Stimulated Emission Devices LASERS - OREA 2011

Stimulated Emission Devices LASERS 

Light Amplification by Stimulated Emission of Radiation 

Erasmus Intensive Program OREA 2011 

4 – 15 July 2011, Department of Electronics 

TEI of Crete, Chania, Crete, Greece 

Dr Kostantinos Petridis


An Introduction – Fifty Brilliant Years




An Introduction – Fifty Brilliant Years 

• Lasers: An invention that many of us take it as granted because we have 

grown up with, and has always been a part of our everyday lives. 

• The development of lasers has changed science forever and its various 

forms continue to have considerable impact in many areas of science, 

technology and industry. 

• 50 th anniversary of the laser invention (16 th of May 1960).


In 1958 A.L.Schalow and C.H. 

Townes published the first theoretical 

work concerning laser operational 

principles. 

They expanded their research from 

microwave frequencies operated 

devices (masers) to optical frequency 

operated devices (lasers) 

The consequent development of the 

laser changed science forever. 

Arthur L. Schawlow Semiconductor lasers have 

revolutionized telecoms. Nd:YAG 

lasers have changed the way we 

perform material processing. Excimer 

lasers drive the semiconductor 

industry. Ti:Sapphire lasers enable 

ultrafast studies of matter 

Charles H. Townes 

Both scientists awarded with the 

Nobel prize for their work. 

7 7/3/11


In 1960 Maiman build the 

first operational laser device. 

It was a ruby laser 

In 1984 he awarded with the 

Nobel prize. 

8 7/3/11



• Schalow & Townes back in 1958 : “The laser is a solution in search of problem” 

• Lasers have gone on to be one of the outstanding success stories in physics 

• Federico Capasso (inventor of QCL’s): “Laser is a solution of many problems” 

• Lasers were changing how we live, sometimes in ways so dramatic that one might 

ask, which is truth and which is the fiction? 

• The impact of the laser on many areas of research is very evident, making it one of 

the greatest achievements in modern science. 

• The use of lasers has become widespread not only in commercial applications but 

also in areas ranging from fundamental physics to chemistry, biology and medical 

research. 

• The laser technology is still rejuvenated in comparison with other great inventions. 

1 Maiman T.H., Nature 187, 493 – 494 (1960)







• It was an idea that there was on human’s mind even before its invention 

• Even after the construction of the laser this technology immediately hosted by the 

Hollywood movie producers (see Goldfinger 1964)



• Lasers have demonstrated a versatility & capability for constant innovation that surpasses many 

other main scientific discoveries: 

(a) Matter Lasers in Bose – Einstein condensates 1 

(b) The phonon laser 2 & nanolasers (10 nm thick) 6 

(c) Attosecond 3 & zeptosecond 4 laser pulses push the extremes of physics 

(d) New types of lasers: “Anything will lase if pumped hard enough”: Organic lasers (Brock 

et.al. 1961) & Single cell biological lasers !!! 

(e) High power lasers (ELI – 200 petawatts – attosecond pulses). 

(f) Laser facilities as PETAL that will drive the European effort to produce energy through laser 

fusion 5 . 

• 14 physics Nobel prizes have been awarded for achievements directly related or linked to 

lasers. 

1 Nature Materials 9, 374 – 375 (2010) 

2 Nature Physics 5, 682 – 686 (2009) 

3 Journal of Physics: Conference Series 194 (2009) 

7 http://www.nature.com/milestones/photons/ 

8 http://www.hiper-laser.org 

9 Science vol 328 14 May 2010



• Lasers in BioPhysics: study of molecules of life such us the proteins, nucleic 

acids, carbonhydrates and other chemicals – see optical tweezers!! 

• High energy laser weapons: ABL project (equipped with a megawatt chemical 

oxygen – iodine laser. The new version will use diode pumped alkali metal 

vapours) (see video) 

• Solid state lasers (CW or pulsed operation 100 kW) can neutralize IED’s 

from a safe distance. 

