# Electromagnetic waves

## Pełen tekst

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particle (photon) - Isaac Newton (1642-1727)

EM wave - Huygens (1629-1695) Fresnel (1788-1827)

dualism - wave-particle duality, De Broglie (1924)

Wave model - electromagnetic theory Photon model - quantum electrodynamics

### particle?

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The electromagnetic wave can be described by Maxwell equations:

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### •Visible light 400 to 700 nm – This is the only form of electromagnetic wave visible to the human eye.

•Ultraviolet – these cause our skin to darken (tan) and can even damage our skin.

The ozone layer protects us from most UV radiation from the sun.

•X-rays – these EM waves pass through much matter. It doesn’t pass through dense bone, so it is useful in determining whether a bone is broken or not.

•Gamma rays – these EM waves are very penetrating and can severely damage cells.

•Infrared – these waves are responsible for the heat we feel. Heat seeking missiles detect infrared sources such as tanks or aircraft.

•Microwaves – these waves are used obviously to heat food. As these waves pass through food, it causes the particles to vibrate resulting in the heating of the food.

Microwaves are also used in communication.

•Radio waves – these waves include both TV and radio waves. Signals are transmitted and devices with antennas can receive the signals.

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In 1865 James Clerk Maxwell created a mathematical theory that joined electricity and magnetism,

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It explained existing experiments with electromagnetism,

### Electromagnetic waves - beginings

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4 equations written in integral form or differential form.

Gauss’s Law (E)

Gauss’s Law (B)

Ampere’s law

### Electromagnetic waves - beginings

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Maxwell’s equations can be used to derive the wave equation for electromagnetic waves.

The wave speed is the speed of light c.

This means that if the a charged particle is accelerating, then EM waves are radiating outward from it.

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Electric field lines of a point charge oscilating in simple harmonic motion (during one period T)

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The arrow shows one kink in the lines of electric field as it propagates outwards from the point charge.

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Their magnetic field (not shown in figure) comprises circles that lie in planes perpendicular to these figures and concentric with the axis of oscilation,

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Also known as Gauss’s flux theorem,

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Formulated by Carl Friedrich Gauss in 1835,

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Relates the distribution of electric charge to the resulting electric field,

The electric flux through any closed surface is proportional to the enclosed electric charge.

A surface integral denoting the electric flux through a closed surface S

total charge enclosed by S divided by the electric constant.

εₒ= 8.854...×10−12 F/m

### Maxwell’s equations – Gauss law

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in differential form:

Divergence of electric field

ρ – charge density ε0= 8.854×10−12 F/m

These two forms are correct for electric charges in vacuum

### Maxwell’s equations – Gauss law

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The contribution of medium can be expressed using the electric induction.

Inside the material media, the electric field causes a dispacement of electric charges, which results in generation of induced charges.

where ΦD,S is the flux of the electric displacement field D through S, and Qfree is the free charge contained in V.

The divergence of the electric displacement field is equal to the free electric charge density ρfree

### Maxwell’s equations – Gauss law

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The magnetic field B has divergence equal to zero

### .

(magnetic monopoles does not exist – only magnetic dipoles)

For each volume element in space, there is exactly the same number of

"magnetic field lines" entering and exiting the volume. No total "magnetic charge" can build up in any point in space.

the net flux of the magnetic field out of the surface

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Relates the integrated magnetic field around a closed loop to the electric current passing through the loop,

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Corrected by Maxwell (by including the displacement current)

Displacement current

The magnetic field is generated by current and electric field changes.

### law

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A changing magnetic field creates an electric field

Due to existence of only partial time derivatives, this equations can be used when the test charge is stationary in a time varying magnetic field.

When a charged particle moves in a magnetic field, it does not account for electromagnetic induction.

