chapter 2. geometrical opticsoptics.hanyang.ac.kr/~shsong/2-geometrical optics.pdf · 2016. 8....
TRANSCRIPT
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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
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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
∑=
=m
iiisnc
t1
1
∑=
=m
iiisnOPL
1
∫=P
SdssnOPL )( ∫=
P
S
dsvcOPL
S
P
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Reflection and RefractionReflection and Refraction
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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.
( ) ( ) ( ) ( )
( )
( ) ( )
( )( )
( ) ( )( )
( ) ( )( )
( ) ( )
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
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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.
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Refraction –Snell’s Law :
???? nn ti 0
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Negative index of refraction : n < 0
RHM N > 1
LHMN = -1
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Principle of reversibility Principle of reversibility
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2-4. Reflection in plane mirrors 2-4. Reflection in plane mirrors
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Plane surface – Image formation Plane surface – Image formation
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Total internal Reflection (TIR) Total internal Reflection (TIR)
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2-6. Imaging by an Optical System 2-6. 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
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.
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Approximation by Spherical SurfacesApproximation by Spherical Surfaces
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2-7. Reflection at a Spherical Surface2-7. Reflection at a Spherical Surface
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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.
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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 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).
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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
= ∞
= −
R < 0f > 0
R > 0f < 0
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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 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’
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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
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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
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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
⎛ ⎞⎛ ⎞− = −⎜ ⎟⎜ ⎟ ′⎝ ⎠ ⎝ ⎠
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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
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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
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Example 2-2 : Concept of imaging by a lensExample 2-2 : Concept of imaging by a lens
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2-9. Thin (refractive) lenses2-9. Thin (refractive) lenses
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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:
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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:
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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 = ∞:
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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.
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Image Formation by Thin LensesImage Formation by Thin Lenses
Convex Lens
Concave Lens
sms′
= −
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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
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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
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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
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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
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Image Formation Summary TableImage Formation Summary Table
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Image Formation Summary FigureImage Formation Summary Figure
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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
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1 2 3P P P P= + + +L
Two more useful equationsTwo more useful equations
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2-12. Cylindrical lenses2-12. Cylindrical lenses
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Cylindrical lensesCylindrical lenses
Top view
Side view