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InterviewSolution
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Derive the equation for acceptanc angleand numerical aperture, of optical fiber. Acceptance angle in optical fibre: |
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Answer» Solution :To ensure the cirtical angle INCIDENCE in the core - cladding boundary INSIDE the optical fibre, the light should be incident at a certain angle at the END of the optical while entering in to it. This angleis called acceptance angle. It dependson the refractiveincidens of the core`n_(1),` cladding `n_(2)` and the outer medium `n_(3)`, Assume the light is incident at an angle called acceptance angle `i_(a)` at the outer medium and core bounadary at A. The Snell.s law in the product form, equation for this refraction at the POINT A.   `n_(3) sin i_(a) = n_(1) sin r_(a)` To have the total interal reflection inside optical, fibre, the angle of incidence at the core-cadding INTERFACE at B should be atleastcritical angle `i_(c)`,. Snell.s law in the product form, equaiton for the refractionat point B is, `n_(1) sin i_(c) = n_(2) 90^(@)` `n_(1) sin i_(c) n_(2) "" therefore sin 90^(@) = 1` `therefore sin i_(c) n (n_(2))/(n_(1))` From the right angle triangle `DeltaABC`, `i_(c) = 90^(@) - r_(a)` Now, equation (3) becomes, `sin (90^(@) - r_(a)) = (n_(2))/(n_(1))` Using trigonometry, `cos r_(a) = (n_(2))/(n_(1))` `sin r_(a) = sqrt(1 - cos^(2) r_(a))` Substituting for `cos r_(a)` `sinr_(a)=sqrt(1-((n_(2))/(n_(1)))^(2))=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(1)^(2)))` Substituting this in equation (1) `n_(3)sini_(a)=n_(1)=n_(1)sqrt((n_(1)^(2)-n_(2)^(2))/(n_(1)^(2)))=sqrt(n_(1)^(2)-n_(2)^(2))` On further simoplificaiton, `sini_(a)=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)))(or)sini_(a)=sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)^(2))` `i_(a)=sini^(-1)(sqrt((n_(1)^(2)-n_(2)^(2))/(n_(3)^(2))))` If outer medium is air, then `n_(3) = 1`. The acceptance angle `i_(a)` becomes, `i_(a)=sini^(-1)(sqrt(n_(1)^(2)-n_(2)^(2)))` Light can haveany angle of incidence from o to `i_(a)` with the normal at the end of the optical fibre forming a conical shape called acceptancecone. In the equation (6), the term `(n_(3)sini_(a))` `NA=n_(3)sini_(a)sqrt(n_(1)^(2)-n_(2)^(2))` |
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