Misfocused microscopy and shadowgraph.

Equations (3.23), (3.26) and (3.35) allow to evaluate the evolution of the signals as $z$, the misfocusing, is increased, for a simple microscope objective, for phase contrast and dark field. It should be noted that we are dealing with images formed by laser light: as $z$ increases, it is possible to recover the original shape of the observed objects; this is completely different from a white light microscope, in which the misfocusing simply smears the images.

Equation (3.30) has not been extended for $z\ne 0$; in all the above mentioned techniques, however, the variation of $z$ strongly influences the relation between the light path $\delta l \left(\vec{q}\right)$ and the signal $i\left(\vec{q}\right)$. This is generally a defect: thick objects, or even thin objects dispersed in a thick volume, are difficult to be analysed. In general, all the above mentioned techniques are applied to samples that are thin, and in the focal plane. No improvement is obtained by misfocusing.

A well known exception is shadograph. Shadowgraph technique consists in sending a plane wave onto a sample, and observing the intensity modulations generated by the sample on a plane placed at a distance $z$ from the sample. Using Eq. (3.23), we can derive the transfer function $T_{shadowgraph}\left(\vec{q},z\right)$ of the shadowgraph technique [15,16,17]:

$$T_{shadowgraph}\left(\vec{q},z\right) = 2k\sin\left[\left(k-\sqrt{k^2-q^2}\right)z\right] \approx 2k\sin\left(\frac{q^2z}{2k}\right)$$ (3.37)

The approximation holds for $q\ll k$. The transfer function is defined as the ratio between the signal and the light path modulation amplitude:

$$i\left(\vec{q},z\right)= T\left(\vec{q},z\right) \delta l\left(\vec{q}\right)$$ (3.38)

For $z=0$, the transfer function vanishes; misfocusing is needed, and is a simple way to make phase modulations evident.

Looking at Eq. (3.26) we can notice that phase contrast transfer function, as a function of $z$, has a cosinusoidal behaviour:

$$T_{phase\,contrast}\left(\vec{q},z\right) = 2k\cos\left[\left(k-\sqrt{k^2-q^2}\right)z\right] \approx 2k\cos\left(\frac{q^2z}{2k}\right)$$ (3.39)

When considering opaque objects, with no phase modulations, the transfer functions are exchanged: cosinusoidal for shadowgraph, sinusoidal for phase contrast.

The shadowgraph image is created by the interference between every scattered beam and the transmitted beam; it's always possible, in principle, to find the value of $\delta l$, in every point, simply by a deconvolution. Shadowgraph allows the measurement of one component of the field, which, in turns, is the convolution of the light path with a particular function. The absolute intensity modulation of the shadowgraph image is proportional to the mean intensity and to the light path modulation; the constant of proportionality is the the transfer function. The transfer function vanishes for some wave vectors, but has maxima for other ones. At the maxima, the sensibility equals the sensibility of phase contrast and Schlieren techniques.

Now we evaluate the effect of misfocusing on a dark field microscope. We obtain an image of a plane a distance $z$ from the cell. Using Eq. (3.19), we can derive the relation between the light path and the measured intensity, at a given $z$:

$$i_{dark\,field}\left(\vec{q},z\right) = \frac{1}{\left(2\pi\right... ...left[\sqrt{k^2-\left(\vec{q}-\vec{q}'\right)^2}-k\right]z} \mathrm{d}\vec{q}' }$$ (3.40)

This expression reduces to Eq. (3.30) for $z=0$. At this point, there's no appearent reason to use a misfocused dark field instead of a focused one.

The knowledge of $i_{dark\,field}\left(\vec{q},z\right)$, for every $z$, can give some informations about the spreading of the scattered light. For the scattering of a single particle, one can measure the intensity on planes with increasing values of $z$, and calculate how fast the light is diverging. This provides informations both on the position of the particle and on the scattered intensity. For a sample composed by a great number of particles, this cannot be done, and a different, statistical approach must be applied.

For Schlieren technique, from Eq. (3.35):

$$T_{Schlieren}\left(\vec{q},z\right) = \left\{ \begin{array}{ll} \... ... i\left(\sqrt{k^2-q^2}-k\right)z} &\vec{q}\cdot\vec{n}\ge 0 \end{array} \right.$$ (3.41)

By evaluating the square modulus of the transfer function, we obtain:

$$\left\vert T_{Schlieren}\left(\vec{q},z\right)\right\vert^2 = 2k^2$$ (3.42)

This means that the power spectrum of the electric field is proportional to the power spectrum of the light path, without any dependence on spatial wavelength and misfocusing.

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