semiconductor lasers

34130 Introduction to Optical Communication

Direct Current Modulation of Semiconductor Lasers

Christophe Peucheret

DTU Fotonik

Department of Photonics Engineering

Technical University of Denmark

[email protected]

Abstract

This note describes the simulation exercise of the course 34130 Introduction to OpticalCommunication on direct modulation of semiconductor lasers. After a review of the theoryof continuous wave operation and small-signal modulation of single-mode laser diodes basedon the resolution of the rate equations for the carriers and photons, the three assignmentsare described. They make use of a numerical model allowing the resolution of the rateequations implemented in the VPIplayer environment. The main goals of the exercise are:1) to calculate the static power-versus-current characteristics of the laser, 2) to calculate itssmall-signal frequency response, and 3) to examine its dynamic behaviour under large-signalmodulation.

1 Introduction

The goal of this simulation exercise is to illustrate a number of important concepts that havebeen described in the lectures on sources and transmitters. This will be achieved by using anumerical simulation model of a single-mode directly modulated laser and comparing the resultswith simple calculations based on the theory developed in the lectures on semiconductor lasers.We will focus on the dynamic properties of the laser and calculate its small-signal frequencyresponse, as well as study its behaviour under large-signal modulation.

2 Theoretical background

Some important results that have been discussed in the lectures and that are detailed in thetextbook are summarised below. Furthermore, some derivations of the small-signal frequencyresponse of single-mode laser diodes, of relaxation oscillations and of frequency chirp are pro-vided. The main results of this section will be used in the preparation of the exercise (Sect. 3)and in the exercise itself (Sect. 4 to 6).

2.1 Rate equations

The rate equations for the photon number P and carrier number N can be written

dP

dt= GP +Rsp −

P

τp, (1)

dN

dt=

I

e− N

τc−GP, (2)

Simulation Exercise 2 14/07/2011 1


where Rsp is the rate of spontaneous emission in the lasing mode, τp the photon lifetime, τc thecarrier lifetime, G the net rate of stimulated emission, I the intensity of the current applied tothe laser diode, and e the elementary charge of the electron.

It has been seen in the lectures that the dependence of the peak material gain on the carrierdensity can be empirically approximated by

gm =σgV

(N −N0) (3)

where σg is the gain cross section, or differential gain, V is the active volume andN0 is the carriernumber at transparency. Due to the spreading of the optical mode outside the active layer, thepower gain at the mode frequency g needs to be corrected according to g = Γgm where Γ is theconfinement factor. The net rate of stimulated emission G can then be calculated according to

G = Γvggm = GN (N −N0) (4)

where GN = Γvgσg/V and vg is the mode group velocity within the cavity.

The feedback condition imposes the following relation between the gain g, the internal lossper unit length αint, the cavity length L and the reflectivities of the two facets R1 and R2

g = αint +1

2Lln

(

1

R1R2

)

= αint + αmir. (5)

The photon lifetime is directly related to the cavity loss αcav = αint + αmir according to

1

τp= vg (αint + αmir) . (6)

Finally, the carrier lifetime accounts for carrier recombination mechanisms that do not con-tribute to the stimulated emission, including spontaneous emission and non-radiative recombi-nation (due to e.g. traps and defects as well as Auger processes),

1

τc= Anr +BN + CN2. (7)

Here, AnrN is the recombination rate due to traps and defects, BN2 is the spontaneous emissionrate, and CN3 is the recombination rate induced by Auger processes. If β is the fraction ofspontaneous emission that falls into the lasing mode, Rsp can be expressed as a function of thecarrier number according to

Rsp = βBN2. (8)

2.2 Steady state operation

The rate equations (1)-(2) can be solved easily under steady-state operation of the laser. Inthis case, the applied current I is constant and

dN

dt= 0,

dP

dt= 0. (9)

If spontaneous emission is neglected (Rsp = 0), (1) leads to

P

(

G− 1

τp

)

= 0. (10)



Apart from the trivial solution P = 0, N = τcI/e corresponding to biasing of the laser belowthreshold, (10) leads to

G =1

τp. (11)

Reporting into (3), it can be seen that the number of carriers is constant and clamped to thethreshold value Nth

N = N0 +1

GN τp≡ Nth. (12)

Hence the number of photons,

P =τpe

(

I − eNth

τc

)

