Phase contrast microscopy

Transcription

1 Phase contrast microscopy CJR Sheppard Division of Bioengineering National University of Singapore 9 Engineering Drive 1 Singapore Synopsis Many biological specimens behave as phase objects, that is, they alter the phase of the light transmitted through them, rather than its intensity. Similarly, reflection from a surface with height variations alters the phase of the reflected light. Observation of phase objects requires special imaging techniques. These include defocusing, dark field, Zernike phase contrast, Schlieren imaging, Hoffmann modulation contrast, differential phase contrast, interference microscopy, shearing interferometry, Nomarski differential interference microscopy, and digital phase retrieval. The imaging performance of these different methods are most easily appreciated and compared by consideration of the transfer function approach. In particular, we can consider weakly scattering objects and slowly varying phase objects as special cases. Key words defocus, dark field, Zernike phase contrast, Schlieren imaging, Hoffmann modulation contrast, differential phase contrast, interference microscopy, shearing interferometry, Nomarski differential interference contrast microscopy, phase retrieval, surface profiling. Published in: Sheppard CJR (2004) Phase contrast microscopy, Encyclopedia of Modern Optics, RD Guenther, DG Steel, L Bayvel, eds, Elsevier, Oxford, ISBN , 3, pp When a thin unstained biological specimen is illuminated, in general the transmitted light suffers changes in amplitude, phase and polarization. The change in polarization is related to the birefringence of the sample, which is neglected in the following. Thus the object [complex] amplitude transmittance can be written, [1] where is real and represents the modulus (also called amplitude) of the light wave on exiting from the sample, and, also real, represents the phase. In a perfect microscope, we observe the intensity of the transmitted light, so the image is given by. [2]

2 Thus the phase information is invisible in the image. A perfect microscope does not image phase variations in the object, but results in an image that depends on the amplitude of the transmittance, called amplitude contrast. Usually the change in modulus is also small, so that the image will exhibit weak contrast, which may not be discernable. In order to render the phase variations visible it is necessary to modify the microscope in one of a number of different ways. This conclusion is also relevant in incident light microscopy of surface structures. Then phase changes result from variations in surface height, and can be made visible by phase contrast microscopy. Quantitative measurement of the phase allows quantitative measurement of the surface profile. But care must be taken to avoid errors caused by phase change on reflection from a conducting material, or from effects of the numerical aperture of the optical system. In transmission microscopy quantitative phase measurement can be useful too, for example giving the concentration of a solute. Some of these phase contrast methods are easily understood using the simple imaging model described above. Others are more easily appreciated by considering imaging in terms of Fourier optics, equivalent to Abbe s theory of microscope imaging. We introduce the angular spectrum of the object amplitude transmittance, [3] where are spatial frequencies in the directions. We can write for the object amplitude transmittance, [4] where is in general complex. For a weak object we have, [5] and the spectrum is, [6] where δ is a Dirac delta function and. The problem with imaging of phase arises because a change in the imaginary part of b produces only a small change in the modulus of t. For a coherent imaging system, with coherent transfer function intensity is the image. [7]

3 where M is the magnification. The coherent transfer function is equal to the scaled pupil function, where f is the focal length of the objective. Substituting [6] into [7] and dropping terms of second order we obtain.[8] Normalizing the transfer function to unity at the origin. [9] For an axially-symmetric aberration-free system, the coherent transfer function is real and even, so that a pure phase object is not imaged (only the Hermitian component of B is imaged). Many phase contrast methods in principle rely on appropriate modification of the coherent transfer function, either by making it complex or asymmetric. However, this theory is an approximation as microscope systems are in practice partially coherent. For a partially coherent system the image intensity is, [10] where is the partially coherent transfer function, sometimes called the transmission cross-coefficient, which can be calculated as an integral over the objective and condenser pupils. The partially coherent transfer function is non-zero over a finite region of space, thus limiting the resolution of the system. Then [8] for a weak object becomes, normalizing so that,, [11] where is called the weak object transfer function. For the image of a general object that is not necessarily weak, [10] is often difficult to interpret. We can gain much insight into the imaging process by considering another special form of object, this time a slowly varying object, [12] where a and φ are assumed slowly varying. Thus the object spectrum, assumed now as slowly spatially varying, is

