october 1, 2008 ...modern physics v h satheeshkumar department of physics sri bhagawan mahaveer jain...

Modern Physics

V H SatheeshkumarDepartment of Physics

Sri Bhagawan Mahaveer Jain College of EngineeringJain Global Campus, Kanakapura Road

Bangalore 562 112, [email protected]

October 1, 2008

Who can use this?

The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is thefirst of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU forthe first-semester (September 2008 - January 2009) BE students of all branches. Anystudent interested in exploring more about the course may visit the course homepageat www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy ofstudying this: this chapter is worth 20 marks in the final exam! Cheers ;-)

Syllabus as prescribed by VTU

Introduction to blackbody radiation spectrum; Photoelectric effect;

Compton effect; Wave particle Dualism; de Broglie hypothesis:de Broglie

wavelength, Extension to electron particle; Davisson and Germer

Experiment, Matter waves and their Characteristic properties; Phase

velocity, group velocity and particle velocity; Relation between phase

velocity and group velocity; Relation between group velocity and

particle velocity; Expression for de Broglie wavelength using group

velocity.

Reference

• Arthur Beiser, Concepts of Modern Physics, 6th Edition, Tata McGraw-Hill Pub-lishing Company Limited, ISBN- 0-07-049553-X.

————————————This document is typeset in Free Software LATEX2e distributed under the terms of the GNU General Public License.

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1 Introduction

The turn of the 20th century brought the start of a revolution in physics. In 1900, Max Planck publishedhis explanation of blackbody radiation. This equation assumed that radiators are quantized, whichproved to be the opening argument in the edifice that would become quantum mechanics. In thischapter, many of the developments which form the foundation of modern physics are discussed.

2 Blackbody radiation spectrum

A blackbody is an object that absorbs all light that falls on it. Since no light is reflected or transmitted,the object appears black when it is cold. The term blackbody was introduced by Gustav Kirchhoff in1860. A perfect blackbody, in thermal equilibrium, will emit exactly as much as it absorbs at everywavelength. The light emitted by a blackbody is called blackbody radiation.

The plot of distribution of emitted energy as a function of wavelength and temperature of blackbodyis know as blackbody spectrum. It has the following characteristics.

• The spectral distribution of energy in the radiation depends only on the temperature of theblackbody.

• The higher the temperature, the greater the amount of total radiation energy emitted and alsoenergy emitted at individual wavelengths.

• The higher the temperature, the lower the wavelength at which maximum emission occurs.

Many theories were proposed to explain the nature of blackbody radiation based on classical physicsarguments. But non of them could explain the complete blackbody spectrum satisfactorily. Thesetheories are discussed in brief below.

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2.1 Stefan-Boltzmann law

The Stefan-Boltzmann law, also known as Stefan’s law, states that the total energy radiated per unitsurface area of a blackbody in unit time (known variously as the blackbody irradiance, energy fluxdensity, radiant flux, or the emissive power), E?, is directly proportional to the fourth power of theblackbody’s thermodynamic temperature T (also called absolute temperature):

E? = σT 4. (1)

The constant of proportionality σ is called the Stefan-Boltzmann constant or Stefan’s constant. It isnot a fundamental constant, in the sense that it can be derived from other known constants of nature.The value of the constant is 5.6704× 10−8 J s−1 m−2 K−4. The Stefan-Boltzmann law is an example ofa power law.

2.2 Wien’s displacement law

Wien’s displacement law states that there is an inverse relationship between the wavelength of the peakof the emission of a blackbody and its absolute temperature.

λmax ∝1

T

Tλmax = b (2)

whereλmax is the peak wavelength in meters,T is the temperature of the blackbody in kelvins (K), andb is a constant of proportionality, called Wien’s displacement constant and equals 2.8978×10−3 mK.

In other words, Wien’s displacement law states that the hotter an object is, the shorter the wavelengthat which it will emit most of its radiation.

2.3 Wien’s distribution law

According to Wein, the energy density, Eλ, emitted by a blackbody in a wavelength interval λ andλ + dλ is given by

Eλ dλ =c1

λ5e(−c2/λT ) dλ (3)

where c1 and c2 are constants. This is known as Wien’s distribution law or simply Wein’s law. Thislaw holds good for smaller values of λ but does not match the experimental results for larger values ofλ. Wien received the 1911 Nobel Prize for his work on heat radiation.

2.4 Rayleigh-Jeans’ law

According to Rayleigh and Jeans the energy density, Eλ, emitted by a blackbody in a wavelengthinterval λ and λ + dλ is given by

Eλ dλ =8πkT

λ4dλ, (4)

where k is the Boltzmann’s constant whose value is equal to 1.381× 10−23 JK−1.It agrees well with experimental measurements for long wavelengths. However it predicts an en-

ergy output that diverges towards infinity as wavelengths grow smaller. This was not supported byexperiments and the failure has become known as the ultraviolet catastrophe or Rayleigh-Jeanscatastrophe. Here the word ultraviolet signifies shorter wavelength or higher frequencies and not theultraviolet region of the spectrum. One more thing to note is that, it was not, as is sometimes assertedin physics textbooks, a motivation for quantum theory.



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2.5 Planck’s law of black-body radiation

Explaining the blackbody radiation curve was a major challenge in theoretical physics during the latenineteenth century. All the theories based on classical ideas failed in one or the other way. Thewavelength at which the radiation is strongest is given by Wien’s displacement law, and the overallpower emitted per unit area is given by the Stefan-Boltzmann law. Wein’s law could explain theblackbody radiation curve only for shorter wavelengths whereas Rayleigh-Jeans’ law worked well onlyfor larger wavelengths. The problem was finally solved in 1901 by Max Planck.

Planck came up with the following formula for the spectral energy density of blackbody radiationin a wavelength range λ and λ + dλ,

Eλ dλ =8πhc

λ5

1

ehc/λkT − 1dλ, (5)

where h is the Planck’s constant whose value is 6.626× 10−34 Js. This formula could explain the entireblackbody spectrum and does not suffer from an ultraviolet catastrophe unlike the previous ones. Butthe problem was to justify it in terms of physical principles. Planck proposed a radically new idea thatthe oscillators in the blackbody do not have continuous distribution of energies but only in discreteamounts. An oscillator emits radiation of frequency ν when it drops from one energy state to the nextlower one, and it jumps to the next higher state when it absorbs radiation of frequency ν. Each suchdiscrete bundle of energy hν is called quantum. Hence, the energy of an oscillator can be written as

En = nhν n = 0, 1, 2, 3, .... (6)

2.5.1 Derivation Wien’s law from Planck’s law

The Planck’s law of blackbody radiation expressed in terms of wavelength is given by

Eλ dλ =8πhc

λ5

1

ehc/λkT − 1dλ.

In the limit of shorter wavelengths, hc/λkT becomes very small resulting in

ehc/λkT � 1.

Thereforeehc/λkT − 1 ≈ ehc/λkT .

This reduces the Planck’s law to

Eλ dλ =8πhc

λ5

1

ehc/λkTdλ

or

Eλ dλ =8πhc

λ5e−hc/λkT dλ.

Now identifying 8πhc as c1 and hc/k as c2, the above equation takes the form

Eλ dλ =c1

λ5e(−c2/λT ) dλ.

This is the familiar Wien’s law.



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2.5.2 Derivation of Rayleigh-Jeans’ law from Planck’s law

The Planck’s law of blackbody radiation expressed in terms of wavelength is given by

Eλ dλ =8πhc

λ5

1

ehc/λkT − 1dλ.

In the limit of long wavelengths, the term in the exponential becomes small. Now expressing it inthe form of power series (ex = 1 + x + x2

2!+ x3

3!+ x4

4!+ ....),

ehc/λkT = 1 +

(hc

λkT

)+

(hc

λkT

)2

2!+ ...

Since hcλkT

is small, any higher order of the same will be much smaller, so we truncate the series beyondthe first order term,

ehc/λkT ≈ 1 +

(hc

λkT

).

Therefore, the Planck’s law takes the form

Eλ dλ =8πhc

λ5

1

(1 + hc/λkT )− 1dλ,

that is,

Eλ dλ =8πhc

λ5

1

(hc/λkT )dλ,

Eλ dλ =8πhc

λ5

λkT

hcdλ.

This gives back the Rayleigh-Jeans Law

Eλ dλ =8πkT

λ4dλ.

3 Photo-electric effect

The phenomenon of electrons being emitted from a metal when struck by incident electromagneticradiation of certain frequency is called photoelectric effect. The emitted electrons can be referred to asphotoelectrons. The effect is also termed the Hertz Effect in the honor of its discoverer, although theterm has generally fallen out of use.

3.1 Experimental results of the photoelectric emission

1. The time lag between the incidence of radiation and the emission of a photoelectron is very small,less than 10−9 second.

2. For a given metal, there exists a certain minimum frequency of incident radiation below whichno photoelectrons can be emitted. This frequency is called the threshold frequency or criticalfrequency, denoted by ν0. The energy corresponding to this threshold frequency is the mini-mum energy required to eject a photoelectron from the surface. This minimum energy is thecharacteristic of the material which is called work function (φ).

3. For a given metal and frequency of incident radiation, the number of photoelectrons ejected isdirectly proportional to the intensity of the incident light.



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4. Above the threshold frequency, the maximum kinetic energy of the emitted photoelectron isindependent of the intensity of the incident light but depends on the frequency of the incidentlight.

5. The photoelectron emission can be stopped by applying the voltage in a reverse way. Thisreverse voltage required to stop the photoelectron emission is called the stopping potential. Thisis independent of the intensity but increases with increase in the frequency of incident radiation.

3.2 Einstein’s explanation of the photoelectric effect

The above experimental results were at odds with Maxwell’s wave theory of light, which predicted thatthe energy would be proportional to the intensity of the radiation. In 1905, Einstein solved this paradoxby describing light as composed of discrete quanta, now called photons, rather than continuous waves.Based upon Planck’s theory of blackbody radiation, Einstein theorized that the energy in each quantumof light was equal to the frequency multiplied by a constant, called Planck’s constant. A photon abovea threshold frequency has the required energy to eject a single electron, creating the observed effect.Einstein came up the following explanation

Energy of incident photon = Energy needed to remove an electron+Kinetic energy of the emitted electron

Algebraically,

hν = φ + KEmax (7)

whereh is Planck’s constant,ν is the frequency of the incident photon,φ = hν0 is the work function where ν0 is the threshold frequency,KEmax = 1

2mv2 is the maximum kinetic energy of ejected electrons,

m is the rest mass of the ejected electron, andv is the speed of the ejected electron.

Since an emitted electron cannot have negative kinetic energy, the equation implies that if the photon’senergy (hν) is less than the work function (φ), no electron will be emitted.

The photoelectric effect helped propel the then-emerging concept of the dualistic nature of light,that light exhibits characteristics of waves and particles at different times. The effect was impossible tounderstand in terms of the classical wave description of light, as the energy of the emitted electrons didnot depend on the intensity of the incident radiation. In his famous paper of 1905, Einstein extendedPlanck’s quantum hypothesis by postulating that quantization was not a property of the emissionmechanism, but rather an intrinsic property of the electromagnetic field. Using this hypothesis, Einsteinwas able to explain the observed phenomenon. Explanation of the photoelectric effect was one of thefirst triumphs of quantum mechanics and earned Einstein the Nobel Prize in 1921.



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4 Compton effect

Compton scattering or the Compton effect is the decrease in energy (increase in wavelength) of an X-rayor gamma ray photon, when it interacts with matter. The amount the wavelength increases by is calledthe Compton shift. The Compton effect was observed in 1923 by Arthur Compton who got the 1927Nobel Prize in Physics for the discovery.

The interaction between electrons and high energy photons results in the electron being given partof the energy and a photon containing the remaining energy being emitted in a different direction fromthe original, so that the overall momentum of the system is conserved. In this scenario, the electron istreated as free or loosely bound. If the photon is of sufficient energy, it can eject an electron from itshost atom entirely resulting in the Photoelectric effect instead of undergoing Compton scattering.

The Compton scattering equation is given by,

λ′ − λ =h

mec(1− cos θ) (8)

whereλ is the wavelength of the photon before scattering,λ′ is the wavelength of the photon after scattering,me is the mass of the electron,θ is the angle by which the photon’s heading changes,h is Planck’s constant, andc is the speed of light.

hmec

= 2.43× 10−12 m is known as the Compton wavelength.The effect is important because it demonstrates that light cannot be explained purely as a wave phe-nomenon. Light must behave as if it consists of particles in order to explain the Compton scattering.Compton’s experiment convinced physicists that light can behave as a stream of particles whose energyis proportional to the frequency.

