# Phase contrast microscope

A phase contrast microscope is a microscope that does not require staining to view the slide. This microscope made it possible to study the cell cycle.

File:Phase condenser.JPG
Phase condenser

As light travels through a medium other than vacuum, interaction with this medium causes its amplitude and phase to change in a way which depends on properties of the medium. Changes in amplitude give rise to familiar absorption of light which gives rise to colours when it is wavelength dependent. The human eye measures only the energy of light arriving on the retina, so changes in phase are not easily observed, yet often these changes in phase carry a large amount of information.

The same holds in a typical microscope, i.e., although the phase variations introduced by the sample are preserved by the instrument (at least in the limit of the perfect imaging instrument) this information is lost in the process which measures the light. In order to make phase variations observable, it is necessary to combine the light passing through the sample with a reference so that the resulting interference reveals the phase structure of the sample.

This was first realized by Frits Zernike during his study of diffraction gratings. During these studies he appreciated both that it is necessary to interfere with a reference beam, and that to maximise the contrast achieved with the technique, it is necessary to introduce a phase shift to this reference so that the no-phase-change condition gives rise to completely destructive interference.

He later realised that the same technique can be applied to optical microscopy. The necessary phase shift is introduced by rings etched accurately onto glass plates so that they introduce the required phase shift when inserted into the optical path of the microscope. When in use, this technique allows phase of the light passing through the object under study to be inferred from the intensity of the image produced by the microscope. This is the phase-contrast technique.

In optical microscopy many objects such as cell parts in protozoans, bacteria and sperm tails are essentially fully transparent unless stained (and therefore killed). The difference in densities and composition within these objects however often give rise to changes in the phase of light passing through them, hence they are sometimes called "phase objects". Using the phase-contrast technique makes these structures visible and allows their study with the specimen still alive.

This phase contrast technique proved to be such an advancement in microscopy that Zernike was awarded the Nobel prize (physics) in 1953.

## Contents

### Implementing phase contrast function using a 4F correlator

We can see how the phase contrast principle works by considering the figure below, which shows a 4F correlator (see Fourier optics) that implements the phase contrast function (in this figure, magnification is unity, so it cannot really be called a microscope in the usual sense).

With reference to this figure, we assume a plane wave incident from the left and a phase transmittance function of the form: ${\displaystyle {\frac {}{}}T(x,y)=e^{j\phi (x,y)}}$

The term, ${\displaystyle \phi (x,y)}$ in the exponent is known as the phase (see phase (waves)) of the transmittance function. If this "phase object" is thin, so that ${\displaystyle \phi (x,y)<<1}$ then, ${\displaystyle T(x,y)=e^{j\phi (x,y)}\cong 1+j\phi (x,y)}$

Film (or detectors) respond to variations in amplitude, not phase. The transmittance function above will have very small variations in amplitude, since the two terms in the transmittance function are in phase quadrature. For maximum contrast, we will prefer to have these two terms in-phase (not in quadrature phase), so that variations in ${\displaystyle \phi (x,y)}$ directly impact the amplitude of the transmittance function. We accomplish this by selectively multiplying one term in the equation above by a factor of j, thus bringing the two terms in-phase.

We can accomplish this using the 4F correlator in the following way. We assume a plane wave field incident on the "input plane" of the correlator (on the far left in the diagram). The Fourier transform (FT) of the phase transmittance function ${\displaystyle {\frac {}{}}T(x,y)=1+j\phi (x,y)}$

is formed in the back focal plane of the first lens as ${\displaystyle {\frac {}{}}T(k_{x},k_{y})=PSF(k_{x},k_{y})+j\Phi (k_{x},k_{y})}$

where ${\displaystyle PSF(k_{x},k_{y})}$ is the Point spread function (PSF) of the lens. The PSF is basically just a small dot in the FT plane (the back focal plane of the first lens), whereas the function ${\displaystyle \Phi (k_{x},k_{y})}$ will be more spread out. Since the PSF is localized to a small region about the optic axis (the horizontal axis) in the FT plane, we may place a small, quarter-wavelength thick dot there. This dot will impart a quarter wavelength phase shift to the ${\displaystyle PSF(k_{x},k_{y})}$ term, while leaving the ${\displaystyle \Phi (k_{x},k_{y})}$ term relatively unaffected.

So, behind the dot, the field has the form: ${\displaystyle {\frac {}{}}E(k_{x},k_{y})=jPSF(k_{x},k_{y})+j\Phi (k_{x},k_{y})}$

and both terms are now in-phase. We now FT this field distribution using the second lens, to produce the following field in the "output plane" (the rightmost plane) of the 4F correlator system: ${\displaystyle {\frac {}{}}E(x,y)=1+\phi (x,y)}$

where we now neglect the factor of j common to both terms. Now the phase function, ${\displaystyle \phi (x,y)}$ directly modulates transmittance amplitude, making for better contrast in the image.