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Lightwave Generation and Modulation
1. Principle of semiconductor laser diodes
Fundamental Elements in Oscillator
Population inversion
(to provide gain mechanism)
External pumping
(to make gain sustainable)
Feedback
(to provide coherence)
Fundamental Elements in LD
Semiconductor band structure
Current injection and P-N junction
FP or DFB/DBR
2. Behavior model for semiconductor laser diodes
Key in the modeling of semiconductor laser diode: interaction between EM field and gain
medium.
Physical process in semiconductor laser diode:
Bias
Carrier
Poisson and
Carrier Continuity
Equations
Gain/Index
Schordinger
Equation and
Lasher’s Model
Photon
Maxwell Equation
Behavior model:
Carrier density balance equation:
Carrier density variation rate = Carrier density generation rate – Carrier density
consumption rate
dN
I

 Rnr  sp  Rst
dt
qV
Rnr  sp  AN  BN 2  CN 3  N /  c
Rst  v g gS / V
where: N is the carrier density, I is the bias current,  is the current capture coefficient
defined as the ratio between the current captured by the active region and the bias current,
q is the electron charge, V is the volume of the active region, A is SRH plus surface
recombination rate, B is bimolecular recombination plus spontaneous emission rate, C is
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Auger recombination rate,  c is the carrier life time,  is optical confinement factor, v g
is the group velocity of the light, S is the photon number, g is the material gain defined
as: a( N  N tr ) for bulk and a ln( N / N tr ) for quantum well active regions, respectively.
Photon number balanced equation:
Photon number variation rate = Photon number generation rate – Photon number
consumption rate
dS
S
 v g gS  Rsp 
dt
p
Rsp  Knsp v g g
where:  p is the photon lifetime defined as the total loss in the cavity 1 /[ v g ( cav   int )] ,
K is Petermann’s factor, n sp is the population inversion factor.
Photon phase rate equation:
d 1
1
  LEF (v g g  )
dt 2
p
where:  LEF is the linewidth enhancement factor defined as  4n /( g ) , n is the
refractive index change due to the gain change g from Kramers-Kroenig relation.
Approximations made:
a) Optical field is spatially homogeneous: optical mode exists and optical field are
spatially and temporally separable
b) Quasi-neutrality in active region
c) Uniform carrier density and transverse optical field distributions in active region
3. Steady-state analysis based on behavior model
At steady state:

qV
(
1
p
I
1
c
N
v g g
V
S
 v g g ) S  Knsp v g g
Approximate (ideal) solution: Rsp  0
2
If S  0 :
N
 c
qV
I
If S  0 :
g  g th 
1
v g  p
N  N th  N tr 
S
 p
I th 
q
qV
 c
1
1
, N tr exp[
]
av g p
av g p
( I  I th )
N th
Since output optical power is related to the photon number by: P  hvvg  cav S , we have:
P(
 cav
hv
) (
)( I  I th )
q
 cav   int
Below threshold, there is no optical power generated and the carrier density grows
linearly with the bias current. Above threshold, optical power grows linearly with the bias
current and the carrier density is clamped at its threshold value.
With spontaneous emission considered, numerical solution can be found by twodimensional root searching for N and S under any given static bias I . In this case,
S  0 as long as N  N tr ( I  I tr  qVN tr /( c ) ) and strict threshold doesn’t exist.
In reality, optical gain is saturated by the optical power (detailed derivation must be from
the density matrix approach). It can be phenomenologically written as: g  g L /(1  S )
where g L is the linear gain given as above. This leads to the spectral hole burning effect.
Also, static bias change causes optical field re-distribution, since optical index changes as
optical gain changes in active region. Field re-distribution again leads to the cavity loss
change. Therefore, the photon lifetime also changes as a function of the static bias. This
is the spatial hole burning effect.
Both spectral and spatial hole burning effects lead to the carrier density increase after
“threshold”, strict carrier density clamping (gain clamping) doesn’t exist.
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Finally, thermal effect is not considered in the above rate equations. Taking
phenomenological approach, we can assume: a  ao exp(T / Ta ) , N tr  N tro exp(T / Ttr )
and  int   int o exp(T / Tint ) , hence “threshold” current increases and output optical power
decreases exponentially as functions of the temperature.
P
N
Thermal Effect
Spectrum/Spatial
Hole Burning
T1
Nth
T2>T1
Spontaneous
Emission
Ith
I
4. Small-signal analysis based on behavior model
Assuming I (t )  I o  I (t ) , where I (t )  I o , we have: N (t )  N o  N (t ) and
S (t )  S o  S (t ) with N (t )  N o and S (t )  S o . From rate equation we obtain:
v g g o
v g S o
dN

