## ABSTRACT

It is shown that the simple theory of phase-contrast microscopy may be extended to take some account of the size of the object by the introduction of an ‘overlap factor’ which expresses the fraction of the diffracted light that passes through the phase-changing region of the phase plate. The extended theory can explain observed effects that cannot be explained even qualitatively if overlap is ignored. The overlap factor is given for disk-shaped objects and an annular phase plate.

## Introduction

The full mathematical treatment of phase contrast is fairly complex and microscopists usually rely on a simplified theory. This considers first a point source of light, and the complex amplitude imposed by the object is followed through the optical system; it is then assumed that an extended source gives a similar result. The treatment is given either in complex algebra or as the Argand or ‘vector’ diagram that represents these complex numbers (Barer, 1952). Many observed results can be explained in this way, but there are obvious limitations, as no account is taken of the size or shape of the object. Thus no explanation is given of the haloes surrounding sharp edges or of the loss of contrast towards the centre of the image when the object is not very small.

Hopkins (1953*a*) has shown that the assumptions on which the simple theory is based reduce to :

(1) all the direct light, and none of the diffracted light, passes through the phase-changing region of the phase plate;

(2) all the diffracted light falls within the aperture of the image-forming system.

The second assumption ignores the limited aperture of the objective and therefore implies that the objective has unlimited resolving power, so that, if no phase system were present, the image would be an exact reproduction of the object. The effects of a limited aperture are similar in both the phase microscope and the ordinary microscope and will not be considered here.

The first assumption has no equivalent in the theory of ordinary microscopes. In many practical cases it is not even approximately true, as shown by the inability to explain the phenomena mentioned above. To explain these results a theory such as that given by Ramsay (1952) is required. He considers the case of an extended object, an optical system with a limited aperture, and a circular phase plate, and his results then agree with observation. However, his theory is too complex for practical use by microscopists.

In this paper we develop a simpler theory that follows the form of the elementary theory, well known to microscopists, and show that this can explain the loss of contrast at the centre of large objects. Such objects diffract an appreciable amount of light through small angles and, as both the source (condenser diaphragm) and the phase-changing region have a finite size, there can be a considerable overlap of diffracted images of the source on to the phase-changing region. This ‘overlap factor’ can be introduced either into the algebra of the simple amplitude theory or into Barer’s Argand diagrams.

## Experimental Results

Although the simple theory suggests that the image contrast obtained is independent of the size of the object, it is well known that objects having the same phase advance or retardation but different sizes give rise to different image contrasts. When the object is also absorbing, the image of a larger object may even show a reversed contrast as compared with that of a smaller one.

A simple experiment can be used to demonstrate these effects. An air-dried smear of an unidentified amoeba mixed with bacteria is stained red with erythrosin and, after drying, mounted in Reichert immersion oil (refractive index at *D* line, 1·515). This preparation provides objects of two distinct sizes, both having refractive indices lower than that of the mounting medium and hence giving small phase advances. The coloration enables the absorption in the objects to be varied by the use of light of different colours; their transmission is almost unity in red light but much lower in green. Although the use of light of different wavelengths gives slightly different properties to the phase system (a quarter wavelength retardation in green light becomes just over a fifth of a wavelength in red), this should make only a negligible’ difference to the results obtained.

The preparation is now examined with a phase system that advances and absorbs the direct light; such a system is known as *positive phase contrast*. The images in red light of both the amoebae and the bacteria appear brighter than the background, as would be expected ; we shall call this *positive image contrast*. In green light, however, although the bacteria still appear with positive but reduced contrast, the larger amoebae appear in negative contrast (darker than the background).

By the simple theory, the images should always have the same contrast if both specimens have approximately the same phase change and absorption. Even if these are not quite the same, no object that gives a small advance of phase should give an image with negative contrast with a positive phase contrast system. Fig. 1 shows the calculated image contrast *C′* as a function of the amplitude transmission *a* of the object for different phase advances *ϕ* of the object; *C′* is defined as *(B′*_{s}*B′*_{F}*) B′*_{F}where *B′*_{s} and *B′*_{F} are the illuminations in the image plane at the image of the specimen and across the background. The results are given for positive phase objectives having amplitude transmissions for the direct light of *k —* 1 or and a phase advance of In no case are negative contrasts predicted.