• Free electron lasers: use special relativity to provide tunable electromagnetic 

radiation from a relativistic free electrons as they move through a transverse 

spatially periodic applied magnetic field. (see video)



• Lasers ( 5- 15 W) in astronomy in combination with adaptive optics



• The ABL project: The HiPER project:



• The free electron laser:



1) O. Svelto: Principles of Lasers 

2) Siegman: Lasers 

3) William T. Silfvast: Laser Fundamentals 

Historical Review 

1) C.H. Townes: “How the laser happened” 

2) M. Bertollotti: “The history of the laser” 

18 7/3/11



The present state of the art includes: 

(a) Peak powers > 10 12 W 

(b) Pulses shorter than 10 -15 sec



• Laser light unique properties: A laser is a specialized light source that should be used only 

when its unique properties are required. 

(a) Monochromatic light: (b) Spatial and temporal coherence: 

(c) Directional light: (d) High intensity light:

• Monochromatic: 



Do not forget: Almost monochromatic broadening homogeneous & inhomogeneous 

The spectral linewidth of a laser beam it will be much narrower than the atomic 

transition. This is because the emission is affected by the optical cavity. 

• Coherence: 

Spatial coherence refers to whether there are irregularities in the optical phase in a 

cross sectional slice of the beam. 

Temporal coherence refers to the time duration over which the phase of the beam is 

well defined. 

t c = 1 

Δν

• Directionality: 



The light comes out as a highly directional beam. 

• Brightness: 

The power per unit area is very high 

• Ultrashort Pulse Generation: 

The power duration of a laser pulse is defined by the uncertainty principle: 

t p ∝ 1 

Δν



10 -21 

Nuclear 

Dynamics 

10-18 Atomic 

Dynamics 

10 -15 

Molecular 

Dynamics 



10 -10 

Photon 

Creation 

10 -6 

Brown 

Motion 

Seconds 

10 -3 

10 -1 

Acoustics 

Earthquakes 

109 Human Life 10 16 

Star Life 

10 

Universe Life 

18



• Gain Medium ( something to amplify the light) 

• Resonator (something to provide suitable optical feedback) 

• Pump Source (something to create the population inversion) 

Pump Source 

100% Reflector Gain Medium 98% Reflector 

Output beam 

25 7/3/11


An Introduction – Fifty brilliant years 

• The device consists of three essential elements: 

(a) External energy source or pump (b) A gain medium (c) An optical cavity or resonator 

• The pump is an external source that produces a population inversion in the laser gain 

medium. 

• Pumps could be: (1) Electrical (diode lasers), (2) chemical, (3) thermal , (4) optical (solid 

and liquid lasers) 

• The participating energy levels in the gain medium determine the wavelength of the laser 

radiation. Only in gases they have been discovered 1000 laser transitions. 

• Gain medium could be solid, semiconductor crystal, gas or liquid. 

• The most important requirement of the amplifying medium is its ability to support a 

population inversion between two energy levels of the laser atoms. 

• Due to different lifetimes of the available atomic energy levels only certain pairs of 

energy levels with appropriate spontaneous lifetimes can be inverted even with vigorous 

pumping.

• The various interactions that occur inside the atom has as a result the creation of the following energy 

states: 

Gross structure effects (1 – 10 eV) 

a) the kinetic energy of the electrons in their orbits around the nucleus 

b) the attractive electrostatic potential between the positive nucleus and the negative 

electrons 

c) the repulsive electrostatic interaction between the different electrons in the multi 

- electron atom 

Fine structure effects (0.001 – 0.01 eV): The spectral lines reveals that often coma as multiplets. 

a) Spin – orbit interaction 

Hyperfine structure effects (10 -6 – 10 -5 eV): Fine structure lines are also split into more multiplets. 

This splitting is due to interaction between the nucleus (spin of nucleus) and the electrons magnetic 

moment due to its orbital motion. 


Gain medium – Energy Levels


Gain medium – Energy Levels 

• Atoms are composed by charged particles so interact with E/M radiation. 

• The atoms and the molecules that constitute matter have discrete energy 

levels (electronic, vibrational, rotational).


