T. Wilson, Department of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, U.K. Tel: +(0)1865 27 3116; Fax: +(0)1865 27 3905; e-mail: email@example.com
In this short review, we present a self-contained discussion of the image formation properties of the fluorescent confocal microscope. The optical sectioning or depth discrimination property is discussed in detail and new analytic formulae are presented, which relate the optical sectioning strength to the wavelength, numerical aperture and pinhole aperture size in a particularly simple fashion.
Fluorescence labelling combined confocal microscopy has become a powerful tool in daily use in bioscience laboratories throughout the world (see, e.g. Wilson & Sheppard, 1984; Wilson, 1990; Pawley, 1995; Diaspro, 2001; Amos et al., 2011). The main reason for the popularity of these instruments is that the confocal microscope, unlike its conventional counterpart, has the ability to image well only detail that arises from the regions of the specimen which lie close to the focal plane. In other words, the confocal system (or, indeed, any other system possessing optical sectioning; see, e.g. Wilson, 2011) is one in which all detail present in the specimen, whether it be fine structure or coarse structure, is imaged with increasingly poor contrast as the microscope is defocused.
A common way to explain the physical basis by which the confocal microscope achieves optical sectioning or depth discrimination is shown in Figure 1, which shows that light emanating from out-of-focus planes is blocked by the limiting pinhole which therefore causes the photodetector to record a reduced image signal. As the pinhole becomes larger in size, the amount of light it can block decreases and the optical sectioning strength becomes weaker. In the limit when no pinhole is used, the optical sectioning property disappears and the imaging becomes equivalent to that of the conventional microscope. This explanation, although superficially attractive, does not allow us to adequately quantify the strength of the optical sectioning nor to discuss properly the differences between the image formation properties of the confocal and conventional instruments. There are a number of formulae describing the different behaviour of the two instruments available on the web as well in manufacturers’ literature. In many cases, these formulae are unattributed and, in other cases, misleading at best. In this self-contained short review, we will discuss the resolution improvement that may be achieved by using confocal instruments and introduce criteria by which the thickness of the optical section may be determined. New analytic formulae are presented which relate the optical sectioning strength to the wavelength, numerical aperture (NA) and pinhole aperture size in fluorescence confocal microscopy.
The image formation properties of both conventional and confocal microscopes may be described mathematically in two complementary ways and it is a matter of analytical convenience as to which method is used. In one approach, the specimen is described as a spatial distribution of fluophore, f(t1, w1), and the image may be thought of as the convolution of this function with an appropriate intensity point-spread function, H. Specifically, therefore, we may write
We have chosen to write the equation in terms of optical coordinates, t, w, rather than real lateral coordinates x, y, because optical coordinates naturally take into account the effects of wavelength and NA (Born & Wolf, 1975). The two coordinate systems are proportional to each other via
where k= 2π/λ, λ specifies the wavelength, n is the refractive index of the immersion medium and n sin α denotes the NA.
where h2(t, w) denotes the amplitude point-spread function of the objective lens evaluated at the fluorescence wavelength. For circularly symmetric pupils, the amplitude point-spread function is related to the pupil function, P(ρ), via
where and J0() denotes a zero-order Bessel function of the first kind. The pupil function may also account for defocus via
where u is again an optical coordinate that is proportional to the actual defocus, z, via
We chose to write Eq. (1) in the spatial domain, but an entirely equivalent expression can be written in the (spatial) frequency domain. In this approach, we describe the fluophore distribution mathematically as the superposition of a number of spatial frequency components of different strength (the actual strength of a specific component being determined by the specimen structure). Here, instead of a point-spread function, we introduce a transfer function, which tells us the strength (efficiency) with which a particular specimen spatial frequency will be present in the final image. Formally, we rewrite Eq. (1) as
where F(m, n) represents the spectrum (Fourier transform) of f(t1, w1), m and n are normalized spatial frequencies in the t and w directions, respectively. The normalized spatial frequencies are related to real spatial frequency f via . C(m, n) denotes the optical transfer function, which is given by the Fourier transform of the intensity point-spread function, H. For a conventional microscope, H= |h2 |2 and the transfer function may be conveniently evaluated as
where the asterisk indicates complex conjugate and the ⊗ symbol denotes the convolution operation. We illustrate the form of this function in Figure 2, where we have noted that and, hence, introduced . We see that the maximum spatial frequency that can be present in the image is given by δmax= 2. The physical reason, of course, is that finer specimen detail – corresponding to higher values of δ– causes the imaging light to be diffracted though such large angles that it is not collected by the objective lens.
