RAY OPTICS FOR DIFFRACTION: A USEFUL PARADOX IN A PATH INTEGRAL CONTEXT

ABSTRACT

L.S. Schulman

Physics Department Technion-Israel Institute of Technology Haifa, Israel

Geometrical diffraction theory uses ray tracing techniques to calculate diffraction and other properties of the electromag­netic field generally considered characteristically wave like. We here study this dualism of the classical electromagnetic field so as to distinguish those aspects of quantum dualism that arise simply as properties of oscillatory integrals and those that may have deeper origins. By a series of transformations the solutions of certain optics problems are reduced to the evaluation of a Feynman path integral and the known semiclassical approximations for the path integral provide a justification for the geometrical diffraction theory. Particular attention is paid to the problem of edge diffraction and for a half plane barrier a closed form solution is obtained. A classical variational principle for barrier penetration is also presented.

I. INTRODUCTION

This article is about what wave~particle dualism isntt, That is, I will discuss a theory of little relevance to the fundamental problems of quantum mechanics and show that it has many charac­teristics of wave-particle dualism. This is being done not only out of general orneriness, but is offered in the spirit of the counterexample in mathematics: any philosophical conclusions one would wish to draw from wave-particle dualism ought not to be based on those aspects of dualism that I shall discuss below.

253

S. Diner et al. (eds.), The Wave-Particle Dualism, 253-272. © 1984 bv D. Reidel PublishinK Company.

254 L. S. SCHULMAN

The propagation of light is historically the first example of people being torn between wave and particle interpretations. The characteristic property of light leading to a particle inter­pretation was its propagation along straight lines or rays, while the wave property was later demonstrated through diffraction and interference. The theory I shall discuss manages to encompass in particle language - or more precisely, in terms of rays - many of the characteristic wave features of light: diffraction, tunneling and the illwnination of Hforbidden" regions near a caustic, It is known by the paradoxical juxtaposition: "Geometrical Diffrac­tion Theory" and is due to J.B. Keller (1). Since the introduc­tion of these ideas in the 1950s they have been developed not only by mathematicians interested in asymptotic expansions but for practical technological reasons as well, for example in the interpretation of radar reflections or in the construction of antennas (2).

As we shall see below, what makes the whole thing work is an oscillatory integral with a large parameter k, the wave nwnber. The same is true of the semiclassical approximation in quantwn mechanics and therein lies the close resemblance. But the quantwn parameter ~ carries more philosophical baggage and it is my purpose here to allow one to distinguish between the deep features of quantwn mechanics and between those that arise merely because of the asymptotic behavior of an oscillatory integral. It may also be worth mentioning that geometrical diffraction theory has nothing to do with photons. The only scale parameter around is the (large) ratio of characteristic distances to the wavelength of light; there would be no change if Planck's constant were ten orders of magnitude smaller or indeed zero.

In this article I will develop the geometrical diffraction theory from the Feynman path integral. Then I will show that many familiar semiclassical techniques lie behind the geometrical diffraction theory. But first I will say a few words about geo­metrical diffraction theory as well as about path integration and its relation to optics.

II. GEOMETRICAL DIFFRACTION THEORY

Consider an electromagnetic plane wave impinging on a per" fectly conducting half plane (Fig. 1). Geometrical optics pre­dicts a direct and reflected ray but there is also a diffracted wave. In fact, Sommerfeld solved this boundary value problem for Maxwell's equation in clo~ed form (3). He found that if the incoming wave is exp(it.x) then to l~ading order in k = \k\ the field in the shadow region is D e1kp / (kp) 1/2 where p is the distance from the edge of the knife edge and D is an angle

RAY OPTICS FOR DIFFRACTION

y

255

DIRECTION OF OBSERVATION OF OUTGOING WAVE

x

Fig. 1. Geometry of waves impinging upon and scattered off an infinite half plane perfectly conducting barrier. All angles are measured positively from the x~axis.

dependent factor of order one. Keller used this formula to suggest a kind of Huygens! principle for diffraction. He pos­tulated rays emanating from the edge and spreading into both the illuminated and shadow regions. As in the usual optics these rays carry a phase; this is the factor exp(ikp). The dropoff p-l/2 is appropriate to a cylindrical wave so that in all respects, once launched from the knife edge this ray is perfectly respectable. Its special chiracter as a diffracted ray is expressed in the factor k- l 2, which is a term that disappears in the k+oo limit. It may be a perfectly respectable ray - once launched - but it has a reduced amplitude due to its origin in a breakdown in the conventional laws of geometrical optics.

