.24 plane.
kZ
i
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0 0.2 0.4 0.6 0.8
_ 0}2 0|4 0.6 0. b)
0 0.2 0.4 0.6 0.8 1
Fig. 5. Effect of the filter (r) = r2
: (a)illuminance (- - - -, YN; , YNF) and (b)
chromaticity in the W2 0 = 0 plane.
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
as \~~~~~~~~~~a
0 0.2 0.4 0.6 0.8
p
/''| ' | (
\(S\\,, j~~~~~~b_
0 0.2 0.4 0.6 0.8 1
Fig. 6. Effect of the filter r(r) = 1 - r2 : (a)illuminance (- - - -, YN; , YNF) and (b)
chromaticity in the SW20 = 0 plane.
butions are symmetrical in the planes before and afterthe paraxial pane (6W2 0 = 0).
In Fig. 2 we show the illuminance and chromaticitydistributions in the paraxial plane (8W2 0 = 0). In thiscase, the paraxial plane coincides with the best imageplane (BIP, the plane where illuminance is maximum).With all the filters that we use in this paper (ampli-tude-only filters) the paraxial plane is the BIP. Withother kinds of filter or if we study an aberrated system,the BIP would be different from the paraxial plane. Inthis figure it may be observed that there is a greatchromaticity variation near the first minimum (p =0.34) and the chromaticity changes slowly where illu-minance is high.
We now study some defocused planes to see howsome image quality parameters change when we moveaway from the paraxial plane.
In Fig. 3 we show the illuminance and chromaticitydistributions in a transverse plane where the maxi-mum illuminance is 0.8. This happens at 6W20 = 0.14.We see that there is a widening of the illuminancedistribution. Although the first minimum is still at p= 0.34, the illuminance is higher at this point (there islower illuminance at p = 0 and higher off-axis). So, theencircled illuminance (in a circle of radius p = 0.34)decreases as we move away from the paraxial plane.There is a reduction in the chromaticity distributiondue to the widening of the monochromatic intensities.
The next plane to be studied is at 6W20 = 0.24 (Fig.4). In this plane the maximum illuminance is 0.5 andthe illuminance distribution is wider. In this case as pincreases the illuminance decays monotonically and at
p = 0.34 there is no minimum. The chromaticity dis-tribution is reduced, although it has the same shape asin the 6W 2 0 = 0 and 6W 2 0 = 0.14 planes.
Thus, as we move away from the paraxial plane themaximum illuminance decreases, the illuminance dis-tribution is wider (the encircled illuminance de-creases), and the chromaticity distribution is reduced.
IV. Aberration Free System with Filters r(r) = r2 and(r) = -r2The use of filters that produce apodizationl' 2 or hy-
perresolution2 3 on the PSF is widely known. Theyalso produce effects along the axis, which we now in-vestigate.
As an example of (transverse) hyperresolving filtersin Fig. 5 we show the PSF of the aberration free systemwith the filter (r) = r. In Fig. 6 we show the PSF ofthe aberration free system with the filter r(r) = 1 - r ,which is an example of a (transverse) apodizing filter.We marked the position of the first minimum in theilluminance distribution and its corresponding chro-maticity coordinates for both filters.
The illuminance and chromaticity distributionsalong the axis are identical for filters r(r) = r2 and r(r)= 1 - r2 (for further details see Ref. 4). Thus, theseaxial distributions are represented in Fig. 7. It may beseen that there is a widening in the axial illuminancedistribution due to the filter, in comparison with thecase when no filter is used. This produces an increasein the depth of focus. To assess the increase in thedepth of focus we use the position of the plane in whichnormalized illuminance is equal to 0.5 (YNF = 0.5, see
Fig. 7. Effect of the filter 1(r) = r2 or T(r) =1 - r2: (a) illuminance (- - - -, YN;
YNF) and (b) chromaticity along the axis.
Table I). In this case the position of this plane (with afilter) is 8W20 = 0.26 and this position was W2 0 = 0.24(without a filter). There is also a strong reduction inthe chromaticity variation (in the white zone) alongthe axis. This also produces an increase in the depthof focus in chromaticity.