The same two methods we have just introduced may also be used to describe the image formation in confocal microscopes. Equation (1) still applies, but the effective intensity point-spread function is now given by (Wilson & Sheppard, 1984),
where h1(t, w) denotes the amplitude point-spread function of the objective used to excite the fluorescence. We note that in writing Eq. (9), we have assumed that the limiting confocal pinhole is infinitely small. We shall return to this point later. If we ignore the wavelength difference between the excitation light and the fluorescence, and also assume that the same objectives are used both to excite and collect the fluorescence, then we may write
We may also use Eq. (7) as an alternative way to describe the image formation but the transfer function is now given by the Fourier transform of the confocal intensity point-spread function, Hconf(t, w), of Eq. (10). This may be conveniently written as
which is also plotted in Figure 2, where we see that the maximum spatial frequency that can be present in the confocal image is given by δmax= 4, which is twice as high as in the conventional case. This suggests that, because the confocal microscope image contains spatial frequencies twice as high as those found in the conventional microscope image, then the images formed in these instruments will be a more faithful representation of the specimen than those obtained in the conventional microscope. This conclusion is correct but, if we compare the actual values of the transfer functions, we are forced to conclude that the higher spatial frequencies present in the confocal case are imaged relatively poorly and that, whilst it could be argued (correctly) that the confocal instrument images all spatial frequencies with higher contrast than the conventional instrument, it has to be admitted that the two instruments are likely to produce images that are substantially similar. Indeed it is important to remember that conventional microscopes are capable of producing high-resolution images.
To further quantify the difference in imaging between the conventional and confocal microscopes, we will consider the image of a simple point object. Equation (1) together with Eqs (3) and (10) permit us to write
where we have again used optical coordinates u and v to represent the actual axial distance, z and radial distance, r. We may write analytic expressions for intensity in the focal plane, u= 0 and along the optic axis, v= 0, using Eqs (4) and (5), as
where J1(.) is a first-order Bessel function of the first kind. Further
If we measure resolution in terms of the width of the image between the points where the intensity falls to 50% of the peak, full-width half-maximum (FWHM), then in the lateral direction the resolution for the conventional microscope is given by 3.24 optical units (OU), whereas, in the confocal case it is found to be 2.32 OU and, hence, the resolution of the confocal microscope could be said to be better than that of the confocal. If we revert to real units, we may write the lateral FWHM (LFWHM) in the two cases as
We may obtain equivalent metrics of axial resolution by considering the half-widths of the intensities given in Eq. (14). In this case, axial resolution in the conventional case is given by 11.2 OU, whereas, for the confocal microscope a value of 8.0 OU is found, which is again an improvement of over the conventional case. We may translate these axial OU into real axial FWHM (AFWHM) values using Eq. (6) together with simple trigonometry to write the expressions in terms of the NA, as
We show comparison images in Figure 3. The immediate observation is that, although the confocal image is sharper, both microscopes image a point reasonably well.
Our consideration of point objects has suggested that resolution in the confocal microscope is roughly times better that in the conventional microscope. Our analysis, based on spatial frequency content, Figure 2, by contrast has suggested that the improvement in resolution by using a confocal microscope is a factor of two. It is, of course, inappropriate to compare these factors directly because they refer to different things, but it is interesting to note that for normalized spatial frequencies greater than the values of the confocal transfer function, although not equal to zero, is very low indeed. It could be argued, therefore, that the effective resolution improvement enjoyed by using a confocal instruments is of the order of .