5 Wave-particle dualism

Albert Einstein’s analysis of the photoelectric effect in 1905 demonstrated that light possessed particle-like properties, and this was further confirmed with the discovery of the Compton scattering in 1923.Later on, the diffraction of electrons would be predicted and experimentally confirmed, thus showingthat electrons must have wave-like properties in addition to particle properties. The wave-particleduality is the concept that all matter and energy exhibits both wave-like and particle-like properties.This duality addresses the inadequacy of classical concepts like ‘particle’ and ‘wave’ in fully describingthe behavior of small-scale objects. This confusion over particle versus wave properties was eventuallyresolved with the advent and establishment of quantum mechanics in the first half of the 20th century.



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In 1924, Louis de Broglie formulated the de Broglie hypothesis, claiming that all matter has a wave-like nature and the wavelength (denoted as λ) of a moving particle of momentum (denoted as p) isgiven by:

λ =h

p(9)

where h is Planck’s constant. de Broglie’s formula was confirmed three years later for electrons withthe observation of electron diffraction and he was awarded the Nobel Prize for Physics in 1929 for hishypothesis.

The above formula holds true for all particles. In most of the laboratory experiments for measuringthe de Broglie wavelength, we accelerate a charged particle using an electric field. When an electron atrest is accelerated by applying a potential difference of V , it will have a kinetic energy given by

1

2mv2 = eV.

Expressing the kinetic energy in terms of linear momentum p(= mv), we rewrite the above equation as

p2

2m= eV,

that isp =

√2meV .

Now plugging this equation into the expression for de Broglie wavelength, we get

λ =h√

2meV.

Substituting the numerical values of the natural constants (h = 6.626× 10−34 Js, m = 9.11× 10−31 kgand e = 1.602× 10−19 C), we get

λ =1.226× 10−9

√V

m. (10)

5.1 Davisson and Germer Experiment

In 1927, while working for Bell Labs, Clinton Davisson and Lester Germer performed an experimentshowing that electrons were diffracted at the surface of a crystal of nickel. The basic idea is thatthe planar nature of crystal structure provides scattering surfaces at regular intervals, thus waves thatscatter from one surface can constructively or destructively interfere from waves that scatter from thenext crystal plane deeper into the crystal. This celebrated Davisson-Germer experiment confirmedthe de Broglie hypothesis that particles of matter have a wave-like nature, which is a central tenet ofquantum mechanics. In particular, their observation of diffraction allowed the first measurement of awavelength for electrons. The measured wavelength agreed well with de Broglie’s equation.

The Davisson-Germer consisted of firing an electron beam from an electron gun on a nickel crystalat normal incidence i.e. perpendicular to the surface of the crystal. The electron gun consisted ofa heated filament that released thermally excited electrons, which were then accelerated through apotential difference V , giving them a kinetic energy of eV where e is the charge of an electron. Theangular dependence of the reflected electron intensity was measured, and was determined to have thesame diffraction pattern as those predicted by Bragg for X-rays. An electron detector was placed at anangle θ = 50◦ and measured the number of electrons that were scattered at that particular angle.

According to the de Broglie relation, a beam of 54 eV had a wavelength of 0.165 nm. This matchedthe predictions of Bragg’s law

nλ = 2d sin

(90◦ − θ

2

),



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for n = 1, θ = 50◦, and for the spacing of the crystalline planes of nickel (d = 0.091 nm) obtained fromprevious X-ray scattering experiments on crystalline nickel.

This was also replicated by George Thomson. Thomson and Davisson shared the Nobel Prize forPhysics in 1937 for their experimental work. This, in combination with Arthur Compton’s experiment,established the wave-particle duality hypothesis, which was a fundamental step in quantum theory.

5.2 Properties of Matter-waves

1. Matter-waves are associated with any moving body and their wavelength is given by λ = hmv

.

2. The wavelength of matter-waves is inversely proportional to the velocity of the body. Hence, abody at rest has an infinite wavelength whereas the one traveling with a high velocity has a lowerwavelength.

3. Wavelength of matter-waves depends on the mass of the body and decreases with increase inmass. Because of this, the wave-like behavior of heavier objects is not very evident whereas thewave nature of subatomic particles can be observed experimentally.

4. Amplitude of the matter-waves at a particular space and time depends on the probability offinding the particle at that space and time.

5. Unlike other waves, there is no physical quantity that varies periodically in the case of matter-waves.

6. Matter waves are represented by a wave packet made up of a group of waves of slightly differingwavelengths. Hence, we talk of group velocity of matter waves rather than the phase velocity.

7. Matter-waves show similar properties as other waves such as interference and diffraction.

6 Phase velocity, group velocity and particle velocity

The phase velocity of a wave is the rate at which the phase of the wave propagates in space. Thisis the speed at which the phase of any one frequency component of the wave travels. For such acomponent, any given phase of the wave (for example, the crest) will appear to travel at the phasevelocity.



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The phase speed is given in terms of the wavelength λ and period T as

vphase =λ

T. (11)

Or, equivalently, in terms of the wave’s angular frequency ω and wavenumber k by

vphase =ω

k. (12)

In quantum mechanics, particles also behave as waves with complex phases. By the de Broglie hypoth-esis, we see that

vphase =ω

k=

E/~p/~

,

vphase =E

p. (13)

The phase velocity of electromagnetic radiation may under certain circumstances (e.g. in the case ofanomalous dispersion) exceed the speed of light in a vacuum, but this does not indicate any superluminalinformation or energy transfer. It was theoretically described by physicists such as Arnold Sommerfeldand Leon Brillouin.

The group velocity of a wave is the velocity with which the variations in the shape of the wave’samplitude (known as the modulation or envelope of the wave) propagate through space. For example,imagine what happens if you throw a stone into the middle of a very still pond. When the stone hitsthe surface of the water, a circular pattern of waves appears. It soon turns into a circular ring of waveswith a quiescent center. The ever expanding ring of waves is the group, within which one can discernindividual wavelets of differing wavelengths traveling at different speeds. The longer waves travel fasterthan the group as a whole, but they die out as they approach the leading edge. The shorter wavestravel slower and they die out as they emerge from the trailing boundary of the group.

Now, we shall arrive at the expression for the group velocity using the concept of superposition oftwo almost similar waves.

Let the two waves be given byy1 = A cos(ωt− kx),

y2 = A cos[(ω + ∆ω)t− (k + ∆k)x].



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When these two waves superimpose, we get

y = y1 + y2,

y = A cos(ωt− kx) + A cos[(ω + ∆ω)t− (k + ∆k)x].

Using the trigonometric relation

cos α + cos β = 2 cos

(α + β

2

)cos

(α− β

2

),

we get

y = 2A cos

([ωt− kx] + [(ω + ∆ω)t− (k + ∆k)x]

2

)cos

([ωt− kx]− [(ω + ∆ω)t− (k + ∆k)x]

2

),

y = 2A cos

((2ω + ∆ω)t− (2k + ∆k)x

2

)cos

(∆ωt−∆kx

2

).

Since ∆ω is too small compared to 2ω, we can write

2ω + ∆ω ≈ 2ω

Now using this in the above equation and rearranging the terms, we end up with

y = 2A cos

(∆ωt−∆kx

2

)cos

((2ω)t− (2k)x

2

).

Further simplifying it, gives us

y = 2A cos

(∆ωt−∆kx

2

)cos (ωt− kx).

Identifying 2 cos(

∆ωt−∆kx2

)as the constant amplitude of the superposed wave, we can write

2A cos

(∆ωt−∆kx

2

)= constant

i.e., (∆ωt−∆kx

2

)= constant

(∆ωt−∆kx) = constant

x =

(∆ωt

∆k

)+ constant

Differentiating the above equation with respect to t, we get the group velocity,

vgroup =dx

dt=

∆ω

∆k

under the limiting condition, we get

vgroup =dω

dk(14)

This is the defining equation of group velocity. In a dispersive medium, the phase velocity varies withfrequency and is not necessarily the same as the group velocity of the wave.



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The particle velocity is the velocity v of a particle in a medium as it transmits a wave. For aparticle of mass m possessing a linear momentum p, the particle velocity is given by

vparticle =p

m. (15)

In many cases this is a longitudinal wave of pressure as with sound, but it can also be a transverse waveas with the vibration of a taut string. When applied to a sound wave through a medium of air, particlevelocity would be the physical speed of an air molecule as it moves back and forth in the direction thesound wave is traveling as it passes. Particle velocity should not be confused with the speed of the waveas it passes through the medium, i.e. in the case of a sound wave, particle velocity is not the same asthe speed of sound.

6.1 Relation between group velocity and phase velocity

The group velocity of a matter wave is given by

vgroup =dω

dκ,

whereas phase velocity is given by

vphase =ω

κ.

From the definition of phase velocity, we can write

ω = vphase κ.

Substituting this in the expression for group velocity, we get

vgroup =d (vphase κ)

dκ.

Differentiating using the product rule, we get

vgroup = vphase + κdvphase

dκ.

We rewrite this in the following form

vgroup = vphase + κdvphase

dλ

dλ

dκ.

Since

κ =2π

λ,

we havedκ

dλ=−2π

λ2.

Plugging these two equations into the vgroup expression ,we get

vgroup = vphase +

(2π

λ

)dvphase

dλ

(−λ2

2π

).

Further simplifying,

vgroup = vphase − λdvphase

dλ. (16)



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6.2 Relation between group velocity and particle velocity


vgroup =dω

dκ,

whereω = 2πν

and

κ =2π

λ.

From Planck’s equation E = hν, we can write

ν =E

h;

and from de Broglie wavelength, we can write

λ =h

p.

Using the above equations, we rewrite the expressions for ω and κ,

ω = 2πE

h

andκ = 2π

p

h.

Now, differentiating the expressions for ω and κ, we get

dω =2π

hdE

and

dκ =2π

hdp.

Substituting the expressions for dω and dκ into the vgroup equation,

vgroup =2πh

dE2πh

dp,

that is

vgroup =dE

dp.

Since we are dealing with the matter-waves E can be the kinetic energy of particle in wave motion.Using the relation

E =p2

2mand differtiating it with respect to p, we get

vgroup =dE

dp=

d

dp

(p2

2m

)=

2p

2m,

sovgroup =

p

m.

Right hand is nothing but the particle velocity vparticle. Therefore

vgroup = vparticle. (17)



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6.3 Relation between phase velocity and particle velocity

The phase velocity of a matter-wave is given by

vphase =ω

κ,

whereω = 2πν

and

κ =2π

λ.

From Planck’s equation E = hν, we can write

ν =E

h;

and from de Broglie wavelength, we can write

λ =h

p.

Using the above equations, we rewrite the expressions for ω and κ,

ω = 2πE

h

andκ = 2π

p

h.

Now, substituting these into the expression for vphase, we get

vphase =2πE

h

2π ph

,

that is,

vphase =E

p.

From Einstein’s mass-energy equivalence relation, we have

E = mc2

and from the definition of linear momentum of a particle, we have

p = mvparticle.

Using E and p expressions in equation of vphase, we get

vphase =mc2

mvparticle

,

or

vphase =c2

vparticle

,

which gives usvphase · vparticle = c2. (18)

Since vgroup = vparticle, we can also write

vphase · vgroup = c2. (19)



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6.4 Expression for de Broglie wavelength using group velocity

Consider particle moving with kinetic energy mv2/2. This can be associated with energy hν. Therefore

hν =mv2

2,

ν =m

2hv2.

Differentiating the above expression with respect to λ,

dν

dλ=

m

2h2v

dv

dλ,

ordν

dλ=

mv

h

dv

dλ.


vgroup =dω

dκ,

whereω = 2πν

and

κ =2π

λ.

Differentiating ω and κ, we getdω = 2πdν

and

dκ =−2π

λ2dλ.

vgroup =2πdν−2πλ2 dλ

,

vgroup = −λ2 dν

dλ,

We can express this in the following way

dν

dλ=−vgroup

λ2,

Equating the two expressions for dνdλ

, we get

mv

h

dv

dλ=−vgroup

λ2.

Simplifying this equation leaves us withdv

dλ=−h

mλ2.

Rewriting this in the following form

dv = − h

mλ2dλ

Integrating the above equation ∫dv = −

∫h

mλ2dλ



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we get

v =h

mλ+ constant.

To fix the constant we use the condition: as λ →∞, v → 0. This makes constant = 0

v =h

mλ

That is

λ =h

mv(20)

***



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Quantum Mechanics

V H SatheeshkumarDepartment of Physics and

Center for Advanced Research and DevelopmentSri Bhagawan Mahaveer Jain College of Engineering

Jain Global Campus, Kanakapura RoadBangalore 562 112, India.

