Download Fresnel`s Theory of wave propagation

Camera Obscura
The Wave Nature of Light
Maxwell’s equations ⟼ waves
Plane and spherical waves
Energy flow / Intensity
The Eikonal and Fermat’s principle
optimum pinhole size
From geometrical to wave optics
contradicts expectations from
geometrical optics
Huygen’s principle
from Hecht, Optics
lecture 1
lecture 2
Maxwell’s Equations
~ ·B
~ =0
r
~ ·D
~ =0
r
Plane and Spherical Waves
~
~ ⇥H
~ = dD
r
dt
~
~ ⇥E
~ = dB
r
dt
~ =
r2 E
⇣ n ⌘ 2 d2
~
E
c
dt2
and
~ =
r2 H
⇣ n ⌘ 2 d2
~
H
c
dt2
plane wave
~
D
~
B
~
✏r ✏0 E
~
= µr µ0 H
⇣ ⌘ 2 d2
~ = n
~
r2 E
E
c
dt2
lecture 2
linear isotropic medium
⇢=0
J~ = 0
=
and
~ r, t) = E
~ 0 ei(~k~r
E(~
!t)
spherical wave
⇣ ⌘2 d2
~ = n
~
r2 H
H
c
dt2
E(~r, t) = E0 ei(kr
lecture 2
!t)
/r
p ✏ rµ r
p ✏ 0µ 0
=
n
⌫
1/
/k =
=
!
=
c
c/n
=
~
vp
~ ⇥H
E
~ =
S
2⇡
= nk0
k=
Scalar Approach ⟼ Eikonal
E(~r, t) = E0 (~r)e
Fresnel Number
i!t
(r2 + n2 k02 )E0 (~r) = 0
E(~r, t) = A0 (~r)ei(k0 L(~r)
F =
a
!t)
b
(r · L)2 = n2
Eikonal
equation
a2
b
F
L3
L2
L1
lecture 2
L0
1
Geometrical optics
F =1
Fresnel diffraction
F ⌧1
Fraunhofer diffraction
wave optics
lecture 2
Wave Nature of Light
Maxwell’s wave equations
Wave Nature of Light
+ k 2 )ur = 0
(
Maxwell’s wave equations
Plane and spherical transverse waves
(
+ k 2 )ur = 0
Plane and spherical transverse waves
Wave propagation
single-frequency wave:
plane or spherical waves:
wave vector:
angular frequency:
intensity:
lecture 3
u(~r, t) = ur (~r)e
u(~r, t) = u0 e
i!t
i(~
k~
r !t)
u0 i(kr
e
or
r
Energy flow & Poynting vector
Eikonal equation
Fermat’s principle (geom. Optics)
!t)
k = 2⇡/
! = 2⇡/T = c/
I / |u(~r, t)|2
Fresnel number
0
⇥ c/n
F =
lecture 3
a2
b
a
b
Huygens’ wavelet
Huygens’ wavelet
ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...
ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...
Huygens’ wavelet
Huygens’ wavelet
Wave Nature of Light
Huygen’s Principle
Wave propagation
Every point on a wave front
can be considered as a source
of secondary spherical waves
Huygen’s principle
Kirchhoff integral
Historic disputes
Fraunhofer diffraction
Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves,
wavelets
recombine to produce the propagating wave front. (C) The diffraction of sound around a corner
Interference and
diffraction
arising from Huygens' wavelets.
Z
~
ik|R
~
r | waves and (B) plane waves,
Figure 2: Huygens' wavelets. Originating along the fronts ofe(A)
circular
~ /wave front.
wavelets recombine to produce the propagating
u(R)
u(~r)(C) The diffraction
dS of sound around a corner
~ ~r|
arising from Huygens' wavelets.
|R
Superposition of waves
Fourier methods
lecture 3
lecture 3
Huygen’s Principle
Huygens’ wavelet
Wave Propagation
ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...
print articles
Huygens’ wavelet
Fresnel’s Theory of wave propagation
From Maxwell’s equations:
~ /
u(R)
Z ⇣
~
G(R
G(~r) =
~ /
u(R)
Z
~ r u(~r)
~r)r
~ r G(R
~
u(~r)r
⌘
~
~r) dS
print articles
1 ik|~r|
e
|~r|
~
1 of 1
⌘(✓i , ✓o )u(~r)
eik|R ~r|
dS
~ ~r|
|R
23.11.2008 19:56 Uhr
Figure 2: Huygens' wavelets. Originating along the fronts of (A) circular waves and (B) plane waves,
wavelets recombine to produce the propagating wave front. (C) The diffraction of sound around a corner
arising from Huygens' wavelets.
