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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 byc = 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
aOPL n ds= ∫ a b
ds
n1
n4
n2
n5
nm
n3
∑=
=m
iiisnc
t1
1
∑=
=m
iiisnOPL
1
∫=P
SdssnOPL )( ∫=
P
S
dsvcOPL
S
P
Reflection and RefractionReflection and Refraction
Plane of incidencePlane of incidence
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.
( ) ( ) ( ) ( )
( )
( ) ( )
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
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
= − + − + − + −
− − −= = +
+ − + −
− −= −
+ − + −
= −
⇒ =x
y
(x1, y1)
(0, y2)
(x3, y3)
θr
θi
A
B
i rθ θ= : Law of reflection
Fermat’s Principle: Law of RefractionFermat’s Principle: Law of Refraction( ) ( ) ( ) ( )
( ) ( )
( ) ( )
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
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
= − + + − + −
− − −= = +
− + − +
− −= −
− + − +
= −
⇒ =
Law of refraction:
x
y
(x1, y1)
(x2, 0)
(x3, y3)
θt
θi
A
ni
nt
i i t tn nθ θ= : Law of refractionin paraxial approx.
Refraction –Snell’s Law :
???? nn ti 0
Negative index of refraction : n < 0
RHM N > 1
LHMN = -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 SurfacesCartesian 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 surfacesbecause they are much easier to manufacture.
Spherical approximationor, paraxial approx.
Approximation by Spherical SurfacesApproximation by Spherical Surfaces
2-7. Reflection at a Spherical Surface2-7. Reflection at a Spherical Surface
Reflection at Spherical Surfaces I Reflection at Spherical Surfaces I
Use paraxial or small-angle approximationfor 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.
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′ ′≅ ≅ ≅ ≅ ≅ ≅′
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 22h h hs s R s s R− = − ⇒ − = −
′ ′
At this point the sign convention in the book is changed !
1 1 2s 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 1s s f+ =
′
2
sRf
= ∞
= −
R < 0f > 0
R > 0f < 0
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 pathRay 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 / 2
? ?
s cm R cm f Rs f s
ss ms
= = + = − = −′
′′ = = − =
Vs s’
Reflection at Spherical Surfaces VI Reflection at Spherical Surfaces VI
CFO
1
23
I
I'
O'
1 1 117 8 / 2s cm R cm f Rs f s
ss ms
= + = − = − = = − =′
′′ = = − =
V
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
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 1 2 2sin sinn nθ θ=Using the small angle approximation we obtain
1 1 2 2n nθ θ=
Substituting for the angles θ1 and θ2 we obtain
( ) ( )1 2n nα ϕ α ϕ′− = −Neglecting the distance QV and writing tangents for the angles gives
1 2h h h hn ns 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 2n n n ns s R
−− =
′
1 2 2 1n n n n Ps s R
−+ = =
′
P : power of the refracting surface
Refraction at Spherical Surfaces III Refraction at Spherical Surfaces III
CO
I
I'
O'
1 2
1 2 2 1
7 8 1.0 4.23
' ?
s cm R cm n nn n n n ss s R
= = + = =−
+ = ⇒ =′
V
θ1
θ2
Example 2-2 : Concept of imaging by a lensExample 2-2 : Concept of imaging by a lens
2-9. Thin (refractive) lenses2-9. Thin (refractive) lenses
The Thin Lens Equation I The Thin Lens Equation I
O
O'
t
C2
C1
n1 n1n2
s1s'1
1 2 2 1
1 1 1
n n n ns 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 ns s R
−+ =
′
For surface 1:
2 1 1 2
2 2 2
n n n ns s R
−+ =
′
For surface 2:
2 1s t s′= −
Object for surface 2 is virtual, with s2 given by:
2 1 2 1,t s s s s′ ′⇒ = −
For a thin lens:
1 2 2 1 1 1 2 1 1 21 2
1 1 1 2 1 2 1 2
n n n n n n n n n n P Ps s s s s s R R
− −+ − + = + = + = +
′ ′ ′ ′
Substituting this expression we obtain:
The Thin Lens Equation III The Thin Lens Equation III
( )2 11 2 1 1 2
1 1 1 1n ns s n R R
− ⎛ ⎞+ = −⎜ ⎟′ ⎝ ⎠
Simplifying this expression we obtain:
( )22
21
1
1 1
1 1 1 1sn n
ss
ss
ns
R R′ ′= =
− ⎛ ⎞+ = −⎜ ⎟′ ⎝ ⎠
⇒
For the thin lens:
( )2 11 1 2
1 11 1n nf n R R
ss
= ∞ ⇒ =′
− ⎛ ⎞= −⎜ ⎟
⎝ ⎠
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 1s s f+ =
′
The focal length of a thin lens is
positive for a convex lens,
negative for a concave lens.
Image Formation by Thin LensesImage Formation by Thin Lenses
Convex Lens
Concave Lens
sms′
= −
Image Formation by Convex LensImage Formation by Convex Lens
1 1 1 5 9f cm s cm ss f s
m s s
′= − = + = + ⇒ =′
′= − =
Convex Lens, focal length = 5 cm:
F
F
ho
hi
RI
Image Formation by Concave LensImage Formation by Concave LensConcave Lens, focal length = -5 cm:
1 1 1 5 9f cm s cm ss f s
m s s
′= − = − = + ⇒ =′
′= − =
FF
hohi
VI
Image Formation: Two-Lens System IImage Formation: Two-Lens System I
1 11 1 1
1 1 1 1 1
2 2 22 2 2
1 2
1 1 1 15 25
1 1 1 15
s f f cm s cm ss f s s f
f cm s ss f sm m m
− ′= − = = + = + ⇒ =′
′= − = − = ⇒ =′
= =
60 cm
Image Formation: Two-Lens System IIImage Formation: Two-Lens System II
1 1 11 1 1
2 2 22 2 2
1 2
1 1 1 3.5 5.2
1 1 1 1.8
f cm s cm ss f s
f cm s ss f s
m m m
′= − = + = + ⇒ =′
′= − = + = ⇒ =′
= =
7 cm
Image Formation Summary TableImage Formation Summary Table
Image Formation Summary FigureImage Formation Summary Figure
Vergence and refractive power : DiopterVergence and refractive power : Diopter
1 1 1s 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 > 0
D < 0
1m
0.5m2 diopter
1 diopter
1 m
-1 diopter
1 2 3P P P P= + + +L
Two more useful equationsTwo more useful equations
2-12. Cylindrical lenses2-12. Cylindrical lenses
Cylindrical lensesCylindrical lenses
Top view
Side view