LASER_U·3 Ligth Amplification

8/12/2019 LASER_U3 Ligth Amplification

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Lasers andApplications

Unit 3: Amplification of Light


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Unit 3 Amplification of Light 2

Contents

Introduction ........................................................................................ 3Learning outcomes ............................................................................ 3

Optical gain ......................................................................................... 4Population inversion.......................................................................... 7

Two-level system................................................................................................9Three-level system..............................................................................................9


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Introduction

The previous units introduced the concept of stimulated emission and therelationship between the Einstein A and B coefficients. In this unit weshall see how stimulated emission can lead to gain (optical amplification).We shall derive an equation that predicts either an exponential growth oran exponential decay of the radiation depending on a set of conditions.These conditions are incorporated into a parameter we call the gaincoefficient. We shall also talk about the concept of population inversion,and show that population inversion is a necessary condition for gain.Well discuss ways to achieve population inversion, and look at examples

of commercially available optical amplifiers.

Learning outcomes

After studying this unit you will be able to

derive an expression for the gain coefficient using the rate equations calculate the gain coefficient under various conditions calculate gain cross section

discuss the concept of population inversion, and calculate thepopulation differences given the required parameters

show that two level atoms cannot produce gain discuss the properties of three and four-level amplifiers, and be able to

describe devices based on these schemes discuss erbium doped fibre amplifiers.


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Optical gain

As stimulated emission produces an additional photon for eachstimulating photon, it is reasonable to assume that one could use this

process to amplify light.

We shall now derive a necessary condition for amplification that willintroduce us to the concept of population inversion. We shall do thisdifferently to what you will find in the textbook.

Assume a substance (solid, gas, liquid) with atoms that have two

energy levels, an upper level, Eu, and a lower energy level El, and thereis a total of

N = N1 + Nu

atoms in this medium.. The optical transition connecting these levelshas a lineshape function G(


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Well now derive an expression for the change of intensity as lighttravels through this material.

The change in intensity over a distance )z is equal to the number of

optical transitions per second times the photon energy:

)I = (dN/dt) h


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Using

Au1=

sp

t

1

where tspis the spontaneous lifetime, we get:

g(


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The other important component of the gain coefficient is the (Nu-Nl)term, which is, as we know, the difference in populations between theupper (u) and lower states (l). It is this term that will play a vital rolein the making of the laser.

Population inversion

Looking at the expression for the intensity:

I(z) = I(0)exp(gz)

we see there are two possibilities: gcan be negative, which means theintensity decaysexponentially as it passes through the medium:

I(z) = I0exp(-*g*z).

In such a case, g(


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The key to amplification is the positive sign of g. In the followingwell discuss ways to get positive g, that is, ways to make anoptical amplifier.

However, before we do that, lets define the overall gain, S( Nu, gis negative,the medium absorbs.

o So, it seems that to make an optical amplifier operating at

Nl condition, and bingo, we'll get more photons out thangoing in. The only small problem is that in thermalequilibrium, Nlis always larger than Nu. How can we be sosure? Well, Boltzmann's distribution provides us with theratio of Nland Nuin thermal equilibrium:

Nl/Nu = exp(Eu-El)/kT.

o According to this expression, Nl is always larger than Nu.

For example, at optical frequencies (>Nu. Thus, in thermalequilibrium, g is positive so matter will always absorblight! So how can we make an optical amplifier? How canwe beat the system?

o The important and slightly overlooked term above was inthermal equilibrium. Therefore, to invert the atomic

population from N1 > Nuto Nu> N1 we have to get awayfrom thermal equilibrium. We have to trick the system

into the Nu> N1situation, a condition called populationinversion. Population inversion is a necessary condition


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for light amplification (or laser action) to occur. However,well also see shortly that it is not a sufficient condition.Other conditions will also have to be met beforeamplification can be achieved.

o By the way, when Nu= N1the medium is neither absorbingnor amplifying. It is perfectly transparent. This is what wecall optical transparency.

Two-level system

So lets try to invert the population of an atomic system. To startwith, lets assume the simplest possibility, only two levels, E2andE1, and see whether we can achieve population inversion byoptically pumping the majority of electrons from E1to E2andthus inverting the population between these two levels.

The question we are asking is the following: Is it possible to shinelight of frequency


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Unit 3 Amplif ication of Light 10

--------------------- E3

--------------------- E2

---------------------- E1

Fig. 3.2 Schematics of a 3 level system used to achieve population inversionbetween levels 2 and 1. Level E1is the ground state of the system. In the figure,the arrow going up from level 1 to level 3 represents the pumping or excitation oflevel 3, while the downward arrows represent the relaxation between theselevels.