=τpe(I − Ith) , (13)

from which the power emitted by the laser can be calculated according to

Pe =1

2vgαmir~ωP. (14)

In order to derive (14), it has been assumed that the two laser cavity facets have the samereflectivities R1 = R2. It will furthermore be assumed that the internal quantum efficiencyηint is equal to 1 (meaning that all injected electrons contribute to stimulated emission abovethreshold), and that the optical coupling efficiency is also equal to 1 (all the emitted opticalpower can be collected at the output of the laser chip). The laser output power can also beexpressed in terms of the external quantum efficiency ηext

Pe =1

2ηext

hνI

e, (15)

which can also be written

Pe =1

2ηd (I − Ith)

hν

e, (16)

where ηd is the differential quantum efficiency, which verifies ηext = ηd (1− Ith/I) and can beexpressed as

ηd =ηintαmir

αmir + αint. (17)

2.3 Small-signal modulation

Remark: A detailed derivation of the small-signal frequency response of the laser is presentedin this section. Its main purpose is to complement the textbook and show how the importantresults on the laser modulation bandwidth can be obtained from the rate equations. Althoughsome of the equations might look scary at first, the calculations do not present major difficulties.In case of emergency, the main results that are necessary for the simulation exercise can be foundin equations (39)-(52).

The laser is now biased above threshold with a current Ib, resulting in a photon numberPb. If a small signal modulation ∆I (t), with |∆I (t)| ≪ |Ib − Ith| for all t, is added to the biascurrent

I (t) = Ib +∆I (t) , (18)

the photon number is expected to deviate from Pb according to

P (t) = Pb +∆P (t) , (19)



and similarly for the carrier number

N (t) = Nb +∆N (t) . (20)

An empirical relation between the gain G and the carrier number N has been introducedin (3). However this relation does not account for observed gain saturation effects, making itnecessary to also introduce a dependence of G on the photon number P . An explicit relationbetween G, N and P will be introduced in (42). For the time being, the dependence of thegain on the carrier and photon numbers will be taken into account by performing a first orderTaylor expansion around the biasing point

G = Gb +

(

∂G

∂N

)

∆N +

(

∂G

∂P

)

∆P, (21)

where the partial derivatives are taken at P = Pb and N = Nb. It should also be rememberedat this point that both the carrier lifetime and the spontaneous emission rate depend on thecarrier number N through (7) and (8). They can therefore be expressed according to Taylorexpansions of type

Rsp = Rsp (Nb) +∂Rsp

∂N∆N. (22)

The rate equations (1)-(2) are verified by the steady state values, leading to

0 = Pb

(

Gb −1

τp

)

+Rsp (Nb) , (23)

0 =Ibe− Nb

τc (Nb)−GbPb, (24)

where the dependence of the spontaneous emission rate Rsp and the carrier lifetime τc on theelectron number has been made more explicit.

Under small-signal modulation, inserting the expressions for the current and the numbersof photons and carriers (equations (18) to (20)), as well as the Taylor expansions for the gain,the carrier lifetime and the spontaneous emission rate into the rate equations (1) and (2), andconserving only first order terms in ∆P and ∆N leads to

d∆P

dt=

[

Gb −1

τp+

(

∂G

∂P

)

Pb

]

∆P +

[(

∂G

∂N

)

Pb +∂Rsp

∂N

]

∆N, (25)

and

d∆N

dt= −

[

Gb +

(

∂G

∂P

)

Pb

]

∆P −[(

∂G

∂N

)

Pb +1

τc+Nb

∂ (1/τc)

∂N

]

∆N +∆I

e. (26)

Introducing the quantities

ΓPP =Rsp

Pb−

(

∂G

∂P

)

Pb, ΓPN = −(

∂G

∂N

)

Pb −∂Rsp

∂N, (27)

ΓNP = Gb +

(

∂G

∂P

)

Pb, ΓNN =

(

∂G

∂N

)

Pb +1

τc+Nb

∂ (1/τc)

∂N, (28)



the system of equations (25)-(26) can be written under the simpler form

d∆P

dt= −ΓPP ∆P − ΓPN ∆N, (29)

d∆N

dt= −ΓNP ∆P − ΓNN ∆N +

∆I

e. (30)

If it is furthermore assumed that the current driving the laser varies sinusoidally with angularfrequency ωm and amplitude Im around the biasing point Ib

I (t) = Ib + Im cos (ωmt) , (31)

then, under the small-signal approximation, one also expects the photon and carrier numbersto vary sinusoidally at ωm around the steady-state values Pb and Nb, respectively. Adoptingcomplex notations, the variations of the current, photon and carrier numbers can be written

∆I (t) = ℜ[

Imejωmt]

(32)

∆P (t) = ℜ[

Pmejωmt]

(33)

∆N (t) = ℜ[

Nmejωmt]

(34)

Here, Pn and Nm are complex in order to account for eventual phase shifts between the photonand carrier numbers and the modulating current.

