LN1_s

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Interaction of light with matter

Basicprocesses (A.Einstein,1916)E2

E1

h = E2 - E1

E2

E1

h = E2 - E1

Absorption Spontaneousemission transitionprobabilityA21 random phaseanddirection

Stimulatedemission

E2

E1

h = E2 - E1h = E2 - E1

transitionprobabilityB21 hasthesame frequencyand

phase asthe incidentlight lightamplification

1Laser fundamentals

transitionprobabilityB12

2

An ensemble of (two-level) atoms in equilibrium with black-body radiation at temperatute T:

kTh

kTE

kTE

ee

e

N

N //

/

1

2

1

2

==

when E2 >E1 N2

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3

B12=B21 and N2

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5

4-level laser

dt

dN

dt

dN

dt

dN

dt

dN

NNdt

dN

NN

dt

dN

NNdt

dN

3210

33203

3322212

2211101

++=

=

+=

+=

0

3

2

1

32

21

10

30,3121

1

3

2

32

21

Population inversion: N2>N1 , when>21

3-level laser

Laser fundamentals

4-levelsystem

Population inversion much easier to achieve

in a 4-level system

0idN

dt=

In equilibrium:

6

In the optical region:

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7

M M

L

z

Laser has an open resonator

in its simlest form: two mirrors facing each other

phase factor due to propagation:

e-ikz (k=2/ =2/c)

pkL =+ 222

one round trip through the resonator:

the phase must change by a multiple of 2 resonance frequencies

Phase change in mirror reflection

p = 0, 1, 2 ...

assume: empty cavity

L

cpL

c 22 =

separationbetweensuccessiveresonances: constant2

==L

c

Laser fundamentals

2

c

L

thephaseconditiondefinestheLONGITUDINAL (ORAXIAL)MODESTRUCTURE

8

Eigenmodes of the laser resonator:

21/2

= c

L2

FL

c 2

42/1

=

(OBS!NOTthelaserlinewidth)

(R= mirror reflectivity)

2)1(

4

R

RF

=

Linewidth

lossesAmplification scattering

diffraction mirrors

Losses

OutputIntensity

Laser fundamentals

AMPLIFYINGMEDIUMplacedinOPTICALRESONATOR LASER

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9

LASERTHRESHOLD(limitwhereround-tripgainexceedstotalloss)

Np =cavityphotonnumberr =normalizedpumprate

Np

Under threshold: (spontaneous emission)

radiation isotropic incoherent, thermal light

broad spectrum

Above threshold: (stimulated emission)

laser output in a directed, narrow beam coherent light narrow spectrum

Laser fundamentals

What happens when pumping is gradually increased?

spontaneous emission staysat its threshold value

Np explodes at the threshold

Above threshold Np increases linearly asa function of pumping

Above threshold the population inversionstays at its threshold value

10

Summary:

PUMPING

Optical pumping

Electron excitation

Inelastic atom-atom or

molecule-molecule collisions

Chemical reaction

Laser fundamentals

LASERMEDIUM solid gas

liquid semiconductor

LASERBEAM

R~100% R

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Unstableresonator

11Laser fundamentals

LASERRESONATORCONFIGURATIONS


M1 M2

(x,y)E1(x,y) (x,y)

E2(x,y)z

R

Fresnel - Kirchhoff :

( ) ( ) [ ]

( ) dxdye

yxEik

dxdyeyxEikyxE

ik

ik

+=

,2

1cos,4','

1

12

R >>x,y,x,yR

yy

R

xxR

''

Integrationlimits

( )

+

+

dxdyeyxER

eik R

yy

R

xxikikR

''

1 ,2E2(x,y)

TRANSVERSEMODESTRUCTURE:CONFOCALRESONATORAtransversemodeisafieldconfigurationonthesurfaceofonereflectorthatpropagatestotheotherreflectorandback,returninginthesamepattern,apartfromacomplexamplitudefactor(thatgivesthetotalphaseshiftandlossoftheroundtrip.

M1 and M2: radius of curvature = R cos ~1

(essentially Huygens principle in mathematical form)

R

2-D Fourier transform

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E1returns to itself after one round trip Symmetry for the main modes: E2 = E1 E1 is its own F-transform:

(cf. Quantummechanics / harmonicoscillator)( ) ( )

2222 ybxanm eyHxHE

For an eigenmode: field distribution is stationary inside the resonator

TEMmn - modes

TEM00 - mode

Ansatz:

+

+

= dyeedxeeR

eik RikyyryRikxxrxikR

/'//'/ 2022

02

2E2(x,y)

+

= 2222 4/ axixa e

adxee

( )222

20

2

''42

02

yxR

rkikR

ereik +

=

E2(x,y)

( ) 2022 / ryxe +=E1(x,y)

