Chapter5 Final

8/4/2019 Chapter5 Final

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Chapter 5 Frequency Response of Injection Locked Semiconductor Ring Lasers

109

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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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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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


dt

j E t

=

+ (5.2)


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112

( ) ( ) ( )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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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


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


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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114

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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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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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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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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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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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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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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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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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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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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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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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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127

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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128

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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129

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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130

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

109

1010

1011

-10

-5

0

5

10


Resp

onse(rad/rad)


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131

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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132

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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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


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


Response(dB)


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134

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


Rpe(adrad


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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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136

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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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


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


Response[10log(ra

d2/sec.m

W)]

-10 GHz

0 GHz

10 GHz

25 GHz

45 GHz


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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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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


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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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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141

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Chapter5 Final

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