4 . [13] Substituting in [10], we then find [14] in which C is called the phase gradient transfer function, and is real. For an ordinary bright field system, C is an even function. Thus a phase gradient results in a change in image intensity, but for an axially symmetric system the sign of the change is independent of the sign of the phase change. 1 Defocused imaging Historically, the oldest method of phase contrast is simply to defocus the object. The transfer function is then complex, and phase detail is imaged. If the system is aberration-free, the contrast of the phase information reverses on the opposite side of focus. The strength of the contrast increases with defocus, until eventually artefacts are introduced when the imaginary part of the transfer function goes negative. If the relative condenser aperture is too large, contrast is reduced. Assume a pure weak phase object so that, [15]. [16] Then. [17] where is the imaginary part of. Thus the phase information is imaged by the imaginary part of the weak object transfer function. 2 Dark field microscopy Dark field microscopy can be used to visualize weak amplitude and phase information. The constant background in [5] is eliminated, so we obtain an image of b. Note that [8] and [11] are no longer valid because we cannot neglect the second order term. For a coherent system, the background can be eliminated by use of a central stop to intercept the non-diffracted light. For a partially coherent microscope, an annular condenser aperture (larger in diameter than the objective pupil) is used

5 (Fig.1). In both cases in practice a range of low spatial frequencies are eliminated, resulting in the appearance of halo artefacts. aperture stop objective object condenser annular mask Fig.1 Schematic diagram of dark field microscopy. For the pure weak phase object of [12], the image in a coherent microscope is [18] so that the spatial frequency is double that in the original object. For the partially coherent microscope, we have, [19] in which the two terms represent difference and sum frequency components. However, it can be shown that for the annular dark field system the transfer function for sum frequencies is zero, so the image intensity is a constant. The system images only difference frequencies of pairs of spatial frequencies, and not sum frequencies. For a slowly varying phase object, zero intensity results from a region of constant phase, but the intensity rises quickly for small values of phase gradient, thus giving strong contrast. For larger phase gradients, the intensity falls off slowly. 3 Zernike phase contrast The Zernike phase contrast method is similar to dark field except that the direct (nondiffracted) light has its relative phase changed by 90º, rather being eliminated. In practice this is achieved using an annular condenser aperture and an objective with a phase ring (Fig.2). The phase change can be, giving positive or negative phase

6 contrast. The phase ring is usually only partially transmitting, which has the effect of enhancing the sensitivity. From [8] or [11], imaging of a weak object is linear in phase. The halo artefact is present, as in dark field microscopy. Weak amplitude information is imaged as in dark field microscopy. phase ring objective object condenser annular mask Fig.2 Schematic diagram of Zernike phase contrast microscopy. For the weak phase object [12], we then have, [20] for positive or negative phase contrast, respectively, where g is the amplitude transmittance of the phase ring. The response of the weak object transfer function is improved as a result of the annular condenser aperture. 4 Schlieren imaging, Hoffman modulation contrast, and differential phase contrast (DPC) In Schlieren imaging with coherent illumination, a half plane mask (a Foucault knife edge) is inserted to eliminate say the negative spatial frequencies in x. Taking now, [8] becomes after normalization. [21] For the weak phase object [15], we then have. [22]

