## List of Physics Optics Formulas, Equations Latex Code

rockingdingo 2023-03-26 #physics #optics

List of Physics Optics Formulas, Equations Latex Code

In this blog, we will introduce most popuplar formulas in Optics, Physics. We will also provide latex code of the equations. Topics include Bending of light, Snell's law, Fermat's principle, Paraxial geometrical, Mirrors, Magnification, Matrix Methods, Reflection and Transmission, Fresnel Equations, Polarization of Optics, Prisms and Dispersion, Diffraction, Birefringence and Dichroism, Retarders: waveplates and compensators, Fabry-Perot interferometer, etc.

### 1. Optics

• #### Bending of light Snell's law

##### Equation

$\Psi(t)=\hat{\Psi}(t)e^{i(\omega t \pm \phi)}=\hat{\Psi}(t)\cos (\omega t \pm \phi) \\ \sum_{i} \hat{\Psi_{i}}\cos(\alpha_{i} \pm \omega t) =\hat{\Phi}\cos (\beta \pm \omega t) \\ \tan (\beta)=\frac{\sum_{i} \hat{\Psi_{i}} \sin (\alpha_{i})}{\sum_{i} \hat{\Psi_{i}} \cos (\alpha_{i})} \\ \hat{\Phi}^{2} = \sum_{i} \hat{\Psi_{i}^{2}} + 2 \sum_{j > i} \sum_{i} \hat{\Psi_{i}} \hat{\Psi_{j}} \cos (\alpha_{i} - \alpha_{j}) \\ \int x(t) dt=\frac{x(t)}{i \omega} \\ \frac{d^{n}(x(t))}{d t^{n}}=(i \omega)^{n} x(t)$
##### Latex Code
            n_i\sin(\theta_i)=n_t\sin(\theta_t) \\
\frac{n_2}{n_1}=\frac{\lambda_1}{\lambda_2}=\frac{v_1}{v_2} \\
n^2=1+\frac{n_{\rm e}e^2}{\varepsilon_0m}\sum_j\frac{f_j}{\omega_{0,j}^2-\omega^2-i\delta\omega} \\
v_{\rm g}=c/(1+(n_{\rm e}e^2/2\varepsilon_0m\omega^2))

##### Explanation

Latex code for the Bending of light, Snell's law. I will briefly introduce the notations in this formulation.

• $n$: refractive index of the material
• $n_{\rm e}$: electron density
• $f_j$: oscillator strength

• #### Bending of light Fermat's principle

##### Equation

$\delta\int\limits_1^2 dt=\delta\int\limits_1^2\frac{n(s)}{c}ds=0\Rightarrow \delta\int\limits_1^2 n(s)ds=0$
##### Latex Code
            \delta\int\limits_1^2 dt=\delta\int\limits_1^2\frac{n(s)}{c}ds=0\Rightarrow \delta\int\limits_1^2 n(s)ds=0

##### Explanation

Latex code for the Bending of light Fermat's principle. I will briefly introduce the notations in this formulation.

• #### Paraxial geometrical optics

##### Equation

$\frac{n_1}{v}-\frac{n_2}{b}=\frac{n_1-n_2}{R} \\ \frac{1}{f}=(n_{\rm l}-1)\left(\frac{1}{R_2}-\frac{1}{R_1}\right) \\ \frac{1}{f}=\frac{1}{v}-\frac{1}{b} \\ \frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2}$
##### Latex Code
            \frac{n_1}{v}-\frac{n_2}{b}=\frac{n_1-n_2}{R} \\
\frac{1}{f}=(n_{\rm l}-1)\left(\frac{1}{R_2}-\frac{1}{R_1}\right) \\
\frac{1}{f}=\frac{1}{v}-\frac{1}{b} \\
\frac{1}{f}=\frac{1}{f_1}+\frac{1}{f_2}-\frac{d}{f_1f_2}

##### Explanation

Latex code for the Paraxial geometrical optics. I will briefly introduce the notations in this formulation.

• $R$: refraction at a spherical surface with radius R
• $|v|$: distance of the object
• $|b|$: distance of the image
• $n_{1}$: refractive index of the lens
• $f$: focal length
• $R_{1}, R_{2}$: curvature radii of both surfaces
• $D:=1/f$: dioptric power of a lens
• $1/f=8(n-1)d/D^2$: Approximation of focal length

• #### Mirrors

##### Equation

$\frac{1}{f}=\frac{1}{v}+\frac{1}{b}=\frac{2}{R}+\frac{h^2}{2}\left(\frac{1}{R}-\frac{1}{v}\right)^2$
##### Latex Code
            \frac{1}{f}=\frac{1}{v}+\frac{1}{b}=\frac{2}{R}+\frac{h^2}{2}\left(\frac{1}{R}-\frac{1}{v}\right)^2

##### Explanation

Latex code for the Paraxial geometrical optics. I will briefly introduce the notations in this formulation.