The system of equations (29)-(30) can then be expressed in a simple form according to

(jωm + ΓPP )Pm + ΓPNNm = 0, (35)

ΓNPPm + (jωm + ΓNN)Nm = Ime , (36)

which can be easily solved, leading to

Pm =−ΓPNIm/e

(jωm + ΓPP ) (jωm + ΓNN)− ΓPNΓNP

. (37)

Rearranging,

Pm =−ΓPNIm/e

(ΩR + ωm − jΓR) (ΩR − ωm + jΓR), (38)

where

ΩR =

[

−ΓPNΓNP − 1

4(ΓNN − ΓPP )

2

]1/2

(39)

and

ΓR =1

2(ΓPP + ΓNN ) (40)

will be related to the relaxation oscillations frequency and damping rate in Sect. 2.4. Hence thesmall-signal frequency response for the photon number and therefore the laser output power,

H (ωm) =Pm (ωm)

Pm (0)=

Ω2R + Γ2

R

(ΩR + ωm − jΓR) (ΩR − ωm + jΓR). (41)

So far no approximation has been made on G, as well as on the various terms involved inthe expression of the Γij coefficients in (27) and (28). The empirical expression for the net rate



of stimulated emission (3) is refined further by taking into account gain saturation due to e.g.spectral hole burning,

G = GN (N −N0) (1− ǫNLP ) . (42)

The gain suppression factor ǫNL accounts for the observed reduction of the gain for largevalues of P . The following approximations can then be made:

(

∂G

∂N

)

P=Pb

= GN (1− ǫNLPb) ≈ GN , (43)

(

∂G

∂P

)

N=Nb

= −ǫNLGN (Nb −N0) ≈ −ǫNLGb, (44)

hence

ΓPP ≈ Rsp

Pb+ ǫNLGbPb, ΓNN ≈ GNPb +

1

τc, (45)

ΓPN ≈ −GNPb, ΓNP ≈ Gb, (46)

and

ΩR ≈[

GbGNPb −1

4(ΓNN − ΓPP )

2

]1/2

≈√

GbGNPb. (47)

Using (11) and (13), the relaxation frequency can be linked to the biasing conditions of thelaser

ΩR ≈[

GN (Ib − Ith)

e

]1/2

. (48)

Finally, the 3-dB modulation bandwidth of the laser can be obtained from (41) by solvingthe equation

|H (2π∆f3dB)| =1

2, (49)

hence

∆f3dB =1

2π

[

Ω2R − Γ2

R + 2(

Γ4R +Ω4

R +Ω2RΓ

2R

)1/2]1/2

, (50)

which can often be approximated by

∆f3dB ≈√3ΩR

2π. (51)

The modulation transfer function can be normalised according to

H (ω) =1

1−(

ωω0

)2+ j 2γ

(

ωω0

)

(52)

where ω20 = Ω2

R + Γ2R and γ = ΓR/ω0. It can easily be shown that the modulus of the transfer

function (52) presents an extremum for ω2 = Ω2R − Γ2

R. The value of this peak is equal to

1/2γ√

1− γ2 which is minimum for γ = 1/√2. This situation occurs for Ω2

R = Γ2R, resulting in

a maximum corresponding to 1 reached for ω = 0. Normalised small-signal frequency responsescorresponding to different values of the damping parameter γ are represented in Fig. 1, showingthe existence of a peak associated to relaxation oscillations for γ < 1/

√2.



0.01 0.1 1 10

0.01

0.1

1

10

γ = 0.05

γ = 0.707

γ = 5

γ = 1

γ = 0.1

|

(ω)|

ω / ω0

γ = 0.2

Figure 1 Magnitude of the small-signal frequency response |H (ω)| as a function of the normalisedfrequency ω/ω0 for several values of the normalised damping parameter γ = ΓR/ω0.