V. Axial Apodizing Filters
Different types of filter that increase the depth offocus have been studied.59"12 The case in which trans-mission is low in the center and in the outer part of thepupil and high in the intermediate zone is interesting ifthey are amplitude-only filters. There are many func-tions that obey these conditions. In this paper westudy two different functions that are especially sim-ple. Type I filters are those of the form r(r) = ar2 - ar4with maximum transmission at r = (1/2)1/2 = 0.7 and
Table 1. Some Image Quality Parameters for the Aberration Free Systemwith Different Filters
AFS 5W2 0 p (First minimum)system (YNF= 0.5) YN (max) (8W20 = 0)
Without a filter 0.24 1 0.34With filter (r) = 1- r2 0.26 0.25 0.46With filter (r) = 0.26 0.25 0.26With filter I-A 0.32 0.44 0.32With filter I-A 0.36 0.32 0.36With filter III-A 0.18 0.1 0.42With filter IV-A 0.21 0.18 0.3
Note: W20 defines the plane in which the illuminance YNF de-creases to 0.5. Maximum illuminance (YNmax). Radius (p) of thefirst minimum is in the best image plane.
0.8
0.6
0.4
0.2
0.00 0.2 0.4 0.6 0.8 I
Fig. 8. Transmission functions of the filters: 1, 1 - 6.75r2 + 13.5r4
zero transmission at r = 0 and r = 1. This kind of filterwas studied before in the monochromatic case7-9 whena = 4. We also study filters of the form r(r) = ar2 -
2ar4 + ar6 (type II filters) with zero transmission at r =Oandr = land maximum transmission atr = (1/3)1/2=0.57. We proposed this type of filter in a previouspaper12 and are now interested in studying its poly-chromatic behavior in defocused planes. The fact thatthese two kinds of filter have maximum transmissionin a different place (see Fig. 8) creates the differencesbetween them.
A. Type I Filters
As an example of type I filters we show the results fora = 4 [filter r(r) = 4r2 - 4r4, which we call filter I-A, seeFig. 8]. In this case maximum transmission is 1. For a> 4 the transmission [(r) = ar2 - ar4] in some placeswould be >1, and the function must be taken equal to 1in these places because they are passive filters [thetransmission function will be '(r) = ar2 - ar4 where7(r) < 1 and '(r) = 1, where r(r) 2 1]. With thesefilters (a > 4) maximum illuminance increases as doesa but the depth of focus decreases and the axial re-sponse approximates the AFS without a filter. For a <4 maximum illuminance decreases as does a. There-fore, the a = 4 case is especially interesting. In Fig. 9we see that in this case the axial illuminance is wider;YNF = 0.5 in the plane given by 8W20 = 0.24 for the AFSwithout a filter (see Table I). For the AFS with thisfilter YNF = 0.5 at 6 W20 = 0.32 (the filter increases thedepth of focus). It may also be observed that the axialilluminance distribution is apodized. For this reason,we call them axial apodizing filters. If we compare theaxial results for the AFS with a type I-A filter (a = 4)and with the filters r(r) = r or r(r) = 1-r 2 , somedifferences may be found. A type I-A filter apodizesthe illuminance distribution more than filter r(r) = r2or r(r) = 1 - r2 . Moreover, maximum illuminance ishigher for a type I filter (YN = 0.44) than for a T(r) = r2or r(r) = 1 - r2 filter (YN = 0.25). The axial chromatic-ity distribution is better for filter r(r) = r2 or r(r) = 1 -
(a) illuminance (- - - -, YN; , YNF) and(b) chromaticity along the axis.
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0 6
0.4
0.2
0.00 0.2 0.4 0.6 0.8
Fig. 10. Effect of the filter r(r) = 4r2 - 4r4 :(a) illuminance (- - - -, YN; , YNF) and
(b) chromaticity in the 6W 20 = 0 plane.
>
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
l | | ( | I I | | W I -I I j(a)
Ns
0 0.2 0.4 0.6 0.8 1
p
0 0.2 0.4 0.6 0.8 1
Fig. 11. Effect of the filter 1(r) = 4r 2- 4r4 :
(a) illuminance (- - - -, YN; , YNF) and(b) chromaticity in the 6W 20 = 0.18 plane.
r2 (it is reduced in the white zone), but for high illumin-ances (YNF> 0.5) it is also good for type I filters (in factthe axial chromaticity distribution is similar to theAFS without a filter). We may thus conclude that thetype I filter (a = 4) increases the depth of focus (eitherin illuminance or chromaticity) producing higher max-imum illuminance than other filters [r(r) = r2 , r(r) = 1- r2] and moreover it has a simple transmission func-tion.
