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The refractive index of carbon dioxide at the wavelengths 509, 534 and 589 nm is equal to 1.647, 1.640, and 1.630 respectively. Calculate the phase and group velocities of light in the vicinity of lambda = 534 nm. |
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Answer» Solution :The PHASE velocity of light in the vicinity of `lambda = 534 NM = lambda_(0)` is obtained as `v(lambda_(0)) = (c )/(n(lambda_(0))) = (3 xx 10^(8))/(1.640) = 1.829 xx 10^(8) m//s` To get the group velocity we need to calaculate `((dn)/(d lambda))_(lambda = lambda_(0))`. we shall use linear interpolation in the two intervals. Thus `((dn)/(d lambda))_(lambda = 521.5) =-(007)/(25) =- 28 xx 10^(-5) pernm` `((dn)/(d lambda))_(lambda = 561.5) =-(01)/(55) =- 18.2 xx 10^(-5) pernm` There `(dn//d lambda)` values have been assigned to the mid-points of the two intervals. INTERPOLATING again we get `((dn)/(d lambda))_(lambda = 534) = [-28 + (9.8)/(40) xx 12.5] xx 10^(-5) per nm =- 24.9 xx 10^(-5) per nm`. FINALLY `u=(c )/(n)-lambda(d)/(d lambda) ((c )/(n)) = (c )/(n) [1+(lambda)/(n) ((dn)/(d lambda))]` At `lambda = 534` `u = (3 xx 10^(8))/(1.640) [1- (534)/(1.640) xx 24.9 xx 10^(-5)] m//s = 1.59 xx 10^(8) m//s` |
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