A simple approach to mode analysis for parabolic waveguides

A simple method of mode analysis for parabolic waveguides, based on one-dimensional analytic continuation, is presented. The method gives essentially exact values for parabolic cylinder functions; i.e., the relative error in the computed result is on the order of the machine round-off. When supplemented with a Newton-Poisson shooting method and simple homotopy techniques, this continuation method can be used to find the transverse electric and transverse magnetic mode eigenvalues, and associated separation constants, for arbitrary parabolic domains. These methods are also used to compute a power-handling efficiency factor for a range of parabolic regions. >

In the following section, we show that solutions to (6) and (7) can be computed via one-dimensional analytic continuation. Section 111 discusses a Newton-Poisson shooting method for finding the separation constants, a, and eigenvalues, k , for fixed and qo. This method is easy to use and reveals some inaccuracies in previously published work. In values of the first nine TM modes and the first eight TE modes for to = 1 and 0.1 G T~ G 1, as illustrated in Figs. 6-9.
These figures show that the TM modal ordering given in [2, pp. 1401-14021 is incorrect (since the TM modes immediately following k31,k13 should be k,,, and k , rather than k22). Highly accurate polynomial/rational approximations of the separation constants and eigenvalues are also given in this section for the first three TM and TE modes for 0.1 Q T~ Q 1 and to = 1.
The last section of this paper considers selecting qo (for to =: 1) so as to optimize the power handling capability [3], [20] of the parabolic waveguide. Fig. 10 shows that there are two local maxima of the power handling efficiency factor, y , for 0 G 770 Q 1. As in [4], both of these maxima occur at qo values for which the second and third TE mode eigenvalues are equal. The largest of these y values is 0.4600, which occiirs at v0 = 1, and gives a symmetrical cross section to the parabolic waveguide. This compares well with y = 0.4653 for the 2: 1 rectangle and y = 0.4698 for an ellipse of eccentricity e = 19.8546.

(17)
where yn is the nth derivative of y at x = 0. We may assume that yo and y, are given (in fact y o = 1, y, = 0 generates the even, cosinelike solution to (16) as illustrated in Fig. 2; y o = 0, y l = l generates the odd, sinelike solution to (16)). The higher values of y, can be found from the recursion 1 4 y, = ---( n -2)(n -3)ynP4.
On finite precision computers, the expansion (17) provides accurate values for y ( x ) if x is small. However, as x increases, destructive cancellation of large positive and negative terms leads to an unnecessary loss of accuracy. For example, Fig. 3 shows that the use of (17) leads to divergence starting at about x = 1.8 for the even solution to (16) with a = l . This problem can be avoided by taking several small Taylor steps rather than one large step. This can be done by using the Taylor expansion of y about an arbitrary point xo: where y, now denotes the nth derivative of y at x = xo. By substituting x = x, + h into (16), the higher order values of yn can be generated by -Differentiating (19) with respect to h gives the series expansion for y,: hn-1 y,(x, + h ) = y 1 + y2h + . . ' + y, -+ . . . . (21) ( n -l ) ! Thus to evaluate y at x, given y(0) and y,(O), set h = x / N for N large enough so that Ihl is small, say Ih( Q 0.1; then evaluate y(h) and y,(h) by (19) and (21). Proceeding sequentially, evaluate y and y, at ih for 1 < i < N. The sums in (19) and (21) are truncated so that the pseudorelative error is less than a prescribed tolerance value E . That is, pick n large enough so that 1 IYn+lh"+ll. The above method is really just a one-dimensional version of analytic continuation [ll] and is sometimes referred to as a constant-step variable-order Taylor method [ 121. The accuracy of the analytic continuation method was checked by using the ordinary differential equation solver LSODE [13], which employs error monitoring procedures such as those described in [ 141. In all computations the analytic continuation method gave at least 11 digits of accuracy and because of its specialized nature was ten to 100 times faster than the LSODE solver.

NEWTON-POISSON SHOOTING METHOD
By combining features of the Poisson shooting method [151 with the vector form of Newton's method 1121, we can solve the second-order ordinary differential equations (12) and For vector valued functions, f: R" + R", Newton's method of solving f ( w ) = 0 consists of an iterative procedure: where O'ER" is given, and Vf =(df,/dw,) is Lie gradient matrix of f. This can be applied to the problem of solving for ( a , k), such that (12)-(15) are satisfied, by setting w = ( a , k) and letting

