Di raction: Boundary Value Problems of Electromagnetic Waves

Transcription

1 Chater 7 Di raction: Boundary Value Problems of Electromagnetic Waves 7. ntroduction n electrostatics, a rescribed otential distribution on a closed surface uniquely determines the otential elsewhere according to the Dirichlet s formulation, s(r )d ; G = on ; rovided G(r; r ), the Green s function, is so chosen that it vanishes on the closed surface : The normal derivative of the Green s can be regarded as a rojection or maing oerator to convert a given distribution of the surface otential s (r ) into the otential at an arbitrary observing osition r: For scalar otentials, the rojection oerator is a scalar function, or more recisely, matrix function. For any vector elds, too, rescribing a vector on a closed surface uniquely determines the vector led elsewhere. However, a vector eld changes its direction as well as its magnitude over a closed surface. A vector eld rescribed on a closed surface is to be converted into a vector eld with de nitive direction and magnitude at the observing osition. A rojection oerator that transforms one vector eld on a closed surface into another elsewhere must be a tensor (or dyadic) comosed of aroriate eigenvectors for the vector eld of concern. For electromagnetic elds, the TE and TM eigenvectors identi ed in Chater 5 can be conveniently used for this urose. n this Chater, a basic formulation will be develoed for vector boundary value roblems of electromagnetic elds, E and B: t will then be alied to some di raction roblems in which boundary electromagnetic elds are known. However, in most ractical roblems, it is di cult to know recisely boundary elds because of the feedback from the di racted elds themselves. A rigorous analysis of di raction roblems usually requires solving integral equations based on the boundary conditions for electromagnetic waves. 7. Vector Green s Theorem For arbitrary vector elds E and F, by exloiting the exansion r (E r F) = r E r F E r r F; (7.)

2 r (F r E) = r F r E F r r E; (7.) the following identity can easily be roven, (E r r F F r r E)dV = (F r E E r F) d: (7.3) ince V E r r F = E [r(r F) r F] (7.4) = r [E(r F)] (r E)(r F) E r F; (7.5) Eq. (7.3) can be modi ed as V (F r E E r F)dV = [(Fr E Er F) n (n r E) F n E r F] d; (7.6) where n is the unit normal vector on the closed surface directed away from the volume V; d = nd: (ee Fig. 7..) This is a mathematical identity and holds for arbitrary vector elds E and F. Figure 7-: For E s ; the electric eld seci ed on the closed surface ; the electric eld o the surface E is uniquely determined. n is the normal unit vector directed away from the region of interest. We now let the vector E be the electric eld and F be one comonent of Green s dyadic. n source free region, the electric eld E satis es the Helmholtz equation, (r + k )E = ; (7.7) and the Green s dyadic satis es the singular inhomogeneous Helmholtz equation (r + k )G = (r r ); (7.8) where is the unit dyadic de ned in terms of aroriate eigenvectors for the electromagnetic elds. Then,

3 Eq. (7.4) yields an exression for the electric eld E in terms of the electric eld seci ed on the closed surface ; E(r) = [(n E)r G + (n r E) G + (n E) (r G)] d; (7.9) where r E = (no charges on ) has been assumed. Note that r G is a vector and r G is a dyadic. For a diagonal Green s dyadic G = G; (7.) it follows that r G = rg; (7.) (n r E) G = (n r E)G; (7.) (n E) (r G) = (n E) rg: (7.3) Therefore, Eq. (7.9) reduces to E(r) = [(n E)rG + (n r E)G + (n E) rg] d: (7.4) The scalar Green s function G satis es and its exlicit exression in terms of sherical harmonic exansion is G(r; r ) = 4 jr r j eikjr r j ( X lx j l (kr ) h () l (kr) = ik j l (kr) h () l (kr ) l= m= l (r + k )G = (r r ); (7.5) ) Y lm (; ) Y lm ; ; ( r > r r < r ) : (7.6) As shown in Chater 6, the solution for the vector otential that satis es the Helmholtz equation r + k A = J; (7.7) can be decomosed into TE and TM modes. n the outer region r > r ; we have the following decomosition, A(r) = ik + i k X lx l= m= l X lx l= m= l l(l + ) h() l (kr)r ry lm (; ) l(l + ) r [h() l (kr)r ry lm (; )] j l (kr )r r Y lm ( ; ) J(r )dv r [j l (kr )r r Y lm ( ; )] J(r (7.8) )dv : This can be written in a more comact form using the Green s dyadic G; A(r) = e ikjr r j 4 jr r j J(r )dv = GJ(r )dv ; (7.9) 3