This example is canonical, that is there is rigorous infor­mation on the diffracted rays. Working from a number of such examples Keller develops a scheme for calculating the diffracted field in geometries not amenable to exact solution. Namely, when the medium is continuous do the usual ray tracing, but when en­countering a discontinuity look for a solved canonical problem which is locally of the same form and use it to determine the properties of the resulting diffracted rays. Then once these rays leave the neighborhood of the discontinuity they are treated

256 L. S. SCHULMAN

like any other rays. The key work above is "locally". The wave nature of the phenomenon is suppressed ~ asymptotically speaking ~ on distance scales large compared to A = 2n/k. This is why ray tracing works for media effectively constant on that scale and why the matchup with a solved geometry is also governed by the same scale. (Of course the linearity of Maxwell's equations has implicitly been used in allowing arbitrary superposition of dif~ fracted and other rays).

Besides diffracted rays Keller's theory includes something he calls imaginary rays. These illuminate classically forbidden regions, as in tunnelling and in the field near a caustic. Barrier penetration seems a strong argument for a wave inter~ pretation of quantum mechanics but we shall see that much of the machinery for dealing with rays in optics or with particle tra~ jectories in mechanics is also used in the apparently different barrier penetration problem. I shall not review the details of Keller's handling of imaginary rays; suffice to say that they agree with the WKB formulas obtained from the path integral below.

III. PATH INTEGRAL FOR OPTICS

Let u(x, t) or its potential. fraction n(x) , u

2 (n(x))2 a u _

c at2

be some component of the electromagnetic field In a medium with a slowly varying index of re­

satisfies

2 'V u = 0 (3.1)

For waves of definite frequency w, u(x,t) = w(x) exp(~iwt), and we suppose that for any direction n(x)~onst as Ixl~. Call this constant IE; for free space E = 1. By substituting and rescaling the spatial coordinate as follows

x = (3yc/w (3.2)

Eq. (3.1) becomes

1 2 ;Z 'Vy W + V(y)W (3.3)

where V(y) ~ E_(n(x))2. Thus a stationary solution of the wave equation satisfies a Laplace equation which in turn can be made to look like Schroedinger's equation by choosing the right letters of the alphabet. Of interest is the one letter for which I have desisted f!~m full alphabetic identification. ~ will play the role of ~ but I draw attention to its independent physical nature. In equation (3.2), c/w is essentially the wavelength

RAY OPTICS FOR DIFFRACTION 257

of the radiation. so that if y has order unity. then 13 is the scale factor measuring x in units of wavelengths. Geometrical optics is the large 13 limit. But then (3.3) tells us that our knowledge of the h ~ 0 behavior of the Schroedinger equation can be used to gain information about geometrical optics.

A convenient way to study semiclassical properties of (3.3) is by the path integral (4). Introduce an artificial "time!! variable T such that ~(y,T) = exp(-iETI3)~(y). We invoke the path integral to write down a propagator for ~(y,T):

~(y,T) = Idy' G(y,T;Y')~(Y',O)

(y,T) G(y,T;Y') I dy(.)exphI3S[y(·))}

(y' ,0)

T S[y(T')] = I dT' [k(;:,)2 - V(y)]

o

(3.4)

(3.5)

(3.6)

S is the classical action corresponding to (3.3) and the sum in (3.5) is over all paths starting at "time" zero at y~ and arriving at "time" T at y. T is eliminated by Fourier transformation. To summarize, the solution to the electromagnetic problem (with fixed w) is provided by the following quantity

00 (b, T) G(b,a;E) I dTexp(iI3ET) I dy(.)exp{iI3S[y(·)]} (3.7)

o (a,o)

As discussed in Ref. 4, Fermat's principle and the usual geometri~ cal optics are obtained by a stationary phase approximation (13~) for the ~l dimensional integral above. In particular, the re~ quirement that S be stationary with respect to variation of y(.) leads to the Euler-Lagrange equations. To get the semiclassical approximation to (3.7) we use the well-known form of this approxi~ mation for (3.5):

G(b,T;a) (3.8)

where ~ runs over classic~l paths only, Sc& is the action evalu­ated along path ~ and ~ S~/~a~b is the derivative of Sa with respect to the endpoints. Such a derivative is meaningful only when the classical paths do not form a continuous family and indeed such continuous families are characteristic of focusing where a2Sa/abaa is singular. The determinant allows for b and a to be vectors, Eq. (3.8) is a major simplification of (3.5) since

258 L. S. SCHULMAN

only paths satisfying the classical equations of motion appear. The object in the square root is the van Vleck determinant and was introduced to path integration by Cecile DeWitt (5). In optics it is deduced from the conservation of power in tubes of rays.