407
The values of f l and f 2 are easily computed by the methods of the previous section, and the derivatives in (27) can be approximated by using second-order central difference formulas. In this method, both the values . $, and vo and the initial conditions U , u x , U , and U , are fixed, so that (25?-(27) do not correspond exactly to the classical Newton-Poisson shooting method [15] in which only some of the initial conditions are specified.
The iteration (24) converges quadratically to the exact solution, provided that the initial guess, wo = ( a o , ko), is sufficiently close to the exact solution. It is the invertibility of the gradient matrix (see the Appendix) which accounts for this rapid convergence. This raises the question of how to select the initial values of a. and k,. A lusid account of this problem is given in [2], which we paraphrase below.
Consider the problem of finding U and k for even TM modes. Let a have an arbitrary fixed value and let U and u satisfy U,, + (; Then U has simple zeros 0 < 20' < 22' < zqf we may define the values . from which (Even subscripts are used to indicate that U is an even function.? Similarly, U has simple zeros 0 < z; < z ; < zq . . ' and we set The values k; and k; vary with a, and if k: = k, for some value a = am,, then for k,, equal to the mutual value of k; and k ; , the pair ( a m n r k m n ) forces f l and f 2 to be zero in (25). See Fig. 4, which illustrates the intersections of these zero curves for the special case to = qO = 1. Similarly, for the odd TM modes, if we let U and U satisfy (28) and (29) with the initial conditions 4 0 ) = 0, u,(O) = l,u(O) = 0, uJO) = 1, then U has simple zeros 0 < z : < z: < z< -. which interlace the even zeros z;~, and U has simple zeros 0 < z ; < z; < 2 ; . . which interlace the even zeros z;". Defining k; and k; as in (30)  (24)1-(27). For the TE modes we use the same procedure but work with the zeros of U , and U , rather than U and U . Plots of these values are similar to the TM case. This was done for the special case 770=to=1 and the results are given in Tables I and I1 [lo], 1191 is given in Table V. IV. POWER HANDLING CAPABILITY OF PARABOLIC WAVEGUIDES In [31, Baum defines the efficiency factor, y, for a planar domain 52 as where 0 < k , Q k , are the two lowest nonzero TE eigenvalues and Using the methods of the previous two sections, we investigated y = y(qo) for confocal parabolic domains, 52, with to = 1 and 0.1 Q 770 Q 1. (The integral on the right-hand side of (3'2) was evaluated numerically by using a 24-point Gaussian product formula, which is exact on polynomials up through order 47.) The results are given in Fig. 10, where we see that the overall maximum occurs at vo = 1 with y(l> = 0.4600. A secondary maximum occurs at qo = 0.14083 with ~(0.14083) P 0.4234. Intermediate between these maxima. Y is the TE eigenfunction corresponding to k , . maxima at q0 = 1 and q0 = 0.14083) or the crossing point for the first and second TE eigenvalues ( q o = 0.31297).
We can interpret these results by comparison with the efficiency factor for a rectangle with aspect ratio r = height/width. In this case the maximum efficiency factor is 0.4653 = (3/64)'14 which is attained at r = 2 and r = 1/2. Between these two maxima, the efficiency factor reaches a minimum of zero at r = 1.
For a confocal parabolic region, 52, with lo = 1, the aspect ratio of the height to the width is given by

(33)
If the efficiency factor depended only on r, we would then expect y to be maximized at r = 2 and r = 1/2 and minimized at r = 1 by analogy with the rectangle. That is, we would expect from (33) to see maximum efficiency at 770 = 1 (for r = 2) and at 770 = 4 -fi = 0.12702 (for r = 1/21. The corresponding minimum would then be at qo = 2 -6 = 0.26795 (for r = 1). Since these values are close to the true values of q0 = 1, T~ = 0.14083, and vo = 0.31297, we conclude that for parabolic waveguides the aspect ratio of the cross section is of prime importance in determining the power handling capability-just as in the case of rectangular and triangular waveguides [ 161, [171, [20].

V. CONCLUSION
Mode analysis for confocal coaxial parabolic regions is greatly simplified by using analytic continuation to evaluate parabolic cylinder functions. When combined with Newton-Poisson shooting and homotopy methods, this continuation technique easily generates the separation constants and eigenvalues of arbitrary parabolic regions, and has been applied to the problem of determining power handling capabilities for such regions.

APPENDIX NONSINGULARITY OF NEWTON'S METHOD
We need two technical lemmas to show that the gradient matrix, Vf, in (24) is nonsingular.

Remark:
Essentially the same proof shows that ak; / a a < Theorem: The gradient matrix, V f , is nonsingular at ( a , k ) Proof: First consider the TM mode case: f l ( a , k ) = u (m(,,, a), f 2 ( a , k ) = U (~T , , ,  a), where the second argument denotes the dependence on a . The gradient matrix is 0 and ak;/aa > 0 for the TE modes. For the TE modes we find det V f = ([o~ou,,u,,/ 8k,,,,,)