1
M.P. Vaughan
PY3101 Optics
Revision
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Course overview
Wave Optics
Electromagnetic Waves
Geometrical Optics
Crystal Optics
2
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Wave Optics
• General physics of waves with application to
optics
• Huygens-Fresnel Principle
• Derivation of Laws of Optical Propagation
• Rectilinear motion
• Reflection
• Refraction
• Diffraction
• Diffraction gratings (use in spectroscopy)
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Electromagnetic Waves
• The wave equation
• The electric susceptibility tensor• Light propagation in isotropic media
• Refractive index and dispersion
• Optical loss
• Polarisation• Polarising optical elements (linear, retardation
plates)
3
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometrical Optics
• Fermat’s Principle (least time)
• Derivation of Laws of Optical Propagation
• Imaging with lenses and mirrors
• Perfect imaging
• Spherical lenses and mirrors
• Paraxial approximation
• Aberrations
• Systems of lenses and mirrors
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Crystal Optics
• The index ellipsoid
• Birefringence
Crystal Optics – light propagation in anisotropic
media
4
Huygens-Fresnel Principle
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to
• Write down the form of a spherical wave
• Define a wavefront
• State Huygens’ Principle
• Using Huygens’ Principle
• Derive the Law of Rectilinear Propagation
• Derive the Law of Reflection
• Derive the Law of Refraction
Huygens’ Principle
5
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical waves
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical waves
Since the intensity of an EM wave is proportional to the squared modulus of the amplitude, by the conservation
of energy, the amplitude must vary as 1/r.
Moreover, the requirement that the amplitude be finite at r= 0 means that the spherical wave must be of the form
( ) .sin, krer
EtrE tir ω−=
6
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometric wavefront
A geometric wavefront is the surface in
space containing all points in an optical
field that have the same phase.
A ray is a path through space that is
everywhere perpendicular to the
wavefront.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometric wavefront - spherical
Wavefronts –contours of
constant phase
7
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometric wavefront - spherical
Wavefronts –contours of
constant phaseRays –
everywhere perpendicular to wavefronts
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometric wavefront - plane
Wavefronts –contours of
constant phase
8
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometric wavefront - plane
Wavefronts –contours of
constant phaseRays –
everywhere perpendicular to wavefronts
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle
Each point on a wavefront acts as a source
of secondary, spherical wavelets.
At a later time, t, a new wavefront is
constructed from the sum of these
wavelets.
9
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle – rectilinear propagation
All points on the wavefront act as sources of spherical wavelets
z0
constant phase over surface of sphere
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle – rectilinear propagation
Since all points on the spheres must have the same phase, the tangent to the leading edge of all the spheres must also be at a constant phase.
z0
10
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - reflection
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - reflection
11
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - reflection
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - reflection
12
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - reflection
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - refraction
13
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - refraction
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - refraction
14
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - refraction
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Huygens’ Principle - refraction
15
The Huygens-Fresnel Principle
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to
• Explain what is meant by coherence and
interference
• State the Huygens-Fresnel Principle
• Explain how this is different to Huygens’
Principle
Huygens-Fresnel Principle
16
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coherence
If two beams of light are coherent with
each other, then there is a fixed relation
between their phases
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Interference
Two coherent beams may add together via
the Principle of Linear Superposition to
obtain interference
17
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Huygens-Fresnel Principle
For light of a given frequency, every point
on a wavefront acts as a secondary source
of spherical wavelets with the same
frequency and the same initial phase.
The wavefront at a later time and position
is then the linear superposition of all of
these wavelets.
Diffraction
18
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to
• Describe the diffraction regimes (near-field
and far field)
• Derive the single-slit diffraction pattern
• Generalise to the multiple-slit case
• Explain the Rayleigh criterion
• Analyse diffraction gratings
• Discuss the applications of diffraction
• Sketch and explain basic monochromator
designs
Huygens-Fresnel Principle
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
What is diffraction?
• Diffraction is the ‘bending’ of
waves around objects or through
apertures
• It is an interference effect
19
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Light passing through a narrow aperture
Huygens-Fresnel Principle
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Light passing through a narrow aperture
Maximum possible path difference
.max DABBPAP ==−=∆
20
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Limiting cases: λλλλ >> D
∆max always less than λ – wavelets add constructively in all directions.
Emergent field looks like point
source.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Limiting cases: λλλλ << D
Both constructive and destructive interference outside shaded region
Wavelets add constructively in this region
21
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Fresnel and Fraunhofer diffraction
Near field (Fresnel diffraction)
Far field (Fraunhofer diffraction)
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Single slit diffraction
EL is the field strength per unit
length
EP is the total field a the point P
22
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Single slit diffraction
Field at x
.dxEdE L=
Contribution to field EP due to dE
( )( )[ ] .sin dxxkrt
xr
EdE L
P −= ω
Total field EP
( )( )[ ] .sin
2/
2/∫− −=D
D
LP dxxkrt
xr
EE ω
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Single slit diffraction
r(x) is given by the cosine rule
( ) ( )θπ −−+=2
222 cos2RxxRxr
x
or
( ) .sin2
1
2/1
2
2
−+= θ
R
x
R
xRxr
To find a closed form solution, we must approximate this expression.
