unit ii - govt.college for girls sector 11...
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UNIT II
Diffraction and Polarization
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Diffraction
Huygen’s principle requires that the waves spread out after they pass through slits
This spreading out of light from its initial line of travel is called diffraction
In general, diffraction occurs when waves pass through small openings, around obstacles or by sharp edges
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Diffraction, 2
A single slit placed between a distant light source and a screen produces a diffraction pattern
It will have a broad, intense central band
The central band will be flanked by a series of narrower, less intense secondary bands
Called secondary maxima
The central band will also be flanked by a series of dark bands
Called minima
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Diffraction, 3
The results of the single slit cannot be explained by geometric optics
Geometric optics would say that light rays traveling in straight lines should cast a sharp image of the slit on the screen
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Fraunhofer Diffraction
Fraunhofer Diffractionoccurs when the rays leave the diffracting object in parallel directions Screen very far from the
slit
Converging lens (shown)
A bright fringe is seen along the axis (θ = 0) with alternating bright and dark fringes on each side
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Single Slit Diffraction
According to Huygen’s principle, each portion of the slit acts as a source of waves
The light from one portion of the slit can interfere with light from another portion
The resultant intensity on the screen depends on the direction θ
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Single Slit Diffraction, 2
All the waves that originate at the slit are in phase
Wave 1 travels farther than wave 3 by an amount equal to the path difference (a/2) sin θ
If this path difference is exactly half of a wavelength, the two waves cancel each other and destructive interference results
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Single Slit Diffraction, 3
In general, destructive interference occurs for a single slit of width a when sin θdark = mλ / a
m = 1, 2, 3, …
Doesn’t give any information about the variations in intensity along the screen
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Single Slit Diffraction, 4
The general features of the intensity distribution are shown
A broad central bright fringe is flanked by much weaker bright fringes alternating with dark fringes
The points of constructive interference lie approximately halfway between the dark fringes
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Diffraction by a Single Slit - Locate the First Minima
D >> a, rays are parallel
Divide the screen into two zones of width a/2
Find the first minima above the midpoint by
pairing up rays from the top point of the top zone
and the top point of the bottom zone
Rays are in phase at the slit but must be out of
phase by by λ/2 at screen
The path length difference = a/2 sin θ
Repeat for other pairs of rays in the upper zone
and lower zone: a/2 sin θ = λ/2
a sin θ = λ first minima
If we narrow the slit the angle must get bigger
- more flaring - - what happens when a = λ?
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Diffraction by a Single Slit -
Subsequent Minima
Repeat the process for paired rays (4) from
corresponding points from each of the zones
Rays are in phase at the slit but must be out of
phase by by λ/2 at screen
second minimum
Dark fringes - minima
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Intensity in Single Slit Diffraction - Derive using
phasors - see text
where
α is just a convenient connection between the angle θ that locates a point
on the screen and the light intensity I(θ)
I = Im occurs at the central maximum (θ = 0) and ϕ is the phase difference
(in radians) between the top and bottom rays from the slit of width a
Studying the expression above reveals that the intensity minima occur at
Plug this into the expression for α above and we find
alpha
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Where are the diffraction Maxima?
minima asin m m=1, 2, 3..
After the central maximum the next maximum occurs about halfway between the first and second minimum.
maxima asin (m 1
2) m =1, 2, 3..
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Exact solution for diffraction maxima
I() I0sin2
2
I0(sin2)(2)
dI
d 0
I02sin cos 2 I02sin2 3 0
cos sin /
tan
Transcendental equation. Solve graphically
To find maxima of a function, take derivative and set equal to 0
Recall
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Diffraction by a circular aperture (lens ...)
First minimum
Compare to the relation for a single slit
A complex analysis leads to
When light from a point source
passes through a small circular
aperture, it does not produce a
bright dot as an image, but
rather a diffuse circular disc
known as Airy's disc surrounded
by much fainter concentric
circular rings. This example of
diffraction is of great importance
because the eye and many
optical instruments have circular
apertures.