Gain Medium – Energy Levels 

• Laser light is the manifestation of a particular interaction between charged particles and E/ 

M fields: An understanding of this necessitates a study of the quantization of energies of E/ 

M fields and atoms. 

• The energy of E/M radiation of frequency ν is quantized in units of hν which are called 

photons. 

• The total energy stored in an E/M field of frequency ν is given by: 

E = 1 

hν + nhν 

2 

where the first right term is energy that corresponds to the electromagnetic vacuum.



Bohr’s theory tried to explain the discrete emission lines of hydrogen spectrum: 

1. The hydrogen atom included a positively charged nucleus (proton) and a negatively charged electron 

orbiting in a circular motion around the nucleus. 

2. The electron could temporarily remain in a particular orbit if its angular momentum is a integral 

multiple of 

3. Radiation is emitted from the atom when the electron jumps from a higher energy to a lower 

energy level 

1. When a radiation is emitted its frequency can be determined by the Einstein’s equation: 

hν 21 = E 2 − E 1 

30 7/3/11



De Broglie suggested that an electron should have wavelike properties. Thus an electron 

is characterised by a wavelength λ e and is rotating in orbits whose radius is an integer 

multiple of the electron’s wavelength. 

λ e = h 

p e 

The double slit experiment showed us that the electrons are not oscillating in the classical 

trajectories that Bohr theory suggested earlier. There is an uncertainty where the electrons 

are. 

Heisenberg’s Uncertainty Principles: 

31 7/3/11



The matter consists from atoms around which the electrons move (Bohr model). 

The matter stability reveals that a) the electrons energy can take only specific 

values (quantized energy levels) and b) they move across specific orbits that 

each one have a specific energy. 

According to Schrodinger, Dirac and Heisenberg the electrons are particles 

with some energy and are accompanied by a probability wave function that 

describes the probability the electron to be in a specific orbit (eigenstate) . 

Each eigenstate is determined fully by three quantum numbers n, l, m 

32 7/3/11



The quantum mechanics provides the appropriate probability distribution functions 

for the quantities of position, momentum, energy and time. 

Wave functions Ψ are used in order to describe various atomic parameters including 

the electron’s properties in an atom. 

The wave function by it self has no physical meaning. Multiplied by its conjugate 

function and a variable under consideration (e.g. position or momentum or energy) 

provides the probability distribution function associated with these variables. 

The insertion of the term of the wave-function agrees with the uncertainty principles 

and the wavelike properties of the electrons. 

33 7/3/11



The time independent Schrodinger equation: 

The solutions of ψ of the Schrodinger equation are known as eigenfunctions. 

The time dependent Schrodinger equation describes the non stationary functions 

and has the following form: 

34 7/3/11



Two kind of quantum states exist: The stationary and the coherent states. 

Each quantum state is related with an energy E n 

The stationary states are described by the Bohr theory. Each stationary energy 

state is described by its wave-function Ψ n that informs for the probability the 

electron to be found on a specific specific orbit around the nucleus. 

The time dependence of a wave function is of the form of 

The wave – function of a stationary state has the following form: 

35 7/3/11



When the electron is in mode from changing from one specific state to lower state, the 

wave function must incorporate both the initial state and the final state. Thus, 

The probability for the electron to be in this coherent state can be represented by 

Ψ * * * * * * 

Ψ = C1 C1ψ 1ψ1 

+ C2C2ψ 2ψ 

2 + C1 C2ψ 1ψ 2e −iω 21t * 

+ C2C1ψ 1ψ 2e −iω 21t The solution of the Schrodinger equation allows to connect the energy of a coherent 

state with the corresponding wave – function. 

36 7/3/11



How strong two eigenstates are related depends on the interaction among them. That 

interaction involves the computation of the electric dipole matrix element M if . 

∞ 

Mif = ∫ 

* 

Ψf erΨidV −∞ 

The above integral is calculated over the entire volume V associated with the radiating 

atom. The factor er represents the electric dipole moment of the electron – proton system. 

The transition probability λ if depends on how well the two eigenstates interact 

(overlap) and on the number of various ways the transition can be occurred (is expressed 

by the density of the final eigenstates ρ f ). 

