CC BY
30
MSC 78A60
ANALYSIS OF THE ELECTRIC FIELD IN A LASER BY THE MULTIPOLE METHOD A.B. Paltsev
Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Vavilov St., 40, Moscow, 119991, Russia, e-mail: [email protected]
Abstract. The multipole method is modified such that it may be applicable to analysis of electric field in the laser of special design. An optimal form of electrodes in the device under consideration is found. Main characteristics of the field are obtained in terms of closed formulae. Data of numerical study which confirm high effectiveness and accuracy of this method are given.
Key words: boundary value problem, multipole method, calculation of electric field in the laser.
Introduction
The multipole method suggested in [1], [2] and developed in [3]- [15] is based on the use of functions Qp, p = 1, 2,..., which satisfy identically a given equation in initial domain g (e.g., the Laplace equation), satisfy homogeneous boundary condition at curve 7 C dg, and constitute a complete and minimum system at the complementary arc r = dg\7. A solution of a boundary value problem is presented as a sequence of linear combinations of functions Qp. Those functions are boundary multipoles for an extension G of initial domain g over arc r; the concept of the boundary multipoles is meant in the sense of [3]. It is important that these functions can be expressed by the simple formula in the case of boundary value problems for the Laplace equation. Thus, if a Dirichlet problem in g with zero condition at 7 is considered, then functions Qp will be given as follows: Qp = Im Fp, where F is a conformal mapping of the above extension G onto the upper half-plane.
The multipole method was substantiated and investigated in [3]. The obtained theoretical estimates show the method enables to calculate effectively the problem solution and all its derivatives both in domain g and at arc 7 even if it has complex shape, contains geometrical singularities or infinities. It should be emphasized , that this method, according to the above estimates, gives the convergence in Cn-norm with arbitrary n in domain, including a part of its boundary, while traditional methods (e.g., finite element method) yield approximation only in Wf-norm (energy norm), and the error of gradient increases when approaching the boundary. The performed numerical experiments confirmed high efficiency of the multipole method; e.g., when the Dirichlet problem was being solved for the Poisson equation in L-type domain with rounded re-entrant corner [5], [7], [9], the use of only 40 degrees of freedom (i.e. functions Qp) ensured the accuracy 10-8 in C-norm for gradient near the rounded corner.
The present work is devoted to modification of this method and to its application to a difficult engineering problem which arises when designing a gas laser of special structure.
The work was financially supported by Russian Foundation for Basic Research (project №13-01-00923).
According to the general principle of laser operation, an active medium which would amplify electromagnetic waves passing through, must be created in the laser [16].
The most effective process of an active medium creation in a gas laser is implemented by a glow electric discharge of sufficient intensity maintained in the gas mixture. Discharge conditions, the speed of this process, and, therefore, effectiveness of laser generation are characterized by some parameters among which electric field intensity E plays a vital role [16]. The laser operation is highly sensitive to the change of E, so very accurate analysis of the field is essentially important.
The laser under consideration is of a complex design what makes this analysis rather difficult. The main feature of this device is in the special electrode structure suggested by researchers from P.N. Lebedev Physical Institute of the Russian Academy of Sciences (Prof.
A.N. Lobanov and his colleagues). Namely, both the anode and cathode inserted into the gas medium are composed of isolated sections provided with a system which enables to vary a potential on each of them independently. Thus, a variable distribution of potential can be fed at the bottoms of the electrodes; there is a constant potential at their side facings. The lasers of such a structure have a supplementary possibility which is that the electric field can be tuned for the most efficient laser operation, owing to redistribution of the potential and variation of the bottoms form.
The experience of above mentioned researches from P.N.Lebedev Physical Institute of the Russian Academy of Sciences showed that application of various numerical methods to the evaluation of field in this laser encountered considerable obstacles and did not yield satisfying results. And all of their efforts to obtain field intensity E, which is a differential characteristic, failed. In the present work we apply the multipole method that enables to get all the characteristics required with high accuracy and efficiency.
A boundary value problem which describes the electric potential in the gas mixture under consideration is stated in Sect. 1. The solution of the boundary value problem is constructed with the help of the multipole method in Sect. 2. Sect. 3 is dedicated to finding such a distance between the electrodes and a form of their bottoms that the given maximum constant field would be kept at the bottoms for a constant potential preassigned at them. Sect. 4 contains general representation of the main field characteristics and data of specific implementation.
The author expresses his gratitude to Prof. A.N. Lobanov from P.N. Lebedev Physical Institute of the Russian Academy of Sciences for the statement of the problem.
1. Statement of the problem
1.1. Domains and Boundary Value Problem. The analysis of discharge conditions which take place in the laser gas mixture can be reduced to solving a Dirichlet problem for the Laplace equation in two-dimensional unbounded domain g. This domain is the upper half-plane from that a half-strip with a curved bottom removed.
To describe domain g accurately, we introduce an auxiliary domain G. Let G in complex plane w = u + iv be half-plane H = {w : v > 0} without two parallel rays:
G = H\{w : u = ±a; v > b},
where a and b are positive numbers. The initial points of rays will be denoted by B = a + ib and B' = —a + ib. The domain G boundary contains three infinities A, M and A' reached as |w| ^ oo with u > a for A, —a<u<a for M and u < —a for A', see Fig. 1.
Now we join points B and B' by a Jordan smooth curve r which lies in G except for the endpoints. This curve divides domain G into two subdomains, one of them with boundary (ABB'A'A) will be considered as domain g, see Fig. 2.
