chapter5 final
TRANSCRIPT
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Chapter 5 Frequency Response of InjectionLocked Semiconductor Ring Lasers
Theoretical study for frequency response and modulation bandwidth of slave SRL in
the master-slave configuration using optical injection locking has been investigated.
It has been proved that the frequency response of injection locked SRL depends on the
detuning frequency and optical injection ratio between the master laser and slave
SRL. Enhancement in the modulation bandwidth of >100GHz is found between
negative to positive detuning and increasing injection power ratio.
5.1 IntroductionIn optical communication systems, one of the most important figures of merit that
decides the achievable data rate is the modulation bandwidth of the semiconductor
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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laser. Recently a number of works have been proposed that suggests that the
modulation bandwidth of strongly injection locked semiconductor lasers can be
significantly enhanced as compared to direct modulation [1], [5]- [7].
Usually an isolator is placed between the master and slave laser to stop the light
coming from the slave laser into the master laser. SRL works in the direction of
receiving optical injection thus it eliminates the need of isolator [13]. The frequency
response and the modulation bandwidth of SRL are investigated in detail
experimentally in chapter 4. Theoretical study of the frequency response of OIL-SRL
is presented in this chapter. Various parameters that affect the 3-dB bandwidth in
injection locked SRL are considered. It has been proved using different modulation
schemes (direct modulation, amplitude modulation and phase modulation) that OIL
SRL as a slave in the master-slave configuration has huge bandwidth. At the end
chirp-to-power ratio and parasitic amplitude modulation due to phase modulation
response are discussed.
The fundamental theory has been developed in the past by a number of research
groups and can illustrate a wide range of benefits from the optical injection locking,
including RIN reduction [4], suppression of non-linear effects [2], and resonance
frequency enhancement [4]. Strong optical injection locking has also been studied for
the enhancement of bandwidth [1]-[6] and the expression for the frequency response
has been derived for various modulation formats such as direct modulation, amplitude
modulation and phase modulation in the optical injection locked semiconductor lasers
[6],[13].
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Semiconductor Ring Laser operates in clockwise (CW) or counter-clockwise (CCW)
directions as discussed in detail in previous chapters. Single transverse mode and
single longitudinal mode operation have been shown experimentally when SRL works
in the unidirectional region [10]. As this chapter focuses on the theoretical analysis of
the frequency response of an injection locked SRL, only single longitudinal operation
in SRL needs to be considered and the two counter-propagating single mode model is
sufficient for the analysis.
In the single mode operation, at the same wavelength SRL supports only two counter-
propagating modes. The basic model for describing two mode dynamics in SRL was
introduced in early parts of the last decade [8]-[10], [13].
5.2 Basic Rate equationsThe rate equations can be obtained using [8] for most of the SRLs as far as they are
not operated in sub- pico-second regime. Subsequently the polarization dynamics can
be removed adiabatically as intra-band processes have to be taken into the account
[11]. The set of differential equations governing a free-running SRL i.e. without any
external injection can be given as:
( )( )
( )
2 211 2 1
1
( ) 1 1(1 ) 1 ( )
2
( )
g n tr s c
p
f th
dE tj v g N N E E E t
dt
j E t
=
+
(5.1)
( )( )
( )
2 222 1 2
2
( ) 1 1(1 ) 1 ( )
2
( )
g n tr s c
p
f th
dE tj v g N N E E E t
dt
j E t
=
+ (5.2)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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( ) ( ) ( )2 2 2 2 2 2
1 2 1 2 1 21 ( ) ( ) ( ) 1 ( ) ( ) ( )
i
N
g n tr s c s c
IdN N
dt qV
v g N N E t E t E t E t E t E t
=
+
(5.3)
where 1( )E t and 2 ( )E t represent the complex fields of the mode 1 (CCW direction)
and the mode 2 (CW direction) respectively. is the frequency of the free-running
longitudinal mode,
is the line-width enhancement factor [12] that accounts for the
phase-amplitude coupling in the semiconductor medium, is the optical confinement
factor which provides the spatial overlap between the active gain volume and the
optical mode volume,gv represents the group velocity, ng is the differential gain at
transparency, N is the carrier density, trN accounts for the carrier density at
transparency, s and c are self-gain saturation and cross-gain saturation coefficients
respectively. p is the photon lifetime in the laser cavity. th is the resonance
frequency at threshold. i represents the injection efficiency of the bias currentI, q is
the electron charge in the volume of active region V, N is the carrier lifetime. The
carrier life N can be given as:
2
1N
th thA BN CN =
+ +
Where thN , A, B and C represent carrier density at threshold, Non-radiative
coefficient, radiative coefficient and Auger recombination coefficient respectively.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Spontaneous emission is neglected due to the fact that SRL is biased high above the
threshold current in the unidirectional region [8].
5.3 Rate equations with external optical injectionAssuming that the external optical injection from the master laser is added in the
lasing CCW direction, the optical field in this direction becomes stronger and its
phase may locked to that of the injected light and finally slave SRL will be locked to
the master laser a soon as the locking conditions are satisfied. The set of differential
equations governing the complex field of the injection locked SRL is similar to that of
free-running SRL in equations (5.1)-(5.3), with addition of the injection terms in the
CCW direction as discussed in [13].