It is remarkable that in nearly all problems in which the time dependent Green's function is known in closed form it is given exactly by (3.8). Since we shall use the propagator for a free particle in two dimensions I give its explicit form

G (+b +) = S [is(+b_ +a) 2] 1 • ; a 4lTi. exp 4. .. (3.9)

IV. EDGE DIFFRACTION

To study knife edge diffraction 1 or the infinite half plane barrier 1 as illustrated in Fig. 11 it is sufficient to restrict attention to a two dimensional cross section. Obliquely arriving plane waves require consideration of the third dimension~ but it turns out that there is a simple transformation that eliminates this complication. As mentioned this was solved by Sommerfeld and we give the solution in the form provided by Lewis and Boersma (6). The geometry and angles are defined in Fig. 1. The incident plane wave is

u i = exp(-ik(xcos~ + ysin~)) = exp(-ikpcos(~-e)) o (4.1)

where (p,e) are the polar coordinates of the observation point. Define the function

t h(t) = n- l / 2 e- ilT/ 4 J exp(it,2)dt' (4.2)

_00

with h( +00) = +1. Then at the point (p,e) the total field is

u = exp[-ikp cos(~-e)] h[/2kpcos ~(~-e)]

- exp[ikp cos(~+e)] h[-l2kp sin ~(~+e)] (4.3) +

where the upper sign (here and in the sequel) is for the boundary condition u=O on the half plane and the lower sign corresponds to au/an = O. Comparing (4.3) and (4.1) we see that when varia­tion of h can be ignored the first term in u is essentially the incident plane wave. Similarly we can define a reflected wave by u~ = exp(ikp cos(~+e)) and this too appears in (4.3).

RAY OPTICS FOR DIFFRACTION

For large t h the sign of t. corrections to

is issentially constant; 0 or I, depending on The asymptotic expansion for h gives finer

u, namely the diffracted wave. For large t

h(t) 8(t) - ~ TI- l / 2 e iTI/ 4 eit2 i ; an(it2)~n (4,4) n=o

259

where 8(t) is the step function, A bit of algebra yields

a::::l and a = (n - 1/2) a 1. o n n~

1 i-I r Deikp 3/2 u'V 8(cos-2 (l/!-8))u + 8(cos2-(l/!+8))u + -- + O(k- ) o 0 IkP (4.5)

where

iTI/4 1 1 D = e [sec Z(l/!-8) + sec Z(l/J+8)]/2 I2IT (4.6)

For the "shadow boundary", namely ljr-8+TI, the form (4,5) is in", valid (t=O) and going back to (4.3) we see that the incident wave amplitude is down by exactly a factor 1/2 whereas the reflected contribution leads only to a diffracted wave given by the second term in D.

1/2The salient features of the diffracted wave are its overall k- amplitude and its form as a cylindrical wave (thinking 3-dimensionally). As mentioned in the introduction, this justifies thinking of the diffracted wave as comprised of rays emanating from the edge of the barrier.

V, SEMICLASSICAL PROPAGATOR FOR EDGE DIFFRACTION

A semiclassical calculation of the propagator for the knife edge may seem pointless given knowledge of the exact solution. But here our attitude is the same as Keller's: if your approxi~ mations can be verified against known solutions then you can have confidence in them when exact solutions are not available. Even more: out approach makes no use of special features of the geometry. It would be valid for a curved edge, or for any geo­metry whose scale is large compared to A. Thus the verification of our approximation against the leading terms in (4.5) is a justification of Keller's method.

First we calculate the propagator for fixed initial and final points and "time" 1" and then take apJ?ropriate limits and transformations to obtain the electromagnetic field. The geometry is shown in Fig. 2. The propagator is given by the path integral

260

a

c ,,-­,,, -...; , " " , " , " I " , , , , ,

L. S. SCHULMAN

y

x

Fig. 2. The initial point is a and the final point b, having polar coordinates (Pa.ljJ) and (Pb.e) respectively; all angles are measured counterclockwise from the positive x axis. The inter~ mediate point of integration is c. The dotted line ac is the direct path from a to c; similarly, for the dotted line cb. The pair of dotted lines aR~Rc form the (specularly) reflected path from a to c and together have the same length as the dotted line act where c' is the reflection of c in the y axis. The polar coordinates of c are (P.~) and the direct~direct path is not blocked by the barrier for e+n~~~-n. Similarly reflection off the barrier can occur for ~~2n-1jJ and the direct path cb is not blocked for ~<e+n.

+ + G(b,T;a)

~ (b, T) T

+1 dy(') exp{iS 1 dT' [~(ti,) 2]} (a,o) 0

(5.1)

summed over all paths that a~oid the barrier. Except for the barrier. n(x) =1 so that V(y) =0 and E=l (notation of Sec. II). We first pick an intermediate time Tc and use the general relation

RAY OPTICS FOR DIFFRACTION 261

~ -to 2 -to -to -to -to G(b,T;a) :::; fd c G(b,T - T ;c) G(C,T ;a) c c (5.2)

As polar coordinates (p,$) respectively. Now we make our *ajor all paths from a to classical paths using Eq. (3.9).