23
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Taylor series expansion of r(x)
The Taylor series expansion for a function (1 + ξ)1/2 is
( ) K+−+=+82
112
2/1 ξξξ
Hence,
( )
++−= Kθθ 2
2
2
cos2
sin1R
x
R
xRxr
and
( ) .cos2
sin 22
K++−= θθR
kxkxkRxkr
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Fraunhofer condition
The third term in the expression for kr(x) takes its maximum
when x ± D/2 and θ = 0. That is
.48
cos2 2
2
2
22
2
R
D
R
kD
R
kx
λπ
θ =→
The condition that this term makes a negligible contribution to the phase is
.4 2
2
πλπ
<<R
D
24
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Fraunhofer condition
Neglecting the factor of 4 in the denominator of the condition just found, it may be re-written as
.DR
D λ<<
This is the Fraunhofer condition for far field diffraction.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Far field approximations
Assuming that the Fraunhofer condition is valid, the third term in the expression for kr(x) may be neglected and we have
( ) .sinθkxkRxkr −≈
The 1/r(x) factor appearing in the integral for EP is less
sensitive to changes in r(x) than the phase and we may simply put
( ).
11
Rxr≈
25
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Integrating over x
Using these approximations, the expression for the total field EP becomes
To perform this integral, we note that
[ ] ( ){ }.Imsinsin sinθωθω kxkRtiekxkRt +−=+−
[ ] .sinsin2/
2/∫− +−=D
D
LP dxkxkRt
R
EE θω
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The total field EP
Integrating over the x-dependent part
where
.sin2
θβkD
=
,sin
sin
2/
2/
sin2/
2/
sin
ββ
θ
θθ D
ik
edxe
D
D
ikxD
D
ikx =
=
−−∫
Hence, the total field EP is
( ).sinsin
kRtR
DEE L
P −= ωββ
26
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for a single slit
Averaging EPover time gives
The squared modulus of this will be proportional to the intensity, i.e.
.sin
2 ββ
R
DEE L
P =
( ) ( ) .sin
0
2
ββ
θ II =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for a single slit
27
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for a single slit
The zeros of the peaks occur at values of
where m is an integer. Hence, the first zeros around the central peak are given by
,sin2
πθβ mkD
==
.sinD
λθ =
Note that this result is only valid for λ < D. In other cases,
there are no zeros from –π to π.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for a circular aperture
28
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Airy disc
Airy pattern by Sakurambo. A computer-generated image of an Airy disk.URL: http://en.wikipedia.org/wiki/File:Airy-pattern.svg
Airy disc
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Airy disc
Laser Interference by Petrov Victor. Diffraction of red laser beam by a circular aperture.
URL: http://en.wikipedia.org/wiki/File:Laser_Interference.JPG
29
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Rayleigh criterion
The first zero of the intensity profile for diffraction from a circular aperture occurs at
.22.1sinD
λθ ≈
This represents the minimum angular separation that two
points can be so that they may be separately resolved.
Using the small angle approximation, this becomes
.22.1D
λθ ≈
This is known as the Rayleigh criterion.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction limited imaging
Intensity profiles for two resolvable distant point sources.
Merged intensity profiles for unresolvable distant point sources.
30
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for multiple slits
( ) ( ) .sin
sin
sin0
22
=ββ
αα
θN
NII
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Intensity profile for multiple slits
31
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction condition
Note that the condition for constructive interference is
.sin λθ ma =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction condition
We can re-write this as
.sin2
mka
πθ =
But this is just,mπα =
( ) ( ) .sin
sin
sin0
22
=ββ
αα
θN
NII
which gives the condition for the local maxima of the intensity
32
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction – wavelength dependence
.sin λθ ma =
Red (longer wavelength) light is diffracted to a greater extent than
blue (shorter wavelength).
(Yellow arrow is incident light and specular reflection)
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction condition
Incident and diffracted wavevectors:
,zk k=
( ).cosˆsinˆ' θθ zxk += k
33
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction condition: off-axis incidence
.sin,sin mi aOBaAO θθ ==
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The grating equation
( ) .sinsin λθθ ma im =+
For off-axis transmission, the diffraction condition is now
This reduces to the case of normal incidence when .0=iθ
This result may be further generalised by taking the incident angle around to the front of the grating – i.e. making the grating into a reflection grating.
34
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection gratings
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection gratings
Light strikes the reflection grating at an angle θi. For certain angles θm, the diffraction condition will
be met:
The path lengths of rays from the incident
wavefront via the successive rulings of the
grating and leaving at the same angle must
differ only be integral multiples of the
wavelength λ.
35
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Derivation of the reflection grating equation
Incident wavefront AC
Path from A to wavefront BD
.sin maAB θ=
Path from C to wavefront BD
.sin iaCD θ=
( ).sinsin imaCDAB θθ −=−Path difference
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The reflection grating equation
The diffraction condition for a reflection grating may then be expressed mathematically as
( ).sinsin imam θθλ −=
This is known as the reflection grating equation.
