Lecture 4: Thin Lenses
Lecture aims to explain:
1. Refraction at spherical surfaces
and paraxial approximation
2. Convex and concave lenses
3. The Lensmaker’s and Thin Lens
equations
Refraction at spherical surfaces
and paraxial approximation
Refraction of light on a spherical surface
A
θo
no
li
γ
lo
O
α
V
β
R
so
Eventually we get:
θi
I
C
si
no ni 1  ni si no so 

+ = 
−
lo li R  li
lo 
ni
Paraxial approximation
Expression
no ni 1  ni si no so 

+ = 
−
lo li R  li
lo 
can be greatly simplified, if we assume that angle θo (or α) is small
Then
lo ≈ so
and
li ≈ si
no ni ni − no
+ =
so si
R
and we obtain:
Concave and Convex Lenses
Convex and Concave Lenses
‘A lens is a refracting device that
reconfigures a transmitted energy
distribution’
Converging, convex or positive lens,
the central section is thicker than the rim
Diverging, concave or negative lens,
central section is thinner than the rim
Materials used for lenses in visible and
near infra-red (IR): various types of glass
and plastic
The oldest lens artifact is the Nimrud lens, which is over three
thousand years old, dating back to ancient Assyria. Lenses were
probably used as magnifying glasses, or as a burning-glass to
start fires by concentrating sunlight.
The Lensmaker’s and Thin Lens
equations
Refraction by a thin lens
Lens with refractive index nl ,
made up from two intersecting
spherical surfaces, surrounded
by medium with nm
First surface:
nm nl nl − nm
+
=
so1 si1
R1
Second surface:
nl nm nm − nl
+
=
so 2 si 2
R2
The Lensmaker’s equation:
1 1 nl − nm  1
1 
 − 
+ =
so si
nm  R1 R2 
For a thin lens d≈0
(so2=-si1) and resulting
formula can be simplified
Focal length
If the object is infinitely far from the lens so=∞ then
the image will be at a distance si=f defined as:
1 nl − nm  1
1 
 − 
=
f
nm  R1 R2 
If the object is at a distance f (so=f) from the lens,
the image will move infinitely far from the lens si=∞
This special distance f is called the focal length.
We can rewrite the Lensmaker’s formula in a form of
the Thin Lens Equation:
1 1 1
+ =
so si
f
Sign conventions
If light is incident from the left (as will be considered in most of the
questions and sketches) the signs of spherical surfaces are as follows:
A convex lens (left) has a positive focal length, a
concave lens (right) has a negative focal length
EXAMPLE 4.1: lens in air and water
Focal length of a glass lens (ng=1.5) in air is 10 cm.
The radius of curvature of the first surface is 10 cm.
1. Find the radius of curvature of the second
surface.
2. What is the focal length of the lens in water
(nw=1.3)?
EXAMPLE 4.2: lens design
Calculate the focal length in air for a glass lens
(n=1.7) with R1=5 cm and R2=2 cm. Sketch the
schematic of this lens.
EXAMPLE 4.3: convex and concave lenses
Show that in the figure opposite the lenses in the
left column are convex (positive f) whereas the
lenses in the right column are concave (negative f).
SUMMARY
We will use paraxial approximation in most cases: angles at
which light propagates with respect to the optical axis are small.
The Lensmaker’s equation:
1 1 nl − nm  1
1 
 − 
+ =
so si
nm  R1 R2 
Can be rewritten in the form
of the Thin Lens Equation:
1 1 1
+ =
so si
f
using the expression for the
focal length:
1 nl − nm  1 1 
 − 
=
f
nm  R1 R2 
The Lensmaker equation requires signs
convention. Figure shows the case for
light incident from the left: