Principles of lasers...Principles of laser sources A. Conditions for oscillation Amplifier with feedback Condition on gain: threshold Condition on phase: longitudinal modes Possible

Principles of lasers

A.  Conditions for oscillation Amplifier with feedback Condition on gain: threshold Condition on phase:

longitudinal modes Possible active modes

B.  Laser gain Laser cross section Rate equations

C. Examples of laser media 3 level systems 4 level systems semi-conductor lasers

D.  Longitudinal modes Possible modes Single mode operation Technical linewidth

E.  Transverse modes Diffraction losses Transverse modes Example: Hermite -Gauss

F.  Laser : concentrated light Concentration in space Concentration in spectrum/time Laser source: all photons in the

same mode

Laser source (ex. ruby laser)

(8 Juillet 1960, New York Times)(8 Juillet 1960, New York Times)

Light emitter

with

a laser amplifier

and

mirrors

1960

Light Amplification by Stimulated Emission of Radiation









same mode

Laser oscillator: amplifier with feedback

Lcav

LA

LASER Amp

M1

M1

Mout TI

RI

I

Amplifier with output fed back to input:

⇒ oscillation is possible

RI

Conditions for oscillation :

•  the gain must be large enough to compensate for the losses

•  the field must be fed back with the right phase

Laser medium: amplifies the field (keeping the phase)

Condition on gain : threshold Gain of the amplifier

G (0) =

Iout

Iin

! exp g (0)LA{ } ≈ 1+ g (0)LA

•  resonant at L cav

L A

Ampli LASER M 1

M 1

M T I

R I

I R I

L cav

L A M 1

1

M out T I

R I

I R I

LASER Amp

g( I ) = g (0)

1+ I / Isat

≤ g (0)

Oscillation condition: threshold (0) (1 )(1 ) 1G T A− − ≥(0)

Ag L T A≥ +

non saturated gain

absorption losses output

coupling

T + A

ωM ω' ω"

g(0)LA Oscillation possible for

ω ω ωʹ ʹ́≤ ≤

ωM =ω 0 =

Eb− E

a

When I << Isat

Decreases with I (saturation)

non-saturated gain

Condition on phase : longitudinal modes Cavity round trip optical length

L cav

L A

Ampli LASER M 1

M 1

M T I

R I

I R I

L cav

L A M 1

1

M out T I

R I

I R I

LASER Amp

Feedback with right phase

longitudinal mode frequency integer

ωM

G(0) cavity longitudinal modes

Lcav = n(r)dr∫

Phase shift for 1 round trip

cav 2L pcω

φ π= =

cav

2pcpL

ω π=

cavLcω

φ =

ω

cav

2 cLπ

refractive index

Condition on phase : longitudinal modes

L cav

L A

Ampli LASER M 1

M 1

M T I

R I

I R I

L cav

L A

LASER Amp M 1

1

M out T I

R I

I R I

Feedback in phase

longitudinal mode frequency integer

ωM

G(0) cavity longitudinal modes

cav 2L pcω

φ π= =

cav

2pcpL

ω π=

ω

cav

2 cLπ

Mode = stationary solution of propagation equations, with boundary conditions

, p integer

Both conditions: possible modes

L cav

L A

Ampli LASER M 1

M 1

M T I

R I

I R I

L cav

L A

LASER Amp M 1

1

M out T I

R I

I R I

ω' ω"

longitudinal modes of cav.

cav

2pcpL

ω π=

modes that can oscillate

T + A

Condition on gain ω ω ωʹ ʹ́≤ ≤

Example (Helium-Neon laser)

