Inverse problems of coherent optics The use of the stationary phase method in problems of focussing radiation onto a curve

169

REFERENCES Shkola, Moscow, 1978. 1. LEPENDIN L.F., Acoustics, Vyssh, 2. REKACH V.G., Guide to the Solution of Problems in the Applied Theory of Elasticity, Vyssh. Shkola, Moscow, 1984. 3. BAIAKRISHNAN A.V., Applied Functional Analysis, Nauka, Moscow, 1980. 4. IL'IN V.A. and POZNYAK E.G., The Foundations of Mathematical Analysis, Part II, Nauka, Moscow, 1973. 5. IADYZHENSKAYA O-A., Boundary Value Problems of Mathematical Physics, Nauka, Moscow, 1973. 6. RIESZ F. and SZOKEFALVI-NAGY B., Lectures on Functional Analysis, Mir, Moscow, 1979. 7. TIKHONOV A.N. and ARSENIN V.YA., Methods of Solving Ill-Posed Problems, Nauka, Moscow, 1979. 8. VASIL'YEV F.P., Methods of Solving Extremal Problems, Nauka, Moscow, 1981. 9. LEONOV A.S., On the use of the generalized principle of the error of closure for the solution of ill-posed extremal problems, Dokl. Akad. Nauk SSSR, 262, 6, 1306-1310, 1982. 10. TIKHONOV A.N. and GLASKO V.B., The use of the method of regularization in non-linear problems, Zh. vychisl. Mat. mat. Fir., 5, 3, 463-473, 1965. 11. GLASKO V.B., GUSHCHIN G.V. and STAROSTENKO V.I., On the use of the regularization method of A.N. Tikhonov to solve non-linear systems of equations, Zh. vychisl. Mat. mat. Fiz., 16, 2, 283-292, 1976. 12. KALITKIN N.N., Numerical Methods, Nauka, Moscow, 1978. 13. SAMARSKII A.A. and NIKOLAYEV E.S., Methods of Solving Net Equations, Nauka, Moscow, 1978. 14. SAMARSKII A.A. and ANDREYEV V.B., Difference Methods for Elliptic Equations, Nauka, Moscow, 1976. 15. TIKHONOV A.N., GONCHARSKII A.V., STEPANOV V.V. and YAGOLA A.G., Regularizing Algorithms and a priori Information, Nauka, Moscow, 1983. 16. HIMMELBLAU D.M., Applied Non-linear Programming, Mir, Moscow, 1975.

Translated

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

vo1.28,No.5,pp.169-177,1988

by E.L.S.

0041-5553/88 $lO.OO+O.OO 01990 Pergamon Press plc

INVERSE PROBLEllSOF COHEREFiTOPTICS THE I;% OF ThE STATIOhAKY FhASE IlETtiOD IN FROBLCtlSOF FOCUSSING RADIATIOII Oi\iTC A CURVE' A.V. GONCHARSKII,

M.V. KLIBANOV

and V.V. STEPANOV

The behaviour of the intensity of radiation focussed into a smooth curve is investigated using the stationary phase method for the Fresnel The concept of the intensity of focussed radiation is integral. introduced and its connection with the intensity of radiation in the approximation of geometrical optics is studied. The asymptotic properties of the intensity of the radiation in the neighbourhood of the ends It is shown that, in the of the focal curve are investigated in detail. neighbourhood of the ends, there is an amplification of the intensity on a certain curve involving a type of build up. 1. Asymptotic behaviour of the intensity of the radiation on the focal curve. Let a scalar wave field, propagating along the Or-axis, have a value a(~, y, O)=A(r, Y)e'"'.Y' in the plane are slowly varying real functions A(z, y) z=o, where and ((1~. Y) In problems of the synthesis of optical phase elements, theamplitude of their own arguments. A(r,y) of the radiation, which is incident upon an optical element lying in the z=O plane, is usually known and is determined by the radiation source used, while the phase (o(z,y) has to be determined starting out from the requirement that the radiation should be focussed in When the phase of the incident radiation is known, there are a number of the plane. i=j technological methods for preparing relief optical phase elements which form an outgoing wave plane (see /l/1. field with a specified phase distribution in the z.=O