One restriction:

### law of induction

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Electric induction D can be expressed as sum of electric constant (ε0) × E and polarization density (P),

Magnetic field intensity H can be expressed as difference of magnetic field density B per magnetic constant and magnetization M

Two fields (E and H) are orthogonal to each other and moves with the speed c

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The EM waves predicted by Maxwell were discovered by Heinrich Hertz in 1887.

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He used an LC circuit with an alternating source.

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This discovery was exploited by Marconi (Radio)

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Hertz’s basic LC circuit

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When the switch is closed, oscillations occur in the current and in the charge on the capacitor,

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When the capacitor is fully charged, the total energy of the circuit is stored in the electric field of the capacitor

– At this time, the current is zero and no energy is stored in the inductor.

S

C L

Qmax +

-

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As the capacitor discharges, the energy stored in the electric field decreases ,

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At the same time, the current increases and the energy stored in the magnetic field increases,

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When the capacitor is fully discharged, there is no energy stored in its electric field,

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The current is at a maximum and all the energy is stored in the magnetic field in the inductor,

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The process repeats in the opposite direction,

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There is a continuous transfer of energy between the inductor and the capacitor.

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An induction coil is connected to two large spheres forming a capacitor,

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Oscillations are initiated by short voltage pulses,

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The inductor and capacitor form the transmitter,

### Hertz’s experimental setup

Input

Induction coil

Transmitter

• When the resonance frequencies of the transmitter and receiver matched,

energy transfer occurred between them,

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Hertz hypothesized the energy transfer was in the form of waves

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These are now known to be electromagnetic waves

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Hertz confirmed Maxwell’s theory by showing the waves existed and had all the properties of light waves

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They had different frequencies and wavelengths

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Hertz measured the speed of the waves from the transmitter

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He used the waves to form an interference pattern and calculated the wavelength, From v = f λ, v was found (very close to speed of light)

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This provided evidence in support of Maxwell’s theory

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When a charged particle undergoes an acceleration, it must radiate energy

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If currents in an ac circuit change rapidly, some energy is lost in the form of EM waves

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EM waves are radiated by any circuit carrying alternating current

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An alternating voltage applied to the wires of an antenna forces the electric charge in the antenna to oscillate.

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Because the oscillating charges in the rod produce a current, there is also a

magnetic field generated

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As the current changes, the magnetic field spreads out from the antenna

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Both the electric field and the magnetic field have energy.

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Waves transport energy.

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The EM wave transports energy at the speed of light!

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Let S be the rate of energy flow per unit area.

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– Energy per unit time per unit area.

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S is called the Poynting vector.

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The time-averaged value of S is the intensity.

x y

z

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EM waves carry momentum too,

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EM waves can exert a pressure on an object,

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The flow rate of momentum is a pressure,

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also known as the light mill,

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consists of an airtight glass bulb, containing a partial vacuum.

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Inside are a set of vanes (white and black) which are mounted on a spindle.

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The vanes rotate when exposed to light, with faster rotation for more intense light,

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It gives possibility to measure the electromagnetic radiation intensity.

### Energy of EM waves

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STOP:

• The idea of work is not related to the pressure of light!

• The darker side of the vane is heated faster than the light one,

• The pressure difference causes the vane to move cold (light) side forward.

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The retina contains two major types of light-sensitive photoreceptor cells used for vision: the rods and the cones.

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Rods

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cannot distinguish colours,

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responsible for low-light, monochrome (black&white) vision,

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they work well in dim light as they contain a pigment (but saturates at higher intensities).

• Cones

– function best in relatively bright light, – less sensitive to light than the rod cells, – allow the perception of color.

– Are also able to perceive finer detail and more rapid changes in images (response times to stimuli are faster than those of rods),

– We have three kinds of cones with different response curves and thus respond to variation in color in different ways (trichromatic vision).

### Eyes – natural EM detectors

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Wavelength [nm]

Normalised intensity [a.u.]

### Eyes – natural EM detectors

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• The frequency of light is very high,

• There is no such detector to measure the electric field changes,

• We are able only to measure the mean value of the square root of the electric field,

• our eyes can detect only intensity of light, not phase.