8cav

cav

0.6 m 5 10 HzcLL

= ⇒ = ×

( ′ω − ′′ω ) / 2π ≈ 2.5GHz⇒ 4 to 5 active modes

ω2π

≈ 5×1014 Hz ⇒ p ≈106

cav

2 cLπ

g(0)LA

ωM

In phase feedback

Linear cavity laser

LASER Amp

M1 MS TI

RI

I LA

L0

Same principles as ring cavity laser

One can use ring laser results, with correspondance

cav 02L L= cavity round trip optical length

( )2ampliG G= gain over 1 cavity round trip









same mode

Laser gain (matter light interaction) nb

na 0 z LA

Iinput Ioutput

{ } ( )0 exp( , ) cosE z t E t kk z zωʹ́ ʹ−−=(1 / 2) 2

kk

kcχ

ω χʹ+

ʹʹ

ʹ

ʹ́=ʹ

2

a bD

2 20 D

( )n dnc

χε δ

Γʹ́Γ +

−=

b a

population invG

eA

IN

rsionn n>⇔

Gain if population inversion

( )b a21 n ndz

kIg dI

ʹ ∝ʹ− −= =

[ ] 1L −[ ] 3L −

na − nb =

na − nb⎡⎣ ⎤⎦

(0)

1+ s

ω ω0

ω

[ ]2b a

Lgn n⎡ ⎤⎢ ⎥−⎣

=⎦

surface : cross section

dimension equation

linear term

quantum term

Laser cross section

laser cross section

( )b a(1 )dII d

g nz

nσ ω −= =a

b

na

nb

M2 2

D

( )( )1 /σ ω

σ ωδ

=+ Γ

Mδ ω ω= −

ωM

Lamb toy model σ (ωM ) = ω

ΓD

d 2

ε0c2

In practice σ (ωM) and ΓD measured data: can be found in tables for known laser lines (Ex.: 2 x 10-20 cm2 for Cr3+ in ruby or Nd3+ in glass)

Easy to use formula (dimensions)

Cross section: useful to write rate equations for photons and atoms

Rate equation for photons Rate equation model : surface σ receives a photon flux

Iω

Number of photons per unit time and surface

dz

S

Number of “collisions” per atom per second :

σ Iω

σ

Absorption in slice S x dz :

d S Iω

⎛⎝⎜

⎞⎠⎟= −σ I

ωnaS dz

⇒ 1

IdIdz⎡⎣⎢

⎤⎦⎥abs

= −naσ

Generalization to stimulated emission, with assumption that stimulated photons add to the beam

Absorbing medium

a

b

na

nb ωM

bsti

1 dI nI dz

σ⎡ ⎤ =⎢ ⎥⎣ ⎦ b a1 ( )dI n nI dz

σ= −

Rate equations for atomic populations

a

b

ΓD

ΓD

λa

λb

naσ

Iω

nbσIω

Absorption and stimulated emission described with atomic populations transfer rates equal to rate for photons: plus relaxation and feeding terms

⇒ Rate equations for atomic populations:

dna

dt= λa − naσ

Iω

+ nbσIω

− ΓDna

dnb

dt= λb + naσ

Iω

− nbσIω

− ΓDnb

d(nb − na )dt

= λb − λa − ΓD + 2σ Iω

⎛⎝⎜

⎞⎠⎟

(nb − na )

d(na + nb )dt

= λa + λb − ΓD(na + nb )

⇒ na + nb⎡⎣ ⎤⎦stat=λa + λb

ΓD

Steady state solution a b( ) 0nd ndt+

=

Stationnary population inversion

d(nb − na )dt

= λb − λa − ΓD + 2σ Iω

⎛⎝⎜

⎞⎠⎟

(nb − na )

a ΓD

ΓD

λa

λb

aIn σωh

bIn σωh

a ΓD

ΓD

λa

λb

aIn σωh

bIn σωh

nb − na⎡⎣ ⎤⎦stat=

λb − λa

ΓD + 2σ Iω

⇒ Steady state:

Result with already found form:

[ ] [ ](0)b ab a stat 1

n nn n

s−

− =+

avec

nb − na⎡⎣ ⎤⎦(0)

=λb − λa

ΓD

and s =2σ Iω Γ

D

sat2 2

D

/1 /I Isδ

=+ Γ

and defining Isat =

!ωΓD

2σ (ωM ) σ =

σ (ωM )1+δ 2 /ΓD

2

Same form as found in lecture 2 with Lamb model

Remembering

non saturated saturation

naσ

Iω

nbσIω

Description of laser amplification by rate equations

We have observed that it is possible to describe quantitatively interaction between the laser medium and light using rate equations for atomic populations and photons.