• $h$: perpendicular distance from the point the light ray hits the mirror to the optical axis

• #### Principal Planes

##### Equation

$h_1=n\frac{m_{11}-1}{m_{12}} \\ h_2=n\frac{m_{22}-1}{m_{12}}$
##### Latex Code
            h_1=n\frac{m_{11}-1}{m_{12}} \\
h_2=n\frac{m_{22}-1}{m_{12}}

##### Explanation

Latex code for the Paraxial geometrical optics. I will briefly introduce the notations in this formulation.

• $N$: nodal points
• $H$: principal points
• $\perp$: plane to the optical axis through the principal points is called the principal plane
• $m_{ij}$: lens is described by a matrix

• #### Magnification

##### Equation

$\displaystyle N=-\frac{b}{v} \\ \displaystyle N_{\alpha}=-\frac{\alpha_{\rm syst}}{\alpha_{\rm none}}$
##### Latex Code
            \displaystyle N=-\frac{b}{v} \\
\displaystyle N_{\alpha}=-\frac{\alpha_{\rm syst}}{\alpha_{\rm none}}

##### Explanation

Latex code for the magnification, optics. I will briefly introduce the notations in this formulation.

• $\displaystyle N=-\frac{b}{v}$: linear magnification
• $\displaystyle N_{\alpha}=-\frac{\alpha_{\rm syst}}{\alpha_{\rm none}}$: The angular magnification
• $\alpha_{\rm sys}$: size of the retinal image in the optical system
• $\alpha_{\rm none}$: size of the retinal image outside the system

• #### Optical System Matrix Methods

##### Equation

$\left(\begin{array}{c}n_2\alpha_2\\y_2\end{array}\right)=M \left(\begin{array}{c}n_1\alpha_1\\y_1\end{array}\right)$
##### Latex Code
            \left(\begin{array}{c}n_2\alpha_2\\y_2\end{array}\right)=M \left(\begin{array}{c}n_1\alpha_1\\y_1\end{array}\right) \\
{\rm Tr}(M)=1

##### Explanation

Latex code for the Optical System Matrix Methods, optics. I will briefly introduce the notations in this formulation.

• $(n\alpha,y)$: A light ray
• $\alpha$: the angle with the optical axis
• $y$: the distance to the optical axis
• $M$: is a product of elementary matrices
• $\displaystyle M_{\rm R}= \left(\begin{array}{cc}1&0\\l/n&1\end{array}\right)$: Transfer along length l
• $\displaystyle M_{\rm T}=\left(\begin{array}{cc}1&-D\\0&1\end{array}\right)$: TRefraction at a surface with dioptric power D

• #### Reflection and Transmission

##### Equation

$r_\parallel\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\parallel~,~~ r_\perp\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\perp~,~~ t_\parallel\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\parallel~,~~ t_\perp\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\perp$
##### Latex Code
            r_\parallel\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\parallel~,~~ r_\perp\equiv\left(\frac{E_{0r}}{E_{0i}}\right)_\perp~,~~ t_\parallel\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\parallel~,~~ t_\perp\equiv\left(\frac{E_{0t}}{E_{0i}}\right)_\perp

##### Explanation

If an electromagnetic wave hits a transparent medium part of the wave will reflect at the same angle as the incident angle, and a part will be refracted at an angle according to Snell's law. It makes a difference whether the E field of the wave is \perp or w.r.t. the surface. When the coefficients of reflection r and transmission t are defined as. I will briefly introduce the notations in this formulation.

• $E_{0r}$: reflected amplitude
• $E_{0t}$: transmitted amplitude

• #### Fresnel Equations

##### Equation

$r_\parallel=\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)} \\ r_\perp =\frac{\sin(\theta_t-\theta_i)}{\sin(\theta_t+\theta_i)} \\ t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)\cos(\theta_t-\theta_i)} \\ t_\perp =\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)}$
##### Latex Code
            r_\parallel=\frac{\tan(\theta_i-\theta_t)}{\tan(\theta_i+\theta_t)} \\
r_\perp =\frac{\sin(\theta_t-\theta_i)}{\sin(\theta_t+\theta_i)} \\
t_\parallel=\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)\cos(\theta_t-\theta_i)} \\
t_\perp =\frac{2\sin(\theta_t)\cos(\theta_i)}{\sin(\theta_t+\theta_i)}

##### Explanation

• $t_\perp-r_\perp=1$
• $t_\parallel+r_\parallel=1$

• #### Polarization of Optics

##### Equation

$P=\frac{I_{\rm p}}{I_{\rm p}+I_{\rm u}}=\frac{I_{\rm max}-I_{\rm min}}{I_{\rm max}+I_{\rm min}}$
##### Latex Code
            P=\frac{I_{\rm p}}{I_{\rm p}+I_{\rm u}}=\frac{I_{\rm max}-I_{\rm min}}{I_{\rm max}+I_{\rm min}} \\
I(\theta)=I(0)\cos^2(\theta)

##### Explanation

• ${I_{\rm p}$: is the intensity of the polarized light
• ${I_{\rm u}$: is the intensity of the unpolarized light
• $I_{\rm max},I_{\rm min}$: the maximum and minimum intensities when the light passes a polarizer
• $I(\theta)=I(0)\cos^2(\theta)$: Malus law applies
• $\theta$: is the angle of the polarizer