( ) ( )2 2
11 2 1
1
( )1 1(1 ) 1 ( )2
( ) ( )
g n tr s c
p
inj inj inj
dE tj v g N N E E E t
dt
E t j E t
=
+
(5.4)
( )( )
( )
2 222 1 2
2
( ) 1 1(1 ) 1 ( )
2
( )
g n tr s c
p
f th
dE tj v g N N E E E t
dt
j E t
=
+
(5.5)
( ) ( ) ( )2 2 2 2 2 21 2 1 2 1 21 ( ) ( ) ( ) 1 ( ) ( ) ( )
i
N
g n tr s c s c
IdN N
dt qV
v g N N E t E t E t E t E t E t
=
+ (5.6)
where ( )in jE t is the injected master optical field, inj is the detuning frequency and
it can be defined asinj M f
= , where is lasing frequency of the master
laser.in j
accounts for the field coupling coefficient of the optical injection into the
SRL cavity and it can be defined as:
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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1in j
l
T
T
= (5.7)
Where T is the power transmission rate of the coupler from the cavity to the output
waveguide (and vice versa) and l is the round-trip time in the laser cavity
(gcavl vL /= ).
Without loss of validity, equation (5.5) may be neglected as it is the field in the non-
lasing CW direction, there is no injection in this direction and it is very small as
observed experimentally. The field equation (5.4) can be split into the field
magnitude and phase by assuming ( )1( ) ( )j tE t S t e , where S (t) is the photon
density of SRL in CCW direction and ( )t is the phase of SRL in CCW direction
relative to nominal master laser phase. Similarly the injection field can be split into
( )( ) ( ) inj
j t
inj injE t S t e , where ( )
in jS t is the photon density of injection light of the
master laser into the slave SRL and ( )inj t is the time-dependant master laser phase.
So the set of equations (5.4)-(5.6) can be re-written as:
( )( )
( )
( ) ( )( ) 1 ( ) ( )
2 ( ) ( ) cos ( ) ( )
g n tr s
p
inj inj inj
dS t S t v g N t N S t S t
dt
S t S t t t
=
+
(5.8)
( ) ( )
( )
( )( ) 1 ( )
2 2
( )sin ( ) ( )
( )
g n tr s
p
in j
inj inj inj
d tv g N t N S t
dt
S tt t
S t
=
(5.9)
( )( )( )( ) ( )
( ) 1 ( ) ( )i g n tr sN
I tdN t N t v g N t N S t S t
dt qV
= (5.10)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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5.4 Steady State SolutionsWhilst solving the rate equations (5.8)-(5.10) for steady state solution, Murakami et
al. [17] has described three equations to solve differential equations in the steady-state
for field magnitude, phase and the carrier density respectively. Similar method is also
used to determine the steady-state solution involving both directions of the SRL [13].
The same technique is to determine the steady-state solution but with slightly different
results in the direction of injection.
The product sS is4
5.4 10 to 38 10 from free-running to strong optical injection
respectively. This product 1sS and has no significant effect. Thus for the steady-
state solution and small signal analysis, the terms involving sS in equations (5.8) to
(5.10) may be neglected.
Solving for the free-running steady state solution, we can set the derivatives and the
injection terms to zero in equations (5.8) to (5.10) thus the analytic solution for
photon density can be given as:
fi
f pN
NIS
qV
=
(5.11)
And
1f tr
p g n
N Nv g
= +
(5.12)
For the external optical injection from the master laser,0inj
is assumed to be zero so
the dc SRL phase 0 represents the total dc phase of the master and slave lasers.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Using the above results and solving equations (5.8)-(5.10) with external optical
injection, the steady state solution can be found in the similar fashion as:
01
p
f
N
p g n
NS
Sv g N
=+
(5.13)
( )
arctan
1
arcsin
2
0
0
0
+
=
S
Sinjinj
inj(5.14)
0
0
0
2cos
inj inj
g n
SN
v g S
=
(5.15)
Where0S , 0 and N are photon density, phase and carrier density difference
respectively at steady state of injection locked SRL. Steady state value of carrier
density is 0N N N = + and 0 /inj f S S is the injection ratio. The convenient
method is to choose injection ratio and phase value. According to Mogensen [18], the
bounds of the phase across the injection locking range are approximately1cot to
/ 2 from negative to positive frequency detuning edge respectively. Using the
value of0 and N(equation 5.15) into (5.13), we can solve easily for 0S as:
3 20 1 0 2 0 3 0A A A = (5.16)
Where
0 0S=
1 02 cosinj p injS =
2S =
3 02 cos
p
inj injN g n
Sv g
=
(5.17)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Detuning frequency can be determined using equation (5.14) as:
( ) ( )2
00 1 sin arctan
inj
inj inj
S
S = + +
(5.18)
Derivation of steady-state equations is given in appendix-I.
5.5 Small Signal AnalysisFor small signal analysis, small perturbations may be added to all time dependant
terms around their steady-state values.
( ) ( )0 expS t S S j t = +
( ) ( )0 expt j t = +
( ) ( )tjNNtN exp0 +=
( ) ( )tjSStS injnjinj exp0 +=
( ) ( )0 expinj inj injt j t = +
( ) ( )tjIItI exp0 += (5.19)
where ( ), ( ), ( )S t t N t represent the output state variables while ( ), ( ), ( )inj injI t S t t are
the input state variables of the system.0 0 0 0 0 0, , , , ,
inj injS N I S may be considered as
the steady-state values and , , , , ,inj injS N I S represent the small-signal
magnitudes. The input perturbation terms are modulated separately so that the
modulated perturbation magnitudes may be considered as real phasors.