In order to optically excite electrons from E1to E3, we need a lightsource whose frequency is


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Unit 3 Amplif ication of Ligh t 11

large), nearly all of the excited atoms from level 3 will arrivequickly to level 2, and only few will relax back to level 1. But thisis not enough to have an inverted population. We have to makesure that the population of level 2 is higher than that of level 1. Inan efficient laser medium, the lifetime of state 2 is long so thatthe population of this state can grow and an inversion can becreated with respect to level 1. That is, to achieve populationinversion between levels 2 and 1, we need to have level 2 to be aslow state. Levels with long lifetimes are called metastablestates. If the parameters are right and we pump hard enough, wecan arrive at a situation where we have more atoms in level 2 thanin level 1. And this situation can be sustained by continuous

pumping. That is, we have achieved a steady state invertedpopulation between levels 2 and 1.

E3 ----------------------- short-lived level

t32 (rapid decay)

E2 ----------------------- long-lived (metastable) level

t31(slow) t21(slow)

E1 ----------------------- ground state

Fig. 3.3 Schematics of the relative lifetimes of a 3 level system used toachieve population inversion between levels 2 and 1.

In terms of lifetimes, the following conditions have to be met to achievepopulation inversion in a three-level system (see Fig 3.3):

1. the spontaneous lifetime of level E2(t21) be large relative to theother lifetimes (i.e. E2is a metastable state)

2. The decay from level 3 to level 2 should be highly probable, whichmeans that t32is short relative to t31.

Under such conditions, a 3 level system may be inverted! In fact, the firstlaser built by Theodore Maiman was a 3 level system, the ruby laser, inwhich t21.10

-3s and t32.10-12s.


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Unit 3 Amplif ication of Light 12

The problem with three-level systems is that the lower stateof the lasing transition is the ground state. The ground state

is always very heavily populated (it is the lowest possiblestate), so it takes a lot of energy to invert the population ofother states relative to the ground state. This is why three-level systems are inefficient, and are rarely used (the rubylaser, which is a three-level laser, is a peculiarity).

To improve the pumping efficiency, it is more efficient touse systems where the lower state of the lasing transition innot the ground state but another excited state. This can beachieved in the so-called four-level systems.

Four-level system

In a four-level system, the final state of the lasing transition is an excitedstate, not the ground state, as shown in the diagram below:

---------------------- E3. t32 (rapid decay)

------------- E2

------------------------- E0 (ground state)

Fig. 3.5 Energy level diagram of a four-level system. Population inversion can beachieved between levels E2and E1.

The population inversion now occurs between levels E2and E1(asshown on the diagram above). In a four-level system, thenecessary condition for population inversion between levels 2 and1 (N2 > N1) entails that t10and t32are much shorter than t21. That is,inan efficient laser medium, the lifetime of state 2 is long so

that the population of this state can grow and an inversion canbe created with respect to state 1.

pumping(E2 metastable state )

E1 (Final state ofamplifier)

t10(rapid decay)


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Unit 3 Amplif ication of Ligh t 13

For example, in the Nd:YAG amplifier /laser which is a four-levelsystem, t32.10

-7s, t10.10-8s and t21.10

-3s. Since four-level devicesare more efficient than three-level devices, most practical lasers are

based around a four-level amplifier.

Examples of commercially available amplifiers based on four-levelschemes:

1. Nd:glass system: This is a typical four level system. The energylevels, which are involved with the amplification process, are thelevels of the Nd3+ion. The glass does not take part directly in thelaser action, its only role is to hold the Nd ions in place. Wecould, and do, incorporate Nd into other hosts, such as yttrium

aluminium garnet (YAG), a transparent crystalline material, andother materials to make useful lasers. In the Nd:glass systemamplification occurs at 8=1.06 :m, in the near infrared spectralregion. The energy difference between E1and the ground state (E0)is quite large (0.26 eV), so at room temperature the population ofthe level E1is small compared to the ground state and thereforepopulation inversion can be achieved at relative ease. Opticalpumping occurs between E0and E3where E3is, in fact, a band ofenergy levels, making it easier and more efficient to pump intothem. Since the energy difference E0-E3lies in the red/near IRspectral region, there are a number of convenient light sources,which can be used for optical excitation (pumping) purposes, suchas various flash lamps or diode lasers.

2. Fibre optic amplifiers: These amplifiers are based on the opticaltransitions of the Er3+(erbium) ion. The optical fibre, made ofSiOx, is the host material that contains the Er dopants.Amplification takes place around 8=1.55:m, which coincides withthe optimum wavelength of modern fibre optic communicationssystems and hence the great interest in these type of fibreamplifiers. Fibre amplifiers can be pumped at several wavelengths

but most commercial systems are pumped in the near infrared(980nm or 1,48:m) by diode lasers. Due to the nature of theoptical fibre, fibre optic amplifiers are not pumped from the side,as we saw with in the ruby laser, but instead, we make use of theintricacies of the fibre itself to couple light into it. We shall learnmore about erbium doped fibre amplifiers in unit 10.

LASER_U·3 Ligth Amplification

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