Geracao.Formatos.Modulacao_02_04

Generation and Detection of Optical Modulation Formats

Christophe Peucheret

DTU Fotonik

Department of Photonics Engineering

Technical University of Denmark

[email protected]

26/03/2012

Abstract

This note is intended as a complement to the lectures in 34150 “Optical Communi-cation Systems” on advanced lightwave modulation and transmission systems. After re-viewing some basic concepts on modulation, it first presents a simple description of theMach-Zehnder modulator used as an intensity or a phase modulator. The concept of differ-ential binary phase-shift keying (DPSK) modulation with interferometric detection is thenintroduced. How Mach-Zehnder modulators can be integrated into a more general modula-tor structure allowing the synthesis of arbitrary modulation formats is presented next. Thegeneration of differential quadrature phase-shift keying (DQPSK) and quadrature-amplitudemodulation (QAM) are taken as examples. The principle of DQPSK modulation used inconjunction with interferometric detection is then briefly outlined. Finally, some hints onhow more advanced modulation formats can be detected in a phase diversity receiver areprovided.

Contents

1 Modulation concepts 2

1.1 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Constellation diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Mach-Zehnder modulator 5

2.1 Principle of operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 General MZM model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Frequency chirping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Phase modulation using a Mach-Zehnder modulator . . . . . . . . . . . . . . . . 13

2.5 Pulse carving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Differential phase-shift keying 17

3.1 Introduction to phase modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Coherent detection of PSK signals . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Self-homodyne DPSK detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Interferometric demodulation of DPSK . . . . . . . . . . . . . . . . . . . . . . . . 21

1

34150 Optical Communication Systems

4 The optical quadrature modulator 26

4.1 Motivations for optical multilevel modulation . . . . . . . . . . . . . . . . . . . . 26

4.2 Principle of the optical IQ modulator . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3 QPSK signal generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.4 Square QAM signal generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

5 Differential quadrature phase-shift keying 30

5.1 Signal generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5.2 Interferometric detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6 Phase diversity receiver 34

Appendix A List of acronyms 37

1 Modulation concepts

1.1 Modulation

In the context of optical communication, modulation consists in imprinting some information,generally available under the form of an electrical signal, on an optical carrier. This can beachieved by modifying one of the physical attributes of the electric field associated with theoptical wave

E (t) = Re{√

P (t)e−jφ(t)ejω0te (t)}. (1)

The quantities that can be modulated according to the information to be transmitted are thepower P (t), the phase φ (t), the carrier frequency ω0, or the state of polarisation e (t).

It has been implicitly assumed when writing (1) that the frequency content of the signal isrestricted to a limited bandwidth around its angular carrier frequency ω0. In other terms, itmeans that the modulation speed (i.e. the rate at which the modulated quantity is made tovary) is much lower than the reciprocal of the carrier frequency T0 = 2π/ω0.

Neglecting the vectorial nature of light (i.e. its polarisation), the arbitrary electric field of(1) can be written1

E (t) = Re{E (t) ejω0t

}, (2)

where

E (t) =√P (t) e−jφ(t), (3)

is known as the complex envelope of the signal. Note that this definition corresponds to that ofa bandpass signal met in the classical theory of signals and systems.

The modulating signal can be of digital or analogue form. This course is mostly concernedwith digital modulation, in which case the complex envelope of a modulated optical wave canbe expressed as

E (t) =√P0

∑

k

ak p (t− kTs) , (4)

1It is important to note that the phase conventions used here are not universal and opposite signs may befound in the fast varying term oscillating at the carrier frequency and/or the phase term. The choice of those signsin the representation of real quantities by complex numbers is arbitrary. However the sign of physical quantitiesshould be right. Different research communities and different individuals tend to favor different conventions. Thisis especially the case here, where the fields of electromagnetism and digital communications converge. Thereforethe reader should be prepared to meet different conventions and should remain alert to ensure they are usedconsistently.

2


where P0 is the signal peak power, p (t) is a pulse shape function whose amplitude is, withoutloss of generality, constrained to [0, 1], and ak is the digital data sequence to be transmitted2.Ts corresponds to the inverse of the rate at which pulses are transmitted, a quantity known asthe symbol rate S. ak takes values in a limited set of complex numbers, which simply expressesthe digital nature of the signal.

If furthermore the digital modulation is binary, the digital sequence can take one of twovalues {a1, a2} and Ts = Tb, where the bit period Tb is the inverse of the bit rate Tb = 1/B. Forinstance, in the case of on-off keying (OOK) modulation the digital sequence can take valueswithin {0, 1} while in the case of binary phase-shift keying (BPSK) modulation the values willbe constrained to

{0, e−jπ

}.

In the more general case of m−ary modulation, ak can take values in a set of m modulationstates {a1, · · · , am}. In this case, each transmitted symbol carries log2m bits and the symbolrate and the bit rate are related according to

S =B

log2m. (5)

The bit rate B is expressed in units of bits per second (bit/s) while the symbol rate S is expressedin bauds or symbols per second. As an example, quadrature phase-shift keying (QPSK) is aquaternary (two bits per symbol) modulation format where the signal can take four phasevalues, hence the ak are taken within

{1, e−jπ/2, e−jπ, e−j3π/2

}. In this case, the symbol rate is

half of the bit rate.

Since E (t) is a complex quantity, it can also be expressed in term of its real and imaginaryparts

E (t) = Re {E (t)}+ jIm {E (t)} . (6)

Re {E (t)} and Im {E (t)} are referred as the in-phase and quadrature components of the signal,respectively.

In order to guarantee the compatibility of the phase convention introduced in (1) withthe definition of the argument of a complex number, z = x + jy = |r| ejθ, and its naturalrepresentation in the complex plane (usual conventions met in digital communications), theI and Q component of the signals will be defined in what follows as I (t) = Re {E (t)} andQ (t) = −Im {E (t)}, leading to

E (t) = I (t)− jQ (t) . (7)

With this convention, a positive phase shift will be represented in the (I,Q) space as as acounter-clockwise rotation, in agreement with the usual conventions met in digital communica-tions.

1.2 Constellation diagrams

In a digital optical communication system, one would like the field to take well defined values,defined hereafter as modulation states depending on the data to be transmitted. These modu-lation states can be simply represented on a constellation diagram, which is a representation inthe complex plane (I,Q) of the allowed values of the complex envelope E (tk) at the samplinginstant tk (i.e. when a decision on the symbol is made. With such a representation, the constel-lation diagram of a m−ary modulation format will consist of m dots in the (I,Q) space whenthe modulation is ideal. During transitions between modulation states, the complex envelope ofthe electric field can take values outside this discrete set, corresponding to trajectories linking

2One may generalise the concept by considering that one waveform sk (t) is associated with each modulationstate. However a description according to (4) is sufficient for all the modulation formats considered in this note.

3


OOK (unchirped) PSK (chirped)PSK (unchirped)OOK (chirped)

QPSK (PM+PM)QPSK (MZM+PM)QPSK (IQ)

Figure 1 Illustration of constellation diagrams for some typical optical modulation formats. The reddots indicate the modulation states, while the blue lines show the trajectory of the electric field duringtransitions between these modulation states.

the modulation states on the constellation diagram. These trajectories are seldom representedin constellation diagrams. However they are important in practice since they provide informa-tion on the deviations of the electric field between the modulation states, hence on frequencychirping, signal peak power and average power, etc. In this sense, once could interpret constel-lation diagram with trajectories as a generalization of the well known concept of eye diagramused for one dimensional modulation.

As an illustration, several constellation diagrams assuming ideal modulation conditions arerepresented in Fig. 1. The red dots indicate the allowed modulation states, while the blue linesshow the trajectory of the complex envelope of the electric field during transitions between themodulation states. For instance, in the case of ideal OOK modulation, the modulation statescorrespond to the two (normalised) power levels 0 and 1, represented as a dot at the centrepoint as well as on the unit circle in the (I,Q) space. It will be seen later that, unless specialprecautions are taken, the signal resulting from OOK modulation may be chirped. This meansthat the instantaneous frequency of the signal, defined with the phase conventions of (1) as

ω (t) = ω0 −∂φ (t)

∂t(8)

is changing when the signal is modulated, or equivalently that its phase is a time varyingfunction. In the constellation diagrams, transitions at constant phase will therefore correspondto unchirped signals, while phase-varying transitions are the fingerprints of chirped signals. Inthe case of BPSK modulation, the modulation states appear as dots on the unit circle separatedby a phase difference of π. Here again, some distinction can be made between chirped andunchirped modulation. The generation of unchirped signals will be discussed in Sec. 2.3 forOOK and in Sec. 2.4 for BPSK.

Finally, several constellation diagrams corresponding to different methods of generation ofoptical QPSK are also represented in Fig. 1. It can be seen that all methods result in thegeneration of the wished modulation states, represented as dots on the unit circle with relativephase offsets of π/2. However, the transitions are fairly different for the three methods. For agiven generation method, some transitions may be chirped while others are not. Furthermoresome methods achieve transitions at constant power (i.e. the corresponding trajectory followsthe unit circle) while others result in variations of the intensity while the phase is modulated.The generation of QPSK signals will be described in more details in Sec. 4.

4


Those examples illustrate how useful constellation diagrams can be to understand the exactnature of the modulation process. In real cases, the modulation states will no longer appearas a single dot in the constellation diagram due to noise and signal distortion resulting frome.g. bandwidth limitations. Furthermore, the transitions may also depart from the ideal curvesdue to imperfections in the modulation process, adding even more practical value to measuredconstellation diagrams.

2 The Mach-Zehnder modulator

In this section the operation of the most common and versatile external optical modulator usedin most of high-speed optical communication systems will be described in details. After outliningits principle of operation and briefly describing its implementation in electro-optic crystals, adetailled and fairly universal model of a Mach-Zehnder modulator (MZM) will be presented.Using this model it will be shown that both chirped and unchirped intensity modulation can beachieved using an MZM. Its use as a phase modulator will then be described. Finally, methodsto generate optical return-to-zero (RZ) signals from non return-to-zero (NRZ) electronics usingcascaded MZM modulators will be presented.

2.1 Principle of operation

The refractive index of some materials can be modified by applying an external electric fieldto them through the linear electro-optic effect, also known as Pockels effect.3 The importantfeatures of this effect is that the refractive index change is proportional to the applied volt-age and that for most practical purposes the effect can be described as “instantaneous”. Byinstantaneous, it is meant that its time scale is of the order of a few femtoseconds (1 fs =1.0×10−15 s), which is much faster than the time scales associated with the modulation, typ-ically of the order of 10-100 ps (1 ps = 1.0×10−12 s) for modulation speeds in the range of10-100 Gbit/s. A straightforward application is the realisation of phase modulators made froman electro-optic waveguide subjected to a time dependent electric field. The applied voltagemodulates the refractive index of the waveguide material, hence the phase shift experienced bya light wave propagating along the waveguide. However, legacy optical communication systemstypically rely on intensity modulation of the light wave. This can be achieved by transformingthe phase modulation induced by the electro-optic effect into intensity modulation using aninterferometric structure.

