Geometrical and Wave Optics

Benoît C. FORGET


Laboratoire neurophotonique
Faculté des Scicences Fondamentales et Biomédicales


  1. Geometrical optics
    1. Rays and Images
    2. Optical Microscope
  2. Wave physics
    1. Wave Propagation
    2. Mathematical Description of Waves

Light propagation and Image formation

Narcisse, par Le Caravage (v. 1595)

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" ?


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

Thin lens

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

More complex systems can be described with focal (and conjugate) planes

Lateral Magnification

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

Afocal system

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

Good online reference

Perfect Two-Lens System

Magnifying Glass, angular magnification

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


Confocal microscopy

Magnification is not enough

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

Energy transport without matter transport

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 \(c\).

Using phase for contrast

Seeing water in water

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.

Improved contrast

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