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Transfer functions of semitransparent and phase annuli
Volume 9. number 2
OPTICS COMMUNICATIONS
October 197"~
TRANSFER FUNCTIONS OF SEMITRANSPARENT AND PHASE ANNULI A.K. JAISWAL Instruments Research and Development Establishment, Dehra Dun, India
Received 9 July 1973 "llle transfer functions for semitransparent and phase annular apertures have been obtained. The performance of these aperture~ with regard to the imagery of extended objects has been discussed. The re ;olution properties of these apertures have been examined m some detail.
It is well known that by decreasing the width of the central maximum of the diffraction pattern of an aperture it is possible to improve tts resolving power and the depth of focus. One of the methods to achieve this is to introduce phase.change coatings on a perfect lens it wa~ first shown by Wiikins [ 1 ] that by intrc lucin~, a phase change of rr on tile central region of the 1,-.~s with respecl to its outer annular region, it is possib ," to achieve a narrowlng of the central bright disc with the addiltonal advanlage that the central m a x i m u m mlensitv is the maxtmum possible l(~r a spectfied radms ~f the central disc. The intensity distribution in the diffraction patterns of semitransparent and phase annuli has also been considered by Thompson [2]. Lit [31, in a recent paper, has investigated the diffraction properties of a phase.zoned lens and has shown that the optimum performance is given by a centrally rr.phased lens. The pupils which give rise to narrower central spots in their diffraction patterns as compared to an uncoated pupil are generally known as super-revolving [4]. Their performance is superior in the sense that they will be able to resolve two equally b r i ~ t obiect points belier on the basis o," the Rayleigh criterion, ilowever, the narrowing of tile primary tnaxmmm ~s always accompanted by an enhancement of the secondary maxmla m the diffraction pattern which is known to be an undesir. able feature in the imagery of extended objects. The resolution also depends to a great exlent upon the objects being examined, in the present work, the tran,qer functions for semitransparent and phase annular apertures have been obtained. The performance of these apertures with regard to lhe imagery of extended ,~b-
jeers has been discussed. The resolving properties of these apertures have also been examined in some detail. The transfer flmction is the convolutioo of the pupil function with its complex conjugate. The normalised transfer function T ( ~ ) may therefore be expressed as +~
f f ax +
co,)') dxdv
T(oJ) . . . . . . .+ o o. .
,
(I)
f f I.ffx,y) 12dxcly where o0 is the normall,,,ed spatial flequency and J~x.y) is the complex atnplitllde distribution over the pupil. (x. y ) are the fractional coordinates over the pupil plane. Let us now consider a pupil in the form of an annulus with outer radius ! and imler radius e. The pupil funclion may be expressed as f i x . y ) = T = te i° . fl)r
x 2 + y2 ,( ¢2 ;
= T' = t ' e w
fore 2 K x 2 + v2 ~< 1"
=0
otherwise,
( 2}
where T and T' are file amplitude transmittances for the two component,; of tile amullus. The transfer function may be evaluated by ust,ag cqs. ( 1) and (2). Although the formal limits of integration in eq. ( i ) are infinite flae integrand is .,ero except over a region common to two circles corresponding to the two displaced pupils. The integrals in eq. ( 1 ) may be expressed in terms of the areas of file segments ~,f circles of radii 1 and e:
161
V o l u m e O, n u m b e r 2
Iio~) . . . . rr{t'2{l
OPTI('S C O M M U N I C A T I O N S
1 -
c2} + t2e 2}
r-
2t'{t'-
X 12t'21(!, to/2)i 1
+I
tcos,(O - 0 ' ) }
~o" -- 1 + e 2
t + 2{t- - _ i t
cos(O
O')+t'2}l(e, co/2e ,j ,
-
(3)
where
,~(r,x) = 0,
ifs~
I"
= r 2 {arccos(s) -- s \ / i --s2}, if-- 1 < s < 1; =l(r,--l),
ifs~<-
1.
(4)
It should be noted that in the expression for T(to), the phases 0 and 0' o f the two concentric regions appear only as the difference (0 - 0 ' ) .
