Reflection & Transmission Sept. 25, 2017

Reflection & refraction: amplitude reflectance (field reflection coef) , amplitude transmittance (field transmission coef)
intensity coef. (power reflectance) and T = {n_2 cos theta_t} / {n_1 cos theta_i} | tau |^2 (power transmittance) with .
Two polarizations -- S ( to the plane of incidence, TE), i.e. & and P ( to the plane of incidence, TM), i.e. & .
Follow a systemic procedure in labeling propagation and field directions, we can write down 's, 's and 's.
Apply boundary conditions to match fields at the boundary
Matching phase -- Snell's law , .
Observation -- totally internal reflection with critical angle for where is the refractive index on the incident side and is the refractive index on the transmission side.
Matching amplitude -- Fresnel equations
, .
, .
Observation -- Brewster's angle for p-polarization at which .

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 .
Ray will seek the path of least time (i.e. smallest optical path), i.e.
Fermat's Principle .
Example -- Snell's law , .

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 (concave mirror R<0) and

Transmission devices;
Planar boundary -- .
Prism -- deflection angle where is the apex angle of the prism.
Spherical boundary -- .
Thin lens -- lens maker formula following book's convention.
Other books have where & > 0 for convex surfaces.
Note that focal length f >0 for converging lens disregarding the convention used.
where is object distance and is image distance.
Light guide -- numerical aperture where is the`max. acceptance angle.
Graded-index (lens like) medium -- , where is pitch length.
Ray trajectory determined by which has solution
For , we have and similarly, . We can represent x or y by r and write .

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

Ray matrices: Each element is represented by a ray matrices .
Basic elements:
space,
lens,
mirrors,
planar dielectric interface,
spherical dielectric interface,