-to -+ -+ for a, band c take (Pa'~)' (Pb,6) and Choose Tc such that Pa/Tc = Pb/(T-Tc). a¥proximation. Instead of summing over

c and t to ~ we sum only over the the semiclassical propagator given in

For the geometry of Fig. 2, specifically for 0<6<; and~ ~<~< 3~/2, there is but a single classical path from C to b and this only for ~<6+~. The propagator on this leg of the journey is therefore approximated by

• -+ -+ 2 = S [lS(b~C)

4~i(T-T ) exp 4(T-T ) c C

(5.3)

-+ -+ To get from a to c there are two~~ssibilities: (1) a direct path with classical action (c-a)2/(4Tc) provided $~~-~ and (2) a reflected path, bouncing specularly off the barrier and which.can e~ist pro~id7d $~2~-~ .. The action for t~e reflected path 1S (laRl + \Rc\)-/4Tc (vert~cal bars denote d1stan~e), which is easy to calculate using c', the reflection of c in the y axis. The semiclassical propagator form ~ to t is therefore

G . D1.rect S is -+ -+ 2

+ GReflected = 4~.T {exp[4T (c-a) ] 1. C C

with the proviso that each term appears only in its classically allowed region. Eqs. (5.3) and (5.4) are inserted in (5.2) to give

-+ -+ Sy oo[ 6+~ iSADD 6+~ il3~D G(b,T;a) = Go 4~i pdp [ [ d~e + [ d~ e ]

o ~-~ 2~-~

- GDD -+ GRD

where we have used the notation

1 1 Y = T + T-T

C C

(5.5)

(5.6)

262

G o

1 Z ~ yp

L. S. SCHULMAN

(5.7)

(5.8)

(5.9)

v is the ~elocity a classical particle would have going from! to 0 to b and the argument of the exponent in Go is the classical action (time is) for that path. It is instructive to examine the quantity ADD. For the special case of thezshadow boundary, namely ~;e+TI, we have identically ADD; yp /4. Goo is now exactly liZ Go. This should be compared to the results of Sec. IV. At the shadow boundary (4.6) does not apply and u is given by liZ u 1 + [diffracted term], The diffracted term is as in (4.5) exceptOthat 0 includes only the second of its contri­butions in (4.6). namely that arising from the "reflected" wave. There is an earlier path integral calculation for the edge dif~ fraction problem (7) in which only the shadow boundary case is considered and which gets the factor liZ but not the additional diffracted term. The latter term for the shadow boundary case in fact arises from GRO ' In general both GOO and GRO contribute to diffraction.

Returning to the general case, we rewrite ADD in terms of

1 1T Wz ; ~(1jJ-e) - :2 (5.10)

For a the limits of integration in GOO are lal < 1T/Z - Wz and ADD becomes ADD = 1/4 ypZ + pv cosa sinwZ' The behavior of GOO for large S is given by the stationary phase approximation for the integral. It is clear that a=O is the only stationary point in a. Then the existence of nonexistence of a stationary point for the p integration depends on the sign of sin wz. For wz > 0 the stationary point in p lies on the negative P axis, outside the range of integration. Examination of Fig. Z shows this to be the situation in which the barrier blocks the straight line path from t to h. This forces the leading asymptotic term to be of smaller order in t3, SpecificallYI when the stationary point falls within the range of integration the two integrals give a factor lis. Having the point fall outside will introduce an additional l/If. This is the price of having a broken line path with a single bend.

RAY OPTICS FOR DIFFRACTION

For Wz ~ 0 the stationary pOint has p > 0; however 1 although the stationary point lies within the range of integra~ tion there is still a diffraction term due to cutting off the integral and this contribution is again l/IS relative to the leading term.

To do the integral use orthogonal coordinates ~1n with the s axis at an angle wI to the x axis. For the case w2 > 0 GOO becomes

263

(5,11)

Extending the n integration to 00 brings GOO to the form

-irr/4 . Z 00 1't2 e -1m = Go --- e f e dt lIT m

(5.lZ)

with

m (5.13)

Consider now the error introduced by extending the upper limits of integration for n to 00 This is problematic for small ~ and from Eq. (4.4) we see that the error is O(l/~{I3). Therefore if it were possible to discard from the ~ integration a range of ~ such that for all remaining ~ ~16» 1, then the replacement of ~/tanwZ by 00 would not affect the asymptotic behavior with 6. Take the range to be discarded to be O<~<~max ~ g(6)/16 where g(6) is a f4nction that goes to 00 with 6 more slowly than 16, for example 61/ 4 . The effect of this is to change the limits of integration in (5.13) to a new value of m' where m' = m + l/Z ~ ~max' Since m is growing with 6, to find the relative effect on the value of GOO we look at (m'-m)/m which is 0(g(6)/I6) and therefore goes to zero. Our arguments do not justify (5.12) as a uniform approximation (for wZ small); how­ever, one notices that for m=O GOO collapses to 1/2 Go, the correct value. In fact (5.13) is uniform since for small Wz the error from extending the limits of integration is O(wz/~IS). Hence ~max can be taken to be w2 g(6)/I6. This cancels the (JJZ appearing in m when one considers the ratio (m'-m)/m.