1.0
~X \,
t=
\ \ ',\ \',
\V/,ooo
Oc o b e r 1973
Let us first consider a semitransparent aperture for which 0 = 0' and t' = 1. The results for varw, us values o f t are shown in fig. 1. The ease t = 0 c o r r e s p o n d s to a completely obscured aperture and the obscuration ratio in this case is taken to be e = 0.5. For t = 0.5 and t = 0.75 the obscuration ratio e = 0.5/x/q - t 2 has been chosen such thai the total light transmittances of the three apertures is the same. The curve for t = i corresponding to a circular aperture has also been given for comparison. It is found that in all the cases there is an improvement in the high.frequency response at a cost of the low-frequency response. In the case o:" t = 0.5, the high.frequency gain is comparable to that of the annular aperture, whereas the low-frequency response is comparalively much betler. We will now consider the transfer functions for pure phase annular apertures, i.e. for t = t' = 1. lu fig. 2 the transfer functions have been plotted t'ol t~ = 0 0' = O, rr/4.7r/2, 3rr/4 and rr fixing lhe value ol e at 0.5. in fig. 3 the curves have been drawn t'o~ a fixed value ¢~t" ~, = rr and for different values of e = 0.0, I).25, 0.50 and 0.75. The cases ~ = 0 in the fCmner and e = 0 in the latter correspond to a clear circular aperlure and have been given for the sake ~)t compaNson, l! is evident from the graphs thal ill :dl l]tc cases the rcsptm,,e l¢~r phase annuli is poorer as ctmlpared Io that of a circular aperlure. In fact it can be shown [51 quite ge~lctally by lhe help o f S c h w a r / l n e ~ l u a l i t v lllat the eflect ol lnire pll:twcoatlng.s ou a pct fccl lolls Is It) produce always a It.~:~s!11 the modulation transfer funclmt; at all frequencies, it is also seen from fig. 2 that the response gets worse with the increase in ~ and spuriaus resolution appears for = 3rr/4 and rr. For ~, = art/4 the first zero ,~t" T(to) occurs at ¢o = 0.77 whereas for O = rr the fir,~t zero occurs at a lower fiequency to = 0.57. All tl~e curves merge with the clear ap,:rture curve at to = 1.5, hei~ce there is no change in tl,,e e x t t e m e high-frequency ~esponse. Similarly it: fig. 3 for a rr-phase aperture, lhe curvc.s l',u e = 0.5 and e = 0.75 show spurmus resolution For c = 0..>, Ilco) vanlshe, at co = 0.57 whereas ft,r t = u.75 it vanishes at a highel frequency w = 0.77.1:¢~1 a rr-phasc filter with c = 0.707, ttle tw~ p~rtions of the a1|ilulus contain equal areas so lhat the amplitude di.~tribution in the diffraction pattern has a zero at thc centre 121. The behaviour ~d" the aperture leverses itself at e = 0.707 in the sense that an increase m c b e y ~ n d this value results in a broadening of the plimary maxtl,mm. We will now examine the questmn of ies¢flution by "~
4------
!
2.0 Fie 1 l-fan',let t u n c t l o n , ~ o f s c m l t r a n s p a r e n t a p e r t u r e , , l o r the t ~alucs ~hc ~ n
,
Volume 9, number 2
OPTICS ('OMMUNICATIONS
October 1973
1,0
1.0
0.00 O. 2 5
I
O5
~s
\\,\ ,..,
.................... ~Q--'-""~ " ~ 7
. . . . . . . . . . .
2,0
-O!
-o.!
0.5
Fig. 2. I ransler tuncllon,; oi phase annular aperture,; for c = and the values ~1 ~ as indicated.
Fig. 3. 1 ransfer funclmns of phase annular apertures for v = rr, for the e values shown.