Domain g corresponds physically to the half section of a volume with gas mixture while the half-strip represents the electrode section. Let a certain potential ^(w) be distributed at electrode bottom r and certain constant potential values <^\ and ^2 be preset at the electrode side facings (B'A') and (AB), respectively. Besides, zero potential is presumed at u-axis (A'A) what follows from antisymmetry of the field with respect to this axis. Then sought potential $ excited in the gas mixture in view of a small density of a volume charge throughout the system is described by the following boundary value problem:
ДФ('ш) = 0, w е д, (1.1)
Ф(эд) = 0, w е (A'A), (1.2)
Ф(м) = ф1, w е (B'A'), (1.3)
Ф(эд) = ф2, w е (AB), (1.4)
Ф(м) = w е Г. (1.5)
Here, boundary function ^ is continuous at r and joins continuously with boundary values at sides (B'A') and (AB), i.e. <^(B') = <^(B) = • Function $(w) is a bounded solution
of problem (1.1)-(1.5) that belongs to C2(g) n C (g\(A U A')).
Note bottom r in practice is chosen smoothly joined with the electrode side facings (B'A') and (AB). Otherwise, a field concentration will occur near the junction of the electrode bottom with the side facings what can cause the unstable discharge and electron breakdown [16].
t V
и
G
ь
S 1 \В
1 1
А» 1 ■ 1 1 | А * ■
777/7///'/////77Г-ъПППТПГ ® Т/7ПТ/7'Г7ГГГ/Т*>
Fig. 1. Domain G.
Fig. 2. Domain g.
1.2. Conformal Mapping of Auxiliary Domain G. When the boundary value problem (1.1)-(1.5) being solved by the multipole method, conformal mapping z = F(w) of domain G onto the half-plane H plays a vital part. However, it is difficult to find an effective global analytical representation for this mapping what makes us turn to an inverse mapping (of H onto G) denoted by f (z) that can be found in this case with reasonable facility. We define the following correspondence between three boundary points of one domain and those of the other: points A, M, and A' of dG correspond to points 1, ro, and —1 of dH, i.e. in terms of mapping f
f (1) = A, f (ro) = M, f (—1) = A'.
We remark that mapping f satisfies relation f(—2) = — f(z), z e H, following from the Schwarz reflection principle [17], [18]. Then points B = a + ib and B' = —a + ib will correspond respectively to real points z = k-1 and z = — k-1 for a certain k e ]0, 1[ which will be determined below.
Function f (z) may be expressed by Schwarz-Christoffel integral [17], [18]
z
w = f (z) = cj (1 — z2 )-3/2 (k-2 — z 2)dZ. (1.6)
0
To find the unknown real k and C, we shall use conditions resulting from the above definition of conformed mapping f (z):
Re f (x) = a, x> 1; f (k-1) = a + ib. (1.7)
Let x be a real number from interval ] — 1,1[. Taking integral (1.6) along real segment [0, x], getting, thus,
f( \ n (k 2 — 1 i • j(x) = C t x + arcsin x
1 — x2
and analytically continuing this function from real interval ] — 1, 1[ to upper half-plane H, we obtain the explicit formula for the required mapping
f(z) = a i2 + ln^2 + ^ ~ + 7~Y'
we consider the main branch of logarithm: Imln z e ] — [, z e H. This expression in
accordance with conditions (1.7) follows C = 2a/n and parameter k = k(a/b) is a root of transcendental equation
Vl — k2 . 1 + \/l — k2 ivb
+ ---------k------= 2Z-
A graph of k as a function of dimensionless parameter a/b is drawn in Fig. 3.
Now we can write out final expression for mapping f (z):
2ai ( k~2 - 1
w = f{z) = —: ^ + In(z + Vz2 - 1) + a;
n Vvz2 — 1 J
Thus, inverse mapping f(z) is completely defined, and the required mapping F(w) can be obtained by inversion of it. A very convenient local inversing procedure will be presented in Subsect. 2.4.
2. Solution of boundary value problem
2.1. Reduction of Problem. We introduce function U0(w) via conformal mapping F(w) defined at the end of the previous section
U0(w) = — Im (<^i In [1 + F(w)} — ф‘2 In [1 — F(w)}). n
(2.1)
Function U0(w) is obviously a bounded harmonic in G and continuous in G\(A U M U A') one, and satisfies the following boundary conditions at dG:
U0(w) = 0, w e (A' A),
U0(w) = 01, w e (mb' A'),
U0(w) = 02, w e (ABM).
Let us present solution $(w) of original problem (1.1) - (1-5) in the form
$(w) = U0(w) + U (w). (2.2)
Taking into account the above properties of function U0 and the inclusion $ e C (g\(A U A')), we find U(w) is a classical solution of the Dirichlet problem
AU(w) = 0, w e g,
U(w) = 0, w e y = 5g\r
U(w) = 0(w) — U0(w), w e r.
Function U(w) will be constructed with the help of the multipole method.
2.2. Multipole Method for Solving Problem. The multipole method which is an analytical-numerical one for solving elliptic boundary value problems in complex-shaped domains was theoretically studied, advanced and generalized in [3]- [15] and some other works, and was successfully used for solving a number of theoretical and applied problems. The basis of the method is an application of functions Qp(w) expressed by the formula
Hp(w) = Im [F(w)]p, p e N.