For the case wZ < 0 we write GOO as the integral over the entire plane minus an integral very much like (5.11) . Thus

co 6+rr 00 00 00 ~/ltanw21 f pdp f dcp ~ f d~ f dn - f d~Z f dn (5.14) 0 1jJ-rr -'" _00 0 0

264 L. S. SCHULMAN

+ + The first term is the £ree propagator from a to b. The second term has the s~ structure as (5.11) and therefore as (5.12) with m" == - (sly v sinw2' Thus

~im 2 . ,,2 -i1T/4 00 . 2

Goo G G ~1m e ! e1t dt == e e

0 0 /-iT mn

~im 2

;:: G e h(~m) (5.15) 0

exactly as in (5.12). It follows that our solution for the direct-direct part of the Green's function has all the simplicity of Sommerfeld's direct ray part of his solution. It did not have to work out this way since our calculation is only valid to lead­ing order in f3 and to the next lower order, or relative size l/IS. It should be borne in mind that we work with the "time" representation of the Green's function whereas Sommerfeld deals with the Fourier transform. Our spatial variable a is also transformed away in his solution. It is quite possible to have WKB give exact answers in one representation while the asymptotic approximation of the Fourier transform is no longer exact, (The hydrogen atom provides an example (4,8).)

Next consider GRD , the term that includes a reflected path. Oe£ining a = ~ - [1T-w2] GRO is brought to a form resembling (5.11) and we find that

with

. 2 -1n GRO ;:: Go e h(-n)

n = s y

(5.16)

(5.17)

Eqs. (5.16) and (5.17) can also be derived by noting that GOO+GRO Under the transformation W+31T-W. This also provides an easy check that GOO-GRO = 0 for W = 31T/2.

To summarize • 2

+ + ,.,l.m G(b,T;a) == G [e o

. 2 h(,.,m) ~ e-1n h(-n)]

+ (5.18)

with m and n given respectively by (5.13) and (5.17), v, y and Go by (5.6) (5.7) and (5.8) and WI and w2 by (5.10). The £unction h is given in (4,2). (Formula (S.18) has turned out to be an exact solution for the time dependent propagator (12)).

To compare this to (4.3)~(4.6) we must get plane wave in~ coming boundary conditions, Fourier transform from T to E and set E ;:: 1. Instead of using Fourier transformation on ! it is

RAY OPTICS FOR DIFFRACTION 265

+ easiest to set a = Pa~. Then let Pa+OO and multiply G by an appropriate factor to match (4.1). Thus

and

00 u = NJdT e if3T

o

m =

S is 2 _im2 -4 -.- exp [-4 (p +Pb) ]{ e h ( -m)

1T1T T a

_ -in2h(_n)} +e

n =

(5.19)

We consider the first term in (5.19) (deriving from GDo) 1 as the second term follows by the replacement m+n. Call thls term uOO' It is convenient to go back to (15.11) but still allow the inte­gral to run to +00. Then

00 00

S Sy uOO = N J dT f dl; 41TiT 41Ti

o 0

2 x exp{if3[T + YJ + t; v sinw2 + 4\ (Pa+Pb)2]) (5.20)

with appropriately substituted values for v and y. integral we do a stationary phase approximation 1 with meter Pa. Since Pa+OO this is an exact evaluation. stationary value of T is

_ 1 1 1;2 1" = 2 Pa + 2 Pb + 4Pb + I; sin w2 + O(l/Pa )

For the T large para­The

(5.21)

and after a bit of tedium and with an appropriate N one obtains

(5.22)

Recall that Pb is related to ~hysical coordinates by Eq, (3.2) which is equivalent to SPb = kl~l. With this correspondence the expression (5.22) is seen to be exactly the same as Sommerfeld's first term.