’ *Zh.vychisl.Mat.mat.F~z., USSR28:5-L

28,10,1540-1550,1988

170 Everywhere below, it is assumed that the magnitude of the focal length, f, of the element is significantly greater than both the dimensions of the element and the dimensions of the image which is focussed in the 2-j plane. Under these conditions, the magnitude of the field in the focal plane can be written using the Fresnel integral /2/: z=f

The integration in (1.1) extends over the domain C which is occupied by the optical element, which we shall assume to be bounded and strictly convex. Let us consider the problem of synthesising an optical phase element which focuses radiation from a domain G onto a smooth curve 9 lying in the plane which is specified by its normal parametric form z=j z=&(e),

~-h(e),

i,v3~+tio*w4,

e=(o,

0.

We shall consider the asymptotic behaviour of the integral (1.1) as k+oo using the stationary phase method /3/ for this purpose. In the case of the fairly continuous functions cp(g, V), the asymptotic form of (1.1) for fixed values of (x, y), isdetermined A(& (1) and by the occurrence of stationary points. Let us use the notation s(e, n, x, I/)-{(z-e)'+(y-?)*I/ 25i-cp(&n). Then, the points of the stationary phase which correspond to the point (x, Y) of plane, are given by the relationships the z=j &(g,rl.r, y)=O, System

(1.2) is equivalent

&(&,t1,r, y)=O.

(1.2)

to the system r=E+fVr(E. rl),

y=rl+h(L

rl).

(1.3)

System (1.3) defines the image 1, of the domain G of the z=o plane in the focal plane z=j. The image (1.3) is identical to the image introduced in /4/ using the approximation of geometrical optics. Hence, the asymptotic form of the Fresnel integral (1.1) is determined at a pointI(x, y) plane by the structure of the prototype I,-'(x,Y). of the z=f In particular, if f,-'(2,Y) represents a single point (to.no) of the domain G, the asymptotic form of the integral (1.1) takes the form

,u(I,y,,),=A(Eo,?o)

(*[-“+0(i)

The leading term in this expression describes the intensity distribution of the radiation which has been transformed by the element in the z-1 plane, if it achieves a non-singular transformation without focusing. On the other hand, at point (x,y) in the z=fplane forwhichI,-'(r,y)=0, the asymptotic form of (1.1) has the form u(x,y,j)=o(l). From this point of view, problems regarding the focusing of the radation are characterized by the fact that a whole manifold of stationary points corresponds to certain points in the focal plane 2-j. For example, focusing of radiation at a point (x,y) is associated with the fact that the set I.-'(2.Y) is a certain manifold of non-zero planar measure. In this case, it is obvious that the asymptotic form of the field at the focal point is of the order of O(k) as li+=. The problem of focusing radiation onto a line, which is considered in this paper, is characterized by the fact that, in the case of the points on the focal curve 9, the sets are smooth one-dimensional manifolds. The asymptotic form of (1.1) at thesepoints I,-'(5 Y) is of the order of O(k%) as k+m. In the case of the remaining points in the focal plane, the set I,-'(x,y) is empty and the asymptotic form has the form of o(l). The properties of the mapping I, which convert G into a smooth curve 2 in such a manner are such that I,-'(x. Y) is either empty or is the smooth curve which has been studied in detail in /4, 5/. In particular, it has been shown in /4, 5/ that, if SpI,'(g.q)PO everywhere in the domain G, then, in the case of the points we), Yde)wz, the prototype r(e)=l,-we), h(e)) is a segment of a line, the normal to which is parallel to the tangent to 2 at the point b(e), y(e)) (s~I;(e, n)-2+f1mu(g, n)~cp~(~, ?)I and 1:(&n) is the linearized image I.(&.tl) at the point (6,n)). The use of the asymptotic form of the Fresnel integral (1.1) enables us to carry out a detailed investigation of the intensity distribution of the radiation in the neighbourhood of the focal curve and thereby to formulate the problem of focusing radiation into a line with a specified intensity distribution. Everywhere below, we shall assume that A, pC'(c) in the closure I?. Let system (1.2) not have any solutions in C with respect to E. and n for all points (x, y)$P (I,-'(x,y)-0). For any points (x, y)~9, let system (1.2) degenerate into the equation S,(g,n,r, y)=O. the set of solutions of which is a continuous curve. We shall assume that the curve 9' is continuous. everywhere in the complement Moreover, let SpZ.‘(k tl)--2+f[wt(&, rl)+ vh(b, 11)I+0 C. Let us fix a point (xo(C3),yo(0))&? on the focal curve. Without loss of generality, we can assume that the coordinate system is chosen in such a manner that i,(e)-0, tiO(e)--i, 2,(e)= We shall be interested inthedistribution of the field of the focussed radiation b(e)-0. (1.1) on a normal to the focal curve. Under the assumptions which have been made, the set of

