Download Edge emitting semiconductor lasers

2443-2
Winter College on Optics: Trends in Laser Development and Multidisciplinary
Applications to Science and Industry
4 - 15 February 2013
Edge emitting semiconductor lasers
I. Montrosset
Politecnico di Torino
Italy
Edge emitting semiconductor
lasers
Ivo Montrosset
Department of Electronics and Telecommunication
POLITECNICO DI TORINO
• Suggested book
• L.A.Coldren, et al., Diode Lasers and
Photonic Integrated Circuits, J.Wiley, 1995
• Or the new edition
OUTLINE
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Introduction
Semiconductor Laser ingredients
Rate equation analysis
DBR tunable lasers and DFB
Laser dynamic modeling with FDTW
Examples of mode locked lasers
Laser diodes structures
Edge emitting laser diode
Red: the active material
The optical waveguide
Vertical cavity surface emitting laser (VCSEL)
The optical cavity
Laser characteristics
Output
power
mW
Typical output power/ current curve for a semiconductor laser
Power spectrum
Fabry Perot laser
Power spectrum for a multimode and single mode laser
Dynamic characteristics: small signal modulation & eye diagram
The electronic oscillator
Reference
section
Amplifier
Noise
s
+
A
s + rE
Signal out
tE
E
rE
rE
Feedback
Ring laser
Condition for stationary solutions
A ( s + rE ) = E
E =
H s
1 − G
,
Oscillation condition:
G
=
1
G
= rA
= Loop gain
Numerator: noise term
Denominator: loop gain characteristics
Edge emitting or VCSEL laser cavity
E+
FL
FR
EReference section
rL
rR
Field reflectivities
Field equations:
E
−
= rR E
+
+ FR
Field equation
E
E
+
= rL E
−
+
+ FL
=
rL F R + F
1 − rL rR
L
Numerator: noise term
Denominator: active cavity
characteristics
Oscillation condition:
rL rR = 1 = G
rL,R can be >/< 1 depending on the
structure
Power spectrum
Power spectrum:
S
E
(  ∝
E(  )
2
=
| Hs
|2
1 − G(  )
2
1
G (ω )
∠
S
1
G (ω )
E
2· (integer)
(ω )
The peaks are the
Modes of the cavity

Laser ingredients
• The longitudinal optical cavity
• The semiconductor optical waveguide to confine
the field in the transversal direction (avoid
diffraction) and to confine the carriers
• Active semiconductor material: photon
amplification and noise by e-h recombination
process
The semiconductor materials
The semiconductor laser structure is realized by epitaxial growth of material
typically with the same lattice constant and with different energy gap
Direct
energy gap
AlGa As/GaAs
InGaAsP/InP
Coldren,Corzine&Masanovic
Radiative transitions in direct Eg semiconductors
E2
E1
Absorption
- photodiode
E2
E1
Spontaneous
emission
- LED
E2
E1
Photon energy:
ω
= E
2
− E
1
Gain -> stimulated emission > absorption
Stimulated
emission
- amplifier
- laser
Direct
Energy
gap
material
Band structure for GaAs and InP
Carrier distributions under injection
E
E
E
fc(E)
Ec0
Ev0
electrons
Efc
Charge neutrality
N=P
Efv
holes
1-fv(E)
Density of states Fermi functions
Density of cb-electrons: N(Ef ) =
Density of vb-holes:

∞
Ec0
P(E f ) = 
E v0
−∞
Carrier distribut.
 c (E)f(E, Ef )dE
 v (E)(1 − f(E, Ef ))dE
Optical gain in semiconductors
Optical gain requires stimulated emission > stimulated absorption
Can be fulfilled for quasi-thermal equilibrium
2 5
2 0
1 5
Efc
quasi-Fermi level
for electrons
Efv
quasi-Fermi level
for holes
1 0
5
0
-4
-3
-2
-1
0
We have gain when
1
2
3
Efc – Efv > Eg
4
Bernard-Duraffour inversion condition
Gain T = 0 K
T>0K
Photon energy
Eg
Efc- Efv
Bulk gain function
g ∝  r (f 2 − f1 )
Measured gain
FP41AB07
200
G/gamma=g-alfa/gamma (cm-1)
0
-200
-400
-600
-800
-1000
-1200
1.52
1.54
1.56
1.58
1.6
Wavelength (m)
Same considerations applies for the others semiconductor materials
1.62
1.64
1.66
-6
x 10
Density of States semiconductor materials
3D
bulk
semiconductor