36
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Dispersion
The dispersion of a grating is defined as
.λθ
θd
dD m=
Differentiating the grating equations, we have
.cos m
m
ad
dm θ
θλ
=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Number of orders
Recall that
So the highest order m is the largest integral value of
( ) .sinsin λθθ ma im =±
( ).sinsin im
aθθ
λ±
.2
max λa
m <
Hence
37
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Resolving power
The resolving power of a grating is defined as
,λλ∆
=R
where, via Rayleigh’s criterion, ∆λ is the minimum resolvable wavelength between the peaks of two wavelengths with midpoint λ.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Common example of a grating
38
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Diffraction around a razor blade
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
X-ray diffraction (non-optical)
(See PY3105)
39
Maxwell’s equations
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to
• Define the electric susceptibility tensor
• Derive the wave equation for an isotropic
medium
• Write down plane-wave solutions of the wave
equation
• Explain the phenomena of dispersion
• Describe the mechanism of optical loss
• Starting with a complex wave vector, derive
the light intensity with distance and the
absorption coefficient
Maxwell’s equations
40
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Maxwell’s equations
.1
and
,
,
,0
,
0
0 MHPED
DjH
BE
B
D
−=+=
∂∂
+=×∇
∂∂
−=×∇
=⋅∇
=⋅∇
µε
ρ
t
t
f
f
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The electric susceptibility tensor
The electrical polarisation P of a medium is given by
,0 EP Eχε=
where χE is the electric susceptibility tensor. χE
characterises the frequency response of the medium to an applied electric field E.
41
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Linear, isotropic and homogeneous media
In a linear, isotropic and homogeneous (LIH) medium
.
00
00
00
0
0
0
=
χχ
χχE
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Wave equation in a LIH medium
In a dielectric the free charge density and free current are zero, so, from Maxwell’s equations
0=⋅∇ D
and
.t∂
∂=×∇D
H
( ) ,00 EEID εεχε =+= E
where ε is the relative permittivity.
42
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Wave equation in a LIH medium
Similarly
,1
0
BHµµ
=
where µ is the relative permeability. Hence
00 =⋅∇=⋅∇ ED εεand
.
,1
00
0
0
t
tt
∂∂
=×∇→
∂∂
=∂∂
=×∇=×∇
EB
EDBH
µεµε
εεµµ
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Wave equation in a LIH medium
From Maxwell’s equations
.2
2
00tt ∂
∂−=
∂×∂∇
−=×∇×∇EB
E µεµε
Using the vector identity
,0=⋅∇ E
and
( ) EEE2∇−⋅∇∇=×∇×∇
43
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Wave equation in a LIH medium
we have
.2
2
00
2
t∂∂
=∇E
E µεµε
The wave speed is
( ) 2/1εµ=n
where
( ) ,2/1
00n
cv == −µεµε
is the refractive index.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Plane-wave solutions
We look for solutions of the form
where E0 is a Jones vector containing information about the polarisation.
.
,
2
2
2
222
ω−→∂∂
=−→∇
t
kk
Now
( ) ( ),, 0
tiet ω−⋅= rkErE
44
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Plane-wave solutions
So
which gives
.nk
vω
=
,00
22
2
2
00
2EE
EE µεµεωµεµε =→
∂∂
=∇ kt
.k
vω
=
We may put where k0 is the free space wave-vector, so
,0nkk →
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Frequency dependence
A simple analysis yields
showing that χE is frequency dependent. This gives rise to the phenomenon of dispersion (different frequencies of light travelling at different speeds in an optical medium).
( )∑ +−=
i ii
iiE
i
mq,
22
2
0 τωωωχε
45
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Frequency dependence
The relative permittivity is
This means
.1 21 εεχε iE −=+=
21 innn −=and
,21 ikkk −=
where .0nkk =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Relative permittivity
46
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optical loss
Using
Substituting for v using the complex refractive index,
( ) ,exp, 0
−=v
ztitz ωEE
( ) .expexp, 210
−
−=c
zn
c
zntitz ωEE
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The absorption coefficient
Now the intensity of the radiation I(z) is proportional to the squared modulus of the field
So
( ) .2
exp 22
0
2
−=∝c
znzI
ωEE
( ) ( ) ,0 zeIzI α−=
where
c
n22ωα =
is the absorption coefficient.
47
Polarisation
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to
• Define the polarisation of light
• Describe and analyse
• plane-polarisation
• circular polarisation
• elliptical polarisation
• Describe
• linear polarisers
• retardation (wave) plates
• Apply the Jones calculus to states of
polarisation and optical elements
Polarisation
48
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Linear polarisation
( ),0
rkEE ⋅−= tie ω
A plane-wave may be written
where E0 is a Jones vector, containing information about the polarisation.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Linear polarisation
.sin
cos00
=
θθ
EE
For linear polarisation at an angle θ to the x-axis E0 is given by
49
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Linear polarisation
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Linear polarisation – special cases
,0
100
= EE x
,1
000
= EE y
,0=θ
,2
πθ =
x-linearly polarised
y-linearly polarised
50
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
y-linearly polarised light
(x-linearly polarised aligned with x-axis)
y-linearly polarised
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The dichroic sheet
51
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The linear polariser
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• The most general form of polarisation
is elliptical polarisation
• the electric field spirals around the
propagation axis tracing out an ellipse.
• This may be understood by resolving
the electric field into orthogonal
components.
• So long as these components remain in
phase, the polarisation will be linear.