This result was definitely not obvious a priori : an atom is a quantum object, described by a state vector, a much richer description than probabilities to be in each level*: oscillating dipole associated with coherence between |a> and |b>. Derivation of rate equations difficult : relaxation must be taken into account: Optical Bloch Equations.

In most cases (T2 << T1), laser amplification can indeed be described by rate equations leading to formulae analogous to the ones found for Lamb toy model: very useful.

* Similarly an electromagnetic wave is more than a flux of photons !

Principles of laser sources








same mode

3-level amplification Ruby laser (0.694 µm) ; Erbium doped fiber laser (1.5 µm)

Ruby : Cr3+ ions substituting some Al3+ ions in alumine crystal

laser 694 nm

a

b

Fast non radiative transitions (10-7 s)

Puming with flash lamp

0.42 µm

0.55 µm

Metastable level

(3 x 10-3 s)

3-level modeling

Wp

a

e b

population inversion if

p baW > Γpumping

Spontaneous de-excitation

Γeb (fast)

Γba (slow)

(8 Juillet 1960, New York Times)(8 Juillet 1960, New York Times)

3-level laser: population inversion Rate equations modelling

3-levels system

Γba (slow)

Γeb fast

Wp

a

e

b

dne

dt=Wp(na − ne )−Γebne

dnb

dt=Γebne −Γbanb

n = na + nb + ne

Rate equations (no laser emission)

= 0

= 0 Steady state

Γeb ≫Wp ⇒ ne "Wp

Γeb

na

⇒ nb =Wp

Γba

na

population inversion if

Wp > Γba

3-level system: the medium must be bleached to be inverted

From 3- to 4-levels 3-level system

fast

Wp

a

e b

Inversion difficult because one must not only feed b but also empty a

slow

fast

Wp

a

e b

slow

4-level system

f very fast

Inversion easy since a always almost empty. Fast relaxation: continuous (cw) laser possible.

Rate equations modeling

Examples of 4-level laser sytems

laser 1.06 µm

b

Fast non radiative transitions

Puming by lamp or

semi-cond laser

Metastable level

fast

Nd:YAG (or glass)

“easy” to double ⇒ green at 530 nm

Electric discharge lasers •  Helium-Neon, •  Ionised Argon, Krypton …

Many different wavelengths, but fixed, not tunable

etc… many other systems

Tunable 4-level laser (dye, Ti:sapphire)

Fast relaxation to band bottom

Fast relaxation to band bottom

Pumping by non tunable

laser Lower end of laser transition anywhere

in lower band

Emission bandwidth equal to lower band width • dye: [565 nm , 595 nm]

(25 x 1012 Hz) •  Ti: Sapphire: [700 nm , 1100 nm]

Wavelength selection by filter in the cavity

Broadband laser amplif.

Semiconductor laser (diode laser) p-n junction between 2 semi-conductors

•  Lack of charge carriers (electrons and holes) •  A photon with energy larger than gap can be

absorbed, with creation of an electron-hole pair: photodetection

•  Conversely, an electron-hole pair can annihilate and emit a photon: photoemission

Eg

el

holes

- +

If injected current density large enough: Stimulated emission dominates (4-levels) Current concentrated with heterostructures

+!

Metal!

Metal!

N! N!

N!

P!

n!

active layer!P-doped!SiO2!