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ResolvabilityThe fact that a lens image is a diffraction pattern is important when
trying to resolve distant point objects
When the angular separation for two objects is such
that the central maximum of one diffraction pattern
coincides with the first minimum of the other we have
a condition called the Rayleigh’s criterion for
resolvability
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Diffraction and Interference by a double slit
I = I (double slit interference) x I(diffraction)
I() Im cos2()sin2
2
d sin
dsin=m m =0,1,2.. maxima
asin
asin = m m=1,2,3.. minima
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Diffraction GratingsLike the double slit arrangement but
with a much greater number of slits, or
rulings, sometimes as many as several
1000 per millimeter
Light passed through the grating forms
narrow interference fringes that can be
analyzed to determine the wavelength
As the number of rulings increases
beyond 2 the intensity plot changes from
that of a double slit pattern to one with
very narrow maxima (called lines)
surrounded by relatively wide dark regions
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Diffraction GratingsPrincipal maxima, where all waves
interfere constructively are found with
what is now a familiar procedure:
each m represents a different line and
the integers are called order numbers
Rewriting the above we find
So the angle to a particular line (say m =3) depends on the wavelength
being used
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DispersionIn order to distinguish different wavelengths that are close to
each other a diffraction grating must spread out the lines
associated with each wavelength. Dispersion is the term used to
quantify this and is defined as
Δθ is the angular separation between two lines that differ by Δλ.
The larger D the larger the angular separation between lines of
different λ.
It can be shown that and D gets larger for higher
order (m) and smaller grating
spacing (d)
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Resolving Power
To make lines that whose wavelengths are close together (to resolve
them) the line should be as narrow as possible
The resolving power is defined by
where λavg is the average of the two wavelengths studied and Δλ is the
difference between them. Large R allows two close emission lines to be
resolved
It can be shown that To get a high resolving power
we should use as many rulings.
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Dispersion and Resolving Power
Three gratings illuminated with light of
λ=589 nm, m = 1
Grating N d(nm) θ D(o/μm) R
A 10000 2540 13.4 23.2 10000
B 20000 2540 13.4 23.2 20000
C 10000 1360 25.5 46.3 10000
Highest dispersion
Highest resolution
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Part I: Different polarization states of light
Light as an electromagnetic wave
Mathematical and graphical descriptions of polarization
Linear, circular, elliptical light
Polarized, unpolarized light
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Light as an electromagnetic wave
Light is a transverse wave,
an electromagnetic wave
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Mathematical description of the EM wave
Light wave that propagates in the z direction:
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
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Graphical representation of the EM wave (I)
One can go from:
to the equation of an ellipse (using
trigonometric identities, squaring, adding):
2
0y
y
0x
x
2
0y
y
2
0x
x sincosE
E
E
E2
E
E
E
E
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
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Graphical representation of the EM wave (II)
An ellipse can be represented
by 4 quantities:
1. size of minor axis
2. size of major axis
3. orientation (angle)
4. sense (CW, CCW)
Light can be represented by 4 quantities...
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Vertically polarized light
If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical).
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
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Polarization at 45º (I)
If there is no phase difference (=0)
and
E0x = E0y, then Ex = Ey
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
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Polarization at 45º (II)
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Circular polarization (I)
If the phase difference is = 90º
and E0x = E0y
then: Ex / E0x = cos , Ey / E0y = sin
and we get the equation of a circle:1sin cosE
E
E
E 22
2
0y
y
2
0x
x
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
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Circular polarization (II)
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Circular polarization (III)
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Circular polarization (IV)
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Elliptical polarization
• Linear + circular polarization = elliptical polarization
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Unpolarized light(natural light)
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Polarization of Light Waves
Each atom produces a wave with its own orientation of
All directions of the electric field vector are equally possible and lie in a plane perpendicular to the direction of propagation
This is an unpolarized wave
E
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Polarization of Light, cont
A wave is said to be linearly polarized if the resultant electric field vibrates in the same direction at all times at a particular point
Polarization can be obtained from an unpolarized beam by
selective absorption
reflection
scattering
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Polarization by Selective Absorption
The most common technique for polarizing light
Uses a material that transmits waves whose electric field vectors in the plane are parallel to a certain direction and absorbs waves whose electric field vectors are perpendicular to that direction
