By modification of the conformal transformation used by Chang in determining uniform field electrode profiles, a large improvement can be achieved in the uniformity of the electric field strength distribution over the surface of the electrodes. When such electrodes are used in a TEA laser system, smaller electrodes can be used for the same gas discharge width

Ernst, Gerard J.

English

Ernst, G.J.

University of Twente Research Information

Volume 47, number 1 OPTICS COMMUNICATIONS 1 August 1983 COMPACT UNIFORM FIELD ELECTRODE PROFILES Gerard J. ERNST Department o f  Applied Physics, Twente University o f  Technology, Enschede, The Netherlands Received 18 April 1983 By modification of the conformal transformation used by Chang in determining uniform field electrode profiles, a large improvement can be achieved in the uniformity of the electric field strength distribution over the surface of the electrodes. When such electrodes are used in a TEA laser system, smaller electrodes can be used for the same gas discharge width. 1. Introduction 2. The conformal transformation In order to obtain high output powers from trans- versely excited pulsed lasers it is important to have a very uniform energy loading of  the active gas medium. That is why there is a need for specially contoured electrodes which produce a very uniform field strength over a certain amount o f  surface. A number o f  authors have given solutions to this problem. Up to now, Rogowski profiles [ 1] are most commonly used, but have the serious disadvantage that three smooth seg- ments have to be joint together in an undefined and rough manner. The best profiles are made by using the analytic formulas derived by Chang [2]. For a realistic Chang profde, however, the width of  a TEA laser electrode, required to produce a square discharge, amounts to approximately 3.5 times the discharge width [3]. For some applications, however, it is des- irable to have a smaller electrode-to-discharge width ratio. This is, for instance, the case for large aperture CO 2 or CO lasers, where the distance of  the UV source from the electrode centre is an  important measure, or for large aperture UV lasers, where the electrode in- ductance contributes significantly to the total circuit inductance. Compacting the electrodes in the way Chang describes in his paper is undesirable because that leads to a decrease in the field strength uniformity at the electrode surface. That is why we looked for compacting in a different way. We start with the same conformal transformation as used by Chang: ~ = w + k ( w ) s i n h  w , (1) where ~ = x + iy and w = u + io, with x and y being the space coordinates and u and o being the flux and po- tential functions, respectively. However, instead o f  k being a constant, we assume k to be a function o f  w. For every value o f  o (Iol < rr) the prof'de of  the corresponding equipotential surface is given by x = u + Re (k)coso sinhu - Im(k)sino coshu , (2) y = o + Re (k) sin o coshu + Im(k)coso sinhu , (3) where u is the running variable. Because the prof'de has to be s~.nmetric with re- spect to the y axis and because the +v and - o  equipo- tentials have to be mirror images with respect to the x axis, the real part of  k, designated by Re (k), has to be an even function with respect to u as well as o, whereas the imaginary part o f  k, designated by Im (k), has to be an odd function With respect to u as well as u. Those conditions are fulfilled when k is an even function of  w. As can be seen from relations (2) and (3), the uni- form field electrode (UFE) profile is not uniquely