The reflected term presents no new challenge and we find complete agreement with the exact solution. In fact we have the remarkable result that the semiclassical approximation to the path integral used at a single intermediate time (liT _II) is enough to recover the exact solution. c

Since we have recovered Sommerfeld's exact solution W~ have a fortiori obtained the aSYIIJptoticform (4.5)-(4,'G). The same asymptotic expansion used there when applied to the time dependent

266 L. S. SCHULMAN

G of Eq. (5.18) shows it to be composed of direct and reflected rays and of diffracted terms that can be considered to have arisen from trajectories from ! to 0 to b. The latter terms are smaller by a factor IS relative to the former. In our cal~ culation we did not get them from a particular path from ! to o to b but rather by summing over paths that were defective at one point. That is, from a to the integration point C they followed a path of stationary action (a straight line or a straight line reflected off the barrier). The same is true from C to t. But at C they violated the stationary action condi~ tion with a single bend, The price of this deviation (when all these paths are summed) is IllS.

VI. INFINITE DIMENSIONAL PERSPECTIVE

In the last section we were involved in the details of a long calculation. Now I'd like to take a more global view. The propagator is the sum over all paths from ~ to t (Fig. 2) of exp(iSS). The usual stationary phase approximation for G is given in (3.10) and consists of a sum 6ver~aths y(.) for which 8S=0. But for the case in which a and b have the barrier between them (w2 > 0 in the notation of Sec. V) there is no such path. In the 2 dimensional integrals of Sec. V this was mani­fested in that the stationary point lay outside the range of in­tegration. We can look at the infinite dimensional integral (3.5) in the same way. The impenetrable barrier restricts the range of integration in function space, eliminating paths that pass through the barrier and in particular eliminating the path for which 8S=0.

We review some features of the stationary phase approximation Let

00 is(x-xo)2 f(S) = J dx e

o

For xo > 0 and xolS» 1, we know that f(S) '" lilT/S. On the other hand, for Xo < 0 and sixo I »1 we have f(S) = 0(1/13). To see this integrate by parts:

f(S)=exp(iSx2);dX[-2iS(X _x)]-l ~ exp[-2iSx(x -x/2)] o 0 oX 0

iSX~ e ---+ 2iSx o

o iSx~ 00

e x a 1 ~ J dx exp[-2iSx(xo - 2)] ax x-:x

o 0

The first term above is 0(1/13) while the second term is (1/13) times an integral of the same form as f itself. Another

RAY OPTICS FOR DIFFRACTION 267

integration by parts on this second integral reduces it by another power of S so that to leading order f is given by the first term alone.

There is no substantial change in the foregoing results when f is a multidimensional integral. Consider

f(S) f dx exp[iS4>Cx)] rl

where x E: Rn and n eRn is a convex domain of integration. If Vo/ vanishes at x within rl, then f = O(S-n/2). This follows by diagonalizing £he matrix of second deri vati ves of <p around the point where its first derivative vanishes (we assume this matrix is nonsingular). Now suppose V<p ~ 0 in n. Then we can integrate by parts using the divergence theorem of vector calculus (see (9))

= eiS<P + e S<P ~ ----:I:13 (6.1)

with ~ = v<P/I~<p12. Therefore f is written as a volume inte­gral plus a surface integral with a factor liS multiplying both, The volume integral can be subjected to another integration by parts (i.e. the divergence theorem) losing another factor of S. The surface integral is of one lower dimension and for a closed volume n (of iqtegration and smooth function <p, it will be of order S- n-lJ/2. This is because V<p when restricted to an (the surface of n) must vanish somewhere since <p must assume its maximum on the boundary. Assembling the various powers of S, we see that just as for the one dimensional case the absence of a stationary point within the volume of integration costs a factor lB.

Application of the foregoing considerations to the functional integral is not immediate since the volume of integration (all paths that do not pass through the barrier) is not closed. However, we know by more prosaic calculations that the diffracted ray con­tribution is down by one factor of IB, so there is reason to believe that we are on the right track. In fact from the previous sections we know that when the Green's function is reduced to a one dimensional integral a factor IS is lost because the station­ary point is outside the range of integration. When we ane not too de~ in the diffracted ray regime, i.e. when the bend in the line aOb is not too great, then in function space there is a point where 8S, the gradient of the classical action, vanishes just outside the domain of integration. Under these circumstances

268

it is possible a point on the surface - does IS dropoff.

L. S. SCHULMAN

to show in linear approximation that there must be surface at which oS - when restricted to the vanish. This is what is needed to get the extra

It is also of interest to look at oS along the path y(t)~ which goes in a straight line from t to 0 and from 0 to b with constant velocity except for its discontinuity (in direction) at 0, This path is on the surface of the domain of functional integration and it is the classical action evaluated along this path that gives the diffracted ray contribution to the Green's function. This leads us to believe that oS when restricted to the surface should vanish. To evaluate oS we note that yet) = (Itl+ltl)(a+S)O(t-Lc)' Since for a general variation n about y we have

L I~I + 2 L 2 S(y+n) = 4 ( a +Ibl) + J (-y(t))n(t)dt+o S + ... o

it follows that oS is proportional to a delta function. This can also be put in a discrete form in which case oS is the vector with components

where j is the index of the discrete time at which y makes its bend~

At present I have not brought everything together by showing oS to be perpendicular to an at y. What is clear is that trivial application of (6.1) will not always work and for some cases the gradient vanishes nowhere on the surface. (Consider two separated barriers where one expects a factor l/~ = (11132).)

