Backscattering patterns of step index optical fibers: angular width

P. S. Ho; M. E. Marhic; M. Epstein

doi:10.1364/AO.14.002598

Applied Optics
Vol. 14,
Issue 11,
pp. 2598-2601
(1975)
•https://doi.org/10.1364/AO.14.002598

Backscattering patterns of step index optical fibers: angular width

P. S. Ho, M. E. Marhic, and M. Epstein

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P. S. Ho, M. E. Marhic, and M. Epstein, "Backscattering patterns of step index optical fibers: angular width," Appl. Opt. 14, 2598-2601 (1975)

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Abstract

Angular widths of laser backscattering patterns from optical fibers with thin claddings due to rays reflected by core-cladding interface are investigated. Calculated and measured values are presented and found to be in good agreement. A summary of different backscattering patterns from clad fibers is given.

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Ray Path	Deviation angle θ	Maximum incident angle i_m	Maximum backscattered angle ϕ_m (= π − θ_m)
I	π + 2i − 4γ	$\frac{2 cos i_{m}}{{({n_{1}}^{' 2} - {sin}^{2} i_{m})}^{½}} = 1$	$4 {sin}^{- 1} [\frac{2}{{n_{1}}^{'} {(3)}^{½}} {(1 - \frac{{n_{1}}^{' 2}}{4})}^{½}] - 2 {sin}^{- 1} [\frac{2}{{(3)}^{½}} {(1 - \frac{{n_{1}}^{' 2}}{4})}^{½}]$
II	π + 2i − 4γ₂ + 4i′ − 4γ₁	$\frac{2 cos i_{m}}{{[{(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - {sin}^{2} i_{m}]}^{½}} - \frac{2 cos i_{m}}{{[{(\frac{{n_{1}}^{'} r_{2}}{r_{1}})}^{2} - {sin}^{2} i_{m}]}^{½}} + \frac{2 cos i_{m}}{{({n_{1}}^{' 2} - {sin}^{2} i_{m})}^{½}} = 1$	$4 {cos}^{- 1} {\frac{2}{{n_{2}}^{'} {(3)}^{½}} {[{n_{2}}^{' 2} - {(\frac{r_{1}}{r_{2}})}^{2}]}^{½}} - 2 {cos}^{- 1} {\frac{1}{{(3)}^{½}} {[{(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - 1]}^{½}} - 4 {cos}^{- 1} {\frac{1}{{n_{1}}^{'}} {[{n_{1}}^{' 2} + \frac{1}{3} {n_{2}}^{' 2} - \frac{4}{3} {(\frac{r_{1}}{r_{2}})}^{2}]}^{½}} + 4 {cos}^{- 1} {\frac{1}{{n_{1}}^{'}} {[{n_{1}}^{' 2} + \frac{1}{3} {(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - \frac{4}{3}]}^{½}}$
III	π + 2i − 4γ₂ + 2i′ − 2γ₁	$\frac{2 cos i_{m}}{{[{(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - {sin}^{2} i_{m}]}^{½}} + \frac{cos i_{m}}{{({n_{1}}^{' 2} - {sin}^{2} i_{m})}^{½}} - \frac{cos i_{m}}{[{(\frac{{n_{1}}^{'} r_{2}}{r_{1}})}^{2} - {sin}^{2} i_{m}] ½} = 1$	$4 {cos}^{- 1} {\frac{2}{{n_{2}}^{'} {(3)}^{½}} {[{n_{2}}^{' 2} - {(\frac{r_{1}}{r_{2}})}^{2}]}^{½}} - 2 {cos}^{- 1} {\frac{1}{{(3)}^{½}} {[{(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - 1]}^{½}} - 2 {cos}^{- 1} {\frac{1}{{n_{1}}^{'}} {[{n_{1}}^{' 2} + \frac{1}{3} {n_{2}}^{' 2} - \frac{4}{3} {(\frac{r_{1}}{r_{2}})}^{2}]}^{½}} + 2 {cos}^{- 1} {\frac{1}{{n_{1}}^{'}} {[{n_{1}}^{' 2} + \frac{1}{3} {(\frac{{n_{2}}^{'} r_{2}}{r_{1}})}^{2} - \frac{4}{3}]}^{½}}$

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Applied Optics