Chapter 7_L14 156

Rate equations

Four-level laser

Three-level laser

Rate equations 157

Rate equations are a set of coupled differential equations, describing the temporal (and spatial) behavoirs of the carriers (populations) and the photons (lasing power). It is a phenomenonlogical description of the Bloch-Maxwell’s equations

Two main types of lasers:

Four-level lasers: Ionic crystal lasers like Nd3+:YAG (Neodymium-doped Yttrium Aluminium Garnet 掺钕钇铝石榴石 1064 nm); Gas lasers like CO2 (10.6 um), He-Ne (632.5 nm)

Three-level lasers: fiber lasers like Er (铒1550 nm), Yb (镱1030 nm) , Ho (钬2000 nm) , Tm (铥2000 nm) doped fiber lasers

Four-level laser 158

Assumptions: The carrier density (population) is uniform inside the gain medium The photon density (electric field) is uniform inside the cavity The light volume is the same as the gain medium volume Only one mode oscillates in the cavity Then, the rate equation is space independent.

The lifetimes of level 3 and level 1 is very fast

3 10 0N N ，

The laser emission occurs at level 3 level 2

2

2

The population inversion

The gain

N N

g N N

fast

pump

fast

N3

N1

N2

Ng

Four-level laser 159

fast

pump

fast

N3

N1

N2

Ng

2 2g

2g

The rate equations

v

v

p P

sp

P PP

p sp

dN NR gN

dt

dN N NgN

dt

Rp is the pump rate

is the spontaneous emission factor

2 1

g g 2

g

= = F

v v

v

P P

P

W N N W N N

N N N N

gN

2 22 1

22 1

3

1

The rate equations

0

0

p

sp

P P

p sp

g

dN NR W N N

dt

dN N NW N N

dt

N

N

N const

2

sp

N is small, and is negligible in most

discussions.

Four-level laser 160

L

R2 R1

l Optical length

o rL L l n l

Loss coefficient

1 2

1 1ln

2T i

L R R

gain coefficient

g N

Round-trip time

2 oLt

c

Photon lifetime 1

p

g Tv

Some relations -2 -1

3

2

Photon flux (cm s )

F=

Energy density (J/cm )

Light intensity(mW/cm )

F

g P

P

g P

g

v N

N h

I v N h

h v

The output power 161

The energy output (loss) rate of the two mirrors

1 2

1 2

1 1 1ln

2

If R =R

1 1 1ln

g m g

m

g

m

v vL R R

vL R

The energy inside the cavity

P pE N h V The total output power from the two mirrors is

out

m

P p g m

EP

N h V v

Examples 7.1

(Quasi-) three-level laser 162

fast

pump

N3

N1

N2

In a (quasi-) three-level laser, the lower laser level is the ground level, or a sub-level of the ground level. All the sublevels are strongly coupled and hence in thermal equilibrium.

2 2g

2g

1 2

3

The rate equations

v

v

0

p P

sp

P PP

p sp

t

dN NR gN

dt

dN N NgN

dt

N N N

N

gain coefficient

22 1

1

=g

g N N Ng

2 2 22 1

1 1

1d N d g g dN

N Ndt dt g g dt

2 2 2 22 1

1 1 1

1 11 t t

sp sp sp

g N g gN N N N N

g g g

2 2 2g

1 1 1

2g

1

The rate equations

11 1 v

1v

p P t

sp

P PP t

p sp

d N g g gR gN N N

dt g g g

dN N ggN N N

dt g

163

Lasing threshold and power

Four-level laser

Three-level laser

Chapter 7_L15

Four-level laser 164

The carrier lifetime

1 1 1

sp nr

Condition for possible continuous-wave lasing

2 1

2 2g

2g

The rate equations

v

=v

p P

sp

P PP

p sp

dN NR gN

dt

dN N NgN

dt

Ignore the spon. emission

g

g

=v

1( ) (0)exp v

P PP

p

p p

p

dN NgN

dt

N t N g t

Thus, the lasing threshold is

2

g g

1 1

v vth th th

p p

g N N

g

1( ) (0) vp p

p

N t N g

To achieve lasing,

Below the threshold 165

Below the threshold, the stimulated emission is negligible (photon number

is small), and there is only spontaneous emission, then the rate equations become