It is interesting to study whether the PSF in thedefocused planes for the AFS with this filter is as goodas it was without a filter. Thus we study the PSF inthe planes where the maximum illuminance is YNF = 1(paraxial plane), YNF = 0.8, YNF = 0.5, and YNF = 0.2.Since it is difficult to establish image quality criteria indefocused planes, especially in chromaticity distribu-tions, we compare the illuminance and chromaticitydistributions of the AFS with and without filters in theplanes where they have equal YNF, and if they aresimilar we conclude that the image quality is good (inthe sense that it is similar to the AFS without a filter).
As mentioned above, these kinds of filter (transmis-sion-only filters) do not change the position of the bestimage plane if the system is without aberrations. So,the BIP is at W 2 0 = 0 (paraxial plane) and YNF = 1according to the normalization defined in Sec. II (seeFig. 10). The effect of this filter is slightly hyperre-solving, the first minimum is at p = 0.32 and without afilter it was at p = 0.34 (see Table I). Since it occurredin the case without a filter the chromaticity changesslowly where illuminance is high and rapidly where it islow. In fact, the illuminance and chromaticity distri-
butions in this case are similar to those of the AFSwithout filters.
In the 61W20 = 0.18 plane the maximum illuminanceis YNF = 0.8 (see Fig. 11). If comparison is made withthe AFS without filters in the plane where illuminanceis also YNF = 0.8 (Fig. 3), we observe that the illumi-nance and chromaticity distributions are similar. Theonly difference is that in the present case the radius ofthe first minimum is somewhat smaller and the sec-ondary maximum is slightly higher. Naturally themaximum illuminance YN is lower (due to the filter)but the plane where YNF = 0.8 is at 6W 2 0 = 0.18 and thisplane was at 5W 2 0 = 0.14 for the AFS without a filter.The results for the AFS with filter r(r) = 4r2 - 4r4 andwithout a filter in the planes where YNF = 0.5 (Figs. 12and 4, respectively) are also similar. But the YNF = 0.5plane is placed at 5W 2 0 = 0.32 when a filter is used and6W 20 = 0.24 when there is no filter. There is not only awidening in the axial illuminance distribution but alsoa good response in the defocused planes. Thus, thereis a real increase in the depth of focus. As an exampleof a plane where illuminance is low, in Fig. 13 we showthe results for the 6W 2 0 = 0.47 plane where illuminanceYNF = 0.2. In this plane the illuminance distributionis much wider, the chromaticity distribution muchmore reduced, and the image quality is poorer.
B. Type II Filters
As an interesting example of type II filters we showthe results for a = 6.75 [filter r(r) = 6.75r2 - 13.5r4 +6.75r 6 which we call filter II-A, see Fig. 8). If weconsider a > 6.75, the filters would have transmission
Fig. 12. Effect of the filter T(r) = 4r 2 - 4r4 :(a) illuminance (- - - -, YN; , YNF) and(b) chromaticity in the .W 2 0 = 0.32 plane.
Fig. 13. Effect of the filter r(r) = 4r 2 - 4r4 :(a) illuminance (- - - -, YN; , YNF) and(b) chromaticity in the W 2 0 = 0.47 plane.
Fig. 14. Effect of the filter r(r) = 6.75r2 -13.5r4 + 6.75r6: (a) illuminance (- - - -, YN;
, YNF) and (b) chromaticity along theaxis.
>1 and the transmission must be taken equal to 1 inthese places. As a increases, maximum illuminance ishigher but the depth of focus decreases and the axialresponse approximates to the AFS without a filter. Ifwe take a < 6.75, maximum illuminance decreases.
