Optics
Table of Contents
1. Reflection
1.1. Retroreflective Material
Material that always reflect light to exactly where it came from
The road signs can be backed up by it.
2. Radiosity Method
The net heat transfer between surfaces can be calculated from the radiosity for each surface \(i\) \[ J_{\mathrm{e},i} &= \varepsilon_i\sigma T_i^4 + (1 - \varepsilon_i) E_{\mathrm{e}, i} \] where \(\varepsilon_i\) is the emissivity of surface \(i\), \(\sigma\) is the Stefan-Boltzmann constant, \(T\) is the temperature of the surface, \(E_{\mathrm{e},i}\) is the irradiance of the surface.
The irradiance of the surface \(i\) can be obtained from the radiosities of other surfaces: \[ A_i E_{\mathrm{e}, i} = \sum_{j=1}^N F_{ji} A_j J_{\mathrm{e}, j} \] where \(F_{ji}\) is the view factor or shape factor from surface \(j\) to surface \(i\).
The system of equations can be solved, and used to calculate the net he at transfer \(\dot{Q}_i\) at surface \(i\): \[ \dot{Q}_i = A_i (J_{\rm e, \it i} - E_{\mathrm{e}, i}) \]
3. Grating
3.1. Grating Equation
\[ d(\sin\theta_i + \sin \theta}) = m\lambda \] where \(d\) is the distance between slits, \(\theta_i\) is the incident angle with respect to normal, \(\theta\) is the angle of the screen point with respect to slit normal, \(m\) is the integer representing the order of interference, \(\lambda\) is the wavelength.
3.2. Dispersion
\[ \frac{\Delta\theta}{\Delta\lambda} = \frac{m}{d\cos\theta} \] the dispersion \(\Delta\theta\) due to difference in wavelength \(\Delta\lambda\).
3.3. Angular Half Width
Angular half width of principal maximum is given by \[ \Delta\theta = \frac{\lambda}{Nd\cos\theta} \] where \(N\) is the number of slits.
3.4. Resolving Power
The maximum resolution occurs when angular half width is equal to the dispersion angle. Combining dispersion angle equation and angular half width equation, \[ \frac{\lambda}{\Delta\lambda} = mN \]
3.5. Vibration Curve
The sum of phasors from each slit in phase space.
4. Stokes Relations
The derivation relies on the time-reversal symmetry, sometimes referred to as the principle of reversibility in this context. Therefore, this only works when their is no absorption.
The relation between the transmission and reflection coefficients in one direction \( r,t \), and the other direction \( r', t' \) are given by:
\begin{align*} &t't + r^2 = 1, \\ &r = -r'. \end{align*}5. Numerical Aperture
- NA
\[ \mathrm{NA} = n \sin\theta \] where \(n\) is the refractive index of the medium between the lens and the object, and \(\theta\) is the half-angle of the maximum cone of light that can enter or exit the lens.
It represents the Range of angles over which the system can accept or emit light, that is conserved across medium.
6. Fraunhofer Diffraction
Diffraction pattern at a long distance away.
Fraunhofer Diffraction Equation \[ U(x,y,z) \approx \frac{e^{ikz} e^{ik(x^2 + y^2)/2z}}{i\lambda z} \iint_{\rm aperture} A(x',y') e^{-\frac{i k}{z}(x'x + y'y) \dd{x'}\dd{y'}} \] where \(k\) is the wave number.
7. Fresnel Diffraction
7.1. Half-Period Zones
Zones in the wavefront such that the distance to the destination point from the boundaries are half-period apart.
7.2. Fresnel Number
\[ F = \frac{a^2}{L\lambda} \] where \(a\) is the characteristic size of the aperture, \(L\) is the distance of the screen from the aperture, \(\lambda\) is the incident wavelength.
7.3. Point Source
7.3.1. Vibration Curve
- Vibration Spiral
The curve slowly spirals in, towards the half maximum amplitude.
7.4. Rectangular Aperture
7.4.1. Vibration Curve
- Cornu's Spiral, Cornu Spiral
The curve spirals in while shifting sideways.
\[ z = \mathcal{C}(v) + i\mathcal{S}(v) = \int_0^v e^{i\pi v^2/2} \dd{v} \] where \(v\) is the length of spiral from the origin.
This equation is derived from the property of Cornu's spiral that the tangential angle at each point on curve is equal to \(\pi v^2/2\).
7.5. Fresnel Diffraction Equation
\[ E(x,y,z) = \frac{e^{ikz}}{i\lambda z} \iint_{\rm aperture} E(x', y',0) e^{\frac{ik}{2z}\left[ (x-x')^2 + (y-y')^2 \right]} \dd{x'} \dd{y'} \]
7.6. Rectangular Aperture
8. Optical Theorem
\[ \sigma = \frac{4 \pi}{k} \operatorname{Im} f(0) \] where \( \sigma \) is the cross section, \( f(0) \) is the scattering amplitude with an angle of zero, \( k \) is the wave vector in the incident direction.
9. Rainbow
9.1. Caustic
The angle of reflection is maximum at certain angle creating a bright spot.
Moreover, the cross section per radius is greater at the edge making it even brighter.
Smaller wavelength light bends more, attaining its maximum at a smaller angle compared to the longer wavelength light.
They merge in the center making white light, but they separates at the edge.
For a normal rainbow the angle is 42 degree away from the shadow of the eyes.
9.2. Alexander's Dark Bend
Between 42 degree and 50 degree, there's dark bend that no light reflects.
9.3. Supernumerary Rainbow
When the droplets are tenth of milimeters wide, interference occurs between lights going out in slightly different angles. Creating dark region at certain angle, letting other light to stands out.
9.4. Glory
The light incident at the edge is also reflected very near the opposite edge. The droplets becomes a ring source of light with a parallel light.
The light at different angles interferes, and it is different among wavelength, creating a rainbow between the angle of 2 degree to 4 degree.
10. Holography
10.1. Holographic Principle
The intensity and phase is recorded on a material, which can be used to reconstruct the exact electromagnetic wave on the entire space.