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Light guidence in an optical fiber can be understood by considering a structure comparising of thin solid glass cyclider of refractive index n_(1) surrounded by a medium of lower refractive index n_(2). The light giudence in the structure takes place due to successive total internal reflections at the interface of media n_(1) and n_(2) as shown in Fig. All rays with the angle of nicidence i less than a particular value i_(m) are confined in a medium of refractive index n_(1). The numerical aperture (NA) of the structure is defined as sin i_(m). For two structure namely S_(1) with n_(1) = (sqrt(45))/(4) and n_(2) = (3)/(5), and S_(2) with n_(1) = (8)/(5) and n_(2) = (7)/(5) and taking refractive index of water to be (4)/(3) and their of air to be 1, the correct option (s) is (are). |
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Answer» `NA` of `S_(1)` immersed in WATER is same as that of `S_(2)` immersed in a liquid of refractive index `(16)/(3sqrt(15))`, As `n_(s) sin i_(m) = n_(1) sin(90^(@) - C)`…(i) ALSO, `sin C = (1)/(mu) = (n_(2))/(n_(1))`…(ii) As `NA = sin i_(m)` From (i), `sin i_(m) = (n_(1))/(n_(s)) sin (90^(@) - C)` `= (n_(1))/(n_(s)) cos C = (n_(1))/(n_(s)) sqrt(1 -sin^(2) C)` `:. NA = (n_(1))/(n_(s))sqrt(1 - (n_(2)^(2))/(n_(1)^(2))) = (1)/(n_(s))sqrt(n_(1)^(2) - n_(2)^(2))` [from (ii)] For `S_(1)` in air, `n_(s) = 1, n_(1) = (sqrt(45))/(4), n_(1) = (3)/(2)` `NA = (1)/(1) = sqrt((45)/(16) - (9)/(4)) = (3)/(4)` `NA = (sqrt(15))/(6)sqrt((45)/(16) - (9)/(4))= (3sqrt(15))/(24) = (sqrt(15))/(8)` For `S_(1)` in water `NA = (1)/(4//3)sqrt((45)/(16) - (9)/(4)) = (3)/(4)((3)/(4)) = (9)/(16)` For `S_(2)` inair , `n_(s) = 1, n_(1) = (8)/(5), n_(2) = (7)/(5)` `NA = (1)/(1)sqrt((64)/(25) - (49)/(25)) = (sqrt(15))/(5)` For `S_(2)` in water, `n_(2) = (4)/(3)` `NA = (1)/(4//3)sqrt((64)/(25) - (49)/(25)) = (3)/(4)(sqrt(15))/(5)` For `S_(2)` in `n_(s) = (16)/(3sqrt(15))` `NA = (3sqrt(15))/(16)sqrt((64)/(25) - (49)/(25)) = (9)/(16)` For `S_(2)` in `n_(s) = (4)/(sqrt(15))` `NA = (sqrt(15))/(4)sqrt((64)/(25) - (49)/(25)) = (3)/(4)` Hence, options 'a' and 'c' are correct. |
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