37 7/3/11


Gain Medium – Rate equations


Gain Medium – Rate equations 

• Rate equations relate the population densities to the properties of the 

electromagnetic field that interacts with the gain medium. 

• Rate equations together with equations representing the effect of a cavity on 

an E/M field, are used to develop a relation that predicts the output irradiance 

as a function of the pump, gain medium and cavity that comprises the laser 

system. 

• The Einstein coefficients are characteristic of the two energy states.



• Absorption 

An incoming photon can be absorbed by the system and an electron will be excited from 

the lower to the upper state. 

The frequency of the photon should be such that 

The rate at which the electrons occupy state 2 due to absorption of ‘suitable’ photon is 

dN 1 

dt = B 12ρ(ν)N 1 

where ρ(ν) is the photon energy density at frequency ν 

Β 12 is the Einstein coefficient for stimulated absorption. 

v 12 = E 2 − E 1 

h 

40 7/3/11


Gain Medium - Rate equations



• Stimulated Emission 

An incoming ‘suitable’ photon can cause an electron in the upper state too relax and an 

additional photon with the same frequency and phase will be emitted. 

The stimulated emission flow of electrons from state 2 to state 1 is given by 

dN 2 

dt = −B 21ρ ν 

( )N 2 

where B 21 is the Einstein coefficient of stimulated emission. 

The stimulated emission is the key to the laser action but not the only action!!!. 

43 7/3/11



Under thermodynamic equilibrium the population of the atomic energy levels is 

determined by the Boltzmann distribution, 

N 2 

N 1 

= g 2 

g 1 

k is known as the Boltzmann constant and its value is equal to 1.38×10 -23 joule/Kelvin 

where g 1,2 are the degeneracies of state 1 and 2 

exp − E ⎡ ⎛ 2 − E1⎞ ⎤ 

⎢ 

⎣ ⎝ kT ⎠ ⎥ 

⎦ 

44 7/3/11



In thermal equilibrium the rate of change of upper and lower populations must be zero i.e. 

dN1 dt = dN2 dt 

Using the rates of the absorption, spontaneous and stimulated emission for the upper 

and lower energy levels in equilibrium we get 

ρ( ν) 

= 

A 2 

B12 B21 = 0 

B 21 

N 1 

N 2 

−1 

46 7/3/11



Replacing the Boltzmann distribution and equating the result with the density of 

states of the Blackbody we get, 

Conclusions: 

A 2 

B 21 

= 8πn 3 hν 3 

c 3 ⇒ A2 B21ρ ν 

g 1 B 12 = g 2 B 21 

(a) In order to increase substantially the stimulated emission over the spontaneous 

the emitted photons should be limited to a very fine spectrum region: increase 

( ) 

⎛ hν⎞ 

= exp 

⎝ kT⎠ 

−1 

the spectral density energy. Contain the medium within a cavity!! 

(b) Increase the population density of the upper energy 

47 7/3/11



• Conclusions: 

(a) For stimulated emission in order to overcome photon absorption we 

need to achieve population inversion . 

(b) For stimulated emission in order to exceed spontaneous emission 

we must have large photon concentration which is achieved by 

building an optical cavity to contain photons. 

• Population inversion means that we depart from the situation of the 

thermal equilibrium. So the laser principle is based on non-thermal 

equilibrium.


Gain medium – Small Gain Coefficient

The photons that are emitted spontaneously are incoherent to the incoming radiation and 

therefore they do not give useful output. The rate of change of the photons number is given 

by 


Gain Medium – Small Gain Coefficient 

dD ν 

dt 

= ρ ν ( )B21 N2 − N [ ] 1 

The change of photons due to the processes of the absorption and stimulated emission is: 

ΔD ν = 

ΔD ν 

dt 

( ) − ρ( x) 

ρ x + dx 

hν 

= −aρ ν 

c 

( ) 

n 

1 

hν 

= dI ν 

dx 

Δx n 

hν c = −aIνΔt 1 

hν ⇔ 

52 7/3/11


Gain Medium – Small Gain Coefficient 

So the small gain or the absorption coefficient can be evaluated thought: 

Conclusion: 

a = B 21hνn 

c 

[ N2 − N ] 1 

(a) Population inversion should be established 

(b) A pumping source is needed 

(c) Gain > Loss


Gain Medium – Pumping Source 

In order to amplify the incident radiation population inversion is required . 

