In confocal microscopy, spherical aberration is introduced when one is focusing deep within the specimen.
This can be compensated for by altering the effective tube length at which the objective is operated. Thelimitations of this approach are investigated.
Introduction
Confocal microscopy allows three-dimensional im-ages to be formed from thick object structures.'However, as the microscope is focused into the speci-men, spherical aberration is introduced, degradingthe imaging performance. It has been shown that thepresence of extremely small amounts of sphericalaberration are sufficient to produce a substantialdegradation of the imaging performance far more inthe depth direction than in the transverse direction.'The spherical aberration can be compensated for byintroducing opposite-sign spherical aberration by al-tering the effective tube length at which the objectiveis operated. This can be achieved in practice byincorporating a correction lens into the optical sys-tem, the position of which can be altered to vary theamount of spherical aberration that must be compen-sated for.3 However, both focusing through a slab ofdielectric and altering tube length produce higherorders of spherical aberration. These in general donot cancel, so aberration balancing is not complete. Inthis paper we investigate the effect on the axialimaging performance of focusing through a layer ofmaterial whose refractive index varies only slightlyfrom that of the immersion medium. In the analysis,all higher-order spherical aberration terms are re-tained. Similarly, the effect on axial imaging perfor-mance of altering the tube length is investigated,again with higher orders of spherical aberrationincorporated. Finally, balancing of these two sourcesof aberration is studied, showing the limitations ofthe method brought about by the residual higher-order terms.
Aberration from a Single Source
We consider first the spherical aberration introducedby a slab of dielectric, of thickness t and refractiveindex n2, immersed into a lens in which the objectspace is filled with a medium of refractive index n.Then for a ray inclined to the axis at angles 0 and 02,
respectively, in the two media (Fig. 1), the phase errorintroduced by the slab is3
= kt(n 2 COS 02 - n cos 0), (1)
where k = 27r/rI. Assuming that the refractive index ofthe slab differs only slightly from that of the surround-ing medium,
n2 = n + An, (2)
then, by Snell's law,
n sin 0 = n2 sin 02, (3)
or, to first order in An,
Ancos 0= cos0 1 +-tan2 0). (4)
Substituting Eqs. (4) and (2) into Eq. (1), we obtain
4' = ktAn sec 0 (5)
or
4' =A sec 0, (6)
The authors are with the Department of Physical Optics, Schoolof Physics, University of Sydney, Sydney NSW 2006, Australia.
Fig. 1. Geometry of a ray passing through a slab of dielectric.
The second term represents merely defocus and aconstant phase term and hence may be neglected. Weare left with
4, = B tan2 0, (14)
with
B = - 1/2 kd2 (i).
spherical aberration, together with defocus and aconstant phase change.
Next, we consider the spherical aberration intro-duced by alteration of the tube length at which a lensis operated. Consider a lens that focuses perfectlyfrom an axial point L to an axial point D (Fig. 2). Thephase introduced by a thick lens for a ray crossing theprincipal planes a distance h from the axis is3
4'=k(l- ,,/2+h nd-n,2+h2). (8)
By differentiation we see that the phase changeproduced if 1/l and d are changed is
A4 = - k [12 A(•) - 1 3 (' - Ad nd Ad\L I h I + 12 \s -d2 + h2J
Introducing
If a perfect mirror is used as the object in a confocalmicroscope that satisfies the sine condition, the inten-sity varies with its axial position as4
I(z) = J exp[2i(4' + kz cos 0)]sin 0 cos Od| . (16)
Here z is the defocus distance, a the semiangularaperture of the object, and + the wave-front aberra-tion of the lens. This is a useful criterion for assessingthe axial imaging capabilities of the system and iseasily investigated both theoretically and experimen-tally.
For the case of imaging through a slab of dielectric,
(9) Iz) = I exp 2i (c + kzc)j cdc |,
with c = cos 0, and for a change in tube length
h = d tan 0, (10)
and the magnification
I = Md, (11)
we obtain
4 = - k [(M2d2 A ) A - nAd (1 - cos o. (12)
If the magnification M is large, the square root can beexpanded by the binomial theorem to give
4 = - kd2A (? tan 2 0 + knAd(1 - cos 0). (13)
I IIh
Fig. 2. Geometry of focusing by a lens.