Physically I denotes the direct modulation of current,inj
S is the amplitude
modulation andinj
represents the phase modulation of the master laser injected into
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
118
the slave SRL. S and represent output modulation magnitude and phase of the
slave SRL respectively.
By linearising equations (5.8)-(5.10) the differential equation system may be placed in
the matrix form as:
1XM
S
N
=
(5.20)
Where M is the state transition matrix and it can be given as:
11 12 13
21 22 23
31 32 33
M
a j a a
a a j a
a a a j
+ = + +
The elements of state transition matrix M can be given as:
11 0
12 0 0
13 0
21 0
0
22 0
23
31 0
32
33 0
cos
2 sin
1sin
2
cos
2
12 cos /
0
1
g n
g n
p
g n
N d
a z
a zS
a v g S
za
S
a z
a v g
a z
a
a v g S
=
=
=
=
=
=
=
=
= +
(5.21)
Where0
inj
inj
S
z S= , andNd is differential carrier life time and it can be given as:
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
119
2
0 0
1
2 3Nd
A BN CN =
+ +
X is the input vector and driven separately in accordance with the modulation scheme.
For direct modulation current (I) is injected and thus X can be given as:
0
X 0
/
DM
i
I
qV
=
(5.22)
When amplitude modulated signal is injected into the SRL, X can be described as:
0 0 0
0 0
/ cos
X sin / 2
0
inj
AM inj inj
zS S
z S S
=
(5.23)
Similarly phase modulated signal injected into the SRL can be given as:
0 0
0
2 sin
X cos
0
PM inj
zS
z
=
(5.24)
5.6 Frequency ResponseThe frequency response of the output perturbation may be found by inverting the state
transition matrix M in equation (5.20) and solving under the respective input
modulated perturbations.
For direct modulation using input modulated perturbation in equation (5.22)
1 1
0
M X M 0
/
DM
i
S
I
N qV
= =
(5.25)
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Similarly using amplitude modulation, frequency response of OIL-SRL using
amplitude modulated input perturbations in equation (5.23)
0 0 0
1 1
0 0
/ cos
M X M sin / 2
0
inj
AM inj inj
S zS S
z S S
N
= =
(5.26)
Frequency response using phase modulation can be found by using equation (5.24), as
phase modulated input perturbations
0 0
1 1
0
2 sinM X M cos
0
PM inj
S zSz
N
= =
(5.27)
Equations (5.25)-(5.27) consist of the same system matrix M so the poles for
frequency response using any modulation scheme will be the same and can be
determined by solving the determinant of the matrix M as:
3 2det( ) j A j B C = + +M (5.28)
where
332211 aaaA ++=
311321123322332211 )( aaaaaaaaaB ++=
223113332112312312332211 aaaaaaaaaaaaC +=
Due to the same determinant all the modulation schemes share the same resonance
frequency and damping factor. The resonance enhancement in the injection locked
laser has been extensively studied [5]. The equation (5.28) can be modified to
determine the resonance frequency and damping factor as:
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
121
( )( )
( )
2 2det( )
1 1
2 2
p R
p R R
M j j
j j j j j
= + +
= + + + +
(5.29)
Where ,p R
and are low frequency roll-off pole, resonance frequency and
damping factor respectively. According to [5]-[6] they can be given as:
( )2 13 31 12 21R a a a a = + (5.30)
and
11 22 33A a a a = = + + (5.31)
5.6.1Response to SRL current modulationIn the case of direct modulation of OIL-SRL, the continuous wave light is injected
from the master laser to the slave SRL. Classically for direct modulation response the
output is photon density modulation and thus the frequency response can be given as
using equation (5.25)
CBjAj
ZjM
I
SjH ddirdir
++
+==
23.)(
(5.32)
where . 13i
dir aqV
=
and dZ is the zero of the system and it may expressed as:
12 23 22 13
13
d
a a a aZ
a
=
5.6.2Response to amplitude modulation in optical injectionFor amplitude modulation, the output is the photon density modulation of the slave
SRL and the input is the amplitude modulated signal from the amplitude modulator.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Thus the frequency response using equation (5.26) for amplitude modulated OIL-SRL
system may be given as:
1 2
3 2
( )( )( ) AM AM AM AM
inj
j Z j Z SH j M
S j A j B C
+ += =
+ + (5.33)
where 00
0
cosAMinj
SM z
S= and 1AMZ and 2AMZ are zeros and they may be expressed
as:
1 33 0
1AM g n
NdZ a v g S = = +
0
2cos
zZAM =
5.6.3Response to phase modulation in optical injectionSimilarly for phase modulated injection locking system, the input is phase modulated
light signal injected from the phase modulator and the output is the phase modulation
of the slave laser and it can be given as by using (5.27):
2
3 2( )PM
inj
a b CH j
j A j B C
+ += =
+ + (5.34)
where
0cosa z =
( )2 233 01 cosb z a = +
Hence, the frequency response of slave SRL can be simply determined using equation
(5.32) for direct modulation, (5.33) for amplitude modulation and (5.34) for the phase
modulation.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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5.7 Simulation Results and DiscussionSimulations and calculations are based on the SRL device very similar to which used
in chapter 4. It is assumed that the device is biased at 100mA. Table 5.1 presents the
basic parameter values used in the simulations of the frequency response of Slave
SRL.