Eg
2D
E

quantum well
E
1D
WHY LOWER
DENSITY OF
STATE ARE
BETTER ?

quantum wire
E
0D

quantum dot
E
Carriers filling – bulk vs. QW
Bulk – 3D:

c
0
Ec0
QW – 2D:
Carrier distribution at T=0 K
E
Ef

Ec0
Fixed E f : N 2D
Diff. gain :
∂g
∂g
>
∂N 2D ∂N 3D
f
) dE
E
Ef
∂Ef
< N 3D 
∂N
 ρ (E ) f (E; E
Let compare 3D and 2D
material at the same Ef
c
0
N =
2D
∂Ef
>
∂N
Easier to invert population
Less current
3D
Higher gain sensitivity to carrier variation ->
higher modulation bandwidth with QW!
Calculated gain spectra
ColdrenCorzine , p.165
The semiconductor waveguide: 1D
• Should be able to maximize the interaction by carriers (e&h) and
photons by confining them in the same as small as possible region
• This is obtained by the realization of a double heterostructure diode
Carriers well
confined in the
low Eg region
Refractive index
increases if Eg
decreases
Potential well
Optical
waveguide
Photons and carriers confined in the same region
Total reflection
Semiconductor waveguides : 2D
confinement of current, carriers and photons
Important to realize in the lateral direction structures able to control not
only the carrier and photon confinement but also the current flow.
Gain guided
I
Current spreading: no control
Lateral carrier diffusion in the
active layer
Lateral guiding only due to
carriers -> gain non uniformity
Index guided
I
Current laterally localized by multi-junctions
No lateral carrier diffusion in the active layer:
higher carrier density N
Lateral guiding due to refractive index
change: higher photon density
Examples of advanced semiconductor waveguides
(UGlasgow)
The heterostructure
Ridge waveguide
Field computed with FEM
MQW waveguide with vertical field optimized
for optical waveguide coupling (UGlasgow)
Band structure
eln
holes
Active region
Dummy waveguide
Semiconductor waveguide analysis
• The semiconductor waveguide can be analyzed as lossless
dielectric waveguide with a perturbation due to the presence
of losses (scattering, doping,..) and of the carriers in the
active region
• The electromagnetic waveguide modes of the lossless
structure are computed than the other effects are added
perturbatively assuming that the field distribution remain
unchanged and the perturbations only affect the modes
propagation constant
• The two steps of the analysis consist in:
– evaluate the modal fields and their propagation constants
– estimate the contribution of the perturbation (losses, gain, etc.) to
the propagation constant variation
Lossless Wave guides: approximate analysis
Wave equation:
∇
~
E + k
2
2
0
~
ε ( x, y )E = 0
For weakly guiding optical waveguides
~
E ≈ e
Quasi-TE modes:
-TM modes
H
U (x, y ) Z (z)
y
≈ h
y
U (x, y ) Z (z)
Applying the separation of variables
Transverse mode equation:
∇
2
t
U(x,
≡ β
y) + k
2
= k
2
0
2
0
n
ε ( x , y ) U(x,
y)
= β
2
U(x,
y)
where  is the mode propagation constant
and n is the mode effective index
2
Longitudinal mode equation:
∂ 2
Z (z) + β
∂z 2
Z ( z ) ∝ exp[  j(  z)]
2
Z (z) = 0
Propagation constant perturbation correction
~
2    = k
From perturbation theory
~
~
 =  +   ≅ k 0n + k 0

xy
=
and
i
=

active
where

n2 W -. Lossless waveguide
~ = n 2 +  ~
w
For ε of the form
 (α
2
0
Δ n~
xy

 ~ | U | 2 dxdy

(gain)
| U | 2 dxdy
| U | 2 dxdy
( x, y )

