Laser Systems and Applications

Cristina Masoller Research group on Dynamics, Nonlinear Optics and Lasers (DONLL)

Departament de Física i Enginyeria Nuclear Universitat Politècnica de Catalunya

cristina.masoller@upc.edu www.fisica.edu.uy/~cris

MSc in Photonics & Europhotonics

mailto:cristina.masoller@upc.eduhttp://www.fisica.edu.uy/~cris

Outline

Block 1: Low power laser systems Introduction

Semiconductor light sources

Models

Dynamical effects

Applications: telecom & datacom, storage, others

Goals

• Acquire a basic knowledge of the rate equation model for the

photon and carrier densities.

– Steady-state solutions and LI curve.

– Laser dynamics with time-varying injection current

• Turn on delay and relaxation oscillations

• Dynamic LI curve and hysteresis

• Current modulation and modulation bandwidth

• Become familiar with the rate equation for the complex

optical field.

– Phase-amplitude coupling

– Laser line-width and intensity noise

• Understand optical injection, feedback & polarization effects

06/01/2016 3

Outline: Models and dynamical effects

• Simplest rate equation model: photon and carrier density – Steady state solutions

– Time dependent solutions (turn on, relaxation oscillations, response to periodic modulation)

• Rate equation model for the complex optical field – Response to optical perturbations

– Polarization dynamics

06/01/2016 4

Semiconductor lasers are class B lasers

06/01/2016 5

• Governed by two rate-equations: one for the photons (S) and one for the

carriers (N) .

• Display a stable output (with only transient relaxation oscillations).

• Single-mode “conventional” EELs diode lasers are class B lasers.

• Other class B lasers are ruby, Nd:YAG, and CO2 lasers.

• Because of the -factor (a specific feature of diode lasers, more latter)

diode lasers display complex dynamics when they are optically perturbed.

Characteristic times for common class B lasers

n = Carrier

lifetime

p = Photon

lifetime

Other types of lasers

• Class A (Visible He-Ne lasers, Ar-ion lasers, dye lasers):

governed by one rate equation for the optical field (the

material variables can be adiabatically eliminated), no

oscillations.

• Class C (He-Ne lasers operating at infrared lines):

governed by three rate equations (N, S, P=macroscopic

atomic polarization), display sustained oscillations and

even a chaotic output. No commercial applications.

06/01/2016 6

06/01/2016 7

Carrier density

Injected electrical current

Photon density

Cavity losses

Diode lasers : electrical to power conversion

Net gain:

emission –

absorption

06/01/2016 8

Rate equation for the carrier density N

I : Injection current (I/eV is the number of electrons injected

per unit volume and per unit time).

N : Carrier lifetime.

G (N,S) : Net gain

GSN

eV

I

dt

dN

N

Pump:

injection of

carriers

Recombination

of carriers

Stimulated

emission &

absorption

06/01/2016 9

Rate equation for the photon density S

N

sp

p

NSGS

dt

dS

Stimulated

emission -

absorption Spontaneous

emission

p : Photon lifetime.

G (N,S) : Net gain

sp : Spontaneous emission rate

Cavity

losses

Simple model for the semiconductor gain

06/01/2016 10

RT InGaASP laser 0NNaG

00 /ln NNaNG

DH

QW

Carrier density

at transparency

Differential gain

coefficient

We assume single-

mode emission at 0.

The differential gain

coefficient a depends

on 0 (multi-mode

model latter).

0

Threshold carrier density

06/01/2016 11

p

thNG

1)(

N

sp

p

NSGS

dt

dS

0NNaG

ththp

NNaNNaNNaG 001

Threshold

condition: net

gain = cavity

loss

01

NNa thp

N

sp

th

NSNNa

dt

dS

• Ordinary differential equations (spatial effects neglected)

• Additional nonlinearities from carrier re-combination and gain saturation

• These equations allow to understand the LI curve and the modulation

response.

• To understand the intensity noise and the line-width (the optical spectrum),

we need a stochastic equation for the complex field E (S=|E|2) (more latter).

• Spatial effects (diffraction, carrier diffusion) and thermal effects can be

included phenomenologically.