2.4 Relaxation oscillations

The parameters ΩR and ΓR have been described earlier as the relaxation oscillations frequencyand damping rate. In this section, this terminology is clarified by looking at the response ofthe laser to a perturbation of its steady state operation condition, for instance in response to acurrent transient.

Assuming a small perturbation from the equilibrium condition, the linearised rate equations(29) and (30) can be used. In the present case, the solutions are assumed to be of the form

∆P (t) = ∆P0 e−λt, (53)

∆N (t) = ∆N0 e−λt, (54)

where λ is a complex constant. Inserting into the rate equations leads to

ΓPP ∆P0 + ΓPN ∆N0 = λ ∆P0, (55)

ΓNP ∆P0 + ΓNN ∆N0 = λ ∆N0, (56)

which can be written in a matrix form(

ΓPP ΓPN

ΓNP ΓNN

)(

∆P0

∆N0

)

= λ

(

∆P0

∆N0

)

. (57)

Consequently λ is an eigenvalue of the matrix

Γ =

(

ΓPP ΓPN

ΓNP ΓNN

)

, (58)

which can be found by solving the second degree equation det (Γ− λI) = 0, where I is the unitymatrix. The solutions are

λ =1

2(ΓPP + ΓNN)± j

[

−ΓPNΓNP − 1

4(ΓNN − ΓPP )

2

]1/2

, (59)



which can be expressed with the notations introduced in (39) and (40) as

λ = ΓR ± jΩR. (60)

Consequently, the carrier and photon numbers exhibit an oscillatory behaviour with a pseudo-period TR = 2π/ΩR, while the amplitude of the oscillations decays according to e−ΓRt.

2.5 Frequency chirping

Remark: In this section, the change in emission frequency when the laser driving currentis varied is derived. The derivation may be skipped, but the mechanism responsible for thisfrequency chirping should be understood and equation (74) will be used in the exercise.

A change in carrier density under current injection will lead to a change in refractive index,which in turn will modify the lasing frequency, whether the laser is of the Fabry-Perot type(resonance condition L = mλ/(2n), where L is the cavity length, n the mode index and man integer), or makes use of a distributed-feedback (DFB) structure (resonance condition Λ =mλ/(2n), where Λ is the period of the grating). When driven by a time varying signal, theinstantaneous lasing frequency will vary as a function of time, an effect known as frequencychirping. The relation between the frequency shift ∆ν (t) induced by frequency chirping andthe waveform at the laser output Pe (t) is derived in this section.

First, the lasing frequency shift induced by a change in carrier number is calculated. For aFabry-Perot laser, the lasing frequencies correspond to

ν =p c

2nL, (61)

where p is an integer and L is the cavity length. The effective index n is frequency dependent(dispersion) and also depends on the carrier number N . Assuming these variations to berelatively small, a first order Taylor expansion of the effective index about its value at thresholdcan be performed, leading to

n (ν,N) = n (νth, Nth) +∂n

∂ν(ν − νth) +

∂n

∂N(N −Nth) . (62)

Inserting (62) into (61) leads to

(ν − νth)

(

nth + νth∂n

∂ν

)

= −νth∂n

∂N(N −Nth) , (63)

where nth + νth∂n/∂ν is equal to the group index ng. Hence the deviation from the thresholdfrequency under the influence of a change in carrier number

ν − νth = −νthng

∂n

∂N(N −Nth) . (64)

Next, a relation between N−Nth and the photon number P is established. For this purpose,the gain G can be expanded according to (21) around a working point just above the laserthreshold. In these conditions, Gb ≈ 1/τp and Pb ≈ 0, hence

G =1

τp+

∂G

∂N(N −Nth) +

∂G

∂PP. (65)



Reporting into the rate equation for the photons (1), the desired relation between N −Nth andP is obtained:

(N −Nth)∂G

∂N=

1

P

dP

dt− Rsp

P− ∂G

∂PP. (66)

If furthermore a dependence of the gain on the carrier and photon numbers according to (42)is assumed, and with the usual approximations (43)-(44),

N −Nth =1

GN

[

1

P

dP

dt− Rsp

P+

ǫNL

τpP

]

. (67)

In a third and final step, the relation between ∂n/∂N appearing in (64) and ∂G/∂N ap-pearing in (66) is established. A general expression for the electric field propagating in the +zdirection in the cavity is

E (x, y, z, t) = E0 U (x, y) e1

2(g−αint)zej(ωt−kz) e, (68)

where U (x, y) is the transverse field distribution, k = 2πλ n, and e is the polarisation vector.

