men IREQi-EXCV AMl'IJIIER 47 



where 



If we assume tliat the tube whicli surrounds the beam be taken as infinitely 

 remote, the appropriate solutions outside the beam are 



//^o = A,K,{yr) (A-21) 



and inside the beam 



j"^^ A,K,M (A-22) 



^vi = Aihi^r) (A-23) 



E,i - -i-'^^.li/oaO (A-24) 



\/L 



where 7- = (S^ - k'~ ^ (3'^ 



(A-25) 



The /'s and i^T's in equations (A-21)"(A-24) are modified Bessel functions.' 

 At the surface of the beam (r = a), one has the following two independent 

 boundary conditions 



B^i = H^o (A-26) 



^zi = -£ro (A-25a) 



which, using equations (A-21)-(A-24), yield 



VLh(^a) K,{ya) ^'''" ^ 



From equations (A-13), (A44), (A-15) and (A-24) one has 



^a = Z/3oa\/Z (A-28) 



ya = ZjSoa (A-29) 



If we now define a beam wavelength, X^ , by the relations 



/3o - f (A-30) 



and assume for the purpose of simplifying the calculation that in the ex- 

 pression for L in (A-20) 



wIbI = wIbI = W^ (A-31) 



^ See A Treatise on the Theory of Bessel Functions, G. N. Watson, Chapter 3. 