06/01/2016 12

GSN

eV

I

dt

dN

N

N

sp

p

NSGS

dt

dS

S

NNaG

1

0

21 CNBNAN

Two coupled nonlinear rate-equations

• Define the dimensionless variable:

• Normalizing the equations eliminates two parameters (a, No)

• In the following we drop the “ ’ ”

06/01/2016 13

Normalized equations

N

sp

p

NSN

dt

dS

1

1

SNNdt

Nd

N

1

thNN

NNN

0

p

th NNa

10

Threshold carrier

density: gain = loss

Pump current parameter:

proportional to I/Ith

Role of spontaneous emission

• If at t=0 there are no photons in the cavity: S(0) = 0

• Then, without noise (sp=0): if S=0 at t=0 dS/dt=0

S remains 0 (regardless the value of and N).

• Without spontaneous emission noise the laser does not turn.

06/01/2016 14

N

sp

p

NSN

dt

dS

1

1

06/01/2016 15

Steady state solutions with sp=0

SNdt

dS

p

11

NSNdt

dN

N

1

1

00

N

S

dt

dS

1 1

00

SN

NS

dt

dN

Above threshold

the carriers are

“clamped”.

th = 1

(Simple

expressions

if sp is

neglected)

S=0

N=

S = -1

N =1

Laser off Laser on

Stable if

1

06/01/2016 16

Graphical representation

0.5 1 1.50

0.1

0.2

0.3

0.4

0.5

S

Pump I

0.5 1 1.5

0.6

0.7

0.8

0.9

1

1.1

Pump I

N

Pump current, Pump current, th = 1

S=0 S=-1

off on

N= N= 1

off on Above

threshold the

carriers are

“clamped”.

Experimental LI curve

06/01/2016 17

Hitachi Laser Diode HL6724MG (A. Aragoneses PhD thesis 2014)

N

sp

th

NSNNa

dt

dS

This LI curve is obtained

from model simulations

when sp is not neglected.

LI curve with sp0

06/01/2016 18

N

sp

p

NSN

dt

dS

1

1

NSNdt

dN

N

1

N

p

N

pspb

bNSNdt

dS 1

'

NSNdt

dN

'

p

tt

'

bS 411

2

1 2

• If >1 S -1

• If

“kink” of the LI curve

06/01/2016 19

• If >1 S -1

• If

Model for a multi-mode laser

06/01/2016 20

Gain coefficient

for mode j:

Carrier density (n) + several photon densities (for each longitudinal mode)

SIMPLEST RATE EQUATION MODEL -STEADY-STATE SOLUTIONS & LI CURVE

-TIME-DEPENDENT SOLUTIONS WHEN THE INJECTION

CURRENT VARIES

06/01/2016 21

• Step (laser turn on): off, on

• Triangular (dynamic LI curve): min, max, Tramp

• Sinusoidal (modulation response): dc, A, Tmod

06/01/2016 22

NSNdt

dN

N

1

(t)

N

sp

p

NSN

dt

dS

1

1

p 1 ps

N 1 ns

sp 10-4

Parameter values

Time variation of the injection current

Current step: turn-on delay & relaxation oscillations

06/01/2016 23 Adapted from J. Ohtsubo

10 20 30 40 50

0

0.5

1

1.5

2

2.5Simulations

Time (ns)

S

Np

RO

1

A linear stability analysis of the rate equations

allows to calculate the RO frequency

Experiments (diode laser)

Variation of the relaxation oscillation frequency with the output power

06/01/2016 24

Np

RO

1 S -1 Laser on:

Np

RO

S

Turn-on transient of a Nd3+:YAG laser

06/01/2016 25 From T. Erneux and P. Glorieux, Laser Dynamics (CUP 2010)

Note the time-scale: for diode lasers is a few ns

Phase-space representation

S, N plane For the Nd3+:YAG laser

06/01/2016 26

What is D? D=(dI/dt)/I+1

Turn-on of a multi-mode laser

06/01/2016 27

Parabolic gain profile:

From Ohtsubo (Semiconductor lasers)

• Winner takes all: after

transient “mode-competition”,

the mode with maximum gain

coefficient wins.

• But non-transient competition

has been observed.