One can interprete function Qp(w) for every natural p in the electrostatic sense [3]: it represents electric potential excited in domain G by pth order boundary multipole which is located in point M. These are harmonic in g functions, which satisfy zero boundary condition at arc y = (B' A' AB) and constitute complete and minimum system in L2(r). The required solution of problem (2.3) - (2.5) can be obtained as the limit
U(w) = lim UK(w) (2.6)
of sequence of functions UK(w) which are Qp(w) linear combinations
K
UK(w) = ^2 aK (w),
p=1
(2.3)
(2.4)
(2.5)
where coefficients aK are defined by the condition that L2(r)-norm of difference of UK (w) and boundary function 0(w) — U0(w) should be minimum
||UK - ф + Uo||i2(r) i—► min . (2.7)
Sequence UK(w) converges uniformly to solution U(w) of problem (2.3) - (2.5) everywhere in every compact lying into set g U y and, moreover, admits differentiation of any order in set gU y\(AU A'); the differentiated sequence converges uniformly in every compact lying into the latter set [3].
Besides, an expansion in the multipole system
X
U(w) = } ap Op(w), ap = lim aK
/ K ^x p
n=1
is an analog of Taylor series in the sense that it can be differentiated any times and it converges with exponential speed everywhere in its convergence set
{w = f (z); |z| < R, Imz > 0}, R = min |F(w)|.
w€r
We remark that the set represents the union of a certain subdomain of g and a part of arc (B' A' AB) adjacent to it.
Condition (2.7) results in linear algebraic system
K
]TCpqaK = Hq, q =1,2,...,K, (2.8)
p=i
where Cpq are elements of Gram’s matrix for system {^(w)}^, and Hp is a projection of boundary function 0 — U0 onto Op :
Cpq = J Qp(w)Qq(w)|dw|, Hp = J Qp(w)[0(w) — U0(w)]|dw|.
By virtue of the fact that function system {Op(w)}pX=1 is complete and minimum in L2(r) as it has been said, algebraic system (2.8) is uniquely solvable.
Thus, summing up the preceding, we can write out the expression for the approximate solution of problem (1.1) - (1.5)
1K
>A (w) = - Im (0i In [1 + F(w)] - ф-2 In [1 - F(w)]) + a^Qp(w)
p=i
converging to $(w) in the closed domain g as K ^ ro.
2.3. Expansion of Solution in Orthonormal System. It may be convenient to use a representation for $(w) in the form of series in an orthonormal system. Following [3], we introduce orthonormal function system {^p(w)} which can be obtained from system Qp(w) by the Schmidt orthogonalization process, see e.g. [19]
1 p
~ “ " [w)
шр(и0 — n i— AnpQn{w),
yDetpDetp-i
where Detp is determinant of matrix {Cmra}m;ra=1 while Anp is the algebraic complement of element Cnp in this matrix. Then exact solution $(w) is presented as follows
1 X
$(w) = - Im (0i In [1 + F(w)] - (p2 In [1 - F(w)]) + ^ hvu}p(w), (2.9)
p=1
where
hp = wp(w)^(w) - Uo(w)]|dw|.
p
/Г
According to [3], series (2.9) converges everywhere in g and admits any order differentiation in g U y\(A U A').
2.4. Constructing of Mapping z = F(w). As we have already said, conformal mapping z = F(w) of the extended domain G onto half-plane H is an important, apparatus for the multipole method. However, the problem of constructive representation for this mapping in the general case is difficult. Therefore, when the multipole method being implemented, one has to use the inverse of F(w) denoted by f (z) which can usually be found easier. If f (z) is determined, then the required mapping could be constructed by inversing f (z). It can be done with the help of the method of successive approximations which is based on the Newton’s method and gives local inversing procedure [20]. Here we formulate this method for a conformal mapping in the general case.
Let function w = ^(z) accomplish a conformal mapping of domain G1 onto domain G2, and the inverse image z0 e G1 for a certain w0 e G2 is known,
f (z0) = w0.
Now we introduce some objects required for the formulation and proof of the method. The distance of point z0 to the domain G1 boundary will be denoted by Rz0; the disk
Dz0(r) = {z : |z — z0| < r}
for every r e]0, Rz0] lies obviously into domain G1. Let us define the function
^'(z0)(z1 — z2)
which is holomorphic in G1 with respect to the both variables by virtue of the fact that ^(z) is holomorphic in G1 and ^'(z0) = 0. A maximum absolute value of W with respect to zi and z2 from the closed disk Dz0(r), r e ]0, Rz0] will be denoted by Q(r),
Q(r) = max |W(z1,z2)|, z1,z2 e Dgz0(r);
and maximum absolute value of |W| of z1 from the same disk, when z2 = z0, will be denoted by q(r)
q(r) = max |W(z1,z0)|, z1 e Dgz0(r).
It is clear that Q(0) = q(0) = 0. Now we will state some other properties of the intoduced functions.
Lemma 2.1. If function ^(z) is not linear, then for every r e ]0, Rz0]
1. The following inequality takes place
0 < q(r) < Q(r);
2. Functions Q(r) and q(r) increase strictly;
3. Functions Q(r) and q(r) are majorized by convergent series
Q(r) < q(r) <
i f;eoir.<00;
|^/(zo)| n=1 n!
n=1
oo
(2.10)
(2.11)
(2.12)
□ Using the principle of maximum for holomorphic function [17], [18], it is not difficult to verify that function |W(z1,z0)| over the disk Dz0(r) reaches its maximum value at the disk boundary, while the maximum of |W(z1, z2)| is reached when both points z1, z2 belongs to the disk boundary. In other words, the relations take place
Q(r) = |W(z0 + reiai, z0 + reia2)|, q(r) = |W(z0 + reia3, z0)|
for certain a = aj(r) e [0, 2n[, j = 1, 2, 3. This follows the validity of inequality (2.10) and the strict monotony of the functions Q(r) and q(r). Further, expanding function W(z1,z2)
W (zi,z2)
1
E
^(n)(zo) (zi - Zo)n - (Z2 - zo)
^/(zo) n=2 n! (zi - Zo) - (Z2 - Zo)
placing zj = z0 + reiQj into this expression and estimating its modulus, we obtain the inequality
Q(r) <
1
|^/(zo)l
E
n=2
|^(n)(zo)|
n!
giai ___ gi^2
„n— 1
Taking into account
giai gia2
n- i
< E | giaim gia2(n-m— 1) | ^
m=o
we obtain the required estimate (2.11). To get (2.12), one can operate analogously. The lemma is proved. ■
Note if function ^(z) is linear, then Q(r) = q(r) = 0 for every r.
