Laboratoire neurophotonique
Faculté des Scicences Fondamentales et Biomédicales
benoit.forget@parisdescartes.fr
Wall painting from the Stanzino delle Matematiche in the Galleria degli Uffizi (Florence, Italy). Painted by Giulio Parigi (1571-1635) in the years 1599-1600.
Microscope Zeiss, circa 1879, see more at Museum optischer Instrumente
Is there a bijective relation between "object" and "image" : can we recover the "object" from the "image" ?
Reflection et Refraction : A at the interface between two media of different index of refraction a ray of light changes it's direction.
Véronese 1585 : Vénus au miroir
The focal point is the conjugate point of infinity
Instrument qui:
In an elastic medium, internal forces tend to bring it back to its original after a perturbation
This perturbation (deformation) moves at a carateristic speed (celerity) which is solely determined by the mechanical proporties of the media
A deformation moving while keeping the same shape
Space and time evolution (variables) are 'coupled':
$$\xi(x,t) = \xi(x-ct) \qquad\left\{\xi(x+ct)\;\textrm{if}\; \vec c= -c \vec u_x \right\}$$
Is there an equation that allows for solution : $$A\xi(x-ct)+B\xi(x+ct)$$
d'Alembert equation ! $$\frac{\partial^2 \xi}{\partial t^2} - c^2 \frac{\partial^2 \xi}{\partial x^2} =0 $$
Newtonian dynamics ... \(\displaystyle \to\quad \frac{\partial^2 \xi}{\partial t^2}-c^2\frac{\partial^2 \xi}{\partial x^2}=0\)
Newtonian dynamics + fluid elasticity ... \(\displaystyle \to\quad \frac{\partial^2 p}{\partial t^2}-c^2\frac{\partial^2 p}{\partial x^2}=0\)
Trig functions (sine, cosine) are solution to d'Alembert's equation.
In the form : $$\xi(x,t)= A\cos(kx \pm \omega t+ \phi ) $$ with : \(\displaystyle c=\frac{\omega}{k}\)
Space (\(\lambda\)) and time (\(T\)) periodicity are explicited : $$\cos\left(2\pi\left(\frac{x}{\lambda} \pm \frac{t}{T}\right)+ \phi \right )$$ $$k=\frac{2\pi}{\lambda} \; ; \; \omega = 2\pi f = \frac{2\pi}{T} \quad\to \quad c=f\lambda$$
detector | response time |
eye | \( \approx 0,1\) s |
photo film | \( \approx 10^{-4} - 10^{-2} \) s |
single electronic detector | \( \approx 10^{-6} - 10^{-2}\) s |
CCD | \( \approx 10^{-2}\) s |
The EM field \(E(x,t)\)is written in complex notation : \begin{align*} E=A\cos(kx \pm \omega t + \phi) & = \Re\left\{\tilde E = Ae ^{j(kx \pm \omega t + \phi)}\right\} \\ & = \Re\left\{\tilde E = Ae ^{j\phi}e ^{j(kx \pm \omega t)}\right\} \\ & = \Re\left\{\tilde E =\tilde Ae ^{j(kx \pm \omega t)}\right\} \end{align*}
Note: Physical (measurable) quantites can only be expressed with real numbers.
For example, a sound wave : $$f=1000\,{\rm Hz} \; ;\; c=330\,{\rm m.s^{-1}} \quad\to\; \lambda = 0,33\,\textrm{m}$$
One particular "point" of phase \(\varphi_0\) travels along the axis \(Ox\) at speed \(c\).
Brightfield reflectance microscopy is based on intensity constrast
$$ R=\frac{\left(n_1-n_2\right)^2}{\left(n_1+n_2\right)^2} $$
The cell is not verry "optically different" than the sourounding medium :
Cell : \(n = 1, 36 \,\to\, R\approx 0, 0233\)
nutriment medium : \(n = 1, 335 \,\to\, R \approx 0, 0206\)
Contrast : \( \displaystyle C=\frac{I_{max}-I_{min}}{I_{max}+I_{min}} \approx 6\% \)
$$\Delta\varphi = k\Delta(nz) = \frac{2\pi}{\lambda}z\Delta n$$ $$z=6 µm\,;\, \Delta n =0,025 \,;\, \lambda=0,5 µm$$ $$\Delta (nz)=\frac{\lambda}{4}\,;\,\Delta\varphi \approx \frac{\pi}{2} $$
Remember the expression of the field (in complex and real notations) : $$ \tilde E = \tilde A e^{j(kr - \omega t)} $$ $$ \Re\{E\} = A \cos (kr - \omega t + \varphi) $$
Only the intensity (average value over tile of the energy of the wave) can be detected. $$ I = \left < E^2 \right > = \frac{1}{2} EE^* = \frac{A^2}{2} $$
Detection of the intensity is phase independant !
The sum of the wave (interference) is also a wave : $$ E_1 = \tilde A e^{j(kr - \omega t)} \qquad E_2 = \tilde A e^{j(kr - \omega t+\Delta \phi)} $$
The intensity of this interference is phase dependant : $$ I_T = \frac{1}{2} (E_1+E_2)(E_1+E_2)^* = I_1+I_2+2\sqrt{I_1I_2}\cos \Delta\phi $$
it is a function of the phase difference \(\Delta \varphi\) between the two intefering waves.