Deriving Bandwidth Using Geometric Optics

1. Deriving Bandwidth Using Geometric Optics Prepared for The Handbook of Fiber Optic Data Communication Third Edition Carolyn DeCusatis The State University of New York, New Paltz

2. Key Concepts • Geometric Optics approximation • Derivation of Numerical Aperture • Derivation of Multipath Time Dispersion • Maximum Bit Rate and Bandwidth-Distance Product

3. A Slab Waveguide air n2 q n1>n2 f n2 a This is a 2 dimensional approximation If α=0, the ray pictured would have been axial. (It isn’t.) The ray pictured here is oblique, and goes bouncing down the slab. The critical ray is the largest a that will propagate down the slab by total internal reflection.

4. air n2 q n1>n2 f n2 a φ= π/2-θ < π/2- θc sin α = n1sin φ = n1cos θ for critical rays: sin α c = n1sin φ c = n1cos θ c n1sin θ c = n2 therefore cos θ c= therefore sin αc= = NA= numerical aperture

5. n1>n2 sin αc= = NA= numerical aperture The numerical aperture is the light gathering power of a microscope, or other lens system. The acceptance cone is half the numerical aperture.

6. Time dispersion in unclad fiber is large. The axial ray travels a distance l in The oblique ray travels a distance l in = = The arrival time difference is ΔT= The multipath time dispersion is = And, to a good approximation, the maximum bit rate, B, is related to the multpath time dispersion, which is related to the Bandwidth Δf

7. To a good approximation, B ≈ 2Δf , And the bandwidth distance product is (Δf)l≈ Fibre Channel distances when using multimode fiber optic cable

8. Conclusions • Geometric Optics can be used to approximate the path of optical rays in a fiber • Axial and oblique rays • Numerical Aperture is the light gathering power of a fiber • Acceptance angle is half the numerical aperture • Tradeoff between numerical aperture and bandwidth-distance product