• More advanced gain models

allow explaining non-transient

competition.

06/01/2016 28

Triangular signal: LI curve

Slow “quasi-static”

current ramp (T=200 ns)

0 500 1000

0

0.5

1

1.5

Time (ns)

0.5 1 1.5

0

0.1

0.2

0.3

0.4

S

Pump parameter,

S

06/01/2016 29

With a fast ramp: turn-on delay

Tredicce et al, Am. J. Phys., Vol. 72, No. 6, June 2004

0 50 100

0

0.5

1

1.5T=20 ns

Time (ns)

S

Experiments

Dynamical hysteresis

Simulations Experiments

06/01/2016 30

0.5 1 1.5

0

0.1

0.2

0.3

0.4

S

Pump

Influence of gain saturation

=0 =0.01

06/01/2016 31

0.5 1 1.5

0

0.1

0.2

0.3

0.4

0.5

0.5 1 1.5

0

0.1

0.2

0.3

0.4

0.5

Pump Pump

S

N

sp

p

NSG

dt

dS

1

1 GSNdt

dN

N

1

S

NSNG

1),(

Gain saturation takes into account phenomenologically several effects

(spatial and spectral hole burning, thermal effects)

Damped

relaxation

oscillations

Injection current modulation

Digital Analog

06/01/2016 32

0 10 20 30 40 500

0.5

1

1.5

0 10 20 30 40 500

0.5

1

1.5

2

Time (ns) Time (ns)

0.5 1 1.5 2

0

0.2

0.4

0.6

0.8

1

S

Direct modulation allows to encode information in the laser

output power (“amplitude modulation”)

Why modulation is important?

06/01/2016 33 Adapted from H. Jäckel, ETHZ

Optical waves can be modulated in Amplitude, Phase and in

Frequency in order to carry information

0 2 4 6 8 100

1

2

3

4

0 1 2 3 40

1

2

3

4

Weak sinusoidal modulation: influence of the modulation frequency

Tmod= 0.1 ns 0.5 ns

06/01/2016 34

The laser intensity (S = photon density) is modulated at the

same frequency of the pump current (), but the phase of the

intensity and the current are not necessarily the same.

Time (ns)

S

= dc + A sin modt

0 2 4 60

1

2

3

4

0.281 ns (=TRO)

For =1.5: RO = 3.56 GHz dc = 1.5, A=0.1

Small-signal modulation response

06/01/2016 35

Oscillation

amplitude

mod/ RO

diode-pumped

Nd3+:YAG laser

Modulation frequency (KHz)

Analytical

expressions

can be

calculated

from the

linearization

of the rate

equations.

Resonance at mod = RO

06/01/2016 36

Np

RO

1

For a semiconductor laser (VCSEL)

Adapted from R. Michalzik (VCSELs 2013)

Large-signal modulation response

06/01/2016 37 Adapted from J. Ohtsubo

Experiments Simulations

tmt mod0 sin1)(

Hysteresis induced by strong modulation

06/01/2016 38

• The figures represents the maxima of the oscillations as a function of the

normalized frequency near the onset of hysteresis (left), and far away from

the onset of hysteresis (right).

• Hysteresis cycle obtained by slowly changing the modulation frequency

forward (full line) and then backward (broken line).

• The additional smaller jump near ωmod/ωRO =1/2 is a signature of another

resonance.

• The laser is a Nd 3+:YAG laser subject to a periodically modulated pump.

Summary

The simple rate equation model for the photon and carrier densities allows to understand the main features of the laser

dynamics when the injection current varies:

─ The turn on delay & relaxation oscillations

─ The LI curve (static & dynamic)

─ The modulation response (small and large signal response)

06/01/2016 39

TF test Semiconductor lasers are described by two rate equations, one for

the photon density and another for the carrier density.

The laser relaxation oscillation (RO) frequency is proportional to the

injected current.

The delay in the laser turn-on is independent of the current ramp.

The modulation bandwidth has a resonance at the RO frequency.

Small and large amplitude modulation result in sinusoidal oscillation

of the output power.

The relative phase of the output power and input signal depends on

the modulation frequency.

In a multimode model with a parabolic gain profile, mode

competition is a transient dynamics.