7 The image exhibits a variation in phase quadrature to the original phase variation, that is, it is related to the x-derivative of the phase. The effect occurs because the coherent transfer function contains an odd (but real) part. For a partially coherent microscope a similar effect can be achieved simply by offsetting the illumination system, but now the transfer function is no longer singlesided. It can be resolved into odd and even parts that result in differential phase contrast and amplitude contrast, respectively. The amplitude contrast component merely contributes to the background for a phase-only object, and can be removed simply by contrast enhancement of a digital image. The offset can be introduced in practice by inserting a half-plane mask into the condenser aperture stop. Imaging of a slowly varying object can be explained by [14], in which the transfer function is not symmetrical. The image shows a pseudo-three-dimensional bas relief effect. A commercial implementation of an asymmetric transfer function is Hoffman modulation contrast, which uses offset illumination together with an amplitude mask in the objective pupil (Fig.3). Advantages of Hoffman contrast are that it uses standard bright-field objectives and avoids problems from birefringent specimens. modulator plate objective object condenser slit mask Fig.3 Schematic diagram of Hoffmann modulation contrast microscopy. A similar method can be used in scanning laser microscopy. In a scanning microscope the effect of the illumination and detection systems are interchanged. Thus differential phase contrast (DPC) can be achieved in a scanning laser microscope by offsetting the detector. In practice this is conveniently done by using a split detector, (Fig.4) a detector divided into two semicircular elements. If the signal from one half is subtracted from that from the other, the amplitude contrast component cancels out, so that pure differential phase contrast results. For weak phase gradients the intensity varies linearly with phase gradient, again showing the bas relief effect. It is thus bidirectional, and the difference signal can become negative. A constant value is therefore usually added to it before display. The sum of the two detector elements results in a conventional bright-field image. From [14], both difference and sum

8 signals are proportional to, so that if the ratio of these two signals is extracted, an image of the phase gradient in the x direction results. The response to either weak phase gradients or fine detail can be enhanced by modifications using annular split detectors. Use of a quadrant detector allows the phase gradients in the x and y directions to be imaged. DPC microscopy has been demonstrated to be sensitive enough to observe monomolecular films and atomic surface steps. + split detector collector lens object objective Fig.4 Schematic diagram of differential phase contrast (DPC) microscopy. 5 Interference microscopy Interference microscopy allows quantitative measurement of the object phase by interference of the object beam with a reference beam. In the transmission geometry a Mach-Zehnder interferometer is used, which is effectively two parallel microscopes, each with matched condenser and objective lenses. In the reflection geometry we can use a Michelson, a Mirau or a Linnik interferometer. The Michelson interferometer has an inclined beam splitter and the reference mirror situated between the objective and object, and hence can be used only with low numerical aperture objectives. The Linnik interferometer has matched objectives in the object and reference beam paths. There is no limit to the numerical aperture that can be used, but it is not a commonpath interferometer and is hence sensitive to vibrations and air currents. Probably the most practicable system for interference microscopy in the reflection geometry is based on the Mirau interferometer that uses a beam splitter with its normal parallel to the optic axis. Until recently interference microscopes used a small condenser aperture giving nearly coherent illumination. The image consists of three components: an object beam term (non-interference image) a reference beam term, and the interference term. The phase (and amplitude) of the object beam can be extracted using phase stepping or heterodyning techniques. The measured phase is wrapped, and requires unwrapping to obtain the absolute phase.

9 If the condenser aperture is opened up, the behaviour is modified. The interference term is given by [23] where is the coherent transfer function for interference imaging, given by a convolution integral of the object path objective pupil with the product of the reference pupil and the source. The first thing to note is that for a high numerical aperture the spacing of the interference fringes is increased by 20-40%, and depends on defocus, tilt and central obscuration. Thus accurate measurement of phase requires careful calibration. The objective pupil exhibits a phase variation when defocused that reduces the modulus of the coherent transfer function, resulting in the fringes being modulated by an envelope the width of which becomes smaller as the aperture becomes larger. The envelope is similar to that of the optical sectioning effect in confocal microscopy. This envelope allows the zero-order fringe to be identified, although simulations have suggested that fringe hopping could occur at phase jumps as a result of diffraction effects. 5.1 Multiple beam interferometry The topography of a surface can be investigated by using a Fizeau interferometer formed by bringing a coated optical flat near to the surface. Then if the surface and flat have high reflectivity, multiple beam interference occurs and the width of the bright fringes can be much reduced. The system can be used with a tilted flat, to give fringes that are substantially straight, or with an almost parallel flat, so that the entire field of view is occupied by a single fringe. In the latter case the sensitivity can be in the sub-nanometre range. Multiple beam interference also enhances the contrast of weak amplitude information. 6 Shearing interferometry In conventional interferometry, the object beam is arranged to interfere with a constant reference beam. In shearing interferometry, on the other hand, it interferes with a shifted version of itself. 6.1 Lateral shearing interferometry An approach that is suitable for quantitative optical path measurements is to use an interferometer that shears the microscope image. The Interphako system uses a Mach- Zehnder interferometer that can measure optical path differences as low as about 3 nm in either the reflection or transmission geometry. The interferometer allows the shear distance to be varied. 6.2 Differential interference microscopy (DIC) The most common form of shearing interferometry is differential interference microscopy, usually associated with the name of Nomarski. DIC microscopy can be performed in either the transmission or the reflection geometry. A plane polarized illumination beam is split into two orthogonally polarized beams travelling at slightly differing angles by a Wollaston prism in the front focal plane of the condenser lens (Fig.5). The beams are thus sheared laterally in the object plane. The light from the