Elliptical polarisation
52
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• If, however, a phase shift is introduced
on to one of the components, the
polarisation will become elliptical.
• This is illustrated in the next slide...
Elliptical polarisation
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Elliptically polarised light
53
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Retardation
( ),sin
cos0
rkEE
⋅−
= ti
i
i
ee
ey
x
ωφ
φ
θθ
.xy φφ −=Γ
This phase shift is known as the between orthogonal components is known as the retardation Γ. For the polarisation
This is
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Retardation
( ).sin
cos0
rkEE
⋅−Γ
−
= ti
i
ie
ee x ωφ
θθ
.sin
cos00
= Γ θ
θie
EE
We may re-write the polarisation as
Since the x-phase factor is arbitrary,
54
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation
iee ii ±== ±Γ 2/π
.1
2
0
0
±
=i
EE
,4
πθ =
In the case
and
we have
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – x component
55
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – ΓΓΓΓ = ππππ/2
,2/ iee ii ==Γ π
( ).1
2
0 rkE
E⋅−
= tie
i
ω
[ ] ( )( )
.sin
cos
2Re
0
⋅−−
⋅−=
rk
rkEE
t
t
ωω
In the case
we have
The real part is then
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – y component
2/πixy eEE =
56
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – left polarised
2/πixy eEE =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – ΓΓΓΓ = –ππππ/2
,2/ iee ii −== −Γ π
( ).1
2
0 rkE
E⋅−
−
= tiei
ω
[ ] ( )( )
.sin
cos
2Re
0
⋅−
⋅−=
rk
rkEE
t
t
ωω
In the case
we have
The real part is then
57
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – y component
2/πixy eEE −=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Circular polarisation – right polarised
2/πixy eEE −=
58
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Elliptical polarisation: general case
.sincos2 2
00
2
0
2
0
Γ=Γ−
+
x
x
y
y
x
x
y
y
E
E
E
E
E
E
E
E
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Elliptical polarisation: general case
xE
yE
E
α
. arbitrary, is 00 yx EE ≠Γ
59
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• A retardation or wave plate is an optical
element that produces some
retardation between the orthogonal
components of the wave.
• The physical origin of this retardation
is due to the phenomenon of
birefringence
Wave plates
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Birefringence is a phenomenon in
which the components of the wave see
a different refractive index depending
on the orientation of the polarisation
within some anisotropic material
• This leads to light with different polarisation
directions having different phase velocities
Wave plates
60
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Fast and slow axes
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• The speed of the waves is determined
by the refractive index that it sees.
• This is determined by a construction
known as the index ellipsoid
The index ellipsoid
61
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Quarter wave plate
• Introduces a phase shift of
• Produces circularly polarised light from
linearly polarised light
• Half wave plate
• Introduces a phase shift of
• Reverses sign of y-component
• Hence, for linearly polarised light at an angle θto the wave plate axis, the light is rotated by
2θ.
Types of wave plate
2/π±
π±
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Polarised light may be analysed by
passing it through a linear polariser
• In the case of initially linearly polarised
light, the emergent intensity follows
Malus’ Law
• Here, we consider the general case
The analysis of polarised light
62
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The analysis of polarised light
( ) ( ),cossincos2sincos 222
0 Γ++= θθθθθ rrII
.0
0
x
y
E
Er =
The power intensity through an analyser is given by
where
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The analysis of linearly polarised light
( ) .cos20 θθ II =
,00 =yE
For x-linearly polarised light, we have
which gives
This is known as Malus’ Law.
63
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The analysis of circularly polarised light
( ) ( ) .sincos 0
22 II =+= θθθ
.2
and1π
±=Γ=r
For circularly polarised light, we have
which gives
In other words, the time-averaged intensity is
constant.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Jones vectors
.sin
cos00
=
θθ
EE
Linear polarisation
General case
x-linearly polarised
,0
100
= EE x
y-linearly polarised
,1
000
= EE y
64
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Jones vectors
Elliptical polarisation
General case
left-circularly polarised
.1
2
0
=+
i
EE
right-circularly polarised
.1
2
0
−
=−i
EE
.sin
cos00
= Γ θ
θie
EE
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Jones matrices
Polarisers
General case
x-linear polariser
.00
01
=xP
y-linear polariser
.10
00
=yP
.sinsincos
sincoscos2
2
=
θθθθθθ
θP
65
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Jones matrices
Retardation plate
General case
Quarter wave-plate
.0
012/
±
=±i
πM
Half wave-plate
.10
01
−
=πM
.0
01
= ΓΓ ie
M
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Jones matrices
The effect of a series of optical elements may be modelled by multiplying the
corresponding Jones matrices together
to form a combined element.
66
Fermat’s Principle
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to• Define the optical path length
• State Fermat’s Principle
• Use Fermat’s Principle to
• derive the Law of Reflection
• derive the Law of Refraction
• Demonstrate perfect imaging
Fermat’s Principle
67
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Geometrical wavefront
These points are all in phase with one another and constitute a geometric wavefront.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The optical path length
Putting T = t - t0, we may then multiply T by c to express the propagation time in dimensions of space
( ) ( ).0ttcr −=Λ
The quantity Λ(r) is known as the optical path length and is a function of distance.
