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# Lecture2 - Wave Equations Guided Modes Step-Index Fibre...

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1 Wave Equations; Guided Modes; Step-Index Fibre; Graded-Index Fibre; Single-Mode Fibre EE4035 Optical Communications Semester A 2010-11 Lecture 2

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2 Intended Learning Outcomes (ILOs) Derive the scalar wave equation from the vector wave equation. Describe the general properties of guided modes. Solve the scalar wave equation for the LP modes of a step-index fibre and express the solutions in terms of normalized parameters. Sketch the field distributions of the LP modes. Determine the effective indices (or mode indices) of the LP modes from the universal dispersion curves. Determine the cutoff conditions of the LP modes of a step-index fibre. Estimate the number of guided modes in a multimode fibre. Explain the concepts of modal noise and mode coupling in a multimode fibre. Determine the single-mode conditions of a step-index fibre and a power- law graded-index fibre. Evaluate the mode-field diameter (or spot size) of a single-mode fibre. Explain the bend loss in a single-mode fibre.
3 Vector Wave Equation Consider only steady-state solutions, i.e., the field quantities vary sinusoidally with time with angular optical frequency ω : () t j z y x E t z y x E ω exp ) , , ( ) , , , ( v v = t j z y x H t z y x H exp ) , , ( ) , , , ( v v = Wave propagation in an optical fibre is in general described by the following vector wave equations: −∇ = + r r E E k n E ε v v v 2 2 2 (1) H H k n H r r v v v × × = + 2 2 2 (2) E v : Electric field in Volt/metre (a vector) H v : Magnetic field in Ampere-turn/metre (a vector) ε r : Relative dielectric constant (a function of space) r n = : Refractive-index profile (a function of space) λ π 2 = k : Free-space wavenumber ( : optical wavelength)

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4 Scalar Wave Equation For most practical fibres, the weak guidance assumption is satisfied: 1 ~ 2 1 2 1 2 1 2 2 2 1 << = Δ n n n n n n (3) n 1 : Peak index in the core n 2 : Cladding index The right-hand side of (1) or (2) is of order , and, therefore, can be neglected for most practical applications. This leads to the scalar wave equations: 0 0 2 2 2 2 2 2 = + = + H k n H E k n E v v v v Each field component satisfies the scalar wave equation. We can thus write 0 2 2 2 = + s s k n ψψ where the scalar field Ψ s can represent any component of the electric or magnetic field. (4)
5 Scalar Wave Equation The wave that propagates in an optical fibre can be expressed as ( ) z j y x z y x s β ψ = exp ) , ( ) , , ( (5) The scalar wave equation becomes ( ) 0 2 2 2 2 T = + ψβ k n (6) where T is the transverse component of and is the propagation constant, which is real for a lossless fibre. Rectangular Coordinates () 0 2 2 2 2 2 2 2 = + + ψψ k n y x (7) x y z r φ ( ) 0 1 1 2 2 2 2 2 2 2 2 = + + + φ k n r r r r Cylindrical Coordinates (8)

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6 Guided Modes A guided mode is a light wave that propagates along the fibre at a constant phase velocity and is characterised by a field distribution across the fibre cross-section. For weakly guiding fibres, the phase velocity and the field distribution can be obtained by solving the scalar wave equation. The propagation constant of a guided mode must
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Lecture2 - Wave Equations Guided Modes Step-Index Fibre...

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