K. Minami

T. Watanabe

V.L. Granatstein

Y. Choyal

National Institute for Fusion Science (NIFS-Repository)

• §3 0. The Range of Val idity of the Rayleigh Hypothes isl) Watanabe, T., Choyal, Y. (Devi Ahi lya Univ. indore), Minami , K. (Niigata Uillv.), Granatstein , V.L. (Uni v. of Maryland) The parameter range over which the Rayleigh hypoth-esis (RR) for optical gratings might be validly applied to analysis of high power bad.-ward wave oscillators (BWOs) has been investigated nnmerically. The field patterns and ei.gen frequency for the slow wave struc-tme (SWS) are solved numerically by using code HIDM (Higher Order Implicit Difference Method) that is free from RH. For a deep corrugation) h [<0 = 5 x 0.448 , using RH is still valid for obtaini ng the dispersion rela-tion , although the Floquet Harmonic Expansion (FHE) fails to correctly represent the field patterns inside the corrugation, here, hand 1(0 are, respectively, the am-pli tude and wave number of the axisymmetric SWS. It has been pointed out that from rigorous mathematical viewpoint , RH was only valid for shallow corrugations of SWS, h 1(0 < 0.448. New analytical Frame for Electromagnetic Field ~I agnetic field B can be expressed by vector potential A = (<»(" , e, z , t)jr, \]) (r, e, z, t)fT, 0) without loss of gen-erality, because of \7 . B = O. We choose Az = O. Fara-day' law . 0 = at! + \7 x E , gives E = - 001" - V ),. Scalar potential ), is possible to be solved from the z component of the Ampere' law, (1) If we substitute this gauge expression for the scalar po-tent ial ), into the (T, e) component of Ampere law, we can get the following equations for the numerical com-putation. a2 <» <0110 at 2 a2 \]) Eotlo at 2 - 2 a\]) J (aJz aJr) L'> <» - ?"-.- - 110" - - - - dz ,.- ae ar az - 2 a<» J (1 aJz aJe) L'> \]) + - - - 110r -- - - dz r·2 ae T ae az The operator L'> is defined by the following expTession. - a (laJ ) 1 a 2 J a 2J L'>J(r'.O,z,t)=r'-a --a +?"an o + a o' r T r r- (7 - z -(2) For the vacuum field case, J = O. Boundary conditions on SWS with inner radius r = X (z) given by E x V (r -X (z)) = 0 are reduced to the following equations, { 1 a2<» a 2<» 1 a2\])} {1 a2\]) a2<» } X' (.) r2 ae2 + az2 - ;: aear = ;: aeaz + araz 206 a 2\]) 1 a 2\]) 1 a2<» <0110 at2 + r2 ae2 -;: araO = O . The boundary condition for the normal component of magnetic field B, V (r' - X (z)) = 0 is automatically satisfied, N ulllerical Treatment for Eigemffiode Analysis "Ve solve eigenmodes of electromagnetic field in SvVSs with periodic inner radius numerically ( periodicity = L. X (z + L) = X (z) ). { <» } = { x(r, z) } exp(tee + ,kll z _ twt) (3) \]) '7(r,z) where (kill l, W) are wave numbers and a frequency of eigenmode ( e is arbitrary integer ) . x(r, z) and ry(r, z) are periodic function in z wi th periodicity L . (4) (5) Similar equations hold for \]). To treat the boundary conditions on the SWS in high accuracYl we introduce a curvilinear coordinate system (~(", TjX (Z),z) instead of (r, z) . The inner wall of SWS is specified by ~ = l. For z-differentiation: we introduce very high order differ-ence scheme. Reduced simultaneous ordinary differential equation system in ~ direction is numerically solved by HID r..II. Numerical example is shown in Fig.l. Fig.l Fie ld pattern of quasi-TM mode (€ = 1) in SWS. Reference 1) Watanabe, T. , et.a!. Physical Review E. 69 (200-!) 056606 , 

The Range of Validity of the Rayleigh Hypothesis

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