68
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The optical path length
Now, if
,n
cv
dt
dS==
then
( ) ,ndScdtrd ==Λ
so
( ) ( ) .,,0∫=Λr
dSzyxnr
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Homogeneous medium
In a homogeneous medium
( ) ,0,, =∇ zyxn
that is:
the refractive index is the same everywhere.
69
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Isotropic medium
In an isotropic medium
( ) ,,, 0nzyxn =
that is:
the refractive index is the same in all directions
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optical path length – LIH medium
We also make the usual assumption that the response of
the medium is linearly proportional to the applied field.
For a linear, isotropic homogenous (LIH) medium
becomes
( ) .0∫=Λr
dSnr
( ) ( )∫=Λr
dSzyxnr0
,,
70
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Fermat’s Principle
The path taken between two points by
a ray of light is the path that can
traversed in the least time
or, equivalently in terms of optical path length,
Light traverses the route between two points for which the optical path
length is a minimum.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Rectilinear propagation
Using the calculus of variations, we can use Fermat’s Principle to derive Law of Rectilinear Propagation
In a LIH medium, light propagates in straight lines.
71
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
We may also apply Fermat’s Principle under constraint.
For instance, we may impose the constraint that light
travelling between A and B in a medium of refractive index
n1 must touch some point on the interface between this
medium and another of refractive index n2.
This is the required constraint for reflection.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
72
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
.11 BA SnSn +=Λ
( )[ ]( ) .
,
2/122
2/122
BBB
AA
yxxS
yxS
+−=
+=
Applying Fermat’s Principle, we have
.01 =
∂∂
+∂∂
=∂Λ∂
x
S
x
Sn
x
BA
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
( )
[ ]( ).
,
2/122
2/122
B
B
BB
BB
AA
A
S
xx
yxx
xx
x
S
S
x
yx
x
x
S
−=
+−
−=
∂∂
=+
=∂∂
But
.sinandsin r
B
Bi
A S
xx
S
xθθ =
−=
73
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
So Fermat’s Principle implies
( ) .0sinsin1 =−=∂Λ∂
rinx
θθ
This is satisfied when
ri θθ sinsin =
or
ri θθ =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Reflection
Thus, Fermat’s Principle reproduces the Law of Reflection
In a LIH medium, the angle of reflection equals the
angle of incidence.
74
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Refraction
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Refraction
.21 BA SnSn +=Λ
( )[ ]( ) .
,
2/122
2/122
BBB
AA
yxxS
yxS
+−=
+=
Applying Fermat’s Principle, we have
.021 =∂∂
+∂∂
=∂Λ∂
x
Sn
x
Sn
x
BA
75
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Refraction
( )
[ ]( ).
,
2/122
2/122
B
B
BB
BB
AA
A
S
xx
yxx
xx
x
S
S
x
yx
x
x
S
−=
+−
−=
∂∂
=+
=∂∂
But
.sinandsin t
B
Bi
A S
xx
S
xθθ =
−=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Refraction
So Fermat’s Principle implies
.0sinsin 21 =−=∂Λ∂
ti nnx
θθ
This is satisfied when
,sinsin 21 ti nn θθ =
i.e. by Snell’s Law.
76
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Refraction
Thus, Fermat’s Principle reproduces the Law of Refraction
In a LIH medium, the Law of Refraction is given by Snell’s Law
,sinsin 21 ti nn θθ =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Perfect imaging – hyperbolic lens
77
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Perfect imaging – elliptical lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Perfect imaging – elliptical mirror
78
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Perfect imaging – elliptical mirror
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Perfect imaging – parabolic mirror
79
Spherical lenses and mirrors
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to• State the paraxial approximation
• Define the focal length
• Recall
• the thin lens equation
• the lens-maker’s equation
• the Gaussian lens formula
• the expression for a series of thin lenses in close
combination
• Recall and apply the rules for image construction
• Calculate transverse magnification
• Define the optical power of a lens
Spherical lenses and mirrors
80
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Cannot obtain perfect imaging
• Reasonable approximation possible
Imaging by a spherical lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Lens sign conventions
81
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Lens sign conventions
• Light is always taken to
propagate from left to right.
• If A is to the left of B, then so is
taken to be positive (and vice
versa).
• If C is to the right of B, then si is
taken to be positive (and vice
versa).
• If the centre of the sphere is to the right of B, R is taken
to be positive. This is a convex lens.
• If the centre of the sphere is to the left of B, R is taken
to be negative. This is a concave lens.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Requirement for perfect imaging is that
all the rays have equal optical path-
length.
• That is, we require Λ to be a constant
Imaging by a spherical lens
82
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Imaging by a spherical lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The optical path length
.1221
o
o
i
i
io l
sn
l
snR
l
n
l
n−=
+
However
( )( ),,
,,
iii
ooo
sll
sll
φ
φ
=
=
so no closed form solutions.