In order to illustrate the operation principle of such a device, the simple interferometricstructure represented in Fig. 2 is considered. It is based on a Mach-Zehnder interferometerincluding one electro-optic material in one of the arms. When used as a data modulator, sucha structure is generally integrated by diffusing some waveguides in the electro-optic materialand depositing the electrodes on top or around the waveguides. Assuming the power is splitor combined equally in the input and output couplers of the Mach-Zehnder interferometer, thepower at the output of the interferometer depends on the difference between the phase shiftsexperienced by the light propagating in the upper and lower arm of the structure, φ (t) and φ0,respectively, according to

Pout = Pin cos2 ∆φ

2, (9)

where ∆φ = φ (t)− φ0.

3This effect and how it can be applied to high-speed light modulation are discussed in details in the course34153 “From Photonics to Optical Communications.”

5


0 1 2 3 40.0

0.5

1.0

Figure 2 Principle of operation of a Mach-Zehnder modulator. a) Mach-Zehnder interferometer con-figuration where the refractive index in one of the arms is modified by an applied voltage through thelinear electro-optic effect. b) Transfer function of the modulator and typical operating condition for nonreturn-to-zero modulation.

The phase shift induced in the upper arm of the interferometer depends on its refractiveindex, which itself depends on the applied external electric field through the electro-optic effect.If a time-dependent voltage V (t) is applied across the upper waveguide of the modulator, itsrefractive index will become time dependent and, in turn, the transmission Pout/Pin of the Mach-Zehnder interferometer will also depend on time. If a continuous optical wave is applied to theinput of the modulator, the output power will thus be modulated according to the electricaldata V (t). The value of the phase shift created by an applied external voltage depends uponmany parameters, including the choice of the electro-optic material, the orientation of theexternal electric field with respect to the principal axes of the crystal, the polarisation of theincoming lightwave, as well as the geometry and dimensions of the waveguide. Lithium-niobate(LiNbO3) is customarily used as a suitable electro-optic material for high-speed modulation inoptical communication systems, owing to its relatively large electro-optic coefficient and wellcontrolled waveguide fabrication processes. In any case it is possible to make abstraction of theactual physical implementation of the modulator and describe the ability of the material andchosen configuration to respond to an applied voltage by introducing a quantity known as thehalf-wave voltage Vπ. Applying a voltage of Vπ to the electrode of an electro-optic waveguidewill result in a voltage-induced phase shift of π. The electro-optically induced phase shift φ (t)can therefore be related to the applied voltage V (t) according to

φ (t) = πV (t)

Vπ. (10)

Through (9) and (10), it is then possible to calculate the transmission of the modulator as afunction of the applied voltage. Such a “transfer function”4, where the applied voltage has beennormalised to the half-wave voltage, is also represented in Fig. 2. The operation conditions ofelectro-optic Mach-Zehnder modulators are detailed further in Sec. 2.2. In particular it will be

4This term is often used to describe the power transmission as a function of applied voltage of an intensitymodulator. It is however improper as it does not correspond to the usual notion of transfer functions used in thetheory of linear systems.

6


Figure 3 Typical layout of a LiNbO3 dual-electrode intensity modulator (Source: Sumitomo OsakaCement Co., Ltd).

shown that, in contrast with this simple description, applying an electrical signal to both armsof the Mach-Zehnder modulator will be required in order to generate an unchirped signal.

Fig. 3 shows the layout of a dual-drive MZM where different modulating signals can beapplied to the two arms of the Mach-Zehnder structure. As previously mentioned, a LiNbO3

crystal is used as a substrate where the Mach-Zehnder interferometer structure is patterned bydiffusing titanium to locally raise the refractive index and form waveguides. Gold travelling waveelectrodes are then deposited to apply the modulation. Even though the physical effect leadingto the refractive index change is nearly instantaneous, some bandwidth limitations neverthelessexist in such device. This is due to the fact that the refractive index change induced by theapplied electric field is quite small. Consequently, a long interaction length between the opticalwave and the electrical radio frequency (RF) driving signal is required in order to create therequired phase shift for switching the modulator between its desired output states. However,since the optical wave is travelling at the light velocity in the medium, efficient interactioncan only be achieved at high frequencies if travelling wave electrodes are used for the drivingsignal. Since the dielectric permittivity of the crystal is different at optical and microwavefrequencies, the velocities of the optical and RF signals are different, which result in bandwidthlimitations5. Consequently a considerable engineering effort is continuously devoted to solutionsable to provide velocity matching between the optical and RF waves in order to enlarge thebandwidth of electro-optic modulators and meet the requirements induced by ever increasingbit rates.

The typical length of electro-optic waveguides in LiNbO3 modulators used for telecommu-nication application is of the order of a few centimeters (e.g. ∼ 4 cm), making such devicesrelatively bulky when fitting on a transponder card together with the RF driver amplifiers andcontrol electronics. Typical switching voltages are of the order of Vπ ∼ 6 V, even though signif-icant effort has been dedicated to decreasing this number over the years. It should be pointedout that obtaining a digital binary signal peak-to-peak voltage of this order of magnitude isfar from being trivial when operating at bit rates of 40 Gbit/s and above, requiring expensivebroadband driving amplifiers.

5These points, as well as the beautiful physics of the electro-optic effect are described in much detail in 34153“From Photonics to Optical Communications”.

7


Figure 4 General structure of a Mach-Zehnder modulator with power splitting and combining ratiosequal to α and β, respectively, and driven with voltages V1 (t) and V2 (t) applied to the upper and lowerarm, respectively.

2.2 General Mach-Zehnder modulator model

The structure of a Mach-Zehnder modulator is depicted schematically in Fig. 4. It consists ofan input waveguide followed by a Y-junction that splits the optical signal between two armswhere the optical field experiences phase shifts equal to φ1 and φ2, respectively. In the mostgeneral case, it is assumed that φ1 and φ2, which depend on the voltages applied to the upperand lower arm, V1 and V2, respectively, can be created independently. The fields propagatingin each of the arms are then combined in an output waveguide via another Y-junction. It isalso assumed that the power splitting ratios of the input and output Y-junctions are α and β,respectively. Furthermore, the loss experienced by the optical field in the upper and lower armof the structure, including excess loss in the input and output Y-junctions, is described by ρ1and ρ2, respectively.

The field at the output of the modulator is the sum of the contributions propagating throughthe upper and lower paths. Therefore, its complex envelope can be expressed as

Eout (t) =(ρ1√α√β e−jφ1 + ρ2

√1− α

√1− β e−jφ2

)Ein (t) . (11)

Introducing the notations,

a = ρ1√α√β, (12)

b = ρ2√1− α

√1− β, (13)

the output field becomes

Eout (t) = e−jφ[a e−j

∆φ

2 + b ej∆φ

2

]Ein (t) , (14)

where

∆φ = φ1 − φ2, (15)

φ =φ1 + φ2

2. (16)

It is possible to find a general expression for the field at the output of the modulator byexpanding (14) into real and imaginary parts

Eout (t) = e−jφ[(a+ b) cos

∆φ

2+ j (b− a) sin

∆φ

2

]Ein (t) , (17)

8


leading to

Eout (t) = Ein (t) e−j(φ+ψ)√

(a+ b)2 cos2∆φ

2+ (a− b)2 sin2

∆φ

2, (18)

where the phase term ψ depends on the power imbalance between the upper and lower armsof the interferometer, as well as on the phase difference experienced by the light propagatingthrough the upper and lower waveguides, according to

tanψ =

[a− b

a+ b

]tan

∆φ

2. (19)

The phase shifts in the upper and lower arms of the structure, φ1 and φ2, can be expressed asa function of the applied voltages V1 and V2, respectively, as well as the half-wave voltage Vπ,which is assumed to be equal in both arms.

φ1 (t) = πV1 (t)

Vπ, (20)

φ2 (t) = πV2 (t)

Vπ. (21)

The voltage applied to each arm of the interferometer can be written as the sum of a d.c. andan a.c. term

V1 (t) = Vdc,1 + Vpp,1 d1 (t) , (22)

V2 (t) = Vdc,2 + Vpp,2 d2 (t) , (23)

where Vpp,i is the peak-to-peak voltage of the signal applied to the arm i, and di (t) is its nor-malised waveform such that di (t) ∈

[−1

2 ,12

].

It is now assumed that the power splitting and combining ratios at the input and outputY-junctions are α = β = 1

2 and the excess loss experienced in the upper and lower paths of theinterferometer are equal (ρ1 = ρ2 = ρ, resulting in a = b = ρ

2 ). Re-writing (18) leads to

Eout (t) =√Pout (t) e

−jφ(t), (24)

where

Pout (t) = Pin (t) ρ2 cos2

[π

2Vπ{Vdc,1 − Vdc,2 + Vpp,1 d1 (t)− Vpp,2 d2 (t)}

], (25)

andφ (t) =

π

2Vπ{Vdc,1 + Vdc,2 + Vpp,1 d1 (t) + Vpp,2 d2 (t)} . (26)

The power at the output of the Mach-Zehnder modulator therefore depends on the differencebetween the d.c. voltages applied to each of the arms, ∆Vbias = Vdc,1−Vdc,2, known asmodulator

bias, as well as on the difference between the driving signals Vpp,1 d1 (t) − Vpp,2 d2 (t). Thisopens up many possibilities for the use of the Mach-Zehnder modulator in optical communicationsystems, a number of which will be described in the following sections. The simple case whereone wants to achieve the commonly used intensity modulation of continuous wave light accordingto an electrical modulating signal d (t) is described first.

Fig. 5 represents the transmission of the modulator Pout/Pin as a function of the differencebetween the voltages applied to each of the two arms ∆V = V1 − V2. For simplicity, it hasbeen assumed that the insertion loss of the modulator can be neglected so that ρ2 = 1. Thetransfer function is obviously periodic with a period 2Vπ. When restricted to a portion limited

9


Figure 5 Illustration of the biasing points and driving voltage for intensity modulation using a Mach-Zehnder modulator.

by a minimum and the nearest maximum, therefore corresponding to monotonous variations ofthe function Pout/Pin, the transfer function of the device can be exploited to perform intensitymodulation of a continuous wave input according to an electrical data stream d (t). In particular,OOK modulation can be be achieved when driving the modulator between a minimum and thenearest maximum in its transfer function, as shown in Fig. 5. Ideally, one would like theextinction ratio of the modulator, defined as

ER =Pmax

Pmin, (27)

where Pmax and Pmin are the maximum and minimum transmitted power, respectively, tobe as large as possible. Obtaining an infinite extinction ratio would require total destructiveinterference at the output of the Mach-Zehnder structure, which can only happen if the splittingand propagation loss are perfectly balanced in the two propagation paths, as can be checkedfrom (18). Such a total destructive interference cannot be achieved in practice, and the staticextinction ratio of MZMs is of the order of 10-15 dB, which is sufficient in practice.

