Ch4Sec2

# Ch4Sec2 - THE FIBER FORUM Fiber Optic Communications...

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THE FIBER FORUM Fiber Optic Communications JOSEPH C. PALAIS PRESENTED BY

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Joseph C. Palais 4.2 2 Section 4.2 MODES IN THE SYMMETRIC-SLAB WAVEGUIDE n 2 n 2 n 1 n 1 > n 2 θ Range of θ for bound waves: ° 90 θ c
Joseph C. Palais 4.2 3 Mode in The Symmetric-Slab Waveguide For an axial ray, θ = 90 ° and n eff = n 1 For a critical angle ray θ = θ c and 2 1 2 1 1 sin n n n n n n c eff = = θ The range of n eff is now (4.9) 1 2 n n n eff Define the effective index of refraction: sin 1 n n eff

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Joseph C. Palais 4.2 4 4.2.1 Mode Condition The two boundaries (upper and lower) form a cavity. A cavity is resonant when the round trip phase shift is 2 π m (where m = 1,2,3 …). Expressing this mathematically, we get ∆ φ = m2 π (4.10)
Joseph C. Palais 4.2 5 Mode Condition ∆ φ is the phase shift for one complete cycle of the zigzag path. It includes the phase shift at the two totally reflected surfaces. Equation (4.10) can be satisfied (for a given λ ) for several distinct ray angles (between θ c and 90º). The waves traveling at these allowed angles are the modes of the waveguide. Waves traveling at other angles will interfere destructively and will not propagate.

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Joseph C. Palais 4.2 6 4.2.2 and 4.2.3 TE and TM Polarization and TE Mode Chart Derivation of fields in the symmetric waveguide: E d n 2 n 2 n 1 z y 1 2 0 2 0 2 1 0 1 k k n k k n μ = = = = TE polarization (Transverse Electric) This is the same as perpendicular polarization (s). z θ
Joseph C. Palais 4.2 7 TE Mode Chart Consider a TE wave in the middle layer where: ( ) ˆ j t k r x E u E e ϖ - = r v y z k 1 θ (1) 1 1 0 1 n c n k k = = z z y y x x k u k u k u k ˆ ˆ ˆ + + = 2 2 2 z y x k k k k + + = z u y u x u r z y x ˆ ˆ ˆ + + =

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Joseph C. Palais 4.2 8 TE Mode Chart Example: ) sin cos ( 1 1 ˆ θ ϖ z k y k t j x e E u E - - + + = This represents the upward traveling wave. If θ = 90º this reduces to the equation of a z- directed wave. (2) sin ˆ cos ˆ 1 1 1 k u k u k z y + = z y x u z u y u x r ˆ ˆ ˆ + + = z k 1 θ y
Joseph C. Palais 4.2 9 y z k 1 θ TE Mode Chart For the downward wave - + - + - - = = E E e E u E z k y k t j x ) sin cos ( 1 1 ˆ θ ϖ (3) Because of total reflection the magnitudes are equal

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Joseph C. Palais 4.2 10 TE Mode Chart The total field is (5) (4) 1 1 1 sin cos cos jk z jk y jk y j t E E E E e e e e θ ϖ + - - - + = + = + ) cos cos( 2 1 sin 1 y k e e E E z jk t j - + = Define β sin , cos 1 1 k k h + E E 2 1
Joseph C. Palais 4.2 11 TE Mode Chart (6) 0 1 0 1 cos , sin h k n k n θ β = = Then 2 2 - d y d (7) ) ( 1 ) cos( z t j e hy E E ϖ - = This equation is valid in the middle layer where

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Joseph C. Palais 4.2 12 TE Mode Chart This is a plane wave traveling in the z-direction with propagation constant β . The amplitude varies spatially sinusoidally in the transverse (y) direction.
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## This note was uploaded on 05/11/2010 for the course EEE EEE-448 taught by Professor Palais during the Fall '09 term at ASU.

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Ch4Sec2 - THE FIBER FORUM Fiber Optic Communications...

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