4 where the Green s dyadic is G(r; r ) = e ikjr r j 4 jr r j = ik X lm + i X k lm Note that the TE eigenvector and TM eigenvector l(l + ) h() l (kr)j l (kr )r ry lm (; )r r Y lm ( ; ) (7.) l(l + ) r [h() l (kr)r ry lm (; )]r [j l (kr )r r Y lm ( ; )]: (7.) h () l (kr)r ry lm (; ); (7.) r [h () l (kr)r ry lm (; )]; (7.3) are orthogonal to each other for given mode numbers l and m and the Green s dyadic is evidently diagonal. For di raction by an aerture in a large, at conducting screen, Eq. (7.4) simli es to E(r) = (n E) rgd: (7.4) aerture This formula was originally found by mythe based on an equivalent layer of magnetic diole (equivalent to a sheet of magnetic current), dm = i! n Ed; (7.5) which roduces a vector otential A (r) = i! (Recall that an oscillating magnetic diole creates a vector otential, rovided kr :) Corresonding electric eld is rg (n E)d: (7.6) A (r) = rg m ' e ikr ik m; E(r) A = rg (n E)d aerture = (n E) rgd: aerture (Exlain why we can use A (r) ; here instead of going through the usual c = r B = r (r in terms of longitudinal and transverse comonents of A:) The tangential comonent of the di racted magnetic eld in the aerture vanishes (that is, in the aerture, the tangential magnetic eld is equal to that 4

5 of the incident wave which is una ected by the aerture) and only the tangential comonent of the electric eld contributes to di raction. Likewise, the normal comonent of the di racted electric eld vanishes in the aerture. These follow from the symmetry roerty of the di racted electric and magnetic eld in front of and behind the conducting late. The di racted electric eld normal to the late is odd with resect to the normal coordinate z, E z (z) = E z ( z); (7.7) and so is the tangential magnetic eld, e z B (z) = e z B ( z) ; (7.8) but the tangential electric eld and normal magnetic eld are even, e z E (z) = e z E ( z) ; B z (z) = B z ( z): (7.9) Therefore in the aerture, the tangential comonent of the di racted magnetic eld should vanish. On the surface of the conducting late away from the aerture, tangential comonent of di racted magnetic eld can exist with a corresonding surface current n H = J s ; (A m ): ince the normal unit vector n changes sign from one side (z < ) to other (z > ) ; J s = n H is even with resect to z; J s (z = ) = J s (z = +) : The surface charges induced on both sides of the conducting late are also equal since = " n E is even with resect to z: We have encountered such symmetry roerties in Chaters and 3 when we analyzed leakage of electric and magnetic elds through a hole in a suerconducting late. n both cases, e ective dioles for elds in regions z < and z > are oosite to each other and erturbed electric and magnetic elds due to the hole have such symmetry roerties which hold in general even for radiation elds as long as the late can be regarded as ideally conducting. The mythe s formula, Eq. (7.4), may be derived urely mathematically as follows. For a at boundary, the normal unit vector n is constant, say n = e z. First, we note (n E) rgd = r (Gn E) d + Gr (n E) d = Gr (n E) d; (7.3) since for a closed surface, r [G (n E)]d = : (7.3) Recalling r E = and exloiting the fact that for a unidirectional vector n; r (n E) = (Er) n (n r) E + n (r E) E (r n) = (n r) E; (7.3) r (n E) = (Er) n+ (n r) E + n (r E) + E (r n) = (n r) E + n (r E) ; (7.33) 5

