Chapter 2

Generalized Phase Contrast

Light intensity is easily quantified by using calibrated detectors that can directly exploit the energy flux from an incident light. Spatial intensity variations can be imaged using an array of such energy detectors, as in a camera, for instance. On the other hand, light phase is invisible to energy detectors and is usually detected indirectly by exploiting phase-dependent phenomena that affect intensity. For example, intercepting light with a lenslet array would generate an array of spots at the common focal plane of the lenslets and any phase perturbations could be deduced from observed changes in the configura-tion of the intensity spots. When using coherent illumination, a common method consists of introducing a reference beam and then analyzing the phase-dependent interference pattern to determine the phase perturbation. Working without the benefit

Fig. 2.1Fig. 2.1Fig. 2.1Fig. 2.1 A simplified schematic of a typical Zernike phase contrast microscope.

8 2 Generalized Phase Contrast

of coherent laser sources, Gabor invented the first holograms capable of interferometri-cally recording phase information by adapting Zernike’s phase contrast configuration, where the reference and the object beams propagate along a common path to ensure coherence [1]. Aside from coherence, common-path interferometry also surmounts typical experimental hurdles that tend to smear out the interference pattern with its relative tolerance to vibrations and fluctuations in the ambient conditions, which becomes a major problem when the reference beam travels along a different path. The accuracy of the extracted phase information from the output of an interferome-

ter is dependent on assumptions about the reference wave, and this is no different for a common-path interferometer. Thus it is vital to examine how a phase contrast method models the reference wave in order to understand its limitations. In this chapter, we examine the assumptions employed in Zernike’s phase contrast method. Although sufficient for very thin phase objects like biological samples, its limited range of validity necessitates a generalized formulation to encompass a wider range and broaden the horizon for possible applications.

2.1 Zernike Phase Contrast

The Dutch physicist Fritz Zernike received a Nobel Prize in 1953 for demonstrating the phase contrast method and inventing the phase contrast microscope. Zernike’s inven-tion paved the way for breakthroughs in medicine and biology by making living biologi-cal samples, like cells or bacteria, clearly visible under a microscope. Being generally colourless and transparent, biological samples are essentially invisible under a regular microscope unless one employs contrast dyes that can potentially harm the cells and prevent the observation of natural biological processes. Zernike’s phase contrast method [2–5] renders vivid details of transparent objects by

converting the phase perturbations introduced by the object into observable intensity fluctuations by the use of a phase shifting filter at the spatial Fourier plane that imparts a relative phase shift on the undiffracted light components. A simplified schematic is shown in Fig. 2.1, which is based on the eventual implementation that uses conical sample illumination and a phase ring filter (Zernike also considered different combina-tions of illuminations and filters, as shown in ref. [4]) . Thin biological specimens are typically weak phase objects that introduce minimal

phase perturbations, ( ),x yφ , onto an incident light. Thus it is sufficient to describe the

incoming phase distribution by a “small-scale” phase approximation where the largest phase deviation is typically taken to be significantly less than 3π [5]. When the input

phase distribution is confined to this limited range, a Taylor expansion to first order is sufficient for the mathematical treatment so that the input wavefront can be written as

( ) ( )exp , 1 ,j x y j x yφ φ ≈ + . (2.1)

2.2 Towards a Generalized Phase Contrast Method 9

For this first order approximation, the constant term represents the undeflected light while the spatially varying second term represents scattered light. The light corresponding to the two terms in this “small-scale” phase approximation can be spatially separated by placing the input phase distribution at the front focal plane of a lens to generate the corresponding spatial Fourier transformation at the back focal plane. In this geometry, light represented by the constant term is focused on-axis while the varying term is scat-tered off-axis, assuming an on-axis plane wave illumination of the phase object‡. Owing to the weak-phase approximation, any unscattered component from the varying term may be reasonably neglected. Zernike realized that it is possible to generate interference between the two phase-quadrature terms in Eq. (2.1) by introducing a small quarter-wave-shifting plate to act on the focused light. As a result, the output intensity becomes