[email protected]

October 12, 2008

Who can use this?

The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is thesecond of eight chapters in Engineering Physics [06PHY12] course prescribed by VTUfor the first-semester (September 2008 - January 2009) BE students of all branches. Anystudent interested in exploring more about the course may visit the course homepageat www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy ofstudying this: this chapter is worth 20 marks in the final exam! Cheers ;-)


Heisenberg’s uncertainty principle and its physical significance,

Application of uncertainty principle; Wave function, Properties and

Physical significance of a wave function, Probability density and

Normalisation of wave function; Setting up of a one dimensional time

independent, Schrodinger wave equation, Eigen values and eigen function,

Application of Schrodinger wave equation : Energy eigen values for a

free particle, Energy eigen values of a particle in a potential well of

infinite depth.

Reference

• Arthur Beiser, Concepts of Modern Physics, 6th Edition, Tata McGraw-Hill Pub-lishing Company Limited, ISBN- 0-07-049553-X.


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1 Introduction

Quantum mechanics is a fundamental branch of physics which generalizes classical mechanics to pro-vide accurate descriptions for many previously unexplained phenomena such as black body radiation,photoelectric effect and Compton effect. The term quantum mechanics was first coined by Max Bornin 1924.

Within the field of engineering, quantum mechanics plays an important role. The study of quantummechanics has lead to many new inventions that include the laser, the diode, the transistor, the elec-tron microscope, and magnetic resonance imaging. Flash memory chips found in USB drives also usequantum ideas to erase their memory cells. The entire science of Nanotechnology is based on the quan-tum mechanics. Researchers are currently seeking robust methods of directly manipulating quantumstates. Efforts are being made to develop quantum cryptography, which will allow guaranteed securetransmission of information. A more distant goal is the development of quantum computers, which areexpected to perform certain computational tasks exponentially faster than the regular computers. Thischapter attempts to give you an elementary introduction to the topic.

2 Heisenberg’s uncertainty principle

We know from the wave-particle duality that every particle has wave-like properties. These wave prop-erties of particles will prevent us from measuring the exact attributes of the particles. This limitationrelated to the measurements at microscopic level is known as the uncertainty principle.

The uncertainty principle states that it is impossible to specify simultaneously the position andmomentum of a particle, such as an electron, with precision. The theory further states that a moreaccurate determination of one quantity will result in a less precise measurement of the other, and thatthe product of both uncertainties is always greater than or equal to Planck’s constant divided by 4π.That is

∆x ·∆px ≥h

4π. (1)

This principle was formulated in 1927 by the German physicist Werner Heisenberg. It is also called theindeterminacy principle.

The Heisenberg’s uncertainty principle can also be expressed in terms of the uncertainties involvedin the simultaneous measurements of angular displacement & angular momentum and energy & time;

∆θ ·∆l ≥ h

4π, (2)

∆t ·∆E ≥ h

4π. (3)

Sometimes h2π

is written as ~. In that case the right had side of the uncertainty relations will have ~2.

2.1 Explanation of uncertainty principle using gamma ray microscope

We use a hypothetical experiment of observing an electron using gamma ray microscope to illustratethe uncertainty principle. Suppose, we look at an electron using light of wavelength λ. Each photon ofthis light has the momentum h/λ. When one of these photons bounces off the electron, the electron’soriginal momentum will be changed. The exact amount of the change ∆p cannot be predicted, but itwill be of the same order of magnitude as the photon momentum h/λ. Hence

∆p ≈ h/λ.

The longer the wavelength of the observing photon, the smaller the uncertainty in the electron’s mo-mentum.



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Because light is a wave phenomenon as well as a particle phenomenon, we cannot expect to determinethe electron’s location with perfect accuracy regardless of the instrument used. A reasonable estimateof the minimum uncertainty in the measurement might be one photon wavelength, so that

∆x ≥ λ.

The shorter the wavelength, the smaller the uncertainty in location. However, if we use light ofshort wavelength to increase the accuracy of the position measurement, there will be a correspondingdecrease in the accuracy of the momentum measurement because the higher photon momentum willdisturb the electron’s motion to a greater extent. Light of long wavelength will give a more accuratemomentum but a less accurate position. Combining the above results gives us

∆x ·∆p ≥ λ.

This agrees well with the uncertainty principle.

2.2 Physical significance of uncertainty principle

The uncertainty principle is based on the assumption that a moving particle is associated with a wavepacket, the extension of which in space accounts for the uncertainty in the position of the particle.The uncertainty in the momentum arises due to the indeterminacy of the wavelength because of thefinite size of the wave packet. Thus, the uncertainty principle is not due to the limited accuracyof measurement but due to the inherent uncertainties in determining the quantities involved. Eventhough, the uncertainty principle prevents us from knowing the precise position and momentum, wecan define the position where the probability of finding the particle is maximum and also the mostprobable momentum of the particle. That means, the uncertainty principle introduces the probabilisticinterpretation of the physical quantities. This is the major difference between the classical physics andquantum mechanics.

2.3 Application of uncertainty principle

The uncertainty principle has far reaching implications. In fact, it has been very useful in explainingmany observations which cannot be explained otherwise. An important one being the proof of thenon-existence of an electron inside the nucleus.

In beta decay, the electrons are emitted from the nucleus of the radioactive element. The radius ofa typical atomic nucleus to be about 5.0 × 10−15m. Assuming that the uncertainty in the position ofthe electron inside the nucleus to be of the same order, we have

∆x = 5.0× 10−15m.

The corresponding uncertainty in the momentum is,

∆px ≥h

4π· 1

∆x,

∆px ≥6.63× 10−34

4× 3.14· 1

5.0× 10−15,

∆px ≥ 1.1× 10−20 kg ms−1.

If this is the uncertainty in a nuclear electron’s momentum p itself must be at least comparable inmagnitude. An electron with such a momentum has a kinetic energy, KE, many times greater than itsrest energy (which is mc2). The kinetic energy of such particle is given by

KE = pc ≥ (1.1× 10−20)× (3× 108)



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KE ≥ 3.3× 10−12 J

KE ≥ 20MeV

This means that the kinetic energy of an electron must exceed 20MeV if it is to be inside a nucleus.Experiments show that the electrons emitted by certain unstable nuclei never have more than a smallfraction of this energy, from which we conclude that nuclei cannot contain electrons. The electron thatan unstable nucleus may emit comes into being only at the moment the nucleus decays.

3 Wave function

In quantum mechanics, because of the wave-particle duality, the properties of the particle can bedescribed as a wave. Therefore, its quantum state can be represented as a wave of arbitrary shape andextending over all of space. This is called a wave function.

The wave function is usually complex and is represented by Ψ. Since the wave function is complex,its direct measurement in any physical experiment is not possible. It is just mathematical function ofx, t etc. Once the wave function corresponding to a system is known, the state of the system can bedetermined. The physical state of system is completely characterized by a wave function.

3.1 Physical significance of a wave function

The wave function contains information about the system it represents. Even though the wave functionitself is not directly an observable quantity, the square of the absolute value of the wave function givesthe probability of finding the particle at a given space and time. This probabilistic interpretation ofwave function was given by Max Born in 1926.

If Ψ is the wave function associated with a particle, the |Ψ|2 is the probability per unit volume thatthe particle will be found at the given point. The probability density is given by

|Ψ|2 = Ψ ·Ψ∗

where Ψ∗ is the complex conjugate of Ψ.For a particle restricted to move only long x− axis, the probability of finding it between x1 and x2

is given by ∫ x2

x1

|Ψ|2 dx.

Since the probability of finding a particle any where in a given voluve must be one, we have∫ +∞

−∞|Ψ|2 dV = 1.

This condition is know as normalization.

3.2 Properties of a wave function

A wave function has the following characteristics.

1. Ψ must be continuous and single-valued everywhere.

2. ∂Ψ/∂x, ∂Ψ/∂y and ∂Ψ/∂z must be continuous and single-valued everywhere.

3. Ψ must be normalizable.



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4 Time independent Schrodinger wave equation in one di-

mension

In quantum mechanics, the Schrdinger equation is an equation that describes how the quantum stateof a physical system changes in time. It is as central to quantum mechanics as Newton’s laws are toclassical mechanics. The equation is named after Erwin Schrdinger, who discovered it in 1926.

Consider a wave function of an arbitrary particle

Ψ(x, t) = Ae−i(ωt−kx). (4)

Using the definitions of ω and k, we write the following

ω = 2πν

and

k =2π

λ.

From Planck’s law we have E = hν and substituting in the ω equation

ω = 2πE

h=

E

h/2π=E

~.

From de Bbroglie’s equation, we have λ = h/p and substituting in the k equation

k =2π

h/p=

p

h/2π=p

~.

Now, we substitute the new expressions for ω and k in the equation of the wave function. This gives us

Ψ(x, t) = Ae−i~ (Et−px).

We re-write the wave function with separate space and time parts

Ψ(x, t) = Ae−iEt

~ · eipx~ ,

Ψ(x, t) = φe−iEt

~ , (5)

whereφ = Ae

ipx~ . (6)

Differentiating the function φ with respect to x twice,

∂φ

∂x=ip

~· Ae

ipx~

∂2φ

∂x2=ip

~· ip

~· Ae

ipx~ ,

that is∂2ψ

∂x2=−p2

~2φ,

From here, we can write

p2φ = −~2∂2φ

∂x2(7)



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The total energy E of a particle of kinetic energy p2

2mand potential energy U is given by,

E = T + U =p2

2m+ U

We multiply Ψ to both the sides of the above equation, to get,

EΨ =p2Ψ

2m+ UΨ

Substituting for Ψ, we get

Eφe−iEt

~ =p2φ

2me−iEt

~ + Uφe−iEt

~

Now inserting for p2φ from previous equations, we get

Eφe−iEt

~ =−~2

2m

∂2φ

∂x2e−iEt

~ + Uφe−iEt

~

Taking all the terms to the left hand side

Eφe−iEt

~ +~2

2m

∂2φ

∂x2e−iEt

~ − Uφe−iEt

~ = 0.

Rearranging the terms~2

2m

∂2φ

∂x2e−iEt

~ + Eφe−iEt

~ − Uφe−iEt

~ = 0,

~2

2m

∂2φ

∂x2e−iEt

~ + (E − U)φe−iEt

~ = 0.

Multiplying throughout by 2m~2 , we get

∂2φ

∂x2e−iEt

~ +2m

~2(E − U)φe

−iEt~ = 0.

Now we absorb e−iEt

~ into the partial differential operator in the first term as it does not affect theequation.

∂2(φe

−iEt~

)∂x2

+2m

~2(E − U)φe

−iEt~ = 0.

Using the relation Ψ = φe−iEt

~ ,, we get

∂2Ψ

∂x2+

2m

~2(E − U)Ψ = 0. (8)

This is the time-independent form of the Schrodinger wave equation in one-dimension. This is alsoknown as Schrodinger’s steady-state equation.

4.1 Eigenvalues and eigenfunctions

Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predic-tions about probability distributions; that is, the probability of obtaining each of the possible outcomesfrom measuring an observable. Naturally, these probabilities will depend on the quantum state at theinstant of the measurement. There are, however, certain states that are associated with a definite valueof a particular observable. These are known as eigenvalues of the observable and the correspondingwave functions are called eigenfunctions. The eigenfunctions are those eigenfunctions which are definiteand single valued. When something is in the condition of being definitely ‘pinned-down’, it is said topossess an eigenvalue. For example, if the position of an electron has been made definite, it is said tohave an eigenvalue of position. The term eigen can be roughly translated from German as inherent oras a characteristic). The German word ”eigen” was first used in this context by the mathematicianDavid Hilbert in 1904.



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4.2 Applications of Schrodinger wave equation

4.2.1 Energy eigen values for a free particle

The time-independent form of the Schrodinger wave equation in one-dimension is given by,

∂2Ψ

∂x2+

2m

~2(E − U)Ψ = 0.

A free particle is defined as the one which is not acted upon by any external force that modifies itsmotion. Hence the potential energy U in the Schrodinger equation is taken to be zero. That is,

∂2Ψ

∂x2+

2m

~2EΨ = 0.

where E is the total energy of the particle and is purely in the form of kinetic energy. The generalsolution of such a differential equation is of the form

Ψ = Asin

(√2mE

~x

)+B cos

(√2mE

~x

)

Its difficult to solve for constants A and B as we cannot impose any boundary conditions on the free

particle. Since the solution has not imposed any restriction on the constant√

2mE~ which we call k, the

free particle is permitted to have any value of energy given by

E =~2k2

2m

The Schrdinger equation, applied to the free particle, predicts that the center of a wave packet willmove through space at a constant velocity, like a classical particle with no forces acting on it. However,the wave packet will also spread out as time progresses, which means that the position becomes moreuncertain.

