Lenses in the Paraxial Limit
Perfect imaging defined:
For a nearby object, light from
every point on the object is
relayed to a corresponding point
on the image.
For very distant objects, the
incoming light is essentially
plane waves, which the lens
focuses to points on the
image.
In general, spherical lenses do
not image perfectly:
But if the rays are confined
near the center of the lens,
imaging approaches
perfect.
The Paraxial Approximation
Using the “sag” form
of the conic surface …
Z
cr 2
1  1  (1  k )c r
2 2
… and the Taylor
f (0) 2
f ( x )  f (0)  f (0)x 
x 
Expansion about the
2!
origin…
We get:
,
f  n  (0) n

x 
n!
1
1
1
2
Z  cr 2  c 3r 4  k  1  c 5r 6  k  1 
2
8
16
The “Paraxial Approximation” will be defined as the
region close enough to the Z-axis (r is small enough)
such that the surface is accurately described as:
1 2
Z  cr
2
The Effect of a Lens on a Wave in the Paraxial Approximation
Consider a thin lens in the paraxial region, with surface curvatures c , c
1
2
and a plane wave incident on its left side:
f
Note:
c1  0
c2  0
The effect of the lens on an incident plane wave is to delay the wave by
(n-1)t, where n is the lens’ index of refraction (and assuming the
surrounding material is of index n=1), and
1
1
t  r   t0  c1r 2  c2r 2
2
2
(where t0 is the thickness at the Z-axis.)
Just to the right of the lens the total delay w.r.t. the wave’s
vertex is:
2
D  r    n  1 t0  t    n  1
r
 c1c2 
2
Which we recognize as a paraxial sphere with curvature C   n  1 c  c  .
1
2
The emerging wave therefore converges to a point 1/C past the lens. This, we
call the ‘focal length’, f, of the lens.
The “focal length” of a lens is the distance from the lens, f, that a plane
wave will be focused to a point. We define the “power” of a lens as:
1
K
f
From the previous calculation, we see that the power of
a thin lens (in the paraxial approximation) is:
1
K    n  1 c1  c2  .
f
This is the “thin
lens” equation.
We can also see, from the calculation of the wave delay caused by a lens, that a
lens of power K will simply add curvature K to the curvature of any incoming wave.
K
i
Consider a wave diverging from
o
point ‘o’, passing a lens of power
‘K’, and converging to point ‘I’:
At the lens, the incoming wave has curvature 1/o, the lens adds curvature K, and the
outgoing wave has curvature 1/i. Hence 1o  K  1i , or:
This is the
1 1 1
“Imaging Equation”
 
f
i
o
Paraxial Ray Tracing
Although we have derived some basic imaging equations using only the
paraxial approximation and the wave nature of light (without even invoking
Snell’s Law), the ray approximation will be more useful than wave analysis for
analyzing most real optical systems, so we will look at rays in the paraxial
approximation.
Expanding sin(θ) about zero gives:
sin     
3 5 7
3!

5!

7!

As a consequence of staying in the region where the spherical surfaces
are well approximated by the first term in the Taylor expansion, the ray
angles will also all be small, so we can likewise approximate the sin with
the first Taylor term, sin 
 , so Snell’s Law becomes: n   n 
 
Additional paraxial assumptions we will use are:
1. For calculating distances, surface sag is ignored as negligible.
2. Angles ≈ sines ≈ tangents
1 1
2 2
Paraxial Ray Trace through a Surface – Variable Definitions:
n
n’
P
i
i’
h
u
u’
a
O
A
z
A’
R
l
l’
Angles of incidence and refraction:
Ray angles with the Z-axis:
iau
u a
(since u  0, a  0)
i   a  u
 u  a
(since u  0, a  0)
a
u
h
h