Quantity Symbol Value
Speed of Light c 83 10 m/sec
Electron charge q 191.602 10 Coulumbs
Refractive Index n 3.41
Line-width enhancement factor 2.52 [24]
Length of cavity L 1406 m
Waveguide width wgW 2 mQuantum-well thickness qwT 6 nm
Number of quantum wells qwN
5
Differential gainng
20 26 10 m
Confinement factor 0.62 Current injection efficiency
i 0.5
Carrier density at transparencytrN
241.25 10 m-3 [24]
Power coupling ratio T 0.5 Non radiative coefficient 82.1 10 sec
-1
[25]
Radiative coefficient B 104.5 10 cm3/sec[25]Auger recombination coefficient C 295.83 10 cm6/s[25]
Table 5.1 Injection locked SRL Parameters
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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5.7.1Stable Locking RangeIn injection locked lasers stable locking can be obtained by using Mogensen locking
range [18]. In SRL it can be proved by considering following important facts.
Firstly, in equation (5.14), the real solution to phase is found if and only if the
absolute value of arcsine is less than and equal to unity. Hence,
0arctan arctan
2 2
(5.35)
Secondly in equation (5.15) for the stable gain, value of the carrier density must be
less than its value at threshold. So,
02
(5.36)
Combining (5.35) and (5.36), we get
1
0 cot
2
(5.37)
Therefore, using equation (5.18) the locking range can be given as:
( )20 0
1inj inj
inj inj inj
S S
S S + (5.38)
The third important fact is the stability check on the region of convergence (ROC).
Considering the system determinant in equation (5.28)
3 2det( ) j A j B C = + +M (5.39)
The solution becomes unstable, when the poles of frequency response i.e. the real
parts of the roots of determinant become positive and can be determined by solving
the equation (5.39) computationally. This reduces the locking region on the positive
side. It also finds out the boundary between stable locking region (SLR) and unstable
locking region (ULR).
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Finally, it must also be taken into account that at high injection ratios the output field
magnitude can diverge significantly from its free-running value. For stable behaviour
0 fS S> , however at high injection ratios and negative detuning frequencies steady-
state field value reduces as compared to free-running field this results in further shrink
of stable locking region on the negative side of the locking region. The injection ratio
used for the simulations is defined as the logarithmic value of Rinj and can be given
as:
10 log 10loginj
inj
SR R
S= =
Figure 5.1 illustrates all these boundaries of the locking range. For the weak injection
FWM can be observed in the unlocked region [26]. The ULR is chaotic region which
may be locked or a sizeable power may be transferred to other side longitudinal
modes [26]-[28]. The boundary between the ULR and SLR is also called Hopf
bifurcation boundary [28]. The focus of this chapter is stable locking region and is
illustrated as the coloured region in the map. Thus the all the simulations are
performed in the stable locking region.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Figure 5.1 Locking range of SRL illustrates stable locking range (SLR) and
unstable locking range (ULR)
5.7.2Frequency response simulationsIn chapter 4, the measurements of frequency response of slave SRL in the master-
slave OIL configuration show its dependence on the optical injection power and the
detuning frequency. Enhancement in the 3-dB bandwidth and the resonance
frequency has been found due to increase in these factors. The frequency response for
different values of the optical injection ratio and the detuning frequency can be
plotted using equations (5.32)-(5.34) for direct modulation, amplitude modulation,
phase modulation, chirp due to phase and chirp due to amplitude respectively.
/ 2 =
1cot =
0 fS S> boundary
ROC
boundary
Increase in
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Free-running response of the SRL is simulated by setting injection parameters to zero
and the SRL is biased at 200mA. The modulation bandwidth of free-running SRL is
2.2 GHz and the response is shown in Figure 5.3 (a).
5.7.2.1 Effects of optical injection ratio
To study the effects of injection ratio on the frequency response of OIL-SRL, the
equations (5.30) and (5.31) are simulated for resonance frequency and damping factor
respectively. The change in the resonance frequency is shown in Figure 5.2 (a) while
Figure 5.2 (b) is about the change in the damping factor with respect to the change in
the optical injection ratio. Figure 5.2 shows very clearly that the resonance frequency
and damping factor increases with the increase in the injection ratio. In Figure 5.2 the
detuning frequency is kept constant at 3.5GHz.
Figure 5.2 Effects of optical injection ratio on (a) resonance frequency (b)
damping factor of OIL-SRL
2 4 6 8 104
7
10
12x 10
10
Injection Ratio (dB)
Resonanc
eFrequency(rad/sec)
2 4 6 8 101.5
2.5
3.5
4.5
5.5x 10
10
Injection Ratio (dB)
Damp
ingFactor(1/sec)
(b)(a)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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To study the effects of injection ratio on the frequency response of SRL using direct
modulation, amplitude modulation and phase modulation, the detuning frequency is
kept constant at 3.5 GHz. Figure 5.3 (a) and (b) show the frequency response curves
to direct modulation and amplitude modulation respectively. Various values of
injection ratio from 2dB to 10 dB as shown in the legend in Figure 5.3(a) are used in
the simulations. The responses are normalised to unity at zero modulation frequency
to compare 3-dB points for the bandwidth.