I-DD'd
d'
IZ) f R exp [2i B (C- 1 + kzcl cdc (18)
The axial response has been computed for these twocases, and representative examples are shown in Fig.3 for a system with a = 60°. In each case as theaberration is increased, the profile moves sidewayswith respect to the origin in z, the maximum intensityis reduced, the central lobe becomes broader, thecurve becomes more asymmetric, the sidelobes on oneside become stronger, and the minima on that sidealso become less pronounced.
The profiles resulting from the two sources ofaberration are similar, and it is seen that a value A =60 is similar in effect to B = 20. These profiles aresimilar to those observed experimentally for changeof tube length' or focusing through a dielectric. Theeffects are summarized in Figs. 4 and 5, which showthe variation of the different properties for values ofthe coefficients A and B. The curves are all symmetri-cal about the axis, and the behavior with A and B issimilar.
It is interesting to compare these effects with theprofiles produced by primary spherical aberrationonly, for which
(a)Fig. 3. Axial response intensity for focusing through a slab of dielectric for (a) various values of the coefficientA and (b) a system used at anincorrect tube length specified by the coefficient B. Numerical aperture 0.866.
equivalent to that proposed previously for high-numerical-aperture systems. 2 A typical profile is illus-trated in Fig. 6(a). Now the response is markedlydifferent, being much more irregular in shape. This istrue even for smaller values of C. It should be noticedthat the profiles are still asymmetric because thesystem is assumed to be of high aperture, satisfying
1
0.8
0.6
0.4
0.2
-1 00 100
20
10
0
A
Fig. 4. Effect of the coefficient A on 1, the Strehl intensity; 2, thewidth of the response at half of the maximum height (half-width)measured in units of kz; 3, the strength of the first subsidiarymaximum relative to the peak intensity; 4, the strength of the firstminimum relative to the peak intensity.
the sine condition. For a system with uniform angu-lar illumination, rather than one satisfying the sinecondition, the expression becomes
I(z) = f L exp[2i (cc2 + kzc)] dc | (20)
and the corresponding profiles are shown in Fig. 6(b).The response is now symmetric and agrees withpreviously published results for the axial intensity fora lens with primary spherical aberration.
The difference between the plots in Fig. 6 and thoseof Fig. 3 leads us to the conclusion that higher ordersof spherical aberration are certainly not negligible inthe cases of imaging through a slab of dielectric or ofalteration of tube length and justifies the approachadopted in this paper.
The behavior can be to some extent explained byconsideration of the asymptotic evaluation of theintegrals. By the principle of stationary phase,' forvalues of A or B much greater than 1 the integrals canbe shown to be equal to the sum of three components,one from each of the limits of the integral and onefrom the stationary point of the phase variation. Byconsideration of the turning points of the phase
28 variation it is seen that the stationary point contrib-utes only in the range
24A < kz <A sec2 a (21)
20
or16
2B < kz < 2B sec' a,12
8
4
0
Fig. 5. Effect of the coefficient B on 1, the Strehl intensity; 2, thewidth of the response at half of the maximum height, (half-width)measured in units of kz; 3, the strength of the first subsidiarymaximum relative to the peak intensity; 4, the strength of the firstminimum relative to the peak intensity.
0.05
0.04
0.03
0.02
0.01
kz
(a)
0.05
0.04 -
0.03
0.02
0.01
60 90 120 150kz
(b)Fig. 6. Axial response intensity for primary spherical aberrationof strength given by the coefficient C: (a) for a system satisfying thesine condition with numerical aperture 0.866, (b) for a system ofuniform angular illumination.
(22)
the lower inequality describing the onset of theoscillation in the profile in each case. For the case ofprimary spherical aberration the corresponding rangeis
-2C cos a < kz < -2C. (23)
For the case when the profile is given by Eq. (20) theterm resulting from the stationary point is of con-stant strength throughout this range, explaining theflat top to the lower profile in Fig. 6(b). On the otherhand, this term decreases in strength through theappropriate range for the cases in Fig. 3.