83
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The paraxial approximation
Small angle approximation
.1cos,sin ≈≈ φφφ
22
oo sl →
So
and
.22
ii sl →
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The paraxial approximation
With these approximations, we obtain
( ).112
21 nnRs
n
s
n
io
−=+
84
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The focal length
In the case
,0 ∞→s
From this, we may define the focal length within the lens fi
( ).112
2 nnRs
n
i
−=
.12
2ii fR
nn
ns ≡
−=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The focal length
Similarly, when
,∞→is
we may define the focal length outside the lens fi
.12
1oo fR
nn
ns ≡
−=
85
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
A thick lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
A thick lens
86
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
A thick lens
For the surface with radius of curvature R1,
( ).112
11
2
1
1 nnRs
n
s
n
io
−=+
( ).112
22
2
2
1 nnRs
n
s
n
oi
−−=+
For the surface with radius of curvature R2,
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
A thick lens
Combining these results
( )
+−−
−=
+
2
2
1
212
2112
1
11111
oioi s
n
snnn
RRssn
Inserting
,12 io sds −=
87
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The thin lens equation
( )( )
.1111
11
212
2112
1
iioi sds
dnnn
RRssn
−−−
−=
+
.1111
211
12
−
−=+
RRn
nn
ss oi
Taking the limit and putting,0→d ,, 22 ooii ssss ==
This is the thin lens equation.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The lens maker’s formula
.111
211
12
−
−=
RRn
nn
f
If either or ∞→is ∞→os
.111
or111
211
12
211
12
−
−=
−
−=
RRn
nn
sRRn
nn
s io
But the term on the RHS is a constant, which we define to be the focal length f
This is the lens maker’s formula.
88
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The Gaussian lens formula
.111
fss oi
=+
We must also have
This is the Gaussian lens formula.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Thin lenses in close combination
In general
.11
∑=i iff
89
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Mirror sign conventions
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Light is always taken to propagate from
left to right.
• The object distance so is positive when it
is to the left of the mirror surface.
• The image distance si is positive when it
is to the left of the mirror surface (real
image).
Mirror sign conventions
• The image distance si is negative when it is to the
• right of the mirror surface (virtual image).
• The radius R is positive if the mirror surface is to the right of the
centre of the sphere (convex mirror)
• The radius R is negative if the mirror surface is to the left of the
centre of the sphere (concave mirror)
90
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical mirrors
From the Law of Reflection
.φαθ +=
( ) .φββπφπθ −=−−−=
From inspection of the figure
Hence
.2 αβφ −=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical mirrors
Hence
.211
Rss io
−=+
Taking limits as before and defining the focal length f, we then have
,111
fss io
=+
which is the same expression as the Gaussian lens
formula.
91
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Sketch a ray from the tip of the object parallel to the
horizontal (the principle axis) to the centre line of the
lens. From there, sketch another ray passing
through the focus associated with the left-hand lens
surface.
• Sketch a ray from the tip of the object directly
through the centre of the lens without deviation.
• Sketch a ray from the tip of the object passing
through the focus associated with the right-hand
lens surface to the centre line of the lens. From there
sketch a line parallel to the principle axis towards
the image.
Image construction for convex lenses
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Convex lens
92
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Convex lens
The image at i is real and inverted.
The magnification of the image M is given by
.o
i
y
yM =
Since yi is negative, so is M (inverted image).
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Sketch a ray parallel to the optical axis of the
lens. Since f < 0, this must pass through f on
the left of the lens (as a virtual ray).
• Sketch a ray passing through the centre of
the lens without deviation
• Sketch a ray following the line through f on
the right of the lens (this extension is virtual
on the right) and emerging parallel to the
optical axis. The parallel line is then
extended to the left as a virtual ray
Image construction for concave lenses
93
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Concave lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Concave lens
In this case, xo is the distance
between the object and f on the right of the lens.
xi is the distance between the image and f on the left of the lens.
94
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Other types of lenses
plano-convex plano-convex doublet
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optical power
.1
f=P
The optical power P of a lens is a measure of the degree to which it converges or diverges light.
P is defined as the reciprocal of the focal length.
The SI unit of P is called the dioptre (m-1).
95
Optical instruments
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to• Discuss the anatomy and function of the
human eye
• Describe common visual impairments
• Derive the angular magnification for a
magnifying glass
• Describe different types of refracting
telescope and their designs
• Describe the design of reflecting
telescopes
Optical instruments
96
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The anatomy of the eye
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The lens
The curvature of the lens may be changed by muscle contractions in the eye.
97
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Near and far points
• The near point is the closest distance for
which the lens can focus light on the retina
• Typically at age 10, this is about 18 cm
• It increases with age, ~ 25 cm for an adult
• The far point of the eye represents the
largest distance for which the lens of the
relaxed eye can focus light on the retina
• Normal vision has a far point of infinity
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Farsightedness – hyperopia
Distant objects may be focussed but not nearby objects.
98
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Correcting farsightedness
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Nearsightedness – myopia
• axial myopia
• Lens too far from retina
• refractive myopia
• Lens-cornea system too powerful to focus properly onto the
retina
99
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Correcting nearsightedness
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular size of unaided image
The object at o subtends an angle of αu at the viewing point.
100
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Magnifying glass - aided image
The image i of an object placed at
o within the focal length of a convex mirror subtends an angle of αa at the viewing point.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular magnification
The angular magnification Mα is defined as the ratio of the aided and unaided viewing angles
.u
aMαα
α =
We shall employ the paraxial approximation to obtain an expression for this.
