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Unformatted text preview: 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 2 2 2 = + ∇ E n k E o 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 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 ≈ = Solutions 3 Then for r < a...
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This note was uploaded on 02/23/2011 for the course EE 474 taught by Professor Lingo during the Spring '11 term at USC.
 Spring '11
 Lingo
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