## Three theoretical "frameworks" for optics

• L'optique géométrique
• "rayon lumineux" pour étudier la propagation de la lumière et la formation d'images ;
• "lois empiriques" de propagation rectiligne, de réflexion et réfraction ;
• permet la conception d'instruments (eg. télescope, microscope, fibres optiques ...)
• L'optique ondulatoire
• Modèle scalaire
• décrit les interférences, la diffraction, la diffusion, etc. ;
• applications : mesures interférométrique (très haute précision) et la spectroscopie
• Ondes électromagnétiques
• notion de polarisation
• L'optique quantique
• émission de la lumière et interaction avec les atomes, les molécules ;
• a permis en particulier le développement du laser.

### geometrical (ray) optics

• Light rays propagate in a rectilinear path as they travel in a homogeneous medium
• Rays bend (and may split in two) at the interface between two dissimilar media
• Allows the design of efficient (and cool) instruments

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

## Image formation and stigmatism

• le système est stigmatique pour les points $$A$$ et $$A'$$ ;
• $$A'$$ est le point conjugué du point $$A$$ : $$A'$$ est l'image $$A$$ ;
• tout rayon lumineux dont le support dans le milieu incident passe par $$A'$$ a son support dans le milieu émergeant passant par $$A'$$ ;
• en l'absence de stigmatisme, l'image est floue

## image nette

Is there a bijective relation between "object" and "image" : can we recover the "object" from the "image" ?

## Dioptrics

Reflection et Refraction : A at the interface between two media of different index of refraction a ray of light changes it's direction.

## Snell - Descartes

• Réflexion : $$i = -r$$

• Réfraction : $$n_1 \sin i_1 = n_2 \sin i_2$$

• Conditions de Gauss : $$\sin i \approx i$$

• Retour inverse de la lumière

### Who is Venus looking at ?

Véronese 1585 : Vénus au miroir

## Descartes' conjugate equation

$$\frac{1}{\overline{OA'}}-\frac{1}{\overline{OA}}=\frac{1}{\overline{OF'}} \left(=-\frac{1}{\overline{OF}} \right)$$

## focal points (planes)

The focal point is the conjugate point of infinity

## Lateral Magnification

$$\gamma_t = \frac{\overline{A'B'}}{\overline{AB}}=\frac{\overline{OA'}}{\overline{OA}}$$

## Afocal system

$$\gamma_t = \frac{\overline{A'B'}}{\overline{AB}}=-\frac{\overline{O_2F_2'}}{\overline{O_1F_1'}}$$

### What is a microscope ?

Instrument qui:

• donne une image grossie d’un petit objet (grossissement)
• sépare les détails de celui-ci sur l’image (résolution)
• rend les détails visibles à l’œil ou avec une caméra

## What is a wave ?

### What is a mechanical wave ?

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

## Mathematical description of waves

A deformation moving while keeping the same shape

Space and time evolution (variables) are 'coupled':
$$\xi(x,t) = \xi(x-ct) \qquad\left\{\xi(x+ct)\;\textrm{if}\; \vec c= -c \vec u_x \right\}$$

### From a mathematical point of view

Is there an equation that allows for solution : $$A\xi(x-ct)+B\xi(x+ct)$$

d'Alembert equation ! $$\frac{\partial^2 \xi}{\partial t^2} - c^2 \frac{\partial^2 \xi}{\partial x^2} =0$$

## From a physical point of view

### Vibrating string

Newtonian dynamics ... $$\displaystyle \to\quad \frac{\partial^2 \xi}{\partial t^2}-c^2\frac{\partial^2 \xi}{\partial x^2}=0$$

## From a physical point of view

### pressure wave in a sound pipe

Newtonian dynamics + fluid elasticity ... $$\displaystyle \to\quad \frac{\partial^2 p}{\partial t^2}-c^2\frac{\partial^2 p}{\partial x^2}=0$$

### What about EM waves ?

\begin{align} \vec \nabla \cdot \vec E &= \frac{\rho}{\varepsilon_0} \\ \vec \nabla \cdot \vec B &= 0 \\ \vec \nabla \times \vec E &= -\frac{\partial\vec B}{\partial t} \\ \vec \nabla \times \vec B &= \mu_0 \vec j + \mu_0\varepsilon_0\frac{\partial\vec E}{\partial t} \\ \end{align}
$$-\nabla^2 \vec E +\mu_0\varepsilon_0 \frac{\partial^2 \vec E}{\partial t^2} = 0$$ $$c=\frac{1}{\sqrt{\mu_0\varepsilon_0}}$$

## trig functions as solutions

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 : $$\cos\left(2\pi\left(\frac{x}{\lambda} \pm \frac{t}{T}\right)+ \phi \right )$$ $$k=\frac{2\pi}{\lambda} \; ; \; \omega = 2\pi f = \frac{2\pi}{T} \quad\to \quad c=f\lambda$$

## Optical Waves

### Visible light waves have "very high" frequency :

$$c=3\times 10^8 {\rm m/s} \qquad 400 {\rm nm} < \lambda < 700 {\rm nm}$$ $$f= \frac{3\times 10^8 {\rm m/s}}{500\times 10^{-9} {\rm m}} = 6\times 10^{14} {\rm Hz}$$
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

### Complex notation

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.

## The traveling wave

$$\tilde{E}=\tilde{A}e^{j( \omega t \pm kr)}$$

For example, a sound wave : $$f=1000\,{\rm Hz} \; ;\; c=330\,{\rm m.s^{-1}} \quad\to\; \lambda = 0,33\,\textrm{m}$$

$$e^{j(\omega t - kx)}$$
$$e^{j(\omega t + kx)}$$

### Propagating wave : the phasor

\begin{align} \xi(x+\Delta x,t) &= \tilde A e^{j(\omega t -k(x+\Delta x))} \\ &= \tilde A e^{j(\omega t -k(x))}e^{-jk\Delta x} \\ &= \xi(x,t) e^{-jk\Delta x} \end{align}
\begin{align} \omega t -kw &= \varphi_0 \\ x &= \frac{\varphi_0}{k}+\frac{\omega}{k}t \\ &= \frac{\varphi_0}{k}+ct \end{align}

One particular "point" of phase $$\varphi_0$$ travels along the axis $$Ox$$ at speed $$c$$.

## Intensity constrast

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 : $$\displaystyle C=\frac{I_{max}-I_{min}}{I_{max}+I_{min}} \approx 6\%$$

## Why not measure phase contrast

$$\Delta\varphi = k\Delta(nz) = \frac{2\pi}{\lambda}z\Delta n$$ $$z=6 µm\,;\, \Delta n =0,025 \,;\, \lambda=0,5 µm$$ $$\Delta (nz)=\frac{\lambda}{4}\,;\,\Delta\varphi \approx \frac{\pi}{2}$$

## Amplitude, phase and intensity

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 !

### D'Alembert's equation is linear

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.

## More to come ...

• Optical Wavefront
• 2D and 3D Waves
• From Wave Optics back to Geometrical Optics
• Wavefront Engineering
• The Point Spread Function (PSF)
• Abbe's theory of image formation
• Diffraction limited focus
• PSF, Convolution and Image Resolution