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Chapter 3.Geometrical Optics
Chapter 3.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 byc = 299,792,458 meters /
second = 186,300 miles / second.
Index of Refraction
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Huygen’s Wavelet Concept Huygen’s Wavelet Concept
Collimated Plane Wave Spherical Wave
At time 0, wavefront is defined by line (or curve) AB. Each
point on the original wavefront emits a spherical wavelet which
propagates at speed caway from the origin. At time t, the new
wavefrontis defined such that it is tangent to the wavelets from
each of the time 0 source points. A ray of light in geometric
optics is found by drawing a line from the source point to the
tangent point for each wavelet.
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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
aOPL n ds= ∫ a b
ds
n1
n4
n2
n5
nm
n3
S
P
∑=
=m
iiisnc
t1
1
∑=
=m
iiisnOPL
1
∫=P
SdssnOPL )( ∫=
P
S
dsvcOPL
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Plane of incidencePlane of incidence
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Fermat’s Principle: Law of ReflectionFermat’s Principle: Law of
ReflectionFermat’s principle: Light rays will travel from point A
to point B in a medium along a path that minimizes the time of
propagation.
Law of reflection:
x
y
(x1, y1)
(0, y2)
(x3, y3)
θr
θi
( ) ( ) ( ) ( )
( )
( ) ( )
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
2 2 2 21 2 1 3 3 2
1 1 3 3
2 1 3 2
2 2 2 22 1 2 1 3 3 2
2 1 3 2
2 2 2 21 2 1 3 3 2
, , ,
1 12 2 12 20
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 ydOPLdy x y y x y y
n y y n y y
x y y x y y
n nθ θ
θ θ
= − + − + − + −
− − −= = +
+ − + −
− −= −
+ − + −
= −
⇒ =
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Fermat’s Principle: Law of RefractionFermat’s Principle: Law of
Refraction
Law of refraction:
x
y
(x1, y1)
(x2, 0)
(x3, y3)
θt
θi
( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
2 2 2 22 1 1 3 2 3
1 1 3 3
2 1 3 2
2 2 2 22 2 1 1 3 2 3
2 1 3 2
2 2 2 22 1 1 3 2 3
, , ,
1 12 2 12 20
0
0 sin sin
sin sin
AB i t
i tAB
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 xd OPLdy 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
= − + + − + −
− − −= = +
− + − +
− −= −
− + − +
= −
⇒ =
A
ni
nt
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Refraction –Snell’s Law
???? nn ti 0
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Negative Refraction : n < 0
LHM
RHM N > 1
N = -1
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Principle of reversibility Principle of reversibility
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Reflection in plane mirrors Reflection in plane mirrors
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Imaging by an Optical System Imaging by an Optical System
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Cartesian SurfacesCartesian Surfaces
• A Cartesian surface – those which form perfect images of a
point object
• E.g. ellipsoid and hyperboloid
O I
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Imaging by Cartesian reflecting surfaces Imaging by Cartesian
reflecting surfaces
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Imaging by Cartesian refracting Surfaces Imaging by Cartesian
refracting Surfaces
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Imaging by Cartesian refracting Surfaces Imaging by Cartesian
refracting Surfaces
Consider imaging of object point O by the Cartesian surface Σ.
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.
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Approximation by Spherical SurfacesApproximation by Spherical
Surfaces
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Reflection at a Spherical SurfaceReflection at a Spherical
Surface
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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
sin3! 5!
cos 1 12! 4!
ϕ ϕϕ ϕ ϕ
ϕ ϕϕ
= − + − ≅
= − + − ≅
L
L
Reflection from a spherical convex surface gives rise to a
virtual image. Rays appear to emanate from point I behind the
spherical reflector.
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Reflection at Spherical Surfaces II Reflection at Spherical
Surfaces II
Considering Triangle OPC and then Triangle OPI we obtain:
2θ α ϕ θ α α′= + = +
Combining these relations we obtain:
2α α ϕ′− = −
Again using the small angle approximation:
tan tan tanh h hs s R
α α α α ϕ ϕ′ ′≅ ≅ ≅ ≅ ≅ ≅′
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Reflection at Spherical Surfaces III Reflection at Spherical
Surfaces III Now find the image distance s' in terms of the object
distance s and mirror radius R:
1 1 22h h hs s R s s R− = − ⇒ − = −
′ ′
At this point the sign convention in the book is changed and the
imaging equation becomes:
1 1 2s s R+ = −
′
The following rules must be followed in using this equation:
1. Assume that light propagates from left to right. Image
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).
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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 1s s f+ =
′
2
sRf
= ∞
= −
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Reflection at Spherical Surfaces V Reflection at Spherical
Surfaces V
CFO I
12
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 axisRay 3:
Enters through O' parallel to optical axis, leaves along
line through F and intersection of ray with mirror surface
1 1 17 8 / 2s cm R cm f Rs f s
ss ms
= = + = − = = − =′
′′ = = − =
V
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Reflection at Spherical Surfaces VI Reflection at Spherical
Surfaces VI
CFO I
1
2
3
I'
O'
1 1 117 8 / 2s cm R cm f Rs f s
ss ms
= + = − = − = = − =′
′′ = = − =
V
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Reflection at Spherical Surfaces VII Reflection at Spherical
Surfaces VII
1 1 1 0
0
s fs f s
sms
> ⇒ = − >′
′= − <
1 1 1 0
0
s fs f s
sms
< ⇒ = − <′
′= − >
Real, Inverted Image Virtual Image, Not Inverted
