Laboratoire neurophotonique

UFR des Scicences Fondamentales et Biomédicales

benoit.forget@parisdescartes.fr

- Abbe's theory of image formation
- Diffraction limit, numerical apperture and resolution
- PSF and convolution
- Measuring and Improving the "real" PSF

Points \(B\), \(B'\) and \(B''\) are conjugate points ; there is a **bijective** relation between them.

This is not the case in "real life" microscopy

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 !

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*}

$$ NA= n\sin\alpha$$

$$I(\theta) = I_0 \left ( \frac{2 J_1(ka \sin \theta)}{ka \sin \theta} \right )^2$$

$$k = \frac{2\pi}{NA}$$

$$k = \frac{2\pi}{NA}$$

$$ \frac{1,22 \lambda}{2 ON} = \frac{0,61 \lambda}{NA}$$

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

Deconvolution is possible *to a certain extent* ...

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

Optical element are not perfect.

- 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