Furthermore, if operated between a minimum and it nearest maximum transmission, thenon-linear nature of the Mach-Zehnder transfer function will result in eventual residual voltageripples of the driving signal to be significantly attenuated in the generated optical signal. Thenon-linear transfer function of the modulator therefore provides a mean to prevent electrical sig-nal imperfections from being transferred to the optical domain in case binary OOK modulationis performed.

2.3 Frequency chirping

Single-arm drive

The most straightforward implementation of the Mach-Zehnder modulator consists in applyinga modulating signal d (t) to only a single arm of the interferometer. The power at the output

10


0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

-10

-5

0

5

10

fre

qu

ency c

hir

p (

GH

z)

time (ns)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

0.0

0.2

0.4

0.6

0.8

1.0

po

we

r (m

W)

Figure 6 Illustration of the frequency chirp generated at the output of a single-drive Mach-Zehndermodulator. The power and chirp at the output of the modulator have been calculated numerically for a10 Gbit/s NRZ signal.

of the device can be directly expressed from (25) with d1 (t) = d (t) and d2 (t) = 0, immediatelyleading to

Pout = Pin ρ2 cos2

[π

2Vπ{∆Vbias + Vpp,1 d (t)}

]. (28)

In this case, the ideal operating conditions to generate an intensity modulated signal with largeextinction ratio and maximum peak power require the modulating signal to have a peak-to-peakvoltage equal to Vpp,1 = Vπ.

Eq. (26) also shows that, the phase shift induced by the modulator is time dependent. Ifthe constant phase shift, equivalent to a propagation delay, is neglected, the time varying phaseshift induced by the modulator can be written

φ (t) =π

2VπVpp,1 d (t) . (29)

Consequently, the generation of an intensity modulated signal using a single arm drive Mach-Zehnder modulator is always accompanied with phase modulation. This is equivalent to achange of the instantaneous frequency of the signal according to (8), hence to frequency chirping.As an illustration, the frequency chirp at the output of a Mach-Zehnder modulator driven froma single arm with a 10 Gbit/s non return-to-zero (NRZ) signal has been calculated and isrepresented in Fig. 6 together with the emitted waveform. It can be seen that, with realisticvalues of the modulator parameters, frequency deviations of the order of ±10 GHz are inducedon the signal.

Even though well controlled frequency chirp can be beneficial for transmission, frequencychirping that follows the variations of the intensity modulated signal is likely to result in trans-mission impairments and is therefore an undesirable feature of the modulated signal. Suchfrequency chirping is a well-known detrimental effect that restricts the use of directly modu-lated lasers (DMLs) at high bit rates to short transmission distances. In the context of DMLs,the instantaneous frequency change δν (t) can be expressed as a function of the emitted power

11


P (t) according to

δν (t) =α

4π

(d

dt[lnP (t)] + κP (t)

), (30)

where α is known as the linewidth enhancement factor and κ is the adiabatic chirp coefficient.The first term in (30), known as transient chirp, only exists when the emitted power varies withtime, for instance during transients of the applied electrical signal, while the second term, namedadiabatic chirp, is responsible for the different emission frequencies observed at the output ofDMLs under steady state when a “1” or a “0” is transmitted. In case the chirp behaviour isdominated by transient effects, the frequency deviation can be expressed as

δν (t) =α

4π

(1

P (t)

dP (t)

dt

), (31)

hence, since the instantaneous frequency change is related to the phase of the complex envelopeof the signal according to (8),

α = −dφdt

12P

dPdt

. (32)

In analogy with directly modulated lasers, it is possible to define a chirp parameter αMZ accord-ing to (32) for Mach-Zehnder modulators. In the case of a single drive modulator, substituting(28) and (29) into (??) immediately leads to

αMZ = cot

[π

2Vπ{∆Vbias + Vpp,1 d (t)}

]. (33)

It is generally considered that frequency chirping is to be avoided in optical transmitters forlong distance transmission, since it broadens the spectrum, which in turn reduces the dispersiontolerance of the signal and limits the spectral efficiency. Consequently, chirp-free operation isa desired feature of any modulation process. Interestingly, this can also be achieved in MZMsprovided the driving conditions are modified, as detailled in the following section.

Chirp-free operation

In the most general case when two distinct driving signals d1 (t) and d2 (t) are applied to thetwo arms of the modulator, (25) and (26) apply. It can especially be seen from (26) that, forchirp-free operation, the phase of the signal at the output of the modulator should becometime-independent, which can be achieved provided the condition

Vpp,1 d1 (t) + Vpp,2 d2 (t) = 0 (34)

is satisfied. If furthermore the two signals that are applied to each of the arms of the modulatorhave the same peak-to-peak voltage,Vpp,1 = Vpp,2 = Vpp, this condition translates into

d1 (t) = −d2 (t) . (35)

This condition is general for chirp-free operation of the modulator, whether the signals to bemodulated are analogue or digital. In the case of digital binary modulation, the digital signal tobe transmitted d (t) should be applied to one of the arms of the modulator, while the invertedsignal, also known as complementary signal, should be applied to the second arm. In the caseof modulation by a sinusoidal signal, the two driving voltages should be phase shifted by 180◦

before being applied to the two arms of the modulator. Such driving conditions are usuallydescribed as push-pull operation of a dual-drive MZM.

12


hot electrodewaveguide

LiNbO3 crystal

Ea field

x-cut

ground electrode

waveguide

ground electrode

buffer layer

Figure 7 Cross section of an x-cut LiNbO3 Mach-Zehnder modulator structure. The light is prop-agating in the y direction. The electrical data is applied to the hot electrode, resulting in oppositeorientations of the applied electric field vectors over the two waveguides, hence electro-optically inducedphase shifts of opposite signs.

If binary intensity modulation is performed, the extinction ratio and modulator outputpower are simultaneously optimised provided the peak-to-peak voltage of each of the modulatingsignals is equal to Vpp,1 = Vpp,2 = Vπ

2 . Since it is usually difficult – and expensive – to obtainlarge peak-to-peak voltages at high bit rates, push-pull operation of the Mach-Zehnder relaxesthe requirement in term of signal amplification compared to the single arm drive case discussedearlier. However, in practice, two driver amplifiers are needed instead of one, even though theirrequired output voltage is only half the value in the single drive case. Furthermore, chirp freeoperation requires (34) to be satisfied, which imposes strict conditions on the voltage imbalanceand delay that can be tolerated between the signals driving each of the arms of the modulator.Consequently, push-pull operation results in an increased complexity at the transmitter.

This complexity can however be alleviated by the use of properly designed MZMs, whicheven though they require only one single driving signal, effectively operate in push-pull operationby inducing refractive index changes of opposite signs in the two arms of the Mach-Zehnderstructure. The cross section of such an x-cut MZM is represented in Fig. 7. The oppositeorientations of the external electric field applied across the two waveguides ensure oppositephase shifts for this particular LiNbO3 crystal orientation. Such a structure does not requirecareful alignment and balancing of two inverted driving signals, at the cost of a higher switchingvoltage.

All the applications discussed later on in this note will by default make use of chirp-freeMZMs and the phase varying term will therefore be ignored.

2.4 Phase modulation using a Mach-Zehnder modulator

It has been reminded in the introduction to this section that performing phase modulationin an electro-optic crystal is straightforward since the phase shift induced by the crystal isproportional to the refractive index change, hence is linearly dependent on the applied voltage.The input-output relation of such a phase modulator can therefore be simply expressed as

Eout (t) = Ein e−jπVπV (t), (36)

and all amounts of phase modulation can be achieved, depending on the available voltage V (t).

This inherent simplicity is however not without practical difficulties. Achieving a specifiedamount of phase modulation ∆φ is not straightforward, especially for digital signals, sincethe phase modulation is not easily “visible”, unlike intensity modulation that can be easilymonitored on an oscilloscope following detection in a photodiode.

13


Figure 8 Phase modulation using an electro-optic phase modulator. Due to the linear dependenceof the electro-optically induced phase shift on the applied voltage, fluctuations of the driving signal aretransferred to the optical phase.

Furthermore, since the phase-versus-voltage transfer of the phase modulator is linear, anyfluctuation present on the driving electrical signal will be transferred to the phase. This pointis illustrated in Fig. 8 in the case of BPSK phase modulation. In the ideal case where thedriving signal has a well defined peak-to-peak voltage of Vπ and the nominal voltage levels arereached for each bit, the desired phase modulation is obtained, as illustrated by the blue tracesand blue dots in the constellation diagram. If now some voltage fluctuations exist from bit tobit, as shown in the red waveforms, some phase deviation δφ is present in the optical signal.This manifests itself on the constellation diagram as an azimuthal spreading of the modulationstates at fixed radius, which is equivalent to phase noise. One can also note that, when a phasemodulator is employed, the linear phase-versus-voltage transfer results in the optical signalbeing chirped, as can be seen in the trajectory on the constellation diagram that follows theunit circle.

Interestingly, unchirped BPSK modulation with exact π phase shifts can be achieved usingan MZM. This may look as an unnecessary complication in the first place since an MZMis essentially made from two phase modulators in a Mach-Zehnder interferometer structure.However the benefit of exact π phase shifts and of unchirped modulation make the use of MZMthe method of choice for BPSK modulation. The principle is illustrated in Fig. 9 where notonly the usual power transfer function, but also the corresponding field transfer Eout/Ein arerepresented. It can be seen that, whenever the power transfer of the modulator reaches aminimum, the field transfer changes sign, which is equivalent to a π phase shift (e−jπ = −1).Therefore, driving the MZM between two consecutive maxima results in the introduction of anexact π phase shift each time the operating point crosses the transmission minimum. This isthe principle of BPSK modulation using an MZM.