This field can also be expressed as

E (x, y, z, t) = E0 U (x, y) ej(ωt−2π

λnz) e, (69)

where n = n + jn is a complex refractive index whose real and imaginary parts are linked byHilbert transformation (also known as Kramers-Kronig) relations,

n = n+ jλ

4π(g − αint) . (70)

The changes induced in the real and imaginary parts of the complex refractive index by vari-ations of the carrier number are approximately proportional and their ratio is defined as thelinewidth enhancement factor α

α = −(

∂n

∂N

)

/

(

∂n

∂N

)

. (71)

Using (70),∂n

∂N= −α

λ

4π

∂g

∂N. (72)

Taking into account the fact that G = gvg (see Sect. 2.1) and reporting (72) and (66) into (64)leads to

ν − νth =α

4π

[

1

P

dP

dt− Rsp

P+

ǫNL

τpP

]

. (73)

Making use of (13) and (16), and neglecting the term Rsp/P , the instantaneous frequency shiftcan be expressed in term of the emitted power Pe according to

∆ν (t) =α

4π

(

d

dt[lnPe (t)] + κPe (t)

)

, (74)

where

κ =2ǫNL

ηdhν(75)



is often referred to as the adiabatic chirp coefficient1. The equation relating the frequency chirp∆ν (t) to the emitted power Pe (t) is valid under large-signal modulation. It can be seen thatthe chirp consists of two terms. The first term, named transient chirp only exists when theemitted power varies with time, for instance during transients of the applied current I (t) andinduced relaxation oscillations, while the second term, named adiabatic chirp, is responsiblefor the different emission frequencies observed under steady state when a “1” or “0” bit istransmitted.

Note that in (74), the frequency shift ∆ν = ν − νth is referred to the emission frequencyat threshold, which is difficult to evaluate in practice. However (74) can be used convenientlyto evaluate the difference in emission frequency between two operating points (correspondingto two different bias currents, hence two different emitted powers) of the semiconductor laser.This remark may be helpful for the resolution of the third assignment, in particular questions6 and 9.

3 Part 0: Preparation

In this exercise, a laser whose physical parameters are provided in Table 1 in Sect. 7 is considered.The following calculations need to be performed before the simulations so that the results canbe compared with those of the numerical simulations. The calculations could be performed ina Matlab programme where the required relations between the different parameters would beimplemented.

Assignment:

1. Calculate the laser threshold Ith.

2. Calculate the slope efficiency ∆Pe/∆Ib assuming an internal quantum efficiency ηint equalto 100%. The laser emission wavelength will be taken equal to 1552.5 nm.

3. The laser is now biased at Ib= 10 mA. Calculate the output power under steady stateoperation. Same question for Ib= 30 mA.

4. Calculate and plot the small-signal frequency response of the laserH (ωm) for bias currentsequal to Ib= 10 mA and 30 mA. The frequency range will be limited to 20 GHz. Whatare the 3 dB modulation bandwidths of the laser at those two biasing points? What arethe corresponding relaxation oscillation frequencies?

4 Part 1: Static characterisation

The static characteristics of a single-mode semiconductor laser are studied in this first assign-ment. The goal is to characterise the emitted power Pe versus bias current Ib curve of the laserand to extract the threshold value Ith and the slope efficiency ∆Pe/∆Ib. This information will

1Note that alternative definitions of the gain suppression factor are often found in the literature, especiallyin connection with rate equations expressed in terms of the photon density S (instead of the photon number Pused in this note), where the nonlinear gain will be expressed as (1− ǫS), leading to an alternative expressionfor the adiabatic chirp coefficient

κ =2ǫΓ

ηdhνV.



be used in later parts of the exercise in order to set the operating point of the laser under smalland large-signal modulation.

Figure 2 VPIplayer environment for the static characterisation of a single-mode semiconductor laser.

