ln1_s
TRANSCRIPT
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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
12Laser fundamentals
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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13Laser fundamentals
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
+ +
14Laser fundamentals
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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15Laser fundamentals
TEM00 TEM10 TEM20
TEM30 TEM60 TEM11
TEM23TEM22TEM21
HIGHERORDERTEMnm MODES
OBS.Toeachtransversemodetherecorrespondsasetoflongitudinal modesspacedbyc/2L
16Laser fundamentals
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)
17Laser fundamentals
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
18Laser fundamentals
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19Laser fundamentals
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
20Laser fundamentals
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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