2 2g

2g

The rate equations

v

=v

p P

sp

P PP

p sp

dN NR gN

dt

dN N NgN

dt

2 2

2=

p

sp

P P

p sp

dN NR

dt

dN N N

dt

Under steady-state condition

2 0; =0PdN dN

dt dt

The steady-state solutions are

2

=

p sp

P p p

N R

N R

RP

N2

RP

NP

Both the carrier density and

the photon density increase

linearly with the pump rate

(Well) above the threshold 166

(Well) above the threshold, the stimulated emission dominates, and the

spontaneous emission is negligible, then the rate equations become

2 2g

2g

The rate equations

v

=v

p P

sp

P PP

p sp

dN NR gN

dt

dN N NgN

dt

2 2g

g

v

=v

p P

sp

P PP

p

dN NR gN

dt

dN NgN

dt

The steady-state solutions are

2

g g

2 2

=0

1 1= ;

v v

;

P

p p

th th

dN

dt

g N

g g N N

Gain clamping:

Above the threshold, the gain does not increase with the pump rate, but remains the same as that at the threshold. (carrier population clamping)

2th

=0

=P p p

sp

dN

dt

NN R

The photon density increases linearly with the pump rate.

The threshold pump rate is (Np>=0)

2th

g

1

v

thp

sp p sp

NR

Below and above the threshold 167

2 2g

2g

The rate equations

v

=v

p P

sp

P PP

p sp

dN NR gN

dt

dN N NgN

dt

Below the threshold

RP

N2

RP

NP

RPth RPth

2

=

p sp

P p p

N R

N R

At the threshold

g

2

g

2th

1v

1

v

= 0

th

p

th

p

thp

sp

pth p p

g

N

NR

N R

Above the threshold

2 2

th=

th

P p p p

N N

N R R

Carrier reservoir

Rsp

Rp

Rst

The output power 168

g

= 1

1= 1

v

p

P pth p

pth

p

sp pth

RN R

R

R

R

The photon density

1

1

out P p g m

p

m p

sp pth

p

m s p

pth

P N h V v

RhV

R

RI V

R

The output power

=P p p pthN R R

out P p g m

p p g m p pth

P N h V v

h V v R R

The output power 169

For the optical pump,

the pump power Pp

For the electrical pump, the

pump current I

fast

pump

fast

N3

N1

N2

Ng

2=p p p p

p

dNP R Vhv Vhv

dt

1

1

p

out m s p

pth

p

m s p

pth

RP I V

R

PI V

P

p

IR

qV

1

1

p

out m s p

pth

m s p

th

RP I V

R

II V

I

The output power 170

The laser slope efficiency: the output laser power versus the input light

power or injected current

(W/W)

(W/A)

m s poutso

p pth

m s poutse

th

I VdP

dP P

I VdP

dI I

The laser quantum efficiency: ratio of output photon numbers and input

photon/ electron numbers

/

/

/

/

outqo

p p

outqe

P hv

P hv

P hv

I q

(Quasi-) three-level laser 171

The threshold population inversion (ignore spon. emission) =0pdN

dt

2 2 2g

1 1 1

2g

1

The rate equations

11 1 v

1v

p P t

sp

P PP t

p sp

d N g g gR gN N N

dt g g g

dN N ggN N N

dt g

1 1= ; th th

g p g p

g Nv v

The threshold pump rate =0; =0p

d NN

dt

2

2 1 1

1 1

1 /pth th t

sp

gR N N

g g g

The photon density =0; =0pdN d N

dt dt

=P p p pthN R R

22 2

t th tth

N N NN

Optimum coupling (max power) 172

1 2

1 2

1 1= ln

2

If R =R

1 1= ln

m

m

L R R

L R

The mirror output coupling (loss)

th

g

g

1

v

1

v

p

p sp

p

i m

R

When T increases, on one hand, the output power tends

to increase; on the other hand, the total photon number in

the cavity tends to decrease.