### Energy of EM waves

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If two monochromatic waves described as:

will overlap in some plane x=const, then:

Responsible for interference

### Interference

For : > 0 constructive interference

= 0

destructive interference

< 0

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The same phases The oposite phases

Constructive interference Destructive interference

-2,5 -2 -1,5 -1 -0,5 0 0,5 1 1,5 2 2,5

0 2 4 6 8 10

-1,5 -1 -0,5 0 0,5 1 1,5

0 2 4 6 8 10

### Interference

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Christian Huygens 1629-1695

All points in a wavefront serve as point sources of spherical secondary waves.

After a time t, the new wavefront will be the tangent to all the resulting spherical waves.

### Huyghens principle

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For plane waves entering a single slit, the waves emerging from the slit start spreading out, diffracting

### Interference

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For waves entering two slits, the emerging waves interfere and form an interference (diffraction) pattern

Young experiment in 1801:

light is a wave phenomenon First plane wave through a small slit yields coherent spherical wave,

Then interposed two slits:

interference of two spherical waves on a screen,

### Young’s double slit experiment

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• Phase difference between two waves can change for paths of different lengths

• Each point on the screen is determined by the path length difference DL of the rays reaching that point

Path Length Difference:

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12

1

1

### Interference

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• Two sources can produce an interference that is stable over time, if their light has a phase relationship that does not change with time: E(t)=E0cos(wt+f).

• Coherent sources: Phase f must be well defined and constant

• Sunlight is coherent over a short length and time range

• Since laser light is produced by cooperative behavior of atoms, it is coherent of long length and time ranges

• Incoherent sources: f jitters randomly in time, no stable interference occurs

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### Interferometers

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Central maximum

Side or secondary maxima

Light

Fresnel Bright Spot.

Bright spot

Light

These patterns cannot be explained using geometrical optics!

### Diffraction

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When light goes through a narrow slit, it spreads out to form a diffraction pattern.

### Single slit diffraction

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X-band: =10cm

You are doing 137 mph on I-10 and you pass a little old lady doing 55mph when a cop, Located 1km away fires his radar gun, which has a 10 cm opening. Can he punish you for fast driving, if the gun Is X-band? What about Laser?

1m 1m

10 m 1000m

w  2

### 

L

a 2  0.1m1000m

0.1m  2000m w 2L

a 2 0.000001m1000m

0.1m  0.02m Laser-band: =1m

### Diffraction

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The Rayleigh Resolution Criterion says that the minimum separation to separate two objects is to have the diffraction peak of one at the

diffraction minimum of the other, i.e., D  1.22 /D.

Example: The Hubble Space Telescope has a mirror diameter of 4 m, leading to excellent resolution of close-lying objects.

For light with wavelength of 500 nm, the angular resolution of the Hubble is D = 1.53 x 10-7 radians.

### Diffraction

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A spy satellite in a 200km low-Earth orbit is imaging the Earth in the visible wavelength of 500nm.

How big a diameter telescope does it need to read a newspaper over your shoulder from Outer Space?

### D

Dx = RD = R(1.22/D)

D = R(1.22/Dx) = (200x103m)(1.22x500x10–9m)/(10X10–3m)

= 12.2m

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### Holography

• Holography is a method of producing a three-dimensional (3-D) image of an object. (The three dimensions are height, width, and depth.)

• Later the object can be reconstructed.

• The hologram is actually a recording of the difference between two beams of coherent light

• Can be used as optical store disk, in information processing,

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Conventional:

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2-d version of a 3-d scene

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Photograph lacks depth perception or parallax

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Film sensitive only to radiant energy

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Phase relation (i.e. interference) are lost

### Holography vs. conventional photo

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Light

Object

Reflected wave

Photographic film:

The intensity is recorded

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Hologram:

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Freezes the intricate wavefront of light that carries all the visual information of the scene

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To view a hologram, the wavefront is reconstructed

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View what we would have seen if present at the original scene through the window defined by the hologram

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Provides depth perception and parallax

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Converts phase information into amplitude information (in-phase - maximum amplitude, out-of-phase – minimum amplitude)

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Interfere wavefront of light from a scene with a reference wave

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The hologram is a complex interference pattern of microscopically spaced fringes

### Holography vs. conventional photo

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Reference wave

Photographic film.