6 we nd for a constant vector n; n (r E) = r (n E) + r (n E) : (7.34) Then Gn (re) d = = Gr (n E) d + Gr(n E)d (n E) rgd (n E)rG d; and Eq. (7.4) reduces to or E(r) = [(n E)rG + (n r E)G + (n E) rg] d = (n E) rg d; whole at surface E(r) = [(n E)rG + (n r E)G] d: whole at surface n the case of di raction by an aerture in a conducting late, n E = excet in the aerture. Therefore, E(r) = (n E) rg d: (7.35) aerture This should be equal to E(r) = [(n E)rG + i!(n B)G] d; (7.36) conducting late since in the aerture where no eld discontinuity exists, both n E and nb vanish. (They are odd functions of z:) Rigorous analysis of di raction by an aerture in a conducting screen is extremely di cult mainly because in Eqs. (7.4) and (7.4), the elds on the surface and in aerture are the di racted elds which are to be found! However, in some limiting cases, sensible analytic solutions can be found. n short wavelength limit ka where a is the size of aerture, geometric otics aroximation may be used. n this case, the eld in the aerture may be aroximated by the incident eld. n long wavelength limit ka ; the eld in the aerture is totally unknown. However, di racted ower can be calculated accurately using e ective dioles. Then the eld intensity in the aerture can be estimated to be kae i E i where E i is the incident electric eld. 7.3 ome Examles of Aerture Di raction 7.3. Long lit Let a lane wave be incident on a at oaque screen having a long slit with an oening width a: The roblem is essentially two dimensional and the di racted wavevector k may be assumed to lie in the x y lane. ince the slit extends from z = to ; we are not allowed to assume r r and the di racted electric 6

7 Figure 7-: Di raction by a long slit of width a in a conducting screen. n short wavelength limit ka ; the electric eld in the slit oening may be aroximated by that of the incident wave. mythe s formula in Eq. (7.4) is alicable. eld has to be calculated from E(r) = (n E) r Gdy ; (7.37) where G is the two-dimensional Green s function satisfying (r + k )G = (r r ) = (x x )(y y ): (7.38) olution for G is G = i 4 H() hk x + (y y ) i ; (7.39) where H () (x) is the Hankel function of the rst kind. ts asymtotic form is H () (x) ' r kx eikx i 4 ; x : (7.4) Using n = n (unit normal directed into the di racted region) and noting r G = rg ; we rewrite Eq. (7.37) as E(r) = r G(n E)dy : (7.4) ubstituting H () hk i x + (y y ) ' = r k eik r k eik i 4 iky i 4 iky sin ; (7.4) 7

8 where = x + y ; (7.43) and is the angle between k and x-axis, we obtain E(r) ' a r k (n E ) k eik i sin 4 ; (7.44) with de ned by = ka sin : (7.45) The function sin = is the familiar Fraunhofer di raction form factor to describe the angular () deendence of the eld intensity. The radiation (di racted) ower er unit length of the slit is P l = = c" jej d: (7.46) = n short wavelength limit ka ; the integration limits may be aroximated by from to ; and we obtain an exected result P = c" je j a; ka : (7.47) l n the oosite limit ka ; the electric eld in the slit oening cannot be relaced by the incident eld because the current induced in the conducting late greatly disturbs the incident eld. The roblem is dual of scattering of electromagnetic waves by a thin, long conducting late which will be analyzed as a searate roblem Circular Aerture Figure 7-3: Geometry of circular aerture of radius a in an oaque screen. Radiation from a circular aerture on a large conducting late is a classical roblem and of ractical imortance. n short wavelength limit ka (a the hole radius), the eld in the aerture may be aroximated 8

9 by that of the incident wave E : The di racted electric eld is given by E(r) = i r k (n E )e ikr e ikr d; (7.48) where in the geometry shown, k r = kr sin sin : The integral can be erformed as follows: a = The di racted electric eld thus reduces to = r dr e ikr sin sin d a J (kr sin )r dr a k sin J (ka sin ): (7.49) E(r) = ia r k (n E )e ikr J (ka sin ) : (7.5) ka sin ince the amlitude of the Bessel function J (x) decreases with x in a manner J (x) _ = x; the electric eld is areciable only at small : n order of magnitude, di racted Poynting ux has an angular sread about the z axis, sin ' ' ka : The rst intensity minimum occurs at the rst root of J (x) = ; ka sin ' 3:83; or ' 3:83 ka = : ; D = a (diameter of the hole). (7.5) D This is the familiar di raction limited resolving ower of circular aertures such as mirrors in telescoes and arabolic microwave antennas. The radiation ower associated with the di racted eld is P = c" r jej d = c" je j a = J (ka sin ) sin sin d: (7.5) ince ka ; the uer limit of the integral may be extended to and sin may be aroximated by : Noting J (ka) d = ; (indeendent of ka); we nd P = c" je j a : (7.53) This is an exected result in the short wavelength limit (ka ) and simly corresonds to the incident ower going through the circular aerture. n long wavelength limit ka ; the above analysis breaks down comletely, for the electromagnetic elds in the aerture are entirely di erent from those associated with the incident wave. For normal incidence on a conducting late, the electric eld vanishes at the late but the magnetic eld is doubled because of comlete 9

