Two coherent sources of light of intensity ratio `beta` produce interference pattern. Prove that in the interferencepattern
`(I_(max) - I_(min))/(I_(max) + (I_(min))) = (2 sqrt beta)/(1 + beta)`
where `I_(max)` and `I_(min)` are maximum and mininum intensities in the resultant wave.

Explanation

If `a_(1), a_(2)` are amplitudes of superposing waves and `I_(1), I_(2)` are intensities, than
`beta = (I_(1))/(I_(2)) = (a_(1)^(2))/(a_(2)^(2))` or `(a_(1))/(a_(2)) = sqrt beta`
`:. I_(max) = a_(1)^(2) + a_(2)^(2) + 2a_(1) a_(2) = (a_(1) + a_(2))^(2)`
and ` I_(min) = a_(1)^(2) + a_(2)^(2) + 2a_(1) a_(2) = (a_(1) - a_(2))^(2)`
`implies(I_(max) - I_(min))/(I_(max) + I_(min)) = ((a_(1) + a_(2))^(2) - (a_(1) - a_(2))^(2))/((a_(1) + a_(2))^(2) + (a_(1) -a_(2))^(2)`
`= (4 a_(1) a_(2))/(2(a_(1) + a_(2)^(2))) =((2(a_(1))/(a_(2))))/(((a_(1)^(2))/(a_(2)^(2)) +1)) = (2 sqrt beta)/((1 + beta)`.