171 is a line points of the stationary phase which corresponds to z-z,(e), y-Pa(8) r(e)-I*-'(0, ee(L,, LX) (see /4/j and, since has a constant St(t,Q, 0, 0)-O on r(e), S(e, qO, 0, 0) 0) : q-h value on r(e). Let the domain

e+L,xd);

G-{(E, q) : v,(t)
the integral

(1.1) at the point

(2,O)

can be written

in the form

(1 C-E)’25+ (1’ +dE,rl)]}drl.

'exp ik

For any values of z and E, the stationary points of the internal integral with respect to n are determined by the relationships S,(b, n, 5.0)~SJE. q, 0, O)=O. Consequently, this point is unique and coincides with no. Let A>0 be a fairly small number. Then, when e=(L,+A, Lx-A) , the stationary point lies within the interval (V,(e),V,(E)) and, for any &E(Lt, LA, the asymptotic form

hold and, moreover, the function R,(E, k) is uniformly bounded in the interval (LI.La,) together with its derivatives and R,(g, k)+O when uniformly on k-cm [L,+A, &-Al together with its derivatives /3/. Precisely the same applies for all ee(L,,L,]. “,(I, 5 AR. rl)exp[~kS(k.rl,t.O) V,lO

and

ldq -

is unifor;y_bounded when &92[,&,,Ll] and uniformly bounded when together with its derivatives. Hence, the asyptotic (1.1) has the form

R&(6. k)

A] tends to zero as integral

ee[L,-A,L,iform of the

‘$1

exp[ik(f+S,))exp -iz (

)

X

(1.4)

L* A(Evrl) exp( -~)dEt(~)‘iexp[ik(f+S.)]X s ‘~ ISPl,'(:.?)I" R(t,k)exp(

and the function R,(E, k) is as together with its k--m of the points L, and L,. For example, it follows the modulus of the field) is pression

-y)dt,

uniformly bounded together with its derivative derivative uniformly on any set not containing

and R(E. k)-+O the neighbourhood

from (1.4) that the intensity of the radiation (the square of described, at points immsdiatelyon the focal line, by the ex-

(1.5)

The use of this expression to determine the distribution of the intensity of the focused radiation along the focal curve has a number of specific features associated with the rapid decay of the magnitude of the field as it becomes more remote from the focal curve and the In fact that the rate of this decay is different for different points on the focal curve. the case of the parameters which are typical of problems concerned with the synthesis of optical elements, the width of the focal line is a quantity with an order of magnitude of 10-100 Pm. In the majority of practical problems, the resolution of the receiving devices exceeds these characteristic dimensions and an intensity is recorded by them whichis averaged over a certain domain, the characteristic dimensions of which will be denoted by A. Hence, the observed intensity of the Focused radiation can be written in the form

172

Let us now consider the asymptotic behaviour of this expression as k-L=. the notation A(E,~r)lSpl,'(&,rlo)I-K--B(b). Then,

We shall use

XI

sin[ kA &-Et)

if)

da,+j;R,&)dElj I,

n(C,-E!)

I.

R,(b)x

sin[kA(Ez-El)lfl dE

n&-b) Using the notation

kA/f+,

*

we have

and R,(&, b) is uniformly bounded on (LI,t,) and uniformly tends to zero as B(t) is doubly continuously differentiable. Then, [L,+A, J.&A], if

I

bb” r,))

A’& d

dz+o(i)

as

B-m

on

k-w.