Huygen‘s wavelets recombine to produce
the propagating wavefront
lecture 3
lecture 3
1 of 1
23.11.2008 19:56 Uhr
Fresnel‘s Theory of Wave Propagation
Fresnel‘s Theory of Wave Propagation
Fresnel-Kirchhoff diffraction integral
i
⇤
up =
(⇥in , ⇥out )
plane-to-plane
u0 ikr
e dS
r
obliquity factor
(⇥in , ⇥out ) =
1
(cos ⇥in + cos ⇥out )
2
Fraunhofer (far field) diffraction is a special case
lecture 3
eikr ⇤ eikr0 · ei(
Huygens secondary sources on wavefront at -z
radiate to point P on new wavefront at z = 0
x x+ y y)
Fresnel‘s Theory of Wave Propagation
Fresnel‘s Theory of Wave Propagation
plane-to-plane
r=
!
up = u0
up =
⇥(⇤in , ⇤out )
Huygens secondary sources on wavefront at -z
radiate to point P on new wavefront at z = 0
u0 ikr
e dS
r
lecture 3
q2
+
⇥2
⇥2
q⇥
2q
Fresnel Diffraction → Talbot Effect
Complementary Models
Near-field diffraction
of an optical grating
Geometric Optics
Fermat’s principle
Light rays
Corpuscular explanation (Newton)
self-imaging at
zT = 2d2 /
Wave Optics
Huygen’s principle
Fresnel-Kirchhoff integral
Interference and diffraction
lecture 3
lecture 3
Wave or Particle? (17th century)
Isaac Newton
Christiaan Huygens
Wave or
Particle?
✤
Wave theory of light
Christiaan Huygens (1690)
✤
Explains:
reflection, refraction,
colours,
diffraction, interference
lecture 3
lecture 3
Wave or
Particle?
✤
Wave or
Particle?
Corpuscular theory of light
✤
Isaac Newton (1704)
✤
Isaac Newton (1704)
Explains:
✤
reflection, refraction,
colours
✤
Explains:
reflection, refraction,
colours
but not:
✤
diffraction, interference
but not:
diffraction, interference
lecture 3
lecture 3
Wave or
Particle?
✤
Corpuscular theory of light
Young‘s Double Slit
Evidence for light waves
intensity
Thomas Young (1803)
✤
light behaves like a wave
Explains:
diffraction
interference
bending light around corners
lecture 3
lit
eS
oubl
D
lecture 3
position
Young‘s Double Slit
Young‘s Double Slit
intensity
light behaves like a wave
intensity
position
position
Interference:
crest meets crest
through meets through
crest meets through
lecture 3
➙ annihilation
lecture 3
Poisson versus Fresnel
particles
} ➙ amplification
Poisson versus Fresnel
waves
particles
waves
Poisson versus Fresnel
Fraunhofer Diffraction
çois
Fran o
g
Ara
y
aperture
θ
Poisson Spot
lecture 4
Diffraction in the far field
= (k sin ⇥) · y
Fraunhofer Diffraction
Fraunhofer Diffraction
A diffraction pattern for which the phase
of the light at the observation point is a
linear function of the position for all
points in the diffracting aperture is
Fraunhofer diffraction
A diffraction pattern for which the phase
of the light at the observation point is a
linear function of the position for all
points in the diffracting aperture is
Fraunhofer diffraction
eikr ⇤ eikr0 · ei(
x x+ y y)
How linear is linear?
lecture 4
Fraunhofer Diffraction
Fraunhofer Diffraction
d
r = |rmax
a2
⇥ ⇥/8
d| ⇤
8d
lecture 4
r=
1
+
2
a2
⇥
8
1
1
+
ds
dp
lecture 4
Fraunhofer Diffraction
Fraunhofer Diffraction
y
y
⇥
⇥/8
f
aperture
θ
aperture
θ
= (k sin ⇥) · y
= (k sin ⇥) · y
Fraunhofer Diffraction
Fraunhofer Diffraction
y
f
A diffraction pattern formed in the
image plane of an optical system is
subject to Fraunhofer diffraction
illumination
aperture
θ
Diffraction in the image plane
what is being imaged?
= (k sin ⇥) · y
lecture 4
Fraunhofer Diffraction
Fraunhofer Diffraction
Fraunhofer
diffraction:
in the image
plane
lecture 4
Equivalent
lens system:
Fraunhofer
diffraction
independent
on aperture
position
lecture 4