(a)
(b)
Figure 5.3 Frequency response (a) direct modulation (b) Amplitude modulation
with changing optical injection and detuning frequency is fixed to
3.5GHz
108
109
1010
1011
-25
-20
-15
-10
-5
0
5
10
15
Frequency (Hz)
Response[10log(mW/mA)]
Free-running
2 dB
4 dB
6 dB
8 dB
10 dB
108
109
1010
1011
-20
-10
0
10
20
Modulation Frequency (Hz)
Respon
se(dB)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Figure 5.3 (a) shows the small signal response of OIL-SRL to direct modulation.
Frequency response broadens with the increase in the injection ratio while its
resonance peak drops. It is because the increase in the damping factor and the
resonance frequency. The dip between the DC and the resonance is mainly due to
the real polep in the system response and is also known as roll-off factor. It is
shown in Figure 5.3 (a) that this low-frequency roll-off factor shrinks the 3-dB
frequency with increasing injection ratio. Figure 5.3 (b) shows the frequency
response of OIL-SRL to the amplitude modulated injection. This response also
depends on the optical injection ratio and the resonance peak broadens with increasing
injection ratio. The benefit of using intensity modulation is that dependence on the
roll-off factor is decreased with the increase in the injection ratio.
Figure 5.4 shows the small signal response of OIL-SRL to phase modulated injection.
The response is simulated as linear function of ratio of output phase of the OIL-SRL
to the phase of the injected light. Thus the response curves are different from the
results shown in work of E.K. Lau et. al. [6]. The response is equal to unity before the
appearance of resonance. It is due to the fact that the output phase of SRL tracks the
changes in the phase of injected light from the master laser. Various values of
injection ratio from 2dB to a high 10 dB as shown in the legend in Figure 5.3(a) are
used in the simulations. With increasing injection ratio the resonance moves to higher
modulation frequency and the resonance peak is broadened because the damping and
resonance frequency both are enhanced. Thus increase in the injection power ratio
causes the slave phase to track more quickly the phase of the master laser, thus the
phase tracking bandwidth is enhanced. .
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Figure 5.4 Frequency response phase modulation with changing optical injection
and detuning frequency is fixed to 3.5 GHz
5.7.2.2 Effects of detuning frequency
It is also very interesting to study the frequency response by changing the detuning
frequency and keeping the injection ratio unchanged. In similar fashion, the
resonance frequency and damping factor are computed using equations (5.30) and
(5.31) respectively by changing detuning frequency and injection ratio is fixed at
10dB. Figure 5.5 (a) shows that the resonance frequency increases with increase in
the detuning frequency in the stable locking range from negative to the positive
region. . The damping factor decreases with increasing detuning frequency from
negative to the positive edge of the locking range as shown in Figure 5.5 (b).
108
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1011
-10
-5
0
5
10
Modulation Frequency (Hz)
Resp
onse(rad/rad)
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Figure 5.5 Effects of detuning frequency on (a) resonance frequency (b) damping
factor of OIL SRL
To study the effects of detuning frequency on the small signal response of OIL-SRL
using direct modulation and amplitude modulation injection ratio is kept fixed to
10dB. Figure 5.6 shows the frequency response curves at various values detuning
frequency from the negative edge to the positive edge of the stable locking range
given in Figure 5.1. The values of detuning frequency used in the simulations are
shown in the legend in Figure 5.6 (b). The responses are normalised to unity at zero
modulation frequency to compare 3-dB points for the bandwidth.
The small signal response of slave OIL-SRL due to direct modulation is shown in
Figure 5.6 (a). When the detuning frequency is increased from the negative edge to
the positive of locking range, due to increase in the resonance frequency enhancement
in the resonance peak is observed. But the resonance peak narrows because the
-15 0 15 30 450
0.5
1
1.5
2
2.5
3
3.5x 10
11
Detuning Frequency (GHz)
Re
sonanceFrequency(rad/sec)
-15 0 15 30 450
1
2
3
4
5x 10
11
Detuning Frequency (GHz)
DampingFactor(1/sec)
(a)(b)
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damping factor decreases with increasing detuning frequency. Similarly when the
detuning frequency is changed from negative edge to the positive edge of the locking
range, the resonance peak becomes narrow and it height is enhanced and moves to the
higher modulation frequency. Figure 5.6 (b) shows the response to the intensity
modulation, when the detuning frequency is -10 GHz, the response is highly damped
and resonance peak is almost flat. But when the detuning frequency is increased, the
resonance peak starts to appear and the response becomes better for the data
modulation applications. The further increase in the detuning frequency enhancement
in the resonance peak is observed and it approaches 18dB higher at 45 GHz detuning
frequency which is very close to the positive edge of the locking range. This high
resonance is suitable for the RF applications. After 48GHz the response enters the
unstable locking region (ULR). The enhancement in the bandwidth of 100 GHz is
predicted.