ISPI/(e.s)I

Let Z,-'(e,, 0,) be the prototype of an arc of the curve 2 and M(O,). Now let us consider

between the points

M(S1)

(1.6)

II .@(E.n)dEdn. I.-'(l),.@,) We introduce the coordinates (0.l) into the domain G is such a way that z(b,d-vfdb. ~6

t+de), +ih(e).

+~+fc,(b,

(1.74

(l.rn)

As the variable 1, we shall use the metric (L,'+A'-!)variable on the curve r(S)= note that this system of coordinates is orthogonal by virtue of the symmetry of the operator

I,-'(zO(EI),yo(0)). We

I,’ -

II=c =nII Yc Y,

.

The Jacobian for the transformation from (5, n) to (I&1) is equal to

(6,

Without loss of generality, we will select a rectangular Cartesian system such that &(0)-O, while &(0)-i and, fromthe condition I&-&?, we get

of

coordinates

q)

I,’ =

II00

0 (1.8)

II ’ SP1.'(6,rl)

and this means that it follows from (1.7) that

riodO-SpZ,‘(g,

q)dtl.

We

determine

the

173 derivatives Jacobian

l&-O, 0,-Sp1,'(C, n) directly

from this.

I-

D&l)

the expression

for the

I

=ISPl.'(e.,)I.

D(b>n)

Next, by changing

Hence, we obtain

to the variables

(6.1)

in the integral

(1.61, we get

and, consequently,

A’(E,tl)dtdrl+o(i),

k--m.

The principle term in the latter expression is naturally used as the determinant of the intensity of the focused radiation as k-m. The use of this concept of the intensity of the focused radiation enables us, as was done in /4/. to determine the phase function 9ce, 9) which ensures that the radiation is focused in the smooth curve 9' with the specified intensity distribution. 2. Asymptotic behaviour of the intensity in the neighbourhood of the ends of the curve 9. Below, we shall assume that the domain C is a unit circle, that the curve 9 is open and is the closure of the (r,,y,),(zz,yd are its terminal Points and ((z,,yl))U((z,,yd)UP=&? curve 9 in the sense of set theory. Let us investigate the asymptotic behaviour of the field strength as k-cm in the case of points (2, y)&?, but lying close_to the points lyinginthe 6For small C-0, we shall denote the set of points (r, Y)_*ip (z,,Y,). (2,. Y.). neighbourhwdof apoint (zf,Y,),j-i, 2 by V,(6). We shall initially investigate the asymptotic behaviour of the intensity at the points and consider the function I, as continuously extended (r,,VA, i-1. 2 V(E. 11) and the mapping on the complement C. By using the results in /4/, it can be shown that, if pm((z, y)+(z,. y,). the limiting direction of the straight line r(r, y)=I,-'(t, y) will be the direction of the tangent to the neighbourhood aG at the point (&,,nj)-I.-'(rjryl), (Ej,n,)EdC, j-i, 2. For example, let us consider the point (b,.q,). Without loss of generality, we shall aG at the assume that (E,,rl,)=(O,O) and the direction of the tangent to the neighbourhood point (0.0) is identical with the direction of the 5 axis (see Fig-l). By analogy with the behaviour of the function S(i,n, r,y) at the internal points of G, we shall assume, in accordance with (1.2), (1.3) and (1.81, that, when (E,q)-(0,O) and (E,n)EC, the function can be represented in the form S(E, 11,=*Y) S(E, % r.Y)--hn*+o($),

h-=sp1,'(0,0)+0.

for small 00. Since the points stationaryphase in the case of the function S(& n, z, y), radiation at the point (z,,y,), when k-w, has the form

Let G.-{(E,(I)%O-=V=~]

A(e,rl)erp[lkb(e,cl,t,y)ld6drl+o(k-’)

Z(r*,YL)'

Let us consider

(2.1)

(8,q)eG\G. are not pints of the intensity of the focused

, I .

(2.2)

the integral H.(k)-5

A(e,Il)exp[ikS(e,tl,t,,y,)ldgdrl. 0.