For the four-level lasers

1

out p p g m p pth

p

m p

i m sp

P h V v R R

Rh V

For a fixed pump rate, the maximum power is at

0out

m p sp i i

dP

dt

R

173

Multimode oscillation

Chapter 7_L16

Multimode (longitudinal) oscillation 174

Lasers usually oscillate on many modes, due to the usual wide gain profile, much broader than the mode spacing (FSR).

Homogeneous broadening medium only oscillates on one mode at the gain peak, due to the gain clamping effect.

gth

Multimode (longitudinal) oscillation 175

In practice, in homo. broadening medium, there are several modes oscillating around the gain peak, when the pumping is high enough.

Electric fields are standing waves in the cavity. For spatial points at extrema of the electric field, the population inversion are clamped. For spatical points at zeros of the electric field, the population inversion remains increasing. Thus, there will be holes formed at different positions of the gain medium. This is called spatial hole burning.

Multimode (longitudinal) oscillation 176

In the homogeneous broadening medium, due to the spatial hole burning, different longitudinal mode can use carriers at different places, so several modes around the gain peak can oscillates. These modes competes for the strongest lasing, and is called (spatial) competition of (longitudinal) modes, which leads to the lasing power fluctuations.

In homo. gas medium, carriers moves fast in the space, spatial holes can hardly formed, so single mode can be obtained. In homo. solid medium, carriers moves slow in the space, spatial holes will be formed, and multimodes around the gain peak can be obtained. In ring cavity with isolator, the wave is traveling rather than standing, so no spatial hole burning effect, and single mode is formed.

Increase pump

177

Inhomogeneous broadening medium oscillates on many modes within in the gain profile, which are independent with each other. Spectral hole burning will be formed on the gain spectrum.

Multimode (longitudinal) oscillation

If the mode spacing is too small, the holes will have some overlap with each other, leading to the mode competition due to the sharing of the same group carriers.

178

Different transverse mode has different diffraction loss, leading to different values of threshold. TEM00 has the lowest loss and thus oscillates firstly.

The electric field distribution of transverse mode is not uniform, leading to the transverse spatial hole burning. Different transverse mode has different electric field distribution, and use different spatial carriers. When the pump is strong enough, there will be many modes oscillating simultaneously.

Multimode (transverse) oscillation

Lamb dip 179

In the laser with Doppler inhomo. broadening medium, the output power increases when the longitudinal mode frequency is tuned towards the central frequency (gain peak). However, when the mode frequency is close to the central frequency, the power will decrease instead. Then, a dip is form on the power spectrum, which is called Lamb dip.

When the mode frequency vm!=v0, the light interacts with two groups of carriers with velocity +vz and –vz. Both groups contribute to the lasing power.

When the mode frequency vm=v0 , only one group of carrier with velocity vz=0 interacts with the light, and contribute to the lasing power. Although the gain is highest, but the contributed carriers are the smallest. Therefore, the lasing power decreases, leading to the Lamb dip on the power spectrum.

Lamb dip is a famous technique to stablize the laser frequency.

180

Single mode selection

Transverse mode selection

Longitudinal mode selection

Chapter 7_L17

Single transverse mode selection 181

TEM00 mode is usually desirable. The selection basis is that The diffraction loss (δ) has an enough portion in comparison with other losses (α).

The diffraction loss increases with the order of the transverse mode.