semitransparent and phase.coated apertures. The absolulte resolving limit is given by the frequency at which the OTF falls t o zero value. This is same for ;ill the cases co1:sitlered above and is given by the normalised frequency co = 2. On the basis of a finite image contrast criterion, the frequency at which the. contrast in the image assumes that value would give us the value of resolving power. Since the magnilude of ()TF is a measure o f the conlras! in the image relalive to lhal In the object, the value of the aclt|al observable resolution may be legatded as the inlelsectitn~ between the OTF curve and the curve representing the mim~num modulation whtch can be detected by the delector. This oilier!on will correspond to tile conventional tests based on lhe bat or anni,lar resolution targets, fllough as pojnted out by ('hamlan I~1. there will be somc quantitative disagreements. According I,I thic or!tel!on, for a system to
have better resolving power its high frequency response sl,ould be larger. This being not the case with phasecoated apertures [cf. figs. 2 and 3] they will not be able t~ resolve any betler as compared to an uncoated aperture, ! i fact in those cases where tile transfer function has a negative lobe the actual resolution will be much less. As menlicmed before, these pupils do have a superior lwo-poi~lt resoluliotl. In general, if the narrowing of the prunaly maximum is achieved by an amplitude va~allon ~v,'r lhe pupil, lhe restdling Iransfer funclion will show an actual tmprovemen! in tile high frequetlcy respo,~sc Tllis is illustrated by the semitransparent and completely obscured apertures Ic.f. fig. I1 which will sh~)w be~ter resolution even when tested by conventional largets. On tile other hand, for apertures with pure phase valiatums, the higher frequencies ;ire exaggerated only as compared to tile lower ones. These t;:atures may be attributed to Ib3
X,,lumc c~, numl×-r 2
OPTICS CO~IMUNICATIONS
tile particular nonnalisation of the transfer function 171 . To conclude, the contrast m tile images of extended periodic objects witl be poorer for phase-coated objectives. A loss ill the contrast will also reduce the edgegradient and hence the sharpness in the images of extended objects. Moreover, these so called super-resolving pupds do not appear to have any superior resolution at all in the conventional sense.
Octt ber 1973
References i I 1 J.E. Wttkms, Jr., J. Opt. Soc. Am. 4(t (1950) 222. [21 B.J. Thompson, J. Opt. Soc. Am. 55 (1965) 145. 131 I.W.Y. Lit, J. Opt. Soc. Am. 61 (1971) 297. [4] P. Jacquinot and B. Roizen-Dossier, Progress it. Optics, Ed. E. Wolf (North Holland Publishing Co., Am~teldam) V 11966) 201. 151 E.L. O'Neill, Introduction to Statistical Optics (Addison Wesley Publishing Co.. Inc., Reading, Mass; 1963). [6] W.N. Charman, Phot. Sci. Eng. 8 11964) 2",3. [7] E. Ingeistam and P. itjelmstrom, Optik 22 (1965) 188.
OPTICS COMMUNICATIONS
October 197"~
TRANSFER FUNCTIONS OF SEMITRANSPARENT AND PHASE ANNULI A.K. JAISWAL Instruments Research and Development Establishment, Dehra Dun, India
Received 9 July 1973 "llle transfer functions for semitransparent and phase annular apertures have been obtained. The performance of these aperture~ with regard to the imagery of extended objects has been discussed. The re ;olution properties of these apertures have been examined m some detail.
It is well known that by decreasing the width of the central maximum of the diffraction pattern of an aperture it is possible to improve tts resolving power and the depth of focus. One of the methods to achieve this is to introduce phase.change coatings on a perfect lens it wa~ first shown by Wiikins [ 1 ] that by intrc lucin~, a phase change of rr on tile central region of the 1,-.~s with respecl to its outer annular region, it is possib ," to achieve a narrowlng of the central bright disc with the addiltonal advanlage that the central m a x i m u m mlensitv is the maxtmum possible l(~r a spectfied radms ~f the central disc. The intensity distribution in the diffraction patterns of semitransparent and phase annuli has also been considered by Thompson [2]. Lit [31, in a recent paper, has investigated the diffraction properties of a phase.zoned lens and has shown that the optimum performance is given by a centrally rr.phased lens. The pupils which give rise to narrower central spots in their diffraction patterns as compared to an uncoated pupil are generally known as super-revolving [4]. Their performance is superior in the sense that they will be able to resolve two equally b r i ~ t obiect points belier on the basis o," the Rayleigh criterion, ilowever, the narrowing of tile primary tnaxmmm ~s always accompanted by an enhancement of the secondary maxmla m the diffraction pattern which is known to be an undesir. able feature in the imagery of extended objects. The resolution also depends to a great exlent upon the objects being examined, in the present work, the tran,qer functions for semitransparent and phase annular apertures have been obtained. The performance of these apertures with regard to lhe imagery of extended ,~b-
jeers has been discussed. The resolving properties of these apertures have also been examined in some detail. The transfer flmction is the convolutioo of the pupil function with its complex conjugate. The normalised transfer function T ( ~ ) may therefore be expressed as +~
f f ax +
co,)') dxdv
T(oJ) . . . . . . .+ o o. .