If the MZM is operated in push-pull mode, the modulation is chirp-free. A drawback ofthe method is that some intensity modulation is introduced simultaneously with the desiredphase modulation. An intensity dip is created in the modulated optical signal each time thephase is flipped between 0 and π. Those effects can be clearly observed in the constellationdiagram where the transitions between the 0 and π phase states follow a straight line that crossesthe origin. The benefits of chirp-free exact π modulation typically outweighs the drawbacks.Furthermore, in case RZ modulation is used, which is often the case for this type of modulation

14


1 32

power / field

Figure 9 Principle of binary phase modulation using a Mach-Zehnder modulator.

format, some pulses will be modulated on top of the phase modulated data, effectively resultingin a suppression of those intensity transitions accompanying the phase shifts. From a practicalperspective, using an MZM to realise binary phase modulation requires a peak-to-peak voltagedifference of 2Vπ, which is twice as large as when a phase modulator is used. Furthermore, thismethod only allows ro reach π phase shifts while the use of a phase modulator is more versatilesince it allows any phase shift to be generated by a proper choice of the driving voltage, howeverwith the inherent difficulty of adjusting the driving voltage, as discussed previously.

2.5 Pulse carving

Until now, the focus has been on the generation of NRZ modulation formats, where the intensityof the light remains at its high level when consecutive “1” bits are transmitted. This legacycoding has been employed to a large extent due to bandwidth limitations in electronic circuitsand electro-optic components. Indeed, conventional electronic circuits are designed to operatewith NRZ signals since RZ-compatible components would require a significant larger bandwidth.

However, in the optical domain, the bandwidth usage has become a major preoccupationonly very recently. On the other hand, the peculiarities of the optical fibre channel, and inparticular optical fibre nonlinearities, have triggered a decade of intense research on opticalmodulation formats presenting a good resilience to transmission impairments. For some systems,having an optical RZ waveform presents some significant advantages. The purpose of the presentnote is not to discuss transmission impairments and their mitigation. However, one can clearlyunderstand the benefit of optical RZ modulation by considering the case of transmission at40 Gbit/s and above. It is well known that an optimum optical transmission system is theresult of a compromise between the competing targets of reaching a high optical signal-to-noiseratio (OSNR), calling for the use of a high transmitted power, and limiting the effect of opticalfibre nonlinearities, which on the other hand requires a “not-too-high” transmitted power. Afavourable trade-off can be reached at high bit rates by employing RZ line coding. Due to group

15


NRZ data

CW

power

MZM1 MZM2

33% 50% 67%

1 bit period

33% RZ 50% RZ 67% RZ

NRZ1 bit

period

Figure 10 RZ pulse carving using an MZM.

velocity dispersion, RZ pulses will disperse very fast at high bit rates. This means that welldefined pulses will cease to exist after only a short length of transmission fibre. The peak powerof the pulses having been considerably decreased, they are no longer vulnerable to self-phasemodulation right after the optical repeaters, as is the case in a traditional optically amplifiedlink. This means that, in turn, the power of the signal can be increased, which results in animproved OSNR at the end of the link.

In general, RZ pulses will experience a better resilience to optical fibre nonlinearities. Thisis also true in the case of BPSK or QPSK modulation where an RZ waveform can be used tocarry the phase information. In this case the intensity of the light is modulated periodically,with a pulse being present in every single symbol slot. The pulse itself does not carry anyinformation, which is encoded onto the phase. However, the fact that all symbols present thesame temporal evolution will also provide benefits in term of nonlinearities mitigation. Similararguments can be applied for more complex modulation formats. An additional benefit is thatthe generation of such advanced modulation formats by interferometric modulation methodsoften present some transients, resulting in unwanted intensity fluctuations, between the symbols.A simple illustration of this point is the use of an MZM to generate BPSK modulation, whereit has been seen that each phase transition between 0 and π is associated with an intensity dip.By employing an RZ intensity on top of the modulation, these transients occurring in-betweensymbols will be greatly suppressed.

Consequently there is a need for generating optical RZ waveforms, even though availableelectrical signals make use of NRZ line coding. This can be achieved by the technique of pulsecarving illustrated in Fig. 10. By driving an MZM with an electrical clock signal, it becomespossible to “carve” some pulses out of a continuous wave (CW) lightwave. If the frequencyfm of the sinusoidal electrical clock signal is equal to the bit rate B, and if the pulse carvingmodulator is driven with a peak-to-peak voltage difference equal to Vπ between a transmissionmaximum and the next minimum, RZ pulses with a full-width at half-maximum (FWHM)duration equal to 50% of the bit slot TB = 1/B are generated. Then a second data modulatoris used to modulate the signal in the modulation format of interest (e.g. OOK, as illustrated

16


here, BPSK, or more sophisticated formats). It is of course important to properly synchronisethe signals driving the pulse carver and the data modulator. Note that the order of the pulsecarver and data modulators can be swapped in such a scheme. Alternatively, RZ modulationwith a duty cycle of 33% or of 67% can be obtained by driving the pulse carver with a 2Vπsignal at fm = B/2 and biasing the modulator at a maximum or a minimum transmission,respectively.

What to remember:

• The linear electro-optic effect can be exploited to modify the refractive index of certaincrystals depending on an external electric field, hence voltage, applied to them. Thechange in refractive index is proportional to the applied voltage and quasi-instantaneous(∼fs) compared to the modulation speeds attainable now.

• This effect can be exploited in a straightforward way to perform phase modulation oflight.

• In order to perform intensity modulation, one can take advantage of an interferometricstructure where the phase modulation is converted to intensity modulation. This istypically done using integrated modulators in a Mach-Zehnder interferometer structure.

• Frequency chirping can be avoided by driving a Mach-Zehnder modulator in push-pullmode.

• In this case, the complex envelope of the modulated signal can be expressed as

Eout (t) = Ein cos[π

2Vπ{∆Vbias + 2Vppd (t)}

](37)

• A Mach-Zehnder modulator can also be used to perform binary phase modulation withexact phase shifts of π between the two modulation states.

• RZ signals can be generated by combining a data modulator with an MZM driven bya sinusoidal clock signal used as a pulse carver.

3 Differential phase-shift keying

3.1 Introduction to phase modulation

Traditionally, binary modulation of the light intensity, such as OOK, is used in optical commu-nication systems due to the inherent simplicity of the signal detection process, where a photodi-ode directly converts the optical intensity variations into an electrical signal, from which binarydecision can be made from a single threshold receiver. However, it is well known from digitalcommunication theory that other formats of modulation such as phase or frequency modulation,might result in a better receiver sensitivity when used in conjunction with a proper detectionscheme, thus potentially allowing an increase in the system power budget, hence in its reach.

This section deals with the transmission of phase modulated signals, which in a digitalcontext are known as phase-shift keying (PSK) signals. It will more specifically focus on binaryPSK, for which the phase can take two values, ideally 0 and π so that the distance between themodulation states on the constellation diagram is maximised for a constant value of the optical

17


PSK signal

local oscillator

signal phase

signal

b

a

Figure 11 Illustration of direct detection (a) versus coherent detection (b) of a phase modulated signal.

power. Such a signal can be represented by

Es (t) = Re{√

Pse−j[φs(t)+ϕs(t)]ejωstes

}, (38)

where Ps is the signal constant power, the deterministic phase term φs (t) takes values of 0 orπ depending on the normalised binary modulating signal a (t)

φs (t) = πa (t) , (39)

and ϕs (t) is a phase noise term due to random phase fluctuations associated with the lightemission process .

The photocurrent generated by a photodiode is proportional to the received optical power,

i (t) = RP (t) , (40)

where R is the responsivity of the photodiode. This value obviously turns out to be constantfor PSK modulated signals, as illustrated in Fig. 11. Consequently, direct detection in a photo-diode discards the phase information and other methods need to be employed to recover phasemodulated data.

3.2 Coherent detection of PSK signals

One solution consists in beating the phase modulated signal with the light generated by a localoscillator (LO) laser in a photodiode, as represented in Fig. 11. Expressing the electric field ofthe light generated by the local oscillator laser as

Elo (t) = Re{√

Ploe−jϕlo(t)ejωlotelo

}, (41)

and assuming that both the phase modulated signal and the local oscillator light have identicalstates of polarisation, the total optical power after the two beams have been combined can beexpressed as

Ptot (t) = Ps + Plo + 2√PsPlo cos [(ωs − ωlo) t+ φs (t) + ϕs (t)− ϕlo (t)] . (42)

18


In (41), Plo, ωlo and elo are the power, angular frequency and polarisation of the local oscillator,respectively, and ϕlo (t) is a phase noise term similar to the one introduced in (38). The a.c.term of the photocurrent is therefore

iac (t) = 2R√PsPlo cos [(ωs − ωlo) t+ φs (t) + ϕs (t)− ϕlo (t)] , (43)

which depends on the value of the information-carrying signal phase φs (t).At this point one should distinguish between homodyne systems, where the frequency of the

local oscillator is equal to that of the original signal, ωlo = ωs, and heterodyne systems, wherethe frequency difference ωif = |ωs−ωlo|, known as intermediate frequency, lies in the microwaveregion. In the latter case, further processing is required in the microwave domain in order toretrieve the original baseband data. A detailled description of those so-called coherent receiversis beyond the scope of this note. Nevertheless, it is clear from the previous analysis that,coherent optical receivers of this form are particularly challenging to implement in practice. Thisis due to the facts that polarisation matching between the LO and the received signal shouldbe ensured, the frequency of the LO should match exactly that of the signal for homodynedetection, and some action should be taken to ensure the relative phases of the signal and theLO, ϕs (t)−ϕlo (t), are somehow locked to each other, either optically for homodyne detection,or electronically under the form of a carrier recovery circuit for heterodyne detection.

These challenges have been the object of intense research in the late 1980’s and early 1990’s.One of the main motivations behind such efforts was the fact that phase modulation schemesassociated with coherent demodulation, whether homodyne or heterodyne, result in a betterreceiver sensitivity than OOK used with direct detection. The picture was radically changedwith the development of the erbium-doped fibre amplifier (EDFA) initiated in the late 1980’s.An efficient and practical all-optical amplification scheme was now available, which could beused as a booster or in-line amplifier, thus relieving the power budget considerations, or evenas a preamplifier with a theoretical quantum limit as low as 38.4 photons per bit for intensitymodulation - direct detection (IM-DD). Once a limitation in the design of optical communica-tion systems, the receiver sensitivity requirements have therefore been considerably alleviatedby the introduction of the EDFA. As a consequence, the interest of the optical communica-tion community shifted away from coherent systems until around 2004 when it was realisedthat the progress made in high-speed electronic digital signal processing (DSP) circuits couldenable compensation of propagation impairments in the digital domain, provided coherent de-tection is employed. Indeed, coherent receivers can allow the detection of both the in-phase andquadrature components of the optical signal, as will be discussed in Sec. 6. It thus becomespossible to map the entire optical field to the electrical domain without any loss of information,unlike when a direct detection receiver, which discards the phase information, is used. Fol-lowing analogue-to-digital conversion of the signal, DSP techniques can be used to recover thetransmitted information after electronic compensation of group velocity or polarisation modedispersion and digital carrier recovery. The need for frequency locking of the local oscillator isalso relieved in this type of receiver using an intradyne scheme.6

3.3 Self-homodyne DPSK detection

In order to circumvent some of the practical limitations associated with coherent systems, inparticular the need for a local oscillator laser, the principle of self-homodyne or interferometric

detection of PSK signals was introduced in the late 1980’s. In this scheme, the received signalis made to beat with a delayed version of itself in a photodiode. This is equivalent to perform-ing phase-to-intensity modulation (PM-to-IM) conversion in an interferometer prior to direct

6Coherent digital receivers are presented at a later point in this course.

