Chapter 2.
Geometrical Optics Chapter 2.
Geometrical Optics
Light Ray : the path along which light energy is transmitted from one point to another in an optical system.
Speed of Light : Speed of light (in vacuum): a fundamental (or a “defined”) constant of nature given by c = 299,792,458 meters / second = 186,300 miles / second.
Index of Refraction
Optical path length Optical path length
The optical path length in a medium is the integral of the refractive index and a differential geometric length:
b
OPL = ∫a n ds a ds b
n1
n4 n2
n5 nm n3
∑
==
mi
i i
s c n
t
1
1
∑
==
mi
i i
s n OPL
1
∫
=
PS
n s ds
OPL ( ) = ∫
PS
v ds OPL c
S
P
Reflection and Refraction
Reflection and Refraction
Plane of incidence
Plane of incidence
Fermat’s Principle: Law of Reflection Fermat’s Principle: Law of Reflection
Fermat’s principle:
Light rays will travel from point A to point B in a medium along a path that minimizes the time of propagation.
( ) ( ) ( ) ( )
( )
( ) ( )
( )( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
2 2 2 2
1 2 1 3 3 2
1 1 3 3
2 1 3 2
2 2 2 2
2 1 2 1 3 3 2
2 1 3 2
2 2 2 2
1 2 1 3 3 2
, , ,
1 1
2 2 1
2 2
0
0
0 sin sin
sin sin
AB
AB
i r
i r
OPL n x y y n x y y
Fix x y x y
n y y n y y
dOPL
dy x y y x y y
n y y n y y
x y y x y y
n θ n θ
θ θ
= − + − + − + −
− − −
= = +
+ − + −
− −
= −
+ − + −
= −
⇒ =
x y
(x1, y1)
(0, y2) (x3, y3)
θ
rθ
iA B
i r
θ θ =
: Law of reflectionFermat’s Principle: Law of Refraction Fermat’s Principle: Law of Refraction
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
2 2 2 2
2 1 1 3 2 3
1 1 3 3
2 1 3 2
2 2 2 2
2 2 1 1 3 2 3
2 1 3 2
2 2 2 2
2 1 1 3 2 3
, , ,
1 1
2 2 1
2 2
0
0
0 sin sin
sin sin
AB i t
i t
AB
i t
i i t t
i i t t
OPL n x x y n x x y
Fix x y x y
n x x n x x
d OPL
dy x x y x x y
n x x n x x
x x y x x y
n n
n n
θ θ
θ θ
= − + + − + −
− − −
= = +
− + − +
− −
= −
− + − +
= −
⇒ =
Law of refraction:
x y
(x1, y1)
(x2, 0)
(x3, y3) θt
θi A
n
in
ti i t t
n θ = n θ
: Law of refraction in paraxial approx.Refraction –Snell’s Law :
????
n
n
i×
t< 0
t t
i
i
n
n sin θ = sin θ
Negative index of refraction : n < 0
RHM
N > 1LHM
N = -1
Principle of reversibility
Principle of reversibility
2-4. Reflection in plane mirrors
2-4. Reflection in plane mirrors
Plane surface – Image formation
Plane surface – Image formation
Total internal Reflection (TIR)
Total internal Reflection (TIR)
2-6. Imaging by an Optical System
2-6. Imaging by an Optical System
Cartesian Surfaces Cartesian Surfaces
• A Cartesian surface – those which form perfect images of a point object
• E.g. ellipsoid and hyperboloid
O I
Imaging by Cartesian reflecting surfaces
Imaging by Cartesian reflecting surfaces
Imaging by Cartesian refracting Surfaces
Imaging by Cartesian refracting Surfaces
Imaging by Cartesian refracting Surfaces Imaging by Cartesian refracting Surfaces
The optical path length for any path from Point O to the image Point I must be the same by Fermat’s principle.
( )
22 2 2 2 2
constant
constant
o o i i o o i i
o i o i
o o i i
n d n d n s n s
n x y z n s s x y z
n s n s
+ = + =
+ + + + − + +
= + =
The Cartesian or perfect imaging surface is a
paraboloid in three dimensions.
Usually, though, lenses have spherical surfaces
because they are much easier to manufacture.
ÆSpherical approximation or, paraxial approx.
