Phase-contrast microscopy

Phase-contrast microscopy converts specimen-dependent phase differences into visible light intensity differences. It can reveal structures in living, unstained biological cells, allowing their changes to be followed without first making an absorbing preparation.^[ref] The key question is how an optical microscope can read a phase when an ordinary detector records light intensity.

Consider a transparent specimen with complex amplitude transmission $t(x,y)=e^{i\varphi (x,y)}$. In ideal in-focus coherent imaging with all relevant spatial frequencies transmitted, its image has $I=|t{|}^2=1$: varying $\varphi$ alone produces no brightness detail. This is a particular model, not a claim that all transparent specimens are invisible. Diffraction, defocus, absorption and a finite optical pupil can already change a real image.

A phase-contrast microscope images an illuminating annulus onto a phase plate ring near the objective's rear focal plane. The aligned ring attenuates and shifts direct light relative to most diffracted light; their interference changes image brightness.^[ref]

The method is beautiful because the bright direct beam becomes its own reference: a small specimen-dependent complex amplitude can interfere with it and change the image at first order. Merely magnifying an invisible phase pattern would leave $|t{|}^2$ unchanged.

A phase-contrast microscope makes small specimen-induced wave phase changes visible through wave interference. An illumination annulus is imaged onto a phase ring in the microscope objective's rear focal plane. The ring changes the relative phase and amplitude of largely undeviated illumination compared with specimen-diffracted light, producing intensity differences in the image.^[ref]

Correctly matching and centering the annulus and ring is an apparatus requirement. A transparent unstained specimen can then acquire strong image contrast, but halos and image contrast reversals can occur. Comparing ordinary illumination with the aligned phase system demonstrates that image brightness is not simply a map of optical absorption.

A specimen whose leading optical effect is a spatially varying phase, rather than absorption. A thin specimen with refractive index $n_s$ in a medium of refractive index $n_m$ has, in a straight-path approximation,Here $\lambda$ is the vacuum wavelength; the integral is an optical path length difference. For the chosen values $\lambda =550\mathrm{nm}$, thickness $1\mu \mathrm{m}$ and $n_s-n_m=0.005$, the phase is $\pi /55\simeq 0.0571$ radians. This calculation does not identify thickness separately from refractive index contrast.^[ref]

A phase object for which the relevant relative phase variations are much smaller than one radian. The Taylor expansion $e^{i\varphi }=1+i\varphi +O({\varphi }^2)$ separates a direct reference term from a small specimen term. A constant overall phase can be removed before making this approximation: phase-contrast microscopy measures spatial variations, not an absolute common phase.

An optical filter that changes selected light's phase relative to other transmitted light. In phase-contrast microscopy, its ring can also attenuate the direct reference; the amplitude transmission and light intensity transmission must be distinguished. An amplitude transmission $a=1/2$ gives an intensity transmission $a^2=1/4$.

The following original model isolates the conversion mechanism. Use a periodic image window, unit illumination, ideal unit magnification, and a phase plate that changes only the zero spatial frequency. Write $c_0=⟨e^{i\varphi }⟩$ for the spatial mean. Multiplying that single Fourier coefficient by $a{e}^{i\theta }$, with $0<a\le 1$, gives the exact image field in this model:This formula follows by splitting $e^{i\varphi }=c_0+(e^{i\varphi }-c_0)$: the second term has zero mean and is left unchanged. A real annular, partly coherent instrument alters a finite band of directions, so this calculation is a mechanism model rather than its full image-formation law.

For a weak phase object with $⟨\varphi ⟩=0$, one has $c_0=1+O({\varphi }^2)$ and henceThe cross term is $2\mathrm{Re}(a{e}^{-i\theta }i\varphi )=2a\varphi \sin \theta$. Thus $\theta =\pi /2$ makes positive $\varphi$ brighter and $\theta =-\pi /2$ makes it darker under this stated phase convention for complex amplitudes. The sign comes from this convention and chosen plate; a manufacturer’s positive/negative contrast labels should not be inferred from the sign of $\theta$ alone.

For example, choose $a=0.5$ and a zero-mean periodic pattern whose peak phase is $0.084$. The first-order peak intensities are $0.25\pm 0.084$, compared with the uniform ideal bright-field microscopy value $1$. The relative change about the reduced background is approximately $2\varphi /a$ for a quarter-cycle shift. Attenuating the reference increases relative contrast while reducing the background photon count; it is not light amplification or a universal improvement of signal-to-noise ratio.

Two checks expose what the model can and cannot measure. A spatially constant phase gives $E_{\mathrm{out}}=a{e}^{i\theta }{e}^{i\varphi }$ and $I_{\mathrm{out}}=a^2$, so no absolute phase is recovered. Orthogonality of the constant and zero-mean parts also givesFor $a\le 1$ this cannot exceed the incident mean light intensity. The image redistributes brightness and loses some reference light rather than creating energy.

A finite phase plate also intercepts diffracted light, altering low spatial frequencies. Halos and shade-off can therefore mimic borders or obscure interiors.^[ref] In the coherent zero-mode model of phase contrast, the separation is perfect by assumption, so that model cannot predict those artifacts.

Frits Zernike traced the idea to unwanted companion lines of a diffraction grating. Around 1930 he found that visible striping disappeared when a telescope focused exactly on the grating. A phase strip placed over the principal spectral line changed its relation to the companion light and restored contrast. He then applied the separation of direct and diffracted light to a microscope and demonstrated it at Zeiss in 1932.^[ref]

His later absorbing and annular strips improved contrast and the appearance of halos; commercial Zeiss accessories followed in 1941.^[ref] The route from a grating defect to biological observation illustrates why apparently unwanted instrumental effects can be productive experimental clues.