4.2.2 Energy eigen values of a particle in a potential well of infinite depth

The time-independent form of the Schrodinger wave equation in one-dimension is given by,

∂2Ψ

∂x2+

2m

~2(E − U)Ψ = 0.



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Consider a particle trapped in a potential well of infinite depth and width L. A particle in thispotential is completely free i.e., potential energy is zero, except at the two ends (x = 0 and x = L),where an infinite force prevents it from escaping;

U = 0 for 0 ≥ x ≥ L.

But within the well the particle does not lose any energy when it collides with the walls and hence thetotal energy of the particle remains constant. Since the article cannot exist outside the box, we have

Ψ = 0 for 0 ≤ x and x ≥ L.

The Schrodinger equation for such case takes the form

∂2Ψ

∂x2+

2m

~2EΨ = 0.

where E is the total energy of the particle and is purely in the form of kinetic energy. The generalsolution of such a differential equation is of the form

Ψ = Asin

(√2mE

~x

)+B cos

(√2mE

~x

)We use the boundary conditions to find out the constants A and B. Applying the condition

Ψ = 0 for x = 0,

the solution becomes0 = Asin (0) +B cos (0)

This impliesB = 0.

Then, the solution reduces to

Ψ = Asin

(√2mE

~x

)Now, we use the second boundary condition

Ψ = 0 for x = L,

Then,

Asin

(√2mE

~L

)= 0

If A = 0, the wavefunction will become zero irrespective of the value of x. Hence, A cannot be takenas zero. Therefore,

sin

(√2mE

~L

)= 0

or √2mE

~L = nπ where n = 1, 2, 3, .....

Here, n cannot be zero as it leads to trivial solution. Hence, the energy eigenvalues may be written as

En =n2π2~2

2mL2where n = 1, 2, 3, .....



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From this equation, we infer that the energy of the particle is discrete as n can have integer values.In other words, the energy is quantized. We also note that n cannot be zero because in that case, thewave function as well as the probability of finding the particle becomes zero for all values of x. Thelowest energy of the particle can possess corresponds to n = 1 is given by

E1 =π2~2

2mL2.

This is called the ground state energy or zero point energy. The first and second excited energies aregiven by

E2 =4π2~2

2mL2,

and

E3 =9π2~2

2mL2.

The energy levels are like E1, 4E1, 9E1, 16E1 ....which indicates that the energy levels are not equallyspaced.

The eigenfunctions corresponding to the above eigenvalues are given by

Ψ = Asin

(√2mEn

~x

)Substituting En = n2π2~2

2mL2 in the above equation, we get

Ψ = Asin(nπLx)

We apply the normalization condition to fix the value of A, that is∫ L

0

|Ψ|2 dx = 1,

∫ L

0

A2 sin2(nπLx)dx = 1

or

A2

∫ L

0

sin2(nπLx)dx = 1.

From standard integrals, we know that∫ L

0

sin2(nπLx)dx =

L

2.

Hence, the above integral becomes

A2L

2= 1

or

A =

√2

L

Now the eigenfunction becomes

Ψ =

√2

Lsin

(√2mEn

~x

)or

Ψ =

√2

Lsin(nπxL

)



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The above figure shows the variation of the wavefunction inside the infinite potential well for differentvalues of n. The probability density is given by

|Ψ|2 =2

Lsin2

(nπxL

)This figure shows the variation of the probability densities of finding the particle at different places

inside the infinite potential well for different values of n. Thus, it suggests that the probability offinding any particle at the lowest energy level is maximum at the center of the box.

***



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Electrical Conductivity in Metals




[email protected]

November 22, 2008

Who can use this?

The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is thethird of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU forthe first-semester (September 2008 - January 2009) BE students of all branches. Anystudent interested in exploring more about the course may visit the course homepageat www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy ofstudying this: this chapter is worth 20 marks in the final exam! Cheers ;-)


Free-electron concept; Classical free-electron theory, Assumptions;

Drift velocity, Mean collision time, Relaxation time and Mean free path;

Expression for drift velocity. Expression for electrical conductivity

in metals. Effect of impurity and temperature on electrical resistivity

of metals. Failure of classical free-electron theory. Quantum

free-electron theory. Fermi - Dirac Statistics. Fermi-energy. Fermi

factor. Density of states (with derivation). Expression for electrical

resistivity / conductivity. Temperature dependence of resistivity of

metals. Merits of Quantum free electron theory.

Reference

• Leonid Azaroff, Introduction to Solids, TMH Edition, Tata McGraw-Hill Pub-lishing Company Limited, ISBN- 0-07-099219-3.


1



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1 Introduction

The fact that electricity can flow through a substance was discovered by 17th century German physicistOtto von Guericke. Conduction was rediscovered independently by Englishman Stephen Gray duringthe early 1700s. Gray also noted that some substances are good conductors while others are insulators.The electron theory, which is the basis of modern electrical theory, was first advanced by Dutch physicistHendrik Antoon Lorentz in 1892. The widespread use of electricity as a source of power is largely due tothe work of pioneering American engineers and inventors such as Thomas Alva Edison, Nikola Tesla, andCharles Proteus Steinmetz during the late 19th and early 20th centuries. Thanks to our understandingof how electric conduction happens, now it plays a part in nearly every aspect of modern technology.

In this chapter, we discuss the theory of conduction, specifically the theory of classical conductionwhose defects were explained by the quantum theory. The modifications that the quantum theory addsto classical conduction not only explains the flaws that arose in the classical theory, but also adds anew dimension to conduction that is currently leading to new developments in the physics world.

2 Classical free-electron theory

Around 1900, Paul Drude improved the theory of classical conduction given by Lorentz. He reasonedthat since metals conduct electricity so well, they must contain free electrons that move through alattice of positive ions. This motion of electrons led to the formation of Ohm’s law. The free-movingelectrons act just as a gas would; moving in every direction throughout the lattice. These electronscollide with the lattice ions as they move about, which is key in understanding thermal equilibrium.The average velocity due to the thermal energy is zero since the electrons are going in every direction.There is a way of affecting this free motion of electrons, which is by use of an electric field. This processis known as electrical conduction and theory is called Drude-Lorentz theory.

The assumptions of the Drude-Lorentz classical theory of free-electrons are the following.

• Metals contain free electrons that move through a lattice of positive ions. These free electrons areresponsible for electrical conduction when an electric potential is maintained across the conductor.

• Electric field produced by lattice ions is considered to be uniform throughout the solid and henceneglected.

• The force of repulsion between the electrons and force of attraction between electrons and latticeions is neglected.

• Free electrons in a metal resemble molecules of a gas and therefore the laws of kinetic theoryof gases are applicable to free electrons. The motion of an electron is completely random. Inthe absence of electric field, number of electrons crossing any cross section of a conductor in onedirection is equal to number of electrons crossing the same cross section in opposite direction.Therefore net electric current is Zero. This random motion of the electrons is due to thermalenergy. Hence, the average kinetic energy of the electron is given by

1

2mev

2th =

3

2kT,

where, me is the mass of the electron, vth is the thermal velocity, k is the Boltzmann constantand T is the absolute temperature. Therefore, the thermal velocity of free-electrons in a metal atgiven temperature is given by

vth =

√3kT

me

.

• Electric current in the conductor is due to the drift velocity acquired by the electrons in thepresence of the applied electric field.



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2.1 Drift velocity, Mobility, Mean collision time, Relaxation time andMean free path

An electric field provides a potential difference along a wire of electrons, which creates a force, F = eE,where e is the charge of an electron and E is the magnitude of the electric field. That force acceleratesthe electrons, as expected by Newtons second law F = mea. The electrons are given a velocity awayfrom the field, which leads to these collisions with the fixed ions. The collisions rid the electrons oftheir kinetic energy momentarily, transferring that energy to the ion lattice in the form of heat. As aresult of this the conduction electrons move with constant velocity. This is called the drift velocity ,usually denoted by vd. We get the expression for vd by equating the above two equations of force,

mea = eE.

The acceleration can be written as vd/τ ,mevd

τ= eE,

vd =eEτ

me

.

In the above expression τ is called the mean collision time . It is the average time taken by anelectron between two successive collisions during its random motion.

The mobility of electrons is defined as the magnitude of the drift velocity acquired by the electronsin unit electric field. That is

µ =vd

E.

Using the expression for drift velocity in the above expression, we get

µ =1

E· eEτ

me

,

µ =eτ

me

.

When an electric field is applied, the electrons move with drift velocity vd. If the electric field isswitched off, the drift velocity decays exponentially to zero after some time and the electrons will bemoving with velocity v0 only because of thermal agitation. The decay process follows the equation,

v0 = vde−t/τr ,

where t is the time counted from the instant the field is turned off and τr is called the relaxationtime . If t = τr,

v0 = vde−1,

v0 =1

evd.

Thus the relaxation time is defined as the time during which drift velocity reduces to 1/e times itsmaximum value after the electric filed is switched off. The relationship between relaxation time andmean collision time is given by

τr =τ

1− < cosθ >,

where θ is the scattering angle and cosθ is the average value of cosθ taken over very large number ofcollisions made by electrons. In most of the situations < cosθ >= 0 and hence τr = τ , that means thatthe mean collision time is same as the relaxation time.

The average distance traveled by electrons between two successive collisions during their randommotion is called mean free path , denoted by λ.



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2.2 Expression for electrical conductivity in metals

Consider a conductor of length L and area of cross section A. The electric current I is proportional tothe voltage drop V across the conductor, which is called Ohms Law and is given by

V = IR.

We rewrite the above equation,

I =V

R,

I

A=

V

AR,

I

A=

L

AR

V

L.

We identify the above quantities as current density J = I/A, electrical conductivity σ = L/RA andelectric field E = V/L. Then,

J = σE,

which is the Ohm’s law in a general form.Since our aim is to find the expression for the electrical conductivity, we use the above equation in

the following form

σ =J

E.

Substituting the definition of J back in the above equation, we get

σ =I

AE.

Now, we need to find the expression for current I in the conductor. Let n be the number of electronsper unit volume of the conductor and vd be the drift velocity of electrons with charge e. Then, thenumber of electrons crossing any cross section per unit time is n(vdA). Therefore the current passingthrough any cross section of the conductor is given by

I = nevdA

Plugging this expression into that of conductivity, we get

σ =nevdA

AE,

σ =nevd

E.

Using the equation of drift velocity vd = eEτme

in the above equation, we have

σ =ne

E· eEτ

me

,

σ =ne2τ

me

.

This is the expression of electrical conductivity that we were looking for.



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2.2.1 Effect of impurity and temperature on electrical resistivity of metals

We know that the electrical conductivity of a metal is given by

σ =ne2τ

me

.

We write τ in terms of thermal velocity vth and mean free path λ

τ =λ

vth,

substituting for vth =√

3kTme

, we get

τ =λ√3kTme

= λ

√me

3kT.

Plugging τ into the expression for conductivity, we get

σ =ne2

me

λ

√me

3kT,

σ =ne2λ√3mekT

.

Hence, resistivity is given by

ρ =

√3mekT

ne2λ.

This equation suggests that the resistivity of a metal must be directly proportional to the square rootof temperature.

The resistivity of metals is attributed to the scattering of conduction electrons. The scatteringof electrons takes place because of two reasons: one due collisions of conduction electrons with thevibrating lattice ions and the other is caused by scattering of electrons by the impurities present in themetal. The resistivity due to scattering of electrons by the lattice vibrations called phonons is denotedby ρp. This increases with temperature. It arises even in a pure conductor and hence called the idealresistivity. Whereas the resistivity of metals caused by scattering of electrons with the impurities isdenoted by ρi. This is independent of temperature and present even at absolute zero of temperatureand hence called residual resistivity. Therefore, the total resistivity of a metal can be written as thesum of the two resistivities. This is called Matthiessen’s rule. Mathematically,

ρ = ρp + ρi

Since ρ =me

ne2τ, we can rewrite the above equation in the following form

ρ =me

ne2τp

+me

ne2τi

,

where τp and τi are the mean collision times of electrons with phonons and impurities respectively. Atlower temperatures ρp tends zero as the amplitude of lattice vibrations becomes small which essentiallymeans that all the resistivity will be due to impurities, i.e., ρ = ρi. At higher temperatures ρp increaseswith temperature however the curve of ρ versus T at room temperature remains more or less linear asshown in the following figure.



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2.3 Failure of classical free-electron theory

While the classical understanding of conduction is useful in constructing Ohms law and providing anunderstanding of the motion of electrons, there is a number of inherent flaws in this theory.