Emitting zone 1 x 10 µm2

Many different wavelenths •  From 1.3 or 1.5 µm (telecom) down to 0.32 µm •  Massive investment, but mass production : low

prices

Linear laser cavity Cleaved faces, perfectly parallel (high n )









same mode

Longitudinal modes of a laser source Gain band-width Lcav c / Lcav

Number possible modes

He-Ne 109 Hz 0.6 m 0.5 x 109 Hz 3

Ti:Sapphire 1014 Hz 1.5 m 0.2 x 109 Hz 5 x 104

diode laser 1012 Hz 3 mm 1011 Hz 10

Narrow lines, separated by cav2cL

ωπΔ

= Number max

N = gain bandwidthΔω

• Frequently, not all the modes oscillate simultaneously: mode competition (cf. lecture 4).

• One can force single mode operation

ω' ω"

modes longit. cavité

oscillation possibleT + A

cav

2 cLπ

g(0)LA

ωM

cavity long. modes possible oscillation

Single longitudinal mode operation

Lcav

LA

LASER Amp

M1

M1

Mout TI

RI

I RI

ωp

cavity long. modes possible oscillation

T + A

Filter (Fabry-Perot)

T ω

1

0

absorption

ω

1

0

absorption

Filter makes losses at all wavelengths except a narrow band

Oscillation possible only in the filter bandpass

Demands a very high selectivity: cascaded filters; frequencies of the filters must be aligned (feedback loops). High tech devices

cav

2 cLπ

(0)Ag L T A≥ +

(0)Ag L

Oscillation if

Technical linewidth (jitter) Single longitudinal mode

ω p = p2π c

Lcav

; p ∈ 6cav 10p

Lpλ

= ≈

Fluctuations of Lcav : ⇒ ωp fluctuates : jitter

ωp

T + A

T

ωp

T + A

T• Vibrations, temperature: length • Pressure (refraction index)

cav cav2 /p pL c Lδ λ δω π= ⇒ =

• Hard to do better with passive methods (temperature controlled within 10-4 °C, pressure within 10-5 atm)

Servo-controlled cavity length • Mirror on PZT (position control) • Error signal on frequency

δω p / 2π <103 Hz

↔ δLcav <10−2 nm !!!

It is enough to have Lcav varied by λ (less than 1 µm) to have one mode replacing its first neighbour

cav

2 cLπ







E.  Transverse modes Diffraction losses Transverse modes Example: Hermite-Gauss


same mode

Diffraction losses in a laser cavity

cav 2/L

a Losses negligible if

λa

Lcav a

Not the case, in general (confined laser amplifiers)

a = 1mm ⇒ a2

λ= 1 m

A solution : stable cavity with curved mirrors

The mirrors impose their curvatures to the laser wave

NB. Semiconductor laser: guided propagation, plane wave

Transverse modes of a stable cavity (cf. complement 3B)

z

x y

Modes: stationary solution of 3D propagation equation, with boundary conditions (mirrors)

Series of solutions, depending on 3 integer numbers , , ( , , ) ep

m n pi tu x y z ω−

, ,cav

: longitudinal index 2p m n pcp pL

ω π ε= +

m, n : transverse indices : number of nodes (zeroes) in transverse profile

Example : Hermite-Gaussian modes

um,n,p (x, y, z) = Aexp − x2 + y2

w(z)2

⎧⎨⎩

⎫⎬⎭

Hm

x 2w(z)

⎛

⎝⎜⎞

⎠⎟Hn

y 2w(z)

⎛

⎝⎜⎞

⎠⎟cos

ω p

cz +φ(z)

⎛

⎝⎜⎞

⎠⎟

z

xy

z

xy

0

12

2

( ) 1( ) 2

( ) 4 2

H uH u uH u u

=

=

= −

Hermite polynomials

1

m!2m Hm(u)⎡⎣ ⎤⎦2e−u2

m = 0

m = 2

m = 1

x

y

0,0

0,1

0,3

0,2

1,3

Another point of view on transverse modes: self consistent propagation with diffraction