Aberration Compensation
Both the introduction of a slab of dielectric and thealteration of tube length give rise to spherical aberra-tion, suggesting that the aberrations produced by theformer can be to some extent canceled by the latter, ifthey are of the appropriate sign. To illustrate thebehavior, Fig. 7 shows the effect of altering the valueof A for a fixed value of B (in this case B = 20). Asexpected, the maximum intensity (the Strehl inten-sity) is increased, the response is sharpened, thesidelobes become weaker, and the minima becomemore pronounced, until a value of A between - 70 and-80 is reached. After that, further increasing thenegative value of A causes the sidelobes to growquickly, until for A near -100 the response is poor.Figure 8 summarizes the effects. The behavior is nownot symmetric with A, and in addition the optimumcompensation depends on the criterion by which theperformance is judged. For example, the half-width ofthe central peak continues to become narrower, andthe Strehl intensity continues increasing for values ofA larger than that needed for the weakest sidelobes.Since it is possible to have a narrow central peak withstrong sidelobe levels, it seems that the best indicatorof performance is in fact the strength of the sidelobes.For B = 20, this optimum occurs for A = - 73.
The relationship for optimum compensation of Aand B according to this criterion is close to linearwithin the range studied, and optimum compensationoccurs at values nearA = -3.74B, although it shouldbe stressed that the optimum values are to someextent subjective.
A theoretical estimate of the optimum compensa-tion can be made in the following way: First, weexpand the wave-front aberrations2 in power series insin(0/2). We obtain
Fig. 8. Effect of the coefficient A, for B = 20, on 1, the Strehl
intensity; 2, the width of the response at half of the maximumheight (half-width) measured in units of kz; 3, the strength of thefirst subsidiary maximum relative to the peak intensity; 4, thestrength of the first minimum relative to the peak intensity.
0.12
0.1 - AB
0.08 -
0.060.040.02
-80 -60 -40kz
0.05
0.04 AB
0.03
0.02
0.01
-100 -80 -60kz
values of coefficient A for a fixed value of B = 20.
-20 0
-20
The advantage2 of expanding in terms of s is that thedefocus is represented as a single term in s2.
It can be argued that compensation is achievedwhen the fourth-order terms, representing primaryspherical aberration, are equal, so that A = -3B.However, there are still some residual higher-orderaberrations, so a better estimate results from balanc-ing the aberrations of orders up to secondary spheri-cal aberration. This occurs (Ref. 6, p. 467) when thecoefficients satisfy the conditions
8A+32B 4A+12B 2A+4B-2kz20 - -30 12
(27)
which yield A = -3.75B and kz = -2.35B. Theformer number agrees well with the computed values,but there is some discrepancy with the latter. This isnot too surprising, as the relationships in Eq. (27)strictly apply for a low-aperture system, in whichthere is no apodization resulting from satisfaction ofthe sine condition.
Figure 9 shows a series of responses for differentpairs of values A and B that approximately satisfy thecondition for optimum performance. The behavior issummarized in Fig. 10, with the performance becom-ing worse for larger aberrations, as expected. It
kz kzFig. 9. Axial response intensity for pairs of values of the coefficients A and B that approximately satisfy the condition for optimumperformance.
should be noted that the range of values ofA shown inFigs. 9 and 10 is much larger than in Figs. 3 and 4.Therefore, over a range of optical thickness mismatchup to - 50 wavelengths, it is possible to achieveacceptable depth imaging performance by altering the
0.8
0.6
0.4
0.2
0-80 -40 0
B40
Fig. 10. Effect of the coefficientB, together with a coefficientA -3.8B, on 1, the Strehl intensity; 2, the width of the response athalf of the maximum height (half-width) measured in units of kz; 3,the strength of the first subsidiary maximum relative to the peakintensity.
tube length at which the objective lens is used. Thiswould suggest that it is possible to focus 150 m intoa watery object with an oil immersion system, whilemaintaining good axial resolution, by using appropri-ate compensation. The compensation reduces thespherical aberration in the system and thus alsoimproves the transverse resolution as well as the axialresponse. However, the effects of the aberrations onthe transverse, rather than the axial, imaging perfor-mance are much less pronounced.
References1. C. J. R. Sheppard and C. J. Cogswell, "3-D image formation in
confocal microscopy," J. Microsc. (Oxford) 159, 179-194 (1990).2. C. J. R. Sheppard, "Aberrations in high aperture conventional
and confocal imaging systems," Appl. Opt. 27, 4782-4786(1986).
3. C. J. R. Sheppard and C. J. Cogswell, "Effects of aberratinglayers and tube length on confocal imaging properties," Optik(Stuttgart) 87, 34-38 (1991).
4. C. J. R. Sheppard and T. Wilson, "Effects of high angles ofconvergence on V(z) in the sacnning acoustic microscope,"Appl. Phys. Lett. 38, 858-859 (1981).
5. J. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, UK,1986), p. 91.
6. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon,Oxford, 1975), p. 467.
