Reflection & Transmission `\ \ \ \ \ \ `Sept. 25, 2017

• Reflection & refraction: amplitude reflectance (field reflection coef) `r = {E_r} / {E_i)`, amplitude transmittance (field transmission coef) `tau = {E_t} / {E_i)`;
intensity coef. `R = |r|^2` (power reflectance) and `T = {n_2 cos theta_t} / {n_1 cos theta_i} | tau |^2` (power transmittance) with `R + T =1`.
Two polarizations -- S (`_|_` to the plane of incidence, TE), i.e. `r_{_|_}` & `tau_{_|_}` and P (`||` to the plane of incidence, TM), i.e. `r_{||}` & `tau_{||}`.
Follow a systemic procedure in labeling propagation and field directions, we can write down `hat E`'s, `hat H`'s and `hat k`'s.
Apply boundary conditions to match fields at the boundary
Matching phase -- Snell's law `theta_r = theta_i`, `n_1 sin theta_i = n_2 sin theta_t`.
Observation -- totally internal reflection with critical angle `theta_c = sin^{-1} (n_2 / n_1 )` for `n_1 > n_2` where `n_1` is the refractive index on the incident side and `n_2` is the refractive index on the transmission side.
Matching amplitude -- Fresnel equations
`r_{_|_} = -{sin(theta_i - theta_t)}/{sin(theta_i+theta_t)`, `tau_{_|_}= r_{_|_} + 1`.
`r_{||} = - {tan(theta_i - theta_t)}/{tan(theta_i+theta_t)`, `tau_{||}= (r_{||} + 1) {cos theta_i}/{cos theta_t}`.
Observation -- Brewster's angle `theta_B = tan^{-1} ( n_2 / n_1 )` for p-polarization at which `r_{||}=0`.

Ray (Geometric) Optics (Ch. 1) `\ \ \ \ \ \ `Sept. 25, 2017

• Ray: Ray travels in straight line in homogeneous media and optical path `= n d`.
Ray travels in curve in inhomogeneous media and optical path `= int_A^B n( vec r ) ds`.
Ray will seek the path of least time (i.e. smallest optical path), i.e.
Fermat's Principle `delta int_A^B n( vec r ) ds =0`.
Example -- Snell's law `theta_r = theta_i`, `n_1 sin theta_i = n_2 sin theta_t`.

• Conventions for optical elements:
+ Each element has an optical axis (horizontal z) through its center and locate at z=0 (origin), i.e. z<0 at input side and z>0 at transmission side. The vertical axis can be x or y or r.
+ For spherical surface, its radius of curvature R < 0 if its center is on the left (input side).

+ For image formation, distance is negative for virtual image, i.e. in front of a lens or distance changes polarity in case of a virtual image behind a mirror.
+ The height is negative (y < 0) for a inverted image.
+ We focus on paraxial ray almost parallel to z, i.e. small incidence angle.

• Reflective devices: You can unfold the ray so that reflected ray is on the transmission side
Planar mirror
Spherical mirror -- focal length `f= -R/2` (concave mirror `R <0`) and `-z_1^{-1}-z_2^{-1} =f^{-1}`, `z_1 < 0` & `z_2 , 0`

• Transmission devices;
Planar boundary -- `theta_2 = {n_1}/{n_2} theta_{1}`.
Prism -- deflection angle `theta_d ~= (n-1) alpha` where `alpha` is the apex angle of the prism.
Spherical boundary -- `n_1 ( y/R + theta_1 ) = n_2 ( y /R + theta_2 )`.
Thin lens -- lens maker formula `n_{air} f^{-1} = ( n_{\l\ens} - n_{air})(R_1^{-1}-R_2^{-1})` following book's convention.
Other books have `n_{air} f^{-1} = (n_{\l\ens} - n_{air})(R_1^{-1}+R_2^{-1})` where `R_1` & `R_2 > 0` for convex surfaces.
Note that focal length `f > 0` for converging lens disregarding the convention used.
`n_{air} f^{-1} = -n_{air} z_1^{-1} + n_{air} z_2^{-1}` where `z_1 < 0` is object distance and `z_2` is image distance.
Light guide -- numerical aperture `NA = n_{air} sin theta_a = sqrt{ n_1^2 - n_2^2}` where `theta_a` is the`max. acceptance angle.
Graded-index (lens like) medium -- `n^2 ( y)=n_o^2 ( 1 - alpha^2 y^2 )`, `n(y) ~= n_o ( 1 - {alpha^2 y^2}/2)` where `{2 pi} / alpha` is pitch length.
Ray trajectory determined by `{d^2 y}/{dz^2} = 1/n {dn}/{dy}` which has solution `y= y_o cos alpha z + {theta_o}/alpha sin alpha z`
For `n^2 (r) = n_o^2 ( 1 - alpha^2 (x^2 + y^2 ) )`, we have `{d^2 y}/{dz^2} = 1/n {partial n}/ {partial y}` and similarly, `{d^2 x}/{dz^2} = 1/n {partial n}/ {partial x}`. We can represent `x` or `y` by `r` and write `{d^2 r}/{dz^2} = 1/n {partial n}/ {partial r}`.

• Specify a ray on a transverse plane by its radial position `r` (`x` or `y`) relative to optical axis and slope `r'` or `theta`.

• Ray matrices: Each element is represented by a `2 times 2` ray matrices `([A, B], [C, D])`.
Basic elements:
planar dielectric interface,
spherical dielectric interface,
Graded-index element.
* Independence of convention, `f>0` for converging lens and mirrors while `f < 0` for diverging lens and mirrors.

Cascading basic elements, we can construct ray matrices for any optical system.
E.g. ray matrix of a cavity with mirrors -- find an equivalent unit cell
Steps: replace mirror by equivalent lens
Unfold the path and form a linear array of lenses
Identify the unit cell which has no identical elements.
Note: Unit cell may not have the length of a round trip.
Pick the beginning coinciding with the point of interest.

Last Modified: Sept. 24, 2017
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