To overcome the thermal distribution of the energy states we create population 

inversion by pumping. 

The various types of pumping are: 

(a) Optical Pumping 

(b) Pumping through the energy that an exothermic chemical reaction releases 

(c) Electrical discharge 

(d) Electrical pumping 

54 7/3/11


Gain Medium – Laser Energy Systems 

Impossible to obtain population inversion with a two energy system !!! Why?


Gain Medium – Laser energy systems 

Transition a: The pump transfers population from the ground state to the higher 

energy level 

Transition b: The excited population through non radiative decay from energy 

level 3 to energy level 2. The lifetime of 3 is very short and all the 

population in state 3 decays to state 2. 

Transition c: Stimulated emission from state 2 to state 1. The lifetime of energy 

state 2 is long so population inversion to be achieved with respect to 

state 1. Once an inversion is obtained, stimulated emission will give 

optical gain 

56 7/3/11



In order to achieve population inversion in a three level laser system we need to 

pump at least the half population of the ground state to the upper laser energy level 

(level 2). 

This means that 

N 2 ≥ N tot 

2 

The minimum rate at which the active medium should be pumped is 

The main disadvantage of the three laser level system is that requires at least the half 

population of the ground state to be excited. 

R th = N tot 

2 

57 7/3/11 

1 

τ 2



4 Energy Level



The terminal state of the laser transition in a four level laser system is not the 

ground state and so is easier, compared to a three level system, to maintain the 

population inversion. 

It is easily observable that population inversion between states 1 and 2 can be 

obtained even for small populations in state 2. 

The required level of pumping is: 

R th = N th 

1 

τ 2 

59 7/3/11


Gain Medium - Threshold Point

Considering that all the losses are included into a coefficient γ the single round trip gain through 

the resonator and gain media is: 

where r 1,2 are the reflectivities of the cavity mirrors and L is the length of the gain medium. 

The parameter γ includes all the losses except the transmittance of the mirrors. 

• Loss sources: 

(a) Transparency of the cavity mirrors 

(b) Absorption and scattering in the gain medium 

(c) Diffraction 

At threshold, where G ain equals unity, the small gain coefficient can be calculated, using the 

above equation , as follows 


Gain Medium – Threshold Point 

Gain = r1r2 exp{ 2(kgain − γ )L} 

k th = γ − 1 

2L ln(r 1r 2) 

61 7/3/11



The exponential gain of a laser beam is 

The exponential loss is defined as: 

Where t c is the loss cavity time. The time t is related to the length that the beam 

propagates through the cavity as follows, 

At the threshold the gain equals unity and so: 

k tht c 

I = Io exp(kx) 

I = Io exp(− t 

) 

τ c 

c 

n = 1 ⇒ k th = n 

ct c 

62 7/3/11



By manipulating the previous relations an expression for the threshold population 

inversion can be extracted: 

ΔNth = ( N2 − N ) 1 = 

where ΔΝ th is the inversion population at threshold. 

1 

B 21 hνt c 

= 8πn 3 ν 2 τ 2 

c 3 t c 

For some lasers the pump power required to attain a threshold can only be achieved for 

short time. These lasers are pulsed lasers. 

If the inversion can be maintained indefinitely the output is continuous, Such lasers are 

called CW lasers. 