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Selective Absorption, cont
E. H. Land discovered a material that polarizes light through selective absorption He called the material Polaroid
The molecules readily absorb light whose electric field vector is parallel to their lengths and transmit light whose electric field vector is perpendicular to their lengths
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Selective Absorption, final
The intensity of the polarized beam transmitted through the second polarizing sheet (the analyzer) varies as
I = Io cos2 θ
Io is the intensity of the polarized wave incident on the analyzer
This is known as Malus’ Law and applies to any two polarizing materials whose transmission axes are at an angle of θ to each other
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Polarization by Reflection
When an unpolarized light beam is reflected from a surface, the reflected light is Completely polarized
Partially polarized
Unpolarized
It depends on the angle of incidence If the angle is 0° or 90°, the reflected beam is
unpolarized
For angles between this, there is some degree of polarization
For one particular angle, the beam is completely polarized
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Polarization by Reflection, cont
The angle of incidence for which the reflected beam is completely polarized is called the polarizing angle, θp
Brewster’s Law relates the polarizing angle to the index of refraction for the material
θp may also be called Brewster’s Angle
sintan
cos
p
p
p
n
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Polarization by Scattering
When light is incident on a system of particles, the electrons in the medium can absorb and reradiate part of the light
This process is called scattering
An example of scattering is the sunlight reaching an observer on the earth becoming polarized
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Polarization by Scattering, cont
The horizontal part of the electric field vector in the incident wave causes the charges to vibrate horizontally
The vertical part of the vector simultaneously causes them to vibrate vertically
Horizontally and vertically polarized waves are emitted
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Optical Activity
Certain materials display the property of optical activity
A substance is optically active if it rotates the plane of polarization of transmitted light
Optical activity occurs in a material because of an asymmetry in the shape of its constituent materials
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Polarizers
Polarizers absorb one component of the
polarization but not the other.
The input is natural light, the output is polarized light (linear,
circular, elliptical). They work by dichroism, birefringence,
reflection, or scattering.
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Wire-grid polarizers (I)[dichroism]
Mainly used in the IR and longer wavelengths
Grid of parallel conducting wires with a spacing comparable to the wavelength of observation
Electric field vector parallel to the wires is attenuated because of currents induced in the wires
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Wide-grid polarizers (II)[dichroism]
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Dichroic crystals[dichroism]
Dichroic crystals absorb one
polarization state over the other
one.
Example: tourmaline.
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Polaroids[dichroism]
Made by heating and stretching a sheet of PVA laminated to
a supporting sheet of cellulose acetate treated with iodine
solution (H-type polaroid). Invented in 1928.
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Crystal polarizers (I)[birefringence]
• Optically anisotropic crystals
• Mechanical model:
• the crystal is anisotropic, which means that
the electrons are bound with different
‘springs’ depending on the orientation
• different ‘spring constants’ gives different
propagation speeds, therefore different indices
of refraction, therefore 2 output beams
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Crystal polarizers (II)[birefringence]
The 2 output beams are polarized (orthogonally).
isotropic
crystal
(sodium
chloride)
anisotropic
crystal
(calcite)
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Crystal polarizers (IV)[birefringence]
• Crystal polarizers used as:• Beam displacers,• Beam splitters,• Polarizers,• Analyzers, ...
• Examples: Nicol prism, Glan-
Thomson polarizer, Glan or Glan-
Foucault prism, Wollaston prism,
Thin-film polarizer, ...
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Retarders
• In retarders, one polarization gets ‘retarded’, or delayed,
with respect to the other one. There is a final phase
difference between the 2 components of the polarization.
Therefore, the polarization is changed.
• Most retarders are based on birefringent materials (quartz,
mica, polymers) that have different indices of refraction
depending on the polarization of the incoming light.
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Half-Wave plate (I)
Retardation of ½ wave or 180º for one of the polarizations.
Used to flip the linear polarization or change the handedness of circular polarization.
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Half-Wave plate (II)
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Quarter-Wave plate (I)
Retardation of ¼ wave or 90º for one of the polarizations
Used to convert linear polarization to elliptical.
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• Special case: incoming light polarized at 45º with respect to
the retarder’s axis
• Conversion from linear to circular polarization (vice versa)
Quarter-Wave plate (II)
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(Back to polarizers, briefly)
Circular polarizers
• Input light: unpolarized ---
Output light: circularly polarized
• Made of a linear polarizer
glued to a quarter-wave plate
oriented at 45º with respect to
one another.
Part III: Optical components, polarizers