6.441 Final Project: Polarization Shift KeyingModulationMatt WillisMay 11, 200611 IntroductionAs demand grows for communications systems with large data banwidths, theuse of a light signal as an information-bearing medium between two distantpoints has become increasingly common. Typical modulation schemes havedealt with one of two properties of light; amplitude and phase. In terms of ampli-tude modulation, perhaps the most common technique is on-off keying (OOK).In terms of phase modulation, two common techniques are phase-shift-keying(PSK) and differential-phase-shift-keying ( DPSK). More recently, efforts havebeen made to vary both amplitude and relative phase simultaneously, and theend result is known as polarization shift keying modulation (POLSK, DPOLSK).The purpose of this project is to describe this novel technique of informationtransfer using the polarization state of light, and explore the noise models thatdescribe its error behavior. POLSK is an attractive (although not widely used)modulation scheme because it enables multi-level transmission (i.e., input al-phab e t ΩX≥ 2), whereas many realizable communications schemes rely onlyon binary input alphabets.This project relies heavily on a published paper on the subject by Benedettoand Poggiolini[1].2 A Mathematical Description of PolarizationThe state of p olarization (SOP) of a fully polarized lightwave can be describedusing either plane wave terminology or Stokes parameters. Let us first considera plane wave description: a beam of light propagating in the ˆz direction is avector quantity which can be characterized as follows:~E = Ex(t)ˆx + Ey(t)ˆy,where Ex(t) and Ey(t) are functions describing the respective electric fieldsalong the ˆx and ˆy directions:Ex(t) = Ax(t)ei(ωt+φx(t))Ex(t) = Ax(t)ei(ωt+φx(t))Although the frequency ω of light can also be varied in time, in this analysis wewill assume it to be constant over time. For notational simplicity, we will ass umethat the time dependence of the amplitude and phase functions is understood.Insofar as we are concerned only with the relative phase (∆ = φy(t) −φx(t)) between Ex(t) and Ey(t), there exists a unique mapping between theaforementioned plane wave representation of p olarization, and its representationas a so-called ”Stokes” vector (~S = (S1, S2, S3)) in three-dimensional space:S1= A2x− A2yS2= 2AxAycos(∆)S3= 2AxAysin(∆)2Figure 1: The Poincare Sphere. Red points demonstrate actual data for avertically polarized light source.An additional Stokes parameter, S0, represe nts the total electromagnetic powerdensity traveling in the ˆz direction:S0= A2x+ A2y. (1)With this additional parameter, the following equation also holds instanta-neously, or for constant optical power over time:S20= S21+ S22+ S23, (2)which is to say that the range of Stokes parameters comprises a sphere offixed radius when the transmitted power is held constant– this is known as the”Poincare Sphere” of radius S0(see Figure 1). Note that we can still compareelectromagnetic waves of differing power densities by considering their normal-ized Stokes vector,~S = (S1S0,S2S0,S3S0), where S21+ S22+ S23= 1; in common usage,”Stokes vector” is interchangeable with ”normalized Stokes vector.” Through-out our discussion this usage will cause no loss of generality. The transmissionsystems of interest are designed to operate in a mode of steady power, but thenormalized Stokes vector can be used for signals that fade or vary in powerover time; one could imagine the scenario where the modulation scheme simul-taneously uses both power density and polarization state to convey information, but this scenario will not be treated here. The geometric interpretation ofpolarization states as points on the unit sphere will provide the framework forunderstanding the encoding and decoding schemes described in subsequent sec-tions.Note that all linear polarization states (i.e., those states with ∆ = 0) lie onthe (S1, S2, 0) equator of the Poincare sphere; the ”North” pole (0,0,1) representsleft-hand circularly polarized light (to an observer, the net transverse electric3field appears to rotate in a clockwise circle ); the ”South” pole (0,0,-1) representsleft-hand circularly polarized light (the net transverse electric field appears torotate in a counterclockwise circle); all other points on the Poincare sphererepresent states of elliptic polarization.Another characteristic of the Poincare sphere worthy of mention is thatSOP’s which are orthogonal according to the hermitian scalar product:h~E ·~E∗0i = E · E0∗e(jωt+φ(t))e(−jωt+φ0(t))map onto anitpodal points on the sphere. This representation of SOP’s is inaccordance intuition: orthogonal polarization states are spaced as far apart fromone another as possible.3 Encoding/Decoding Schemes for n-POLSKThe state of polarization as defined by the Stokes vector is a continuous randomvariable that can take on any value on the surface of the Poincare sphere. Thesurface area of the Poincare sphere is partitioned into n disjoint regions; withinthe ithregion of this partition, one state of polarization~Siis chosen to representto represent this region. The encoder assigns each member of the input alphabet{1, 2, ..., n} to one of the representative SOPs:E : {1, 2, ..., n} → {~S1,~S2, ...,~Sn}.Note that in this analysis, there are no ”preferred” encodings; in the absenceof phase noise considerations, and under random white Gaussian processes, anySOP will have similar noise properties, so the user is allowed to choose anydisjoint partition of the sphere’s surface. Obviously the optimal partitions willbe those that produce that greatest distance between points in the constellation.The decoder takes the scalar product of the received vector with the set ofreference vectors, and selects the index of the reference vector that maximizesthis scalar product, i.e.:D :~S → i ∈ {1, 2, ..., n}, where i is such thatSi= maxj∈{1,2,...,n}~S ·~Sj,where the scalar product, ~a·~b, between vectors ~a = (a1, a2, a3) and~b = (b1, b2, b3)is defined as:~a ·~b =XiaibiAs a example, consider 2-POLSK with input alphabet {0, 1}, and the fol-lowing encoding:E : 0 →~S0= (0, 0, 1)E : 1 →~S1= (0, 0, −1)4Figure 2: Decoding example with binary input alphabet.In this case the decoding is very simple; the decoder interprets a received SOPvector~S as a zero if the sign of the scalar product~S ·~S0is positive, and as azero if the sign is