10 re ection. This does not mean there exists a tangential magnetic eld of H =H in the aerture because the aerture should not a ect the incident magnetic eld and the tangential comonent of the magnetic eld in the aerture is H = H :What we can do is to nd an e ective magnetic diole moment of the hole using H as the unerturbed eld as we did in analyzing the leakage of magnetic eld through a hole in a suerconducting late. The diole moment of the hole is m = 8a3 3 H = 6a3 3 H ; (7.54) which radiates at a ower P =! 4 m 4" 3c 5 = 64 7 (ka)4 a jh j ; (7.55) where the factor is due to radiation into half sace (behind the screen). The transmission cross-section is t ' P jh j = 64 7 (ka)4 a _ k 4 a 6 : (7.56) The deendence _ k 4 (or! 4 ) is the common feature of Rayleigh scattering of electromagnetic waves by small objects (in this case an aerture). _ a 6 indicates extremely sensitive deendence of the cross section on the hole radius. To order k 6 ; the cross section is ince the radiation ower is roortional to ' 64 7 (ka)4 a + 5 (ka) + ; ka : (7.57) (n E a )e ikr d ' E aa 4 ; where E a is the electric eld in the aerture, it is evident that the eld is of the order of E a ' kae E ; being much smaller than the incident eld E : This is exected since the incident wave is essentially shortcircuited at the late and only a small eld can leak through the aerture. The electric eld in the aerture has been worked out by Bouwkam, E x (x; y) = i 4k xy 3 a x y E ; (7.58) E y (x; y) = i 4k x + y a 3 a x y E = i 4k + sin a E : (7.59) 3 a E x is an odd function of x and y and does not contribute (integrates to ). The eld diverges at the rim = a but is integrable. ntegration over the aerture yields a d de y (x; y) = i 8ka3 3 E : (7.6)

11 The di racted electric eld can then be found using the mythe s formula which is equivalent to the eld due to an e ective magnetic diole moment in Eq. (7.54). Di raction in the regime ka ' O() is di cult to analyze and requires numerical analysis. The transmission cross-section eaks at ka ' :56 and its value is max ' :8 a. This behavior is similar to the case of scattering by a conducting shere Radiation from an Oen End of a Rectangular Waveguide Figure 7-4: Radiation from an oen end of a rectangular waveguide. The electric eld in the oening is E y (x) = E sin(x=a) and the tangential comonent of the magnetic eld is H x (x) = E(x)= where is the imedance of the TE mode. The electric eld of the TE mode in a rectangular waveguide is given by where a is the width of the waveguide and E(x; z; t) = E sin a x e i(kzz!t) e y ; (7.6) k z =! c (!c =!) ; (7.6) is the axial wavenumber with! c = c a ; (7.63) being the cuto frequency of the TE mode. The accomanying magnetic eld transverse to the axis is H(x; z; t) = E sin a x e i(kzz!t) e x ; (7.64) where = r " (!c =!) ; (7.65) is the imedance of the TE mode. (The axial magnetic eld H z associated with the TE mode does not

12 contribute to radiation from the aerture.) f a waveguide is truncated, the oen radiates electromagnetic waves. The mythe s formula is not alicable because it only ertains to radiation from aertures in a large conducting late. n the general di raction formula where E(r) = we assume r r. Then, and Also, Then, E(r) ' ieikr 4r (n E)r G + G(n r E) + (n E) r G d; (7.66) G(r; r ) = eikjr r j 4 jr r j ; G(r; r ) ' 4r eikr ikr ; r G = rg ' ik 4r eikr r = i!b: ikr : k (n E) +!n B k(n E) e ikr d; (7.67) where n = n is the normal unit directed toward the region where the electric eld is to be evaluated. t is not obvious that the contribution from the second and third terms in the integrand,!n B k(n E) e ikr d; (7.68) is exlicitly transverse (erendicular to k) which should be the case since we are calculating radiation eld. To rove this, we rst show that r (n B) G (r; r ) d; can be exressed in terms of the normal comonent of the electric eld as follows. r G (n B) d = r G (n B) d = r G (B n) d = r G B nd = r (GB) nd = + i! c G (E n) d; Gr B nd where use is made of the identity r A nd = :

