Download Waves, Light & Quanta PHYS1011

Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
Categories of optical polarization
• linear (plane) polarization
• non-equal components in phase
• circular polarization
• equal components 90° out of phase
• elliptical polarization
• all other cases
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Polarizing components
LINEAR
CIRCULAR
POLARIZER
(filter/separator)
Tx  Ty
TL  TR
WAVEPLATE
(retarder)
x   y
L  R
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Polarization notation
• circular polarization
• right- or left-handed rotation when
looking towards source
• traces out opposite (right- or left-)
handed thread
RCP
plane of
incidence
perpendicular
parallel
• linear (plane) polarization
• parallel or perpendicular to plane of
incidence
• plane of incidence contains
wavevector and normal to surface
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Characterizing the optical polarization
• wavevector insufficient to define
electromagnetic wave
• we must additionally define the
polarization vector

a  ax , a y


• e.g. linear polarization at angle 

i
a   cos, esin
 
sin
k


x
z
y
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Jones vector calculus
• if the polarization state may be represented
by a Jones vector

a  ax, a y

• then the action of an optical element
may be described by a matrix
 ax   a11 a12  ax 
 
 

 a  a
a 
a
y
   21 22  y 
JONES MATRIX
 a11 a12 
A

a
a
 21 22 
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Jones vector calculus
 state may be represented
transmission by
• if thepolarization
Aby

1 a
horizontal polarizer

Jones vector
1 0
0 0 
ai a x, a0y
exp

 retardation by
x
A
 the action of an optical element
• 2then
 waveplate
0
exp
i

y
may 
be described by a matrix
11 a12 
 cosA  asin
projection onto
a

A3  

rotated axes
21 a22 


sin

cos




 ax   a11 a12  ax 
 
 

 a  a
a 
a
y
   21 22  y 
JONES MATRIX
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Birefringence
• asymmetry in crystal structure
causes two different refractive
indices
• opposite polarizations follow
different paths through crystal
• birefringence, double refraction
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Linear polarizers (analyzers)
• birefringence results in different angles of
refraction and total internal reflection
• many different designs, offering different
geometries and acceptance angles
o-ray
e-ray
38.5º
e-ray
o-ray
s-ray
• a similar function results from
multiple reflection
p-ray
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Waveplates (retarders)
• at normal incidence, a birefringent
material retards one polarization
relative to the other
• linearly polarized light
becomes elliptically polarized
 
2

0 e  l

WAVEPLATE
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Compensators
• a variable waveplate uses two wedges
to provide a variable thickness of
birefringent crystal
adjust
• a further crystal, oriented with the fast
and slow axes interchanged, allows the
retardation to be adjusted around zero
variable
• with a single, fixed first section, this is a
‘single order’ (or ‘zero order’) waveplate
for small constant retardation
SOLEIL
COMPENSATOR
fixed
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Unpolarized light
• for any system
• intensity
 Ex  a b  Ex 
 
 
 E   c d  E 
 y 
 y 
E x2  E y2  aEx  bE y 2  cEx  dE y 2







 a 2  c 2 Ex2  b 2  d 2 E y2  2ab  cd Ex E y
• if no correlation between
E x and E y ,

 a 2  c 2 Ex2  b 2  d 2 E y2
• if
Ex2  Ex2  12 E02,

Tx  Ty
2
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Electromagnetic waves
• light is a transverse wave: E perpendicular to k
x
Ex
z
By
Ex
• Faraday
• Ampère

y
B
.
d
S
t 
E 

 B.ds  0   J   0 t .dS
E
 B.ds  0 0  t .dS
 E.ds  
x
y
z
By
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Dielectrics
• atomic electrons move in
response to electric field
• resulting atomic
dipole radiates field
which adds to original
• Faraday
• Ampère
z

B.dS

t
EE


B
.
d
s


J



.dS
0  
0 r .dS

t t 

 E.ds  
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Waves, Light & Quanta
Tim Freegarde
Web Gallery of Art; National Gallery, London
Diffraction
• irridescence of feathers (Grimaldi, 1665)
S Yoshioka & S Kinoshita, Forma 17 169 (2002)
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Diffraction
x
d

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Diffraction
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Diffraction
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Huygens’ wave construction
• propagation from a point source
Christiaan Huygens (1629-1695)
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Huygens’ wave construction
• reflection at a plane surface
Christiaan Huygens (1629-1695)
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Huygens’ wave construction
• refraction at a plane surface
Christiaan Huygens (1629-1695)
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Huygens’ wave construction
• mirages by refraction in the atmosphere
Christiaan Huygens (1629-1695)
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Huygens’ wave construction
• Fresnel integral
• phasors shorter / rotate more quickly
at distance to give spiral
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Arago’s bright spot
• M A Fresnel, La diffraction de la lumière (1818)
• S D Poisson: Let parallel light impinge on an opaque disk, the
surrounding being perfectly transparent. The disk
casts a shadow - of course - but the very centre of
the shadow will be bright. Succinctly, there is no
darkness anywhere along the central perpendicular
behind an opaque disk (except immediately behind
the disk).
• F Arago:
One of your commissioners, M Poisson, had
deduced from the integrals reported by [Fresnel] the
singular result that the centre of the shadow of an
opaque circular screen must, when the rays
penetrate there at incidences which are only a little
more oblique, be just as illuminated as if the screen
did not exist. The consequence has been submitted to
the test of direct experiment, and observation has
perfectly confirmed the calculation.
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