Interference of reference and reflected waves is recorded

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Film is developed,

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Hologram illuminated at same angle as reference beam during original exposure to reveal holographic image,

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y

###     

Vertically (y axis) polarized wave having an amplitude A, a wavelength of  and an angular velocity (frequency * 2) of , propagating along the x axis.

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Vertical

Horizontal

y

z

### Polarization of EM waves

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• superposition of two waves that have the same amplitude and wavelength,

• are polarized in two perpendicular planes and oscillate in the same phase.

• Oscillating in the same phase means that the two waves reach their peaks and cross the zero line in the same moments

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Right circular

Left circular

y

z

y

z

### Polarization of EM waves

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• two circularly polarized waves can meet as well.

• In that case, the fields are added according to the rules of vector addition, just as with plane-polarized waves.

• The superposition of two circularly polarized light beams can result in various outcomes.

Any linearly polarized light wave can be obtained as a superposition of a left circularly polarized and a right circularly polarized light wave, whose amplitude is identical.

### Polarization of EM waves

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Different polarization of light get reflected and

refracted with different amplitudes (“birefringence”).

At one particular angle, the parallel polarization is NOT reflected at all!

This is the “Brewster angle” B, and B + r = 90o. (Absorption)

2 2

1

o

###  

Polarizing Sunglasses

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Opaque

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absorbs or reflects all light

Transparent

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allows light to pass through completely

Translucent

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allows some light to pass through

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y

x

###    

Material with an extinction coefficient 

The light gets weaker (its amplitude drops)

In Out

The process by which EM radiant energy is absorbed by a molecule or particle and converted to another form of energy

### Absorption

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In Out

• The intensity of light decreases

exponentially inside the shown piece of material.

• After the light exits the medium, its field vector rotates as before but its length is lower than the original value.

### Absorption

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In Out Material with an index of

refraction n

The light slows down inside the material, therefore its wavelength becomes shorter and its phase gets shifted

y

### Refraction

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In Out Material having different

extinction coefficients for right and left circularly polarized lights: R and L

Plane-polarized light becomes elliptically polar

y Rx

Lx

z

Rx

Lx

### Circular dichroism

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In Out Material having different

refraction indices for right and left circularly polarized lights: nR and nL

The plane of polarization of plane- polarized light gets rotated

R

L

y

###      

) /

2 sin(

) /

2

sin( nR x t A n x t

A

Ez

L

### Circular birefringence

(70)

In Out Material having different

extincion coefficients AND refraction indices for right and left circularly polarized lights: R and L AND nR and nL

Plane polarized light gets elliptically polar, with the great axis of the ellipse being rotated relative to the original plane of polarization

R

L

y Rx

Lx

R

L

z

Rx

Lx

(71)

Rayleigh

Mie

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Geometric

The process whereby EM radiation is absorbed and immediately re-emitted by a particle or molecule – energy can be emitted in multiple-directions

The type of scattering is controlled by the size of the wavelength relative to the size of the particle

### Light scattering

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NOON

• less atmosphere

• less scattering

• blue sky, yellow sun

SUNSET

• more atmosphere

• more scattering

• orange-red sky & sun

• Molecules in atmosphere scatter light rays.

• Shorter wavelengths (blue, violet) are scattered more easily.

### Light scattering

Blue sky and red sunset

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Wavelength of light is much larger than scattering particles,

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Blue light ~4000 Angstroms, scattering particles ~1 Angstrom (1A=10-10 m)

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Occurs when the wavelength ≅ particle size,

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Explains scattering around larger droplets such as Corona around the sun or moon, Glory and similar phenomena.