1. From the equation for resistivity, we have

ρ =

√3mekT

ne2λ,

which means that the resistivity of a metal must be directly proportional to the square root oftemperature. But it is observed by experiments that resistivity has a linear relationship withtemperature. Furthermore, the above stated equation will give a value that is about seven timesthe measured value of resistivity at a temperature of 300 K.

2. According to the classical theory, the molar heat capacity of free electrons in a metal is3

2R.

However, the experimentally determined molar heat capacity of metals is 10−4RT .

3. As seen from the expression for electrical conductivity,

σ =ne2τ

me

,

the conductivity is directly proportional to the electron density n. Hence, divalent and trivalentmetals should possess much higher electrical conductivity than monovalent metals. This is con-trary to the experimental observation that silver and copper are more conducting than zinc andaluminum.

4. There are some flaws from a statistical perspective as well. Applying kinetic theory of gases to

the electrons will give an average kinetic energy of3

2kT . But the observed kinetic energy is kT.

5. According to quantum mechanics, it is known that electrons share wave-like properties as well,and the classical theory makes no mention of such properties.

3 Quantum free-electron theory

By altering the classical theory through the application of the wave properties of electrons, the quantumtheory of conduction was formed by Arnold Sommerfeld in late 1920s. It is with this theory that thephysics world is constantly moving forward by determining new utilizations of the scattering of electron-wave properties throughout a material. The underlying assumptions of the theory are the following.

• The electrons can have only discrete values of energy. Those allowed energies for the electron arecalled the energy levels.

• The electrons are distributed among the energy levels according to Pauli’s exclusion principle,which states that there cannot be more than two electrons in any given energy level.

• The electrons experience a constant electric potential due to ions located at the lattice points andremain within the solid.

• The electron-electron and electron-lattice ion interactions are neglected.



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3.1 Density of states

The energy levels of electrons in the case of single atom are sharp as we know in the case of Hydrogenatom. Whereas in the case of a solid, energy levels of electrons will spread out over a range called energyband due to presence of huge number of atoms. Each energy band consists of a number of closely spacedenergy levels. To describe the number of states at each energy level that are available to be occupiedby the electrons, we introduce the concept of the density of states of a system . The density of states,g(E), is defined as the number of energy levels available per unit volume per unit energy centered at E.The number of states per unit volume between the energy levels E and E + dE is denoted by g(E)dE.

From the discussion of an electron in one-dimensional potential well of width L, we know that theallowed energies are given by

En =n2π2~2

2mL2where n = 1, 2, 3, .....

or

En =n2h2

8mL2.

Since the free-electrons in a solid experience a three dimensional potential well, the above equationtakes the form

E =h2

8mL2(n2

x + n2y + n2

z),

where nx,ny and nz are non-zero positive numbers. Each set of (nx, ny, nz) indicates the permittedenergy vale. Taking

E0 =h2

8mL2

andR2 = n2

x + n2y + n2

z

the above equation will be reduced toE = E0R

2.

The equation R2 = n2x + n2

y + n2z represents a sphere of radius R formed by the points (nx, ny, nz) with

nx,ny and nz as the three mutually perpendicular coordinate axes. Since nx,ny and nz can take only

the positive integers, the above equation represents only1

8of the sphere called the octant as shown in

the diagram.

The number of allowed energy values N(E) in a small energy range between E and E + dE is equal

to the product of the1

8

th

of the volume of the sphere between the shells of radius R and R + dr, and



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the number of points per unit volume which is 1. Mathematically,

N(E)dE =1

84πR2dR× 1,

N(E)dE =1

2πR2dR.

Since, each such energy value can accommodate two states of electrons according to Pauli’s exclusionprinciple, the number of allowed energy states between E and E + dE is,

N(E)dE = 2× 1

2πR2dR,

N(E)dE = πR2dR.

In the above equation R is abstract quantity which we cannot measure, so we wish to express the entireright hand side in terms of energies using the equation E = E0R

2. We can express R as√E

E0

= R

and differentiating it, we havedE = 2E0R dR.

Now, multiplying the above two expressions give us√E

E0

dE = 2E0R2 dR,

or

R2 dR =1

2

√E

E30

dE.

Substituting this in the expression for N(E)dE, we get

N(E)dE =π

2

√E

E30

dE,

Using E0 = h2

8mL2 in the above equation,

N(E)dE =π

2

√E

( h2

8mL2 )3dE,

further simplification gives

N(E)dE =

(8√

2m3πL3

h3

)√

E dE.

In this equation L3 represents the volume of the solid and since the density of states is the number ofenergy states per unit volume,

g(E)dE =N(E)dE

L3,

hence the equation

g(E)dE =

(8√

2m3π

h3

)√

E dE.

The plot of g(E) versus E is shown below.



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3.2 Fermi-energy and Fermi factor

In a single atom there will be many allowed energy levels whereas in a solid each such energy level willspread over a range of few eV ′s. If there are N number of atoms, there will be N closely spaced energylevels in each energy band of the solid. According to Pauli’s exclusion principle, each such energy levelcan accommodate two electrons. At absolute zero temperature, two electrons with least energy withopposite spins occupy the lowest available energy level. The next two electrons with opposite spinswill occupy next energy level and so on. Thus, the top most energy level occupied by electrons atabsolute zero temperature is called Fermi energy level. The energy corresponding to that energy levelis called Fermi energy. Fermi energy, EF , is defined as the energy at absolute zero corresponding tothe highest filled energy level, below which all energy levels are completely occupied and above whichall the energy levels completely empty. Thus Fermi energy represents maximum energy that electronscan have at absolute zero temperature.

At absolute zero all energy levels below Fermi energy are completely filled and above it are completelyempty. But at any given temperature, the electrons get thermally excited and move up to higherenergy levels. As a result there will be many vacant energy levels below as well as above Fermi energylevel. Under thermal equilibrium, the distribution of electrons among various energy levels is given bystatistical function f(E). The function f(E) is called the Fermi factor and this gives the probabilityof occupation of a given energy level under thermal equilibrium. The expression for f(E) is,

f(E) =1

e(E−Ef )/kT + 1.

3.2.1 Variation of Fermi factor with energy and temperature

Variation of Fermi factor with energy and temperature is discussed below.

Case 1: For T = 0K and E < EF

The Fermi factor is given by

f(E) =1

e(E−Ef )/kT + 1.

For T = 0K and E < EF , the above expression becomes

f(E) =1

e−∞ + 1=

1

0 + 1

f(E) = 1.

This implies that at absolute zero, all the energy levels below EF are 100% occupied which is true fromthe definition Fermi energy.



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Case 2: For T = 0K and E > EF

The Fermi factor is given by

f(E) =1

e(E−Ef )/kT + 1.

For T = 0K and E > EF , the above expression becomes

f(E) =1

e∞ + 1=

1

∞

f(E) = 0.

This implies that at absolute zero, all the energy levels above EF are unoccupied which is true fromthe definition Fermi energy.

Case 3: For T > 0K and any E

At ordinary temperatures for E = EF , we get

f(E) =1

e(E−Ef )/kT + 1=

1

e0 + 1. =

1

2

For E � EF , the probability starts decreasing from 1 and reaches 0.5 at E = EF , and for E > EF , itfurther falls off as shown in the figure below.

In conclusion, the Fermi energy is the most probable or the average energy of the electrons in asolid.

3.2.2 Variation of Fermi energy with temperature

We want to find out how the Fermi energy varies with temperature. To achieve this, we consider thefollowing. The number of electrons per unit volume of the solid, n, is nothing but the product of densityof states available and the probability of occupancy of electrons among various energy levels up to theFermi level EF , that is,

n =

∫ EF

0

f(E) · g(E) dE.



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Case 1: For T = 0K

We know that, for T = 0K, we have f(E) = 1 and the Fermi energy for such a case is taken as EF0 .Hence the equation of n takes the form

n =

∫ EF0

0

g(E) dE.

Substituting for density of states g(E) dE, we have

n =

∫ EF0

0

(8√

2m3π

h3

)√

E dE,

n =

(8√

2m3π

h3

)∫ EF0

0

√E dE,

n =

(8√

2m3π

h3

)×[2

3E3/2

]EF0

0

,

n =

(8√

2m3π

h3

)×[2

3EF0

3/2

],

Further simplification gives us

EF0

3/2 =

(3h3

(8m)3/2π

)n

or

EF0 =

(h2

8m

)(3

π

)2/3

n2/3

Case 1: For T > 0 K

We know that, for T > 0 K, we have f(E) 6= 1 and the Fermi energy for such a case is taken as EF ,which is given by

EF = EF0

[1− π2

12

(kT

EF0

)2]

.

For smaller temperatures, the second term of the above equation vanishes giving,

EF = EF0 ,

implying that at ordinary temperatures EF and EF0 essentially the same.

3.3 Fermi - Dirac Statistics

A metal has very large number of free electrons and these electrons are distributed among variousenergy levels in the energy bands. The statistics which governs how the free electrons are distributedamong various energy levels is called Fermi-Dirac statistics . This obeys Pauli’s exclusion principle

and is applicable to any indistinguishable particles of spin1

2. The distribution of electrons among

various available energy levels according Fermi-Dirac statistics under thermal equilibrium is calledFermi-Dirac distribution . Mathematically

N(E) dE = f(E)× g(E) dE,

where N(E) dE is the number of electrons in unit volume possessing the energies between E and E+dE.The plot of N(E) dE versus E is shown in the following figure for various temperatures.



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3.4 Expression for electrical resistivity / conductivity

Using the concepts of density of states and Fermi-Dirac statistics Sommerfeld arrived at the followingexpression for electrical conductivity in metals,

σ =ne2λ

m∗vF

,

where λ is the mean free path; m∗ is the effective mass and vF is called the Fermi velocity. The Fermivelocity can be found out by equating the Fermi energy to the kinetic energy of the electrons in a metal.That is,

1

2mv2

F = EF ,

vF =

√2EF

m.

The resistivity of the metal is given by

ρ =1

σ.

Therefore,

ρ =m∗vF

ne2λ.

3.5 Merits of Quantum free electron theory

The quantum theory of free electrons solves the flaws of the classical theory which is discussed below.

1. Temperature dependence of resistivity of metals: The resistivity of a metal is given by

ρ =m∗vF

ne2λ.

In the above expression only the mean free path λ is the temperature dependent quantity. Inthe classical theory, the collision was seen as a particle bouncing off another. In the quantumunderstanding, an electron is viewed as a wave traveling through a medium. If r represents theamplitude of the oscillation of the lattice ions, then the λ is inversely proportional to the area ofcross section, i.e.,

λ ∝ 1

πr2.

But the area of cross section is directly proportional to the absolute temperature, i.e.,

r2 ∝ T.



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Therefore,

λ ∝ 1

T.

Henceρ ∝ T,

which is exactly the same as the experimental prediction.

2. Specific heat of free electrons: From quantum theory of free electrons, the specific heat of freeelectrons is given by

Cv =2k

EF

RT.

For a typical value of EF = 5 eV , we get

Cv = 10−4RT,

which is in agreement with the experimental results.

3. Dependence of electrical conductivity on electron concentration: The electrical conduc-tivity in metals is given by

σ =e2

m∗λ

vF

n.

It is clear from this equation that the electrical conductivity depends both the electron concen-tration and λ

vF. Now by taking above expression, if we calculate the σ, we get the observed

conductivities of the metals.

***



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Lasers




[email protected]

November 23, 2008

Who can use this?

The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is thefifth of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU forthe first-semester (September 2008 - January 2009) BE students of all branches. Anystudent interested in exploring more about the course may visit the course homepageat www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy ofstudying this: this chapter is worth 20 marks in the final exam! Cheers ;-)


Principle and production. Spontaneous and stimulated emission and

induced absorption. Einsteins coefficients. Requisites of a Laser

system. Condition for Laser action. Principle, Construction and

working of He-Ne and semiconductor Laser. Applications of Laser: Laser

welding, cutting and drilling. Measurement of atmospheric pollutants.

Holography, Principle of Recording and reconstruction of 3-D images.

Selected applications of holography

Reference



1



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1 Introduction

The term LASER is an anachronism for Light Amplification by Stimulated Emission of Radiation. Thefirst working laser was demonstrated in 1960 by Theodore Maiman at Hughes Research Laboratories.Since then, lasers have become a multi-billion dollar industry. By far the largest single application oflasers is in optical storage devices such as compact disc and DVD players. The second-largest applicationis fiber-optic communication. Other common applications of lasers are bar code readers, laser printersand laser pointers. In manufacturing, lasers are used for cutting precise patterns in glass and metal, andwelding metal. Lasers are used by the military for range-finding, target designation, and illumination.Lasers have also begun to be used as directed-energy weapons. Lasers are used in medicine for surgery,diagnostics, to reshape corneas to correct poor vision and therapeutic applications. But we also uselasers as very precise light sources in supermarket checkout lines. In this chapter, we describe theprinciples behind the lasers and some of its uses.