10 sample travels through the objective and a second Wollaston prism, which recombines the two beams into one. In the reflection geometry, a single lens and Wollaston prism are used. An analyser set at 45º allows the two beams to interfere. The shear is arranged to be smaller than the width of the point spread function. Different prisms are supplied for use with different magnification objectives. The system results in differential phase contrast, but there are differences as compared with the DPC microscope described above. analyser Wollaston prism objective object condenser Wollaston prism polarizer Fig.5 Nomarski differential interference contrast (DIC) microscopy. There are two different implementations of DIC in use. In one implementation, the illumination polarizer and analyser are crossed. The relative phase difference between the two beams (the bias retardation) is adjusted by moving the Wollaston prism laterally. It is claimed that rotation of the analyser can be useful when observing mixed phase/amplitude objects. In the second arrangement (the Sénarmont compensator) the position of the Wollaston prism is fixed, and the analyser rotated to change the relative phase. A recent variation of DIC uses circularly polarized illumination. The Wollaston prism still produces two plane polarized components, but with an additional phase difference of. The advantage of using circular polarization is that the prism can then be rotated to change the direction of shear. If the shear is 2Δ, and the phase difference is (i.e. if then the two beams are in anti-phase), then the effective object spectrum is Thus the effective transfer function can be written. [24]

11 [25] It is the final term of the three in braces that, because it is an odd function, results in differential phase contrast. We can now examine imaging of special forms of object. For a weak object, putting as in [11], The first term in braces corresponds to an amplitude contrast background component because it is real and even. The second term results in DPC because it is real and odd. For zero bias retardation, dark field imaging results. As the bias retardation is increased from zero, the strength of both the amplitude contrast and the DPC components increase, but that of the DPC component increases more rapidly. For visual observation we require good contrast, and hence we choose a small value of bias retardation, just large enough so that the background is large enough to avoid artefacts in the DPC image. In digital microscopy, contrast is not so important as it can be enhanced digitally, and hence we can increase to a value of, which maximizes the strength of the DPC component. Further increase in bias retardation decreases the DPC signal, until for pure amplitude contrast results. The actual form of the weak object transfer function depends on the magnitude of the shear. The condition, where is the cut-off in spatial frequency, gives optimum performance for the imaging of weak objects. The phase gradient transfer function is [26]. [27] For the bright-field microscope the phase gradient transfer function falls off symmetrically from its value at zero phase gradient. The effect of the pre-multiplying factor in [27] is to increase the fall-off for positive phase gradients and reduce it for negative ones. A bas-relief effect occurs, with highlighting for negative phase gradients for some values of the parameters. Increasing the value of the shear increases the highlighting effect, but, unlike in DPC, the behaviour is not antisymmetric. Because interference is dependent on the wavelength of the light, useful and beautiful effects can be seen with DIC microscopy using white light. Usually it is regarded as a qualitative, rather than a quantitative, imaging method. However, as from [14] the intensity can be written, [28]

