Reflection & Transmission Sept. 25, 2017

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

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

[equation] Ray: Ray travels in straight line in homogeneous media and optical path = n d.
Ray travels in curve in inhomogeneous media and optical path [equation].
Ray will seek the path of least time (i.e. smallest optical path), i.e.
Fermat's Principle [equation].
Example -- Snell's law [equation], [equation].

[equation] 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.

[equation] Reflective devices: You can unfold the ray so that reflected ray is on the transmission side
Planar mirror
Spherical mirror -- focal length [equation] (concave mirror R<0) and [equation]

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

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

[equation] Ray matrices: Each element is represented by a [equation] ray matrices [equation].
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.

[equation] 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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