For filter II-A (see Fig. 14) the axial illuminancedistribution is wider than for filter I-A. Table I showsthat the position of the plane in which YNF = 0.5 was8W20 = 0.32 (filter I-A) and now it is r3W20 = 0.36 (filterII-A). Thus, the increase in depth of focus is higher forfilter II-A, but the maximum illuminance is lower inthis case. The axial chromaticity distribution withthis filter is significantly better (it is more reduced inthe white zone) than with filter I-A or without a filter.This is interesting but, at the points where illuminanceis high (for example, up to YNF = 0.5), the chromaticresponse is also good with filter I-A or without a filter.This reduction in the axial chromaticity distributiondoes not happen in systems with chromatic aberra-tions.12
In Fig. 15 we observe the PSF in the 6W 20 = 0 plane(BIP). The illuminance distribution is similar to theAFS but somewhat wider (radius of the first minimumis 0.36 now and it was 0.34 without a filter, see Table I)and with the secondary maximum a little higher. Thechromaticity distribution is similar to that of the AFSwith filter I-A and without a filter. In Figs. 16, 17, and18 we show the results in the W20 = 0.2, W20 = 0.36,and W2 0 = 0.52 planes in which maximum illumi-nance is YNF = 0.8, YNF = 0.5, and YNF = 0.2, respec-tively. As it occurred with filter I-A, when we moveaway from the paraxial plane maximum illuminance
1.0
0 8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
'X I''' ''I' a )~
0 0.2 0.4 0.6 0.8
p
' |r| ' ' '| ' ' | '' b)-A~~~~~~~~~~~
0 0.2 0.4 0.6 0.8
Fig. 15. Effect of the filter T(r) = 6.75r2-
13.5r4 + 6.75r6: (a) illuminance (- - - -, YN;, YNp) and (b) chromaticity in the 6W2o
13.5r4 + 6.75r6: (a) illuminance (- - - -, YN;, YNF) and (b) chromaticity in the 5W 20
= 0.2 plane.
l
Z
0.8
0.6
0.4
0.2
0.0C
1.0
0.8
0.6
0.4
0.2
0.0
i
0.8
0.6
0.4
0.2
0.00.2 0.4 0.6 0.8 1
1.0
0.8
0.6
0.4
0.2
0.00 0.2 0.4 0.6 0.8
Fig. 17. Effect of the filter (r) = 6.75r2-
13.5r4 + 6.75r6: (a) illuminance, (- ---,YN; , YNF) and (b) chromaticity in the
6W20 = 0.36 plane.
I....I
. . .. ... . . . . l
0 02 0.4 0.6 0.8 1
p
r | | || § If| |(b): -~.,- - I,, i
-~~~~~~~.. k1 ...0 0.2 0.4 0.6 0.8 1
Fig. 18. Effect of the filter (r) = 6.75r2 -13.5r 4 + 6.75r 6 : (a) illuminance(- - -- , YN; -, YNF) and (b) chromatic-
ity in the W20 = 0.52 plane.
decreases and there is more energy outside the princi-pal maximum. The chromaticity distributions are re-duced as we move away from the paraxial plane. But ifwe compare the illuminance and chromaticity distri-butions in the YNF = 0.8 and YNF = 0.5 planes whenusing filter II-A with the distributions for filter I-A andwithout a filter, we find that they are similar (theimage quality is good in this sense). As these planesare at a greater distance from the paraxial plane thanwhen there is no filter, it may be concluded that bothtypes of filter produce a real increase in the depth offocus (see Table I).
VI. Axial Hyperresolving Filters (Multifoci Filters)
In some cases it may be interesting to use filters thatproduce axial hyperresolution.9 -1 2 If we want to useamplitude-only filters, it is easy to find transmissionfunctions to produce hyperresolution if we observe thefilters discussed in Sec. V. Clearly, we must use filterswith opposite conditions. So, we study filters withminimum transmission in the intermediate zone of thepupil and the maxima at r = 0 and r = 1. We investi-gate filters of the form r-(r) = 1 - ar2 + ar4 (see Refs. 9and 12) with minimum transmission at r = (1/2)1/2 =0.7 and a transmission which is 1 at r = 0 and r = 1 (wecall them type III filters). We also study filters (seeRef. 12) of the form (r) = 1 - ar2 + 2ar 4 - ar6 (we callthem type IV filters). This type of filter produces anaxial hyperresolution (decrease in the depth of focus).It also produces an increase in the illuminance of theaxial secondary maxima so we may consider them mul-
tifoci filters. This kind of behavior was studied beforein the monochromatic case.11'16 We are especially in-terested in studying whether the PSF in the secondarymaximum is good. Once again we compare theseplanes with others of similar illuminance to obtain animage quality criterium.