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Occurs with particles that are actually 0.1 to 10 times the size of the wavelength

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Primary Mie scatterers are dust particles, soot from smoke

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Mie scatterers are found lower in theTroposphere

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The dependence of wave speed and index of refraction on wavelength is called dispersion,

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The index of refraction depends on the frequency of the light: the higher the frequency, the higher the index of refraction,

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Because white light is a mixture of frequencies, different wavelengths travel in different directions.

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Dispersion is the cause of chromatic aberration in a simple lens:

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Different colours focus at different points,

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It’s a common defect of simple lenses,

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Sometimes you see a fringe of colours around an image seen through a lens or a telescope,

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Chromatic aberration can be corrected by combining two or more lenses.

### Dispersion

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• 1665-1666 – First experiment of Isaac Newton with dispersion, experimentum crucis

• For this distribution of colours Newton coined the term spectrum,

• White light can be dispersed into different wavelengths,

• Dispersed ray (monochromatic) cannot be divided into other wavelengths

### *

Slit Slit

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• White light is a “Heterogeneous mixture of different refrangible Rays”

• Colours of the spectrum cannot be individually modified.

• Colours are “Original and connate properties, which in divers Rays are divers. Some Rays are disposed to exhibit a red colour and no other; some a yellow and no other, some a green and no other, and so of the rest”.

(79)

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In the presence of dispersion, wave velocity is no longer uniquely defined, giving rise to the distinction of phase velocity and group velocity.

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A well-known effect of phase velocity dispersion is the color dependence of light refraction that can be observed in prisms and rainbows.

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Dispersion may be caused either by geometric boundary conditions (waveguides, shallow water) or by interaction of the waves with the transmitting medium.

(80)

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is the phenomenon in which the phase velocity of a wave depends on its

frequency, or alternatively when the group velocity depends on the frequency.

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Media having such a property are termed dispersive media. Dispersion is sometimes called chromatic dispersion to emphasize its wavelength- dependent nature, or group-velocity dispersion (GVD) to emphasize the role of the group velocity.

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Dispersion is called normal when the refractive index decreases with the wavelength,

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For materials with selective absorption the refractive index can increase in some selective wavelength ranges…

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This kind of dispersion is called anomalous.

### Dispersion

Refractive index (n)

(81)

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A rainbow is caused by the dispersion of light in droplets of rain.

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When sunlight enters a drop, it is separated into its coloured components.

• The final direction of light is quite opposite to its incident direction.

• Violet light changes its direction by 320°.

• Red light changes its direction by 318°.

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In a range of absorption maximum, the refractive index can be even <1,

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it means that for 0<n<1 the wave speed:

• There is no paradox in it.

• Phase velocity is the rate at which the phase of the wave propagates,

• It this does not indicate any superluminal information or energy transfer,

Phase velocity

(83)

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The group velocity of a wave is the velocity with which the overall shape of the wave's amplitudes — known as the modulation or envelope of the wave

— propagates through space.

• The group velocity is often thought of as the velocity at which energy or information is conveyed along a wave.

• In most cases this is accurate, and the group velocity can be thought of as the signal velocity of the waveform.

• However, if the wave is travelling through an absorptive medium, this does not always hold.

(84)

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various experiments have verified that it is possible for the group velocity of laser light pulses sent through specially prepared materials to significantly exceed the speed of light in vacuum,

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However, superluminal communication is not possible in this case, since the signal velocity remains less than the speed of light.

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It is also possible to reduce the group velocity to zero, stopping the pulse, or have negative group velocity, making the pulse appear to propagate backwards.

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However, in all these cases, photons continue to propagate at the expected speed of light in the medium

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1999 Rowland Institute for Science, Cambridge,

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2000 NEC Research Institute, Princton,

Group velocity (equal to an electron's speed) should not be confused with phase velocity (equal to the product of the electron's frequency multiplied by its

wavelength).

