Overview

Stigmatism and the microscope

Points $$B$$, $$B'$$ and $$B''$$ are conjugate points ; there is a bijective relation between them.

This is not the case in "real life" microscopy

Abbe's theory of image formation

$$d=\frac{\lambda}{2n\sin \alpha}$$

An image is a spatial distribution of intensity, the intensity of the total field reaching the image plane

This total field can be expressed as the sum of individual fields and the image can "viewed" as the interference pattern genrated by these individual fileds !

Fourier series

Any periodic function $$f(t)$$ of period $$T$$ (pulsation $$\omega$$, $$\omega T=2\pi$$) can be expressed as :
\begin{align*} f(t) &= a_0 + a_1\sin\omega t + b_1 \cos\omega t + a_2\sin 2\omega t + b_2 \cos 2\omega t + \ldots \\\ &= a_0 + \sum_n a_n \sin n\omega t + \sum_n b_n \cos n\omega t \end{align*}

Abbe's diffraction limit

Numerical aperture

$$NA= n\sin\alpha$$

Airy pattern and Rayleigh criterion

$$I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2$$
$$k = \frac{2\pi}{NA}$$
$$\frac{1,22 \lambda}{2 ON} = \frac{0,61 \lambda}{NA}$$

PSF, Impulse response ... Convolution

The imaging system (microscope) is linear invariant system analogous to a LTI : $$I(x,y)=PSF(x,y)*O(x,y)$$

Deconvolution

Deconvolution is possible to a certain extent ...

Real PSF

A 3D problem

In fluorescence microscopy, one needs to consider excitation and detection: $$PSF=PSF_{exc}*PSF_{det}$$

Aberrations

Optical element are not perfect.

Conclusion

• Determine the required PSF for your needs
• What is the smallest feature (x,y,z) you need to image ?
• What is the sample thickness
• Estimate the PSF of your system
• What is the NA of your objective ?
• Measure and if necessary improve the real PSF
• Image small, well separated, beads
• Consider Wavefront Engineering (call The Wavefront Engineering Group !) ...
• ... or superresolution techniques