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
133
(a)
(b)
Figure 5.6 Frequency response of OIL-SRL using (a) direct modulation (b)
Amplitude modulation with changing detuning frequency with 10 dB
fixed injection ratio value
Figure 5.7 shows the effects of detuning frequency on response of the phase of the
slave laser (SRL) to the phase of the master laser in the injection locked system. In
the simulations same values of detuning frequency are used as shown in the legend in
Figure 5.6 (a) and injection ratio is fixed to 10 dB. As the detuning frequency is
increased the enhancement in the resonance peak is observed. When the detuning
108
109
1010
1011
-25
-20
-15
-10
-5
0
5
Modulation Frequency (Hz)
Response[10log(mW/m
A)]
- 10 GHz
0 GHz
10 GHz
25 GHz
45 GHz
108
109
1010
1011
-10
-5
0
5
10
15
20
Modulation Frequency (Hz)
Response(dB)
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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frequency is -10 GHz, the response is highly damped and resonance is not in the
picture. But when the detuning frequency is increased, the resonance peak starts to
appear and the response becomes better for the data modulation applications. When
the detuning frequency is made 45 GHz the big rise ( 10 ) in the resonance is
observed and it is shown in Figure 5.7. The resonance frequency and damping factor
can be controlled by a maintaining a detuning frequency to get the maximum phase
tracking bandwidth.
Figure 5.7 Frequency response of OIL-SRL to phase modulation with changing
detuning frequency and 10 dB fixed injection ratio value.
It is discussed above that OIL-SRL system depends on two factors i.e. injection ratio
and detuning frequency. When the injection ratio is increased the resonance
frequency and the damping factor both are enhanced, as a result the resonance peak of
the response broadens thus 3-dB bandwidth of the slave SRL is enhanced. However,
109
1010
1011
-10
-5
0
5
10
Modulation Frequency (Hz)
Rpe(adrad
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
135
when the detuning is increased the increase in the resonance frequency and decrease
in the damping factor is observed.
The above results show that the frequency response of OIL-SRL can be tuned for
various applications. Narrow and huge resonance peak can be utilised for the RF
applications when the response is tuned near the positive edge of the stable locking
range. When the detuning frequency is set towards the negative edge of the locking
range the damping is very high and the modulation bandwidth falls as well. However,
when the injection parameters (injection ratio and detuning frequency) are managed in
between the positive and negative edge, the system becomes very suitable for data
modulation schemes with huge modulation bandwidth. In SRL enhancement in the
modulation bandwidth of 100GHz is predicted. In addition the resonance can be
tuned at any modulation frequency by changing the injection ratio and detuning
frequency.
5.8 Frequency ChirpThe frequency chirp is defined as the instantaneous change in the frequency of
modulated output light of the laser. It is one of the most severe limitations along with
the fibre chromatic dispersion to the maximum attainable value of the length-bit rate
product in the data transmissions at 1550nm wavelength [20]. Although injection
locking systems have provided sufficient reduction in the chirp [21]-[23] for high
speed communication systems but most of the research is concentrated on the direct
modulation in injection locked semiconductor lasers [22].
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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However, in the intensity modulation (amplitude modulation) the significant measure
that affects the system performances is the ratio between the fundamental frequency
and the intensity modulation photon densities [21] and it is also referred as CPR
(Chirp to power ratio). CPR indicates the amount of chirp when trying to achieve
certain amount of amplitude modulation. It provides a frequency shift for a given
small signal modulation power, thus it is a convenient measure of frequency
modulation.
G. Yabre has shown significant reduction in the CPR by strong injection locking in a
directly modulated semiconductor laser [23]. In similar sense it is also reported by E.
K. Lau et. al . have also described CPR as the magnitude of ratio of deviation in the
output phase of the OIL-Laser to the deviation in its output intensity [7].
In this thesis, it is attempted to derive expression for the frequency response of
parasitic modulation in the output phase of OIL-SRL due to change in the intensity
modulated input signal. So the frequency response for chirp to input modulated
power can be given as by using equation (5.26) and normalised by multiplying
modulation frequency:
2
1 1 1
3 2( )chirp chirp
in j
a j b cH j M
S j A j B C
+= =
+ + (5.40)
Where
( )
( ) ( )
0
1 0
1 11 33 0 21 0 0
1 31 13 11 33 0 0 31 23 21 33 0
2
sin
sin 2 cos
sin 2 cos
chirp
inj
zM
S
a
b a a a S
c a a a a S a a a a
=
=
= + +
= +
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
137
This expression is different to the CPR found in the literature as it is described as the
magnitude of change in the output phase due to change in the output amplitude.
(a)
(b)
Figure 5.8 Normalised chirp response with change in (a) injection ratio (b)
detuning frequency
Chirp response is simulated at different injection power and the detuning frequency is
kept constant to 3.5GHz as shown in Figure 5.8(a). As discussed earlier, the
resonance frequency and the damping factor increase with the increase in the injection
ratio. Chirp response becomes almost flat and low in amplitude at high injection
108
109
1010
1011
120
140
160
180
200
Modulation Frequency (Hz)
Response[10log(rad2/sec.m
W)
2 dB
4 dB
6 dB
8 dB10 dB
108
109
1010
1011
120
130
140
150160
170
180
190
Modulation Frequency (Hz)
Response[10log(ra
d2/sec.m
W)]
-10 GHz
0 GHz
10 GHz
25 GHz
45 GHz
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
138
ratio. Figure 5.8(b) shows the chirp response at different detuning frequencies and
fixed injection ratio of 10dB, resonance becomes narrow and enhanced with increase
in detuning frequency. It is due to the fact that the damping factor decreases with the
increase in the increase in the detuning frequency from the negative to the positive
edge of the locking range.
