Useful Formulas
Paraxial Optics:
n’
n
i
i’
h
u
u’
a
l
R
l’
Sign Rules:
1.
2.
3.
4.
The (local) origin is at the intersection of the surface and the z-axis.
All distances are measured from the origin: Right and Up are positive.
All distances are directed and add as such.
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
5. 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
NOTE: If “angle variables” (e.g., ui, ui’) are considered to be tangents of the ray
angles, then Gaussian ray tracing works for finite apertures and represents
“perfect” optical surfaces/lenses.
Notation Conventions:
1. Unprimed values are before the surface (or lens/system).
2. Primed values are after passing the surface (or lens/system).
3. Some variables have only one (unprimed) value:
a. Curvature, c, and power, K
b. Distance, dn is from the nth surface to the (n+1)th surface or lens.
Paraxial Ray Tracing Equations:
Refract at a surface, with index n before refraction and n’
nu  nu  hK
after
u  u  hK
hi 1  hi  diui
Refract through a thin lens in air
Transfer between surfaces separated by distance d
Paraxial Imaging Equations:
n n
 K
l l
1 1
 K
l l
li 1  li  di
Image location from a surface
Image location from a thin lens in air
Image to next lens’ object transfer
Graphical Ray Tracing Through Paraxial lenses or surfaces
I)
Incident rays parallel to the axis, pass through the focal point.
II)
Rays that pass through the focal point emerge parallel to the axis.
III)
Rays through the center of the lens are undeviated.
IV)
Once the image has been identified (by the intersection of any two of the
above rays, say; or by using the thin lens imaging equation), any ray from
a given object point will arrive at the corresponding image point.
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:
d
2
K  n  1c1  c2   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:
1
WFNO 
,
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.
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 Fnumber, 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 
“Optimum” is somewhat subjective here – D is the diameter where the first null
ring of the Airy Pattern for a point source (seen through the pinhole) equals the
diameter of the pinhole at the “focal” distance, 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(  ) ,
(eq 14a)
where  is the half-angle of a light cone entering or leaving an optical system
NA
f MAX 
(eq 14b)
,

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)
Dispersion
Standard wavelengths for dispersion correction in the visible:



F = 0.48613 µm
d = 0.58756 µm
C = 0.65627 µm
(F line of hydrogen)
(d line of helium)
(C line of hydrogen)
Abbe number (V-number):
𝑉𝑑 =
(eq 15)
𝑛𝑑 −1
𝑛𝐹 −𝑛𝐶
Power change of a lens between the F and C wavelengths (from the thin lens
equation):
𝐾𝐹 − 𝐾𝐶 =
(eq 16)
𝐾𝑑
𝑉𝑑
Hence, for two thin lenses to have zero combined power change between F and
C wavelengths, the following relations must be simultaneously met:
(eqs 17)
𝐾 = 𝐾1 + 𝐾2 ,
𝐾1
𝑉1
𝐾
+ 𝑉2 = 0
2
Leading to the equations for achromatic doublets:
(eqs 18)
𝐾1 =
𝑉1 𝐾
𝑉1 −𝑉2
, 𝐾2 = −
𝑉2 𝐾
𝑉1 −𝑉2
The Sag equation for conic sections with the vertex at the origin:
Z
(eq 19)
cr 2
1   k  1 c r 
1
2
2
Z
r
R
Where:
 r is the distance from the Z-axis
 Z is the ‘Sag’ of the surface (Z-distance from the vertex)
 c is the surface curvature: c = 1/R
Value of k
Surface Described
k>0
k=0
-1 < k < 0
k = -1
k < -1
Oblate ellipsoid
Sphere
Ellipse
Parabola
Hyperbola
Radiometry:





Flux (Φ): Just another term for power; J/s, but referring to
propagating light.
Intensity (I): Flux per unit solid angle; W/sr (sr = steriadian)
Irradiance (E): Flux per unit area, incident; W/m2
Exitance (M): Flux per unit area, exiting; W/m2
Radiance (L): Flux per unit projected area per unit solid angle;
W/(A∙sr)
Solid Angle:
The solid angle of a circular cone with half angle θ is:
(eq 20):
  2 1  cos θ 
For small θ such that
(eq 21):
θ
sin    , cos   1
   2
Camera Equation:
E
Ω

R
4 F/# 2
Where:
 E is the irradiance of the image (W/m2 )
 R is the radiance of the object (W/(m2 ∙ sr)
 And F/# is the F-number of the lens.