( ) ( ) ( )2

', ' ', ' 1 2 ', 'I x y j j x y x yφ φ≈ + ≈ + , (2.2)

which enables phase visualization characterized by a linear phase-to-intensity trans-formation within the valid regime of the small-scale approximation. An approximately linear phase-to-intensity conversion is therefore achieved by phase contrast microscopes when studying thin and transparent biological specimens. It should be noted that a three-quarter waveplate works equally well to produce

phase contrast, but the plus sign in Eq. (2.2) is negated, leading to so-called negative phase contrast. Although linear, the phase-to-intensity mapping only applies to weak phase objects, which makes the second term in Eq. (2.2) significantly smaller than the constant term. This results in a very restricted intensity modulation depth. A substantial improvement in the visibility can be achieved in a Zernike phase contrast visualization, at the expense of light efficiency, by strongly dampening the focused light in addition to the phase shift required to generate the contrast [5].

2.2 Towards a Generalized Phase Contrast Method

In the general case, the input phase modulation is not limited to a small-scale pertur-bation and, hence, a first-order series expansion, as in the Zernike approximation will insufficiently represent the phase-modulated input. In this generalized regime, higher-order terms in the expansion need to be taken into account, so the expansion takes the form:

( ) ( ) ( ) ( ) ( )2 3 41 1 12 6 24exp , 1 , , , , ...j x y j x y x y j x y x yφ φ φ φ φ ≈ + − − + + (2.3)

‡ Most modern phase contrast microscopes do not use on-axis plane wave illumination but a superposi-tion of plane waves incident at a cone of illumination angles, similar to the schematic in Fig. 2.1. In this case, the lens focuses the undiffracted light, the constant term in Eq. (1), into a ring at the back focal plane where a phase ring introduces a quarter-wave phase shift to produce interference and phase visualization at the output.

10 2 Generalized Phase Contrast

The spatially varying terms can potentially contribute to the undiffracted light even

for weak phase objects, ( ), /3x yφ π≤ , but they are much smaller compared to the

constant first term in the Taylor series expansion and can, thus, be reasonably neglected without serious errors. However, the spatially varying terms can significantly contribute to the on-axis light for inputs with larger modulation depths. In this case, they can no longer be neglected or considered as separate from the focused light, as is customary in the Zernike approximation. The contributions from the spatially varying terms can result in a significant modulation of the focal spot amplitude on the back focal plane of the lens. These terms can, in fact, result in either constructive or destructive interference with the on-axis light, although the net result, based on conservation of energy, will be an attenuation of the focused light amplitude which can only achieve a maximum value for a perfect, unperturbed plane wave at the input. However, examples abound in the literature where the contribution of the higher-

order terms in the Taylor expansion is neglected and it is assumed that only the first term in the Taylor series expansion contributes to the strength of the focused light [8–15]. In particular, some frequently cited derivations of phase contrast [12, 14], whilst correct within the small-scale Zernike approximation, generate significant errors if extended to cover larger-scale phase perturbations. However, the fact that certain results derived for small-scale phase contrast imaging (including for example the dark-field

method) are expressed by use of a general phasor notation, ( )exp ,j x yφ , may explain

why some flawed analyses continue to propagate in the phase contrast literature. For phase contrast or dark field imaging of large-scale phase objects, the use of a first-order Taylor expansion based analysis, which is actually only valid within the regime of the small-scale phase approximation, is unacceptable. For phase objects breaking the first-order Zernike approximation we must identify

an alternative mathematical approach to that of the Taylor expansion given by Eq. (2.3). A Fourier analysis of the phase object provides for a more suitable technique for com-pletely separating the on-axis and higher spatial frequency components. This gives the

following form for ( )exp ,j x yφ , where ( ),x y ∈Ω defines the spatial extent of the

phase object:

( ) ( )φ φ−

Ω Ω

= +

∫∫ ∫∫

1

exp , d d exp , d d “higher frequency terms”j x y x y j x y x y (2.4)

