Stilwell 



1 - 1^ ^ P(t) 9 = 1 -2^ D p(;|,) cp (9) 



The light amplitude can be seen to consist of a constant term plus 



a term proportional to the normal angle of the ocean wave. The 



constant amplitude term will transform into a finite aperture 



equivalent of a delta function and will contribute to the transform 



light intensity only near the region of zero spatial freq^uency and 



can be excluded from further consideration. The Fourier transform 



is then performed on the normal angle as is desired. The higher 



order terms ignored in writing equation (9) are less than a tenth 



the first order term for usual values of D with wave angles up to 



9 

 about ^0 degrees. The resultant error in the final spectrum will 



then be less than 5^. 



Taking the light amplitude in the output plane as propor- 

 tional to the Fourier transform one can write 



.^ '- f^ (10) 



where the constant q. relates to the proportionality of the trans- 

 form (the light amplitude distribution being proportional to the 

 Fourier transform) and the constant e relates to the proportionality 

 of power density and light amplitude, u^n is the intensity which 



will expose the film placed in the output plane to record the 

 spectra, q can be evaluated by knowing a transform pair for the 

 optical system. A convenient pair is just the gaussian shape of 

 the laser amplitude with off -axis position which transforms into 

 a gaussian shape. Then invoking the requirement that the power 

 flow in the input and output planes must be identical, one obtains 



2jt CO 2jt <» 



J J ^ exp [- ^ } rdrde = J J q2(2«)2 !o ^4 

 e^ 00 ^ 



exp f- 72^2} ^d^de (11) 



? = (I^)""- r (12) 



where 



and then 



q. = (L\)"i 



175 