13 Therefore, r r G (n B) d = rr = i! c r i! = c G (n B) d + k G (n B) d G (E n) d + k G (n B) d (E n) r Gd + k G (n B) d; and the di racted eelctric eld can alternatively be calculated from E(r) = = (n E)r G + G(n r E) + (n E) r G d rg (n E) + i r r (Gn H) d:!" Then Eq. (7.67) may be rewritten as E(r) = ieikr 4r k n E k (n H) e ikr d; (7.69)!" which is exlicitly transverse to k: As mentioned in Chater 5, this formula was rst derived by chelkuno in term of a ctitious magnetic current to relace the tangential comonent of the electric eld on a boundary. The rice to be aid for introducing a magnetic current is that the Maxwell s equation, r B = ; has to be violated. n the derivation resented here, no magnetic current is assumed. Denoting the surface integral over the oen end of the waveguide by (; ) = = a dx b dy sin a x e ik(x sin cos +y sin sin ) i =a k sin sin (=a) k sin cos ( + e ika sin cos )( e ikb sin sin ); (7.7) we nally nd the electric eld radiated from the oen end of a rectangular waveguide, E(r) = " ikeikr (; ) 4r cos (!c =!) +! sin e + Calculation of the radiation ower is left for an exercise.! # (!c =!) + cos cos e : (7.7) cattering by a Conducting here Revisited This roblem has been solved rigorously in the receding Chater in terms of sherical harmonic exansion. However, the result is not very illuminating hysically articularly in the short wavelength limit ka wherein summation of a large number of functions involving sherical Bessel functions and their derivatives must be erformed. n this limit, geometrical otics aroximation should be able to yield the scattered electric eld rovided the boundary conditions for the electromagnetic elds are aroriately incororated. The incident lane wave sees the cross-section of the shere a which corresonds to the scattering crosssection due to scattering by the illuminated hemisherical surface facing the incident wave. cattering by the illuminated surface is nothing but re ection by a sherical convex mirror. The Poynting ux associated with re ection is uniform (isotroic) and indeendent of the angles and. The total correct cross-section 3

14 Figure 7-5: cattering by a conducting shere in short wavelength limit ka : as revealed from the rigorous analysis was a : The additional a is due to shadow scattering from the hemisherical surface in the shadow of the incident wave where the eld vanish. This vanishing eld can be interreted as cancellation between the eld of the incident wave and that of the scattered wave. hadow scattering is dual of di raction by a circular aerture in a conducting screen which radiates a ower P sh = c" je j a ; ka : (7.7) Although the scattered ower through shadow scattering is identical to that by the illuminated surface, the Poynting ux is sharly eaked behind the shere as in the case of di raction with an angular sread of order ' =a : f the shere is ideally conducting, the boundary conditions for the electric and magnetic elds are E t = ; B n = : (7.73) Let the electric eld of the incident wave be E i and that of scattered eld be E s : The vanishing tangential comonent of the total eld requires that The vanishing normal comonent of the magnetic eld requires that n the sirit of geometrical otics valid in the short wavelength regime, n (E i +E s ) = : (7.74) n (B i +B s ) = : (7.75) E i = c k k ib i ; E s = c k k sb s ; 4

15 where k i is the wavevector of the incident wave and k s is that of the scattered wave. From n (E i +E s ) = n (k i B i +k s B s ) = ; (7.76) and it follows that Likewise, from we nd n k i = n k s ; n k i = n k s ; (7.77) n (B i B s ) = : (7.78) n (k i E i k s E s ) = ; n (E i E s ) = : With these rearations, we now aly Eq. (7.67) to the illuminated and shadow surfaces searately. The contribution from the illuminated surface is E ill (r) = ieikr 4r (k k ) (n E i ) + (k k )(n E i ) e ikr d; where the incident magnetic eld has been eliminated through!b i = k E i : n the geometry shown in Fig. 7-5, the incident wave roagates in the negative z-direction, E i (r) = E e ikz = E e ikr cos : (7.79) Therefore, the electric eld di racted by the illuminated surface is E ill (r) = ieikr 4r (k k ) (n E ) + (k k )(n E ) e ikr cos ikr d: (7.8) imilarly, the contribution from the shadow surface is E sh (r) = ieikr 4r (k + k ) (n E ) (k k )(n E ) e ikr cos ikr d: (7.8) The total di racted eld is given by the sum of E ill and E sh : However, the angular deendence of the eld intensities is entirely di erent as exlained in the introduction, namely, je ill j is insensitive to and ; while je sh j sharly eaked in the direction = (forward scattering). Therefore, the scattered ower can be calculated searately as if the elds were incoherent. Noting r = a on the shere surface, we nd the hase function reduces to kr cos + k r = ka ( + cos ) cos + sin sin cos( ) = ka f(; ; ; ): (7.8) 5