63 7/3/11


Gain Medium - Saturation 

If in the rate equations of the upper and lower laser levels include the effect of pumping 

through the pumping rate R the population inversion under thermal equilibrium is : 

ΔN = N 2 − N 1 = 

R 1− τ ⎛ ⎞ l 

⎜ ⎟ 

⎝ τ u ⎠ 

Iν 1 

Bul + 

c τ u 

As now the beam travels through a long length gain medium the intensity of the beam 

according to the Beer law will increase continuously. This means that the left factor of the 

denominator will continuously take higher and higher values. At a specific length 

(saturation length L sat ) it will reach a value that is comparable to the value of the right 

denominators’ parameter. At this distance from the beginning the beams’ intensity is equal 

to I sat (saturation intensity). 65 7/3/11 

≈ 

B ul 

I ν 

c 

R 

+ 1 

τ u



• When saturation occurs the upper energy level population would decrease by a factor of 

two to a value: 

• From Beer’s law we know that the exponential is analogous to the population inversion. 

So at the saturation length the intensity of the beam will be reduced since the population 

inversion reduced. 

• Further increase in I would further decrease the gain in regions where the beam 

propagates further into the medium!! 

• As the saturation intensity we define the beam’s intensity where the stimulated emission 

rate becomes equal to the spontaneous emission rate, so: 

I sat = 

N u = R uτ u 

2 

c 

nB ul (ν)τ u 

66 7/3/11



67 7/3/11


Gain medium – Saturation


Gain medium – Spectral Broadening 

• The energy levels of atoms is not exactly discrete but they have a finite energy width 

ΔΕ. This width arises from the interaction of the atom with its environment. 

• Typical value of the energy level width is 10 -7 eV. 

• This energy width relaxes the restriction for a photon in order to interact with an 

electron should have a frequency that matches exactly the energy difference between 

these energy levels. 

• So in order to interact a photon should have an energy within the range: 

hν = E n − E m − ΔE n + ΔE ( m ) 

2 

to hν = E n − E m + ΔE n + ΔE m 

( ) 

2



• From the last relationship is easily can be extracted that: 

ν = E n − E m 

h 

± ΔE ( n + ΔE m ) 

2 

= ν o ± Δν 

2 

• The likelihood of the atom with energy levels E n and E m interacting with a photon of 

frequency ν is proportional to a lineshape function g(ν). It is valid that: 

∞ 

∫ g( ν) 

dν = 1 

−∞


Gain medium – Spectral Broadening

In a homogeneously broadened laser all the 

atoms contribute to the gain at all frequencies. As 

the population inversion is reduced the gain is 

reduced at all frequencies. 

The laser will oscillate at the frequency 

corresponding to the longitudinal mode closest 

to the gain maximum. When the steady state is 

reached only one mode will oscillate. 

Homogeneously broadened lasers oscillate with a 

single longitudinal mode. 



72 7/3/11



In a inhomogeneously broadened 

laser transition the gain saturation is 

stronger near the wavelength of the 

laser beam that is amplified 

through the laser gain medium. 

73 7/3/11


Gain medium – Spectral Broadening



75 7/3/11

From equation is easily observed that in order to increase the stimulated emission rate 

we need to feedback into the gain medium some of the previously emitted light e.g. 

increase ρ ν . 


Optical Resonator 

In order to provide this feedback the laser gain medium is enclosed into a laser cavity. 

76 7/3/11



Usually in the design of laser cavities curved mirrors are used instead of plane mirrors 

since the former laser cavities are more stable. 

77 7/3/11

What are the conditions for a) mirror curvature and b) mirror separation for a stable cavity? 

The condition, known as the stability criterion, that should be satisfied for a stable laser 

cavity is the following: 



0 ≤ g g ≤ 1 ⇒ 0 ≤ 1− 1 2 d ⎛ ⎞ 

⎜ ⎟ 1− 

⎝ ⎠ 

d ⎛ ⎞ 

⎜ ⎟ ≤ 1 

⎝ ⎠ 

where R1,2 are the radii of curvature of the cavity mirrors, d is the length of the laser 

cavity and f1,2 are the focal lengths of the cavity mirrors. 

R 1 

R 2 

79 7/3/11



80 7/3/11

• Why Gaussian Beams? 

The light propagation within an cavity corresponds to a propagation 

through apertures that are in a distance d. The diameter of each aperture equals 

the mirror diameter. The beam propagation includes diffraction effects every time 

the beam incidents on a mirror. These diffraction affects add each other. 