A. Type IlIl Filters
As an example of type III filters we show the resultsfor a = 4 [filter r(r) = 1 - 4r2 + 4r4 that we call filter III-A, see Fig. 8]. For a < 4 the axial hyperresolution isless important (similar to the case without a filter) butwith higher maximum illuminance. For a > 4 mini-mum transmission would be negative in some placesand transmission must be taken equal to 0 in theseplaces [the transmission function will be r'(r) = 1 - ar2+ ar4 with r(r) > 0 and -r'(r) = 0 where r(r) < 0, becausethey are amplitude-only filters]. In this case as aincreases maximum illuminance decreases. We showthe results for a = 4 in which minimum transmission isequal to 0. In Fig. 19 we observe the axial illuminanceand chromaticity distributions. Table I shows thatthere is a decrease in the depth of focus due to thefilter. We also observe that the secondary maxima aremuch higher than in the previous cases. The illumi-nance coordinates are in the white zone at the pointswhere illuminance is high (for example, in the second-ary maxima). The maximum illuminance is not veryhigh because of the great loss of energy due to theintroduction of the filter. This might be a problem insystems in which the loss of light is not desired.
. . . . . . . . . ,I, ..I .0 I .0 l .0 I I . I I I ' ' �a')-_
.., A , _--,_4-_ .�.. �I
I
-r ' ' ' I .I ...I ..I ' ' r7 1 a):0- I I 1 1 1.0
0 8
0 6
0 4
0.2
0 0
1.0
0.8
0.6
0.4
0.2
0 0
- I -0 5 0 0.5C. Woo
L
1
08
0.6 .
04
0.2
0.0
1.0
0.8
0 6
0 4
0.2
0.00 0.2 0.4 0.6 0.8
Fig. 19. Effect of the filter r(r) = 1 - 4r2 +4r4: (a) illuminance (--- - -, YN;
YNF) and (b) chromaticity along the axis.
0 0.2 0.4 0.6 0.8 1
p
i,,, ,, . | || | | | (b)
e \]i-1s \ . IIII 0
0 0.2 0.4 0.6 0.8
Fig. 20. Effect of the filter r(r) = 1 - 4r2 +4r4: (a) illuminance (- - - -, YN;YNF) and (b) chromaticity in the W20 = 0
plane.
I.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0 0.2 0.4 0.6 0.8 1
p
l l l | ~~~(b)-
0 I I 0 0 0.8 10 0.2 0.4 0.6 0.8 1
Fig. 21. Effect of the filter r(r) = 1 - 4r2 +4r4 : (a) illuminance (--- - -, YN;YNF) and (b) chromaticity in the W20 = 0.1
plane.
In Fig. 20 we show the PSF in the 6W 20 = 0 plane.The principal maximum of the illuminance distribu-tion is somewhat wider than without a filter (see TableI) and the secondary maximum is a little higher. As inall cases the chromaticity coordinates are good (in thewhite zone) in places where illuminance is high. InFigs. 21 and 22 we show the illuminance and chroma-ticity distributions in the 6W20 = 0.1 and 3W20 = 0.18planes in which maximum illuminance is YNF = 0.8 andYNF = 0.5, respectively. We observe that as 6W20increases maximum illuminance decreases and theprincipal maximum is wider. The chromaticity distri-bution is reduced in the defocused planes. In Fig. 23we observe the PSF in the secondary maximum (6W20= 0.68). If we compare the distributions in the sec-ondary maximum with the plane where maximum illu-minance is similar (6W20 = 0.18, YNFmax = 0.5) it may beobserved that the illuminance distribution is better inthe secondary maximum (the principal maximum ismore compressed) and the chromaticity distribution issimilar. If we now compare the image in this second-ary maximum with the AFS without a filter in a similarplane (plane where illuminance is YNF = 0.5) (Fig. 4), itis better with the filter (the principal maximum in theilluminance distribution is more compressed). Wemay thus conclude that with this filter a good image (inthe sense that we have established) in the axial second-ary maxima (note the symmetry before and after 6W20= 0) is obtained. Thus, it produces a multifoci effect.
In Fig. 24 we observe the PSF at 6W2 0 = 0.35 in whichthere is the first minimum in the axial illuminance.
Obviously the image in this plane is very poor. Thereis minimum illuminance on-axis and a secondary max-imum off-axis, so the image here approximates a ringmore than a point.