Let us define numbers R]0 and rz0 as follows. If Q(r) < 1 for all r from [0,Rz0], then R^ = Rz0, otherwise, by R^ will be denoted the point at which function Q takes value 1. Then rz0 e [0, R]0] is a point at which function r[1 — q(r)] reaches its maximum,
giain gia2n
giain gia2n
rzo[1 - q(rzo)] _ max (r[1 - q(r)]) , r £ [0,
Proposition 2.1. The function sequence
^o(w) = zo, ^n+i(w) = ^n(w) -
j>(Vn(w)) - w
Ф'Ы
n
0,1,2,...
(2.13)
converges to function z = ^(w), the inverse of ^(z), uniformly inside disk Dw0(rw0), where
rwo = rz0[1 - q(rzo)] 1^'(zo)|.
If |w — wo| = d < rwo, then the convergence rate of successive approximations (2.13) is estimated as follows
d
M - Vn(w)\ < , (2.14)
for
Qo = Q(d*) < 1, d* =
1 - Qo |^'(zo)| d
I1 - 9(rz0)]K(z>)l
(2.15)
^(z) - w ^(z) - ^(zo) wo - w
V’'(Zo) - zo < - -° m \ F (zo) + ф'Ы
□ Let d be an arbitrary number from the interval ]0, rwo[. We define quantity d* by relation (2.15); it is clear that 0 < d* < rzo. Now we will prove that for w such that |w — wo| = d, all the approximations ^n(w) belong to disk Dzo(d*). As to ^o(w) = zo, it belongs obviously to this disk; let ^n(w) be known for a certain n to lie into this disk, we show that ^n+1(w) is from Dzo(d*) too. Indeed, denoting z = ^n(w), we write
|^ra+1(w) — Z01 —
the former summand is, by definition of q, not greater than q(d*)|z zo| and, thus, it is less than q(d*)d*, the latter summand equals to d/|^'(zo)| = [1 — q(rzo)]d*, that follows
|^n+i(w) — zo| < q(d*)d* + [1 — q(rzo)]d* < d*.
By virtue of the principle of mathematical induction, we obtain that ^n(w) G Dzo(d*), n = 0,1, 2 ...
Further, according to the definition of function Q(r), the following inequality takes place
Zl - Z2 -
^(Zi) - ^(z2)
Ф' (z0)
<Q(d*)|zi - Z2|, Q(d*) < 1,
for every z1, z2 from Dzo(d*). We can place z1 = ^n(w), z2 = ^n-1(w) in this formula, what, taking into account (2.13), follows
|^n+i(w) - ФпМ| < Q(d*)^„(w) - Фга-1(ш)|.
and for every N > n
Е|фт+1(«о-фт.Н1<
1 - Q(d*)
^(zo) - w
Ф' (zo)
This means sequence ^n(w) converges uniformly in every closed disk Dw0(d), d < rw0 to function ^(w), with ^ o ^(w) = w. Tending the upper limit N in the last inequality to the infinity, we obtain estimate (2.14). According to Weierstrass’ theorem [18], ^(w) constructed in this manner is a holomorphic function in the open disk Dw0(rw0). The proposition is proved. ■
3. Optimum Electrode Shape
3.1. Formulation of Problem. We remind domain g from the class under consideration is uniquely defined by arc r (the electrode bottom shape) and parameters a (the electrode half-width) and b (the altitude of electrode sides over (A'A)-axis).
We will consider the following statement: let a certain constant potential 00 > 0 be preset at the whole electrode surface (A B B1 A) with the electrode width 2a being given. It is required to find bottom r and parameter b that provide a constant magnitude of electric intensity along the whole bottom, equal to the preset value E0.
The arc r shape satisfying this statement is an optimum one in the sense that any other arc with the same endpoints contains a some place where the magnitude of electric intensity exceeds E0.
To be specific, we shall solve the following boundary value problem in domain g with free boundary arc r
(3.1)
(3.2)
(3.3)
(3.4)
quantities a, 00 and E0 are assumed to be preset.
An explicit analytical expression for r as a complex-valued parametrical function r(t) = = ri(t) + ir2(t) will be found below in Subsect. 3.3. The corresponding potential $(w) can be obtained by means of the method presented in Sect. 2.
3.2. Preliminary Notes. The problem (3.1)-(3.4) of constructing an optimal rounding curve r is solved below by the hodograph method [21]- [24].
Problem (3.1)-(3.4) is reduced to the question of the existence of such a conformal mapping Z = ^(w) of domain g onto strip
{( : 0 < ImC < 00}
that points A and A would be mapped into the right and left infinities of the strip, respectively, and
|^'(w)| = E0
for every point w of the unknown arc r. Then function $(w) which corresponds to statement (3.1)-(3.4) will be expressed as follows
ДФ(^) = = 0, w e g>
Ф(и>) = = 0, w e (A A),
Ф(и>) = = ф0; w e A cc DC (A
|gradФ(w)| = = E0; w e Г = (BB'),
Ф(и>) = Im$(w).