(85)

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In a dispersive medium, the phase velocity varies with frequency and is not necessarily the same as the group velocity of the wave, which is the rate at which changes in amplitude (known as the envelope of the wave) propagate.

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In some specific cases the phase velocity has a negative sign as group velocity.

moves with the phase velocity, moves with the group velocity

(86)

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In such medium single light pulse will spread (dispersion)

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Due to the dispersion there are limits in signal transmission (ex. optical fiber telecommunication)

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Are there such media in which signal will not change the shape?

(87)

Wavelength

Group velocity

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### •

a soliton is a self-reinforcing solitary wave (a wave packet or pulse) that maintains its shape while it travels at constant speed.

### •

Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (The term "dispersive effects" refers to a property of certain systems where the speed of the waves varies according to frequency.)

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Single soliton behaves like a particle

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Travels with constant shape and velocity

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### •

The soliton phenomenon was first

described by John Scott Russell (1808–

1882)

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Observation in 1834 in canal near Edinburgh

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He reproduced the phenomenon in a wave tank and named it the "Wave of Translation".

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Large “wave of translation” which did not dissipate

“Happiest day of my life”

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When two solitary waves get closer, they gradually deform

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Finally merge into a single wave packet

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This packet soon splits into two solitary waves with the same shape and velocity before "collision".

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LASER acronym Light Amplification by Stimulated Emission of Radiation

### •

They produce narrow beams of intense light,

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They often have pure colors,

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They are dangerous to eyes,

(96)

### •

Excited atoms normally emit light spontaneously

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Photons are uncorrelated and independent

Incoherent light

(97)

### •

Excited atoms can be stimulated into duplicating passing light

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Photons are correlated and identical

Coherent light

(98)

### •

Stimulated emission can amplify light

### •

Laser medium contains excited atom-like systems

### •

Photons must have appropriate wavelength, polarization, and orientation to be duplicated

### •

Duplication is perfect; photons are clones

(99)

### •

Laser medium in a resonator produces oscillator

### •

A spontaneous photon is duplicated over and over

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Duplicated photons leak from semitransparent mirror

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Photons from oscillator are identical

(100)

### •

Coherent – identical photons

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Controllable wavelength/frequency – nice colors

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Controllable spatial structure – narrow beams

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Controllable temporal structure – short pulses

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Energy storage and retrieval – intense pulses

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Giant interference effects

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Apart from these issues, laser light is just light

(101)

### •

Gas (HeNe, CO2, Argon, Krypton)

### •

Solid state (Ruby, Nd:YAG, Ti:Sapphire, Diode)

### •

Liquid (Dye, Jello)

Chemical (HF)

Nuclear

### Types of lasers

(102)

• Many wavelengths

• Multidirectional

• Incoherent

• Monochromatic

• Directional

• Coherent

(103)

### 1.

Energy is applied to a medium raising electrons to an unstable energy level.

### 2.

These atoms spontaneously decay to a relatively long-lived, lower energy, metastable state.

### 3.

A population inversion is achieved when the majority of atoms have reached this metastable state.

### 4.

Lasing action occurs when an electron spontaneously returns to its ground state and produces a photon.

### 5.

If the energy from this photon is of the precise wavelength, it will stimulate the production of another photon of the same wavelength and resulting in a cascading effect.

### 6.

The highly reflective mirror and partially reflective mirror continue the

reaction by directing photons back through the medium along the long axis of the laser.

### 7.

The partially reflective mirror allows the transmission of a small amount of coherent radiation that we observe as the “beam”.

### 8.

Laser radiation will continue as long as energy is applied to the lasing medium.

### Lasing action

(104)

Energy Introduction

Ground State Excited State

Metastable State

Spontaneous Energy Emission

(105)

105

(106)

### •

Beyond some maximum incident angle the ray of light cannot pass through the boundary of the two materials and the ray is completely reflected.