12 by recording images at different bias retardations, conventional phase stepping methods can be used to extract the phase gradient. 6.3 Axial shearing interferometry Instead of shearing in the transverse direction, a shear in the axial direction can be used. In the reflection geometry, the complex amplitude from the point of observation is compared with the average over a defocused spot. This method has proved to be extremely sensitive, and can detect height changes smaller than 0.1 nm. 7 Phase retrieval 7.1 Digital phase retrieval The phase of a single two-dimensional (2-D) measurement of intensity can be retrieved by an iterative procedure if constraints such as non-negativity and bandlimitedness can be assumed. If the modulus of both an object and its Fourier transform are known, then the phases can also be recovered. In this case, the two 2-D real data sets have information content sufficient to reconstruct the 2-D complex object. This approach has been extended to the case of measurement of two defocused images, using the method of phase diversity. 7.2 Imaging using the transport of intensity equation The transport equation of intensity can be used to recover the phase of a partially coherent wave field directly [i.e. non-iteratively] from two 2-D intensity measurements. If the wave field is represented in terms of its intensity I and phase φ in a plane, the transport equation can be written in the paraxial approximation and in the absence of phase singularities,, [29] where is a 2-D operator acting in the plane. This expression can be inverted to give φ quantitatively, if I and are known. However, it gives the phase of the image field, rather than the phase of the object, so that obtaining quantitative values for a strong object may be difficult. The transport of intensity method is valid with partially coherent illumination: if the relative condenser aperture is too great then contrast becomes weak. Further, an effect similar to the variation in fringe spacing with aperture in interference microscopy is introduced because oblique illuminating rays traverse the sample at an angle to the axis. In practice a series of three images are usually taken at different focus positions. Then from the quantitative phase data, various different contrast mechanisms such as DIC, Zernike phase contrast, and so on, can be simulated. 8 Surface profiling methods: Triangulation and coherence domain methods Although strictly they are not phase imaging methods, other surface profiling methods can be used to measure optical path. These could also be adapted to also measure

13 optical path in transmission. It is important also to be aware of these methods, as they could affect measurements made using other phase contrast techniques. Apart from the phase measurement techniques described above, surface profiling methods fall into one of two categories: triangulation and coherence domain methods. An example of a technique that relies on triangulation is confocal microscopy. Confocal microscopy exhibits an optical sectioning effect similar to that described for interference microscopy. The effect is stronger for higher numerical aperture. As a reflector is scanned through focus the intensity goes through a maximum. By locating the position of the maximum, the local surface height can be obtained. In a similar way, in a confocal transmission system, introduction of an object with some optical thickness has the effect of altering the axial position of the confocal pinhole, with a resultant drop in intensity. This effect can be used to image the optical thickness of the sample. Coherence domain methods rely on the finite coherence length of the light. In an interferometer, this results in an envelope that multiplies that arising from the nonzero relative condenser aperture. Measurement of the peak of the envelope allows local surface height to be measured. Unlike the confocal approach, the width of this coherence envelope is independent of the numerical aperture of the system. In reflection, both the confocal and the coherence domain methods have the advantage over direct phase imaging techniques that they allow measurement of surface height changes without errors caused by the phase change on reflection from a conducting medium. Bibliography Bennett AH, Jupnik H, Osterberg H, Richards OW (1952) Phase Microscopy, Wiley, New York. Born M, Wolf E (1999) Principles of Optics, 7th edition, Cambridge University Press, Cambridge. Bradbury S, Evenett PJ (1996), Contrast Techniques in Light Microscopy, Bios Scientific Publishers, Oxford. Françon M (1950) Le Contraste de Phase en Optique et en Microscopie, Revue d Optique, Paris. Françon M (editor) (1952) Le Contraste de Phase et le Contraste par Interférences, Revue d Optique, Paris. Lessor DL, Hartman JS, Gordon RL (1979) Quantitative surface topography determination by Nomarski reflection microscopy, J. Opt. Soc. Am. 69, Inoue S, Spring KR (1997) Video Microscopy: The Fundamentals, 2nd edition, Plenum, New York. Pluta M (1989) Advanced Light microscopy, Volume 2: Specialized Methods, Elsevier, Amsterdam Sheppard CJR, Wilson T (1980) Fourier imaging of phase information in conventional and scanning microscopes, Phil. Trans. Roy. Soc. A 295, Wilson T, Sheppard CJR (1984) Theory and Practice of Scanning Optical Microscopy, Academic Press, London.