Fiber Optic Communication Systems. Unit-04: Theory of Light.

Transcription

1 Unit-04: Theory of Light Department of Telecommunication, MUET UET Jamshoro 1

2 Limitations of Ray theory Ray theory describes only the direction a plane wave takes in a fiber. In reality, plane waves interfere with each other. Therefore, only certain types of rays are able to propagate in an optical fiber. Mode theory is used to describe the types of plane waves able to propagate along an optical fiber. In addition, a modal analysis is necessary when analyzing the coupling of power between modes at waveguide imperfections. Another discrepancy between the ray optics and approach and the modal analysis occurs when an optical fiber is uniformly bent with a constant radius of curvature. Wave optics correctly predicts that every mode of the curved fiber experiences some radiation loss. Ray optics, on the other hand, erroneously predicts that some ray congruences can undergo total internal reflection at the curve and, consequently, can remain guided without loss. Department of Telecommunication, MUET UET Jamshoro 2

3 Maxwell s Equations First equation E = B (1) t The equation gives a differential form of Faraday s law of electromagnetic induction. A circulating electric field is produced by a magnetic field that changes with time. Second equation H = D (2) t A circulating magnetic field is produced by an electric current and by an electric field that changes with time. Department of Telecommunication, MUET UET Jamshoro 3

4 Maxwell s Equations Third equation. D = 0 (3) The equation describes the divergence of electric flux in absence of free charges. Fourth equation. B = 0 (4) The equation describes the divergence of magnetic field in absence of free charges. Where D=εE and B=μH. The parameter ε is permittivity (or dielectric constant) and μ is permeability of the medium. Department of Telecommunication, MUET UET Jamshoro 4

5 Maxwell s Equations A relationship defining the wave phenomenon of the Electromagnetic fields can be derived from Maxwell s equations. Taking the curl of Eq. (1) and making use of Eq. (2) yields E = μ H = μ D = εμ 2 E (5) t t t t 2 Using the vector identity: E =. E 2 E From Eq. (3): Eq. (5) becomes:. D =. εe = 0 2 E = εμ 2 E t 2 (6) Similarly, by taking the curl of Eq. (2), it can be shown that 2 H = εμ 2 H t 2 (7) Equations (6) and (7) are the standard wave equations. Department of Telecommunication, MUET UET Jamshoro 5

6 Waveguide Equations* Consider electromagnetic waves propagating along the cylindrical fiber shown in Fig A cylindrical coordinate system {r, φ, z} is defined with the z axis lying along the axis of the waveguide. Fig Cylindrical coordinate system used for analyzing electromagnetic wave propagation in optical fiber. Department of Telecommunication, MUET UET Jamshoro 6

7 Waveguide Equations* If the electromagnetic waves are to propagate along the z axis, they will have a functional dependence of the form: E = E 0 r, e j ωt βz (8) H = H 0 r, e j ωt βz (9) Which are harmonic in time t and coordinate z. The parameter β is the z component of the propagation vector and will be determined by the boundary conditions on the electromagnetic fields at the core cladding interface. Curl in cylindrical coordinates: Department of Telecommunication, MUET UET Jamshoro 7

8 Waveguide Equations* E = rƹ 1 r E = E z E z 1 r e 1 r e r e z r z E r re E Z + E r z E z r + zƹ 1 r r E r 1 r When Eqs. (8) and (9) are substituted into Maxwell s curl equations, we have, from Eq. (1): 1 r E z + jrβe = jωμh r (10) E r Department of Telecommunication, MUET UET Jamshoro 8

9 Waveguide Equations* jβe r + E z r = jωμh (11) 1 re r r E r and from Eq. (2) 1 r = jμωh z (12) H z + jrβh = jεωe r (13) jβh r + H z r = jεωe (14) 1 r r rh H r = jεωe z (15) Department of Telecommunication, MUET UET Jamshoro 9

10 Waveguide Equations* By eliminating variables these equations can be rewritten such that, when E z and H z are known, the remaining transverse components E r, E φ, H r and H φ can be determined. where E r = j q 2 E = j q 2 H r = j q 2 H = j q 2 β E z r + μω r β r H z E z μω H z r β H z ωε r r β r E z H z + ωε E z r (16) (17) (18) (19) q 2 = ω 2 εμ β 2 = k 2 β 2 Department of Telecommunication, MUET UET Jamshoro 10

