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Appendix A: Useful Formulas
Sign and Variable Conventions:
n’
n
i
i’
h
u
u’
a
l
R
l’
1)
2)
3)
4)
The origin is taken as the intersection of the surface and the axis.
Lengths measured to the right of the origin are positive; to the left are negative.
Lengths measured above the axis are positive, below are negative.
Ray angles with the axis are measured from the axis to the ray, through the
smallest of the two angles formed by the intersection.
5) Ray angle of incidence is measured from the surface normal to the ray, through
the smallest angle formed by the intersection.
6) Angles measured Counter Clockwise (CCW) are positive, CW are negative.
Eq 1: Power of a surface:
K  cn  n  
1
f
where c is the curvature (1/radius) of the surface,
n is the index of refraction before the surface, and
n’ is the index after the surface.
Eq 2: Power of a thin lens, with material index = n, in air:
K  (n  1)c1  c2 
Eq 3: Power of two thin lenses, separated by distance, d, with a material between that has
index, n:
d
K  K1  K 2  K1 K 2
n
Eq 4: “Lensmakers equation” for the power of a thick lens, in air, with material index =n:
ECEN 4616/5616
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K  n  1c1  c2  
d
2
c1c2 n  1
n
Eqs 5: Paraxial ray trace formulas:
niui  ni ui  hi K i
hi 1  hi  d i ui
Eq 6: Imaging (conjugate) equation for a lens (or system) with image-space index = n,
object-space index = n’:
n n n
  ,
l l
f
where f is the focal length of the lens (system) in air.
Eq 7: Petzval Curvature sum:
k
Kj
1
 nk 
rP
j 1 n j 1n j
where:
rP is the radius of curvature of the image plane,
Kj is the power of the jth surface,
nj-1 is the index before the jth surface, and
nj is the index after the jth surface.
Eq 8: Alternate combination of power formula:
If a paraxial ray parallel to the axis is traced through a sequence of thin lenses, the total
system power can be written in terms of the individual lens powers and the ray intercept
heights on each lens as:
h
h
K  K1  2 K 2  3 K 3  
h1
h1
Eq 9: The Optical Invariant:
H  nuI  nuI 
Where n is the index, u is the paraxial angle variable, and I is the image or object
height. Primed variables are after the surface, lens, or system; and unprimed
variables are before. The ray angles are for a marginal ray traced from the axis at
the object distance.
Eq 10: Working F# (WFNO):
For a system with the image space in air (n=1), the working F# is defined as:
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1
,
2 sin(  )
Where  is the image-space angle of the marginal ray from the axis at the object distance.
Note that, for objects at  and optical systems of moderate aperture, WFNO  F#  f/D,
where f is the system focal length and D is the clear diameter of the lens.
WFNO 
Eqs 10: Depth of Focus/Field:
a) Depth of focus, dof = 2*b*F#
b) Depth of Field, DOF = dof/M2
Where b is the “acceptable blur circle diameter”, F# is the system working F-number, and
M is the magnification.
Eq 11: Hyperfocal distance:
The hyperfocal distance is the point of focus (in object space) for a lens such that all
objects from  to the hyperfocal near-point are adequately in focus.
f2
lH  f 
, (hyperfocal distance)
bF #
f2
, (hyperfocal near-point)
2bF #
Where f is the system focal length, b is the acceptable blur circle diameter, and F# is the
system working F-number.
lN  f 
Eq 12: Optimum Pinhole Diameter for a pinhole camera:
D  1.56 f 
Eqs 13: Relations between the Point Spread Function (psf), Pupil wavefront error
function (P), Optical Transfer Function (OTF), and Modulation Transfer Function
(MTF). ‘F{x}’ indicates the Fourier Transform of x.
psf  FP
2
OTF  Fpsf 

 F FP

2


 F FP FP
 P  P (autocorre lation the orem)
and

MTF  OTF
Eqs 14: Resolution and Aperture:
NA  sin(  ) ,
where  is the half-angle of a light cone entering or leaving an optical system
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f MAX 
NA
,

where fMAX is the maximum spatial frequency a light cone can produce on an image (or,
conversely, the minimum cone angle needed to resolve an object with maximum spatial
frequency = fMAX)