B. Type IV Filters
As an example of type IV filters we show the resultsfor a = 6.75 [filter -r(r) = 1 - 6.75r2 + 13.5r4 - 6.75r6
which we call filter IV-A, see Fig. 8]. In this caseminimum transmission = 0. The discussion aboutwhat happens for a 5 6.75 is the same as in the preced-ing case (type III filters). In Fig. 25 we observe thatthe maximum illuminance is higher for filter IV-A thanfor filter III-A (the total transmission energy is higherfor filter IV-A). It may also be observed that the firstminimum in the axial illuminance distribution is nowhigher. As a result the chromaticity distribution isbetter in this case, since the chromaticity is better inthe places where illuminance is high. It is interestingto note that these differences and others that we showarise from the different positions of the minimum inthe transmission of the filters.
In Fig. 26 we observe the illuminance and chromatic-ity distributions at 6W20 = 0. The maximum illumi-nance is higher in this case and there is a hyperresolv-ing effect (see Table I). There is also higherilluminance in the third maximum than in the secondone. The chromaticity distribution is similar to thecase without a filter. Thus, the transverse behaviorseems to be better in this case than with filter III-A,but the axial response is slightly different. In Figs. 27
YN; -, YNF) and (b) chromaticity in the6W20 = 0.21 plane.
V0.4
0.2
0.0
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8 1
Fig. 29. Effect of the filter T(r) = 1 - 6.75r2
+ 13.5r4-6.75r6: (a) illuminance (- ---,YN; , YNF) and (b) chromaticity in the
5W 2 0 = 0.67 plane.
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.8
0.6
0.4
0.2
0.0
1.0
0 8
0.6
0.4
0.2
0.0
0.2 0.4 0.6 0.8
p
0 0.2 0.4 0.6 0.8 1
Fig. 30. Effect of the filter r(r) = 1-6.75r 2
+ 13.5r4-6.75r6 : (a) illuminance (- ---,YN; , YNF) and (b) chromaticity in the
6W20 = 0.39 plane.
and 28 we observe the PSF in the 5W 2 0 = 0.12 and 6W 2 0
= 0.21 planes in which maximum illuminance is YNF =
0.8 and YNF = 0.5, respectively. It may be observedthat, as we move away from the paraxial plane, maxi-mum illuminance decreases, the secondary maximaincrease, and the chromaticity distribution is reduced.In Fig. 29 we observe the PSF in the 6W20 = 0.67 planein which there is a secondary maximum in the axialilluminance. If we compare this with a similar plane(for example, the 6W 20 = 0.21 plane, in which maxi-mum illuminance is similar), it may be observed thatthe illuminance distribution is better in the 5W 20 =
0.67 plane because the secondary maximum is lowerand it is at a greater distance from the minimum in thiscase. Thus, as in the case of filter III-A, it may beconcluded that there is a good image in the axial sec-ondary maxima.
In Fig. 30 we show the results in the BW2 0 = 0.39plane in which there is the first minimum in the axialilluminance distribution. If we compare this with theequivalent plane for filter III-A (Fig. 23), we observethat the image is also poor although slightly different(a ring in the former case and a ring with a maximum inthe center in this case).
VII. Conclusions
The behavior of filters whose most important effectsare on-axis has been shown. First, we showed thepolychromatic effects of typical (transverse) apodizingor hyperresolving filters. These also produce an in-crease in the depth of focus.
Second, the effects of axial apodizing filters havebeen shown. These filters are almost neutral in atransverse section (see Table I). They produce a high-er depth of focus along the axis than the other filters(see Table I) with higher maximum illuminance. Westudied different defocused planes to show the goodimage quality in those planes for illuminance and chro-maticity. Thus, we may conclude that these axialapodizing filters produce a real increase in the depth offocus.
Third, we studied the effects of axial hyperresolvingfilters. They produce lower maximum illuminancethan the axial apodizing filters. Nevertheless, theyreduce the depth of focus (see Table I) and increase thesecondary maxima in the axial illuminance distribu-tion. We studied the image quality in defocusedplanes, especially in the secondary maxima. Wefound that the image in these secondary maxima iseven better than in equivalent planes near the princi-pal maximum. Thus, it may be concluded that thesefilters produce a multifoci effect.
For both axial hyperresolving and apodizing filtersthe position of the minimum (maximum) transmissionseems to play an important role in the effects producedby the filters.
So, we may change the axial response of an opticalsystem by introducing one of these filters withoutchanging other parts of the design.
This work was partially supported by the DireccionGeneral de Investigacion Cientifica y Tecnica (projectPB87-0779).
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