,
(I)
f f I.ffx,y) 12dxcly where o0 is the normall,,,ed spatial flequency and J~x.y) is the complex atnplitllde distribution over the pupil. (x. y ) are the fractional coordinates over the pupil plane. Let us now consider a pupil in the form of an annulus with outer radius ! and imler radius e. The pupil funclion may be expressed as f i x . y ) = T = te i° . fl)r
x 2 + y2 ,( ¢2 ;
= T' = t ' e w
fore 2 K x 2 + v2 ~< 1"
=0
otherwise,
( 2}
where T and T' are file amplitude transmittances for the two component,; of tile amullus. The transfer function may be evaluated by ust,ag cqs. ( 1) and (2). Although the formal limits of integration in eq. ( i ) are infinite flae integrand is .,ero except over a region common to two circles corresponding to the two displaced pupils. The integrals in eq. ( 1 ) may be expressed in terms of the areas of file segments ~,f circles of radii 1 and e:
161
V o l u m e O, n u m b e r 2
Iio~) . . . . rr{t'2{l
OPTI('S C O M M U N I C A T I O N S
1 -
c2} + t2e 2}
r-
2t'{t'-
X 12t'21(!, to/2)i 1
+I
tcos,(O - 0 ' ) }
~o" -- 1 + e 2
t + 2{t- - _ i t
cos(O
O')+t'2}l(e, co/2e ,j ,
-
(3)
where
,~(r,x) = 0,
ifs~
I"
= r 2 {arccos(s) -- s \ / i --s2}, if-- 1 < s < 1; =l(r,--l),
ifs~<-
1.
(4)
It should be noted that in the expression for T(to), the phases 0 and 0' o f the two concentric regions appear only as the difference (0 - 0 ' ) .
1.0
~X \,
t=
\ \ ',\ \',
\V/,ooo
Oc o b e r 1973
Let us first consider a semitransparent aperture for which 0 = 0' and t' = 1. The results for varw, us values o f t are shown in fig. 1. The ease t = 0 c o r r e s p o n d s to a completely obscured aperture and the obscuration ratio in this case is taken to be e = 0.5. For t = 0.5 and t = 0.75 the obscuration ratio e = 0.5/x/q - t 2 has been chosen such thai the total light transmittances of the three apertures is the same. The curve for t = i corresponding to a circular aperture has also been given for comparison. It is found that in all the cases there is an improvement in the high.frequency response at a cost of the low-frequency response. In the case o:" t = 0.5, the high.frequency gain is comparable to that of the annular aperture, whereas the low-frequency response is comparalively much betler. We will now consider the transfer functions for pure phase annular apertures, i.e. for t = t' = 1. lu fig. 2 the transfer functions have been plotted t'ol t~ = 0 0' = O, rr/4.7r/2, 3rr/4 and rr fixing lhe value ol e at 0.5. in fig. 3 the curves have been drawn t'o~ a fixed value ¢~t" ~, = rr and for different values of e = 0.0, I).25, 0.50 and 0.75. The cases ~ = 0 in the fCmner and e = 0 in the latter correspond to a clear circular aperlure and have been given for the sake ~)t compaNson, l! is evident from the graphs thal ill :dl l]tc cases the rcsptm,,e l¢~r phase annuli is poorer as ctmlpared Io that of a circular aperlure. In fact it can be shown [51 quite ge~lctally by lhe help o f S c h w a r / l n e ~ l u a l i t v lllat the eflect ol lnire pll:twcoatlng.s ou a pct fccl lolls Is It) produce always a It.~:~s!11 the modulation transfer funclmt; at all frequencies, it is also seen from fig. 2 that the response gets worse with the increase in ~ and spuriaus resolution appears for = 3rr/4 and rr. For ~, = art/4 the first zero ,~t" T(to) occurs at ¢o = 0.77 whereas for O = rr the fir,~t zero occurs at a lower fiequency to = 0.57. All tl~e curves merge with the clear ap,:rture curve at to = 1.5, hei~ce there is no change in tl,,e e x t t e m e high-frequency ~esponse. Similarly it: fig. 3 for a rr-phase aperture, lhe curvc.s l',u e = 0.5 and e = 0.75 show spurmus resolution For c = 0..>, Ilco) vanlshe, at co = 0.57 whereas ft,r t = u.75 it vanishes at a highel frequency w = 0.77.1:¢~1 a rr-phasc filter with c = 0.707, ttle tw~ p~rtions of the a1|ilulus contain equal areas so lhat the amplitude di.~tribution in the diffraction pattern has a zero at thc centre 121. The behaviour ~d" the aperture leverses itself at e = 0.707 in the sense that an increase m c b e y ~ n d this value results in a broadening of the plimary maxtl,mm. We will now examine the questmn of ies¢flution by "~
4------
!
