Interferential contrast in the case of a transmission microscope
Of two Wollaston prisms identical to the one studied previously, one is positioned at the level of the object focal point of the condenser, and the other one at the level of the image focal point of the objective. These two prisms are placed between crossed polarizers, and at \(\pm 45^{\circ}\) to the prisms' axes.
Question
Express the lateral shift \(d\) between the extraordinary and the ordinary ray.
Solution
The angle \(\epsilon\) being small, the distance d between the two rays is roughly equal to: \(f'_{obj} \cdot \epsilon\),
Attention :
\(f'_{obj}\) being the condenser's focal distance. Distance \(d\) is always very small, it remains less than inferior to the resolution limit of the objective in order not to perceive the shearing of the image, which is of the order of a fraction of a micrometer.
Question
Show that the system visualizes the gradient of the phase variations of the specimen. To make things simpler, we will consider that the source is monochromatic.
Solution
At any point \(x\) of the object where a path difference exists, the variation of the path difference between the ordinary ray and the extraordinary ray shifted from \(d\) has a value of:
\(\Delta (x) = \delta (x+ \frac{d}{2}) - \delta (x- \frac{d}{2}) = \delta'(x) \cdot d\)
where \(\delta '(x)\) is the derivative of the path difference. To illustrate this, we consider a homogeneous medium of index \(n\), of constant thickness \(e\) , except on an area of thickness \(e'\) where the index is equal to\( n' > n\) . Figure 6 shows the path difference in function of the position \(x\) at the level of the object. The same path difference for the extraordinary ray is depicted in red and in blue for the ordinary ray, shifted of \(\pm d/2\) ; the difference of the two path differences corresponding to the two rays that interfere at the level of the image plane is depicted in green.
Attention :
Therefore, the path difference highlighted by the interferences is a linear function of the slope of the path difference introduced by the object. This means that we do not visualize the phase variations, but its gradient variations.