Function ^(w) is called complex potential; its derivative is well known to be related to field intensity E(w) by the formula
E(w) = i ^'(w).
where E(w) is the complex conjugate of E(w). A general representation of function ^(w) for problem (1.1)-(1.5) will be given in Subsect. 4.1.
Existence and uniqueness of arc r which would bring into being the mapping, can be proved with the help of the general variation principle [21], [25]. Note arc r and, therefore, the whole domain g are, in our case, symmetrical with respect to v-axis. The segment of this axis joining the middle point O' of r and the origin of coordinates O divides g into two
symmetrical subdomains g+ = {w G g : u > 0} and g- = {w G g : u < 0}, see Fig. 2.
Note conformal mapping C = ^(w) has one degree of freedom that we will fix with condition ^(O) = 0. It is not difficult to demonstrate, using the reflection principle already mentioned, that this unique mapping transforms domain g+ into half-strip
{C : 0 < Re C; 0 < Im C < 00} (3.5)
with the correspondence of boundary points
tf(0) = 0, tf(A) = TO, tf(O' ) = i00. (3.6)
It is obvious that (ABB'A') is mapped into line ImC = 00, and real axis (A'A) is mapped into real axis Im C = 0. The image of point B is a certain point 3 + i00; positive quantity 3 which depends on the problem parameters will be determined below. Along with normalization (3.6) for ^ we shall use another correspondence which follows from (3.6):
^(A) = to, ^(B) = 3 + i00, ^(O' ) = i00 (3.7)
with unknown 3.
Let us consider now a conformal mapping conditioned by the derivative of function ^(w). The domain g+ image under the mapping W = ^'(w) is readily verified to be a sector
{W : |W| < E0; -n/2 < argW< 0} (3.8)
in the complex plane W (so-called hodograph plane); boundary points are transformed as follows:
tf'(A) = 0, tf'(B) = -iE0, tf'(O') = E0, (3.9)
and ^'(0) is a real number from interval ]0, E0[.
We introduce now function W = Q(C) which accomplishes a conformal mapping of halfstrip (3.5) from complex potential plane C onto sector (3.8) in the hodograph plane with the following normalization
Q(to) = 0, Q(3 + i00) = —iE0, Q(i00) = E0. (3.10)
Then, from definition of ^(w) and Q(C), and in view of correspondences (3.7), (3.9), (3.10), we get ^'(w) = Q o ^(w), whence it follows, owing to C = ^(w), the equality
(3.11)
Integrating right-hand side of (3.11) along straight segment [t + i00, 3 + i00], when 0 < t < 3, in the complex potential plane and the left-hand side along the corresponding part of arc r in initial plane w, we obtain the expression for the required arc in terms of parametrical function
r(t) = a + ib —
dx
Q(x + гфо)
t G [0, в],
(3.12)
where the parameter values t = 0 and t = 3 correspond to points O' and B of arc r.
Therefore, the mapping which satisfies relations (3.10) remains to be found, then everyone can determine the arc sought, applying formula (3.12). Unknown parameters b and 3 will be found in Subsect. 3.3.
3.3. Representation of Optimum Bottom. Mapping Q(C) is obtained effortlessly
(
) = Eo
\
cosh
1 +
К
20o
cosh
sinh
7Г/3
20o
ъС\
20o
sinh
7Г/3
200 у
(3.13)
Replacing C by x + i00 in this relation and substituting the result into (3.12), we find the following formula
Г(t) = a + ib- -j-Eo
(
sinh
1
200
sinh
sinh
+ i
nx \
200
20o
sinh
7Г /3 20o /
dx.
(3.14)
The imaginary part of (3.14) can be written via elemental functions while the real part, by means of substitution of integration variable x = (200/n) arsinh x', can be reduced to expression in terms of the incomplete elliptic integrals [26]:
r(t) = a + ib —
20o
пЕ0т
Ei{0, r) - E2(6i, r) + i (л/l - t2 cosh - Л
V 2фо )
(3.15)
The following designations are accepted in the last formula:
Ei(0,r)
dx
\/l — r2 sin2 x
Е2(в,т) = \/l — r2 sin2 xdx
are the first and second type incomplete elliptic integrals (since letters F and E which are their commonly accepted designations are already engaged in our paper, we have to use unconventional designations), and
nt
sinh
0 = 0{t) = arccos ■
2 00
sinh
7Г /3 20o
т = tanh
7Г /3 20o
(3.16)
в
в
б
Note 0 < 9 < n/2, 0 < t< 1, and expression £1(9, t ) — E2(9, t ) can be represented in the
form of series
E:(0, t) - E2(0, r) = f) (3.17)
n= 1 ' '
here and below (a)m denotes the Pochhammer’s symbol [27]; for values
e
/„<9) ^ I Sin2" Xd.
t
a recurrence formula is valid
/„(«)=«, /.(«) = + (l - ±) 4-.W- (3.18)
Series (3.17) converges for every 9, owing to the fact that t is always less than 1 and quantities I"(9) can be estimated as follows
lni.0) < In{7t/2) = ^ = o (n_1/2) , n ->■ oo.
2 n.