11 Waveguide Equations* Substitution of Eqs. (18) and (19) into Eq. (15) results in the wave equation in cylindrical coordinates: 2 E z + 1 E z r 2 r and substitution of Eqs. (16) and (17) into Eq. (12) leads to r + 1 r 2 2 E z 2 + q2 E z = 0 (20) 2 H z r r H z r + 1 r 2 2 H z 2 + q2 H z = 0 (21) Department of Telecommunication, MUET UET Jamshoro 11

12 Optical mode An optical mode is a specific solution of the wave equation that satisfies the appropriate boundary conditions and has the property that its spatial distribution does not change with propagation. Mathematically, it can be described as mode is an electromagnetic field which results as a solution of Maxwell s equation in a specific geometry and material. A set of guided electromagnetic waves is called the mode of an optical fiber. Mode is a spatial distribution of energy on cross sectional surface of cylindrical waveguides or optical fibers. Department of Telecommunication, MUET UET Jamshoro 12

13 Optical mode The fiber geometry and composition determine the discrete set of electromagnetic fields, or fiber modes, which can propagate in the fiber. If the fiber core is large enough, it can support many simultaneous guided modes. Each guided mode has its own distinct velocity and can be further decomposed into orthogonal linearly polarized components. The optical wave is effectively confined within the guide and the electric field distribution in the x direction does not change as the wave propagates in z direction. Department of Telecommunication, MUET UET Jamshoro 13

14 Optical mode Department of Telecommunication, MUET UET Jamshoro 14

15 Mode Types Three types of modes: Guided modes-modes that are trapped in the core. Radiation modes or cladding modes-modes that are trapped in cladding. The radiation field basically results from the optical power that is outside the fiber acceptance angle being refracted out of the core. As the core and cladding modes propagate along the fiber, mode coupling occurs between the cladding and the higherorder modes. Leaky modes- these are partially confined to the core region and attenuate by continuously radiating their power out of the core as they propagate along the fiber. Department of Telecommunication, MUET UET Jamshoro 15

16 Order of the modes It is equal to the number of field zeros across the guide. Department of Telecommunication, MUET UET Jamshoro 16

17 Propagation Constant Propagation vector is defined as: k=2π/λ, where is λ the optical wavelength in vacuum. It is also called wave number. The term wave number refers to the number of complete wave cycles of an electromagnetic field that exist in one meter (1 m) of linear space. Wave number is expressed in reciprocal meters (m -1 ). The propagation constant of an electromagnetic wave is a measure of the change undergone by the amplitude of the wave as it propagates in a given direction. The propagation constant of a mode in a waveguide (e.g. a fiber), determines how the amplitude and phase of that light with a given frequency varies along the propagation direction z. Department of Telecommunication, MUET UET Jamshoro 17

18 Propagation Constant As the refractive index within the guide is n 1, the optical wavelength in this region is reduced to λ/n 1, while the vacuum propagation constant is increased to n 1 k. The plane wave can be resolved into two component plane waves propagating in the z and x directions, as shown in figure on next slide. The component of the plane wave in the x direction is reflected at the interface between the higher and lower refractive index media. Order of mode: It is equal to the number of field zeros across the guide. Department of Telecommunication, MUET UET Jamshoro 18

19 Optical mode Figure 2.8 The formation of a mode in a planar dielectric guide: (a) a plane wave propagating in the guide shown by its wave vector or equivalent ray the wave vector is resolved into components in the z and x directions; (b) the interference of plane waves in the guide forming the lowest order mode (m = 0) Department of Telecommunication, MUET UET Jamshoro 19

