**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 --
( to the
plane of incidence, TE), i.e.` `
&
and ( to the plane of incidence, TM),
i.e.` `
&
.` `

Follow a systemic
, we can write down
's, 's and 's.` `

Apply boundary conditions to match fields at the boundary

Matching phase --
,
.` `

Observation --
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 --
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:

+ ` `

+ 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 so that reflected ray is on the transmission side

Planar mirror

-- focal length (concave mirror
R<0) and

Transmission devices;

Planar boundary -- .` `

-- deflection angle where
is the apex angle of the prism.` `

-- .` `

-- 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 --
where is the`max. acceptance angle.` `

--
,
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:

,

,

,

,

,

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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