VII. TUNNELLING AND CAUSTICS

Tunnelling occurs in optics as well as in quantum mechanics. In fact R. Dicke when teaching quantum mechanics brought in a pair of prisms to demonstrate that total internal reflection with­in one of the prisms could break down if the other prism is placed close enough (13). One can think of total internal reflection as due to a potential barrier (c f. Eq, (3.3)) and bringing the second prism close means that an allowed region is not far away and tun~ nelling through the forbidden region is possible.

The path integral propagator G(b1 L;!) is not a suitable object for study of barrier penetration since there is a path

RAY OPTICS FOR DIFFRACTION 269

through any finite potential. So we Fourier transform, as is re­quired for the optics problem anyway. Consider therefore the quantity G(b,a;E) as given in (3.7). Inserting the semiclassi­cal expression (3.8) for the time dependent propagator yields

G(b,a;E) 00 a2s

l: fdT[det(iS ~)]1/2 21T abaa

a 0 exp{iSTE+iSS (b,a;T)}

a (7.1)

where we explicitly indicate the dependence of S~ on the end­points. That is, for a particular classical path Ya('), with Ya(O) = a, Ya(T) = b, Sa(b,a;T) = S[Ya(')] with S[·] given by (3.6). For fixed E (7.1) is evaluated by the stationary phase approximation and the important value of T is that for which

From classical mechanics we know that this T is given by

T = ~ ~ dy [E _ V(y)]-1/2 a

and that for this path

b Sa(b,a;E) = f IE-V(y) dy - ET

a

(7.2)

(7.3)

(7.4)

We combine these formulas and for convenience ignore the deter­minant in (7.1) (see Sec. 18, Ref. 4, for proper treatment of this factor) to get

b G(b,a;E) ~ exp[iS f IE - V(y) dy]

a (7.S)

The integral in (7.S) is one dimensional and we neglect the possibility of there being several paths (these extra bounces off walls are unimportant for barrier penetration) .

Now suppose that somewhere on the path of integration V ex­ceeds E. Then there is no classical path from a to band (7.1) breaks down. But consider the expression (7.S). It is an analytic function ofE in the upper half E plane even if V(y) is not particularly smooth. The formula for quantum mechanical barrier penetration comes from just this analytic continuation. Suppose E>V(y) for a<Y<Yl and for Y2<y<b, while E<V(y) for Yl<y<Y2' Then taking the branch of G that leaves it bounded for

270 L. S. SCHULMAN

E -+ iao we have

)j b yz G (b , a; E) = exp [ is Cf + J ) IE - V]

a yz exp[-S J lV(y)-E dy]

Yl

(7.6)

the usual barrier penetration formula.

My purpose in presenting this material is to comment on the extent to which we can ascribe a classical path to the quantities in (7.6). From (7.3) note that the L that gives rise to (7.6) is complex and has an imaginary contribution from the range Yl<y<yZ' Thus for the L integral in (7! 1) there is no stationary phase point on the contour of integration. However, one can deform the contour and bring this contribution to G. Is this an "imaginary ray"? I lean to McLaughlin's view (10) of this as an analytic continuation in T or, in the present description, in E. In particular, I am conservative about letting y be complex. My reason is that the barrier penetration formula holds with no particular analyticity assumptions on V(y) so there is no a priori way to extend V into the complex y plane - nor is there any need to.

Nevertheless, one can still think of the G of (7.6) as arising from the solution of a variational problem. For a pair 0;1; points a and b pick a real number E such that E > yea) and E > V(b) but there is an interval between a and b for which V(y) > E. Let L be the complex number given in (7.3). Let yet) be a real valued function of the complex parameter t with

LIZ yeO) ::; a and y(L) = b, and S = J [~(dy/dt) - V(y)]dt. We want

o S to be stationary relative to variations in y. First imagine that the contour of integration for t in the complex t plane (from 0 to L) is specified. Vary y in the usual way to get the Euler-Lagrange equations. Thus any yet) making S stationary will satisfy these equations and as a particular consequence the quantity

E' ::; ! C~)Z + V(y) 4 dt

will be a constant of the motion. Moreover, since the path yet) is presumed to be stationary we have the usual relation

L = Jb dY[E'_V(y)]-1/2 But L was picked so that this relation a

was true for a particular number E. and V are real so must be (dy/dt)Z. t was allowed to wander arbitrarily path yet) cannot stationary unless:

Therefore E' = E. Since E This shows that even though

in the complex t plane the (1) t moves parallel to

RAY OPTICS FOR DIFFRACTION

the real t axis for y such that E > V(y) and (2) parallel to the imaginary t axis for y such that

t moves E<V(y).