The simulation set-up is represented in Fig. 2. A dc current source is used to bias thesingle-mode laser diode. The value of the intensity of the current can be varied between 0and 100 mA by using the cursor in the control area at the bottom of the set-up. The outputpower of the laser is monitored as a function of time using an oscilloscope and as a functionof wavelength (or more properly frequency in the present case; the commonly used referencefrequency of 193.1 THz corresponds to a wavelength of about 1552.5 nm) using an opticalspectrum analyser. Note that such an oscilloscope is a simulation convenience that allows us todirectly display the optical power as a function of time, since in real life the light would needto be detected by a photodiode first in order to be displayed on a conventional oscilloscope.Furthermore, the total average power at the output of the laser is measured using an opticalpower meter. A multimeter is used to monitor the bias current and check that it has been setto the desired value. Note that in a real implementation, such a multimeter would have to beconnected in series between the current source and the laser in order to measure the intensityof the bias current. The values of the bias current and the total average emitted power arecollected so that they can be displayed in a table in units of mA and mW, respectively. Thesimulation set-up is run simply by adjusting the bias current cursor to the desired value (notethat the values are displayed in A) and pressing the start simulation button. Three windows willopen after completion of the calculations: a text display, an optical spectrum analyser displayand an oscilloscope display, from which it is possible to gather information to plot the Pe versusIb curve, monitor the emission wavelength, and check the output waveform, respectively.



Assignment:

1. Set the bias current to values between 0 and 100 mA.

2. Monitor the output waveform and optical spectrum. Comment the observed behaviour.

3. Write down in a table the values of the optical power and lasing frequency for each valueof the bias current.Hint: the spectrum can be integrated in a given resolution bandwidth, which will give a smoother display and

enable to monitor the lasing frequency more easily. A value of 1 GHz for the resolution bandwidth will be suitable

for the present investigation. What is the corresponding value for this bandwidth expressed in terms of wavelength?

4. Plot the emitted power Pe versus bias current Ib curve.

5. Extract the values for the threshold Ith and the slope efficiency ∆Pe/∆Ib of the laser.Compare with the values calculated analytically while preparing the exercise. Comparethe values of the output power for bias currents of 10 and 30 mA to the ones you had alsocalculated previously.

6. Plot the laser emission frequency (or wavelength) as a function of bias current. Explainthe observed frequency (wavelength) shift.

5 Part 2: small-signal modulation

Figure 3 VPIplayer environment for the small-signal characterisation of a single-mode semiconductorlaser.

The laser is now driven at a bias Ib by a sinusoidal signal whose amplitude is small comparedto Ib, in analogy with the theory detailed in Sect. 2.3. In the simulation set-up represented inFig. 3, a sinusoidal signal source whose frequency can be adjusted by the corresponding cursoris input to a laser driver. The laser driver amplifies the sinusoidal signal to the desired peak-to-peak value and adds a dc offset that corresponds to the desired bias. An oscilloscope isused to monitor the waveform of the current that is used to drive the laser diode. A second



oscilloscope is used to monitor the output of the laser. As long as the biasing point is in thelinear operation range of the laser (i.e. sufficiently larger than the threshold and before gainsaturation is observed at high bias current), an input sinusoidal electrical signal with amplitudeIm (peak-to-peak current 2Im) and angular frequency ωm will result in an output optical signalwith power amplitude Pem and the same angular frequency ωm, as illustrated in Fig. 4. Thesmall-signal frequency response is defined as in Sect. 2.3 as H (ωm) = Pe (ωm) /Pe (ωm → 0).

Figure 4 Principle of the characterisation of the laser small-signal frequency response. Note that, inthis example, the peak-to-peak current and emitted optical power have been magnified for illustrativepurpose.

Assignment:

1. Set the laser bias to Ib = 10 mA.

2. Set the electrical signal amplitude to the desired value. This amplitude is not critical, butyou should ensure that you comply with the small-signal modulation criterion.

3. For different values of the modulation frequency fm, monitor the amplitude of the modu-lation of the optical power. Hint: use horizontal markers in the oscilloscope display. You will also notice that

the value of the amplitude of the modulation of the optical signal is somehow different for small time values. This

is due to convergence issues in the numerical resolution of the rate equations. Consequently, ensure that the results

have converged when you monitor the amplitude of the emitted power. Furthermore, it is easier to monitor the peak-

to-peak power using markers than just the peak (or peak-to-mean) value. The peak-to-peak power is obviously twice

the value of |Pem|, as defined in Fig. 4. Since the small-signal frequency response H (ωm) = Pe (ωm) /Pe (ωm → 0)

is normalised, the peak-to-peak power value can be used directly.