2 Principle and production

Under normal circumstances when light interacts with matter, electrons in the matter may absorb thelight energy and go to the higher energy level and come back to the ground state by emitting the lightof the same frequency. But in any given situation the number of electrons in the ground state is morethan that in the excited state. The Laser is based on the principle of light amplification by stimulatedemission of radiation. This is achieved by having excess concentration of electrons in the higher energystates and stimulating the system by radiation to bring about the de-excitation process.

2.1 Induced absorption, spontaneous emission and stimulated emission

An atom in the ground state may absorb a photon of suitable energy and go to an excited state. Thisis know as the induced absorption and the process is represented by the following equation.

atom + photon → atom∗

An atom which is already in an excited state will be unstable and it falls back to the ground stateby emitting a photon of appropriate energy. This is know as the spontaneous emission and theprocess is represented by the following equation.

atom∗ → atom + photon

An atom which is already in an excited state will emit a photon on its own while making itstransition to the ground state . But by passing a photon of just right energy we can stimulate theexcited atom to emit a photon of the same phase, energy and directed in the same direction as that ofthe introduced one. This is know as the stimulated emission and the process is represented by thefollowing equation.

atom∗ + photon → atom + photon + photon



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2.2 Einsteins coefficients

Consider a system of atoms with two energy states E1 and E2 with N1 and N2 number of atoms perunit volume in each energy states respectively. The N1 and N2 are called the number densities of theatoms. Let a radiation of energy density Eν of frequency ν be incident on the system.

In the case of induced absorption, an atom in the ground state E1 goes to an excited state E2 byabsorbing a suitable photon of energy hν = E2 − E1. The number of such absorptions per unit time,per unit volume is called the rate of induced absorption. This depends on the number density N1 ofthe ground state and the energy density of the incident radiation Eν . That is,

Rate of induced absorption ∝ N1Eν .

By introducing the constant of proportionality B12, we get

Rate of induced absorption = B12N1Eν .

In the case of spontaneous emission, an atom in the excited state E2 makes a transition to theground state E1 by by emitting a photon of appropriate energy hν = E2 − E1. The number of suchspontaneous emissions per unit time, per unit volume is called the rate of spontaneous emission. Thisdepends only on the number density N2 of the excited state. That is,

Rate of spontaneous emission ∝ N2.

By introducing the constant of proportionality A21, we get

Rate of spontaneous emission = A21N2.

In the case of stimulated emission, an atom in the excited state E2 makes a transition to the groundstate E1 upon incidence of a photon of suitable energy hν = E2 − E1, by emitting a photon of sameenergy. The number of such stimulated emissions per unit time, per unit volume is called the rateof stimulated emission. This depends on the number density N2 of the excited state and the energydensity of the incident radiation Eν . That is,

Rate of stimulated emission ∝ N2Eν .

By introducing the constant of proportionality B21, we get

Rate of stimulated emission = B21N2Eν .

In the above discussion, the constants of proportionality are called the Einstein’s coefficients. Underthermal equilibrium,

Rate of induced absorption = Rate of spontaneous emission + Rate of stimulated emission



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that is,B12N1Eν = A21N2 + B21N2Eν .

Taking Eν terms on to the left hand side, we get

B12N1Eν −B21N2Eν = A21N2,

(B12N1 −B21N2)Eν = A21N2,

Eν =A21N2

B12N1 −B21N2

.

By rearranging the terms, we get

Eν =A21

B21

(1

B12N1

B21N2− 1

).

From Boltzmann’s law, we know that the ratio of the population densities N1 and N2 in the groundstate E1 and excited state E1 respectively is given by,

N1

N2

= e(E2−E1)/kT .

But, E2 − E1 = hν, thereforeN1

N2

= ehν/kT .

Using this in the equation of Eν , we get

Eν =A21

B21

(1

B12

B21ehν/kT − 1

).

This equation reminds us of the Planck’s law, which is given by

Eν =8πhν3

c3

(1

ehν/kT − 1

).

By comparing the above two equations, we get

A21

B21

=8πhν3

c3

andB12

B21

= 1 ⇒ B12 = B21.

The equality B12 = B21 implies that the probability of induced absorption is equal to that of stimulatedemission. Hence we can use simply A and B to denote the Einstein’s coefficients. With this our equationfor the energy density at thermal equilibrium takes the form,

Eν =A

B (ehν/kT − 1).



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2.3 Requisites of a laser system

The principal components of a laser system are

1. Gain medium or optical medium which supports population inversion

2. Source for pumping action

3. Fully silvered mirror called High reflector

4. A partially transparent mirror called Output coupler

5. Laser beam

A laser consists of a gain medium inside a highly reflective optical cavity, as well as a means tosupply energy to the gain medium. The gain medium is a material with properties that allow it toamplify light by stimulated emission. In its simplest form, a cavity consists of two mirrors arrangedsuch that light bounces back and forth, each time passing through the gain medium. Typically one ofthe two mirrors, the output coupler, is partially transparent. The output laser beam is emitted throughthis mirror. Light of a specific wavelength that passes through the gain medium is amplified (increasesin power); the surrounding mirrors ensure that most of the light makes many passes through the gainmedium, being amplified repeatedly. Part of the light that is between the mirrors (that is, within thecavity) passes through the partially transparent mirror and escapes as a beam of light. The process ofsupplying the energy required for the amplification is called pumping. The energy is typically suppliedas an electrical current or as light at a different wavelength.

2.4 Condition for laser action

The two conditions to achieve lasing actions are

1. Population inversion

2. Availability of metastable states

Population inversion is a state of a system in which the population of an excited state is morethan that of a ground state. Naturally, we find the number density to be more in the ground statethan in the excited one, hence the name. We can achieve population inversion only in those systemswhich posses a special kind of excited state called metastable state . The electrons will remainonly for about 10−8s in the excited state and after that they make transitions to the ground state byemission of a photon. Whereas, the electrons can stay in the metastable state as long as 10−3s. Ifthe excited state happens to be a metastable state, the atoms can stay excited for longer durationresulting in steady increase in the population of the excited or metastable state and one stage we canachieve the population inversion. Once this happens, the number of stimulated emissions outnumber thespontaneous emissions. The photons from stimulated emission will have the same wavelength, phaseand direction. Once the intensity of the photons is sufficient to pass through the partially silveredmirror, we get the laser light.



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3 Types of Laser

Many thousands of kinds of laser are known, but most of them are not used beyond specialized research.The chief types of lasers are solid state lasers, gas lasers and liquid lasers. A solid, liquid, gas orsemiconductor can act as the laser medium. Here we discuss only Helium-Neon and semiconductorlasers.

3.1 Helium-Neon Laser.

A Helium-Neon laser, usually called a He-Ne laser, is a gas laser of operation wavelength is 632.8 nm,in the red portion of the visible spectrum. This was invented by Ali Javan, William Bennett Jr. andDonald Herriott at Bell Labs. The gain medium of the laser, as suggested by its name, is a mixtureof helium and neon gases, in a 5 : 1 to 20 : 1 ratio, contained at low pressure in a glass tube. Theenergy or pump source of the laser is provided by an electrical discharge of around 1000 V through ananode and cathode at each end of the glass tube. The optical cavity of the laser typically consists ofa plane, high-reflecting mirror at one end of the laser tube, and a concave output coupler mirror ofapproximately 1% transmission at the other end. He-Ne lasers are typically small, with cavity lengthsof around 15 cm up to 0.5 m, and optical output powers ranging from 1 mW to 100 mW .

The laser process in a He-Ne laser starts with collision of electrons from the electrical dischargewith the helium atoms in the gas. This excites helium from the ground state to the 23S and 21Smetastable excited states. Collision of the excited helium atoms with the ground-state neon atomsresults in transfer of energy to the neon atoms, exciting neon electrons into the 3s and 2s levels. Thisis due to a coincidence of energy levels between the helium and neon atoms. This process is given bythe reaction equation:

He + electric energy → He∗



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He∗ + Ne → He + Ne∗

where ∗ represents an excited state. The number of neon atoms entering the excited states builds up asfurther collisions between helium and neon atoms occur, causing a population inversion. Spontaneousand stimulated emission between the 3s and 2p states results in emission of 632.82 nm wavelength light.After this, fast radiative decay occurs from the 2p to the 1s ground state. Because the neon upperlevel saturates with higher current and the lower level varies linearly with current, the He-Ne laser isrestricted to low power operation to maintain population inversion.

With the correct selection of cavity mirrors, other wavelengths of laser emission of the He-Ne laserare possible. There are infrared transitions at 3391.2 nm and 1152.3 nm wavelengths, and a variety ofvisible transitions. The typical 633 nm wavelength red output of a He-Ne laser actually has a muchlower gain compared to other wavelengths such as the 1152.3 nm and 3391.2 nm lines, but these can besuppressed by choosing cavity mirrors with optical coatings that reflect only the desired wavelengths.

It is used in interferometry, holography, spectroscopy, barcode scanning, alignment, optical demon-strations.

3.2 Semiconductor Laser.

A semiconductor laser has the active medium as a semiconductor similar to that found in a light-emitting diode. The most common and practical type of semiconductor laser is formed from a p-njunction and powered by injected electric current.

A semiconductor laser diode, like many other semiconductor devices, is formed by doping a verythin layer on the surface of a crystal wafer. The crystal is doped to produce an n-type region and ap-type region, one above the other, resulting in a p-n junction, or diode. Just as in any semiconductorp-n junction diode, forward electrical bias causes the two species of charge carrier - holes and electrons- to be “injected” from opposite sides of the p-n junction into the depletion region, situated at itsheart. Holes are injected from the p-doped, and electrons from the n-doped, semiconductor. Thecharge injection is a distinguishing feature of semiconductor lasers as compared to all other lasers.When an electron and a hole are present in the same region, they may recombine or annihilate withthe result being spontaneous emission i.e., the electron may re-occupy the energy state of the hole,emitting a photon with energy equal to the difference between the electron and hole states involved. Ina conventional semiconductor junction diode, the energy released from the recombination of electronsand holes is carried away as phonons rather than as photons.

In the absence of stimulated emission conditions, electrons and holes may coexist in proximity toone another, without recombining for a certain time called the “recombination time”. Then a nearbyphoton with energy equal to the recombination energy can cause recombination by stimulated emission.This generates another photon of the same frequency, traveling in the same direction, with the samepolarization and phase as the first photon. As in other lasers, the gain region is surrounded with anoptical cavity. As a light wave passes through the cavity, it is amplified by stimulated emission resultingin a laser beam.



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Semiconductor lasers are numerically the most common type of laser. They find wide use intelecommunication, barcode readers, CD players, DVD and Blu-ray technology, material processing,range-finding, laser target designation, surgery, holography and tattoo removal.

4 Applications of Laser

There are many scientific, industrial, military, medical and commercial laser applications. The co-herency, high monochromaticity, focussability, directionality and ability to reach extremely high powersare all properties which allow for these specialized applications. We discuss a few of them below.

• Laser cutting: In Industry lasers are used for the precise cutting of flat materials. Lasershave the advantage that there is no physical contact with the material so there is no chance ofcontamination, also there is less chance of the material warping as the laser energy can be focusedon a very small area so the whole material is not heated. Even a three dimensional profile can becut using lasers. Laser cutting is also employed in tailoring industry.

• Laser welding: In laser welding, a beam of laser is focused on to the spot to be welded. Due tothe heat generated, the material melts over a tiny area and upon cooling the material becomeshomogeneous solid structure. Laser welding is a contact-less process and thus no outside materialgets into the welded region. Since the heat affected zones are very small, laser welding is ideal formany microelectronic devices.

• Laser drilling: Laser drilling of holes is achieved by subjecting the material to powerful laserpulses of about millisecond duration. The intense heat generated over a short duration by thepulses evaporates the material locally leaving a hole. Very fine holes of the dimensions one tenthof millimeter can be drilled. Since there is mechanical stress involved in laser drilling, even thebrittle materials can be drilled.

• Measurement of pollutants in the atmosphere: There are various types of pollutants in theatmosphere. In the measurement of pollutants, laser is used in the way a radar system is used.Hence it is called LIDAR meaning Light Detection and Ranging. This can evaluate the distance,altitude and angular coordinates of the object.