06/01/2016 40

-ALPHA FACTOR, LINEWIDTH & INTENSITY NOISE

-OPTICAL PERTURBATIONS (INJECTION, FEEDBACK) -POLARIZATION INSTABILITIES

RATE EQUATION MODEL FOR THE OPTICAL FIELD

06/01/2016 41

Schematic representation of the change of magnitude and

phase of the lasing field E due to the spontaneous emission of

one photon.

06/01/2016 42

Laser linewidth

2ES

yx iEEE Ex

Ey

• The line-width of gas and solid-state lasers is well

described by the classic Schawlow-Townes formula (f1/P)

Schawlow-Townes formula

Physical interpretation:

• In each round-trip, some noise (spontaneous emission) is added to the

circulating field

• It changes the amplitude and the phase of the field.

• Amplitude fluctuations are damped: the power returns to values close

to the steady state.

• For phase fluctuations, there is no restoring force.

• Therefore, the phase undergoes a random walk, which leads to phase

noise, which causes a finite line-width.

• But the line-width of semiconductor lasers is significantly higher.

06/01/2016 43

c=half-width

of the cavity

resonance

out

cST

P

h2

• In diode lasers the enhancement of the line-width is due to

the dependence of the refractive index (n) on the carrier

density (N)

SNn

• Henry introduced a phenomenological factor (α) to account

for amplitude–phase coupling.

• The linewidth enhancement factor α is a very important

parameter of semiconductor lasers. Typically =2-5

06/01/2016 44

ST 21

Rate equation for the slowly-varying optical field

Photon density Complex field

06/01/2016 45

yx iEEE

N

sp

p

NSN

dt

dS

1

1

N

sp

p

NENi

dt

dE )1)(1(

2

1

Langevin

stochastic

term:

complex,

uncorrelated,

Gaussian

white noise

factor

2ES

yyx

y

xyxx

DEENkdt

dE

DEENkdt

dE

))(1(

))(1(yxi

N

sp

p

NDk

0 ,

2

1

Derivation of the equations: Ohtsubo Cap. 2

tietEt 0)()(

Rate equation for optical phase

06/01/2016 46

yyx

y

xyxx

DEENkdt

dE

DEENkdt

dE

))(1(

))(1(

)(12

)(11

tNdt

d

tDSNdt

dS

p

S

p

iyx eSiEEE

• The instantaneous frequency

depends on the carrier density.

• is a “slave” variable.

• The noise sources are not

independent.

Ex

Ey

GSNdt

dN

N

1

These equations allow to understand the optical spectrum of a semiconductor laser

• In EELs: L= 200–500 μm =100–200 GHz. Because the

gain bandwidth is 10–40 THz 10–20 longitudinal modes.

• The line-width of each longitudinal mode depends on the alpha-

factor and is of the order of 10 MHz.

06/01/2016 47

Multimode and single-mode

optical spectra (from J. M. Liu, Photonic devices)

FFT of E(t)

These equations also allow to understand the Relative Intensity Noise (RIN): FFT of S(t)

06/01/2016 48

The laser output intensity is detected by a photo-detector, converted to

an electric signal and sent to a RF spectrum analyzer. The RIN is a

measure of the relative noise level to the average dc power.

Adapted from R. Michalzik (VCSELs 2013)

Peak at the relaxation

oscillation frequency NpRO

1

Experimentally measured (VCSEL) Numerical expression

Adapted from J. Ohtsubo

-DIODE LASER LINEWIDTH & INTENSITY NOISE

-RESPONSE TO OPTICAL PERTURBATIONS

-INJECTION

-FEEDBACK

RATE EQUATION MODEL FOR THE OPTICAL FIELD

06/01/2016 49

06/01/2016 50 Adapted from M. Sciamanna PhD thesis (2004)

Optical injection Optical feedback

Optical isolator

“Solitary” diode lasers display a stable output (only transient

oscillations) but they can be easily perturbed by injected light

and can display sustained periodic or irregular oscillations.