The mode selection requirement is 00 00 01 01 10 10 and , T T Tg g g

For the same diffraction loss portion, the larger the diffraction loss difference ( δ10/δ00) between the high-order mode and the fundamental mode, the easier the mode selection. (Note: absolute loss value must be considered)

00 01

00 01

20,

0.1, 1,

20.1, 21;T T

00 01

00 01

10,

10, 15,

20, 25;T T

00 01

00 01

10,

10, 30,

20, 40;T T

Mode discrimination versus Fresnel number

Single transverse mode selection 182

The diffraction loss decreases with Fresnel number, while the mode discrimination increases. The confocal cavity has the lowest loss, but the largest discrimination ability. The concentric cavity and the plane cavity have the highest loss, but the smallest dscrimination ability. In practice, it is easier to use the concentric and plane cavities to select the fundamental mode, due to the high mode losses.

Single transverse mode selection 183

Design a proper aperture inside the cavity

Design a proper parameter g and Fresnel number N of the cavity

Design a proper unstable cavity

Slight adjust the parallelization of mirrors

For plane cavity, TEM00 will suffer the largest loss, and high-order modes oscillate. For stable cavity, due to the small volume of TEM00, and the large volume of high-order modes, the latter suffers high loss, and TEM00 oscillates.

Single longitudinal mode selection 184

Short cavity

This is appliable when the gain broadening linewidth is not large, like in gas (He-Ne) lasers (1.7 GHz in textbook). Piezo-electric transducer is needed to adjust the cavity length carefully, with accuracy on the order of wavelength.

/ 2FSR gb

gb

v v

cL

v

The traveling wave together with homogeneous broadening medium wont form any hole burning effects.

gbv

gain

FSRv

Ring cavity with isolator

Single longitudinal mode selection 185

Selective mode loss using grating, external cavity, and Fabry-Perot etalon …

F-P etalon to select mode 186

The transmission maxima of F-P etalon is at

2 'cos 'm

r

mcv

n L

; ; 2 2 'cos '

ETLS ET ET FSRFSR FSR c

r

c c vv v v

L n L F

A small angle adjustment is needed to tune one transmission peak close to the gain peak.

' 0

F-P etalon to select mode 187

The etalon half peak linewidth must be smaller the mode spacing of the laser cavity.

1

2

ET LS

c FSRv v

The etalon FSR must be larger than half gain linewdith

1

2

ET

FSR gbv v

Consequently, the requirement

12

2

ET LS

gb FSR FSRv v F v

The laser cavity length requirement can be 2F times larger than that without etalon.

2gb

cL F

v

Examples 7.8

'2

r

gb

L cn L

F v

188

Frequency pulling effect

Laser linewidth

Chapter 7_L18

Frequency pulling effect 189

Laser frequency pulling effect: for an optical resonator with a cavity mode at vc, and a gain medium with the peak at v0. The lasing frequency vL will deviate from vc, and is pulled toward the gain peak v0.

0 0

0

/ /

1/ 1/

c cL

c

v v v vv

v v

Usually , so the frequency pulling is generally very small. 0 cv v

Laser linewidth 190

The linewidth of the lasing mode originates from the spontaneous emission noise, which leads to both random phase fluctuation (dominating) and amplitude fluctuation (small) of the electric field.

0( ) sin 2 ( )L ranE t A v t t

The spectral linewdith (Lorentzian shape) of a passive resonant cavity is 1

2c

p

v

The spectral linewidth (Lorenzian shape) of an active resonant cavity (laser linewidth) is

1p

g Tv

1

2L a

p

v

1a

p

g Tv g

Laser linewidth 191

Above threshold, the « active photon lifetime » is not zero due to the spontaneous emission.

2g

2g

2

The rate equation

=v 0

1v

1 1

P PP

p sp

T

sp P

a

p P sp

dN N NgN

dt

Ng

N

N

N

1

0a

p

g T

th T

v g

g

21

2L

P sp

Nv

N

Then, the laser linewidth is

This is the Schawlow-Townes linewdith, which gives the quantum limit of laser linewidth, 1mHz<typically <1 Hz.

2

22 L c

L

hv vNv

N P

The laser linewidth decreases with the lasing power.

Laser linewidth 192

The spectral (optical) linewidth of solid-state lasers, fiber lasers, and gas

lasers are on the order of kHz,while the linewidth of semiconductor lasers is on the oder of MHz.