5 Holography

Holography is a technique that allows the light scattered from an object to be recorded and laterreconstructed so that it appears as if the object is in the same position relative to the recordingmedium as it was when recorded. The image changes as the position and orientation of the viewingsystem changes in exactly the same way as if the object was still present, thus making the recordedimage (hologram) appear three dimensional. The technique of holography can also be used to opticallystore, retrieve, and process information. Holography is commonly used to display static 3-D pictures.Holography was invented in 1947 by Hungarian physicist Dennis Gabor for which he received the NobelPrize in Physics in 1971.

5.1 Recording of a hologram

In holography, some of the light scattered from an object or a set of objects falls on the recordingmedium. A second light beam, known as the reference beam, also illuminates the recording medium,so that interference occurs between the two beams. The resulting light field is an apparently randompattern of varying intensity which is the hologram. There are a variety of recording materials whichcan be used, including photographic film.



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A very simple hologram can be made by superimposing two plane waves from the same light source.One beam (the reference beam) hits the photographic plate normally and the other one (the objectbeam) hits the plate at an angle θ. The relative phase between the two beams varies across thephotographic plate as 2πysinθ where y is the distance along the photographic plate. The two beamsinterfere with one another to form an interference pattern. The relative phase changes by 2π at intervalsof d = λ/sinθ so the spacing of the interference fringes is given by d. Thus, the relative phase of objectand reference beam is encoded as the maxima and minima of the fringe pattern.

5.2 Reconstruction of three dimensional image

The process of producing a holographic reconstruction involves the phenomenon of diffraction of light.When the photographic plate is developed, the fringe pattern acts as a diffraction grating and whenthe reference beam is incident upon the photographic plate, it is partly diffracted into the same angleθ at which the original object beam was incident. Thus, diffraction grating created by the two wavesinterfering has reconstructed the object beam and when we look into the hologram, we sees the objecteven though it may no longer be present.



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5.3 Applications of holography

Some of the applications of holography are discussed in brief below.

1. Holographic interferometry: The most successful application of holography, however, is ininterferometry. If two holograms of the same object are recorded on the same plate, then uponreconstruction the two holographic images will interfere. If the object has undergone a deformationbetween the two recordings, phase differences in certain parts of the two images will result, creatingan interference pattern that clearly shows the deformation.

2. Holographic diffraction gratings: Holography can be used to make gratings. In this methodtwo laser beams are made to interfere on a recording medium which produces the rulings muchmore uniformly than any other method.

3. Holographic storage of digital data: The digital data can be recorded as bright and darkspots in holographic images. A hologram can contain a large number of ‘pages’ that are recordedat different angles relative to the plate, thus allowing the storage of a very large amount of dataon one hologram. By illuminating the hologram with a laser beam at different angles, the pagescan be read out one by one.

4. Acoustic holography: It is a method used to estimate the sound field near a source by mea-suring acoustic parameters away from the source. Measuring techniques included within acousticholography are becoming increasingly popular in various fields, most notably those of transporta-tion, vehicle and aircraft design.

5. Security holograms: They are very difficult to forge because they are replicated from a masterhologram which requires expensive, specialized and technologically advanced equipment. Theyare used widely in many currencies, credit and bank cards as well as books, DVDs and sportsequipment.

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Superconductivity and Optical Fibers




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Who can use this?

The lecture notes are tailor-made for my students at SBMJCE, Bangalore. It is thesixth of eight chapters in Engineering Physics [06PHY12] course prescribed by VTU forthe first-semester (September 2008 - January 2009) BE students of all branches. Anystudent interested in exploring more about the course may visit the course homepageat www.satheesh.bigbig.com/EnggPhy. For those who are looking for the economy ofstudying this: this chapter is worth 20 marks in the final exam! Cheers ;-)


Temperature dependence of resistivity in superconducting materials.

Effect of magnetic field (Meissner effect). Type I and Type II

superconductors. Temperature dependence of critical field. BCS theory.

High temperature superconductors. Applications of superconductors:

Superconducting magnets, Maglev vehicles SQUIDS. Propagation mechanism

in optical fibers: Angle of acceptance. Numerical aperture. Types of

optical fibers and modes of propagation. Attenuation. Applications:

Block diagram discussion of point to point communication

Reference



1



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1 Introduction

Of all the scientific breakthroughs of the last millennium which have made our daily life so comfortable,the major ones are electric and electronic gadgets and communication. A great hope in improvingupon these areas of technology comes from our better understanding of electromagnetic properties ofthe materials and quick transmission of information. The two topics which fall into this category aresuperconductivity and fiber optics. In this chapter we study them in a bit of detail.

2 Superconductivity

Superconductivity is a phenomenon observed in several materials that demonstrate no resistance to theflow of an electric current when cooled to temperatures ranging from near absolute zero (0 K) to liquidnitrogen temperatures (77 K). The temperature below which electrical resistance is zero is called thecritical temperature Tc and this temperature is a characteristic of the material. Superconductivity occursin a wide variety of materials, including simple elements like tin and aluminium, various metallic alloysand some heavily-doped semiconductors. Superconductivity does not occur in noble metals like goldand silver, nor in pure samples of ferromagnetic metals. Another striking property of superconductorsis that they are perfectly diamagnetic.

Superconductivity was discovered in 1911 by Heike Kamerlingh Onnes, who was studying the re-sistance of solid mercury at cryogenic temperatures using liquid helium as a refrigerant. He got 1913Physics Nobel Prize for his work.

2.1 Temperature dependence of resistivity in superconducting materials

The resistivity of metals is attributed to the scattering of conduction electrons. The scattering ofelectrons takes place because of two reasons: one due collisions of conduction electrons with the vibratinglattice ions and the other is caused by scattering of electrons by the impurities present in the metal.The resistivity due to scattering of electrons by the lattice vibrations called phonons is denoted byρp. This increases with temperature. It arises even in a pure conductor and hence called the idealresistivity. Whereas the resistivity of metals caused by scattering of electrons with the impurities isdenoted by ρi. This is independent of temperature and present even at absolute zero of temperatureand hence called residual resistivity. Therefore, the total resistivity of a metal can be written as thesum of the two resistivities. This is called Matthiessen’s rule. Mathematically,

ρ = ρp + ρi.

However, some metals show a remarkable behavior. They lose their electrical resistance completelybelow a certain temperature, called critical temperature Tc. Below the critical temperature these su-perconducting materials can carry large amounts of electrical current for long periods of time without



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loosing energy as ohmic heat. The variation of electrical resistance versus temperature of ordinarymetals and superconductors is shown the graph.

2.2 Meissner effect

When a material makes the transition from the normal to superconducting state, it actively excludesmagnetic fields from its interior; this is called the Meissner effect. Walther Meissner discovered thephenomenon in 1933 by measuring the flux distribution outside of tin and lead specimens as they werecooled below their transition temperature in the presence of a magnetic field. He found that belowthe superconducting transition temperature the specimens became perfectly diamagnetic, canceling allflux inside. The experiment demonstrated for the first time that superconductors were more than justperfect conductors and provided a uniquely defining property of the superconducting state.

One of the theoretical explanations of the Meissner effect is given by the brothers Fritz and HeinzLondon. The London equation shows that the magnetic field decays exponentially inside the super-conductor over a small distance called the London penetration depth, decaying exponentially to zerowithin the bulk of the material. For most superconductors, the London penetration depth is on theorder of 100 nm.

The Meissner effect is sometimes confused with the kind of diamagnetism one would expect in aperfect electrical conductor: according to Lenz’s law, when a changing magnetic field is applied to aconductor, it will induce an electrical current in the conductor that creates an opposing magnetic field.In a perfect conductor, an arbitrarily large current can be induced, and the resulting magnetic fieldexactly cancels the applied field. The Meissner effect is distinct from this because a superconductorexpels all magnetic fields, not just those that are changing. Suppose we have a material in its normalstate, containing a constant internal magnetic field. When the material is cooled below the criticaltemperature, we would observe the abrupt expulsion of the internal magnetic field, which we would notexpect based on Lenz’s law.

2.3 Temperature dependence of critical field

The Meissner effect breaks down when the applied magnetic field is too large. The strength of magneticfield required to just switch a material from superconducting state to normal state is called the criticalfiled Hc. The temperature dependence of the critical field is given by the expression,

Hc = H0

[1− T 2

T 2c

], (1)

where,Hc is the critical field; H0 is the field required to turn the superconductor to normal at 0 K; T isthe temperature of the superconducting material and Tc the critical temperature. This equation says



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that if the strength of the applied magnetic field is greater than H0, the material can never becomesuperconducting however low the temperature may be. This is all depicted in the following graph.

2.4 Types of superconductors

Superconductors are divided into type I and type II depending on their characteristic behavior in thepresence of a magnetic field. The critical temperature at which the resistance vanishes in a super-conductor is reduced when a magnetic field is applied. The maximum field that can be applied to asuperconductor at a particular temperature and still maintain superconductivity is called the criticalfield Hc. The dependence of magnetic moment −M on the external magnetic field H for both type Iand II superconductors is shown in the graph.

In Type I superconductors, superconductivity is abruptly destroyed when the strength of theapplied field rises above a critical value Hc. The maximum critical field Hc in any Type I superconductoris about 2000 gauss (0.2 tesla). At fields greater than Hc, the conductor reverts to the normal stateand regains its normal state resistance. Most pure elemental superconductors are Type I.

Type II superconductors have two critical fields. Raising the applied field past a critical value Hc1

leads to a mixed state in which an increasing amount of magnetic flux penetrates the material, butthere remains no resistance to the flow of electrical current as long as the current is not too large. At a



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second critical field strength Hc2 , superconductivity is destroyed. The mixed state is actually caused byvortices in the electronic superfluid, sometimes called fluxons, because the flux carried by these vorticesis quantized. In Type II materials, superconductivity can persist to several hundred thousand Gauss(Hc2). At fields greater than Hc2 , the conductor reverts to the normal state and regains its normal stateresistance. Almost all impure and compound superconductors are Type II.

2.5 BCS theory

The understanding of superconductivity was advanced in 1957 by three American physicists - JohnBardeen, Leon Cooper, and John Schrieffer. Their microscopic theory of superconductivity is knownas BCS theory for which they received the Nobel Prize in Physics in 1972.

The BCS theory explains superconductivity at temperatures close to absolute zero. According tothis theory a superconducting current is carried by bound pairs of electrons called Cooper pairs,which move through a metal without energy dissipation. Cooper realized that atomic lattice vibrationsforced the electrons to pair up into teams that could pass all of the obstacles which caused resistancein the conductor. As one negatively charged electron passes by positively charged ions in the latticeof the superconductor, the lattice distorts. This in turn causes phonons to be emitted which forms atrough of positive charges around the electron. Before the electron passes by and before the latticesprings back to its normal position, a second electron is drawn into the trough. It is through thisprocess that two electrons, which should repel one another, link up. The forces exerted by the phononsovercome the electrons’ natural repulsion. The electron pairs are coherent with one another as theypass through the conductor in unison. The electrons are screened by the phonons and are separated bysome distance. When one of the electrons that make up a Cooper pair and passes close to an ion in thecrystal lattice, the attraction between the negative electron and the positive ion cause a vibration topass from ion to ion until the other electron of the pair absorbs the vibration. The net effect is that theelectron has emitted a phonon and the other electron has absorbed the phonon. It is this exchange thatkeeps the Cooper pairs together. It is important to understand, however, that the pairs are constantlybreaking and reforming. Because electrons are indistinguishable particles, it is easier to think of themas permanently paired.

2.6 High temperature superconductors

A new era in the study of superconductivity began in 1986 with the discovery of high critical temperature(high-Tc) superconductors by Karl Mller and Johannes Bednorz for which they won the Nobel Prize inPhysics in 1987. Until then it was thought that BCS theory ruled out superconductivity at temperaturesabove 30 K.

High-temperature superconductivity allows some materials to be superconducting at temperaturesabove the boiling point of liquid nitrogen (77 K or −196◦C). Indeed, they offer the highest transitiontemperatures of all superconductors. The ability to use relatively inexpensive and easily handled liquidnitrogen as a coolant has increased the range of practical applications of superconductivity. As of now,the current world record of superconductivity is held by a ceramic superconductor doped with thallium,mercury, copper, barium, calcium and oxygen which has Tc = 138 K.

The critical magnetic field that destroys superconductivity tends to be higher for materials witha high-Tc and in magnet applications this may be more valuable than the high Tc itself. All knownhigh-Tc superconductors are of Type-II and the best known are BSCCO and YBCO. The search for atheoretical understanding of high-temperature superconductivity is an important unsolved problem inphysics.