de- termined. For the case that k is a real constant, k 0, two independent variables, k 0 and o, can be chosen, both of  which determine the form of  the prof'de as well as the electric field strength distribution. To find 0 030-4018/83/0000-0000/$  03.00 © 1983 North-Holland 47 Volume 47, number 1 OPTICS COMMUNICATIONS 1 August 1983 the op t imum UFE profile,  an expression is needed for the electric field strength: E -2  = Id~'/dwl 2= I I + k c o s h w  + ( d k / d w ) s i n h w l  2 . (4) When the electric field strength is expressed as a power series expansion in u, E = E o ( k , o ) + E 2 ( k , o ) u 2 + E 4 ( k , o ) u 4  + ... .  (5) the opt imum profile can be found by requiring the lower coefficients (except,  of  course, for the first one, E0)  to vanish, or, if  this is impossible, to be max- imum. In the case of  k being a constant ko,  the sec- ond coefficient has to be zero or -- [ f (O)f  (2) (13) - g (1)2 (0)]/.f3 (0) = O,  (6) where f ( u )  = 1 + k0coso  coshu , (7) g(u)  = k 0 sin v sirth u , (8) and where the exponent  between brackets denotes the number o f  differentiations with respect to u. From (6) the following condit ion follows: v = arccos ( - k 0 )  , (9) as already has been derived by  Chang. We shall now treat two specific cases of  improved UFE profiles. In the first case a fourth-power func- tion in w will be chosen for k and in the second case k will be an eighth-power function in w. (A) k is a fourth-power function in w. As already mentioned,  improvements of  UFE pro- files can be expected if k is allowed to be some even function o f  w. Although,  in principle, many types of  functions can be t r ied,  the easiest way is probably to take some terms o f  a power series expansion of  k around w = 0. In this section we will consider the case where k has the following form: k = k 0 + l k 2 w 2  + 2-~k4 w4 . (10) For  the coordinates equations (2) and (3) hold where Re(k)  = k 0 + ½ k2(u2 - 0 2 )  + "24 k4( u4 + o4-6 t t2o2) ,  (11) Im(k)  = k2uo + ~ k4(u3o - uo 3) . (12) Now we have four free parameters,  k0,  k2,  k 4 , and o, to optimize the profile. This means that it is not only possible to require the coefficient E2, but  also to re- quire the coefficients E 4 and E 6 from eq. (5) to vanish. So equation (6) must be satisfied together with the following two equations: _ [f(0)f(4)(0) + 3f(2)2(0) + 4g(1)(0)g(3)(0)]/f3 (0) =0 ,  (13) - [f(0)f(6)(0)  + 15f(2)(0)f(4)(0) + 6g(1)(0)g(5)(0) + 10g (3)2 (0)]/) '3(0) = 0 ,  (14) where f ( u )  = 1 + Re(k)  coshu cos v - Im (k)s inhu sin v + Re(1)(k) sinhu coso - Im(1)(k)coshu s i n v ,  (15) g(u )  = Re (k) sinhu sin v + Im(k)  coshu cos v + Re(1)(k) cosh u sin v + Im(1)(k) sinh u cos o, (16) and where the exponent  between brackets denotes the number  of differentiations with respect to u again. The computer  calculations show a rather capricious behaviour of  the coefficient E 6 as a function of  v for E 2 and E 4 both zero. This behaviour is shown in fig. 1 for a k 0 value of  0.02. The dot ted  part of  the curve denotes negative values o f E  6. Two zero points are found in this case, one for v = 1.621403 and one for o = 3.036043, the latter one being unimportant .  Figs. 2 and 3 show the results of  our computer  calculations when all three coefficients E2,  E4, and E 6 equal zero. In order to make the use of  this type of  profile convenient to everyone, the opt imum / /  / I o ~ i i , 10 -4 10 -3 ro -2 10 -~ 10 0 Fig. 1. The coefficient E 6 as a function of o for E 2 and E 4 both zero. k is a fourth-power function of w and the k o value is 0.02. The dotted part in the figure denotes negative values. 