**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 --
(`_|_` to the
plane of incidence, TE), i.e.` `
`r_{_|_}`
&
`tau_{_|_}`
and (`||`
to the plane of incidence, TM),
i.e.` `
`r_{||}`
&
`tau_{||}`.` `

Follow a systemic
, 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 --
`theta_r = theta_i`, `n_1 sin theta_i = n_2 sin theta_t`.` `

Observation --
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

.` `

.` `

Observation --
`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
.` `

•
Conventions for optical elements:

+ ` `
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

Planar mirror

•
Transmission devices;

Planar boundary -- `theta_2 = {n_1}/{n_2} theta_{1}`.` `

` `

` `

` `

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 -- ` `

` `

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:

Graded-index element.` `

*
Independence of convention, `f>0` for converging lens and mirrors
while `f < 0` for diverging lens and mirrors.` `

•
` `

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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