U(x,
) U(x,
y)
y)
2
− j
1
2

i
is the waveguide confinement
factor when assuming χ ( gain )
constant in the active region
region

| U | 2 dxdy
dxdy
2
dxdy
are the internal losses
Propagation constant, group velocity and LEF
 is usually expressed as:
β ≅ k0 n + j 12 (Γxy g −αi )
Expanding the propagation constant around a reference point (r, Nr)
we obtain:
∂β
∂β
β ≈ β (ω r , N r ) +
(ω − ω r ) +
(N − Nr ) .
∂ω
∂N
where
∂ β
∂ω
and
where
≅
∂β
∂N
α
1
v g
≅ k 0Γ
≡ −
∂
∂N
xy
∂ Re
∂N
n~
Δ n~
( gain
∂ Im
∂N
n~
)
=
j
1
2
= − 2 k
(1 +
0
jα )Γ
xy
∂g
.
∂N
∂n
∂N
∂g
∂N
Where  is the active material linewidth enhancement factor -> LEF -> H
The optical cavities
Fabry-Perot
3S-DBR
A
P
P
A
Distributed reflector
DFB
A
VCSEL
Transmission matrix
General complex cavity
P
A
P
The Fabry Perot cavity
Oscillation condition
=m
FP– Field power spectrum below threshold
FP41AB07
-35
measured
-40
β ≅ k0 n+ j 12 [Γ(1+ jαH ) g −αi ]
The difference in the peaks is
due only to difference in the
gain function
Measured optical power
(dBm)
-45
-50
-55
-60
Ith
-65
0.5Ith
-70 0.3Ith
-75
-80
-85
1.45
0.1Ith
1.5
1.55
1.6
1.65
1.7
-6
x 10
Wavelength (m)
Injection ->
carrier density increase -> gain increase-> peaks increase
Peaks shift -> LEF  0 -> refractive index decrease
Peaks separation-> Free Spectral Range
 =
vg
2L
The DBR cavity
|rR|
Effective grating length
Bragg grating reflectivity 
Oscillation condition
Coldren&Corzine
DBR resonance condition and gain margin
Coldren&Corzine
In FP &DBR cavities g is the same
In FP cavity =0 (constant mirror loss) Weak mode selectivity
In DBR cavity >> g strong mode selectivity -> Stronger peaks difference
SLM-> Single Longitudinal Mode
How to correlate previous results with the
current injection: the rate equations (RE)
• The variables are:
– The average photon density in the active region: NP
– The average carrier density in the active region: N
– The injected current: I
• The RE can be written:
– In each sections of the cavity after its longitudinal discretization
– Averaging the variables in all the cavity
• The first case is more accurate (DFB) and the other
more simple numerically and reasonably accurate
• We consider now the second case
Carrier balance in the active region
Rate of stimulated emission: Rst
Total number of generated photons
v g (N pout − N pin ) A =
vggNp V
g depends on (N, ); the value to
be used is that at the  of operation
of the laser -> g(N, op):
Semiconductor gain approximations
For bulk material
g ≈ g peak ≈ a ( N − N tr )
For QW material
g(N, N p ) =
 N + Ns 
g0