16 By assumtion ka : Therefore, the exonential function e ikaf ; (7.83) raidly oscillates as and are varied. The slow angular deendence in the amlitude function (k k ) (n E ) + (k k )(n E ) can be ignored in the integration over and and can be taken out of the integral, E ill (r) ' i eikr (k k ) (n E i ) + (k k 4r )(n E i ) a ; e ikaf( ; ) sin d d ; where and indicate the angular location on the shere surface which makes the dominant contribution to the hase integral e ikaf( ; ) sin d d : (7.84) ince ka ; major contribution to the integral comes from the angular location where the function f( ; ) = : This determines = ; = : This is a trivial result well exected from otical re ection. At an observing angular location (; ); only the wave re ected at ( = =; = ) can be detected. Let us Taylor exand the hase function f( ; ) about the stationary hase oint = = = and = = ; f(; ; ; ) ' cos! sin ( ) + : (7.86) ntegration over yields " = ex ika cos " ' sin ex ika cos = sin # sin d # d ika cos(=) : (7.87) Note that the integration limits can be extended to because only the region ' = contributes to the 6

17 integral. imilarly, ' ' ex ika cos sin ( ) d ex ika cos sin ( ) d (7.88) ika cos(=) sin(=) : (7.89) Therefore, e ikaf( ; ) d = a i ex ( ika cos(=)) : (7.9) ka cos(=) At the articular angular location (=; ); the magnitude of the vector jk k cos and the normal vector n is in the direction of k k ; n = k j is ; (7.9) ; ; : (7.9) Then, (k k ) (n E ) + (k k )(n E ) = k cos [n (n E ) + n (n E )] = k cos [n (n E ) E ] : (7.93) The vector has a magnitude of E ; that is, n (n E ) E ; (7.94) jn (n E ) E j = E ; and is erendicular to the scattered wavevector k. (Prove this statement.) Therefore, the electric eld scattered (or re ected) by the illuminated surface is E ill (r) = a r eikr ika cos(=) [n (n E ) E ] : (7.95) The amlitude je ill (r)j is indeendent of the angles and as exected from isotroic re ection and scattered ower can readily be found as P ill = c" Er a 4 r = c" E a : (7.96) 7

18 The scattering cross-section due to scattering (essentially re ection) by the illuminated surface is therefore ill = a : (7.97) cattering by the shadow surface is identical to di raction by a circular aerture on a conducting screen analyzed earlier, for the boundary conditions E i + E s = ; B i + B s = ; (7.98) indicate that the elds of the incident wave can be used on the surface of the shadow hemishere. (The sign inversion does not a ect the di racted ower.) The shae of the shadow surface is irrelevant as long as its area rojected normal to the incident wave is circular with an area a : To see this, let us consider the electric eld scattered by the shadow surface in Eq. (7.8), E sh (r) = ieikr 4r (k + k ) (n E ) (k k )(n E ) e i(k k)r d : (7.99) ince there is no stationary hase oint in the shadow (it only occurs in the illuminated surface) and the function e i(k k)r raidly oscillates because of the assumtion ka ; the integral will be nonvanishing only if k k ' which means redominantly forward scattering. The change in the wavevector k = k k is thus erendicular to k and k ; k = k sin e = k sin (cos e x + sin e y ); (7.) where is olar angle of scattered wavevector k now measured form the negative z axis, e is the radial unit vector in the cylindrical coordinates and is the azimuthal angle about the z axis. The hase function becomes (k k) r = ak sin sin cos ; (7.) and the eld amlitude may be aroximated by which is indeendent of and. Then (k + k ) (n E ) (k k )(n E ) ' k cos E ; (7.) (k + k ) (n E ) (k k )(n E ) e i(k k)r d ' ke a sin d d e iak sin sin cos( ) cos a = ke d = k (e z E ) d de ik sin cos( ) a d e ik sin cos( ) ; (7.3) where = a sin ; n is the unit vector normal to the circular at surface, and k E ' k E = is 8