The same result can be seen on Barer’s ‘vector circle’ (1952) as shown in fig. 2. For a phase system giving a phase advance to the direct light, the reference origin *O′* for the image lies between *M* and *P*, so that *O′M/PM=k*. The complex transmission of the object is represented by *OF* of length *a* and angle ϕ/ϕ is known to be small. It is obvious that, for positive, the amplitude in the image, represented by the length *O′F*, is always greater than the background amplitude *O’M* (unless ϕ is greater than , which is not true in the experiment above).

The failure of the simple theory to explain such observed results suggests that at least one of the assumptions of this theory is not satisfied. A clue to a better explanation is given by studying the theoretical image given by a micro- scope with no phase change but only absorption in the phase region. In this case negative contrasts are predicted, as shown in fig. 3. If a theory which assumes no phase contrast gives a better prediction of the image of fairly large objects (such as the amoebae) obtained with a phase microscope than does a phase-contrast theory, it is apparent that, in such cases, so much of the diffracted light from these objects passes through the phase-changing region, along with the direct light, that very little effective phase difference is introduced between the two beams. The first assumption of the simple phase-contrast theory therefore needs modification.

## Extended Theory

The discussion above suggests that the correct result is given by a theory intermediate between the usual phase-contrast theory and that of the ordinary microscope. To obtain this theory an ‘overlap factor’ e is introduced to represent the fraction of the diffracted amplitude that passes through the phase-changing region.

The object is assumed to consist of a small region of amplitude transmission *a* (a≤1) and phase advance on ϕ a background of unit amplitude. The complex amplitude leaving the object plane can be broken up into an amount i from the background, which gives the direct light, and an amount *ae*^{iϕ}*–*1 giving the diffracted light.

All the direct light and a fraction e of the diffracted light passes through the phase-changing region whose complex transmission is *ke*^{iθ}. The remaining fraction 1—ϵ of the diffracted light passes through the phase plate with unit transmission. The complex amplitude at the image is then

The illumination at the image is given by |*A*′|^{2} or

The illumination across the background is *k*^{2}, so the image contrast is

These expressions have been used to calculate the image contrasts shown in fig. 4 which are based on , and *k =* 1 and . It is seen that, with this amount of overlap, negative image contrasts may be predicted for objects with a small phase advance and some absorption.

## Modified Argand Diagram

Just as the algebraic theory may be extended to include overlap, a similar extension may be represented on Barer’s ‘vector diagram’. The steps used are shown in fig. 5. As usual, the unit circle of centre *O* represents the domain of the object complex amplitude *ae*^{iθ} with *OM* representing the direction of zero phase. The reference centre for image amplitudes is *O′*, lying a distance *k* from *M* along a line *MP* which makes an angle *OMP = θ* with *OM, θ* being the phase advance of the phase plate, here shown as .

The diffracted light from the object is represented by *MF*. This is divided in the ratio *ϵ* to 1—ϵ by the point *N*. The fraction 1—ϵ, represented by *MN*, passes through a region with no phase-changing material and is transmitted unchanged, but the remaining portion *NF* passes through the phase-changing and absorbing region and is rotated in phase by the same angle *θ* to *NG* and reduced in length by the factor *k* to give finally *NF′*. The image complex amplitude finally obtained is represented by the vector sum *O′M+MN+NF′* and has an amplitude given by the length *O′F′*.

It is seen that this amplitude can be less than the amplitude *O′M* of the background and that negative contrasts can be obtained. Further, a combination of values of object phase ϕ and phase-region transmission *k* that gives negative contrasts for small values of *a*, the object transmission, will give positive contrasts for *a* large. This explains why the image of the amoebae has positive contrast in red light *(a* near unity) but negative contrast in green light *(a* small).

## Overlap Factor

The value of the overlap factor ϵ that should be used to interpret any particular observation depends on the size and shape of the object and of the phase system (condenser diaphragm and phase-changing region). Phase systems are usually annular, with their inside and outside diameters in the ratio 3 to 4 approximately.