101
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular magnification
.,i
i
o
oa
ou
d
h
d
h
D
h=== αα
From the diagrams
So
.ou
a
d
DM ==
αα
α
where D is the distance to the object in the unaided case.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular magnification
.fd
h
f
h
i
io
+=
From the diagram
,i
o
i
o
d
d
h
h=
.111
oi ddf=+
From the expression for αa,
Leading to
102
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular magnification
From
,od
DM =α
.11
+=
idfDMα
We then have
Thus the shorter the focal length, the greater the angular
magnification.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Terrestrial
• Produces upright image
• Employs a concave lens for the eyepiece
• Astronomical
• Inverts the image
• Employs a convex lens for the eyepiece
Types of refracting telescope
103
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Galilean (terrestrial) telescope
Gives upright image. Note the focal points coincide.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Keplerian (astronomical) telescope
Gives inverted image. Note the focal points coincide.
104
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Angular magnification
Galilean telescope Keplarian telescope
From the system matrix of each telescope, both are found to have an angular magnification of
.e
o
f
fM −=α
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Newtonian telescope
105
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Newtonian telescope
.e
o
f
fM −=α
Essentially, we have the same magnification system as for the astronomical telescope.
Hence, the angular magnification is
Aberrations
106
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to• Discuss what is meant by third order
aberration
• List and describe common forms of
monochromatic aberration
• Explain the origin of chromatic aberration
and strategies to correct it
Aberrations
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Third order correction
θθ ≈sin
L++−=!5!3
sin53 θθ
θθ
Previously, we employed the paraxial approximation
In reality
The second term is referred to as the third order correction. The second term in the expansion of cos is also used in corrections to this order.
and.1cos ≈θ
107
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Third order correction
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Third order correction
.1111
2
21
22
1221
−+
++
+−=
+
iioo
io
sRs
n
sRs
nR
nnRs
n
s
n
φ
108
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Third order correction
Note that the Rφ2 term gives a measure of the displacement of the intersection of the ray with the lens from the optical axis.
Thus, in the third-order treatment, the new term increases in proportion with the square of the angular displacement.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Spherical Aberration
• Coma
• Astigmatism
• Field Curvature
• Distortion
Third order corrections
109
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical Aberration
• Due spherical curvature of lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Longitudinal and transverse spherical aberration
The longitudinal spherical aberration LSA is defined as the distance between the intersection of a ray with the optical axis and the paraxial focus.
'.ooSA ssL −=
The transverse spherical aberration LSA is defined as the perpendicular distance above (or below) the paraxial focus that a ray actually passes.
110
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Spherical Aberration
• Due spherical curvature of lens
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Soft image focusing
111
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coma
• Due to off-axis object points
• Transverse magnification is a function of ray height
• Pattern looks like a comet
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coma in a lens
112
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Coma in a parabolic mirror
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Corrective lenses for Newtonian
telescopes with f numbers less than f/6
have been designed
• These employ a dual lens system of a
plano-convex and a plano-concave lens
fitted into an eyepiece adaptor
• An example of a correction strategy for
coma is Baader Rowe Coma Correction
Coma
113
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Baader Rowe Coma Correction
Comparison of the coma in an uncorrected f/3.9 Newtonian telescope vs the affects of coma with the Baader Rowe Coma Corrector.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Astigmatism
• Vertical plane is the ‘tangential’ plane
• Horizontal plane is the ‘sagittal’ plane
• Astigmatism results in different focal length in each plane
114
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Field Curvature
• A thin lens images a spherical surface onto a spherical surface
• Image is distorted in the image plane
• Important in lens design for close objects
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Distortion
• All points in the object plane are imaged to pointsin image plane
• Distortion arises when the magnification of off-axis image is a function of the distance to the lens center
115
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Barrel distortion
• magnification decreases with distance from the optical axis
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Pincushion distortion
• magnification increases with distance from the optical axis
116
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Moustache distortion
• Moustache distortion, in which initially the magnification decreases with distance from the optical axis, whilst at further distances, the magnification increases with distance.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Chromatic Aberration
• Blue refracts more than red (greater refractive index for normal dispersion
117
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Achromatic doublet
• An achromatic doublet (achromat) is often used to compensate for the chromatic aberration.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Achromatic doublet
011
)(11
)(32
12
21
11 =
−−+
−−
RRnn
RRnn RBRB
BR
BRff
ff11
=→=
We require
Thus, we need to choose parameters such that
118
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Example of the use of an achromatic doublet, using a doublet as the objective.
Achromatic doublet
The index ellipsoid
119
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• You should be able to• Describe the modes of vibration of the light
• Explain the modes of vibration in terms of the index ellipsoid
and the k-vector direction
• Explain how the optic axes of the crystal are determined.