There is little new in this description of the barrier penetration ray. Perhaps I have been a bit more explicit than previous authors in stating the variational problem.

271

Does being able to speak of a ray with complex time para~ meter allow one to think of quantum particles as having definite trajectories on their way through a barrier? If one sticks closely to the Newtonian picture of particles and trajectories the answer must be no, But if one takes the Hamilton Jacobi equations as the basis of classical particle mechanics then the relation of G of (7.6) to the solution of variational problem suggests an extension of the notion of particle. One can for example give physical meaning to the idea of the "time spent in the barrier" and to the extent such meaning can be assigned that time coincides with Im~ given above (11).

There is another situation where two points are not connect­ed by any classical path but the leading asymptotic approximation to G is given by an analytic continuation of the classical action function. This occurs for caustics. Keller deals with this in terms of imaginary rays and in Sec. 15 of Ref. (4) ~n particular p. 126) I show how these terms arise from analytic continuation of the action to regions where no classical paths exist. It is an exercise in complex variables and classical mechanics to relate this statement to analytic continuation in ~ or E. The point of the exercise is that once again the leading asymptotic behavior of G is given by a generalized kind of classical mechanics.

VIII. SUMMARY

Electromagnetism ~ a wave phenomenon for the purposes of this paper - lends itself to a particle or ray description in a number of situations considered characteristic of the wave picture. This is the theme of Keller's geometric diffraction theory and it is a theme which the path integral context of this paper makes explicit.

Much of the paper is concerned with calculating knife edge diffraction and recovering the exact solution from what starts as a semiclassical approximation to the path integral. The full propagator then contains term identifiable as a "diffracted ray" which gives rise to traditional diffraction terms but which be­haves like an ordinary ray once it is launched from the knife edge. We also discuss in Sec. VI how this diffracted term arises in the path integral from a stationary point of the action

272 L. S. SCHULMAN

that lies outside the range of functional integration.

Sec. VII deals with two other cases where a sufficiently broad view of classical mechanics would allow one to include other characteristical wave phenomena in the category of particle behavior. The cases are barrier penetration and illumination in the caustic shadow.

ACKNOWLEDGEMENTS

I wish to thank Ady Mann and David McLaughlin for useful conversations on this work.

REFERENCES

1. J.B. Keller, A Geometrical Theory of Diffraction in "Calculus of Variations and its Applications", Proceedings of a Sym­posium in Applied Mathematics", ed. L.M. Graves, McGraw, New York, 1958.

2. R.G. Kouyoumjian, The Geometrical Theory of Diffraction and its Application, in "Numerical and Asymptotic Techniques in Electromagnetics" , ed. R. Mittra, Springer-Verlag, Topics in Applied Physics, Vol. 3, Springer, Berlin, 1975.

3. A.J.W. Sommerfeld, "Optics", Academic Press, New York 1954. 4. L.S. Schulman, "Techniques and Applications of Path Integra­

tion". Wiley. New York 1981. 5. C. Morette (DeWitt), On the Definition and Approximation of

Feynman's Path Integrals, Phys. Rev. 81,848 (1952). 6. R.M. Lewis and J. Boersma, Uniform Asymptotic Theory of Edge

Diffraction, J. Math. Phys. 10, 2291 (1969). 7. S.W. Lee, Path Integrals for Solving some Electromagnetic

Edge Diffraction Problems, J. Math. Phys. 19, 1414 (1978). 8. M.C. Gutzwiller, Phase-Integral Approximation in Momentum

Space and Bound States of an Atom, J. Math. Phys. 8, 1979 (1967).

9. N. Bleistein and K.A. Handelsman, "Asymptotic Expansions of Integrals", Holt, Rinehart and Winston, New York, 1975.

10. D.W. McLaughlin, Complex Time, Contour Independent Path Integrals, and Barrier Penetration, J. Math. Phys. 13, 1099 (1972) .

11. M. Buttiker and R. Landauer, Traversal Time for Tunneling, IBM preprint, 1982.

12. L.S. Schulman, Exact Time-Dependent Green's Function for the Half-Plane Barrier, Phys. Lev. Lett. 49, 599 (1982).

13. See also R.S. Longhurst, "Geometrical and Physical Optics," Longmans, Green and Co. London 1957; p. 439. I thank Michael Berry for bringing this reference to my attention.