Download Fermat `s principle: Light travels along the paths such that the optical

Fermat ’s principle: Light travels along the paths
such that the optical path length is an extremum. In
general, light travels along the path of least time.
τ=
nd
c0
Reflection and
refraction laws
Total internal reflection
 n2 

 n1 
θ c = sin −1 
Spherical Mirror
1 1
2
+ =−
z1 z 2
R
M=
y2
z
=− 2
y1
z1
Refraction at a spherical boundary
n1 n2 n2 − n1
+ =
z1 z 2
R
z1
z2
M=
y2
nz
=− 1 2
y1
n2 z1
Thin lens
1 1 1
+ =
z1 z 2 f
1
1
1 

= (n − 1) − 
f
 R1 R2 
M=
y2
z
=− 2
y1
z1
GRaded INdex Optics

d  dr 
 n  = ∇n
ds  ds 
d  dx  ∂n
n  =
dz  dz  ∂x
Ray equation
Paraxial ray equation
d  dy  ∂n
n  =
dz  dz  ∂y
Ex:
n 2 ( y ) = n02 (1 − α 2 y 2 )
Matrix Optics
1

0

1 d 
0 1 


0
n1 
n2 
n1 n2 n2 − n1
+ =
z1 z 2
R
 1
 n1 − n2
 nR
 2
0
n1 
n2 
1 1 1
+ =
z1 z 2 f
 1
 1
− f

0

1

1 0
0 1 


1 1
2
+ =−
z1 z 2
R
1
2
 R
0

1

2

2 
u
∂
1
2
I (r ) = 2 < u (r , t ) >
Wave equation
∇ u− 2 2 =0
c ∂t



u (r , t ) = A(r ) cos(2πνt + ϕ (r )) Monochromatic wave

 jϕ ( r ) j 2πνt
 j 2πνt
Notations
= U ( r )e
U (r , t ) = A(r )e
e

 2
I (r ) =| U (r ) |
∇ 2U + k 2U = 0
Helmholtz equation


− jk . r
U (r ) = Ae
Plane wave

A − jkr
Spherical wave
U (r ) = e
r


∂A
<< kA
U (r ) = A(r )e − jkz
x2 + y2

A − jkz − jk 2 z
∂z
U (r ) = e e
z
∂A
2
Paraxial wave
∇ ⊥ A − 2 jk
=0
Paraboloidal wave= Fresnel
∂z
equation
approximation to a spherical wave
Refraction law
k3
θ3
U ( x, y , d )
t ( x, y ) =
U ( x, y,0)
t ( x, y ) = h0 e − j ( n −1) k0α x
t ( x, y ) = h0 e − j ( n −1) k0 d ( x , y )
t ( x , y ) = ∑ cq e
− jq
2π
x
Λ
q
t ( x, y ) = h0 e
x2 + y2
jk 0
2f
t ( x, y ) = h0 e − jn ( x , y ) k0 d 0
Interferences
of two waves
Multiple Interferences
M waves, same intensity,
constant phase difference
I = I1 + I 2 + 2 I1 I 2 cos ϕ
sin 2 ( Mϕ / 2)
I = I0
sin 2 (ϕ / 2)
Infinite number of waves,
progressively decreasing
intensity, constant phase
difference
I=
I max
2
1 + (2 F / π ) sin 2 (ϕ / 2)
Interference of two waves travelling in the same direction
ϕ = kd =
2πnd
Michelson
Interference of two spherical waves
ϕ=
Young
2πxθ
λ
λ0
Polychromatic and pulsed light
+∞
u (t ) = ∫ v(ν )e j 2πνt dν
+∞
U (t ) = 2 ∫ v(ν )e j 2πνt dν
−∞
Ex: Pulse
0
U (t ) = A(t − )e
z
c
j 2πν 0 ( t − cz )
Interference of waves with different frequencies
Interferences
of two waves
Light beating
Multiple Interferences
FOURIER OPTICS
+∞
f (t ) = ∫ F (ν )e j 2πνt dν
−∞
f ( x, y ) = ∫∫ F (ν x ,ν y )e
− j 2π (ν x x +ν y y )
dν x dν y
Spatial harmonic functions- Plane Waves
f ( x, y ) = e
− j 2π (ν x x +ν y y )
U ( x, y , z ) = e
k x = 2πν x
− j (kx x+k y y+kz z )
k y = 2πν y
k z = k 2 − k x2 − k y2 = λ−2 −ν x2 −ν y2
θ x = sin (λν x ) = sin (λ / Λ x )
−1
−1
Λx
θx
λ
θx
Fraunhofer approximationFT in the far field
g ( x, y ) ≈ h0 F ( λxd , λyd )
x 2 , y 2 << λd
NF =
2
a
<< 1
λd
x'2 , y '2 << λd
b2
N 'F =
<< 1
λd
FT using a lens
g ( x, y ) ∝ F ( , )
2
x
λf
y
λf
2
DIFFRACTION
Fraunhofer approximationFT in the far field
g ( x, y ) = I i
j
λd
e − jkd P( λxd , λyd )
FT using a lens
IMAGE FORMATION
4f-imaging
f ( x, y )
F ( λxf , λyf )
f ( x, y )
Single lensimaging
h( x, y ) = h0 P ( λdx 2 , λdy 2 )
d2
ρ s = 1.22λ
D
Interferences: Holography
t = I o + I r + U r*U o + U rU o*
Ref= plane wave // z, z=0
Off axis holography
tU r = I oU r + I rU r + I rU o + U r2U o*
U ∝ I o ( x, y ) + I r + I r U o ( x, y ) + I r U o* ( x, y )
U o ( x, y ) = f ( x, y )e − jkx sin θ
U r*
U ∝| f ( x, y ) |2 + I r + I r f ( x, y )e jkx sin θ + I r f * ( x, y )e − jkx sin θ