It is obvious

that the representation

H.(k)-,f drl j

A(E,rl)exp[fkS(E,rl,z,,yl)lde,

(2.3)

holds for the function H.(k) bee Fig.11, where t'(n)EC"[-e, e], t(n)-[rl[i+~(i)]]'~ as exists such that p(O)-0,$(O)-i,t(p(y))-yK Vj'". Consequently, a function p(y)=@?[-e,e] Hence, by changing the variables of integration in (2.3) using when y>O (see /3, p.39/). the formula (rl,E)+(P(v),E), we get P-‘(G)

H.(k)-

I0

4s

p’(y)&/

s

A(g,p(Y))exp~ikS(e,p(Y),z,,Y,)]dt,

--/n

(2.4)

174 where p-' is the function which is the inverse of p.

Fig.1

Fig.2

After obvious replacements of variables, the integral (2.4) can be reduced to the form I

/,P,.)l

(2.5)

It

follows from (2.1) and the properties of the function p that, when z++O, S(az,

p(z’),

21,

y,)=)c~‘Il+JJ(z)l.

(2.6)

By using (2.5), (2.6) and the stationary phase method, we get /3, p.103/ fI.(k)=k_‘!*-

4r('1‘) h" -4(0,0)X

(2.7)

exp[~+ikS(o.o.=,,y,)][f+O(k-“‘)I, where r is the Euler gamma function. It follows from (2.2) and (2.7) that the intensity on k--oo,has the asymptotic form the ends of the curve 9 when

Hence, by virtue of (1.4), (1.51 and (2.8), attenuation of the intensity compared with the interior points of 4 is observed on the ends of the curve 9 for large values of k. Let us now investigate the asymptotic behaviour of the intensity I(r,Y) for the points that is, for the points (t,Y), lying close to the ends of the curve 0 but not (I,idEVJt61, does not have stationary points (E,(I)& when (z, lying on 0. The function S(&n,t, y) when k--a, t Hence, taking account of (l.l), we get /3/, for (r,Y)& Y)CST

where v is the direction of the exterior normal to the neighbourhood aGand do is an element of the length of the arc of the unit circle aG, o=[O, 2s). Let us consider the integral P(s,y,k)-jF(z.y,o)exp[ikQ)tr,Y,o)lda, 80

(2.10)

where

@(r.Y*(r)==S(E,T), I,Y)) $z$'

(2.41)

Let us next consider the system of equations $+,Yd=o,

By virtue of (2.11, we have

(2.12)

175

(2.13)

Here, it is assumed

By taking account

that

(Ej,~)=(e(@),

q(Q)), j=l, 2. I h+,Y,o)=+Y,o):

(2.11),

of (1.1) and

Let r=l,2.

we get

Consequently,

By transforming

to polar coordinates

with the origin

at the centre of the unit sphere dG, we

get hi, h,, h z= h,,

-

f-+0.

Hence, it follows from (2.13) and the theorem on implicit functions that the system specifies the curve ~j-{~(a),Y(a)}, j=1, 2 in a small neighbourhood of the point (9, By using the stationary phase method and (2.9)-(2.12), we obtain, when IL-~, Z(x, y)=O(Pl'"), Moreover,

(x, Yj'E%\Z

m=m(x,

(2.12) ?h)

.

(2.14)

y)>3.

if (z,Y)=(x(a),Y(o))Erl\~,

when

then the number m=3in (2.14). Letusnow assume that(s, y)=II,(G)\yl. #The function (D(x,y, 0) has at IS', o"= [O,~IC),a maximum point and a minimum point. However, least two stationary points then, by virtue of (2.12), since (z, y)= Vj(6)\yj, $J

at each stationary point co. Hence, by taking account

of

(x,y,oa)+O

(2.91, we get

I(x, y)=O(IP),

(I, Y)EV1(G)\Yj,

k-+m,

j=l,

2.

Let us now elucidate the structure of the curves 1, and, also, when the set r,\2 non-empty which is important by virtue of (2.14). We will first present some facts from catastrophy theory /6-9/ which are relevant to the case being considered here. For small let V,.=(s=R", /51
f: U,-R,

V,.=(s=P, f=CWe),

is e>O,

ISI<&), m>l.

