Phase-contrast imaging

Phase-contrast imaging is a method of imaging that has a range of different applications. It measures differences in the refractive index of different materials to differentiate between structures under analysis. In conventional light microscopy, phase contrast can be employed to distinguish between structures of similar transparency, and to examine crystals on the basis of their double refraction. This has uses in biological, medical and geological science. In X-ray tomography, the same physical principles can be used to increase image contrast by highlighting small details of differing refractive index within structures that are otherwise uniform. In transmission electron microscopy (TEM), phase contrast enables very high resolution (HR) imaging, making it possible to distinguish features a few Angstrom apart (at this point highest resolution is 40 pm).

Phase-contrast imaging is commonly used in atomic physics to describe a range of techniques for dispersively imaging ultracold atoms. Dispersion is the phenomena of the propagation of electromagnetic fields (light) in matter. In general, the refractive index of a material, which alters the phase velocity and refraction of the field, depends on the wavelength or frequency of the light. This is what gives rise to the familiar behavior of prisms, which are seen to split light into its constituent wavelengths. Microscopically, we may think of this behavior as arising from the interaction of the electromagnetic wave with the atomic dipoles. The oscillating force field in turn causes the dipoles to oscillate and in doing so reradiate light with the same polarization and frequency, albeit delayed or phase-shifted from the incident wave. These waves interfere to produce the altered wave which propagates through the medium. If the light is monochromatic (that is, an electromagnetic wave of a single frequency or wavelength), with a frequency close to an atomic transition, the atom will also absorb photons from the light field, reducing the amplitude of the incident wave. Mathematically, these two interaction mechanisms (dispersive and absorptive) are commonly written as the real and imaginary parts, respectively, of a Complex refractive index.^{[citation needed]}

Dispersive imaging refers strictly to the measurement of the real part of the refractive index. In phase contrast-imaging, a monochromatic probe field is detuned far away from any atomic transitions to minimize absorption and shone onto an atomic medium (such as a Bose-condensed gas). Since absorption is minimized, the only effect of the gas on the light is to alter the phase of various points along its wavefront. If we write the incident electromagnetic field as

$${\displaystyle \mathbf {E} _{i}={\hat {\mathbf {x} }}E_{0}e^{i(\omega _{0}t-kz)}}$$

then the effect of the medium is to phase shift the wave by some amount ${\displaystyle \Phi }$ which is in general a function of ${\displaystyle (x,y)}$ in the plane of the object (unless the object is of homogenous density, i.e. of constant index of refraction), where we assume the phase shift to be small, such that we can neglect refractive effects:

$${\displaystyle \mathbf {E} _{i}\to \mathbf {E} _{PM}={\hat {\mathbf {x} }}E_{0}e^{i(\omega _{0}t-kz+\Phi )}}$$

We may think of this wave as a superposition of smaller bundles of waves each with a corresponding phase shift ${\displaystyle \varphi (x,y)}$:

$${\displaystyle \mathbf {E} _{PM}={\hat {\mathbf {x} }}{\frac {E_{0}}{A_{o}}}\int _{(x,y)}e^{i(\omega _{0}t-kz+\varphi (x,y))}\,dx\,dy}$$