ECEN 4616/5616
2/20/2013
Paraxial Optics Review
h
Coordinates:
2
1
z
-3
-2
0
-1
1
2
3
-1
-2

Coordinates for ray tracing are local Cartesian coordinates, centered on
the intersection of the surface with the z-axis.

All distances are directed (signed), and measured from the coordinate
origin. (i.e., points above and to the right of the origin are positive
distances, points to the left and below are negative distances.)

Rays are always directed lines. Ray angles are either measured w.r.t. the
z-axis, or the surface normal (as appropriate).

All angles are acute. If the rotation direction from the reference (z-axis or
surface normal) to the ray is CCW, the angle is positive. If CW, the ray is
negative.

All angles are equivalent to slopes (tangents) for purposes of ray
propagation, and sines, for purposes of refraction.
pg. 1
ECEN 4616/5616
2/20/2013
Example:
n
n’
P
i
i’
h
u
u’
a
O
A
z
A’
R
l
l’
Notation Rules:
Unprimed rays, angles and distances are before the ray is refracted by the
surface, primed values are after.
Paraxial Assumptions:
1. Surface sag is ignored as negligible.
2. angles  sines  tangents
Positive Values:
 Angles: 𝑢, 𝑖, 𝑖 ′
 Lengths: 𝑙 ′ , ℎ, 𝑅
Negative Values:
 Angles: 𝑎, 𝑢′
 Lengths: 𝑙
Geometric relationships (taking the sign rules into account):
𝑖 = |𝑎| + |𝑢| = 𝑢 − 𝑎,
𝑖 ′ = |𝑎 | − |𝑢 ′ | = 𝑢 ′ − 𝑎
(𝑠𝑖𝑛𝑐𝑒 𝑢 > 0,
(𝑠𝑖𝑛𝑐𝑒 𝑢′ < 0,
𝑎 < 0)
𝑎 < 0)
|ℎ|
ℎ
=−
|𝑅|
𝑅
|ℎ|
ℎ
𝑢=
=−
|𝑙|
𝑙
|ℎ|
ℎ
𝑢′ = − ′ = − ′
|𝑙 |
𝑙
𝑎=−
pg. 2
ECEN 4616/5616
2/20/2013
Snell’s Law (Paraxial)
Relates the quantities to the left of the surface with those to the right:
𝑛𝑖 = 𝑛′ 𝑖 ′
𝑛(𝑢 − 𝑎) = 𝑛′ (𝑢′ − 𝑎)
Ray Trace Equations:
If we substitute for the incident and refracted angles in Snell’s Law their
expressions in terms of lengths, but leave the ray angles, we arrive at:
𝑛′ 𝑢′ − 𝑛𝑢 = −ℎ𝐾,
Where 𝐾 ≡ 𝑐(𝑛′ − 𝑛) is the surface power
1
and 𝑐 ≡ 𝑅 is the surface curvature
If we trace through sequential surfaces numbered 𝑖 = 1,2, ⋯ we can derive the
ray transfer equation from the ray’s direction and height and the distance, di,
between surface i and surface i+1:
ℎ𝑖+1 = ℎ𝑖 + 𝑑𝑖 𝑢𝑖′
Thin Lens in Air: If we trace a ray through two surfaces, assuming the distance
between them is negligible (d=0), we get the thin lens equation:
𝑢′ − 𝑢 = −ℎ𝐾
Where 𝐾 = (𝑛 − 1)(𝑐1 − 𝑐2 ) is the power of the thin lens
Imaging Equations:
If we again start with the Paraxial Snell’s Law expression, but substitute for all
the angles in terms of the relevant distances, we can derive a similar set of
equations which trace the position of images through a surface, lens, or series of
lenses:
Imaging through a surface:
𝑛′ 𝑛
− =𝐾
𝑙′ 𝑙
Imaging through a thin lens:
1 1
− =𝐾
𝑙′ 𝑙
and image transfer between lenses:
𝑙𝑖+1 = 𝑙𝑖′ − 𝑑𝑖
Analysis of the image equations leads to the concept of focal length (=1/K).
pg. 3
ECEN 4616/5616
2/20/2013
Multiple Lenses, Thick Lenses, etc:
Iteratively applying the above equations to multiple surfaces or lenses allows the
calculation of the thick lens equation, the lens combination equation, and the
derivation of the concept and usage of Principle Planes.
See notes from 1/18/2013 and 1/23/2013 for details.
pg. 4