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2.7 Applications of superconductors

Superconducting magnets: These are some of the most powerful electromagnets known and arealready crucial components of several technologies. Magnetic resonance imaging (MRI) is playingan ever increasing role in diagnostic medicine. Superconducting magnets are used in NuclearMagnetic Resonance (NMR) machines, mass spectrometers, and the beam-steering magnets usedin particle accelerators. They can also be used for magnetic separation, where weakly magneticparticles are extracted from a background of less or non-magnetic particles, as in the pigmentindustries. Powerful new superconducting magnets could be made much smaller than a resistivemagnet,because the windings could carry large currents with no energy loss. Generators woundwith superconductors could generate the same amount of electricity with smaller equipment andless energy. Once the electricity was generated it could be distributed through superconductingwires. Energy could be stored in superconducting coils for long periods of time without significantloss.

Power transmission: The ability of superconductors to conduct electricity with zero resistance can beexploited in the use of electrical transmission lines. Currently, a substantial fraction of electricity islost as heat through resistance associated with traditional conductors such as copper or aluminum.A large scale shift to superconductivity technology depends on whether wires can be prepared fromthe brittle ceramics that retain their superconductivity at 77 K while supporting large currentdensities. Promising future applications include high-performance transformers and power storagedevices.

Digital electronics: The field of electronics holds great promise for practical applications of supercon-ductors. The miniaturization and increased speed of computer chips are limited by the generationof heat and the charging time of capacitors due to the resistance of the interconnecting metal films.The use of new superconductive films may result in more densely packed chips which could trans-mit information more rapidly by several orders of magnitude. Superconducting electronics haveachieved impressive accomplishments in the field of digital electronics. Superconductors have alsobeen used to make digital circuits and microwave filters for mobile phone base stations.

MagLev vehicles: The use of superconductors for transportation in magnetic levitation (MagLev)vehicles has already been established using liquid helium as a refrigerant. Prototype levitatedtrains have been constructed in Japan by using superconducting magnets.

SQUIDs: Superconducting Quantum Interference Devices are very sensitive magnetometers used tomeasure extremely small magnetic fields, based on superconducting loops containing Josephsonjunctions. They are sensitive enough to measure fields as low as 5× 10−18T .

The traditional superconducting materials used for SQUIDs are pure niobium or a lead alloy with10% gold or indium, as pure lead is unstable when its temperature is repeatedly changed. Tomaintain superconductivity, the entire device needs to operate within a few degrees of absolutezero, cooled with liquid helium.

There are two main types of SQUID: DC and RF. RF SQUIDs can work with only one Josephsonjunction, which might make them cheaper to produce, but are less sensitive. The DC SQUID wasinvented in 1964 by Arnold Silver, Robert Jaklevic, John Lambe, and James Mercereau of FordResearch Labs. The RF SQUID was invented in 1965 by James Edward Zimmerman and ArnoldSilver at Ford.

Some of the applications of SQUIDs are listed below.

• The extreme sensitivity of SQUIDs makes them ideal for studies in biology. Magnetoen-cephalography (MEG), for example, uses measurements from an array of SQUIDs to make



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inferences about neural activity inside brains. Because SQUIDs can operate at acquisitionrates much higher than the highest temporal frequency of interest in the signals emitted bythe brain (kHz), MEG achieves good temporal resolution.

• Another area where SQUIDs are used is magnetogastrography, which is concerned withrecording the weak magnetic fields of the stomach.

• A novel application of SQUIDs is the magnetic marker monitoring method, which is used totrace the path of orally applied drugs.

• SQUIDs are being used as detectors to perform Magnetic Resonance Imaging. While highfield MRI uses precession fields of one to several tesla, SQUID-detected MRI uses measure-ment fields that lie in the microtesla regime. Since the NMR signal drops off as the squareof the magnetic field, a SQUID is used as the detector because of its extreme sensitivity.

• Another application is the scanning SQUID microscope, which uses a SQUID immersed inliquid helium as the probe. The use of SQUIDs in oil prospecting, mineral exploration,earthquake prediction and geothermal energy surveying is becoming more widespread assuperconductor technology develops; they are also used as precision movement sensors in avariety of scientific applications.

3 Optical fibers

Optical fibers are thin rods of glass or some other transparent material of high refractive index. If lightis admitted at one end of a fiber, it can travel through the fiber with very low loss, even if the fiber iscurved. A fibre optic cable is made from a glass or plastic core that carries light surrounded by glasscladding that (due to its lower refractive index) reflects ‘escaping’ light back into the core, resulting inthe light being guided along the fibre. The outside of the fibre is protected by cladding and may befurther protected by additional layers of treated paper, PVC or metal. This required to protect thefibre from mechanical deformation and the ingress of water.

3.1 Propagation mechanism in optical fibers

The principle on which this transmission of light depends is that of total internal reflection. Lighttraveling inside the core strikes the outside surface at an angle of incidence greater than the criticalangle, so that all the light is reflected toward the inside of the fiber without loss. Hence, the opticalfibre behaves as a waveguide and thus light can be transmitted over long distances by being reflectedinward thousands of times.

Let η0, η1 and η2 be the refractive indices of surrounding medium, core and cladding of the opticfibre respectively. Now, for refraction at the point of entry of the ray into the core, we have from Snell’s



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lawη0 sin θ0 = η1 sin θ1.

At the point where this ray touches the interface of core and cladding, the angle of incidence is 90− θ1.Therefore, from Snell’s law, we have

η1 sin(90− θ1) = η2 sin 90,

η1 cos θ1 = η2,

orcos θ1 =

η2

η1

.

From the first equation, we have

sin θ0 =η1

η0

sin θ1,

orsin θ0 =

η1

η0

√1− cos2 θ1.

Substituting for cos θ1 from the previous equation, we get

sin θ0 =η1

η0

√1− η2

2

η21

,

or

sin θ0 =

√η2

1 − η22

η0

.

The angle θ0 is called the waveguide acceptance angle or the acceptance cone half-angle and sin θ0 iscalled the numerical aperture of the optic fibre. The numerical aperture represents the light gatheringcapacity of the optical fibre. Therefore,

Numerical Aperture =

√η2

1 − η22

η0

.

If the surrounding medium is air, then η0 = 1 and hence

Numerical Aperture =√

η21 − η2

2.

In general, if θi is angle of incidence, then th ray will propagate only if

θi < θ0

i.e.,sin θi < sin θ0

or

sin θi <√

η21 − η2

2.

Hence, the condition for light propagation is

sin θi < NA



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3.1.1 Fractional Index Change

The Fractional Index Change ∆ is the ratio of the refractive index difference between the core andcladding to the refractive index of the core. That is,

∆ =η1 − η2

η1

.

We derive the relation between the Fractional Index Change and Numerical Aaperture. We know that

NA =√

η21 − η2

2,

NA =√

(η1 + η2)(η1 − η2).

From the expression for ∆, we haveη1∆ = η1 − η2.

Therefore, the above equation takes the form

NA =√

(η1 + η2)η1∆.

For the case η1 ≈ η2, (η1 + η2) = 2η2. Therefore

NA =√

2η21∆,

orNA = η1

√2∆.

3.1.2 Modes of propagation and V-number

Even though it is expected that all such rays which enter into the core at an angle less than the angleof acceptance should travel in through the core, it is not so. The number of modes supported forpropagation in the fibre is determined by a parameter called V − number.

V =πd

λ

√η2

1 − η22

η0

,

where d is core diameter; η1 is the refractive index of the core ; η2 is the refractive index of the claddingand λ is the wavelength of the light propagating in the fibre.

If the surrounding medium is air, then the V−numberis given by

V =πd

λ

√η2

1 − η22,

or

V =πd

λNA.

For V � 1, the number of modes supported by the fibre is approximatelyV 2

2.



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3.1.3 Attenuation

Attenuation is the loss of power suffered by the signal as it propagates. In optical fibers, attenuationis the rate at which the signal light decreases in intensity. The attenuation coefficient α describes theextent to which the intensity of an energy beam is reduced as it passes through optical fiber. If aninput signal of power Pin after passing through cable of length L, reduces to a signal of strength Pout,then the attenuation coefficient α is given by,

α = −10

Llog1 0

(Pout

Pin

)dB/km.

The attenuation happens for the following three main reasons.

1. Atomic Absorption: The atoms of any material are capable of absorbing specific wavelengthsof light because of their electron orbital structure. This absorption can be observed if you lookinto the edge of a pane of glass. The light which emerges has a green colour because so much redand blue light have been absorbed by the atoms of the glass. In the same way, as light passesalong an optical fibre, more and more light is absorbed by the atoms as it continues on its path.

2. Scattering by Flaws and Impurities: This type of scattering is called Rayleigh Scattering.The amount of Rayleigh Scattering which takes place depends on the relative size of the scatteringparticle and the wavelength of the light. If the wavelength of the light is large compared to thesize of the scattering particle then little light is scattered. If the wavelength of the light is smallcompared to the scattering particles then a lot of light is scattered. So long wavelengths arepreferred in fibre optics because of the lower absorption. Thus 1500 nm is better than 1300 nmwhich is better than 850 nm.

3. Reflection by Splices and Connectors: In a long fibre cable there may be many spliceswhich join the individual lengths of fibre together. In a Local Area Network there will be manyconnectors because of the number of subscribers to the system. At each connector and/or splicesome light will be reflected back along the fibre in the opposite direction. This will happen evenfor the most perfect splice or connector. Light reflected backwards does not leave the fibre but isno longer usefully available for the rest of the fibre, i.e. it is no longer part of the ongoing light.

3.2 Types of optical fibers and modes of propagation

Optical fibres are categorized into two main groups based on the modes of propagation. We havedescribe them below with a diagram and step-index profile. For an optical fiber, a step-index profileis a curve showing the variation of refractive index with radial distance within the core and a sharpdecrease in refractive index at the core-cladding interface so that the cladding is of a lower refractiveindex.

3.2.1 Single mode fiber

A singlemode cable has a small core (3 to 10 microns where 9 microns is the most common) thatforces the light to follow a more linear single path down the cable, as opposed to the multipath reflections



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of multimode cables. The singlemode cable has the highest bandwidths and distance ratings. Single-mode fiber gives you a higher transmission rate and up to 50 times more distance than multimode,but it also costs more. Single-mode fiber has a much smaller core than multimode. The small coreand single light-wave virtually eliminate any distortion that could result from overlapping light pulses,providing the least signal attenuation and the highest transmission speeds of any fiber cable type.Single-mode optical fiber is an optical fiber in which only the lowest order bound mode can propagateat the wavelength of interest typically 1300 to 1320 nm.

3.2.2 Multimode fiber

A multimode cable has a relatively large diameter core (50 to 400 microns where 62.5 is the mostcommon one) and a total diameter of 125 microns. Multimode cables are available in two categories;these are graded index and step index. In a step index fibre, as modes reflect through the cable, somehave to travel further than others and in doing so the light pulse will spread. This is one disadvantage,which means the fibre has a lower bandwidth. The solution to this problem is graded index. In thesecables the refractive index reduces gradually from the cores centre towards the cladding. This meansthat a light beam travelling mainly in the centre of the cable. This means higher bandwidth and lowerattenuation.

3.3 Applications

• The simplest application of optical fibers is the transmission of light to locations otherwise hardto reach, for example, the bore of a dentist’s drill. Image transmission by optical fibers is widelyused in medical instruments for viewing inside the human body and for laser surgery, in facsimilesystems, in phototypesetting, in computer graphics, and in many other applications.

• Optical fibers are also being used in a wide variety of sensing devices, ranging from thermometersto gyroscopes. Fibers have also been developed to carry high-power laser beams for cutting anddrilling.

• Local area networks are another growing application for fiber optics. These systems connect manylocal subscribers to expensive centralized equipment such as computers and printers. This systemexpands the utilization of equipment and can easily accommodate new users on a network.

3.3.1 Point-point communication

Fiber-optic communication is a method of transmitting information from one place to another by send-ing pulses of light through an optical fiber. An optical communication system consists of a transmitter,



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which encodes a message into an optical signal, a channel, which carries the signal to its destination,and a receiver, which reproduces the message from the received optical signal.

Fiber-optic laser systems are being used in communications networks. Many long-haul fiber commu-nications networks for both transcontinental connections and, through undersea cables, internationalconnections are in operation. Optical fiber is used for the following advantages.

• Due to much lower attenuation and interference, optical fiber has large advantages over existingcopper wire in long-distance and high-demand applications.

• The information-carrying capacity of a signal increases with frequency, the use of laser light offersmany advantages

• One advantage of optical fiber systems is the long distances that can be maintained before signalrepeaters are needed to regenerate signals.

The main disadvantages of optic fibre communication are,

• Splicing is very difficult task and should be done only by skillful persons.

• Preventing the cable from bending and physical tampering.

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october 1, 2008 ...modern physics v h satheeshkumar department of physics sri bhagawan mahaveer jain...

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