R
R
h
h
h
h
  , u    
l
l
l
l
Snell’s Law in the Paraxial Approx:
ni  ni 
Paraxial Ray Trace Equations: Imaging
Since we don’t want to keep calculating angles of incidence and refraction all
the time, we will eliminate those angles from Snell’s Law. There are two paths
we can take: First, we will eliminate all ray angles and only track object and
image distances (the ray intercepts with the Z-axis).
ni  ni 

n  u  a   n  u   a 
 h h
 h h
n      n    
 l R
 l R 
Hence


 h h
 h h
n      n    
 l R
 l R 
n
n
  nc    nc
l
l
1

where
c



R

n n
  c  n  n   K
l l

n n
  c  n  n   K

l
l
The Gaussian surface
imaging equation
Paraxial Ray Trace Equations: Refraction and Transfer
When tracing rays through multiple lens systems, it is often more convenient
to keep track of a ray’s angles and height (h,u, u’) rather than its Z-axis
intercept points (l,l’). Starting with Snell’s Law and eliminating references to
the intercept distances, we get:
ni  ni 
n  u  a   n  u   a 
h
h


n  u    n  u   
R
R


n  u  hc   n  u   hc 
nu   nu  hc  n  n   hK
Hence
nu  nu  hK
The Gaussian Refraction Equation for
a surface whose power is defined as:
c  n  n   K
Tracing Through Multiple Lenses
In order to trace either rays or images through multiple lenses (or surfaces), we
need transfer equations which calculate the input to the next lens from the output
of the previous lens.
Lens1
Lens2
h1
u1
l1
l2
u’1
u2
l’1
l’2
h2
u’2
d1
ui 1  ui'
For rays, the input angle of a surface is simply the output
angle of the previous surface:
For the imaging equation, the output image distance for surface i,
adjusted by the distance between surfaces, becomes the input
object distance for surface i+1:
For the refraction equation, the output height modified by
the output slope times d gives the input height for the next
surface:
l i 1  l i  di
hi 1  hi  di ui
Summary of Paraxial Ray Trace
Equations
1 1
 K
l l
Imaging through a thin lens:
Image transfer between lenses:
l i 1  l i  di
Refraction of a ray through a thin lens:
Ray transfer between lenses:
This assumes the
index before and
after the lens = 1
u  u  hK
ui 1  ui'
hi 1  hi  di ui
Also assumes the
index before and
after the lens = 1
Using Gaussian ray traces to generate lens formulas:
Trace a horizontal, ray through two lenses (why a horizontal ray?):
Location of equivalent lens:
n1
n1’=n2
n2’
u1=0
h1
u1’=u2
u2’
h2
f
d1
K1
K2
Note that: u2’ = -h1/f
(definition of f for the
combination)
The goal is to find the equivalent focal length for the combination of
two lenses. The equivalent focal length is defined as the focal
length of a single (perfect) lens that would produce the same image.
Ray trace through lens pair:
Refract through first element:
n1u1  n1u1  h1K1
u1  
h1K1
, since u1  0
n1
 u2  
h1K1
, u2  u1, n2  n1
n2
Transfer to second element:
h2  h1  u1d1
 h2  h1 
h1K1
d1
n2
Refract at second element:
n2 u2  n2u2  h2K 2
 h1K1  
h1K1 
 n2  
  h1 
d1  K 2

n2
 n2  

KK
 h 
n2   1   h1K1  h1K 2  h1 1 2 d1
n2
 f 
Eliminating h1 and using K 
1
where K is the
f
equivalent power of the combination, we get:
General formula for combination of two surfaces:
n2 K  K1  K 2 
K1K 2
d1
n2
For two lenses in air, spaced by d, then the indices
after and between the lenses is n=1, so the two
lens formula becomes:
K
1
 K1  K 2  K1K 2d1
f
Suppose that K1 = 1/d1: What is the power of the combination then?
“Graphical Ray Tracing” Through Thin Lenses
We will assume, without further proof, that the paraxial effect of a thin lens on the curvature of
wavefronts is, to first order, unchanged by tilting the lens through a “paraxial” angle.
Given this, there are a number of rays that can be immediately traced through a thin lens, whose
power is known, without further calculation:
Incident rays parallel to the axis, pass through the focal point.
(The corollary is that incident rays that pass through the focal point emerge parallel to the axis.)
Rays through the center of the lens are undeviated.
In the paraxial approximation, the lens is an infinitesimally thin plane-parallel plate at the center.
Any parallel bundle of rays will meet at the focal plane at the point where the central (undeviated)
ray in the bundle meets that plane.
The “focal plane” is a plane perpendicular to the axis at the focal distance. The central ray of a
bundle is often called the “chief ray”.
For object and image distances determined by the thin lens imaging equation, a bundle of rays
diverging from any point on the object plane will converge to a point on the image plane at the
point where the chief ray from the object point intersects the image plane.