2.0 Fie 1 l-fan',let t u n c t l o n , ~ o f s c m l t r a n s p a r e n t a p e r t u r e , , l o r the t ~alucs ~hc ~ n
,
Volume 9, number 2
OPTICS ('OMMUNICATIONS
October 1973
1,0
1.0
0.00 O. 2 5
I
O5
~s
\\,\ ,..,
.................... ~Q--'-""~ " ~ 7
. . . . . . . . . . .
2,0
-O!
-o.!
0.5
Fig. 2. I ransler tuncllon,; oi phase annular aperture,; for c = and the values ~1 ~ as indicated.
Fig. 3. 1 ransfer funclmns of phase annular apertures for v = rr, for the e values shown.
semitransparent and phase.coated apertures. The absolulte resolving limit is given by the frequency at which the OTF falls t o zero value. This is same for ;ill the cases co1:sitlered above and is given by the normalised frequency co = 2. On the basis of a finite image contrast criterion, the frequency at which the. contrast in the image assumes that value would give us the value of resolving power. Since the magnilude of ()TF is a measure o f the conlras! in the image relalive to lhal In the object, the value of the aclt|al observable resolution may be legatded as the inlelsectitn~ between the OTF curve and the curve representing the mim~num modulation whtch can be detected by the delector. This oilier!on will correspond to tile conventional tests based on lhe bat or anni,lar resolution targets, fllough as pojnted out by ('hamlan I~1. there will be somc quantitative disagreements. According I,I thic or!tel!on, for a system to
have better resolving power its high frequency response sl,ould be larger. This being not the case with phasecoated apertures [cf. figs. 2 and 3] they will not be able t~ resolve any betler as compared to an uncoated aperture, ! i fact in those cases where tile transfer function has a negative lobe the actual resolution will be much less. As menlicmed before, these pupils do have a superior lwo-poi~lt resoluliotl. In general, if the narrowing of the prunaly maximum is achieved by an amplitude va~allon ~v,'r lhe pupil, lhe restdling Iransfer funclion will show an actual tmprovemen! in tile high frequetlcy respo,~sc Tllis is illustrated by the semitransparent and completely obscured apertures Ic.f. fig. I1 which will sh~)w be~ter resolution even when tested by conventional largets. On tile other hand, for apertures with pure phase valiatums, the higher frequencies ;ire exaggerated only as compared to tile lower ones. These t;:atures may be attributed to Ib3
X,,lumc c~, numl×-r 2
OPTICS CO~IMUNICATIONS
tile particular nonnalisation of the transfer function 171 . To conclude, the contrast m tile images of extended periodic objects witl be poorer for phase-coated objectives. A loss ill the contrast will also reduce the edgegradient and hence the sharpness in the images of extended objects. Moreover, these so called super-resolving pupds do not appear to have any superior resolution at all in the conventional sense.
Octt ber 1973
References i I 1 J.E. Wttkms, Jr., J. Opt. Soc. Am. 4(t (1950) 222. [21 B.J. Thompson, J. Opt. Soc. Am. 55 (1965) 145. 131 I.W.Y. Lit, J. Opt. Soc. Am. 61 (1971) 297. [4] P. Jacquinot and B. Roizen-Dossier, Progress it. Optics, Ed. E. Wolf (North Holland Publishing Co., Am~teldam) V 11966) 201. 151 E.L. O'Neill, Introduction to Statistical Optics (Addison Wesley Publishing Co.. Inc., Reading, Mass; 1963). [6] W.N. Charman, Phot. Sci. Eng. 8 11964) 2",3. [7] E. Ingeistam and P. itjelmstrom, Optik 22 (1965) 188.