4. Plot the corresponding small-signal frequency response of the laser.

5. Compare with the small-signal frequency response you had calculated while preparing theexercise.Hint: Plot the calculated and simulated responses on the same graph, preferably on a logarithmic scale, i.e.

10 log10 |H| versus fm.

6. Optional: repeat steps 1 to 5 above with Ib= 30 mA.



6 Part 3: large-signal modulation

In this last part of the exercise, the same single-mode laser as the one studied in Sect. 4 and 5 isnow driven by a large non return-to-zero (NRZ) data signal. Pseudo-random data is generatedin the upper left block and is encoded into the NRZ format with perfectly rectangular pulseswhose normalised amplitude is between 0 and 1. In order to make the data more realistic inthe simulations, the rectangular NRZ data is low-pass filtered resulting in a rise and fall timesequal to about one quarter of the bit rate B. The filtered data signal d (t) is then input toa laser driver, where it is amplified to the desired peak-to-peak current value and where a dcoffset is added. An oscilloscope is used to monitor the electrical driving signal. Note that thelaser driving signal is now defined as

I (t) = Ib + Ipp d (t) (76)

where d (t) is the data signal that is normalised to the range [0,1] and Ipp is the peak-to-peakcurrent. Consequently the laser is driven with a current equal to Ib when a “0” bit is transmittedand Ib + Ipp when a “1” bit is transmitted. The simulation set-up shown in Fig. 5 enables toadjust the biasing point Ib, the peak-to-peak current of the modulating signal, as well as thebit rate B. An oscilloscope is used to monitor the current waveform that is used to drive thesingle-mode laser, whose output is characterised thanks to a second oscilloscope and an opticalspectrum analyser.

Figure 5 VPIplayer environment for the study of large-signal modulation of a single-mode semicon-ductor laser.

In this part of the exercise, optical signals will be generated by directly modulating thelaser at bit rates of 2.5 and 10 Gbit/s. The influence of the bit rate and the biasing point onthe laser waveform and spectrum will be studied qualitatively and related to the small-signalfrequency response determined in Sect. 5. The chirping behaviour of the laser will also bestudied and its temporal and spectral features will be related. Furthermore, it will be shownhow the measured waveforms and spectra can be used to retrieve essential parameters such asthe relaxation oscillations frequency and the adiabatic chirp factor.



Assignment:

1. First, set the bit rate to 2.5 Gbit/s, bias the laser at 10 mA and drive it with a peak-to-peak current value of 20 mA.

2. Observe and comment the waveform generated at the output of the laser.

3. Can you retrieve the relaxation oscillation frequencies corresponding to bias values of 10and 30 mA?Hint: use vertical markers in the oscilloscope display.

4. Set the bias current to 30 mA while keeping the bit rate to 2.5 Gbit/s and the peak-to-peakcurrent to 20 mA.

5. Compare the waveform and spectrum with the ones obtained for a bias current of 10 mAin point 1. above.

6. What is the physical origin of the two spikes that appear in the laser spectrum at a biascurrent of 30 mA?

7. Why do those spikes are less pronounced when the laser is biased at 10 mA?

8. The oscilloscope display enables to visualise the chirp of the emitted signal. Select Power:X, then tick Show chirp. By performing this action, only the signal power aligned withthe X-polarisation is displayed. The X-polarisation actually corresponds to the stateof polarisation at the output of the laser (consequently no power is carried along theY-polarisation). It then makes sense to consider the frequency chirp of the signal bycalculating the derivative of its time-varying phase with respect to time. Visualise thefrequency chirp for bias currents equal to 10 and 30 mA. Compare the chirping behaviourat the two biasing points and relate your observations to the corresponding laser outputspectra.

9. Retrieve the gain compression factor ǫNL from the signal emitted by the laser for a biascurrent of 30 mA. Compare with the corresponding value in Table 1.

10. Generate the spectra and waveforms at the output of the laser under the same drivingconditions as in points 1. and 4. above, but this time at a bit rate of 10 Gbit/s.

11. Compare the results with the corresponding waveforms and spectra at 2.5 Gbit/s.

12. Relate your observations to the small-signal frequency responses obtained in Sect. 5.



7 Laser parameters and physical constants

Parameter Symbol Value Unit

Active volume V 3.6 × 10−17 m3

Cavity length L 300× 10−6 m

Gain cross section σg 3.0 × 10−20 m2

Carrier number at trans-parency

N0 72× 106

Confinement factor Γ 0.25

Group effective index ng 4

Linewidth enhancement α 6

Mirror reflectivities R1, R2 0.3

Photon lifetime τp 2.6 × 10−12 s

Carrier lifetime τc 3.2× 10−9 s

Gain suppression factor ǫNL 1.4× 10−7

Table 1 Laser physical parameters.

