ECEN 4616/5616
1/23/13
Paraxial Optics
Sign Rules:
1. The (local) origin is at the intersection of the surface and the z-axis.
2. All distances are measured from the origin: Right and Up are positive
3. All angles are acute. They are measured from:
a. From the z-axis to the ray
b. From the surface normal to the ray
c. CCW is positive, CW is negative
4. Indices of refraction are positive for rays going left-to-right; negative for
rays going right-to-left. (Normal is left-to-right.)
Paraxial Assumptions:
1. Surface sag is ignored as negligible.
2. angles  sines  tangents
(Using angle = tangent allows “paraxial” rays to be of finite height)
Postulates for Ray Propagation:
1. Rays are normal to the wavefronts and vice-versa.
2. Rays indicate the direction of energy propagation.
3. Rays satisfy the laws of reflection and refraction:
a. 𝑛 sin(𝑖) = 𝑛′ sin⁡(𝑖 ′ ) (refraction)
b. 𝑖 = −𝑖 ′ (reflection)
4. The Optical Path Length (OPL) along a ray is the distance light would
have traveled in the same time in free space.
i.e., Given a physical length, l, and index of refraction, n, the time of
𝑙
𝑙
𝑛𝑙
travel is 𝑡 = 𝑣 = 𝑐⁄ = 𝑐 , so the OPL = n∙l
𝑛
5. The irradiance at any point is proportional to the ray density at that point.
(Irradiance is defined as incident power per unit area.)
6. After reflection, indices are negative.
(Note on Problem 6: For an electromagnetic wave in non-magnetic media, the
2
𝜀0
power is 𝑃 = 𝐸⁡𝐻 = 𝑛𝐸 ( )
1⁄
2
𝜇0
, where E, H are the amplitudes of the electric and
magnetic fields, n is the index of refraction, and 𝜀0 , 𝜇0 are the permittivity and
permeability of free space.)
From the above postulates, a number of useful rules can be derived:
1. All rays from an object point to an image point have the same OPL.
(Why? Consider that an object point is the source and it’s
corresponding image point is the sink of the same spherical wave.)
2. Two different rays from an object point can only cross at an image point.
3. The OPL along any rays between two given wavefronts are equal.
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Using the paraxial assumptions and ray postulates, we have derived the raytracing equations for thin lenses. (A ‘thin’ lens is one whose thickness can be
neglected.):
1. The diffraction equation giving the change in angle of a ray through a
surface:
nu  nu  hK .
In the case of a thin lens in air, this reduces to:
u  u  hK
2. The transfer equation, giving the ray’s height at the next surface):
hi 1  hi  diui
3. And the imaging equation:
1 1 1
= −
𝐹 0 𝐼
The focal length, F, of a thin lens is defined as the distance from the lens at
which an object point at infinity is imaged.
Graphical Ray Tracing (thin lenses):
Given the above ray tracing equations, there are a number of rays that can be
drawn by inspection, without need of calculation for thin lenses:
1. An incident ray parallel to the axis crosses the axis at F’ after
refraction.
2. A ray crossing the axis at F will emerge parallel to the axis after
refraction.
3. A ray crossing the axis at the origin (i.e., where the lens is) will be
undeviated.
Graphical Ray Tracing for Mirrors:
1. A ray parallel to the axis crosses the axis at F’ after reflection.
2. A ray crossing the axis at F’ will emerge parallel to the axis after
reflection.
3. A ray crossing the axis at the center of curvature will be reflected back
on itself.
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Focal length of a mirror:
(C is the radius of curvature)
i
i’
h
a
b
F’
C
Note that:




i>0
i’ < 0
a = i = h/C
b = i-i’ = 2i = h/F’
hence, F = C/2
pg. 3