The remarks we have just made relate to the case of an ideal confocal microscope, where the pinhole aperture is assumed to be infinitely small. Naturally, as the pinhole size is increased, the behaviour deviates from the ideal and tends towards that of the conventional instrument. It is straightforward to include the effects of pinhole size, D, in our analysis (Wilson, 1989) because it simply modifies the form of the effective intensity point-spread function of Eq. (9) via
We note that for an infinitely small detector |h2eff(t, w)|2 = |h2(t, w)|2, whereas, for a very large detector |h2eff |2∼ constant and Hconf(t, w) ∼Hconv(t, w).Figure 4 shows the deterioration of the resolution in confocal instruments as the pinhole size is increased. We note that the two curves proceed smoothly between the limits suggested by Eqs (15) and (16). We have measured the size of the pinhole diameter in Airy units (AU). These units are normalized units such that one AU corresponds to the diameter of an in-focus focal spot (often also called the Airy disc) measured between the first zero points. As such a pinhole diameter, Dreal, in real units corresponds to . We note, for completeness that OU are related to AU via . Finally, Figure 4(c) also shows the transition of the transfer function between the limiting confocal and conventional cases as a function of pinhole size.
Although the image of a simple point object is sharper in the confocal case than the conventional case, this alone does not adequately account for the importance of the confocal microscope. As we have seen, both instruments image point objects reasonably well, and the same is true for extended objects in general because the transfer functions are broadly comparable, as can be seen for Figure 2. The images in the confocal case are generally superior to those formed in the conventional instrument, but the principal difference between the two becomes apparent when defocus is considered. To illustrate the crucial differences between the two microscopes, let us begin by considering an object whose fluorescence distribution might be modelled as
where m denotes a modulation depth and v is a spatial frequency. We emphasize that any general specimen may be thought of being composed of a superposition of suitably weighted spatial frequency components – Fourier analysis – which would simply add extra terms to Eq. (18). However, it will be sufficient for our purposes to consider the simplified form of Eq. (18). We note that the first term is constant and represents the average background fluorescence, whereas, the second term represents the spatially varying part of the fluophore distribution. It is probably most convenient to write the image intensity in this case by using the transfer-function approach. Equation (18) together with Eq. (7) yields
where we have introduced the notation C(δ, u) to describe the value of the transfer function for the spatial frequency, δ, in the presence of defocus, u. We note that the transfer function for the conventional microscope is given by Eq. (8), with the pupil function given by Eq. (5) to take account of defocus. The confocal transfer function, by contrast, is given by Eq. (11). The form of these functions is shown in Figure 5. Figures 5(a) and (b) present the data in the usual way where spatial frequency is plotted along the abscissa and the function is plotted separately for a variety of values of defocus. It is, perhaps, also instructive to present these curves in the alternative form where defocus is plotted along the absicca and the curves are plotted for specific spatial frequencies, Figures 5(c) and (d).
The important observation here is that, in the conventional case, all spatial frequencies apart from zero attenuate with defocus. This means that as the defocus increases all spatially varying parts of the object will become less well-imaged, although the background does not change and remains constant. In essence the image of the (any) object will tend, with sufficient defocus to a featureless mid-grey level. In the confocal case, by contrast all spatial frequency components including zero attenuate with frequency, Figure 5(d).
The significance of these observations may be seen by considering the image of the one-dimensional (bar pattern) specimen, which is described by Eq. (19) and shown in greyscale representation in Figure 6. The figure is drawn for the arbitrary choice of m= 0.9 and v= 0.8. In the absence of defocus, the conventional and confocal images are similar, although, as expected, the image contrast is higher in the confocal case, Figures 6(a) and (d). As the defocus increases the contrast in the conventional microscope image is seen to decrease as the transfer function C(v, u) falls in value, Figures 5(a) and (c), whereas C(0, u) remains constant and unchanged. Indeed, at sufficient defocus, the value of C(v, u) becomes so small that the final image appears featureless and uniformly mid-grey to the eye. The confocal images of Figures 6(d)–(f), by contrast, behave quite differently. The main effect of defocus appears as a reduction in the overall image intensity but, insofar as the intensity is sufficient, the object features are still visible and reasonably well resolved. At sufficiently large defocus the image appears uniformly black. This, of course, is the key property of the confocal microscope in that it images efficiently only detail arising from the plane where the microscope is focussed.
As defocus increases it is tempting to say that image contrast disappears in the confocal case. It might be thought that the same could also be said for the conventional case and that the only difference is that the final, contrast free, image is totally black in the confocal case and a uniform mid-grey in the conventional case. In this regard it is instructive to rewrite Eq. (19) as
where we have introduced an effective contrast transfer function, Γ (v, u), given by
In the conventional case, C(0,u) = 1 and hence Γ (v, u) =C(v, u), which is shown in Figures 5 (a) and (c) and is seen to become smaller and smaller as defocus increases until eventually it becomes insignificant. In the limit of large defocus, the second term in Eq. (20) becomes insignificant compared with the first and I(t) ≈ C(0, u) = 1, which corresponds to the uniform background seen in Figure 6(c). The confocal microscope, by contrast, behaves quite differently.