Parameter Symbol Value Unit

Speed of light in vacuum c 299792458 m·s−1

Elementary charge e 1.60217653 × 10−19 C

Planck’s constant h 6.6260693 × 10−34 J·s

Table 2 Essential physical constants.

8 Recommended bibliography

One will first and foremost consult the lecture material:

[1] J. Mørk, “Sources and Transmitters”, Lecture notes in 34130 “Introduction to OpticalCommunication”, Department of Photonics Engineering, Technical University of Denmark,2011.

[2] G. P. Agrawal, “Fiber-Optic Communication Systems”, 3rd edition, Wiley, New-York, 2002.

The notations adopted in this note, as well as the choice of the expressions of the rate equations follow this ref-erence, which is the textbook adopted for the course. The relevant material can be found in:

• Section 3.3: “Semiconductor lasers”

• Section 3.5: “Laser characteristics”



Other classic references that have been used for the preparation of this note include:

[3] A. Yariv, “Optical Electronics in Modern Communications”, 5th edition, Oxford UniversityPress, New York, 1997.

• Chapter 15: “Semiconductor lasers - Theory and Applications”, and more specifically:

• Section 15.5: “Direct-current modulation of semiconductor lasers”

• Section 15.6: “Gain suppression and frequency chirp in current-modulated semiconductor lasers”.

[4] K. Petermann, “Laser Diode Modulation and Noise”, Kluwer Academic Publishers, Dor-drecht, 1988.

In particular:

• Chapter 2: “Basic laser characteristics”

• Chapter 4: “Intensity-modulation characteristics of laser diodes”

• Chapter 5: “Frequency modulation characteristics of laser diodes”

The derivation of the frequency shift (74) in Sect. 2.5 follows the approach presented in this reference.

[5] G. P. Agrawal and N. K. Dutta, “Semiconductor lasers”, 2nd edition, Kluwer AcademicPublishers, Dordrecht, 1993.

The notations and definitions are fully consistent with reference [2] and some of the derivations are carried outfurther than in the textbook, in particular:

• Chapter 6: “Rate equations and operating characteristics”, and more specifically:

• Section 6.4.3 “Relaxation oscillations” is the basis of the derivation of the relaxation oscillations in Sect. 2.4

• Section 6.6 “Modulation response”.

For the specially enthusiastic student, the original references on frequency chirp (without and with the gain saturationterm leading to adiabatic chirp) can be found in:

[6] T. L. Koch and J. E. Bowers, “Nature of wavelength chirping in directly modulated semi-conductor lasers”, Electronics Letters, vol. 20, no. 25/26, pp. 1038–1040, 1984.

[7] T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laserwavelength chirping”, Applied Physics Letters, vol. 48, no. 10, pp. 613–615, 1986.

Finally, apart from the textbook, the following references have been consulted in order to define parameters for thedirectly modulated laser:

[8] S. Mohrdiek, H. Burkhard, F. Steinhagen, H. Hillmer, R. Losh, W. Schlapp, and R. Gobel,“10-Gb/s standard fiber transmission using directly modulated 1.55-µm quantum well DFBlasers”, IEEE Photonics Technology Letters, vol. 7, no. 11, pp. 1357–1359, 1995.

[9] I. Tomkos, D. Chowdhury, J. Conradi, D. Culverhouse, K. Ennser, C. Giroux, B. Hallock,T. Kennedy, A. Kruse, S. Kumar, N. Lascar, I. Roudas, M. Sharma, R. S. Vodhanel, andC.-C. Wang, “Demonstration of negative dispersion fibers for DWDM metropolitan areanetworks”, IEEE Journal of Selected Topics in Quantum Electronics, vol. 7, no. 3, pp. 439–460, 2001.

[10] J. D. Downie, I. Tomkos, N. Antoniades, and A. Boskovic, “Effects of filter concatena-tion for directly modulated transmission lasers at 2.5 and 10 Gb/s”, Journal of LightwaveTechnology, vol. 20, no. 2, pp. 218–228, 2002.


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