Optical perturbations

51

• Two Parameters:

o Injection ratio

o Frequency detuning = s- m

• Dynamical regimes:

o Stable locking (cw output)

o Periodic oscillations

o Chaos

o Beating (no interaction)

m s

Optical Injection

06/01/2016 Adapted from J. Ohtsubo

Detection system

(photo detector,

oscilloscope,

spectrum

analyzer)

=m

52

)( )1)(1(2

1inj tDPEiENi

dt

dE

p

2 1 Eμ-N-Ndt

dN

N

optical injection

from master laser

Pinj: injection strength

=s-m: detuning

spontaneous

emission

noise

: pump current parameter Typical parameters:

= 3, p = 1 ps,

N = 1 ns, D=10-4 ns-1

Model for the injected laser

N

spND

0

DENi

dt

dE

p

)1)(1(2

1Without injection:

With injection:

Optical field (t) = E(t) exp (ist); E(t) = slowly varying amplitude

Injection locking

06/01/2016 53

= s- m

Beat

frequency

=0

Experimental verification

Nd 3+:YAG laser Model prediction:

the locking range is

proportional to the relative

injection strength.

06/01/2016 54

Injection locking increases the resonance frequency and the modulation bandwidth

Outside the injection locking region: regular intensity oscillations

06/01/2016 55 S-C Chan et al, Optics Express 15, 14921 (2007)

The frequency of

the intensity

oscillations, f0,

can be controlled

by tuning the

injection strength

and the detuning.

C. Bonatto et al, PRL 107, 053901 (2011), Optics & Photonics News February 2012,

Research Highlight in Nature Photonics DOI:10.1038/nphoton.2011.240 06/01/2016 56

Also outside the locking region: ultra-high intensity pulses (“optical rogue waves”)

Distribution of pulse amplitudes

for different injection currents Time series of the laser intensity

Outline

Block 1: Low power laser systems Introduction

Semiconductor light sources

Models

Dynamical effects

Applications: telecom & datacom, storage, others

Brief summary to come back after the winter holiday

• SCLs have many applications (in dollars, diode lasers are about 50% of

laser market).

• Operating principle: electron-hole stimulated recombination (LEDs:

spontaneous recombination)

• Direct compound semiconductors (GaAs, InGaAs, etc.) are used: the

bandgap determines the emitted wavelength.

• Two main geometries: edge-emitting & VCSELs.

• SCLs are well described by simple “rate equation” models.

06/01/2016 58

Diode laser model (& other class B lasers)

06/01/2016 59

N

sp

p

NSN

dt

dS

1

1

NSNdt

dN

N

1

Np

RO

1

Model for single-mode diode laser: stochastic rate equation for the slowly-varying optical field

06/01/2016 60

Ex

Ey

tietEt 0)()(

2

ES yx iEEE

DENidt

dE

p

)1)(1(2

1

yx i

SNn

N

spD

Influence of optical perturbations

• Injection from another laser

– Injection locking

– Regular oscillations

– Chaos & extreme optical pulses

06/01/2016 61

Isolator

)( )1)(1(2

1inj tDPEiENi

dt

dE

p

• Self optical feedback

Optical feedback induced regimes

• Regime I: line-width narrowing/

broadening (depending on the

phase of feedback),

• Regime II: mode-hopping,

• Regime III: single-mode narrow-

line operation,

• Regime IV: coherence collapse,

• Regime V: single-mode operation

in an extended cavity mode.

06/01/2016 62 Adapted from S. Donati, Laser and Photonics Rev. 2012

Tkach and Chraplyvy diagram

Lext

Feedback induced instabilities

06/01/2016 63 Adapted from J. Ohtsubo

negligible small feedback

periodic oscillations

with weak feedback

chaotic oscillations with

strong feedback

(coherence collapse,

Regime IV)

I/Ith = 1.3

Optical feedback effects on the LI curve

Coherent feedback Incoherent feedback

06/01/2016 64

Adapted from R. Ju et al,

Feedback-reduced threshold: the

amount of reduction quantifies the

strength of the feedback.