48 Volume 47, number 1 1 0  - I  . l+I 1o-! OPTICS COMMUNICATIONS 1 August 1983 Table 1 k = ko + ~k2w 2 + "~k+w 4 ko k2 k4 o 0.001 -0A894729 E-6 0.3526636 E-6 1.574247 0.0015 -0.1078868 E-5 0.7874861 E-6 1.575911 0.002 -0.1880627 E-5 0.1389751 E-5 1.577540 0.003 -0A073689 E-5 0.3083106 E-5 1.580695 0.005 -0.1052878 E-4 0.8340576 E-5 1.586644 0.007 -0.1928212 E-4 0.1595262 E-4 1.592177 0.01 -0.3577091 E-4 0.3148105 E-4 1.599834 0.015 -0.6956990 E-4 0.6741250 E-4 1.611253 0.02 -0.1082389 E-3 0.1147659 E-3 1.621403 0.03 -0.1912708 E-3 0.2399266 E-3 1.639043 0.05 -0.3465792 E-3 0.5945713 E-3 1.667832 0.07 -0.4561044 E-3 0.1066652 E-2 1.691792 0.1 -0-5005615 E-3 0.1958405 E-2 1.723087 0.15 -0.2169977 E-3 0.3842812 Eo2 1.769531 0.2 0.5203643 E-3 0.6117696 E-2 1.813299 0.3 0.3302783 E-2 0.1148132 E-1 1.900163 0.5 0.1374477 E-I 0.2383356 E-1 2.085975 0.7 0.3039475 E-1 0.3665473 E-1 2.297876 1.0 0.6755833 E-1 0-5572676 E-1 2.692438 , i i 10 -3 10 -2 10- 1 , k o Fig. 2. The optimized values ofk 2 and k 4 as a function ofk o. k is a fourth-power function of w. values of  k 2, k 4, and o are given in tabular form in table 1 for a range ofko-values .  (B) k is an eighth-power function in w. We now consider the case where k has the follow- 3 0 '  v 1 2.5 2 0  1 . 5  i i I0 -a 10 -2 I0-  , ko 1 Fig. 3. The optimized value ofo as a function o fko .k  is a fourth-power function of w. ing form: _ 1 2 l 4 1 6 k - k o + ~ k 2 w  +~k4w +-~k6w + ~ k s w 8 .  (17) For  the coordinates equations (2) and (3) hold again where now Re (k) = k 0 + ½ k2(u2 - 02)+ I k4(t/4 + o4 - 6 u 2 o  2) + ~-6 k6(u 6 - o6 _ 15u4o2 + 15u2o 4) (18) + ~ k8(u8 + 08 - 28u6u 2 - 28U206 + 70u4o4),  Im (k) = k2uo + ~ k4(u3o- u o 3) + ~k6(3uSo  - 10u3o3 + 3uo 5) + s-~k8(u7v - 7u5o 3 + 7u3o 5 - uo7). (19) Six parameters,  k0,  k2,  k4,  k6,  k8,  and o, are free to optimize the profile.  This means that it is now possible to require all coefficients,  E2,  E4,  E6,  E8,  and E l0  from eq. (5) to vanish. So equations (6), (13) and (14) must be satisfied together with the following two equations: - -  [f(O)f(8)(O) + 28#2) (0 )#6) (0 )  + 35 I -(4)2 ((3) (20) + 8g(1)(O)g(7)(O) + 56g(3)(O)g(5)(O)]/f3(O) = O, 49 Volume 47, number 1 OPTICS COMMUNICATIONS 1 August 1983 10 -I 10-: 10-: io-e 10 i 10 -3 1 ~ k  o k k 6 g I i 10 -2 I 0  Fig. 4. The optimized values of k2, k 4, k 6, and k s as a func- tion o fk  o. k is an eighth-power function ofw.  2 5  2 0  15 i I0 -3 10 -2 1 / I I 10 ~ ko Fig. 5. The optimized value ofv  as a function o f k o . k  is an eighth-power function of w. -- [ f ( 0 ) f ( 1 0 ) ( 0 )  + 45f(2)(O)f(8)(O)  + 210/~4)(0)f(6)(0) + lOgw(o)g(9)(o) (21)  + 120g(3) (0)g(7) (0)  + 126g(  5)2 (0)]  If3 (0) = O, where  f(u) a n d  g(u) are given b y  eqs.  (15)  and  (16)  t o g e t h e r  w i t h  ( 1 8 )  a n d  (19)  and  where  the  e x p o n e n t  b e t w e e n  b racke t s  deno t e s  the  n u m b e r  o f  d i f fe rent ia -  t ions  w i t h  respect  to  u again. The  resul ts  o f  ou r  c o m p u t e r  ca lcu la t ions  are s h o w n  in figs. 4 and  5. In o rde r  to  make  c o n v e n i e n t  use o f  th is  t ype  o f  prof i le ,  t he  resul ts  are given again in t abu l a r  f o r m  in tab le  2.  Table 2 1 .  