第三章 光纤系统中的偏振效应

1. 第三章 光纤系统中的偏振效应

2. 内容提要 • 背景介绍 • 偏振光的表述 • 琼斯矢量、斯托克斯矢量、帮加球 • 琼斯矩阵、缪勒矩阵 • 偏振光在双折射晶体中的传播 • 光纤系统中的主偏振态(PSP) • 光纤系统中的偏振模色散(PMD)

3. 参考书 • D S Kliger, J Lewis, and C Randall, “ Polarized Light in Optics and Spectroscopy,” Academic Press, 1990, Chapters 4&5; • S Huard, “Polarization of Light,” John Wiley & Sons, 1996. • A Yariv and P Yeh, “Optical Waves in Crystals,” John Wiley & Sons, 1984, Chapter 5; • H Kogelnik, R Jopson, and L Nelson, “Polarization-Mode Dispersion,”Optical Fiber Telecomm IVB, 2002, Chapt 15. • Gordon and Kogelnik, “PMD fundamentals,”www.pnas.org. • C Poole and J Nagel, “Polarization effects in lightwave systems,”Optical Fiber Telecom IIIA, I Kaminow and T Koch, Eds. Academic, 1997, pp.114-161.

4. PMD: Polarization Mode Dispersion • Ideal single mode fiber: the two HE11 modes are degenerate. The two orthogonal polarized modes have the same group delay. • Real fibers contain some amount anisotropy owing to loss of circular symmetry.

5. Origins of PMD I

6. Origins of PMD II • Optical Birefringence: • Intrinsic Perturbations: • Imperfections in the manufacturing process: noncircular core. • Extrinsic Perturbations: • Lateral stress, bending, or twisting. • External environment changes.

7. Evolution of Polarization in Birefringent Fiber • 理想的单色光； • 脉冲。

8. Time-Domain PMD Effect in Short Fibers • Pulse splitting due to birefringence • Differential Group Delay (DGD): group-delay between the slow and fast modes. DGD linear dependence of length

9. Beat Length Lb • Beat Length: • SSMF: Lb~10m, Dn~10-7; • PMF: Lb~3mm; • Linear length dependence of DGD. • DGD for a single beat length: • tb=5.2fs@1550nm.

10. Frequency-Domain PMD Effect • Output polarization varies with frequency, traces out a circle on the Poincare Sphere. • Characteristic Frequency

11. Polarization-Mode Coupling • Terrestrial and submarine transmission systems ~1000’s of km, Random variations in the axes of the birefringence along the fiber length; • Concatenation of birefringent sections, Polarization-mode coupling; • DGD increases with the square root of distance.

12. Correlation Length Lc • Any polarization state can be observed for a fixed input polarization at large lengths due to random polarization coupling; • Assuming <px>=1 and <py>=0 at the input, <px>-<py> evolves from 1 to 0 at large lengths; • Lc as that length where the power difference has decayed to <px>-<py>=1/e2; • Fiber transmission distance L<<Lc, the fiber is the short-length regime, DGD increases linearly with distance. • Fiber transmission distance L>>Lc, the fiber is the long-length regime, DGD increases with the square root of distance.

13. Frequency-Domain PMD

14. PSP: Principal States of Polarization • Concatenation of birefringent sections with birefringence axes and magnitudes that change randomly along the fiber. • PSP: two special orthogonal polarization states at the fiber input that result in an output pulse that is undistorted to first order.

15. Transmission Distances Limited by PMD • PMD is an ultimate limitation for ultra-high speed transmission systems!

16. 光的偏振特性 • 光的偏振态（SOP）； • Jones矢量、Stokes矢量、Poincare球； • Jones矩阵、Mueller矩阵、Pauli矩阵；

17. State of Polarization (SOP) I • 线偏振光

18. State of Polarization (SOP) II • 圆偏振光

19. State of Polarization (SOP) III • 椭圆偏振光 • 方位角a(azimuth); • 椭偏度e(ellipticity); • 手性（handedness).

20. 琼斯矢量与琼斯矩阵I • 电场矢量 • 琼斯矢量 • 线性偏振 • 垂直线偏振态 • X-线偏振光、Y-线偏振光

21. 琼斯矢量与琼斯矩阵II • 圆偏振 • 变换矩阵

22. 琼斯矢量与琼斯矩阵III • 椭圆偏振 • 变换矩阵

23. Stokes Parameters II • 偏振光

24. Jones Matrix • Linear interaction:

25. 琼斯矢量与琼斯矩阵I • 电场矢量 • 琼斯矢量 • 线性偏振 • 垂直线偏振态 • X-线偏振光、Y-线偏振光

26. Stokes Parameters I • 定义:

27. Jones Matrix • Linear interaction:

28. Jones Calculus of Birefringent Crystal I • o-xyz system:

29. Jones Calculus of Birefringent Crystal II

30. Jones Calculus of Birefringent Crystal III

31. Mueller Matrix • 4x4 Matrix

32. Partially Polarized Light • 完全非偏振光 • 部分偏振光 • 偏振光

33. Degree of Polarization (DOP) • 偏振度(DOP) • 偏振光 DOP=1; 部分偏振光 DOP<1; 完全非偏振光 DOP=0.

34. Jones Vector and Stokes Vector • Jones Vector • Stokes Vector

35. LCP RCP Elliptical Stokes Vectors of the Fundamental States of Polarization

36. Poincare Sphere I • 以s1, s2, s3为坐标的球;

37. Poincare Sphere II • 赤道上的点表示线偏振; • 北极点表示右旋圆偏振光; • 南极点表示左旋圆偏振光; • 同纬度点表示同椭偏度; • 同经度点表示同方位角; • 上半球表示右旋椭偏光; • 下半球表示左旋椭偏光.

38. Polarization Evolution I

39. Polarization Evolution II

41. Jones Calculus of Birefringent Crystal I • o-xyz system:

42. Jones Calculus of Birefringent Crystal II

43. Jones Calculus of Birefringent Crystal III

44. Jones and Stokes Vectors I • Jones space T: transmission Matrix, U: Jones Matrix. • Stokes space: T, U, R IN OUT R: 3x3 rotation matrix in Stokes space.

45. Pauli Spin Matrix • Pauli spin matrix • Hermitian and unitary

46. Birefringence Vector • Pauli spin vector in Stokes space • Birefringence vector in Stokes space • Birefringence vector in Jones space

47. Jones and Stokes Vectors II

48. Expansion of Matrix • Any 2x2 matrix M may be expanded

49. Connection between Jones Vector and Stokes Vector • Unitary transmission matrix • Write the components of the Stokes vector corresponding to

50. Principal States of Polarization (PSP) • Input polarization for which the output state of polarization is independent of frequency to first order. • In the absence of PDL, the PSPs are orthogonal. • The input and output PSPs C Poole and R Wagner, “Phenomenological approach to polarization dispersion in long single-mode fibers”, Elect Lett 22 1029 (1986),