The vector s is conveniently called the variables of state while the vector z is the controlling parameters. Catastrophy theory studies the local structure of a function f close to its critical points, that is, close to the points for which V,f(z, s)=O. (x7s) Let r,(s), r2(s): Ir,,+R’ be diffeomorphisms and ~~(2) : V,,-R” be a local diffeomorphism rl,7X and y, be m times which depends on the parameter s=VZI, let the vector functions continuously differentiable with respect to their arguments, and let the vector function also be m times differentiable with respect to the parameters s,,...(s.. The function Y,(X) is locally equivalent to the function f (see /9/l. g(s, s)=f(y,(.r), II(S))+%(S) The function f(z, S) is said to be stable in the neighbourhood of the point (O,O) if, which is sufficiently small in the and for any function p(x,s) for sufficiently small E>O is locally equivalent to f. norm of P(U-.) the function f(r,s)+p(x,s) Now l&t n=2 and I.=1. We will denote by F the set of all functions foCm(ci,) and, by F' the set of all stable functions (in the neighbourhood of the point (O,O)), from F. According to Whitney's theorem, F: is an open set in F which is everywhere dense in the norm of P(D.) (see /6/). Hence, the stable case is "typical" while the unstable case is "untypical" and every unstable function may be converted into a stable function by a small Let us now assume that f=F’ and perturbation.

176

2 to,o) -$O,O)-0. It follows from Whitney's theorem that the function f in the neighbourhood of the point is equivalent to the function g(y,, y,,p) of the form (p is the variable of state, and and y, are the control variables) k?(Yl, Yl,P)'~(P'+YIP'I2+YrP), The projection

of the set of points

(2.15)

IA lyil,ly~l
(t,s)Elj,, for which

onto the space X is the curve which is assembled. That is, during the mapping of the equivalence, which transforms the function f into the function g, this curve transforms a curve of points of the form r-((y,, yt)~4y,‘+27y+O)=(yt(s), The smooth curve r is, correspondingly, which

onto the y space.

(0,O) y,

into

Y%(S),Y,=S'4-"',Y,~S'27-').

the projection

of the set of points

(Y,,Y,.P), for

We note that

5

(YhYr,PI 20

when

(~a,Y&mr\{(o,O)).

An example of the form of the assembly is shown in Fig.2. Let us now consider the function (see (2.11) in the 6-neighbourhood of the (D(r,y, a) 00 is fairly small, We shall assume that the variable point (ri,Y,,04, where the number (J is the state variable and the variables z and y are the controlling parameters. We shall subsequently consider the function ID as being stable in the neighbourhoods of the points that is, we shall assume that we are dealing with a typical case (the proof (4, y>,0,).i-1, 2, of stability is usually extremely difficult /6/ and it was not possible to do this for the actual case which is being considered). It follows from Whitney's theorem and from (2.1), (2.12)-(2.15) that a curve y, is an assembly and that the points (r,,y,) are critical accumulations (see Fig.2). bet us initially assume that, in the integral (2.10), the function @(z, y, a) has the form of the function g in (2.15) when z-t,=y,, y-y,5y,,o-a,=p. Then, on the basis of (2.12), (2.13) and (2.15) and the stationary phase method, we get that I(t,y)-U(k-") when k-fm,(r, Y)~l,\p,j-i,2. It is known that this asymptotic formula does not change on changing from the function 9 to the function Q, which is equivalent to it. (see /6, 7/). Consequently, the number m==3 in (2.14) and it is independent of the point (s,Y)~~~\p,j=l,Z. The following theorem is thereby proved.

Theorem 1. Let the function be stable in the neighbourhood of a point (2,.yI, @(r, Y, 0) al),i-t, 2. Then, the smooth curve, yl, which is defined by equations (2.12) in a sufficiently small neighbourhood of the point (2,.y,) is an accumulation while the critical accumulations which coincide with the point (t,,Y,). We also postulate that the piece of the curve 9 d-neighbourhood of the point (r:,Y,) is not an accumulation. Then, the set lies in the 1,\z%fiV,(6)

is non-empty

and the asymptotic I(r, Y)--O(k),

relationships

(2,Y)EP,

1(2j,y,)==O(I;'$),

I(z,y)-ff(k-%), (2,Y)%fiV,(W hold when k-m. Hence, there is a "reinforcement"

of the intensity

compared with the intensity at the remaining YJF neighbourhood of a point (z,,yr),j-1, 2.

