The Stream Guide, a Simple, Low-Loss Optical Guiding Medium

In this equation, a is the radius of the pipe, no the refractive index at atmospheric pressure p0, * the ratio of the specific heats {cp/cv), R (^0.90) the so-called recovery factor,* p x , M x and M(z) the pressure *Sre p. 211 of Ref. 4. 761 762 T H E BELL SYSTEM T E C H N I C A L J O U R N A L , M A Y - J U N E 1968 and the Mach number at the beginning and along the pipe, respectively. Provided that the absolute velocity is small compared with the velocity of sound, and that simultaneously the relative change of the velocity along the guide also is kept small, it can be shown t h a t the eikonal equation yields as ray path r{z): r(z) = (1 + /te)»[c, | j-j (1 + /to)1) + CJ^! | (1 + /3*)1)] (2) with a = Ml |(n{) _ 1)(k _ T and p = h Ml a L PoJ a whereby JI/3(.T) and Ni/a(x) are the Bessel and Neumann functions of fractional order 1/3, Ci and C 2 are integration constants and / represents the average friction coefficient. Equation 2 describes an undulating trace with steadily decreasing amplitude and period as the result of the ever increasing velocity and, consequently, focusing strength downstream. If one considers the flow parameters to be locally independent of z one obtains for the period of the beam size variation Lb(z) (with R = 0.90; see Ref. 6): Lb(z) = 3.3a[(n 0 - 1)(k - 1 )M,M(z) (3)