19


XOR

CW PMDI

PD

Figure 12 Principle of optical DPSK transmission.

XOR

Figure 13 Truth table of differential encoding and differential encoder implementation.

detection in a photodiode. It thus becomes possible to employ phase modulation in a way thatis compatible with direct detection, provided a suitable optical discriminator, typically a twobeam interferometer such as a Mach-Zehnder structure, is inserted at the receiver before thephotodiode. The electrical circuitry used to drive the phase modulator also needs to be adaptedin order to generate a slightly modified version of digital binary phase modulation known asdifferential phase-shift keying (DPSK).

The main idea behind DPSK modulation is to encode the data to be transmitted not onabsolute values of the optical phase, but onto phase differences between consecutive bits takingvalues 0 or π. If the value of the bit to be transmitted is “0”, then no phase change willbe imposed onto the signal. If, on the other hand, the bit to be transmitted takes the value“1”, then the phase of the signal will be shifted by π. It is this feature that enables phasedemodulation by comparison of the phase of one bit to that of the previous bit in a delayinterferometer at the receiver. An important consequence is that the absolute phase of thesignal may be allowed to drift over time, as would occur due to the phase noise of the lasersource. This may not be of consequence as long as the phase drift between consecutive samplingtimes of the received signal, spaced by 1/B, is significantly smaller than the expected π jumpsbetween consecutive bits.

A schematic representation of an optical DPSK transmission system is shown in Fig. 12.The data to be transmitted ak is first differentially encoded into a new sequence dk, which isthen used to drive the phase modulator. The differential encoding rule is such that the state ofdk changes (from “1” to “0” or from “0” to “1”) each time ak = 1, while it remains the samewhenever ak = 0. It can easily be checked that dk can be obtained by performing exclusive OR

(XOR, also known as modulo-2 addition) operation between ak and a delayed version of dk thathas been fed-back to the input of the XOR gate with a delay corresponding to one bit, accordingto

dk = ak ⊕ dk−1, (44)

where ⊕ denotes the XOR operation. The differential encoder structure as well as the truth tableof the differential encoding process are represented in Fig. 13.

Once the differentially encoded sequence has been generated, it is used to drive the phase

20


ak 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1

dk 0 1 0 1 1 0 0 1 0 0 0 1 1 1 1 0

φk 0 π 0 π π 0 0 π 0 0 0 π π π π 0

∆φk 0 π −π π 0 −π 0 π −π 0 0 π 0 0 0 −πrk 1 0 0 0 1 0 1 0 0 1 1 0 1 1 1 0

rk 0 1 1 1 0 1 0 1 1 0 0 1 0 0 0 1

Table 1 Illustration of the principle of optical DPSK signal generation and demodulation. ak is theoriginal bit sequence to be transmitted and dk is the differentially encoded sequence. φk is the phaseof the transmitted signal while ∆φk is the phase difference between two consecutive bits. Finally, rk isthe value of the recovered bit after direct detection of the output of the delay interferometer, and rk itscomplementary.

modulator, or equivalently the MZM, resulting in phase shifts of 0 or π. After transmission,the signal is first fed to a delay interferometer (DI), typically in a Mach-Zehnder configuration,where the relative delay between the two arms is made equal to one bit period T = 1/B. Thepower at the output of the Mach-Zehnder delay interferometer (MZDI) can therefore simply beexpressed as

PMZDI,out = PMZDI,in cos2

(φ (t)− φ (t− T )

2

). (45)

This power is then detected in a conventional photodiode. It is clear from (45) that the detectedpower takes high value whenever the phase of two consecutive bits is equal, and is null otherwise.Therefore a phase change can be detected, hence the value of the original sequence bit ak,which has triggered this phase change according to the differential encoding rule. In orderto illustrate the operation of a DPSK system with interferometric detection, the case of anexample bit sequence is considered in Table. 1, where it can be seen that the original pattern isrecovered after differential encoding, phase modulation and interferometric detection. The DI,which enables the demodulation of DPSK signals by converting the phase difference informationbetween consecutive bits into an intensity modulated signal is therefore a key component forthis type of system. Its operation is detailled further in the following section.

3.4 Interferometric demodulation of DPSK

The structure of a DI based on a Mach-Zehnder interferometer is represented in Fig. 14. Theincoming signal is first split along two paths in an input 3-dB coupler. A propagation delayof τ is introduced between the two paths before the two light contributions propagating ineach of the paths are recombined in an output 3-dB coupler. The eventual extra phase shiftdifference that may be experienced by the light contributions propagating in the two arms ofthe interferometer is represented by Φ0. In the following analysis the two 3-dB couplers areassumed to be ideal so that their transfer characteristics can be described by

[Eout,1

Eout,2

]=

[ √α −j

√1− α

−j√1− α

√α

] [Ein,1

Ein,2

], (46)

where α is the power splitting ratio of the coupler and the notations of the input and outputfields are introduced in Fig. 15. For a 3-dB coupler, α = /2.

The electric field at the two outputs of the DI can then be expressed as

Eout,1 (t) =1

2

[Ein (t− τ)− Ein (t) e

−jΦ0], (47)

Eout,2 (t) = − j2

[Ein (t− τ) +Ein (t) e

−jΦ0], (48)

21


Figure 14 Delay interferometer based on a Mach-Zehnder structure. τ represents the extra propaga-tion delay in the upper arm, while Φ0 is the extra phase shift difference that may be experienced betweenthe fields propagating in the two arms.

Figure 15 Schematic representation of a directional coupler with power splitting ratio α.

hence

|Eout,1 (t)|2 =1

4

[|Ein (t− τ)|2 + |Ein (t)|2 − 2Re

{Ein (t− τ)E∗

in (t) ejΦ0

}], (49)

|Eout,2 (t)|2 =1

4

[|Ein (t− τ)|2 + |Ein (t)|2 + 2Im

{Ein (t− τ)E∗

in (t) ejΦ0

}]. (50)

where it has been assumed all along that the two contributions Ein (t− τ) and Ein (t) have thesame state of polarisation. Assuming the usual notations for the input field

Ein (t) = Re{√

Pin (t) e−jφ(t) ejω0t

}, (51)

the power at the upper output port of the DI can be expressed as

Pout,1 (t) =1

4

[Pin (t− τ) + Pin (t)− 2

√Pin (t− τ)Pin (t) cos (φ (t− τ)− φ (t) + Φ0 − ω0τ)

].

(52)Assuming a phase modulated signal for which the power envelope is constant, Pin (t− τ) =Pin (t) = P0, the power at the upper output port of the delay interferometer becomes

Pout,1 (t) =1

2P0 [1− cos (φ (t− τ)− φ (t) + Φ0 − ω0τ)] . (53)

The power at the lower output port of the delay interferometer can be derived in a similarfashion, leading to

Pout,2 (t) =1

2P0 [1 + cos (φ (t− τ)− φ (t) + Φ0 − ω0τ)] . (54)

22


-2 -1 0 1 2

-50

-40

-30

-20

-10

0

tra

nsm

issio

n (

dB

)

to destructive port

to constructive port

Figure 16 Transfer functions to the two output ports of a delay interferometer with delay τ = T = 1/B.The delay interferometer is tuned so that Φ0 − ω0T = 2kπ, resulting in destructive and constructiveinterference at ω0 to output ports ¬ and , respectively. The frequency axis is normalised to the bit rateB and the transfer functions are represented on a logarithmic scale (i.e. the quantities 10 log |TDI,i (ω)|2,where i = 1, 2, are represented.)

If furthermore the delay is equal to the bit period τ = T and the phase shift Φ0 is tuned sothat Φ0 − ω0T = 2kπ, the power at the two outputs of the DI can be simply expressed as

Pout,1 (t) = P0 sin2

(φ (t− T )− φ (t)

2

), (55)

Pout,2 (t) = P0 cos2

(φ (t− T )− φ (t)

2

). (56)

If φ (t− T )− φ (t) = 0, i.e. the phase of the bit at t is the same as that of the previous bit,

Pout,1 (t) = 0, (57)

Pout,2 (t) = P0. (58)

Constructive interference occurs at output port while destructive interference occurs at outputport ¬. This is the reason why port ¬ and port are customarily described as destructive andconstructive ports of the delay interferometer, respectively. If φ (t− T ) − φ (t) = ±π, i.e. thephase of the bit at t differs from that of the previous bit by π,

Pout,1 (t) = P0, (59)

Pout,2 (t) = 0, (60)

and constructive interference occurs at output port ¬ while destructive interference occurs atoutput port . If Φ0 − ω0T = (2k + 1) π, it can easily be checked that the two outputs areswapped. The conclusions are still valid if RZ-DPSK modulation formats are used, in whichcase pulse carving is implemented at the transmitter and the power envelope of the signal isperiodic with period T , hence Pin (t− T ) = Pin (t).

The operation of the delay interferometer can also be analysed in the frequency domain.Fourier transforming (47) immediately leads to

Eout,1 (ω) =1

2Ein (ω)

[e−jωτ − e−jΦ0

], (61)

23


hence the transfer function of the DI to output port ¬

TDI,1 (ω) =Eout,1 (ω)

Ein (ω)= sin

(ωτ − Φ0

2

)e−j

(

ωτ+Φ0+π

2

)

. (62)

Similarly, starting from (48), the transfer function to output port of the delay interferometercan be easily obtained as

TDI,2 (ω) =Eout,2 (ω)

Ein (ω)= cos

(ωτ − Φ0

2

)e−j

(

ωτ+Φ0+π

2

)

. (63)

The power transfer functions to the constructive and destructive ports of the DI, illustrated inFig. 16, are therefore given by

|TDI,1 (ω)|2 = sin2(ωτ − Φ0

2

), (64)

|TDI,2 (ω)|2 = cos2(ωτ − Φ0

2

). (65)

From these transfer functions it can be seen that:

1. The frequency response of the delay interferometer is periodic. This period is known asfree spectral range (FSR) and is equal to ∆fFSR = 1/τ . It has been seen earlier that forDPSK demodulation the delay of the DI should ideally be set to the bit slot durationτ = T . In this case, the value of the FSR is equal to that of the bit rate ∆fFSR = B.