The equation (5.41) describes the frequency response of parasitic modulation in the
output amplitude of injection locked SRL due to change in the phase of the modulated
input signal. It is normalised with the output photon density and has been derived
using equation 5.27.
0 33/ 3 2
0
/ ( )1( )
injS
inj
S S j j aH j
S j A j B C
+= =
+ + (5.41)
The frequency response of change in amplitude due to phase modulation in equation
(5.41) is simulated under change in injection ratio, change in detuning frequency and
change in the injection ratio near the positive edge of the locking range. The
simulation results in Figure 5.9 (a) are with increase in the injection ratio and
detuning frequency value of 3.5 GHz is used. As discussed earlier the damping factor
enhances while the resonance drops off with increase in the injection ratio, change in
amplitude due to phase modulation becomes flat. The simulation results show that the
response is very small at low modulation frequencies. Similarly its resonance
increases with the increase as shown in Figure 5.9(b).
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
139
(a)
(b)
Figure 5.9 Frequency response of modulation change in amplitude due to phase
modulation with change in (a) injection ratio (b) detuning frequency
5.9 ConclusionIn this chapter, the modelling for frequency response and modulation bandwidth of
SRL in the master-slave configuration using optical injection locking is discussed in
detail. Frequency response of master laser modulated OIL-SRL under direct
(current), amplitude and phase modulation are derived and simulated. It is observed
108
109
1010
1011
-25
-15
-5
5
15
25
Modulation Frequency (Hz)
Response[10log(1/rad)]
2 dB
4 dB
6 dB
8 dB
10 dB
108
109
1010
1011
-30
-20
-10
0
10
20
30
Modulation Frequecy (Hz)
Response[10log(1/rad)
-10 GHz
0 GHz
10 GHz
25 GHz
45 GHz
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
140
that the frequency response of OIL-SRL depends on detuning frequency and external
optical injection ratio between the master laser and the slave SRL. The linear
response of OIL-SRL under phase modulation has shown that the slave-SRL phase
tracks the phase of the master laser and this tracking bandwidth increases with the
increase in the injection ratio and maintaining detuning frequency between the
negative and positive edge of the locking range. Enhancement in the modulation
bandwidth of to be 100GHz is found between negative to positive detuning and
increasing injection power ratio. In chapter4, the 3-dB bandwidth of slave SRL has
been measured to be 40GHz. This scheme readily lends itself for monolithic
integration due to the unidirectional lasing characteristics of the SRL as already
demonstrated in [29]. This scheme may leads to a low-cost source in optical
communication systems with high speed.
The parasitic phase modulation response due to amplitude modulation (chirp
response) is derived and simulated. Similarly parasitic amplitude modulation due to
phase modulation response is also investigated. They are found to be not very high at
low modulation frequencies but with the increase in the resonance frequency there is
resonance in both the responses respectively. The chirp response investigated in this
chapter is different from CPR found in the literature [7], [23].
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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
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Bibliography[1] T. B. Simpson, J. M. Liu, and A. Gavrielides, Bandwidth enhancement and
broadband noise reduction in injection-locked semiconductor lasers,IEEE
Photon. Technol. Lett., vol. 7, no. 7, pp. 709-711, Jul. 1995
[2] X. J. Meng, T. Chau, and M. C. Wu, Improved intrinsic dynamic distortionsin directly modulated semiconductor lasers by optical injection locking,
IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 1172-1176, Jul. 1999.
[3] X. J. Meng, T. Jung, C. Tai, and M. C. Wu, "Gain and bandwidthenhancement of directly modulated analog fiber optic links using injection-
locked gain-coupled DFB lasers," International Topical Meeting on
Microwave Photonics (MWP), pp. 141-144, Nov. 1999.
[4] L. Chrostowski, X. Zhao, C. J. Chang-Hasnain, R. Shau, M. Ortsiefer, andM. C. Amann, "50-GHz Optically Injection-Locked 1.55-m VCSELs,"
IEEE Photon. Technol. Lett. , vol. 18, no. 2, pp. 367-369, Jan. 2006
[5] E. K. Lau, H.-K. Sung and M. C. Wu, Frequency Response Enhancement ofOptical Injection-Locked Lasers, IEEE J. Quant. Electron., vol. 44, no. 1,
pp. 90-99, Jan. 2008
[6] Erwin K. Lau, Liang Jie Wong, Xiaoxue Zhao,Young-Kai Chen,Connie J.Chang-Hasnain, Ming C. Wu, Bandwidth Enhancement by Master
Modulation of Optical Injection-Locked LasersIEEEJ. of Light wave Tech.vol. 26, no. 15, pp. 2584-2593, Aug. 2008
[7] E. K. Lau, L. J. Wong, M. C. Wu, Enhanced Modulation Characteristics ofOptical Injection-Locked Lasers: A Tutorial,IEEE J.OF Sel. Top. Qunatum.