Thus, we have found function (3.15) which describes sought arc r, and it remains only to determine quantities /3 and b in this expression. For this purpose we consider two representations for the middle point O' of arc r. On the one hand, as it has been remarked, O' = r(0), hence, placing t = 0 in (3.12), we find
[' dx
u = a + ib —
0
On the other hand, O' can be got by integrating the left-hand side of (3.11) along the straight segment [O, O'] of v-axis (while the right-hand side is integrated along the corresponding straight segment [0, i0o]),
<j>0
qi i f dx
E0 J Q(ix) ‘ o
Inserting already known function Q defined by formula (3.13) into the two last relations, separating real and imaginary parts and equating them, we obtain two equations which define the required quantities completely:
в
a~~kJ
0
\
1------------^§-dx = 0, (3.19)
sinh2 4-20o
and
1
Е0
Фо
cos2
1 -
20о
cos
sinh
2
20о
+
nx \
200
sinh
7Г/3 20о /
dx = b-^~ tanh -jf n Eo 40o
тг- (3-2°)
Equation (3.19) is an implicit expression for ^ which is modified in terms of hypergeometrical function F =2 F1, see e.g. [26], [27], as follows:
^F(1/2,3/2;2;t2) = ^. 2 0o
(3.21)
where t has been defined in formula (3.16).
In view of equation (3.21) we can draw a conclusion that auxiliary parameter t G (0,1) depends only on dimensionless quantity A = aEo/0o that can be varied in the range (0, to), and the required parameter /3 depends on A and 0o.
We remark that for great values of A = aEo/0o it would be convenient for finding parameter /3 to use the following equation, equivalent to (3.21),
}/±Е± у (1/2).,(3/2)^ _ hi t|£„ _ a,E>
тг f ^ (r^W2
n=0
(n!)2
00
(3.22)
here small parameter e is related to /3 as follows
1 2 1-2
6=1 — r = cosh
2ф0
and the recurrence formula for coefficients is valid
(3.23)
^o — 2(ln 4 — 1), — ^ra-l —
n(2n — 1)(2n + 1)
2
Using expressions (3.21) and (3.22), we find asymptotic behavior of parameter /3 in the limiting cases A ^ 0 and A ^ to for every fixed 0o:
/3 = ^A + 0(A2), A —> 0 n
and
/3 = 0OA - 8 ~ ^ + O (Ae^A) , A ^ to.
n
Besides, it is not difficult to demonstrate that /3 as a function of A (for a fixed 0o) increases monotonically while its derivative will decrease monotonically from 40o/7r when A = 0, down to 0o when A ^ to. The graph of /3 as a function of aEo/0o for different values of potential 0o is presented in Fig. 4.
Finally, we are coming to a determination of parameter b. We return to equation (3.20) where all the quantities, except for b, have already been found. Inserting (3.13) with Z = ix into (3.20), we obtain
1
Eq
(
In
cos
1 -
nx
2фо
cos
sinh2
7V(3
2фо
+
nx \
2 ф0
sinh
7Т(3
2фо J
dx = b-^~ tanli W'3
4фо
We rearrange the last formula and have finally a sought expression for b
b = -p- (F{-1/2,1/2,1; 1 - T2) + tt/2) .
Eo t
Thus, all the sought parameters of domain g which satisfy the statement (3.1)-(3.4) are determined completely. The next subsection deals with a study of characteristics of the obtained optimum bottom. As to solution $(w) of problem (3.1)-(3.4), it can be constructed according to the method outlined above, in Sect. 2. A graphical representation of this solution will be given below for a certain set of the domain g parameters, see Subsect. 4.2, Example 3.
Fig. 4. Parameter ft versus aEo/0o f°r different 0o•
3.4. Electrode Curvature and Field Distribution. It is of interest to find such characteristics of the optimum electrode form as its bottom curvature and electric field distribution at side facings (the field magnitude is constant at the electrode bottom, according to condition (3.4)).
We will search for curvature K and field magnitude |E| = |grad^| as functions of coordinate s G (—to, to), which is the arc length measured from point O' along electrode contour with values s = —to , s = 0, s = to corresponding to points A', O' and A. We will find first a relation between parameter t used in representation of arc r (see the previous subsection) and new coordinate s. To do this, we will use equality (3.11): integrating absolute value of its left- and right-hand sides along the corresponding straight segments [0, s] and
[i0o,t + i0o], we obtain just the arc length. Further, taking into account expression (3.12), we find
s(t)
dx
|Q(x + гфо)|
(3.24)
where
|Q(x + гфо)| |Q(x + гфо)|
1
Eo
1
Eo
\\
sinh2
nx
20o
sinh
sinh
2
-1 +
nx \
200
2ф0
sinh
7Г/3 200 /
if |x| < в,
if |x| > в-
Taking integral (3.24), we come to the final expression for s(t): if |t| < /3,
, . t
Ф) = -ЕГ, E0
(3.25)
while if |t| > в;
*® = -ET + — E0
/
6 nt
cosh —— 1 +
— 6 20o
I \
l
1
1
sinh2
nt
200 )
1 + Е2(0(*),^)
(3.26)
Note values |t| < /3 and, therefore, |s| < ,5/Eo correspond to points at the electrode bottom r; values t < —^ (i.e. s < —,5/Eo) and t > ^ (i.e. s > ^/Eo) correspond to the left (B'A') and right (AB) electrode side facings. We remind that quantity e < 1, the solution of equation (3.22), is related to /3 by formula (3.23), parameter 9(t) has been introduced by equality (3.16). Besides, the second type elliptic integral E2(9, \ft), see Subsect. 3.3., can be expanded into a Taylor series in terms of powers of small parameter e:
E2(0,y^) =
E
n=0
( —1/2)n
n!
where factors In(9) are subject to recurrence formula (3.18).
We remark also that expression (3.25) for arc coordinate s follows, in particular, the whole length of the electrode bottom r equals to 2^/Eo.