Close to the laser threshold: Low Frequency Fluctuations (LFFs)

65 I. Fischer et al, PRL 1996

I/Ith = 1.03

LFFs: Complex dynamics, several time-scales

06/01/2016 66

From M. Sciamanna (PhD Thesis 2004)

Recovery after a dropout: in steps

of with relaxation oscillations c

Lext2

If Lext = 1 m = 6.7 ns

Lext

67

DetEEGikdt

dE i

0)( )1)(1(

feedback 2 1 EGNdt

dN

N

noise

R. Lang and K. Kobayashi, IEEE J. Quantum Electron. 16, 347 (1980)

06/01/2016

= feedback strength = feedback delay time = pump current (control parameters)

Single-mode Lang and Kobayashi Model

DENikdt

dE )1)(1(Solitary laser

With optical feedback

c

Lext2

Optical field (t) = E(t) exp (i0t); E(t) = slowly varying amplitude

p

k2

1

N

spD

External cavity modes (ECMs)

• (t) = E(t) exp (i0t)

E(t) = E0 exp [i(-0)t] ; N(t) = N

Monocromatic solutions: (t) = E0exp (i t)

– Stable modes (constructive interference)

– Unstable models (destructive interference)

06/01/2016 68

21 C

arctansin0 C

• The number of ECMs increases with:

– The feedback strength

– The length of the external cavity

cos1k

N

N

NE

2

0

Good agreement model-experiments

69

With a “fast” detector: pulses; with a “slow” detector: dropouts

0.4 0.45 0.5 0.55 0.60

0.5

1

1.5

2

2.5

Time (microseconds)

Lase

r in

tensit

y (a

rb.

unit

s)

Experiments Stochastic simulations

Deterministic simulations

I. Fischer et al, PRL 1996

A. Torcini et al, Phys. Rev. A 74, 063801 (2006)

J. Zamora-Munt et al, Phys Rev A 81, 033820 (2010)

Beyond single-mode equations: model for a multi-mode laser

06/01/2016 70

Koryukin and Mandel, PRA 70, 053819 (2004)

Other models have been proposed to explain observations.

Mode switching with weak optical feedback

06/01/2016 71 Furfaro et al, JQE 41 609 (2005)

Research at our lab: optical neurons

06/01/2016 72

Optical spikes

characterise spikes and compare

to those of biological neurons

Extreme feedback-induced optical pulses

06/01/2016 73

Time (in units of )

Simulations

I. Fischer et al, PRL 87 243901 (2001)

J. M. Aparicio Reinoso et al, PRE 87, 062913 (2013)

Observations

An example of an application of feedback-induced chaos: random bit generation

06/01/2016 74 Adapted from I. Kanter et al, Nature Photonics 2010 & OPN

After processing the

signal, arbitrarily

long sequences can

be generated at the

12.5-Gbit/s rate.

06/01/2016 75

06/01/2016 76

Laser synchronization

06/01/2016 77

POLARIZATION INSTABILITIES

06/01/2016 78

EELs: linearly (TE) polarized output

06/01/2016 79

threshold current

Log scale

Polarization

switching can

be induced by

polarization-

rotated

feedback.

In some VCSELs: polarization switching (PS)

0.9 1.0 1.1 1.2 1.3 1.4 1.5-0.1

0.0

0.1

0.2

0.3

0.4

0.5 Drive current X Polarisation

Y Polarisation

Inte

nsity (

a. u.)

Time (ms)

06/01/2016 80

Adapted from Hong and Shore, Bangor University, Wales, UK

• Circular cavity geometry:

two linear orthogonal

modes (x, y).

• Often there is a

polarization switching

when the pump current is

increased.

• Also hysteresis: the PS

points for increasing and

for decreasing current are

different.

• The total output power

increases/decreases

monotonically with pump.

Adapted from J. Kaiser et al, JOSAB 19, 672 (2002)

Polarization-resolved LI curve

06/01/2016 81

Current-driven PS Stochastic PS

Pump current, I/Ith -1 Pump current, I/Ith -1

Stochastic PS

• Anti-correlated fluctuations of the two polarizations.

