A beam of light of intensity `I_(0)` falls normally on a transparent plane-parallel plate of thickness `l`. The beam contains all the wavelengths in the interval from `lambda_(1)` to `lambda_(2)` of equal spectral intensity. Find the intensity of the transmitted beam if in this wavelength interval the absorption coefficient is a linear function of `lambda`, with exterme values `x_(1)` and `x_(2)`. The coefficient of reflection at each surfcae is equal to `rho`. The secondary reflections are to be neglected.

Explanation

The spectral density of the incident beam (i.e. intwnsity of the components whose wave length lies in the interval `lambda & lambda + d lambda)` is
`(I_(0))/(lambda_(2) - lambda_(1)) d lambda_(1) le lambda le- lambda_(2)`
The absorption factor for this component is
`e^(-[chi_(1)+(lambda+lambda_(1))/(lambda_(1)-lambda_(1))(chi_(1) - chi_(1))]l`
and the transmission factor due to reflection at the surfcaes is `(1 - rho)^(2)`. Thus the intensity of the transmitted beam is
`(1 -rho)^(2)(I_(0))/(lambda_(2)-lambda_(1)) underset(lambda_(1))overset(lambda_(2))int d lambda e^(-l[chi_(1)+(lambda+lambda_(1))/(lambda_(1)-lambda_(1))(chi_(1) - chi_(1))]`
`= (1 - rho)^(2) (I_(0))/(lambda_(2) - lambda_(1)) e^(-chi_(1)l) ((1-e^(-(chi_(2)-chi_(1))l))/((chi_(2)-chi_(1))'me)) chi(lambda_(2) - lambda_(1)) = (1- rho)^(2)I_(0) (e^(-chi_(1)l) -e^(-chi_(2)l))/((chi_(2)-chi_(1))l)`