We show in Figure 7, the form of the effective contrast transfer function, Γ (v, u), for the confocal case and a number of values of defocus where we see that the curves are essentially indistinguishable until the defocus has reached a value at which the premultiplying envelope function, C(0, u), has reduced in value by more than 50%.
The discussion above has illustrated that the confocal microscope does indeed possess optical sectioning, whereas, the conventional does not. It might reasonably be asked, therefore, what is the thickness of the optical section that can be recorded. It is clear from Figure 5(d) that there is no definitive answer to this question because the contrast with which each spatial frequency appears in the final image attenuates at a different rate with defocus. However, because the zero-order spatial frequency attenuates more slowly than other frequencies and particularly because C(0, u) is the ‘envelope function’, the attenuation of which is essentially responsible for the optical sectioning, Eq. (20), it is reasonable to consider the variation of this function with defocus as a measure of the optical sectioning strength. We may further advance the claim of C(0, u) as an appropriate ‘optical sectioning’ function by noting that because the confocal microscope images well only detail lying close to the focal plane it might be thought appropriate to consider the variation of the total energy in an image slice as a function of defocus. Thus we might consider
which may be most conveniently evaluated using Eq. (7) as
where we have again used the notation C(0, u) to represent the value of the transfer function at zero spatial frequency as a function of defocus. We note that F(0, 0) is a constant that represents the average value of the fluophore distribution. The important observation is C(0, u) that again describes the rate with which falls off with defocus. In the conventional case, C(0, u) = 1 always and hence, . This result is, of course, nothing more than a statement of conservation of energy. For the confocal case C(0, u) decreases in value as the defocus increases. Energy is not conserved in a confocal microscope because the limiting pinhole physically prevents all the light reaching the photodetector. It is worth finally noting that consideration of C(0, u) with defocus is equivalent to the variation in image intensity as a zero spatial frequency object – a thin fluorescent sheet – is scanned through focus. The ‘axial’ response has already been presented previously in as the δ= 0 curve in Figure 5(d) but is repeated in Figure 8 together with the corresponding behaviour in the conventional microscope.
The half width of the curve in the confocal case may be measured as 4.181OU and, hence, in real-world units, the FWHM is given by
The curves shown in Figure 8 represent, of course, the limiting cases of infinitely small (ideal confocal) and infinitely large (conventional) pinholes. For finite-sized pinholes, a family of curves may be plotted between these extremes (Wilson, 1989). These curves, then, permit a curve of half-widths to be extracted that may be used to estimate the thickness of the optical section that may be recorded by the microscope as a function of pinhole size. Figure 9(a) presents this data, where we see that the optical sectioning strength remains constant for pinhole diameters less than about 1 AU. Thereafter, the width of the optical section tends to grow linearly with pinhole size. Because Figure 9(a) requires computer calculation, it is tempting to look for readily tractable analytic expressions that fit the calculated curves reasonably well. At first glance it might be through that a portion of a hyperbola is a good candidate. Figure 9(b) shows the fit obtained by approximating the actual curve to
which is a reasonable fit. However, an even better approximation over the limited values of pinhole diameters considered may be achieved by using a cubic approximation, such as
We finally end by showing the optical sectioning strength to be expected when using a number of dry, water and oil immersion objectives in Figure 10.
In this short review, we have presented a self-contained discussion of the resolution improvement that may be achieved by using confocal microscopes, as well as discussing the optical sectioning property in some detail. New analytic formulae have been presented that relate the optical sectioning strength to the wavelength, NA and pinhole aperture size in a particularly simple fashion.
The author would like to thank Dr WB Amos for causing him to revisit basic confocal theory. This work was also undertaken in the context of the CAMINEMS project, 'Integrated Micro-Nano-Opto Fluidic systems for high-content diagnosis and studies of rare cancer cells' (http://www.caminems.eu). CAMINEMS is a collaborative project – medium-scale focused research project supported by the European 7th Framework Programme, contract number: 228980.