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07 Optical Fibers and Dispersion

07 Optical Fibers and Dispersion - Circular Step Index...

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Circular Step Index Waveguides – Optical Fibers Core n 1 Cladding n 2 n 1 > n 2 The solution to this problem is best done in cylindrical coordinates The z component of E does not couple to the other components as the wave propagates, while the r and φ components are strongly coupled. We can write an equation for the z component of the electric field 0 2 2 2 = + E n k E o
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Solutions 1 ( , , ) ( ) ( ) ( ) ( ) and ( ) . . z j z j E r z R r Z z Z z e Ae c c β νφ φ φ φ = Φ = Φ = + The general solution for E z is of the form where Inside the core of the waveguide where is a Bessel function of the first kind of order ν and Outside the core where is a Bessel function of the second kind of order ν and ( ) ( ) R r J r ν κ = 2 2 2 2 o k n κ β = - ( ) ( ) R r K r ν γ = 2 2 2 2 o k n γ β = - ( ) J r ν κ ( ) K r ν γ
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Solutions 2 2 for large ( r) ( ) cos( ) 2 4 for large ( r) ( ) . 2 For the solutions to oscillatory we require r o core o clad J r r r e K r r k n k n ν γ ν νπ π κ κ κ πκ γ γ πγ β - - - =
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Solutions 3 Then for r < a ( ) + c.c.
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