Electron. vol. 15, no. 3, pp. 618-633, 2009
[8] M. Sorel, G. Giuliani, A. Scir, R. Miglierina, S. Donati and P. J. R.Laybourn, Operating Regimes of GaAsAlGaAs Semiconductor Ring
-
8/4/2019 Chapter5 Final
34/36
Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
142
Lasers: Experiment and ModelIEEE J of Quant. Electron., Vol. 39, No. 10,
pp. 1187-1195, OCT 2003
[9] T. Numai, Analysis of Signal Voltage in a Semiconductor Ring Laser GyroIEEE J of Quant. Electron., vol. 36, no. 10, pp. 1161-1167, oct. 2000
[10] M. Sorel, P. J. R. Laybourn, A. Scir, S. Balle, G. Giuliani, R. Miglierina, S.Donati, Alternate oscillations in semiconductor ring lasers, Opt. Lett.,
vol.27, no. 22, pp. 1992-1994, 2002.
[11] W. E. Lamb, Theory of an optical maser, Phys. Rev.A, Gen. Phys., vol.134, pp. A1429-A1450, 1964.
[12] C. Henry, "Theory of the linewidth of semiconductor lasers," IEEE J ofQuant. Electron., vol. 18, no.2, pp. 259-264, 1982.
[13] Guohui Yuan, Siyuan Yu, Bistability and Switching Properties ofSemiconductor Ring Lasers with External Optical Injection, IEEE J of
Quant. Electron., vol. 44, no.1, pp. 41-48, 2008.
[14] R. S. Tucker, High speed modulation of semiconductor lasers, IEEEJ. ofLight wave Tech. vol. 3, no. 6, pp. 1180-1192, Dec. 1985.
[15] T. L. Koch and R. A. Linke, Effect of nonlinear gain reduction onsemiconductor laser wavelength chirping, Appl. Phys. Lett. , vol. 48, no. 10,
pp. 613-615, Mar. 1986.
[16] G. P. Agrawal, Gain Nonlinearities in Semiconductor Lasers: Theory andApplication to Distributed Feedback Lasers, IEEE J of Quant. Electron.,
vol. 23, no. 6, pp. 860-868, Jun 1987.
[17] A. Murakami, K. Kawashima, and K. Atsuki, Cavity resonance shift andbandwidth enhancement in semiconductor lasers with strong light injection,
IEEE J. of Quant. Electron., vol. 39, no.10, pp. 1196-204, 2003.
-
8/4/2019 Chapter5 Final
35/36
Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
143
[18] F. Mogensen, H. Olesen, and G. Jacobsen, Locking conditions and stabilityproperties for a semiconductor laser with external light injection,IEEE J of
Quant. Electron., vol. 21, no.7, pp. 784-793, 1985.
[19] J. Wang, M.K. Haldar, L. Li, F.V.C. Mendis, Enhancement ofmodulation bandwidth of laser diodes by injection locking, IEEE Photon.
Techno. Lett., vol. 8, no1, pp. 34-36, JAN 1996.
[20] J. M. Osterwalder and B. J. Rickett, Frequency Modulation of GaAlAsInjection Lasers at Microwave Frequency Rates, IEEE J of Quant.
Electron., vol. 16, no.3, pp. 250-252, 1980.
[21] T. L. Koch and J. E. Bowers, Nature of wavelength chirping in directlymodulated semiconductor lasers, Electron. Lett., vol. 20, no. 25/26, pp.
1038-1040, Dec. 1984.
[22] S. Piazzolla, P. Spano, M. Tamburrini, Small Signal Analysis of FrequencyChirping in Injection-Locked Semiconductor Lasers, IEEE J of Quant.
Electron., vol. 22, no.12, pp. 2219-2223, 1986.
[23] G. Yabre, Effect of Relatively Strong Light Injection on the Chirp-to-PowerRatio and the 3 dB Bandwidth of Directly Modulated Semiconductor
Lasers, IEEEJ. of Light wave Tech. vol. 14, no. 10, pp. 2367-2373, Oct.
1996.
[24] www.iolos.org, IOLOS EU FP6, 2009.[25] Guohui Yuan, Switching characteristics of Bistable Semiconductor Ring
Lasers,PhD Thesis, University of Bristol, Bristol, UK, Dec. 2008.
[26] Junji Ohtsubo, Semiconductor lasers: stability, instability and chaos,Springer series in optical sciences, 2nd Edition, Springer
[27] T. B. Simpson, J. M. Liu, A. Gavrielides, V. Kovanis, and P. M. Alsing,"Period-doubling route to chaos in a semiconductor laser subject to optical
injection,"Applied Physics Letters, vol. 64, pp. 3539-41, 1994.
-
8/4/2019 Chapter5 Final
36/36
Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers
[28] J. M. Liu, H. F. Chen, X. J. Meng, and T. B. Simpson, Modulation Bandwidth, Noise, andStability of a Semiconductor Laser Subject to Strong Injection Locking, IEEE Photon.
Techno. Vol. 9, no. 10, pp. 1325-1327, Oct. 1997
[29] S. Frst, S. Yu, and M. Sorel, Fast and digitally wavelength-tunablesemiconductor ring laser using a monolithically integrated distributed Bragg
reflector , IEEE Photon. Tech. Lett., vol. 20, no. 23, pp. 1926-1928, Dec,
2008.