2. The frequencies at which the maxima and nulls of the transfer function occur are deter-mined by the phase shift Φ0. For instance the transmission nulls to output port ¬ occurfor ωkτ = Φ0 + k2π, where k is an integer, while they occur at ωlτ = Φ0 + π + l2π, withl integer, to output port . Obviously, a transmission null to output port ¬ correspondsto a transmission maximum to output port and vice-versa. More generally, the sum ofthe two power transfer functions |TDI,1 (ω)|2 + |TDI,2 (ω)|2 is equal to 1, which is a directconsequence of power conservation.

3. The condition Φ0−ω0T = 2kπ introduced in the time domain analysis of the DI thereforeimplies that the carrier frequency ω0 corresponds to a transmission null to output port¬ and a transmission maximum to output port , justifying further the denomination ofdestructive and constructive ports, respectively.

4. In practice, the phase shift Φ0 can be expressed as Φ0 =2πλ nL where L corresponds to the

physical length of a portion of one path in the delay interferometer and n is the refractiveindex of the material used to make the DI. Consequently, the delay interferometer, ormore precisely the frequencies at which maxima or nulls occur to a specific output port,can be tuned by adjusting the value of the refractive index n. This is usually performedby applying a heating element to one arm of the DI, therefore changing the refractiveindex and realising thermal tuning.

Finally, another useful relation can be derived for the complex envelope of the field atthe output of the DI. Assuming that the DI is tuned so that Φ0 − ω0T = 2kπ, substitutingE (t) = E (t) ejω0t into (47) and (48) immediately leads to

Eout,1 (t) =1

2[Ein (t− T )− Ein (t)] e−jΦ0 , (66)

Eout,2 (t) = − j2[Ein (t− T ) + Ein (t)] e−jΦ0 , (67)

24


power

frequency

power

frequency

power

frequency

DPSK

DB

AMI

Figure 17 Spectra and eye diagrams of DPSK demodulation in a 1-bit DI and balanced detection.A DPSK signal generated in an MZM is sent to an MZDI with delay equal to 1/B, resulting in thegeneration of an AMI signal at the destructive port and a DB signal at the constructive port. Thesignals at the two outputs of the DI are detected in two balanced photodiodes, resulting in the balancedeye diagram shown to the right. The dashed line shows the 0 current level. In this illustration, nobandwidth limitation is present in the photodiodes.

where the constant phase term Φ0 can be ignored in most cases of practical interest. It cantherefore be seen that the fields at the two outputs of the DI are the sum and the difference ofthe input field and a delayed version of itself.

The operation of an MZDI on a DPSK signal is illustrated in Fig. 17, where eye diagramsand spectra at various stages of the demodulation and detection process are represented. ADPSK signal generated in an MZM is input to the MZDI whose delay is exactly equal to thereciprocal of the bit rate 1/B. Conversion to OOK is performed in the MZDI. The OOK signalsgenerated at the constructive and destructive ports exhibit different temporal and spectralfeatures. A format known as duobinary (DB) is generated at the constructive port, while anotherformat known as alternate mark inversion (AMI) is generated at the destructive port7. TheAMI spectrum exhibits a clear notch at the centre frequency, corresponding to the destructiveinterference regime of the DI at this frequency. Demodulation of a DPSK signal can indeedbe seen as a linear process resulting in filtering of the signal input spectrum by the transferfunctions represented in Fig. 16 for the constructive and destructive ports, respectively.

Since the constructive and destructive port signals are OOK with opposite polarities, bal-anced detection, where the photocurrents generated by a pair of identical photodiodes are sub-tracted, is generally used for this type of receiver. This results in a balanced eye diagram (i.e.with symmetric amplitudes around the baseline), as shown in Fig. 17. The balanced detectionprocess is known to result in an improved receiver sensitivity of ∼ 3 dB compared to OOK forDPSK systems with interferometric demodulation.

7The reasons for these denominations are beyond the scope of this note. Both signals are OOK, obviouslywith opposite polarity due the destructive-constructive nature of the interference at the two ports, but exhibitdifferent bit-to-bit phase relations.

25


What to remember:

• Other quantities than the power of a lightwave can be modulated to transmit informa-tion. The phase of an optical carrier can, for instance, be modulated between 0 and π,a modulation scheme known as binary phase-shift keying.

• Recovering the phase information can be done in a coherent receiver. However, unlessmore modern concepts briefly outlined in Sec. 6 are used, traditional optical coherentreceiver designs are difficult to implement in practice.

• Instead, one can perform differential phase-shift keying modulation, where the binarydata is encoded into π phase differences between the bits. Such a modulation can beeasily demodulated by a delay interferometer that effectively converts the phase differ-ence between consecutive bits in intensity modulation, which can in turn be detectedby a photodiode.

• Such systems have been investigated first until the early 1990’s, where their attractive-ness stemmed from the fact that they present an improved receiver sensitivity comparedto on-off keying when balanced detection is employed. Their investigation has seen arenewed interest around 2000 due to their resilience to nonlinear effects in optical fibresthanks to their constant (or periodic in case return-to-zero pulse shaping is used) powerenvelope. The focus has now shifted to more sophisticated multilevel modulation for-mats used in conjunction with novel coherent receiver concepts, as discussed in Sec. 4and 6.

4 The optical quadrature modulator

4.1 Motivations for optical multilevel modulation

In the previous section, it has been shown how a simple waveguide made in an electro-opticmaterial can be used to achieve phase modulation. By employing a Mach-Zehnder interferometerstructure, both intensity modulation (e.g. OOK) and binary (taking the values of 0 or π, i.e.BPSK) phase modulation can be achieved. It is well known from the general field of digitalcommunications that those are extremely simple modulation formats. In our everyday life,mobile communications, digital radio and television, broadband access etc are making use ofsignificantly more sophisticated modulation formats in order to adapt to the limited bandwidthsof wireless and wireline channels, to deal with its peculiarities (e.g. multi-path interferencein wireless channels) and to share the limited bandwidth among users. Those applications,however, typically operate at a rather modest baud rate compared to optical communicationsystems where binary OOK modulation had been pushed up to 40 Gbit/s per wavelengthchannel before the question of multilevel optical modulation formats started motivating theresearch community.

The motivations behind the development of multilevel optical modulation formats are asfollows. At equal symbol rate, multilevel modulation can transport a higher bit rate, since itencode several bits per symbol according to (5). Since it is the symbol rate that determinesthe bandwidth of the signal, the spectral efficiency (SE), expressed in units of bit/s/Hz, can beincreased. The SE is defined as

SE =B

∆ν, (68)

where B is the bit rate carried by each individual wavelength channel (assumed to be equal)

26


and ∆ν is the frequency spacing between the centre frequencies of each wavelength channel8.Consequently, in the same bandwidth, more data can be transmitted by employing multilevelmodulation. To this respect, the use of binary OOK modulation, as done to a large extent untilnow, is a fairly inefficient use of bandwidth.

However, the SE cannot be increased arbitrarily. A fundamental limit to the channel ca-pacity C, which is defined as the maximum information rate that can be transmitted over thechannel, is given by Shannon’s theorem

C =W log2 (1 + SNR) , (69)

where W is the channel bandwidth and SNR is the signal-to-noise ratio (assuming additivewhite Gaussian noise). The spectral efficiency can be identified with C/W . Consequently,increasing the SE can only be achieved at the expense of a higher signal-to-noise ratio (SNR)requirement. This also holds true for multilevel modulation, which, at the same symbol rate,require a higher SNR due to a lower noise tolerance, which can be understood intuitively fromthe smaller geometrical distance between the modulation states on the constellation diagramfor a fixed average power.

From a practical perspective, multilevel optical modulation formats allow to generate highbit rate optical signals from lower bit rate electronics since the electronics only need to operateat the symbol rate. This offers the opportunity to generate optical signals at a bit rate largerthan the current limits of electronics, which, even though constantly progressing overt time,may appear as a bottleneck with respect to what can be achieved optically. Furthermore,the fact that high-bandwidth electronic components are expensive make such a scheme alsoeconomically attractive.

The first deployed optical multilevel modulation is QPSK at 40 Gbit/s, carrying two bitsper transmitted symbol, and therefore requiring only 20 Gbit/s electronics. Using polarisa-tion multiplexing, it even becomes possible to generate an 100 Gbit/s signal from 25 Gbit/selectronics. Such systems, relying on a digital coherent receiver, are currently being deployed.

Therefore it is important to be able to generate complex multilevel optical modulationformats. In the remaining of this section, it will be shown how, by integrating MZMs into aninterferometric structure, it becomes possible to generate arbitrary optical modulation formats.

4.2 Principle of the optical IQ modulator

The modulator structure represented in Fig. 18 is now considered. This type of optical modula-tor is referred to in the literature under different names, including optical quadrature modulator,nested Mach-Zehnder modulator, dual parallel Mach-Zehnder modulator, or optical IQ modula-tor. It consists of two MZMs that are themselves embedded into a Mach-Zehnder interferometerstructure. Each modulator is driven by an independent signal, uI (t) and uQ (t), respectively.An additional static phase shift of π/2 between the two arms of the interferometric superstruc-ture is furthermore introduced. If each MZM is biased in a condition corresponding to thatillustrated in Fig. 9, i.e. ∆Vbias = Vπ, and assuming chirp-free modulation, then the outputfield Eout can be easily calculated using (37), immediately leading to

Eout =1

2Eout

[cos

(π

VπuI (t)

)+ cos

(π

VπuI (t)

)ej

π2

]. (70)

8One can in principle also define how “spectrally efficient” a given modulation format is by considering thespectral width of the modulated signal instead of ∆ν. This however requires a common definition of the spectralwidth of a signal (e.g. 3 dB bandwidth -or FWHM- or 10 dB bandwidth?) in order to enable fair comparisons.The fact that optical communication systems typically make use of wavelength division multiplexing (WDM) andthat they operate on a fixed frequency spacing grid according to the International Telecommunications Union(ITU) standards make the definition of (68), which does not have this ambiguity, more meaningful in practice.

27


Figure 18 Principle of operation of the optical IQ modulator. Two electrical signals uI and uQ aredriving each MZM in the interferometric structure. The reachable points in the complex (I,Q) planeare indicated by the blue square to the right.

A direct comparison with (7) shows that arbitrary in-phase and quadrature components of theoptical signal can be synthesised in this way.

