Introduction to Optical Networks – Light propagation VII – Polarisation in Optical Fibres

1. Introduction to Optical Networks – Light propagationVII – Polarisation in Optical Fibres Bruno Vinouze, Michel Morvan, Michel Gadonna : GET ENST Bretagne, firstname.name@enst-bretagne.fr

2. Outline • Polarisation state of an electromagnetic wave • The Poincaré sphere and the Stokes formalism • The Jones formalism • Birefringence • Polarisation Mode Dispersion in fibres • Origin • Consequences • Modelling real fibres • PMD statistics and system accommodation

3. Polarisation state of an electromagnetic wave • The electric and magnetic fields of a plane wave are perpendicular to each other and to the propagation axis • The orientation of the electric field defines the state of polarisation (SOP) of the wave. • Ex and Ey are in phase : linear polarisation • Ex and Ey are phase shifted : elliptic polarisation

4. The polarisation ellipse and its parameters b y c a • Azimuth : angle y • Ellipticity : angle c • tan c = b/a • If c < 0 left rotating • If c > 0 right rotating

5. The major polarisation controlling devices • Polariser : selects a linear state of polarisation • Wave plate or retarder : applies a φ = 2π (Δn ) e/λ phase shift between two perpendicular polarisation components. • Half-wave plate φ = π rotates the polarisation ellipse • Quarter-wave plate : φ = π/ 2 transforms the SOP • Polarisation controller:Allow to get any SOP at the output whatever the SOP at in the input • Association of plates : polariser + λ/4 + λ/2 λ/4 + λ/4 + λ/2 • Fibre loops with defined curvature radii

6. The Poincaré Sphere S3 2c S2 2y S1 • The Poincaré sphere is a convenient mode to represent the SOP of a electromagnetic wave with intensity I. • The Cartesian coordinates of the point on the Sphere are called the Stokes parameters. • The angles y and c are the polarisation ellipse parameters. • The equator of the sphere represent the linear SOPs and its poles the circular ones.

7. Polarised and non polarised light • Polarised light can be assigned a specific SOP, i.e. a specific point on the Poincaré sphere : all the wave power is located in this SOP. • Unlike the monochromatic plane wave which is quite artificial (eg. the laser light), natural light is usually partially or non polarised because it is not coherent. • Non coherent light come from independent atoms so it does not exhibit any phase, spatial nor polarisation coherence. • This means that we can not assign a specific state of polarisation to non polarised light. On the Poincaré sphere, the SOP is not represented as a spot but as an area or even as the complete sphere itself.

8. The Degree of Polarisation • The Degree of Polarisation (DOP) indicates the proportion of polarised light within a specific wave : • A perfectly polarised light (like a monochromatic plane wave) has a DOP of 1 or 100% • A non polarised light has a DOP of 0 or 0% • A partially polarised light that is the sum of fully polarised and non polarised light has a DOP between 0 and 1. • In case of partially polarised light, only the polarised component is represented « on » the Poincaré Sphere. Hence, the Stokes parameters become : • The SOP point is now inside the sphere.

9. The Stokes formalism and the Mueller matrix • An linear optical element can be defined by the transformation it applies to the Stokes parameters. • As for the Jones matrices, we can define a linear relation between the input and output Stokes parameters : • The 4x4 matrix [M] is called a Mueller matrix

10. Some simple examples of Mueller matrices • Horizontal linear polariser • Vertical linear polariser • Angled linear polariser • Quarter-wave plate with horizontal fast axis

11. The Jones formalism y x • Fully polarised light can be represented by a two dimensional vector called Jones vector: • The Jones vector is normalised if absorption is not taken into account. • Linear optical components can be represented by 2x2 matrices called Jones matrices. They allow to compute the output field in function of the input field. E

12. Some simple examples of Jones matrices • Horizontal linear polarizer • Vertical linear polarizer • Angled linear polarizer • Half-wave plate with horizontal fast axis • Quarter-wave plate with horizontal fast axis

13. Anisotropy and birefringence in solids • An anisotropic material is a material whose optical properties depends on the direction and on the SOP of the light propagation. This characteristic is due to the crystalline system of the material. • In general, the optical index seen by a linearly polarised light wave depends on the direction of polarisation, except for one (or two) direction(s) which are called the axes of the crystal. • The trigonal, tetragonal and hexagonal crystalline systems have only one axis and are called uniaxial. They exhibit two optical indexes along two principal axes. • Monoclinic, triclinic and orthorhombic crystalline systems have two axes and are called biaxial crystals. They exhibit three optical indexes along three principal axes.

14. Induced anisotropy and birefringence • Birefrigence may also occur in isotropic materials (e.g. an optical fibre) when • they are stretched or bent (photo elastic effect) • they are under an electric field (Kerr or Pockels effect). • they are under a magnetic field (e.g. Faraday effect) • For optical fibres, induced birefringence is mainly due to : • Mechanical and thermal stresses introduced during manufacturing and resulting in asymmetries in the fibre core geometry. • Mechanical stresses induced by cabling process and vibrations in cable environment.