Now we start finding curvature K(s) and field magnitude |E| at the electrode contour. The curvature of the electrode side facings is obviously equal to zero, i.e. K(s) = 0 for |s| > ^/Eo. Curvature K(t) of an arc r set parametrically r(t) = r1(t) + ir2(t) is expressed by the well-known formula
K(t)
\т)гт - тш)\
|F(t)|3
t
Curvature of the same arc parametrized with the coordinate s will be found by the relation
K(s) = K о t(s),
where, in our case, t(s) = Eos for |s| < ^/Eo. Using formula (3.14) for r(t), we obtain
К(s) = cosh 7T^S (sinh2 — sinh
V ' 20o 20o V 20o
7Г[3 . n 2 kE0s\ 1^
207;
|s| < e/E0.
(3.27)
Here we observe the singular behavior of the bottom curvature at the points of the junction with side facings. Namely, it is not difficult to verify that when coordinate s approaches ^/Eo from the left (i.e. w ^ B as w G r), the following asymptotic relation is valid
(!rs) + 0(lrs)
and a similar relation takes place for s ^ —^/Eo + 0 (i.e. w ^ B' as w G r).
To obtain an expression for field intensity magnitude |E(t)| along the electrode contour at a point corresponding to a certain value of parameter t, we observe first that this magnitude coincides with the modulus of mapping Q(Z) when Z = t + i0o, according to Subsect. 3.2. Using expression (3.13) for Q, we find
|E(t)| = E0, if |t|< в;
(3.28)
( nt sinh ——
______20o
sinh
пв
20o \
sinh2
nt
20o
1
sinh2
7Г/3 20o у
if | t| > в.
(3.29)
One can determine field magnitude at a point which lies in the electrode contour at distance s (measured along the contour) from point O . For this purpose, parameter t in formula (3.29) should be substituted by arc coordinate s in accordance with relation (3.26).
Figures 5 and 6 give the graphs of the electrode curvature (3.27) and field magnitude (3.28), (3.29) at the electrode contour versus arc coordinate s.
Fig. 5. Curvature K of optimum electrode contour versus arc coordinate s.
2' Iff
*0
1 9
-3 -/*/*. ( ?/*• *
Fig. 6. Field magnitude ^| at the contour versus arc coordinate s.
4. Main Field Characteristics
4.1. Analytical Representations. The multipole method enables also to obtain analytical representations for any derivative of the boundary value problem (1.1)-(1.5) solution $(w) as well as for the harmonic conjugate of this solution. Thus, function <F(w), such that function ^(w) = <F(w) + i^(w) is holomorphic and *F(0) = 0 (this function is unique in view of simple connectivity of domain g [2, 3]), can be expressed similarly as (2.1):
<J(w ) = u7q (w) + U (w), here the formulae for U70 and 7(w), analogous to (2.2) and (2.6), take place
U0(w) = - (0! In |1 + F(w)I - 02 In |1 - F(w)I), п
and
K
7(w) = lim V afff^w), Qp(w) = Re [F(w)]p;
p=i
the last limit exists for every w £ g. Summing the functions $(w) and i$(w), we find expression for complex potential
1 K
Ф(w) = — (0i ln[l + F(w)] — 02 ln[l — F(w)]) + lim [F(w)]p. (4.1)
п ^—/ p
p=i
Furthermore, the derivative of holomorphic function ^(w) with respect to complex variable w can be readily obtained because the multipole method admits differentiation of any order. Differentiating (4.1), we get
Ф'(ги) = F'(w) I — п
1 + F (w) 1 — F (w)
K
+ paK [F (w)]p 1
^ p=i
Modulus of the last function coincides with the field intensity magnitude |E(w)| = |^'(w)|.
4.2. Specific Implementation. The specific implementation for the obtained solution was performed for various sets of domain g parameters (quantities a, b and forms of arc r) and various distributions 0(w) of boundary potential. For range of ratio a/b from 0.2 to
5, for sufficiently smooth arcs r and distributions 0(w) chosen in accordance with physical reasons, it was sufficient to use 20 multipoles in order to reach global relative error for field intensity E less than 10-3 everywhere in closed domain g.
Figures 7-9 demonstrate numerical results for Examples 1-3, respectively. For these three examples are presented:
a) equipotentials {w : $(w) = const},
b) lines of force {w : $(w) = const},
c) lines of equal intensity magnitude {w : |E(w)| = const}.
Example 1. The solution of problem (1.1)-(1.5) with the following input parameters is considered: a = 1, b = 0.9; arc r is specified as a graph of dependence
v(u) = 0.9 - 0.2(1 - u2)1/2.
Potential distribution at arc r was prescribed as function of «-coordinate
0(u) = 0.75(u + 0.2)2 + 0.62;
note the continuity condition for the potential along the electrode contour follows: 0i = 1.1, 02 = 1.7.
a.) distribution of equipotentials
c) lines of equal intensity magnitude
Fig. 7. Illustrations for Example 1.
Example 2. The solution of problem (1.1)-(1.5) with the following input parameters is under consideration: a = 1, b = 1; arc Г is specified as a graph
v(u) = 1 — 0.4(1 — u2)1/4 + 0.15(1 — u2) exp(-5u/6).
Potential distribution at arc Г was prescribed as follows
ф(и) = -0.175м3 + 0.525m + 0.35 + 1.35 cos4 -u
2
and ф1 = 0.8, ф2 = 1.5.
Example 3. The solution of optimum problem (3.1)-(3.4) with parameters a = 0.8, ф0 = 1, E0 = 1.3 is under consideration.
b) lines of force 5) lines of force
c) lines of equal intensity magnitude c) lines of equal intensity magnitude
Fig. 8. Illustrations for Example 2. Fig. 9. Illustrations for Example 3.