of the radiation

points

(z, y)&g

at the points

(r,Y)E

lying in a Small

REFERENCES 1. GONCHARSKII A.V., DANILOV V.A., POPOV V.V. et al., Devices for focusing radiation which is incident at an angle, Kvantovaya Elektronika, 11, 1, 116-118, 1984. of Systems and Transformations in Optics, Mir, Moscow, 1971. 2. PAPULIS A., The Theory 3. FEDORYUK M.V., The Method of Steepest Descent, Nauka, Moscow, 1977. 4. GOWCRARSKII A.V. and STEPANOV V.V., on the existence of smooth solutions in problems of the focusing of electromagnetic radiation, Dokl. Akad. Nauk SSSR, 279, 4, 768-792, 1984. 5. GONCIiARSKII A.V. and STEPANOV V.V., Inverse Problems of Coherent Optics. Focusing in a line, Zh. vychisl. Mat. mat. Fiz., 26, 1, 80-91, 1986.

177 6. ARNOL'D V.I., VARCHENKO A.N. and GUSEIN-ZADE S.M., The Singularities of Differentiable The Classification of Critical Points, Caustics and Wave Fronts, Nauka, Mappings. Moscow, 1982. 7. ARNOL'D V.I., VARCHENKO A.N. and GUSEIN-ZADE S.M., The Singularities of Differentiable Mappings. Monodromy and the Asymptotic Form of Integrals, Nauka, Moscow, 1984. 1984. 8. GILMOPE R., Catastrophe Theory for Scientists and Engineers, 1, 2, Mir, Moscow, 9. POSTON T. and STEWART I., Catastrophe Theory and its Applications, Mir, Moscow, 1980.

Translated

Vo1.28,No.5,pp.177-181,1988

U.S.S.R. Comput.Maths.Math.Phys., Printed in Great Britain

by E.L.S.

0041-5553/88 $lo.Cx3+0.00 @1990 Pergamon Press plc

EXACT SOLUTIONS OF THE EQUATIONS OF NON-LINEAR DIFFUSION OF A MAGNETIC FIELD* S.M. PONOMAREV

The non-linear mathematical models which describe the magnetic field in a conducting medium under the action boundary mode, when liberation of Joule heat is taken considered. Exact similarity solutions are found for

diffusion of the of a yiven into account, are some of the models.

In present-day mathematical physics it is often important to devise and study non-linear mathematical models for describing a process which cannot be described in terms of linear The computing experiment /l/, which is a combination ofanalytic andnumerical approximations. methods, is one of the main ways of studying a non-linear model. In the present paper we prove the existence of a finite rate of magnetic-field propagation in a conductor with zero background and show that this rate is greatly affected by We find the class of boundary modes (S-mode and LS-mode) /2-4/, the displacement current. the action of which gives rise to localization of the magnetic field; the domain in which the field is non-zero then remains fixed during a certain time interval, and the field in this domain can rise to values as large as desired. One of the simplest mathematical models that describe the non-linear diffusion of a magnetic field H(z,t) in a half-space (current-conducting but not heat-conducting z>o incompressible medium) under the action of a given boundary mode, in which the displacement current is negligible compared with the conduction current, is the problem (in MKSA units)

aH

-x==UE

for

t=-0,

otzcm,

(W

l?H 8E -jy--P'"x>

(lb) (lc)

o=i/AQ. H(0.t)=H,t".H(G O)=O, Q(r.o)=o, o
(la)

where uo=4nx IO-' v. set/A. m, a is the conductivity of the medium, E(.r,t) is theelectric 6 and A are constants field strength, a and Ho are positive constants, [(7]=~/m3, and which are found as the means of the experimental or theoretical dependences of the electrical resistance on the temperature T or the enthalpy. For most metals, f~=l for TD GTGT m, where TD is the Debye temperature and Tm is the melting point. For TGU.1 TD we have B='I,, since a-T-' J=-"1:. and Q-T'. For a plasma, Eliminating 3 from (11, we obtain

aH

1

dx

AQb aH

---_-_E

aE

-=-PO--, a.t 01 *Zh.vychisl.Nat.mat.Fiz.,28,10,1551-1557,1988

for

OCz
aQ E -=_a dt AQ

c-0,

(24 (2b)