Fig. 18 also shows the points in the complex plane (I,Q) that can be reached using this typeof modulator. It thus becomes possible to generate any optical field, and thus any modulationformat, provided the driving signals uI (t) and uQ (t) can be synthesised. For this purpose thenon-linear relation between the electric driving signals and the coordinate along the I or Qdimension should obviously be taken into account. In principle, arbitrary electric signals can besynthesised using an arbitrary waveform generator (AWG). However, the speed at which suchdevices can operate is still relatively limited. In order to achieve the highest possible symbolrates, one is often in practice limited to binary electronic signals, imposing some constraintson the way such a modulator can be operated. The generation of QPSK and 16-quadratureamplitude modulation (QAM) using such an IQ modulator structure are taken as examples inwhat follows.

4.3 QPSK signal generation

The application of such a modulator to the generation of QPSK signals is straightforward andis illustrated in Fig. 19. Each MZM in the structure is driven with a binary electrical signal asa BPSK modulator as discussed in Sec. 2.4. The relative π/2 phase shift in one of the armshas the effect of rotating the BPSK constellation generated in this arm. After recombinationof the two signals in the output Y-junction, a QPSK constellation is obtained. Since one ofthe benefits of binary phase modulation using an MZM is to result in exact π phase shifts,this property is maintained for both generated BPSK signals. However, one practical difficultyis to adjust the relative phase shift to π/2 to guarantee the orthogonality between the twomodulations. Recalling that using an MZM to perform BPSK modulation results in unwantedintensity dips each time the phase is flipped between 0 and π, it can easily be shown that theoutput intensity of a QPSK signal generated in this way also presents two levels of intensity dips,one corresponding to a transition between 0 and π in only one of the BPSK modulator, as wellas one occurring when such transitions are simultaneous in the two BPSK modulators. Theseintensity dips can easily be visualised in the constellation diagram with transitions associatedwith this type of modulation, as shown in Fig. 1, where it can be seen that transitions betweenmodulation states can either be achieved along a path parallel to the I or Q axes, or via theorigin.

28


CW

data stream 1

data stream 2

power

t

phase

t

Figure 19 Principle of optical QPSK signal generation. Two binary electrical signals are used to drivethe two MZMs of the structure as binary phase modulators, resulting in phase modulation states of 0and π. The phase states are rotated by a static value of π/2 in one of the arms of the interferometersuperstructure before the two signals are recombined, leading to an optical QPSK signal.

QPSK signals generated in this way can be detected and demodulated using either a coherentreceiver, or an interferometric detection technique similar to the one used for DPSK, followingappropriate differential encoding. The interferometric detection of differential quadrature phase-shift keying (DQPSK) signals will be described in more details in Sec. 5.

4.4 Square QAM signal generation

QAM formats are widely used in a number of digital transmission applications, including e.g.digital video broadcasting. They are an extension of QPSK, which can alternatively be consid-ered as a 4-QAM format, utilising an increasing number of modulation states in the in-phaseand quadrature dimensions.

As an example of a QAM modulation format that can still be generated at high baud ratesusing binary electronics, the case of 16-QAM is considered. The constellation diagram of 16-QAM consists of 16 states, hence four bits per symbol, on a square grid. One possible way togenerate optical 16-QAM modulation is to drive an IQ modulator with two four-level electricalsignals, also known as 4-pulse amplitude modulation (PAM) signals, as shown in Fig. 20. Thesesignals can be synthesised by adding two binary OOK-NRZ signals of different amplitudes. Forinstance, adding two a.c. coupled OOK signals, one having a peak-to-peak voltage of Vpp, andthe other with a peak-to-peak voltage of Vpp/2, results in a 4-PAM signal with equal spacingbetween consecutive levels. By driving the two MZMs in the IQ modulator structure, eachbiased at minimum transmission, the four levels of the 4-PAM signals are mapped to fourelectric field values for each quadrature of the signal, resulting in a 16-QAM optical signal.Driving each MZM between two consecutive transmission maxima will result in a maximumaverage optical power at the modulator output. However, in this case, due the nonlinear natureof the field versus applied voltage of the MZMs, the spacing between the levels in the 4-PAMsignals should be optimised, as shown in Fig. 20. Due to the difficulty of finding high-bandwidthdriving amplifiers able to generate a sufficiently high peak-to-peak voltage without distortionof the 4-PAM signals in order to drive the MZMs between consecutive maxima, the MZMs may

29


power / field

electrical

4PAM signal

4PAM

4PAM

optical 16QAM

4PAM electrical signal

generation:

4 bits per symbol

Figure 20 Principle of optical 16-QAM signal generation using an IQ modulator. First, two 4-PAMelectrical signals are generated by combining two binary digital bit streams each. Then the 4-PAMsignals are used to drive the two MZMs in the IQ modulator structure, resulting in an optical 16-QAMsignal.

be driven with a lower peak-to-peak voltage and appropriate level spacings in practice.

Fig. 21 shows an example of 14 Gbaud (hence 56 Gbit/s) 16-QAM signal generation, in-cluding the eye diagram of the generated electrical 4-PAM signal, the intensity at the outputof the IQ modulator, and the constellation recovered in a digital coherent receiver.

5 Differential quadrature phase-shift keying

The basic principle of optical DQPSK (differential quadrature phase shift keying) modulationis to represent each couple of two bits of the information sequence to be transmitted by opticalphase differences between consecutive symbols taking values into

{−π

2 , 0,π2 , π

}. Each transmit-

ted symbol therefore corresponds to two bits of information, meaning that the symbol rate isequal to half of the bit rate.

a b c

Figure 21 Generation of a 16-QAM signal at 14 Gbaud using an optical IQ-modulator. a) Eye diagramof the 4-PAM signal used to drive one of the Mach-Zehnder modulators. b) Measured intensity eyediagram at the output of the modulator. c) Constellation obtained at the output of the IQ modulator.Results obtained in DTU Fotonik’s systems laboratory.

30


5.1 Signal generation

ππππ/2

CW

ττττ

ττττ

uk

vk

Ik

Qk

ππππ/4

-ππππ/4

+_

+_

i1

i2

i3

i4

ττττ

ττττ

rk

sk

pre-coder

encoder

decoder

Figure 22 Configuration of an optical DQPSK transmitter and receiver.

The structure of an optical DQPSK system with interferometric detection is shown in Fig. 22.The transmitter consists of an electrical pre-coder requiring logic circuits functioning at half ofthe bit rate, and an electro-optical encoder. The receiver is made from a pair of Mach-Zehnderinterferometers and balanced receivers. The principle of an optical DQPSK system is describedin details in what follows. The outputs of the different stages are illustrated with a concreteexample in Table 2.

The bit sequence bk, at a bit rate of B bit/s, is first divided into two sets of odd and evennumbered bits, uk and vk, respectively, each at a bit rate of B/2 bit/s.

uk = b2k−1, vk = b2k. (71)

The sets of bits (uk, vk) are then differentially encoded into (Ik, Qk) according to

Ik = (uk ⊕ vk) (uk ⊕ Ik−1) + (uk ⊕ vk) (vk ⊕Qk−1) , (72)

Qk = (uk ⊕ vk) (vk ⊕Qk−1) + (uk ⊕ vk) (uk ⊕ Ik−1) . (73)

The pre-coded sequences (Ik, Qk) are therefore calculated based on the values of (uk, vk) to betransmitted, and on their previous values (Ik−1, Qk−1) taken at a sampling interval equal toτ = 2/B.

The DQPSK transmitter consists of one IQ modulator where the two chirp-free MZMs areused as phase modulators and are therefore biased at a null transmission point and driven bythe differentially encoded sequences Ik and Qk with a voltage corresponding to full swing (i.e.between two consecutive extrema of their transfer functions). The complex representation ofthe optical field at the output of the transmitter can be written

E (t = tk) = E0 cos

[(Ik −Qk) π + π

2

2

]e−j

[

(Ik+Qk)π+π2

2

]

, (74)

from which the following notation for the values of the amplitude and phase at the samplingtime are introduced

E (t = tk) = |E (t = tk)| e−jφ(t=tk) = |Ek| e−jφk . (75)

31


00

11

01

10

Figure 23 Constellation for the complex representation of the electric field at the output of the opticalDQPSK encoder. The phase assignment for the pre-coded dibits (Ik, Qk) is represented.

bk 01 10 10 00 11 10 11 10 01 01

uk 0 1 1 0 1 1 1 1 0 0vk 1 0 0 0 1 0 1 0 1 1

Ik 1 0 0 0 1 0 1 1 1 0Qk 0 0 1 1 0 0 1 0 1 1

|Ek| /E0

√22

√22

√22

√22

√22

√22

√22

√22

√22

√22

φk7π4

π4

3π4

3π4

7π4

π4

5π4

7π4

5π4

3π4

∆kπ2

π2 0 π π

2 π π2 −π

2 −π2

rk 1 1 0 1 1 1 1 0 0sk 0 0 0 1 0 1 0 1 1

Table 2 Illustration of optical DQPSK signal generation and decoding based on the transmission ofan example sequence.

The values taken by the amplitude |Ek| and phase φk of the optical field at the transmitteroutput are shown in Table 2 for the specific bit pattern example considered in this section. Itcan be seen that the field amplitude is constant (away from transitions) while its phase is takingvalues within

(π4 ,

3π4 ,

5π4 ,

7π4

). Each couple of pre-coded bits (Ik, Qk) is therefore assigned an

absolute optical phase value φk. The constellation diagram for the modulated field is representedin Fig. 23 as a function of the pre-coded bits (Ik, Qk). It can be seen that the phase valuesare assigned to the pre-coded bits according to a Gray code, meaning that only one bit valuechanges between two consecutive phases. As it is most likely that in the detection process errorswill be introduced between adjacent phases (i.e. where the distance between the points in theconstellation diagram of Fig. 23 is minimum), this attribution has the effect of minimising theprobability of errors (i.e. only one of the 2 bits (Ik, Qk) will be erroneously detected for a givensymbol detection error).