It is impossible to give values of ϵ for every irregularly shaped object that may occur in practice. But, as a rough approximation, the values for circular disk objects of the same area as the natural object may be used. This is reasonable with annular phase systems, since these average the overlap of the diffracted light for all orientations and thus should not be greatly affected by irregular shapes of the object.

Fig. 6 gives the overlap factor e for uniform circular disk objects in terms of their diameter. These are calculated from the theory given in the appendix for a source and a phase region, both of which have their inner diameter three-quarters of their outer diameter. The results also depend on the value of *b* shown in fig. 7; this is the ratio of the objective aperture at the phase plate to the phase-ring aperture. The diameter of the disk object is given by a value *ω* which depends on *b* and also relates the object size to the limit of resolution of the objective, given by 0·61λ /N.A. If the disk has a diameter *n* times this limit of resolution, the value of *w* to be used in fig. 6 is given by *ω = n/b*.

## Conclusions

It has been shown that the usual elementary theory of phase contrast in the form familiar to microscopists can be extended rather simply to take account of the size of the object by the introduction of an overlap factor which expresses the fraction of the diffracted light from the object that passes through the phase-changing region. This factor is given for circular disk objects for a phase annulus of particular proportions, but, as the whole theory is a very rough approximation, the results could be used for most normal phase objectives and any objects of the same area but different shapes.

To derive the overlap factor, a more complete treatment is given in the appendix. At the same time an indication is given of the size of the errors involved in the simple overlap theory. Although these are not negligible, they are much smaller than the errors of the usual simple theory.

To derive the overlap factor e, a fuller treatment of the theory of image formation is required. The notation used is that of Steel (1959); points at the source are represented by position vectors u and v, in the object plane by x, y, at the pupil or phase plate by u′, v′, and in the image plane by x′. All of these are ‘reduced vectors’ as defined in that paper.

In this notation, the illumination at x’ in the image plane is given by Hopkins (1953*b*) as

where :

(1)

*g′(v)*is the brightness distribution across the effective source;(2)

*L*(x) is the complex amplitude transmission in the object plane,*l*(u′) its two- dimensional Fourier transform, and*l**(u′) the complex conjugate; and(3)

*f*(*u*′) is the complex amplitude transmission of the image-forming system, including the phase plate.

For this expression to give the same result as the simple theory, B(x′) should be expressible as a squared modulus such as, to a constant factor,

This is not true in general, but, if certain assumptions are made as to the form of the object, source, and phase system, an approximate agreement can be found.

We make these assumptions :

(1) The source is of uniform brightness over a region p(v) and, when imaged at the phase plate, it covers the same area as the phase-changing region, which is then p(u′). Thus

where *b* is a constant and *p*(v) is unity over the source and zero for other values of v. Obviously p^{2}(v) *= p(v)*.

*ae*

^{iϕ}

*=*α+1, where S(x) = 1 over the object and zero elsewhere. This object lies in a uniform field of unit transmission. Then where δ(u′) is a two-dimensional Dirac delta function and s(u′) is the Fourier transform of s(x).

(3) The pupil transmission is *ke*^{iθ} = *k* + I over *p*(**u**′) and unity elsewhere, the pupil being taken as unlimited.

If these values are inserted in the general equation, the result can be expressed in terms of the following integrals :

Then

If this result is compared with that given by the simple theory, the first term has the required form of a squared modulus and agrees with the earlier result if ϵ = H(x′). It was this result that was calculated for fig. 6 for the centre of the image (x′ = o). However, for the simple theory to be even approximately true, the extra term containing *W—H*^{2} should be small. It seems reasonable that this should be so, for the integrals *G, H*, and *W* should decrease in value in this order as the regions of integration become smaller.

It is difficult to evaluate *W—H*^{2} for annular phase plates, but it has been computed for circular phase and a circular disk object whose diameter is again expressed in terms of *tv*. The results are given in fig. 8. It can be seen that, although the error is not negligible when compared with unity, it is appreciable for only a small range of object sizes.

## REFERENCES

*Le Contraste de phase et le contraste par interférences*