• #### Prisms and Dispersion

##### Equation

$\delta=\theta_i+\theta_{i'}+\alpha \\ n=\frac{\sin(\frac{1}{2}(\delta_{\rm min}+\alpha))}{\sin(\frac{1}{2}\alpha)} \\ D=\frac{d\delta}{d\lambda}=\frac{d\delta}{dn}\frac{dn}{d\lambda} \\ \frac{d\delta}{dn}=\frac{2\sin(\frac{1}{2}\alpha)}{\cos(\frac{1}{2}(\delta_{\rm min}+\alpha))}$
##### Latex Code
            \delta=\theta_i+\theta_{i'}+\alpha \\
n=\frac{\sin(\frac{1}{2}(\delta_{\rm min}+\alpha))}{\sin(\frac{1}{2}\alpha)} \\
D=\frac{d\delta}{d\lambda}=\frac{d\delta}{dn}\frac{dn}{d\lambda} \\
\frac{d\delta}{dn}=\frac{2\sin(\frac{1}{2}\alpha)}{\cos(\frac{1}{2}(\delta_{\rm min}+\alpha))}

##### Explanation

A light ray passing through a prism is refracted twice and aquires a deviation from its original direction.

• $\alpha$: is apex angle
• $\theta_i$: is the angle between the incident angle and a line perpendicular to the surface
• $\theta_{i'}$: is the angle between the ray leaving the prism and a line perpendicular to the surface
• $n$: is refractive index of the prism
• $D$: is dispersion of a prism

• #### Diffraction

##### Equation

$\frac{I(\theta)}{I_0}=\left(\frac{\sin(u)}{u}\right)^2\cdot \left(\frac{\sin(Nv)}{\sin(v)}\right)^2$
##### Latex Code
            \frac{I(\theta)}{I_0}=\left(\frac{\sin(u)}{u}\right)^2\cdot \left(\frac{\sin(Nv)}{\sin(v)}\right)^2 \\
u=\pi b\sin(\theta)/\lambda \\
v=\pi d\sin(\theta)/\lambda

##### Explanation

A light ray passing through a prism is refracted twice and aquires a deviation from its original direction.

• $N$: is the number of slits
• $b$: is the width of a slit
• $d$: is distance between the slits
• $d\sin(\theta)=k\lambda$: is the maxima in intensity
• $\frac{I(\theta)}{I_0}=\left(\frac{J_1(ka\sin(\theta))}{ka\sin(\theta)}\right)^2$: The diffraction through a spherical aperture with radius \alpha
• $\frac{I(x,y)}{I_0}=\left(\frac{\sin(\alpha')}{\alpha'}\right)^2\left(\frac{\sin(\beta')}{\beta'}\right)^2$: The diffraction pattern of a rectangular aperture at distance R with length \alpha in the x -direction and b in the y-direction

• #### Birefringence and Dichroism

##### Latex Code

##### Explanation

• $\vec{D}$ is not parallel with $\vec{E}$ if the polarizability $\vec{P}$ of a material is not equal in all directions. There are at least three directions, the principal axes, in which they are parallel. This results in three refractive indices ð???ð??? which can be used to construct Fresnelâ??s ellipsoid. In the case $n_2=n_3\neq n_1$, which happens e.g. in trigonal, hexagonal and tetragonal crystals there is one optical axis in the direction of n1. Incident light rays can now be split up in two parts: the ordinary wave is linearly polarized the plane through the transmission direction and the optical axis. The extraordinary wave is linearly polarized in the plane through the transmission direction and the optical axis. Dichroism is caused by a differential absorption of the ordinary and extraordinary wave in some materials. Double images occur when the incident ray makes an angle with the optical axis: the extraordinary wave will refract, the ordinary will not.

• #### Retarders: waveplates and compensators

##### Latex Code

##### Explanation

Incident light will have a phase shift of $\Delta\varphi=2\pi d(|n_0-n_{\rm e}|)/\lambda_0$ if an uniaxial crystal is cut in such a way that the optical axis is parallel with the front and back plane. Here, $\lambda_0$ is the wavelength in vacuum and $n_0$ and $n_e$ the refractive indices for the ordinary and extraordinary wave. For a quarter-wave plate: $\Delta\varphi=\pi/2$.

• #### The Fabry-Perot interferometer

##### Equation

$\frac{I_t}{I_i}=\left[1-\frac{A}{1-R}\right]^2\frac{1}{1+F\sin^2(\theta)}$
##### Latex Code
            T+R+A=1 \\
\frac{I_t}{I_i}=\left[1-\frac{A}{1-R}\right]^2\frac{1}{1+F\sin^2(\theta)} \\
F=4R/(1-R)^2

##### Explanation

For a Fabry-Perot interferometer, it follows for the intensity distribution as above.

• $T$: is the transmission factor
• $R$: is the reflection factor
• $A$: is the absorption factor
• $[1+F\sin^2(\theta)]^{-1}:={\cal A}(\theta)$: Airy function