5.2 Interferometric detection

After transmission, the signal is split into two and input to two Mach-Zehnder interferometersmade of 3-dB couplers and an unit delay τ = 2/B. A relative phase shift equal to ±π/4 is

32


introduced between the two arms of each interferometer. Each Mach-Zehnder is followed by abalanced photo-detector where the two output fields are detected, and where, after subtractionof the generated photocurrents, the output signal is generated. The photocurrents generatedby the first pair of photodiodes can be written as a function of the optical field input to theupper Mach-Zehnder interferometer Eu according to

i1 (t) = R1 |Eu|2 cos2[φ (t)− φ (t− τ) + π

4

2

], (76)

i2 (t) = R2 |Eu|2 cos2[φ (t)− φ (t− τ)− 3π

4

2

]. (77)

Assuming the same responsivities R1 = R2 for the two photodiodes, as expected in a balanceddetector, the output current of the upper receiver can be expressed according to

r (t) = i2 (t)− i1 (t) = R1 |Eu|2√2

2[sin∆ (t)− cos∆ (t)] , (78)

where∆ (t) = φ (t)− φ (t− τ) . (79)

The output current of the lower receiver can be calculated in a similar way, leading to

s (t) = i3 (t)− i4 (t) = −R3 |El|2√2

2[sin∆ (t) + cos∆ (t)] , (80)

It can be observed that the optical phase differences ∆k = ∆(t = tk) take values into{−π

2 , 0,π2 , π

}, and that each dibit of the original information sequence (uk, vk) is encoded into

a value of ∆k.From equations (78) and (80), it becomes possible to evaluate the recovered bit sequences

rk and tk. It is found from Table 2 that

rk = uk, tk = vk. (81)

The original dibits (uk, vk) are therefore recovered at the output of the optical DQPSK decoder.One of the practical benefits of the optical DQPSK scheme is that it only requires elec-

trical and optoelectronic components having a bandwidth compatible with half of the bit rateeffectively transmitted over the optical link, as all the electronic processing, modulation anddetection are performed on the tributary bit streams uk and vk.

What to remember:

• Multilevel modulation can be used to increase the spectral efficiency of optical systemssince each transmitted symbol, whose duration determines the spectral width, can carryseveral bits of information.

• A first successful implementation of multilevel modulation that has been presentedfirst in 2002 is differential quadrature phase-shift keying (DQPSK), making use of fourdifferential phase levels, hence transmitting two bits per symbol.

• Similarly to differential binary phase-shift keying, DQPSK can be detected in pho-todiodes following demodulation by a set of delay interferometers, making the schemehighly compatible with legacy on-off keying systems. This requires differential encodingto be applied to the data at the transmitter.

33


a b

Figure 24 Phase diversity coherent receiver. (a) Realisation of an optical 90◦ hybrid circuit, which,followed by balanced receivers, allows detection of the in-phase and quadrature components of the opticalsignal, hence the mapping of the optical field into the electrical domain. (b) Reminder of the transfermatrix of a 3-dB directional coupler, as used in the optical 90◦ hybrid circuit shown in (a).

• More recently, DQPSK used in conjunction with polarisation multiplexing and thedigital coherent receiver concept briefly introduced in Sec. 6 has turned out to be apopular modulation format allowing the implementation of 100 Gbit/s systems basedon 25 Gbit/s electronics. Such systems are being deployed commercially.

6 Phase diversity receiver

It has been seen in Sec. 3 and 5 that BPSK as well as DQPSK can be detected using aninterferometric demodulation scheme, where some phase-to-intensity modulation conversion isperformed first in a delay interferometer, before detection in balanced receivers. The schemecan be extended to modulation formats making use of more phase levels such as 8-PSK and even16-PSK, however at the price of a significantly higher complexity, as well as a reduction of theperformance. For instance, 8-PSK (with proper differential encoding) typically requires four DIand balanced receivers for its detection. It is clear that it would be challenging to increase thenumber of bits per symbol further and still use some kind of interferometric detection scheme.Furthermore, such a technique would not allow the detection of more advanced modulationformats such as QAM, as introduced in Sec. 4.4.

On the other hand, coherent detection allows the detection of arbitrary modulation formatand can be used to map the complex field of the signal from the optical to the electrical domain.This can be achieved using a so-called phase diversity receiver whose key component is an optical90◦ hybrid coupler represented in Fig. 24.

After recalling the transfer matrix of a 3-dB directional coupler, as shown in Fig. 24(a), itcan easily be shown that, if a signal field Es and a local oscillator field Elo are input to thefirst and fourth input ports of the optical 90◦ hybrid, respectively, then the following fields arerecovered at the four outputs

E1 =1

2(Es − Elo) , (82)

E2 = − j2(Es + Elo) , (83)

E3 = − j2(Es − jElo) , (84)

E4 = − j2(−jEs +Elo) . (85)

34


It is assumed first that the states of polarisation of the signal and the LO are identical at thereceiver input. Expressing the electric field of the signal as

Es = Re{√

Ps e−jφs e−jωst

}, (86)

and similarly for the local oscillator

Elo = Re{√

Plo e−jφlo e−jωlot

}, (87)

the fields at the output of the 90◦ hybrid circuit as expressed in (82)-(85) can be easily expanded.The general case of the linear superposition of two fields

E = Es + aElo (88)

is considered for simplicity. The total field E can be expanded as

E = Re{(√

Ps e−jφs + a

√Plo e

−jφlo ej(ωlo−ωs)t)ejωst

}, (89)

which can be reformulated in terms of the complex envelopes of the signal (with respect to thesignal frequency) and the local oscillator (with respect to the local oscillator frequency), Es andElo, respectively,

E = Re{(

Es + a Elo ej(ωlo−ωs)t)ejωst

}(90)

Provided the frequency difference ωlo−ωs is small with respect to ωs, the term within parentheses() can be considered as the complex envelope of a field at ωs. This condition will be verified inpractice, where the local oscillator will be tuned close to the signal frequency, according to thedistinction between homodyne and heterodyne detection introduced in Sec. 3.2. In this case,the power can be simply expressed as

P =∣∣∣Es + a Elo ej(ωlo−ωs)t

∣∣∣2, (91)

hence

P = |Es|2 + |a|2 |Elo|2 + 2Re{Es E∗

lo a∗ ej(ωs−ωlo)t

}. (92)

Applying this general equation to the cases of (82)-(85), the optical power at the four outputsof the hybrid can be expressed as

P1 =1

4

[|Es|2 + |Elo|2 − 2 Re

{Es E∗

lo ej(ωs−ωlo)t

}], (93)

P2 =1

4

[|Es|2 + |Elo|2 + 2 Re

{Es E∗

lo ej(ωs−ωlo)t

}], (94)

P3 =1

4

[|Es|2 + |Elo|2 − 2 Im

{Es E∗

lo ej(ωs−ωlo)t

}], (95)

P4 =1

4

[|Es|2 + |Elo|2 + 2 Im

{Es E∗

lo ej(ωs−ωlo)t

}]. (96)

By subtracting P1 and P2 and P3 and P4, one gets

P2 − P1 = Re{Es E∗

loej(ωs−ωlo)t

}, (97)

P4 − P3 = Im{Es E∗

loej(ωs−ωlo)t

}. (98)

35


Therefore, from (97) and (98) it can be seen that, if the outputs of the 90◦ hybrid aredetected and subtracted in a balanced detector, the generated photocurrents iI and iQ areproportional to the real and imaginary part of the quantity same quantity Es E∗

loej(ωs−ωlo)t.

Considering the case of ideal homodyne detection first, ∆ω = ωs − ωlo = 0, and assuming aCW laser with negligible phase noise is used as the LO, Elo =

√Plo,

iI = R√Plo Re {Es} , (99)

iQ = R√Plo Im {Es} , (100)

where R is the responsivity of the photodiodes in the two balanced receivers, assumed to beequal. Therefore it is clear that it is possible to recover both the in-phase and the quadraturecomponents of the signal in the electrical domain, hence to detect arbitrary modulation formats.

In the more general case of a signal and LO having different frequencies ∆ω 6= 0, and if thesignal and LO lasers present some random variations of the phase ϕs (t) and ϕlo (t), accordingto (38) and (41), respectively, then the photocurrents can be expressed as

iI (t) = R√Ps (t)Plo cos [∆ωt− φs (t)− ϕs (t) + ϕlo (t)] , (101)

iQ (t) = R√Ps (t)Plo sin [∆ωt− φs (t)− ϕs (t) + ϕlo (t)] . (102)

If the photocurrents are sampled at some time instants tk in a digital-to-analogue converter(DAC), not necessarily synchronous to the symbol rate, and if the frequency offset ∆ω and thephase of the signal laser with respect to that of the LO laser can be estimated, then it becomespossible to recover the in-phase and quadrature components of the signal, and thus to detectany modulation format. This is the principle behind intradyne receivers making use of DSPfor the recovery of the transmitted data. For those receivers, the LO is tuned close, but doesnot have to be locked to the signal frequency and some DSP functionalities are implementedfollowing the optical hybrid, balanced detectors, and analogue-to-digital conversion. It is alsounnecessary to mach the polarisation of the LO to that of the received signal if a phase and

polarisation diversity receiver is implemented. In this configuration, both the received signaland the LO are resolved onto orthogonal polarisation components before being processed bytwo sets of 90◦ hybrid and balanced receiver prior to analogue-to-digital conversion. Sincethe whole optical field is mapped to the electrical domain, it becomes possible to compensateelectronically for transmission impairments such as group-velocity dispersion. Those systemshave been intensively researched since around 2004 and the first generation of commercialproducts are being deployed.

What to remember:

• The detection of advanced modulation formats can be performed in a phase diversityreceiver that effectively maps the optical field to the electrical domain.

• In this way the in-phase and quadrature components of the signal can be recovered.

• Modern coherent receivers employing this technique no longer rely on precise matchingof the frequencies of the transmitter and local oscillator lasers, nor on complex opticalor electrical phase locked loops, which had hindered the earlier generation of coherentreceivers.

• Instead, analogue-to-digital converters followed by digital signal processing techniquesenable the signal to be processed in the digital domain, where the carrier can be recov-ered and some transmission impairments can be compensated.

36


• Today, systems of this type are being deployed for polarisation-multiplexed quadraturephase-shift keying and future generations using quadrature-amplitude modulation arebeing intensively investigated.

Appendices

A List of acronyms

AMI alternate mark inversion

AWG arbitrary waveform generator

BPSK binary phase-shift keying

CW continuous wave

DAC digital-to-analogue converter

DB duobinary

DI delay interferometer

DML directly modulated laser

DPSK differential phase-shift keying

DQPSK differential quadrature phase-shift keying

DSP digital signal processing

EDFA erbium-doped fibre amplifier

FWHM full-width at half-maximum

FSR free spectral range

IM-DD intensity modulation - direct detection

LO local oscillator

MZDI Mach-Zehnder delay interferometer

MZM Mach-Zehnder modulator

NRZ non return-to-zero

OOK on-off keying

OSNR optical signal-to-noise ratio

PAM pulse amplitude modulation

PSK phase-shift keying

QPSK quadrature phase-shift keying

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QAM quadrature amplitude modulation

RF radio frequency

RZ return-to-zero

SE spectral efficiency

SNR signal-to-noise ratio

WDM wavelength division multiplexing

38

Geracao.Formatos.Modulacao_02_04

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