Thus the form of the initial plane wave is altered and the initial intensity drops 

off due to the diffraction. 

Using the Fresnel – Kirchoff theory results that the above behavior is described 

by Gaussian shape beams. 

The beam maintains its Gaussian shape even during its propagation outside of the 

resonator. 


Optical Resonator – Stable Resonators & Gaussian Beams

A wave with a transverse Gaussian intensity propagates in a different fashion to a 

spherical beam. Two are the key parameters a) the beam size and b) the wave-front 

curvature. 


Optical Resonator – Stable Resonators & Gaussian Beams 

82 7/3/11



The position that the beam has got the minimum transverse half width, w = w o , is 

called the Beam waist. 

The half width of a Gaussian beam across the path of propagation can be calculated 

through the relationship 

w(z) = wo 1 + λz ⎛ 

⎜ 

⎝ 

2 

πwo where the distance z r is called the Rayleigh range where the beam waist equals the 

⎞ 

⎟ 

⎠ 

2 

83 7/3/11



Other useful relationships that describe the parameters of a Gaussian beam are 

Radius of curvature R w : 

Rayleigh range z r : 

Far field divergence angle θ: 

Rw ( z) 

= z 1 + z 2 

r 

z 2 

⎛ ⎞ 

⎜ ⎟ 

⎝ ⎠ 

zr = πw 2 

on 

λo θ = 

λ o 

πw o n 

84 7/3/11


Optical Resonator – Stable Resonators & Gaussian 

Beams 

For stable cavity configuration, the mirrors should be placed in a such way 

that they reflect the Gaussian wave-front without changing the beam 

parameter. 

The curvature of the mirrors should fit with the wavefront curvature. In 

other words 

The mirror curvature must match with the mode curvature!! 

85 7/3/11

Within a resonator they do not oscillate all the frequencies emitted from the 

transitions in the laser gain medium. 

The reflections back and forth result in a standing wave field to be set up within the 

resonator. The wavelength of the standing wave has to ‘fit’ exactly within the cavity, 

e.g. 


Optical Resonator - Longitudinal Modes 

m λ 

= L 

2 

86 7/3/11



The allowed cavity frequencies are defined from equation and are 

The frequency spacing between adjacent longitudinal modes of a cavity, called the 

Free Spectral Range (FSR) is 

ν m = mc 

2L 

Δν = 

c 

2nL 

87 7/3/11



88 7/3/11



The various mechanisms of broadening mean that the gain medium will give 

optical gain over continuous range of frequencies. However, the resonator will only 

provide feedback at the cavity mode frequencies. 

The output will be at one or more of the specific frequencies dictated by the cavity 

modes within the gain profile of the laser. 

A laser in which only one longitudinal mode oscillates is called SLM laser. 

89 7/3/11



90 7/3/11


Optical Resonator - Longitudinal Modes

The intensity distribution across the width of the cavity is the Transverse Mode pattern. 

The transverse mode determines the beam shape. The transverse mode type depends on 

the cavity dimensions, reflector sizes, and other limiting apertures that may present in the 

cavity. 

The general form for the transverse electric modes is: TEM pq where p is the number on 

nodes in x- direction and q the respectively number of nodes in y direction. 

The proffered transverse mode that oscillates within the cavity depends and on: 

the radial dependence of the gain. 


Optical Resonator - Transverse Modes 

A single transverse mode laser is restricted to give TEM 00 output. 

92 7/3/11


Optical Resonator - Transverse Modes


Optical Resonator - Transverse Modes 

94 7/3/11


Stimulated Emission and Photon Amplification


Pulsed Operation 

• Pulsed Laser Operation: 

(a) Observation of fast inter - atomic processes (lifetime of an excited state) 

(b) High Intensities (NIF & HiPER projects)


Pulsed Operation – Q Switching 

• Q –Switching is a technique to produce laser pulses. 