06/01/2016 82

• Bistability + noise induced switching

Current-driven PS

• Type 1: from the Y (low freq) X (high freq) polarization

• Type 2: from the X (high freq) Y (low freq) polarization

06/01/2016 83

• Several models have been proposed to explain these PS

Adapted from B. Ryvkin et al, J. Opt. Soc. Am. B 1997

Thermal shift of the gain curve

Adapted from M. Sciamanna PhD Thesis (2004)

X Y

84

When the pump current increases Joule

heating different thermal shift of the

gain curve and of the cavity modes

It explains Y (low freq) X (high freq) Type I PS only

Birefringence: the

polarizations have

different optical

frequencies

Adapted from Y. Hong et al,

Elec. Lett. (2000)

DEiENi

dt

dEpa )( )1)(1(

Martin Regalado et al, JQE 33, 765 (1997).

dichroism birefringence

Assumes a four-level system in which e/h

with spin down (up) recombine to right

(left) circularly polarized photons:

Carrier

recombination

spin-flip

rate Stimulated

recombination Pump:

carrier

injection

2|| 2)()( ENNNN

dt

dNNjN

VCSEL spin-flip model

2/)(

2/)(

EEiE

EEE

y

x

a >0 and large

birefringence: Y X

a

The SFM model also predicts

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• Polarization bi-stability

• Polarization instability

• PS induced by optical injection

• Irreversible PS

Martin-Regalado et al, JQE 33, 765 (1997)

Experimental observation of irreversible PS

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Parallel optical injection

Orthogonal optical

injection

A. A. Qader et al, PTL 25, 1173 (2013)

Polarization switching can be exploited for generating all-optically regular square-waves

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The switching rate is controlled by the coupling delay time.

Also, in EELs, regular square-wave switching can be

generated via orthogonal (polarization-rotated) feedback.

Two mutually coupled lasers

From C. Masoller et al, PRA 84, 023838 (2011)

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• The first example of a

free-running diode laser

(QD VCSEL)

generating chaos.

• The underlying physics

comprises a nonlinear

coupling between two

elliptically polarized

modes.

2.0 mA

2.6 mA

Models beyond ordinary differential equations (ODEs)

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Larsson and Gustavsson, Chapter 4 in VCSELs Editor: Michalzik (2013)

Partial differential equations (pdfs) take into account transverse effects

Transverse effects and polarization

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When the first-order

transverse mode

starts lasing, it is, in

general, orthogonally

polarized to the

fundamental

transverse mode.

The models that take into account polarization and transverse effects depend on the transverse

optical and current confinement

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Gain guided

Index guided

Index-guided model

• In Index-guided VCSELs the mode transverse profile is fixed by the build-in

optical cavity

• Assuming that only two radially symmetric modes can lase:

– j=0 LP01

– j=1 LP11

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i=0 x

i=1 y

And the carrier equation has to be modified to include a carrier diffusion term.

If the VCSEL is gain-guided, 3D modeling is needed (for details: Chapter 3 of

book VCSELs).

ijijij

ijFEgik

dt

dE,,,

,)1)(1( One ODE for each mode:

Take home message • The dynamics induced by optical perturbations (injection from other laser,

optical feedback) can be useful for a number of applications.

For appropriated parameters:

- Optical injection induces “injection locking”: the slave laser emits at the same

frequency as the master laser.

- Optical injection increases the relaxation oscillation frequency (and thus, the

laser modulation bandwidth).

- Optical injection can induce regular oscillations.

- Optical feedback induces single-mode emission and reduces the laser line width.

• However, both, optical feedback and optical injection can lead to chaotic

intensity oscillations with parameter regions where rare and extreme

intensity pulses can occur.

• VCSELs can display a complex interplay of transverse and polarization

effects; polarization instabilities can also be exploited for applications (for

example, all-optical square waves).

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VF Test

In semiconductor laser models, the alpha-factor takes into account

phenomenologically the change in the refractive index induced by

the variation of the carrier density.

In the injection locking regime the laser emits its natural wavelength

but with larger output power.

Strong optical feedback can be used for achieving single-mode

emission.

The external cavity modes are coexisting monochromatic steady-

state solutions, the emitted wavelength depends on the feedback

parameters.

In VCSELs the polarization switching can be due to thermal effects.

In VCSELs a PS always occurs when the pump current is increased,

and is accompanied by a change in the transverse optical mode.

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THANK YOU FOR YOUR ATTENTION !

Universitat Politecnica de Catalunya

http://www.fisica.edu.uy/~cris/