Hermitepolynomials:

)12(2)(

2)(

1)(

22

1

0

=

=

=

xxH

xxH

xH

( ) ( )' '

2 1', ' ,2

xx yyikR ikR Rik eE x y E x y e dxdy

R

+ +


E1 E2 : k

Rr

R

kr

r

2

2

10

0

0

==

Gaussianintensitydistributiononthemirrors:( )2 2 20' ' / 2

2 1

i kRx y rikRE ie e e E

+ + = =

( )222

20

2

''42

02

yxR

rkikR

erR

eik +=

( ) 2022 /ryxe +

Phasefactor: e-i(kR+/2) for one round trip: 2 - - 2Rk = 2p[ foraplanemirrorresonator: 2 - 2Lk = 2p]

2r0

2

2 0r

R

x

OBS. At x=r00

2

2

2 10 == = xrx E

eE

Wavefront Intensitydistribution E.g. = 633 nm, R = 1 m 2r0 = 0.9 mm (small !!)

The mode is completely

determined by theresonatorgeometry(and )Spot size on mirrors:

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TEM00 TEM10 TEM20

TEM30 TEM60 TEM11

TEM23TEM22TEM21

HIGHERORDERTEMnm MODES

OBS.Toeachtransversemodetherecorrespondsasetoflongitudinal modesspacedbyc/2L


Higher order transverse modes can be killedwith a suitable aperture in the cavity

TEM00,TEM10,TEM20modes:

Intensitydistributioninthetransverseplane

-3 -2 -1 0 1 2 30.0

0.2

0.4

0.6

0.8

1.0

1.2

Intensity

(a.u.

)

Transversedistance(a.u.)

SINGLETRANSVERSEMODEOPERATION

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L

c

2=

lossAmplifica

tion

1) by shortening L, the modes get further apart 2) Lower amplification (reduced pumping)

loss

Amplification

1) and2) allow only low powers to be obtained (of no practical use)


SINGLELONGITUDINALMODEOPERATION

LONGITUDINALMODE L

z

,2

L m

= m=integer2

7

L=

26L=

3) Generate additional losses for the extra modes by placing frequency selective optical elementsin the laser resonator

OBS. The lasing mode gets some of the gain of the killed modes

higher power/mode


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Wavelengthrange10 - 15 nm 100 - 500 m (100 eV 0.01 eV) tunable lasers: dye laser, diode laser, Ti:Sapphire laser

Monochromaticity

typically ~ 1 MHz - 1 GHzat best

Directionality (d= beam diameter)

typically ~1 mrad, with extra collimation 1rad

=

1 10010 10

15 12Hz

5 10 Hz14 ~

d,

LASERPROPERTIES

Coherence

coherence time = 1/ e.g. = 1 MHz = 1 s

coherence length z = c

e.g. z = c = 3 108

m/s 1s = 300 m

Spectralbrightness = P / A [W/cm

2-sr-Hz] Sun ~ 1.5 10

-12W/cm2-sr-Hz

HeNe-laser (1 mW) ~ 25 W/cm2-sr-Hz

Nd:glass-laser (10 GW) ~ 2 108W/cm2-sr-Hz

Operationmode CW (continuous wave)

pulsed operation shortest pulses < 10 fs (10-14 s) peak power at best tens of TW


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0.4

Arion0.2 -0.306Aror rion0.33-0.36Ne0.33-0.3

Pulseddye0.32-1.0

Arion0.4 -0. 2Xeion0.4 -0. 4

InGaAlPdiode0.63-0.66

Alexandrite0. 2-0.GaAlAsdiode0. - 0.9

Ti:Sapphire0.6 -1.13

0.

0.

1.0

0.1 Molecular luorine( 2)0.1Ar excimer0.192

rClexcimer0.222r excimer0.24

XeClexcimer0.30He-Cd0.32

N20.33Xe excimer0.3 1He-Cd0.442Cuvapor0. 1HeNe0. 43

Cuvapor0.HeNe0. 94HeNe0.612 Auvapor0.62HeNe0.633

He-Cd0.636GaInPdiode0.6Ruby0.694HeNe0. 3

InGaAsdiode0.9Nd:( AG,Glass, L )1.06

1.

2.0

10.0

1.0

InGaAsPdiode1.2-1.6

Colorcenter1.4-1.6

Colorcenter2.3-3.3H chemical2.6-3.0chemical3.6-4.0

CO -6CO29-11

N2O10-11LeadSaltdiode3.3-29

HeNe1.1

Nd: L 1.313I21.31Nd: AG1.32

HeNe1. 23Er-amplifier1. 4

Holmium2.1

Er: AG2.94HeNe3.39

Wavelength Wavelength

[m [m

Most common laser lines

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