Hence, describe the different optical classes
• Derive the expression for the index ellipsoid from the energy
density
• Explain birefringence and apply to problems in uniaxial
crystals
• Describe the use of birefringence in wave plates
• Explain double refraction in anisotropic crystals
The index ellipsoid
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The wave equation
We look for solutions of the form
( ) .2
2
00
2
t
EE
x
iii
i ∂∂
−=∇−⋅∇∂∂
µεεµE
( ).ti
ieEE ω−⋅= rk
120
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The wave equation
We obtain an eigenvalue problem with the characteristic equation
.ω
κ ii
ck=
where we have defined
,0222
222
222
=
−−−−
−−−−
−−−−
zzzyzx
zyyyyx
zxyxxx
nn
nn
nn
κκκκκκκκκκκκκκκ
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The wave equation
In the general case
and
( ),222222
zzyyxx ananana ++−=
( ) ( ) ( )222222222 111 xzyyzxzyx annannannb −+−+−=
.222
zyx nnnc −=
121
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• The solutions for n2 correspond to two
modes of vibration
• Different components of the
polarisation see different refractive
indices
• These modes of vibration may be
visualised by means of the index
ellipsoid
The index ellipsoid
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
The index ellipsoid
122
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Consider some arbitrary wavevector k
• Taking the intersection of the plane
perpendicular to k with the index
ellipsoid defines an ellipse
• The semi-axes of this ellipse give
refractive indices n’ and n’’, which
correspond to the two modes of
vibration D’ and D’’
The index ellipsoid
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• In general, an ellipsoid has two circular
cross-sections
• In the case of just two distinct semi-
axes, we have a spheroid and there is
just one circular cross-section
• The normals to these cross-sections
are known as the optic axes of the
crystal
Optic axes
123
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optic axes
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Biaxial crystals
• two optic axes (these are shown as N1 and N2
in the previous Fig.)
• Uniaxial crystals
• only one optic axis (taken, convention, to be
along the z-axis)
• Isotropic crystals
• No optic axis – refractive index the same in all
directions
Optic axes – optical classes
124
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optic axes
For certain special k directions, the quadratic will have repeated roots.
.042 =− acb
In these cases, the optical field will only see one refractive index (the cross-section with the index ellipsoid is circular)
These directions are therefore the optic axes of the crystal and determined by the condition
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystals
125
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystals
In a uniaxial crystal, we have (by convention) nx = ny = no
and nz = ne. These are known as the ordinary and extraordinary refractive indices respectively. The coefficients of the quadratic equation for n2 are then
and
( )[ ],2222
zoeo annna −+−=
( ) ( )[ ]2222
0
2
0 11 zez anannb −+−=
.24
eo nnc −=
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Optic axes
The discriminant is
( ) ( ) .142222242
eozo nnanacb −−=−
For
the discriminant is zero when az = 1, i.e. when k is parallel
with the z-axis. Thus, this is the optic axis of the crystal
,22
eo nn ≠
126
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystals
The solutions for n2 are then
and
1
2
2
222 11
−
−+= z
o
ee a
n
nnn
.202 nn =
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystals
k
127
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystals
we may obtain the explicit angular dependence
( ) .cos11
1
2
2
222
−
−+= θθ
o
ee
n
nnn
Putting
,cosθ=za
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Index ellipsoid (easier derivation)
The energy density due to the electric field is given by
This may be re-written
( ).21
21
zzyyxxE EDEDEDu ++=⋅= ED
.2
1
0
2
0
2
0
2
++=
z
z
y
y
x
xE
DDDu
εεεεεε
128
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Index ellipsoid (easier derivation)
Making the change of variable (associated with a scaling)
,2 0
22
εE
ii
u
Dx =
we then have
.12
2
2
2
2
2
=++zyx n
z
n
y
n
x
This is the equation of the index ellipsoid.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystal
For a uniaxial crystal, we have
Taking x = 0 with no loss of generality, from the figure,
.12
2
2
2
2
2
=++eoo n
z
n
y
n
x
( )( ) .sin
,cos
,0
θθθθ
nz
ny
x
=
−=
=
129
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Uniaxial crystal
Substituting these expressions into the index ellipsoid
Rearranging this, we obtain
( ) ( ).1
sincos2
22
2
22
=+eo n
n
n
n θθθθ
( ) ,cos11
1
2
2
222
−
−+= θθ
o
ee
n
nnn
as found earlier.
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Birefringence in a uniaxial crystal
The birefringence is defined as
For a wave with extraordinary and ordinary components, Ee
and Eo, propagating in a direction r, we may write
( ) .0nnn −=∆ θ
( )
−=c
rntiEEe
θωexp0
.exp0
−=c
rntiEE o
o ω
and
130
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Birefringence in a uniaxial crystal
The second of these equations may be re-written
( ) ( )[ ] .expexp0
−
−= rnnc
ic
rntiEE oo θ
ωθω
Hence, after a distance r, the ordinary wave acquires a retardation
( ) ( ).r
cr
θω∆=Γ
This provides the physical basis for retardation plates (see
Polarisation).
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• Further definition according to the
relative sizes of ne and no.
• A negative uniaxial crystal has ne < no
• E.g. calcite CaCO3 and ruby Al2O3.
• For ne > no, we have a positive uniaxial
crystal
• E.g, quartz SiO2
Uniaxial crystal
131
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
• In negative uniaxial crystal, the
extraordinary axis is aligned with the
fast axis of the plate, since c/ne > c/no.
• For a positive uniaxial crystal, we have
the opposite case and the extraordinary
axis is aligned with the slow axis.
Uniaxial crystal – wave plates
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Birefringence - double refraction
132
Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland
ROINN NA FISICE
Department of PhysicsPY3101 Optics
Double refraction – example: calcite