Approximation by Spherical Surfaces
Approximation by Spherical Surfaces
2-7. Reflection at a Spherical Surface
2-7. Reflection at a Spherical Surface
Reflection at Spherical Surfaces I Reflection at Spherical Surfaces I
Use paraxial or small-angle approximation for analysis of optical systems:
3 5
2 4
sin 3! 5!
cos 1 1
2! 4!
ϕ ϕ
ϕ ϕ ϕ
ϕ ϕ ϕ
= − + − ≅
= − + − ≅
L
L
Reflection from a sphericalconvex surface gives rise to a virtual image.
Rays appear to emanate from point I behind the spherical reflector.
Reflection at Spherical Surfaces II Reflection at Spherical Surfaces II
Considering Triangle OPC and then Triangle OPI we obtain:
θ α ϕ = + 2 θ α α = + ′
Combining these relations we obtain:
α α − ′ = − 2 ϕ
Again using the small angle approximation:
tan h tan h tan h
s s R
α ≅ α ≅ α ′ ≅ α ′ ≅ ϕ ≅ ϕ ≅
′
Reflection at Spherical Surfaces III Reflection at Spherical Surfaces III
Image distance s' in terms of the object distance s and mirror radius R:
1 1 2
h h 2 h
s − s = − R ⇒ s − s = − R
′ ′
At this point the sign convention in the book is changed !
1 1 2
s + s = − R
′
The following sign convention must be followed in using this equation:
1. Assume that light propagates from left to right.
Object distance s is positive when point O is to the left of point V.
2. Image distance s' is positive when I is to the left of V (real image) and negative when to the right of V (virtual image).
3. Mirror radius of curvature R is positive for C to the right of V (convex), negative for C to left of V (concave).
Reflection at Spherical Surfaces IV Reflection at Spherical Surfaces IV
The focal length f of the spherical mirror surface is defined as –R/2, where R is the radius of curvature of the mirror. In accordance with the sign convention of the previous page, f > 0 for a concave mirror and f < 0 for a convex mirror.
The imaging equation for the spherical mirror can be rewritten as
1 1 1
s + s = f
′
2 s
f R
= ∞
= −
R < 0 f > 0
R > 0 f < 0
Reflection at Spherical Surfaces V Reflection at Spherical Surfaces V
C F
O I
1
2
3
I' O'
Ray 1: Enters from O' through C, leaves along same path Ray 2: Enters from O' through F, leaves parallel to optical axis Ray 3: Enters through O' parallel to optical axis, leaves along
line through F and intersection of ray with mirror surface
1 1 1
7 8 / 2
? ?
s cm R cm f R
s f s
s m s
s
= = + = − = −
′
′ = = − ′ =
V
s s’
Reflection at Spherical Surfaces VI Reflection at Spherical Surfaces VI
C F
O 1
2 3
I
I' O'
1 1 1
17 8 / 2
s cm R cm f R
s f s
s m s
s
= + = − = − = = − =
′
′ = = − ′ =
V
Reflection at Spherical Surfaces VII Reflection at Spherical Surfaces VII
1 1 1
0
0
s f
s f s
m s
s
> ⇒ = − >
′
= − ′ <
1 1 1
0 0
s f
s f s
m s
s
< ⇒ = − <
′
= − ′ >
Real, Inverted Image Virtual Image, Not Inverted
2-8. Refraction at a Spherical Surface 2-8. Refraction at a Spherical Surface
n2 > n1
At point P we apply the law of refraction to obtain
1
sin
1 2sin
2n θ = n θ
Using the small angle approximation we obtain
1 1 2 2
n θ = n θ
Substituting for the angles θ1 and θ2 we obtain
( ) ( )
1 2
n α ϕ − = n α ϕ ′ −
Neglecting the distance QV and writing tangents for the angles gives
1 2
h h h h
n n
s R s R
⎛ ⎞
⎛ − ⎞ = ⎜ − ⎟
⎜ ⎟ ′
⎝ ⎠ ⎝ ⎠
Refraction by Spherical Surfaces II Refraction by Spherical Surfaces II
n2 > n1
Rearranging the equation we obtain
Using the same sign convention as for mirrors we obtain
1 2 1 2
n n n n
s s R
− = −
′
1 2 2 1
n n n n
s s R P
+ = − =
′
P : power of the refracting surface
Refraction at Spherical Surfaces III Refraction at Spherical Surfaces III
C O
I
I' O'
1 2
1 2 2 1
7 8 1.0 4.23
' ?