ln
1 +  N p  N tr + N s 
 N 
g0

g(N, N p ) ≅
ln
1 +  N p  N tr 
Gain saturation
The Rate Equation for the carriers density is:
dN
I
= i
− R(N) − v g gN p
dt
qV
Rate equation for the lasing mode average
photon density in the active region
The effects to be considered are:
-The stimulated emission
-The spontaneous emission coupled into the lasing mode
-The losses in the cavity: intrinsic and mirrors
For what concern the Stimulated emission we can write the relation between
the total rate of stimulated emission (RstV), the variation of the total number of
carriers (NV) and the variation of the total number of photons NPtot
d ( NV ) d ( N Ptotal )
RstV = −
=
dt
dt
NP
N Ptotal
=
VP
Been relevant for the carrier rate
equation the Photon density NP
we can define:
where VP is the “volume” occupied
by the photons
V 
d (N P )
V
= Rst
= v g g   N P = v g (g )N P
dt
VP
 VP 
 g is the cavity
average modal gain
Losses and Spontaneous emission
Average Cavity Losses:
characterized by photon decay rate (p=1/(vg T)) where (T) are the cavity modal
losses: intrinsic (i) and mirror (m=-ln(|r1 r2|)/L)
Average Spontaneous Emission density coupled to the cavity mode.
Can be obtained from the total spontaneous emission ( Rsp V) normalized respect
to the volume occupied by the photon (Vp) and taking into account that only the
small part (sp) is coupled into the spectral interval of the lasing mode:
sp Rsp V/ Vp =  R’sp
The rate equation for the average photon density in the active region is:


1
=  v g g −  N p + R 'sp


dt

p 

dN p
RE summary:
Rate Equations for the total number of carriers and photons
dN
dt
= 
T
i
I
− R(N)V
q
− v

1 

=
v g g −
N pT


dt
 p 

Static solution (see Coldren Corzine)
dN
pT
m
i + m
P0 ≅ h  F
N p Vp
or
p
g
gN
'
sp
+ R
N
pT
=
p
V

1 
sp
− v
p
Multimode Rate Equations
dN
dt
I
− (R
i
qV

=  Γ m v gm g


= η
dN
pm
dt
N
pm
=

pm
n .r
) −

v
g
gm
m
N
m
− v
1
−
m

1
+ R
sp
τ
R
sp
g
g
pm

 N


pm
+ Γ
pm
)
spm
m
(N,
N
m
R
R
'
spm
pm
sp
g
g(N)
FP lasers P-I & Spontaneous emission factor effect
sp=10-4
CS=sp
sp=3 10-4
SMSR
Measurement of modulation response
I (t) = I
0
+ | I 1 | cos(
 t)
Network analyzer
| P1 |
PD
2
/ | I1 |
2
i pd (t ) ∝ P1 (t )
LD
ω
P ( t ) = P 0 + P1 ( t )
P 1 ( t ) = Re[
jω t
P1 ( ω ) e
] = | P 1 ( ω ) | cos(
ω t + θ
p
)
Usually is measured the so called electrical modulation response
H
H
(ω
(ω )
= 0
2
) dB
= 20 Log
P1 ( ω )
P1 ( ω = 0 )
Small-signal modulation response
From the RE linearization (see Coldren & Corzine) and
assuming a small harmonic current excitation over the
bias, the system of differential equation can be solved
analytically and the impulse response function is
obtained:
P1 ( ω )
ω R2
 2R
H ( ) =
=
= 2
P1 ( ω = 0 )
Δ
 R −  2 + j γ
whose parameters are:
Relaxation oscillation frequency

2
R
≡ γ
NP
γ
PN
Damping rate  ≡ γ
and f3dBmax= 2 2 / K
K
+ γ
NN
= 4 π
γ
NN
+ γ
2
τ
p
PP
PP
v
≈.
aN

. Kf
=
Γ a

1
+

a

g
p
2
R

 ,

p
+ 0
p
,
Experimental results small signal modulation
Coldren &Corzine
 = Kf
2
R
+ 0
The DBR laser: tunability
|rR|
3S DBR
Oscillation condition
Bragg
grating
reflectivity 
Tuning of a DBR laser
Wavelength of mode m:
m λ m ≈ 2 ( n a L a + n p L p + n DBR L eff )
Relative change m due to changes in refractive index na, np, nDBR
Δλm
λm
=
Δ n a L a + Δ n p L p + Δ n DBR L eff
n a L a + n p L p + n DBR L eff
Coldren&Corzine
Index change due to current injection:
For active section the carrier density is clamped Δ n a ≈ 0
For passive sections
ηi I j
qV
= R(N j ) ,
j = p , DBR
j
∂n
Δn j =
N (I j ) ,
∂N
j = p , DBR
Tuning of the DBR grating
Relative change of the Bragg wavelength:
Δ λ Bragg
λ Bragg
λBragg = 2 nDBR Λ
Δ n DBR
1 ∂ n DBR
=
=
Δ N ( I DBR )
n DBR
n DBR ∂ N
Δ λ
7-10 nm
Grating
IBRAGG
current
Discontinuous line for grating current injection only
Continuous line with proper grating and phase section currents injection:
synchronous shift of modes and Bragg wavelength
DBR with wide tunability:SG DBR
(UCSB)
Change in the aligned peaks
Coldren & Corzine
DBR with wide tunability:GCSR
KTH
Selection of the peak by current
injection in the coupler
Respect to SG-DBR
Similar tunability
More output power
Reduced spectral characteristics
The DFB laser
I
A1=0
B2=0
The periodic structure (grating of
period ) produce a distributed
coupling between the forward (A)
and backward (B) propagating
waves represented by

