# Athermalization Model - of a Simple Imaging Module

###### By Yuval Sharon M.Sc.

**Introduction**

Thermal effects can reduce the quality of an optical image,
and should therefore be considered in the design process.
Thermal changes may affect the focusing as well as positioning
drifts that may cause a boresight error. In the following discussion
we exemplify one method to athermalize a simple PCX lens.

**Athermalization Principle**

Consider the following geometry:

Thermal changes cause:

a) The lens’ radius of curvature- R, to increase with temp,
causing the focal distance to increase

b) The refractive index - “n”, to decreases with temp,
causing the focal distance to increase

c) The housing to expand with temperature – causing the image
distance to the focal plane to increase,
thus tending to cancel out the effects of a and b.

d) The distance between the lens’ vertex to its clamping point
to the housing to increase (if within the central thickness) – causing the image distance
to the focal plane to increase, thus tending to cancel out the effects of a and b.

In the following analysis we assume that stress effects caused by thermal changes
are not dominant. This means that that the lens will expand and contract independent
of its outer housing. We will first relate to a front clamping of the lens as in
the above figure – then we will derive other cases.

The focus of a PCX lens is given by:

Where R is the radius of the convex surface.
Thus,

Or:

Since
where
is the thermal expansion coefficient and
is the radius
at the middle temperature range, we get:

**Examples**

Example 1: An all PC solution

Substituting for PC: αe=α=6.8E-5 °C-1, dn/dT= -13E-5, Ro=2.77mm, Ct=3.6mm, Wo≈f+Ct

We obtain a defocus of: 1.1μ / oC (a drift of 33μ per 30 oC)

Example 2: Lens from PC and body from PE – lens is clamped near the Aperture

Substituting for PE: αe = 20x10-5 °C-1, α=6.8E-5 °C-1,
dn/dT= -13E-5, Ro=2.77mm, Ct=3.6mm, Wo≈f+Ct.

We obtain a defocus of: -0.14μ / oC (a drift of -4.2μ per 30 oC)

Example 3: Lens from PC and body from PE – Lens is clamped around vertex

Substituting: Ct=0, Wo≈L≈f, PE αe = 20x10-5 °C-1, α=6.8E-5 °C-1, dn/dT= -13E-5, Ro=2.77mm,

We obtain a defocus of: 0.46μ / oC (a drift of 13.8 μ per 30 oC)

**Conclusions**

Athermalization could be achieved in theory,
although more degrees of freedom may most
likely be involved than in the above simplified example.
The design should relate to the packaging materials and
to the clamping points in addition to the optical design.

**About the Author**

See Yuval Sharon’s profile at: http://il.linkedin.com/in/ysharon
Yuval is one of the owners of CSTM Ltd, a Jerusalem based engineering company,
which undertakes projects involving Optics, Mechanics and Electronics.

Yuval has earned his B.Sc. (1984) in Electro-Optics from the JCT,
and his M.Sc. (1992) from the Hebrew university.

Yuval has been working in the industry since1983,
and has experience in plastic optics, light sources,
diffractive optics, QA, manufacturing,
process control and electrochemistry.

Website: www.cstm.co.il, Email: yuval@cstm.co.il

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