Diffraction model of a step-out transition for a sheet beam in planar geomet~

III. Outlook Using a diffraction model, we derive the longitudinal highIf the step d is not small, we must include the interference of frequency impedance of a small step-out transition for a sheet the two waves diffracted at the lower and upper edge and also beam in planar geometry. the multiple reflections. In this case, the bound~ conditions can still be satisfied in a diffraction model, namelv bv intro1. Sheet Beam and Planar Impedance Consider an infinitely wide ‘sheet beam’ between two perfectly conducting planar surfaces, each having a discontinuous step outward. Denoting the current and the power loss per horizontal unit length by dJ/dz = d~/dx exp (–iw (t – SIC)) and dP/dx, respectively, we define the real part of the longitudinal planar impedance 21 by the equation: dP () dJ 2 — = Re2) ~ dx (1) In our convention the units of the longitudinal planar imPe&nce;, Z{, are Qm. (The tilde over 2} emphasizes these peculiar units.) The electromagnetic fields accompanying the beam are: BOZ = –Eov = *2(T/c) dJ/dx, where the two signs correspond to the regions above and below the sheet beam, respectively. The fields are independent of the vertical beam size, and are identical to those of a plane wave. The incident energy flux is F. = c(B~Z + E~V)/(8m) = n/c (dJ/dx)2. II. Planar Impedance for a Small Step Out . . ducing an infinite set of image currents of alternating polarity, which are spaced a distance A s (2b + 2d) apart. The calculation is simplified by assuming that the step-out is symmetric and that the beam is centered between the two conducting surfaces. Note that, according to the planar w~e theorem [4], the energy loss of the beam is independent of its vertical offset. Instead of the situation just described, Babinet’s principle allows us to consider the complementary problem of an infinite number of plane waves of alternating phase impinging on an infinite grid of slits of width 2d separated by opaque screens of size 2b. The amplitude of the difiacted wave is now proportional to a(y, $) N ~ (-1)” /b+2d dy’ eiWDnfC (3) . n=—m t, where D. = ~s2+(nA+y–y’)2. (4) Noting that at sufficiently large distances s behind the step we have Iy – y’ I << ~s2 + n2 A2, the function D. in Eq. (3) can . be approximated as Dn=~+ 2nA(y – y’) + (y – y’)2 . (5) 2~s2 + n2A2 Let b be the initial transverse distance between the sheet It is not obvious how to fufiher simplify the resulting expresbeam and the upper (or lower) boundary surface. Suppose that sions at location s = O both surfaces undergo a step-out transition Finally, it is interesting to notice that the conventional of size d. If d is small (d << b), we cm ignOre multiple rediffraction model for a cylindrical georne~ [2,3]gives correct flections and the interference of waves diffracted at the upper results for the high-frequency impedance of a deep cavity, aland lower edge, and we can directly apply the reSUltSof Ref. thought~s model does include neithermultiple r flections nor [1]. There, we extended the conventional diffraction model for interference of waves diffracted at different azimuthal locations. a round beam passing a cavity in a circular pipe [2, 3] to the case of a small step-out, by introducing a single image current of opposite polarity at a transverse distance A = (2b + 2d) from the beam. From Ref. [1] we infer that the beam loss power for [1] our sheet beam is dP/dx w 4dFo. Comparison with Eq. (1) yields Re2} = Zod (2) [2] where 20 = 4T /c (= 377Q) denotes the vacuum impedance.This may be compared with the high-frequency impedance of a[3]small transition step for a circular beam in a cylindrical beam [4]11 Zod/(mb) [1].pipe of radius b, which is ReZo NREFERENCES A. Chao and F. Zimmerman, “Diffraction Model of a StepOut Transition”, presented at EPAC96, Sitges (Barcelona), andSLAC-PUB-7140 (1996).J.D. Lawson, Rutherford Lab. Report RHEW 144 (1968), Part.Accel. 25, 107 (1990).K. Bane and M. Sands, Part. Accel. 25,73 (1990).A.W. Chao and K.L.F. Bane, these Snowmass proceedings(1996).