Download Fresnel`s Theory of wave propagation

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Birefringence wikipedia, lookup

Phase-contrast X-ray imaging wikipedia, lookup

Surface plasmon resonance microscopy wikipedia, lookup

Reflection high-energy electron diffraction wikipedia, lookup

Magnetic circular dichroism wikipedia, lookup

Retroreflector wikipedia, lookup

Optical aberration wikipedia, lookup

Light wikipedia, lookup

Harold Hopkins (physicist) wikipedia, lookup

Diffraction topography wikipedia, lookup

Christiaan Huygens wikipedia, lookup

Superlens wikipedia, lookup

Fourier optics wikipedia, lookup

Nonlinear optics wikipedia, lookup

Airy disk wikipedia, lookup

Thomas Young (scientist) wikipedia, lookup

Diffraction grating wikipedia, lookup

Wave interference wikipedia, lookup

Low-energy electron diffraction wikipedia, lookup

Powder diffraction wikipedia, lookup

Diffraction wikipedia, lookup

Transcript
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