In semiconductor lasers, spontaenous emission noise, on one hand, directly

induces phase fluctuations of electric field. On the other hand, induces carrier (electrons and holes) density fluctuations, leading to refractive index changes of the laser medium, and thereby the laser frequency fluctuation, broadening the spectral linewidth. This is called the phase-amplitude coupling effect, and is characterized by the linewidth enhancement factor (Henry factor).

Noise orignating from

spontaneous emission and carrier noises is called quantum noise, while that from mechanical vibrations and thermal fluctuations is called technical noise.

21OL STv v

Examples 7.9

193

Laser frequency (phase) noise and fluctuation

Laser intensity noise

Chapter 7_L19

Laser frequency/phase noise 194

In practice, the laser frequency is not constant, but fluctuates randomly due to the laser noise. Short-term noise (high frequency) is due to the quantum noise (spontaneous emission, carrier noise). Long-term noise (low frequency) is due to technical noise (temperature fluctuation, driven current noise, mirror vibration…).

0( ) sin 2 ( )L ranE t A v t t

The laser electric field with random phase:

Phase noise and frequency noise relation:

( )( )= ran

L

d tv t

dt

The instantaneous laser frequency:

( )= + ( )L L Lv t v v t

( ) ( )L ranv f jf f

Laser frequency/phase noise 195

The frequency/phase noise is usually characterized by the power spectral density (Fourier transform of autocorrelation function):

2 2

2 2

( )= ( ) ( ) exp( )

( ) (Hz /Hz)

( )= ( ) ( ) exp( )

( ) (rad /Hz)

vS f v t v t j d

v f

S f t t j d

f

2( ) ( )vS f f S f

*

0

1( ) ( ) = lim ( ) ( )

T

Tv t v t v t v t dt

T

Laser frequency/phase noise 196

Spontanenous emission noise is a kind of white noise (Sv(f)=const.) in the frequency (electrical) domain, which leads to a Lorentzian-shape spectral (optical) linewidth. The relation between the laser linewidth and the white frequency noise is

Frequency (Hz)

Freq

ue

ncy

no

ise

(Hz2

/Hz)

Flicker (1/f) noise

White noise

( )L vv S f

Besides the white noise due to quantum noise, there is flicker noise at low frequency due to technical noises.

1 MHz

Frequency fluctuation and stablization 197

The frequency stablization can be realized by using a frequency discriminator (F-P etalon) together with a piezoelectric transducer

2

1 1

m

r

m m r

r

cv m

n L

v v n Ln L

The frequency stability is usually characterized by /m mv v

The laser mode frequency fluctuates due to cavity length change and to refractive index change:

transducer medium

discriminator

Power monitor

Voltage

Examples 7.10

Frequency stablization using Lamb dip 198

f

-f f

2f

The laser is modulated at a low frequency f.

If the lasing frequency v=v0, the lasing power modulation frequency is 2f.

If the lasing frequency v>v0, the lasing power modulation frequency is f, and it is in phase with the voltage modulation. Then, the transducer will enlarge the cavity length to move the lasing frequency to v0.

If the lasing frequency v<v0, the lasing power modulation frequency is f, and it is out of phase with the voltage modulation. Then, the transducer will reduce the cavity length to move the lasing frequency to v0.

199

Laser frequency (phase) noise and fluctuation

Laser intensity noise

Chapter 7_L19

Laser intensity noise 200

2

2

2

( ) ( ) exp( )( )=

( ) (dB/Hz)

P t P t j dRIN f

P

P f

P

The laser intensity noise is usually characterized by the relative intensity noise (RIN), which is defined as the Fourier transform of the autocorrelation of the instantaneous lasing power, with respective to the square of the average lasing power.

The peak in the RIN spectrum is around the laser’s relaxation resonance frequency. The RIN can be reduced by applying an optical/optoelectronic feedback loop.

Homework 201

7.3

7.4

7.5

7.6

7.8

7.16

7.17

7.18