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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 can't really be called a microscope in the usual sense).

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  Implementing Phase Contrast Using a 4F Correlator

With reference to this figure, we assume a plane wave incident from the left and a phase transmittance function of the form:

${\frac {}{}}T(x,y)=e^{j\phi (x,y)}$

If this "phase object" is thin, so that $\phi (x,y)<<1$ then,

$T(x,y)=e^{j\phi (x,y)}\cong 1+j\phi (x,y)$

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 $\phi (x,y)$ 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

${\frac {}{}}T(x,y)=1+j\phi (x,y)$

is formed in the back focal plane of the first lens as

${\frac {}{}}T(k_{x},k_{y})=PSF(k_{x},k_{y})+j\Phi (k_{x},k_{y})$

where $PSF(k_{x},k_{y})$ 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 $\Phi (k_{x},k_{y})$ 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 $PSF(k_{x},k_{y})$ term, while leaving the $\Phi (k_{x},k_{y})$ term relatively unaffected.

So, behind the dot, the field has the form:

${\frac {}{}}E(k_{x},k_{y})=jPSF(k_{x},k_{y})+j\Phi (k_{x},k_{y})$

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:

${\frac {}{}}E(x,y)=1+\phi (x,y)$

where we now neglect the factor of j common to both terms.