handbook of modern sensors

4.6 Fresnel Lenses 139
(A)
(B)
Fig. 4.14. Constant-step (A) and constant-depth (B) Fresnel lenses.
from the center. A central portion of the lens may be flat if its diameter is at least 20
times smaller than the focal length. For the shorter focal lengths, it is good practice
to maintain a spherical profile of a central portion. The slope angle of each step may
be determined from the following formula, which is valid for small values of h:
(
ϕ = arctan
hn
f(n− 1)
)
, (4.28)
where f is the focal length.
For a constant-depth lens, both the slope angle ϕ and the step distance r vary
with the distance from the center. The following equations may be useful for the
lens calculation. The distance of a groove from the center may be found through the
groove number ξ (assuming the center portion has number 0);
√
h = 2f(n− 1)ξ d − ξ 2 d 2 , (4.29)
and the slope angle is
(
h
ϕ = arcsin
(n − 1)f
)
. (4.30)
The total number of grooves in the lens may be found through a Fresnel lens aperture
(maximum dimension) D:
Ɣ = (n − 1)f − √ f 2 (n − 1) 2 − D 2
. (4.31)
d
The Fresnel lens may be slightly bent if it is required for a sensor design. However, a
bend changes the positions of the focal points. If a lens is bent with its groves inside
the curvature, the focal distance decreases.