1

I  N 
S 
g
dt
qV
c
V
V
dS
1
 (v g g  )S  (v g S o  Knsp v g )g
dt
p
g
g
| N o , So N 
| N , S S . By applying Fourier transform to
N
S o o
the above equation, we obtain the intensity modulation response:
where g o  g | N o , So , g 
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p
S
q

2
I  j 
j

 
1
d
 r 
where:
r 
d 
v g S o g
v g Po

g
| N o , So 
(1  int )
| N ,S
 pV N
hvV
 cav N o o
 2r
Knsp
 p So
 v g S o
g
1 v g S o g
| N o , So  
| N ,S
S
c
V N o o
are the relaxation oscillation and the damping frequencies, respectively. Obviously, the
intensity response of semiconductor laser diode behaves as a 2nd order LPF with  r as its
relaxation oscillation frequency and  d /  r as its quality factor.
In reality, additional parasitic elements in the laser chip packaging will introduce a 1st
order roll-off at the pass-band edge. Therefore, the total transfer function will be the
cascade of the above 2nd order LPF (from chip) and a 1st order LPF (from chip
packaging).
Similarly, we can obtain the frequency modulation response:
1
j
g
v  LEF p  g

4qS o  j  2 j
I

 
1

d
 r
g 
Knsp
 p So
 v g S o
g
| N ,S
S o o
The frequency response of semiconductor laser diode thus behaves as a 2nd order LPF
with the same relaxation oscillation frequency and quality factor as the intensity
response, but with an extra dip inside the pass-band.
It is also clear that for any given bias current (as the input signal) change, not only the
output optical power from the laser diode (as the output signal intensity) changes, the
lasing frequency of the laser diode (as the output signal phase) also changes. This effect
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is known as the parasitic modulation and it is introduced by the accompanied material
refractive index change due to the material gain change.
In reality, an extra LPF has to be cascaded to the above frequency response in
considering the thermal effect, since the material refractive index of the laser diode also
changes significantly as a function of the temperature.
5. Large-signal analysis based on behavior model
Rate equations can be solved directly in time domain by numerical integration. However,
under step function injection, some features in output optical power and lasing frequency
varying as functions of time can be directly observed from steady state and small signal
equations, such the turn-on delay time, the ringing phenomena (damped relaxation
oscillation) etc.
6. Noise characteristics
The noise characteristics of laser diodes such as the relative intensity noise, the mode
partition noise and the phase noise will be discussed later in following sections.
7. External modulators
Light generated from non-coherent source (such as LED) can only be used as the carrier
for intensity modulation format (or OOK for digital system).
Light generated from coherent source (such as laser) can be written as: E sin( t   ) .
When it is used as the carrier, amplitude, frequency and phase modulation schemes (or
ASK, FSK and PSK/DPSK for digital system) can be used.
Intensity modulator:
a) Electro-Absorption (EA) Modulator: change band-gap through bias voltage, V  0
transparent, V  0 absorption. Advantage: fast speed.
b) Amplifier Modulator: switch on/off gain through external current, I  0 absorption,
I  0 transparent or amplification. Advantage: additional optical gain. Problem: low
speed.
Amplitude modulator:
c) Mach-Zehnder Modulator: split wave into two arms, let the two waves travel through
different arm and combine at the output end. The two waves will interfere
destructively (off) or constructively (on) depending on the propagation delay
difference between the two arms. The propagation delay difference is controlled by
the externally applied voltage through EO effect.
Phase modulator:
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d) Electro-Refraction (ER) Modulator: a straight waveguide with EO effect. The phase
shift is related to the index change directly through:   (2L /  )n .
Frequency modulator:
e) ER Modulator under a Saw-tooth Like Modulation Waveform: ER modulator produce
a phase shift proportional to the applied voltage through EO effect, a linear voltage
change yields a linear phase change, which corresponds to a frequency shift.
f) Surface Acoustic Wave Modulator: makes use of Bragg scattering. Problem: low
speed.
g) Parasitic Frequency Modulator: direct modulation of laser by current injection leads
to changes in both the amplitude and the frequency of emitted light, since gain change
in semiconductor material inevitably introduce a large parasitic index change.
Problem: non-uniformity of the FM response introduces distortion.
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