s cm R cm n n
n n n n
s s R s
= = + = =
+ = − ⇒ =
′
V θ1
θ2
Example 2-2 : Concept of imaging by a lens
Example 2-2 : Concept of imaging by a lens
2-9. Thin (refractive) lenses
2-9. Thin (refractive) lenses
The Thin Lens Equation I The Thin Lens Equation I
O O'
t C2
C1
n1 n1
n2
s1
s'1
1 2 2 1
1 1 1
n n n n
s s R
+ = −
′
V1 V2
For surface 1:
The Thin Lens Equation II The Thin Lens Equation II
1 2 2 1
1 1 1
n n n n
s s R
+ = −
′
For surface 1:
2 1 1 2
2 2 2
n n n n
s s R
+ = −
′
For surface 2:
2 1
s = − t s′
Object for surface 2 is virtual, with s
2given by:
2
,
1 2 1t s s ′ ⇒ s = − s ′
For a thin lens:
1 2 2 1 1 1 2 1 1 2
1 2
1 1 1 2 1 2 1 2
n n n n n n n n n n
P P
s s s s s s R R
− −
+ − + = + = + = +
′ ′ ′ ′
Substituting this expression we obtain:
The Thin Lens Equation III The Thin Lens Equation III
(
2 1)
1 2 1 1 2
1 1 n n 1 1
s s n R R
− ⎛ ⎞
+ ′ = ⎜ ⎝ − ⎟ ⎠
Simplifying this expression we obtain:
(
2)
2
2 1
1
1 1
1 1 1 1
s n n
s s
s s
s n
R R
′ ′
= = − ⎛ ⎞
+ ′ = ⎜ ⎝ − ⎟ ⎠
⇒
For the thin lens:
(
2 1)
1 1 2
1 1
1 n n 1
f n R R
s = ∞ ⇒ s =
′
− ⎛ ⎞
= ⎜ − ⎟
⎝ ⎠
The focal length for the thin lens is found by setting s = ∞:
The Thin Lens Equation IV The Thin Lens Equation IV
In terms of the focal length f the thin lens equation becomes:
1 1 1
s + s = f
′
The focal length of a thin lens is positive for a convex lens,
negative for a concave lens.
Image Formation by Thin Lenses Image Formation by Thin Lenses
Convex Lens
Concave Lens
m s
s
= − ′
Image Formation by Convex Lens Image Formation by Convex Lens
1 1 1
5 9
f cm s cm s
s f s
m s s
= − = + = + ⇒ ′ =
′
= − ′ =
Convex Lens, focal length = 5 cm:
F
F ho
hi RI
Image Formation by Concave Lens Image Formation by Concave Lens
Concave Lens, focal length = -5 cm:
1 1 1
5 9
f cm s cm s
s f s
m s s
= − = − = + ⇒ ′ =
′
= − ′ =
F F hohi VI
Image Formation: Two-Lens System I Image Formation: Two-Lens System I
1 1
1 1 1
1 1 1 1 1
2 2 2
2 2 2
1 2
1 1 1
15 25
1 1 1
15 s f
f cm s cm s
s f s s f
f cm s s
s f s
m m m
− ′
= − = = + = + ⇒ =
′
= − = − = ⇒ ′ =
′
= =
60 cm
Image Formation: Two-Lens System II Image Formation: Two-Lens System II
1 1 1
1 1 1
2 2 2
2 2 2
1 2
1 1 1
3.5 5.2
1 1 1
1.8
f cm s cm s
s f s
f cm s s
s f s
m m m
= − = + = + ⇒ ′ =
′
= − = + = ⇒ ′ =
′
= =
7 cm
Image Formation Summary Table
Image Formation Summary Table
Image Formation Summary Figure
Image Formation Summary Figure
Vergence and refractive power : Diopter Vergence and refractive power : Diopter
1 1 1
s + s = f
′
'
V V + = P
reciprocals
Vergence (V) : curvature of wavefront at the lens
Refracting power (P)
Diopter (D)
: unit of vergence (reciprocal length in meter)
D > 0D < 0
1m
0.5m 2 diopter
1 diopter
1 m -1 diopter
1 2 3
P = + P P + +L P
Two more useful equations
Two more useful equations
2-12. Cylindrical lenses
2-12. Cylindrical lenses
Cylindrical lenses Cylindrical lenses
Top view
Side view