### •

When the angle of incidence exceeds the maximum angle or Critical Angle, we have Total Internal Reflection.

### •

Total Internal Reflection is the property that allows fiber optic communication to occur.

n1 n2

n1 > n2

c r

c = r Incident Wave Reflected Wave

Transition Boundary

1

2

C

Critical Angle

(107)

### •

Optical Fiber is a cylindrical waveguide made of a high purity fused silica.

### •

The core has a refractive index slightly higher than the cladding which allows the propagation of light via total internal reflection.

### •

A single-mode core diameter is typically 5-10m.

### •

A multimode core diameter is typically over 100 m.

1

### > n

2

Total Internal Reflection

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## *

(110)

(111)

### fundamental) mode to propagate.

n2

n1 n

Diameter

n2

n1 n

Diameter

Single Mode Fiber Multi-Mode Fiber

(112)

lm lm lm

lm

### TM component is larger than TE component.

l= # of variation cycles or zeros in direction.

m= # of variation cycles or zeros in r direction.

1

2

0m 1m 1m 0m

0m

0m

01

11

(113)

### Mode propagation constant

 

ligth of wavelength

NA a V

: :

/ 2

2.4 05

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Definition: the wavelength below which multiple modes of light can be propagated along a particular fiber, i.e., l>=lc, single mode, l<lc, multi-mode

### Cut-off wavelength

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In general, a linearly polarized mode is a combination of both of the

degenerate modes. As the modal wave travels along the fiber, the difference in the refractive indices would change the phase difference between these two components & thereby the state of the polarization of the mode. However after certain length referred to as fiber beat length, the modal wave will produce its original state of polarization. This length is simply given by:

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### •

To better describe some optical phenomena, it is important to remember that light is actually a traveling electromagnetic wave.

### •

As light propagates through a fiber, it creates a “standing wave” across the diameter of the fiber core. This is called waveguide propagation.

### •

A small portion of the power also penetrates into the cladding.

Core

Propagating Wavefronts

### Mode propagation constant

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850 940 1030 1120 1210 1300 1390 1480 1570 1660 1750 0

0.25 0.5 0.75 1 1.25 1.5 1.75 2

Water Impurities (OH-)

IR Absorbtion

Rayleigh Scattering

Wavelength (nm)

Attenuation (dB/km)

### Fiber attenuation

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3 main types of Photonic Crystal Fibers

Isotropic with solid core Hollow-core Birefringent with solid core

### •

2-dimmensional photonic crystals with deffect along the fiber length inside the core region,

### •

made of one type of glass material with periodic matrix of air micro-holes forming a structure of photonic crystal,

### Photonic Crystal Fibers

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D d Λ

•Pitch, 

• Large hole diameter, D

• Small hole diameter, d

• Diameter of holey region

• Filling factor, d/

 4.4 m

 4.5 m

 2.2 m

 40 m

 0.5

Commercially available birefringent PCF from NKTPhotonics

Parameters

### Photonic Crystal Fibers

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m-TIR (modified Total Internal Reflection) – characteristic for PCFs with solid core,

PBG (Photonic Band Gap) – characteristic for „hollow-core” PCFs,

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### •

Low sensivity for temperature influence,

### •

Birefringence is generated by introducing 2-fold the symmetry in the periodic structure of the PCF,

### •

Group and phase birefringence depends on wavelength,

### •

Group and phase birefringence may have different values, and even signs, Blazephotonics 1550-01

### Properties of PCFs

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The highest level of tunability of propagation and polarization properties by external fields

Thermal, external ac & dc

fields, optical field sensitivity Variety of LC materials and LC

structures; influence of molecular ordering

mTIR and PBG phenomena Variety of PCF structures

(birefringence, SM, nonlinearity, etc.)

Liquid Crystal Photonic Crystal Fiber Liquid Crystals Photonic Fibers

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