Cartesian Sign Convention
- Light travels from left to right
- Object distance (u) is always negative
- Focal length (f) is positive for convex lenses and negative for concave lenses
- Image distance (v) is positive for real images and negative for virtual images
- Height above principal axis is positive, below is negative
- Radius of curvature (R) is positive if center is to the right of the vertex
Geometrical Optics
Spherical Refracting Surface
\[ \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \]
Relates object and image distances for refraction at a single spherical surface. Uses the Cartesian sign convention.
Lens Maker's Formula
\[ \frac{1}{f} = (n_{\text{lens}} - n_{\text{med}}) \left( \frac{1}{R_1} - \frac{1}{R_2} \right) \]
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
Calculates focal length of a lens based on its geometry and refractive index. The thin lens formula relates object and image distances to focal length.
Magnification
\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]
Ratio of image height to object height, also equal to ratio of image distance to object distance.
Newton's Lens Equation
\[ x_o \cdot x_i = -f^2 \]
Relates distances from object to front focal point and image to rear focal point.
Wave Optics
Wave Equation & Solutions
\[ \frac{\partial^2 \psi}{\partial x^2} = \frac{1}{v^2} \frac{\partial^2 \psi}{\partial t^2} \]
\[ \psi = A \sin(k(x \mp vt)) \]
\[ \psi = A \sin(2\pi (kx \mp \nu t)) \]
\[ \psi = A \sin(kx \mp \omega t) \]
The wave equation describes how waves propagate. Common solutions include harmonic waves with different parameterizations.
Phase & Group Velocity
\[ v_p = \frac{\omega}{k} \]
\[ v_g = \frac{d\omega}{dk} = \frac{\omega_1 - \omega_2}{k_1 - k_2} \]
Phase velocity is the speed of wave phase, group velocity is the speed of wave envelope or modulation.
Dispersive Medium
\[ k(\omega) = \frac{\omega}{c} n(\omega) \]
\[ v_g = \frac{c}{n(\omega) + \omega \frac{dn}{d\omega}} \]
In dispersive media, the wave number depends on frequency. The group velocity formula accounts for this dispersion.
Polarization
Jones Vectors
Linear polarization along x: \( \begin{bmatrix} 1 \\ 0 \end{bmatrix} \)
Linear polarization along y: \( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
Linear at 45°: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix} \)
Right circular: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ -i \end{bmatrix} \)
Left circular: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ i \end{bmatrix} \)
Linear polarization along y: \( \begin{bmatrix} 0 \\ 1 \end{bmatrix} \)
Linear at 45°: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ 1 \end{bmatrix} \)
Right circular: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ -i \end{bmatrix} \)
Left circular: \( \frac{1}{\sqrt{2}}\begin{bmatrix} 1 \\ i \end{bmatrix} \)
Jones vectors represent different polarization states of light.
Jones Matrices
Linear polarizer (x): \( \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix} \)
Quarter-wave plate: \( \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \)
Linear polarizer at 45°: \( \frac{1}{2} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \)
Quarter-wave plate: \( \begin{bmatrix} 1 & 0 \\ 0 & i \end{bmatrix} \)
Linear polarizer at 45°: \( \frac{1}{2} \begin{bmatrix} 1 & 1 \\ 1 & 1 \end{bmatrix} \)
Jones matrices represent optical elements that modify polarization.
Malus's Law
\[ I = I_0 \cos^2\theta \]
Intensity of polarized light after passing through a polarizer at angle θ.
Waveplates
\[ \Delta \phi = \frac{2\pi}{\lambda} d |n_e - n_o| \]
Phase retardation of a waveplate depends on thickness and birefringence.
Maxwell's Equations & EM Waves
In Matter (ρ=0, J=0)
\[ \nabla \cdot \vec{D} = 0 \]
\[ \nabla \cdot \vec{B} = 0 \]
\[ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} \]
\[ \nabla \times \vec{H} = \frac{\partial \vec{D}}{\partial t} \]
Maxwell's equations in linear, isotropic, homogeneous media with no free charges or currents.
Wave Solutions
\[ \vec{E} = \vec{E_0} e^{i(\vec{k} \cdot \vec{r} - \omega t)} \]
\[ \vec{B} = \vec{B_0} e^{i(\vec{k} \cdot \vec{r} - \omega t)} \]
\[ k = \frac{\omega}{v} = \frac{n\omega}{c} \]
\[ n = \sqrt{\frac{\epsilon \mu}{\epsilon_0 \mu_0}} \]
Plane wave solutions to Maxwell's equations. The refractive index depends on material properties.
Poynting Vector
\[ \vec{S} = \vec{E} \times \vec{H} \]
Represents the directional energy flux (power per unit area) of an electromagnetic field.
Reflection & Refraction
Snell's Law
\[ n_1 \sin\theta_i = n_2 \sin\theta_t \]
Relates angles of incidence and refraction to the refractive indices of the two media.
Fresnel's Equations
Parallel polarization (p): \\
\[ r_p = \frac{n_2 \cos\theta_i - n_1 \cos\theta_t}{n_2 \cos\theta_i + n_1 \cos\theta_t} \]
\[ t_p = \frac{2 n_1 \cos\theta_i}{n_2 \cos\theta_i + n_1 \cos\theta_t} \]
Perpendicular polarization (s): \\
\[ r_s = \frac{n_1 \cos\theta_i - n_2 \cos\theta_t}{n_1 \cos\theta_i + n_2 \cos\theta_t} \]
\[ t_s = \frac{2 n_1 \cos\theta_i}{n_1 \cos\theta_i + n_2 \cos\theta_t} \]
Fresnel equations give the reflection and transmission coefficients for light at an interface.
Reflectance & Transmittance
\[ R = |r|^2 \]
\[ T = \frac{n_2 \cos\theta_t}{n_1 \cos\theta_i} |t|^2 \]
The fraction of incident power that is reflected and transmitted.
Special Angles
Brewster's angle: \[ \theta_B = \tan^{-1}\left(\frac{n_2}{n_1}\right) \]
Critical angle: \[ \theta_c = \sin^{-1}\left(\frac{n_2}{n_1}\right) \]
At Brewster's angle, reflected light is perfectly polarized. Beyond the critical angle, total internal reflection occurs.
Evanescent Wave
\[ \alpha = \frac{2\pi}{\lambda} n_2 \sqrt{\sin^2\theta_i - \sin^2\theta_c} \]
Decay constant for the evanescent wave during total internal reflection.
Waveguides
V-Number
\[ V = \frac{2\pi a}{\lambda} \sqrt{n_1^2 - n_2^2} \]
Determines the number of modes in a waveguide. Single-mode operation when V < 2.405.
Propagation Constant
\[ b = \frac{\beta^2 - k_0^2 n_2^2}{k_0^2 (n_1^2 - n_2^2)} \]
Normalized propagation constant. A mode is guided if 0 < b < 1. Cut-off occurs at b = 0.