Camera Obscura

optimum pinhole size

contradicts expectations from geometrical optics

lecture 1

from Hecht, Optics

Maxwell’s equations ⟼ wavesPlane and spherical wavesEnergy flow / IntensityThe Eikonal and Fermat’s principleFrom geometrical to wave opticsHuygen’s principle

The Wave Nature of Light

lecture 2

Maxwell’s Equations

~r · ~B = 0 ~r⇥ ~H =d

dt~D

~r · ~D = 0 ~r⇥ ~E = � d

dt~B

linear isotropic medium⇢ = 0 ~J = 0

~D = ✏r✏0 ~E~B = µrµ0

~H

r2 ~E =⇣nc

⌘2 d2

dt2~E and r2 ~H =

⇣nc

⌘2 d2

dt2~H

lecture 2

Plane and Spherical Waves

r2 ~E =⇣nc

⌘2 d2

dt2~E and r2 ~H =

⇣nc

⌘2 d2

dt2~H

~E(~r, t) = ~E0ei(~k~r�!t)

E(~r, t) = E0ei(kr�!t)/r

plane wave

spherical wave

n =p ✏rµr

c = 1/p ✏0µ0

vp= c/n

= !/k= ⌫�

~S =~E ⇥ ~H

lecture 2

k =2⇡

�= nk0

Scalar Approach ⟼ Eikonal

lecture 2

E(~r, t) = E0(~r)e�i!t

(r2 + n2k20)E0(~r) = 0

E(~r, t) = A0(~r)ei(k0L(~r)�!t)

(r · L)2 = n2

L3

L2

L0L1

Eikonalequation

Fresnel Number

lecture 2

F =a2

�b

b

a

Geometrical optics

Fresnel diffraction

Fraunhofer diffraction

F � 1

F = 1

F ⌧ 1wave optics

Wave Nature of Light

Maxwell’s wave equationsPlane and spherical transverse waves

lecture 3

single-frequency wave: u(~r, t) = ur(~r)e�i!t

plane or spherical waves: u(~r, t) = u0ei(~k~r�!t)

or

u0

rei(kr�!t)

wave vector: k = 2⇡/�

angular frequency: ! = 2⇡/T = c/�0

intensity: I / |u(~r, t)|2 ⇥ c/n

(�+ k2)ur = 0

Wave Nature of Light

Maxwell’s wave equationsPlane and spherical transverse wavesWave propagation

Energy flow & Poynting vector Eikonal equationFermat’s principle (geom. Optics)

Fresnel number

lecture 3

F =a2

�bb

a

(�+ k2)ur = 0

Wave Nature of Light

Wave propagation Huygen’s principleKirchhoff integral Historic disputesFraunhofer diffraction

Interference and diffractionSuperposition of wavesFourier methods

lecture 3

Huygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...

1 of 1 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.

Huygens’ wavelet

print articles

Huygen’s Principle

Huygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...

1 of 1 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.

Huygens’ wavelet

print articles

Every point on a wave frontcan be considered as a sourceof secondary spherical waves

u(~R) /Z

u(~r)eik|

~R�~r|

|~R� ~r|dS

lecture 3

Huygen’s Principle

From Maxwell’s equations:

G(~r) =1

|~r|eik|~r|

u(~R) /Z ⇣

G(~R� ~r)~rru(~r)� u(~r)~rrG(~R� ~r)⌘d~S

u(~R) /Z

⌘(✓i

, ✓o

)u(~r)eik|

~

R�~r|

|~R� ~r|dS

lecture 3

Fresnel’s Theory of wave propagation

Wave PropagationHuygens’ wavelet ebcid:com.britannica.oec2.identifier.AssemblyIdentifier?assemblyId=...

1 of 1 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.

Huygens’ wavelet

print articles

Huygen‘s wavelets recombine to producethe propagating wavefront

lecture 3

Fresnel‘s Theory of Wave Propagation

Fresnel-Kirchhoff diffraction integral

up = � i

⇤

��(⇥in, ⇥out)

u0

reikrdS

�(⇥in, ⇥out) =12(cos ⇥in + cos ⇥out)

obliquity factor

eikr �⇤ eikr0 · ei(�xx+�yy)

Fraunhofer (far field) diffraction is a special case

lecture 3

Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0

Fresnel‘s Theory of Wave Propagation

plane-to-plane

Huygens secondary sources on wavefront at -z radiate to point P on new wavefront at z = 0

Fresnel‘s Theory of Wave Propagation

plane-to-plane

up = �

�⇥(⇤in, ⇤out)

u0

reikrdS

up!= u0

Fresnel‘s Theory of Wave Propagation

�r =�

q2 + ⇥2 � q ⇥ ⇥2

2q

lecture 3

Fresnel Diffraction → Talbot Effect

Near-field diffractionof an optical grating

zT = 2d2/�

self-imaging at

lecture 3

Complementary Models

Geometric OpticsFermat’s principleLight raysCorpuscular explanation (Newton)

Wave OpticsHuygen’s principleFresnel-Kirchhoff integralInterference and diffraction

lecture 3

Wave or Particle? (17th century)

Isaac Newton Christiaan Huygens

lecture 3

Wave or Particle?

✤ Wave theory of light Christiaan Huygens (1690)

✤ Explains: reflection, refraction, colours, diffraction, interference

lecture 3

Wave or Particle?

✤ Corpuscular theory of light Isaac Newton (1704)

✤ Explains: reflection, refraction, colours

✤ but not: diffraction, interference

lecture 3

Wave or Particle?

✤ Corpuscular theory of light Isaac Newton (1704)

✤ Explains: reflection, refraction, colours

✤ but not: diffraction, interference

lecture 3

Wave or Particle?

✤ Evidence for light waves Thomas Young (1803)

✤ Explains: diffractioninterference

bending light around corners

Double Slit

lecture 3

Young‘s Double Slit

position

intensity

light behaves like a wave

lecture 3

Young‘s Double Slit

position

intensity

light behaves like a wave

lecture 3

Young‘s Double Slit

position

intensity

Interference:crest meets crestthrough meets through

crest meets through ➙ annihilation

➙ amplification}

lecture 3

Poisson versus Fresnel

particles waves

Poisson versus Fresnel

particles waves

Poisson versus FresnelFrançois

Arago

Poisson Spot

Fraunhofer Diffraction

� = (k sin ⇥) · y

y

θ

aperture

Diffraction in the far field

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

Fraunhofer Diffraction

eikr �⇤ eikr0 · ei(�xx+�yy)

lecture 4

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

How linear is linear?

Fraunhofer Diffraction

lecture 4

Fraunhofer Diffraction

d

�r = |rmax � d| ⇤ a2

8d⇥ ⇥/8

lecture 4

Fraunhofer Diffraction

�r = �1 + �2 ⇥a2

8

�1ds

+1dp

⇥� ⇥/8

lecture 4

Fraunhofer Diffraction

� = (k sin ⇥) · y

y

θ

aperture

Fraunhofer Diffraction

� = (k sin ⇥) · y

y

θ

aperture

f

Fraunhofer Diffraction

� = (k sin ⇥) · y

y

θ

aperture

fillumination

Diffraction in the image plane

A diffraction pattern formed in the image plane of an optical system issubject to Fraunhofer diffraction

Fraunhofer Diffraction

what is being imaged?

lecture 4

Fraunhoferdiffraction:in the imageplane

Fraunhofer Diffraction

lecture 4

Equivalentlens system: Fraunhofer diffraction independent on aperture position

Fraunhofer Diffraction

lecture 4