• There are two types of Q-Switching: Passive & Active Q-Switching 

Initially: 

Finally:


Pulsed Operation – Q Switching Configurations 

(a) Cavity dumping configuration: 

(b) Pocket Cell configuration:


Pulsed Operation – Q Switching Configurations 

• Acousto – optic modulator: 

• Passive Q- Swiching:


Pulsed Operation – Q Switching


Pulsed Operation – Q Switching


Pulsed Laser devices – Beyond ultrafast


Pulsed Laser devices – Beyond ultrafast 

• Laser systems have steadily improved due to progress to material science 

and mirror technology. 

• Incredibly powerful lasers have been developed or are under development 

for laser fusion and other lasers with ultra - short pulses (attoseconds 10 -18 

sec) are constructing in order to monitor fast procedures within the atom!! 

• Attosecond technology can be used in order to monitor the charge carrier 

movement inside a photovoltaic cell or a transistor. This will aid to the 

improvement of the photovoltaic cell efficiency and the transistor speed.



• The pulsed lasers in comparison with the continuous wave systems emit 

over a broad range of frequencies – billions in fact. These frequencies are 

timed exactly so that their electric fields nearly cancel each other except for 

during one tiny period of time when they constructively interfere.





• The frequencies of light that exist in a laser’s resonant cavity are determined by: 

(a) Lasing transitions within the laser medium: is selected a medium with many 

different lasing transitions. 

From the Heisenberg’s energy time uncertainty principle, a broad band width 

of laser energies or frequencies is required to produce a short pulse: the greater 

the bandwidth, the shorter the pulse. 

(b) The cavity dimensions only allow light frequencies for which the electric 

field has nodes at the cavity’s end mirrors.





• Since the pulsed lasers are alternate between a a short burst of light, known as 

pulse, and a much longer downtime in between, pulsed lasers can produce high 

peak powers (10 12 watts) : since a fixed amount of energy can be emitted 

within 10 -18 s 

• Very fast stroboscopic photography (attosecond) 

• Pulsed laser technology is hand to hand with non-linear optics. 

• Pulsed operation techniques: Mode locking & Q – Switching 

• In a pulsed laser system light appears at the output as pulses separated by the 

cavity round trip.





• Shorter pulses means shorter wavelengths!! 

• Light pulses can not be shorter than one oscillation of the 

carrier electric field. At 750 nm the shortest pulse has duration 

equal to 2.5 fs and at XUV (12 nm) the pulse duration is 

smaller than 100 attosecond!! 

• In order to produce high power laser pulses the latter must be 

amplified from the nanojoules to milijoule level: Chirped 

Pulse Amplification




Pulsed Laser devices – Attosecond Science 

From 1964 laser pulse duration has been 

decreased by about three orders of 

magnitude.



• Attosecond science arises from research of the early 1990s into intense ultrashort – pulse atomic 

physics. 

• There are three important characteristics of the current generation of attosecond technology that imply 

new directions in science: 

(a) The technology of the electron physics can be integrated with optical 

technology. Spatial resolution of an electron is of the order of 1 A. 

(b) Attosecond photon or electron pulses are accompanied by a synchronized 

visible pulse of control waveform. Thus attosecond technology extends 

conventional ultrafast spectroscopy. 

(c) Attosecond photon or electron pulses have energies of 10 eV to 1 keV or beyond. Core level and 

multi – electron dynamics or even nuclear dynamics can be time – resolved.



Step 1: (see figure a) Ionization induced by 

strong NIR laser field. 

Step 2: (see figure b) An electron will gain 

kinetic energy equal to 50 – 1000 eV from the 

field during its first femtosecond of freedom. 

The kinetic energy of the electrons will determine 

the maximum photon energy in the emitted 

attosecond photon pulse. The gained kinetic 

energy is proportional to laser wavelength and the 

laser electric field amplitude. 

Step 3: (see figure e) Recollision. On re- 

encountering the core, the electron scatters from 

its parent ion. In addition , the recollision electron 

may give rise to the emission of an attosecond 

burst



• Controlling the shape and the wavefrom of the optical pulse controls the 

attosecond pulse. 

• The above effect is depicted below:

Stimulated Emission Devices LASERS - OREA 2011

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