~
 dA(ω , z )
=
−
j
− β 0 A(ω , z ) − jkB(ω , z )
β
 dz

 dB(ω , z ) = + j β~ * − β B(ω , z ) + jkA(ω , z )
0
 dz
(
(
)
)
k the coupling coefficient and
0=2/
For lasing: A1=B2=0
0 = T11 A2
B1 = T21 A2
Lasing condition T11=0
The DFB laser - II
Similarly for more complex cavities
cascading the transmission matrix of
each element
R1
R2
0 = T11 A0
Bm = T21 A0
DFB above threshold
The non uniform field
distribution along the cavity
significantly changes the 
parameter of the propagation
equation that depends on the local
power density
QWS-DFB
~
 dA(z )
=
−
j
 (N, N p ) − β 0 A(z ) − jkB( z )
 dz

 dB ( z ) = + j ~
 * (N, N p ) − β 0 B(z ) + jkA(z )
 dz
(
(
)
)
A longitudinal segmentation of the cavity is needed both for the static and
dynamic analysis.
For the dynamic analysis the previous equation can be transformed in a
time domain equation using a Fourier transform technique and linearizing
the frequency dependence of the propagation constant .
~
β (ω , N ) =
dβ (ω , N )
ω0 ~
neff (ω 0 , N ) +
(ω − ω0 )
c
dω
ω =ω
0
DFB dynamic propagation equations
after Fourier transform
 ∂
 +
  ∂z

 ∂
  ∂z −
 
0
1 ∂
 g (N ) − 
(
)
[n eff (N ) − n eff,0 ]a(z, t ) − jkb(z, t )
a
z,
t
=
−
j


v g ∂z 
2
c



1 ∂
 g (N ) − 

+ j 0 [n eff (N ) − n eff,0 ]b (z, t ) + jka(z, t )
b (z, t ) = 
v g ∂z 
2
c


ω
Γg(ω0, N) −α 
n~eff (ω0, N) = 0 neff (N) + j

c
2


Boundary conditions
a(0, t ) = R 0 b(0, t )
b(L, t ) = R L a(L, t )
Carrier rate equation
d
I (t )
N (z, t ) =
− AN (z, t ) + BN
dt
eV
[
2
2
S(z, t ) = a(z, t ) + b(z, t )
2
(z, t ) + CN 3 (z, t )]−
v g g (N )S (z, t )
1 + S
How to include the physical effects
The gain:
The dependence with  can be included in
the time domain using a numerical filter
~
~
 f z,t = (1− A) f z,t + Af z,t −Δt