4 1 _ 6 1 k=ko+½k2w2 ÷ ~-~x4w +-~-6K~w + 4 ~ k a  w8 ko k2 k4 k 6 k 8 o 0.001 0.1344918 E-6 0.2685221 E-6 0.2176708 E-6 0.2138423 E-6 0.0015 0.3021345 E-6 0.6037009 E-6 0.4889647 E-6 0.4806909 E-6 0.002 0.5361709 E-6 0.1072431 E-5 0.8678578 E-6 0.8538265 E-6 0.003 0.1201322 E-5 0.2409479 E-5 0.1946312 E-5 0.1918264 E-5 0.005 0.3301244 E-5 0.6675282 E-5 0.5370816 E-5 0.5317301 E-5 0.007 0.6382663 E-5 0.1305242 E-4 0.1045596 E-4 0.1041026 E-4 0.01 0.1270177 E-4 0.2655204 E-4 0.2111624 E-4 0.2124108 E-4 0.015 0.2711895 E-4 0.5945364 E-4 0.4663559 E-4 0.4790848 E-4 0.02 0.4527998 E-4 0.1051987 E-3 0.8125040 E-4 0.8555060 E-4 0.03 0.8800870 E-4 0.2342309 E-3 0.1747778 E-3 0.1945888 E-3 0.05 0.1766461 E-3 0.6324751 E-3 0.4385705 E-3 0.5505032 E-3 0.07 0.2569693 E-3 0.1195166 E-2 0.7754077 E-3 0.1091081 E-2 0.1 0.3912534 E-3 0.2294400 E-2 0.1384272 E-2 0.2248266 E-2 0.15 0.8167910 E-3 0.4673770 E-2 0.2685508 E-2 0.5098914 E-2 0.2 0.1671438 E-2 0.7616743 E-2 0.4427084 E-2 0.9067361 E-2 0.3 0.4999499 E-2 0.1492337 E-1 0.9510110 E-2 0.1999371 E-1 0.5 0.1861874 E-1 0.3433195 E-1 0.2651986 E-1 0.5037162 E-1 1.571336 1.571606 1.571878 1.572426 1.573543 1.574696 1.576511 1.579810 1.583495 1.592047 1.613038 1.636992 1.674224 1.733464 1.787605 1.885634 2.069868 50 Volume 47, number 1 OPTICS COMMUNICATIONS 1 August 1983 Y 0.5 2 3 ~ X / / y ,  Fig. 6. The shape of the optimized prof'fles for three different cases. For the curves A, B, and C the k function is respectively a constant, a fourth-power and an eighth-power function of w. The ko value is 0.02. 3. Comparison of different prof'des; discussion In fig. 6 the shape is shown of the profdes for the three following cases: all three profiles have k 0 = 0.02; curve A is optimized for k being a constant, curve B for k being a fourth-power function in w and curve C for k being an eighth-power function in w. Curve C has a width that is 10 to 15% smaller than curve A. The field distribution at the surface of the prolrfles is very much different. Fig. 7 shows the field-strength distribution at the electrode surface for the corre- sponding profdes of fig. 6. At the vertical axis the normalized difference has been plotted between the field strength at the position x/y 0 and the electrode centre. A great improvement of the field-strength uni- formity is found. The question, however, is to which part the field-strength distribution in the space be- tween the electrodes determines the discharge width. Therefore the field-strength distribution is also cal- culated at the midplane between the electrodes, where o = 0. The results are plotted in fig. 7, curve A'. The distribution is essentially the same for all three cases. Experiments have to be carried out to determine the discharge width for the different profiles. 0 1 2 3 Fig. 7. The deviation of the electric field strength at the sur- face of the optimized electrodes from the central value as a function of the normalized position at the electrode surface. For the curves A, B, and C the k function is respectively a con. stant, a fourth-power function and an eighth-power function of w. The ko value is 0.02. The curve A' gi¢es the deviation of the electric-field strength at the midplane between the elec- trodes, also from its central value. References [1] W. Rogowski, Arch. Electrotech. 12 (1923) 1. [2] T.Y. Chang, Rev. Scient. Instr. 44 (1973) 405. [3] G.J. Ernst and A.G. Boer, Optics Comm. 27 (1978) 105. 51 

Compact uniform field electrode profiles

By modification of the conformal transformation used by Chang in determining uniform field electrode profiles, a large improvement can be achieved in the uniformity of the electric field strength distribution over the surface of the electrodes. When such electrodes are used in a TEA laser system, smaller electrodes can be used for the same gas discharge width. 1. Introduction 2. Th

Compact uniform field electrode profiles

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