Lcav

M1

M1

MS TI x

y The field profile

( , )x yE

becomes,after a round trip

( , )x y→ ʹE

( , ( ,Mo )e )d : x y x yα=ʹ EE

0 00 0 00( , ), )( ) ( ,xx x y x y dx dyy y− −ʹ = ∫∫ PEEKirchhoff integral (diffraction)

Round trip propagator

Each point of the profile is coupled to all other points (diffraction over a round trip) ⇒ locking of the phase of the field over a transverse plane: transverse coherence

( , )x yE( , )x yʹE









same mode

What is so wonderful about laser light? Certainly not the price per Watt of light

He-Ne 5 mW 100 € 20 k€ / W

Ar+ 2 W 40 k€ 20 k€ / W

CO2 1 kW 150 k€ 0.15 k€ / W

Diode laser 1 mW 1 € 1 k€ / W

Diode laser 500 mW 500 € 1 k€ / W

Laser sources

Light bulb 100 W 1 € 0.01 € / W

discharge 40 W (light) 4 € 0.1 € / W

Standard sources

?

Concentration in space: laser vs. standard source

S' S u’ u

Standard (incoherent) source

′S ⋅sin2 ′u ≥ S ⋅sin2 uEnergy conservation: Irradiance E (W / m2) in image cannot exceed π x L

E ≤ 500 W / cm2 (tungsten wire) E ≤ 5 kW / cm2 (xenon arc)Fundamental limit (2nd principle of thermodynamics)

Laser beam (coherent) : addition of amplitudes

f

2w 2w0

φ

D ′Smini ≈ λ 2

⇒ Emax ≈

φλ 2

φ = 10 mWλ = 0.6 µm

⇒ E 3×106 W / cm2

Concentration at diffraction limit

L = source luminance (W m-2 Sr-1)

⇒ ′Smini ≈ S ⋅sin2 u

L E

E = image irradiance (W m-2)

Concentration in direction of laser light Position and angle: complementary variables (conjugate)

Divergence of a laser beam of diameter D

θ λ

D= 10−6 for D = 0.6 m 300 m wide on moon

x and kx =

2πλ

sinθ x

Lunar reflectors Télémétrie au CERGA (Grasse)

Documents Observatoire Côte d’Azur

more precisely θkx

z

x

Concentration in direction possible if large size

Concentration in spectrum (or time) Bulb lamp (3500 K)

Power spread over all visible range (and IR)

Δω2π 1014 Hz

Maximum irradiance per unit bandwidth

dEdν

≈ 5×10−12 W cm−2 Hz-1

Laser 10 mW Linewidth : Δω2π 106 Hz

Maximum irradiance per unit bandwidth

2 -13 W cm Hz dEdν

−≈

12 orders of

magnitude more !

Conjugate variable : time. Ultra short lasers (< 10-15 s). Energy concentrated in time (giant peak power)

Laser light: concentrated light • Concentration in space (position / direction) • Concentration in spectrum (frequency / time)

Laser: energy concentrated in a single mode of radiation ⇒ Incoherent source: energy diluted over many modes

Number of photons per mode

Laser : N ≈ 1010 to 1020 photons / mode

Thermal source (blackbody radiation)

N = 1

exp!ω

kBT

⎧⎨⎩

⎫⎬⎭−1" exp − !ω

kBT⎧⎨⎩

⎫⎬⎭

0.1 photon / mode @ λ = 0.6 µm @ 3000 K

1xx kΔ ⋅Δ =

Laser beam: all photons in the same mode of the electromagnetic field

Photons are bosons : it is possible to accumulate as many as one wants in the same quantum state (actually they tend to accumulate by bosonic stimulation). A laser beam can be considered as a kind of Bose-Eintein Condensate of photons (not in thermal equilibrium)

All photons in the same mode: •  Same direction •  Same frequency •  Same phase •  Same polarisation

indistinguishability: coherence