In this Fourier decomposition the first term is a complex valued constant linked to

the on-axis focused light from a phase object defined within the spatial region, Ω , and

the second term describes light scattered by spatially varying structures in the phase object. Comparing Eq. (2.3) and Eq. (2.4), it is apparent that the first term of Eq. (2.3) is a poor approximation to the first term of Eq. (2.4) when operating beyond the Zernike small-scale phase regime. A key issue to keep in mind when analysing the effect of spatial filtering on the in-

coming light diffracted by phase perturbations is the definition of what spatially consti-

References 11

tutes focused and scattered light. In the previous description of Zernike phase contrast it was assumed that the focused light is spatially confined to a somewhat unphysical delta function, which is evident when taking the Fourier transform of Eq. (2.1):

( ) ( ) ( ) exp , , ,x yj x y f f j x yφ δ φ ℑ ≈ + ℑ (2.5)

where ( ),x yf f indicates coordinates in the spatial frequency domain. Ensuring that

only the focused light is subjected to the quarter-wave phase delay requires a filter with unphysical delta function dimensions. As we know, any aperture truncation that typically occurs within a practical optical system will lead to a corresponding spatial broadening of the focused light. It is therefore essential that we define the terms “focused light” and “scattered light” explicitly for such a system. Furthermore, the finite aperture effects from a physical phase-shifting filter must be accounted for so as to accurately describe its influence on the observed output intensity. We will also need to re-examine the required phase shift for the filter since the quarter-wave delay was derived based on the “small-scale” phase approximation under the unphysical assump-tions of delta functions for the focused light and apertures. In this context it is necessary to look more carefully at the sequence of apertures

confining the light wave propagation through a typical optical set-up – one that we shall describe in detail in the succeeding chapters. After properly accounting for the aperture effects, we carry the analysis to a level that allows us to determine the appropriate filter parameters. This combined analysis forms the core of the generalized phase contrast method.

References

1. D. Gabor, “Holography, 1948-1971,” from Nobel Lectures, Physics 1971-1980, Editor Stig Lundqvist, World Scientific Publishing Co., Singapore, 1992

2. F. Zernike, “How I discovered phase contrast”, Science 121,121,121,121, 345-349 (1955). 3. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects. Part I,” Physica 9999, 686-698 (1942).

4. F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects. Part II,” Physica 9999, 974-986(1942).

5. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, 2nd

ed., 1996). 6. H. B. Henning, “A new scheme for viewing phase contrast images”, Electro-optical Systems Design 6,6,6,6, 30-34 (1974).

7. G. O. Reynolds, J. B. Develis, G. B. Parrent, Jr., B. J. Thompson, The New Physical

Optics Notebook: Tutorials in Fourier Optics, (SPIE Optical Engineering Press, New

York 1989) Chap. 35.

12 2 Generalized Phase Contrast

8. H. H. Hopkins, “A note on the theory of phase-contrast images”, Proc. Phys. Soc. B., 66666666, 331-333 (1953).

9. S. F. Paul, “Dark-ground illumination as a quantitative diagnostic for plasma density”, Appl. Opt., 21212121, 2531-2537 (1982).

10. R. C. Anderson and S. Lewis, “Flow visualization by dark central ground interfer-ometry”, Appl. Opt. 24242424, 3687 (1985).

11. M. P. Loomis, M. Holt, G. T. Chapman and M. Coon, “Applications of dark central ground interferometry”, Proc. of the 29th Aerospace Sciences Meeting, AIAA 91919191----0565056505650565, 1-8 (1991).

12. D. Malacara, Optical shop Testing, 302-305 (John Wiley & Sons, New York 2nd ed.,

1992). 13. A. K. Aggarwal and S. K. Kaura, “Further applications of point diffraction interfer-ometer”, J. Optics (Paris) 17171717, 135-138 (1986).

14. M. Born and E. Wolf, Principles of Optics, 426-427 (Pergamon Press, 6th ed., 1980).

15. C. A. Mack, “Phase contrast lithography”, Proc. SPIE 1927192719271927, 512-520 (1993). 16. Y. Arieli, N. Eisenberg and A. Lewis, “Pattern generation by inverse phase contrast”, Opt. Comm. 131313138,8,8,8, 284-286 (1997).