References
1. Vlasov V.I. The Multipole Method // Technical report. / M.: Computing Centre of the
Academy of Sciences of the USSR, 1976 (in Russian).
2. Vlasov V.I. On the method of solution of certain plane mixed problems for the Laplace equation // Dokl. Acad. Nauk SSSR. - 1978. - 237;5. - P.1012-1015.
3. Vlasov V.I. Boundary value problems in domains with curved boundary / M.: Computing Centre of the Academy of Sciences of the USSR, 1987 (in Russian).
4. Vlasov V.I., Volkov D.B. Solution of the Dirichlet problem for the Poisson equation in a
domain bounded by a polygon with a rounded corner // Soviet Math. Dokl. - 1989. - 39;3. -
P.606-609.
5. Vlasov V.I. Boundary value problems in domains with curvilinear boundary / Doctoral dissertation / M.: Computing Centre of the Academy of Sciences of the USSR, 1990 (in Russian).
6. Paltsev A.B., Vlasov V.I. A method of solving the Dirichlet problem for domains with a narrow slit // Russian Academy of Sciences. Doklady on Mathematics. - 1993. - 47;3 -P.404-409.
7. Vlasov V.I., Volkov D.B. The multipole method for Poisson’s equation in regions with rounded corners // Comput. Maths. Math. Phys. - 1995. - 35;6. - P.687-707.
8. Vlasov V.I. Multipole method for solving some boundary value problems in complex-shaped domains // Zeitshr. Angew. Math. Mech. - 1996. -76. Suppl. 1. - P.279-282.
9. Vlasov V.I., Volkov D.B. Analytic-numerical method for solving the Poisson equation in domains with rounded corners // Zeitshr. Angew. Math. Mech. - 1996. - 76. Suppl. 1. -P.573-574.
10. Vlasov V.I., Volkov-Bogorodsky D.B. Block multipole method for boundary value problems in complex-shaped domains // Zeitshr. Angew. Math. Mech. - 1998. - 78. Suppl. 1. - P.1111-1112.
11. Vlasov V.I. and Skorokhodov S.L. Multipole method for the Dirichlet problem in doubly connected domains of complex geometry: a generaldescription of the method // Computational Mathematics and Mathematical Physics. - 2000. - 40;11. - P.1567-1581.
12. Vlasov V.I., Paltsev A.B. Asymptotics of the solution to the Dirichlet problem for Poisson’s equation in domains with a narrow slit // Computational Mathematics and Mathematical Physics. - 2003. - 43;12. - P.1718-1737.
13. Vlasov V.I., Paltsev A.B. An analytical-numerical method for conformal mappings of complex-shaped domains // Doklady Mathematics. - 2009. - 80;3. - P.790-792.
14. Bezrodnykh S.I., Vlasov V.I. An inverse problem for the Grad-Shafranov equation with nonlocal condition // Belgorod State University Scientific bulletin. Mathematics and Physics. - 2011. - 23(118);25. - P.21-40 (in Russian).
15. Bezrodnykh S.I., Vlasov V.I. On a certain problem of constructive theory of harmonic mappings // Contemporary Mathematics. Fundamental Issues. - 2012. - 46. - P.5-31 (in Russian.)
16. Svelto O. Principles of lasers / 3rd ed. / New York: Plenum Press, 1989.
17. Berenstein C.A., Gay R. Complex variables. An introduction / New York: Springer-Verlag, 1991.
18. Henrici P. Applied and computational complex analysis. V.1 / New York: John Wiley & Sons, 1974.
19. Lusternik L.A., Sobolev S.L. Elements of function analysis / Moscow: Nauka, 1965 (in Russian).
20. Bakhvalov N.S., Zhidkov N.P., Kobelkov G.M. Numerical methods / Moscow: Nauka, 1987 (in Russian.)
21. Lavrentiev M.A., Shabat B.V. Problems of hydrodynamics and their mathematical models / Moscow: Nauka, 1977 (in Russian).
22. Birkhoff G., Sarantanello E. Jets, wakes and cavities / N.-Y.: Academic Press, 1957.
23. Gurevich M.I. The jet theory of ideal liquid / Moscow: Nauka, 1979.
24. Lavrentiev M.A., Shabat B.V. Methods of the theory of functions of complex variable / Moscow: Nauka, 1973.
25. Lavrentiev M.A. Variational methods in boundary value problems for systems of elliptic type equations / Moscow: Academy of Sciences of USSR, 1962 (in Russian).
26. Janke E., Emde F. and Losch F. Tafeln hohrer Funktionen. 6 Auflage / Stuttgart:
B.G.Teubner, 1960.
27. Bateman H. and Erdelyi A. Higher transcendental functions. V.1 / New York: MaGraw-Hill Book Co., 1953.
ИССЛЕДОВАНИЕ ЭЛЕКТРИЧЕСКОГО ПОЛЯ В ЛАЗЕРЕ С ПОМОЩЬЮ МЕТОДА МУЛЬТИПОЛЕЙ
А.Б. Пальцев
Учреждение Российской академии наук Вычислительный центр им. А.А.Дородницына РАН, ул. Вавилова, 40, Москва, 119991, Россия, e-mail: [email protected]
Аннотация. Работа посвящена модификации метода мультиполей и его применению к исследованию электрического поля в лазере специальной конструкции. Найдена оптимальная форма электродов в этом приборе. Для основных характеристик поля найдены явные формулы. Полученные численные результаты подтверждают высокую эффективность и точность используемого метода.
Ключевые слова: краевые задачи, метод мультиполей, расчет электрического поля в лазере.
