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 themax. 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:
space,
lens,
mirrors,
planar dielectric interface,
spherical dielectric interface,
* Independence of convention, f>0 for converging lens and mirrors while f < 0` for diverging lens and mirrors.