15. The Differential Group Delay (DGD) Dt of birefringent fibre section A birefringent fibre section with index difference Dn and length L presents a differential group delay Dt between its two eigenmodes which is given by : High birefringent fibres (Hi-Bi) are also made on purpose as they allow to maintain the SOP of the propagated signal over relatively long distances. They are used to interconnect polarisation sensitive components (example: a laser diode module to a Lithium Niobate modulator). For a typical polarisation maintaining fibre, Dt is about 2 ps/m

16. Beat length • The beat length of a birefringent optical waveguide is defined as the minimum length for which the SOP returns identical. • It is the length for which the phase shift between the two eigenmodes is p. 1 beat length Slow axis Output SOP Fast axis Input SOP

17. Polarization Mode Dispersion in fibres nx ny • There are induced birefringence in optical fibres that can be caused by : • Mechanical and thermal stresses introduced during manufacturing result in asymmetries in the fibre core geometry. • Mechanical stresses induced by cabling process and vibrations. Two group velocities for two orthogonal polarizations

18. Polarisation Mode Dispersion (PMD) in fibres P(t) DtR -3 dB t P(t) DGD DtT -3 dB t • Line fibres may exhibit some birefringence. Hence the propagation is such fibres will be carried on along two orthogonal modes resulting in a two-path propagation with different group delays and hence a DGD. • When a light pulse is injected into a fibre core, it is decomposed into two orthogonal polarised pulses that propagate with different propagation characteristics. The two pulses arrive at the output with a differential delay. For they are orthogonal, the pulse are added at the receiver side hence resulting in a broadened pulse. • The order of magnitude of the DGD is about several ps to several tens of ps depending on the fibre quality. Received pulse Transmitted pulse

19. Modelling of PMD in real fibres Fast PSP Slow PSP • A real fibre that exhibits PMD is not homogeneous. It can not be seen as a single birefringent fibre section with stable DGD and linearly polarised eigenmodes. Furthermore, thermal and mechanical drifts will affect and modify the fibre with time. • A real fibre be modelled by a concatenation of birefringent fibre sections whose eigenmodes are linearly polarised and arbitrarily oriented with respect to those of the other sections. • It can be shown that the concatenation of all these sections can be reduced in a single section with Principal States of Polarisation (instead of eigenmodes) that are not necessarily linearly polarised. The DGD between the two PSP can be computed for some cases. • The DGD and the PSPs depend on the wavelength.

20. DGD and PSP variations in a mode coupled real fibre • The Mode coupling between birefringent sections vary with time depending on mechanical stresses and thermal drifts. The fibre then changes with time and hence its DGD and PSP. • If the fibre is buried, the time constant is about hours. • If the fibre is mounted on poles, the time constant is about minutes or even second (wind). • The DGD is then a random variable. • In the case of strong mode coupling (i.e. many sections), the DGD follows a Maxwell probability law :

21. The Maxwell statistical distribution of the DGD 0.08 0.06 pdf of DGD 0.04 0.02 0 0 5 10 15 20 25 30 DGD • Note that the statistical distribution has only one free parameter, which is the mean <DGD> of the random variable. Probability destiny function for a mean DGD =10 ps

22. Definition of the PMD of a given fibre section • The Polarisation Mode Dispersion of a fibre section is the root mean square of the DGD : • In the case of a strong mode coupled fibre where the DGD obeys the Maxwell distribution, the mean and the rms are linked by the following relation :

23. The Hinge model • The concatenation of randomly oriented birefringent fibre sections can be modelled using Jones matrices and graphically represented as a random walk in a two dimensional birefringence vector space. • After N steps of representing the propagation through N arbitrarily oriented sections with DGD Dt, we can prove that : • t is the lineic birefringence and l is the length of a birefringent fibre section y DGD x

24. PMD characteristic of a fibre • The average value (or the quadratic value) of the DGD increases proportionally to the square root of the distance. • Hence, the lineic PMD is expressed in ps/km (ps.km-½) • The lineic PMD is now specified in fibre datasheets. • For a homogeneous fibre with length L : Example: The PMD of a 100 km fibre section with 0,5 ps.km-1/2 exhibits a PMD of 0.5 x 100 = 0.5 x 10 = 5 ps.

25. PMD of a composite fibre PMDtotal PMD1 PMD2 PMD3 PMD4 PMD5 • If a fibre line is composed of N fibre sections, each presenting a PMD equal to PMDi, the total PMD of the line is : 

26. A simple rule to compute the PMD limitation • The ITU-T rule : The total PMD (rms of the DGD) of a given fibre section has to be less than 10% of the pulse width for a NRZ digital signal. • Example : At 10 Gbit/s, the pulse width for a NRZ signal is 100 ps. Hence the total PMD of the fibre used for transmission can not excess 10 ps. • Hence, the maximum distance over which you can transmit is : • Example : for a fibre with lineic PMD equal to 0,5 ps/km Lmax=400 km for a 10 Gbit/s NRZ transmission.

27. Main PMD parameters and definitions • Second order PMD : the PMD vector depends on the frequency - Chromatic dispersion due to DGD change - Depolarisation due to the change of the PSPs

28. How to minimize PMD in optical fibres • Due to its random nature, PMD represents the major obstacle to the TDM bit rate increase using short pulses in fibre transmission systems. • The PMD influence can be mitigated by : • Fabricating low PMD fibres using the spinning technique • Optimising cable installation without stress nor torsion • Mitigate PMD in systems using PMD compensators (per channel operation)