20 Optical mode Department of Telecommunication, MUET UET Jamshoro 20

21 TE and TM modes Maxwell's equations describe electromagnetic waves or modes as having two components. The two components are the electric field, E(x, y, z), and the magnetic field, H(x, y, z). The electric field, E, and the magnetic field, H, are at right angles to each other. Modes traveling in an optical fiber are said to be transverse. The transverse modes propagate along the axis of the fiber. In TE modes, the electric field is perpendicular to the direction of propagation and hence E z =0. In TM modes, the magnetic field is perpendicular to the direction of propagation (H z =0) and a component of electric field is in the direction of propagation. Department of Telecommunication, MUET UET Jamshoro 21

22 Plane Waves The mode theory suggests that a light wave can be represented as a plane wave. A plane wave is a wave whose surfaces of constant phase are infinite parallel planes normal to the direction of propagation. A plane wave is described by its direction, amplitude, and wavelength of propagation. The planes having the same phase are called the wavefronts. The wavelength λ of the plane wave is given by: c nf Department of Telecommunication, MUET UET Jamshoro 22

23 Plane Waves Where c is the speed of light in a vacuum, f is the frequency of the light, and n is the index of refraction of the plane-wave medium. Department of Telecommunication, MUET UET Jamshoro 23

24 Wavefront A wavefront is defined as the locus of all points in the wave train which have the same phase. Wavefronts are drawn passing through maxima or minima of the wave, such as the peak or trough of a sine wave. Thus the wavefronts are separated by one wavelength Department of Telecommunication, MUET UET Jamshoro 24

25 Cutoff Wavelength Propagation constant (β) as already defined is function of the wavelength and mode. nk n (2 ) / As the wavelength (λ) changes, the value of the propagation constant must also change. For a given mode, a change in λ can prevent the mode from propagating along the fiber. The mode is no longer bound to the fiber. The mode is said to be cut off. Modes that are bound at one wavelength may not exist at longer wavelengths. Department of Telecommunication, MUET UET Jamshoro 25

26 Cutoff Wavelength The wavelength at which a mode ceases to be bound is called the cutoff wavelength for that mode. Or the wavelength that prevents the next higher mode from propagating is called the cutoff wavelength of the fiber. However, an optical fiber is always able to propagate at least one mode. This mode is referred to as the fundamental mode of the fiber. The fundamental mode can never be cut off. Single mode fiber operates at above cutoff wavelength. Multimode fiber operates below the cutoff wavelength. Department of Telecommunication, MUET UET Jamshoro 26

27 Cladding Mode The modes are not confined to the core of the fiber but extend partially into cladding. Low-order modes penetrate the cladding only slightly. In low-order modes, the electric and magnetic fields are concentrated near the center of the fiber. However, high-order modes penetrate further into the cladding material. In high-order modes, the electrical and magnetic fields are distributed more toward the outer edges of the fiber. This penetration of low-order and high-order modes into the cladding region indicates that some portion is refracted out of the core. The refracted modes may become trapped in the cladding due to the dimension of the cladding region. The modes trapped in the cladding region are called cladding modes. Department of Telecommunication, MUET UET Jamshoro 27

28 Mode Coupling As the core and the cladding modes travel along the fiber, mode coupling occurs. Mode coupling is the exchange of power between two modes. Mode coupling to the cladding results in the loss of power from the core modes. A mode remains bound if the propagation constant β meets the following boundary condition: 2 n 2 n 2 1 Department of Telecommunication, MUET UET Jamshoro 28

29 Normalized Frequency The normalized frequency (a dimensionless quantity) determines how many modes a fiber can support. Normalized frequency (also called V parameter) is defined as: 2 a N n n f Where n 1 is the core index of refraction, n 2 is the cladding index of refraction, a is the core radius, and λ is the wavelength of light in air. The number of modes that can exist in a fiber is a function of N f. As the value of N f increases, the number of modes supported by the fiber increases. Department of Telecommunication, MUET UET Jamshoro 29

30 Normalized Frequency Department of Telecommunication, MUET UET Jamshoro 30

31 Example- No. of modes at 850 nm Diameter (microns Step- Index modes Graded- Index modes E3 22E3 92E3 2.4E E3 46E3 1.2E6 From the V parameter, we see that we can reduce the number of modes in a fiber by reducing: (1) NA (2) diameter (w.r.t. λ) This is exactly the case in single mode fibers. Department of Telecommunication, MUET UET Jamshoro 31