rz ,t + A~
rz,t −Δt
rz,t = (1− A)~
The refractive index variation with the carriers
neff ( z, t ) = neff , 0 − Δn[N ( z , t ) − N 0 ]
The spontaneous emission in each section
2
Esp = β sp BN ( z , t ) Δt
with a random phase
Δn = α LEF
λ0 ∂g (N ( z, t ), t )
2π ∂N (z, t )
The dynamic propagation equations
The previous equations are very general and can be used to study the
characteristics not only of a DFB but also of a quite general guided wave
optoelectronic component when assuming:
- the parameters describing the propagation may change in the longitudinal
direction considering both active, passive, with and without grating sections
- different terminal boundary conditions (with reflection or not)
- static or dynamic excitation.
In practice all the structures discussed before and many others as:
- Lasers for pulse generation: Q-and gain switched, mode locked, ..
- SOA : semiconductor optical amplifiers
- SLED: super luminescent LED based on waveguide configuration
- etc…
Integrated semiconductor mode locked (ML)
lasers for short pulse generation
A short pulse is generated when the longitudinal modes of a laser
are locked in phase .
From Fourier analysis the pulse duration depends
on the number of locked modes and the repetition
rate from the mode frequency separation.
Ex. if
In the time domain:
where the repetition period is
How mode locking take place
The carrier pulsation at a frequency close to the cavity modes
FSR helps the power transfer and the phase locking between
the cavity modes.
How Mode Locking can be obtained:
- by current modulation at the FSR frequency: ACTIVE ML
- by a self induced modulation in a 2 sections lasers: PASSIVE ML
- by a combination of active and passive ML: HYBRID ML
- by self mode locking due to carrier modulation induced by mode
beating
Active ML
G: gain section
R1
P: passive section
R: reflector
f m ≈ FSR
The active section allow the lasing
The passive section can be used to set the FSR and to modify it by
current tuning
The reflector if is a grating it allows to define and tune the pulse
wavelength by current injection
The repetition rate is precisely defined by the modulation frequency
when around the FSR
Passive ML
R1
The active (G) section allow the cavity to lase and contributes with its
saturation to the ML pulse formation
The saturable absorber (A) reverse biased contribute to the pulse
formation with its fast recovery
The passive section (P) to set the FSR and to modify it by current tuning
The reflector (R) if a grating allow to define and tune the pulse wavelength
The repetition rate is reasonably stable around the cavity FSR
Mode locking take place if the
cavity operates in the condition:
dg
dN
A
dg
>
dN
G
Simulation results for passive ML
Hybrid ML






Has the advantage of:
- it requires a smaller modulation
current
4.5
Pulse width [ps]
-repetition rate stability and
accuracy of active ML when it
operates around the cavity FSR.
4
3.5
3
2.5
2
1.5
21.85 21.9 21.95
f [GHz]
22
22.05 22.1
59
Ex.1 of Active ML simulation results
Potenza in uscita
• Modulazion at
22.03GHz
• Pulse width
FWHM 1.74ps
180
160
140
Potenza [mW]
120
100
80
60
4
40
3.5
20
Larghezza degli impulsi [ps]
4.5
3
0
2.865
2.87
2.875
2.88
t [s]
2.5
2
2.89
2.895
−8
x 10
Spettro di potenza (faccetta dx)
1
21.9
21.95
22
22.05
Frequenza di modulazione [GHz]
10
22.1
0
10
-1
10
-2
Spettro [mW]
1.5
21.85
2.885
10
-3
10
-4
10
-5
10
-6
10
-7
10
-6
-5
-4
-3
-2
-1
f [Hz]
0
1
2
3
4
11
x 10
60
Ex.2 of Active ML simulation results
Potenza in uscita a destra
90
• Modulazion at
21.9 GHz
• Multiple peaks
80
70
Potenza [mW]
60
50
40
30
4.5
Larghezza degli impulsi [ps]
20
4
10
3.5
2.1
2.102
2.104
2.106
2.108
2.11
t [s]
2.112
2.114
2.116
2.118
2.12
−8
x 10
3
Potenza nel tempo
2.5
150
2
1.5
21.85
21.9
21.95
22
22.05
Frequenza di modulazione [GHz]
22.1
• Modulazion at
22.1GHz
• Narrow peaks
with strong
modulation
Potenza [mW]
100
50
0
2.42
2.43
2.44
2.45
2.46
2.47
t [s]
2.48
2.49
2.5
2.51
2.52
−8
x 10
61
Example of a realized device
• Hybrid ML laser with a saturable absorber
modulated at 40GHz
• H. Hertz Institute Berlin












62
Examples of self ML
Easy to be found in
long cavity or
external cavity lasers
with high LEF -> 4-6
SP= self pulsation
ML=mode locking