19 noted. The electric eld scattered by the sherical surface in the shadow is thus given by E sh (r) = ieikr r k (e z E ) a d e ik sin cos( ) d: (7.4) This is the negative of the eld di racted by a circular aerture given in Eq. (7.48). ntegrations over and yield E sh (r) = ia r k (e z E ) J (ka sin ) k sin ' ia r k (e z E ) J (ka) ; : (7.5) k The radiation ower associated with shadow di raction is identical to that in the case of circular aerture, Therefore, the total ower reradiated by the shere is and the total scattering cross-section in the short wavelength limit is P sh = c" E a : (7.6) P = P ill + P sh = c" E a ; (7.7) = a ; ka : (7.8) This is consistent with the asymtotic (ka! ) value of the general cross-section derived earlier, where (ka) = k A l = X h (l + ) ja l j + jb l j i ; (7.9) l= j l (ka) h () l (ka) ; B l = d da [aj l(ka)] d (ka)] da [ah() l : (7.) The summation over l in Eq. (7.9) requires u to l ' O(ka) for su cient accuracy and thus becomes cumbersome in otical regime where ka can be huge. Also, it should be noted that the assumtion of ideally conducting shere (" e = ; e = ) entirely breaks down in high frequency (short wavelength) regime. The e ective relative ermittivity of ordinary metals in otical frequency regime is in general comlex and remains of the order of unity. (The relative magnetic ermeability in the regime may be assumed to be unity.) 7.4 cattering by a Knife Edge n scattering by a knife edge, there is no geometrical scale size to seak of excet for k where is the radial distance from the edge ti. n short wavelength otical regime, assuming an incident electric eld along the lane above the edge may rovide the lowest order aroximation. n rigorous aroach, the boundary conditions for the electromagnetic elds must be incororated. We assume an observing oint at (; ): The 9

20 Figure 7-6: Geometry for analyzing scattering by a conducting knife edge. incidence. Bottom: general incidence angle. To: secial case of normal distance between a oint on the y axis and the observing oint is = ( sin y ) + cos ' y sin + y : (7.) Therefore, for small ; the ath di erence between and is y ; and corresonding hase di erence is The electric eld at (x ; y) is thus given by ky : (7.) The function, shown in Fig. (??), y E(y) = E q k F (x) = x is known by Fresnel s integral and its secial values are: i ex t dt: (7.3) i ex t dt; (7.4) F ( ) = ; F () = ( + i); F () = + i:

21 At large ositive y (well above the edge); the incident eld E is recovered as exected. Well below the edge in the shadow region, y! ; the eld vanishes also as exected. At y = ; the eld is E and intensity is 4 of the incident intensity. Fig. (7-8) shows the intensity as a function of the normalized vertical distance q y! y x : F (x) = x i ex t dt Figure 7-7: Fresnel integral F (x) = R x ex i t dt: F ( ) = ; F () = + i: y 4 6 q Figure 7-8: ntensity = as a function of normalized vertical distance y x. (y = ) = 4 ; max ' :37 : The analysis above is rather rimitive because no consideration was given to the boundary conditions of the electric and magnetic elds. For incident electric eld arallel to the conducting late, E = E e z (E-mode), the total eld (incident + scattered) must vanish at the surface of the conductor late. f the

22 incident wave aroaches the late at an angle ; the total wave eld is given by E z (; ) = i k= cos E e ik cos( ) e i t dt i E e ik cos(+ ) k= sin + e i t dt: (7.5) f the incident wave is so olarized that its magnetic eld is arallel to the late, H = H e z (H-mode), the boundary condition is This yields E = at = ; 3 : H z (; ) = i k= cos H e ik cos( ) e i t dt + i H e ik cos(+ ) k= sin + e i t dt: (7.6) The reader should verify that the solutions ( rst formulated by ommerfeld) satisfy the Helmholtz equation and the resective boundary conditions.

23 Problems 7. An electric eld E(; ) = E (; )e + E (; )e is seci ed on the surface of a shere of radius a: Determine the radiation electric eld. 7. A magnetic eld B(; ) = B (; )e + B (; )e is seci ed on the surface of a shere of radius a: Determine the radiation magnetic eld. 7.3 A lane wave is incident normal to a long conducting cylinder having a square cross-section of side a:the electric eld is axial (E z only) and the incident wave falls normal to one of the rectangular surfaces. Derive an integral equation for the surface current density J z (r): (The integral equation can be solved numerically following the rocedure develoed for nding the caacitance of a square conducting late.) 7.4 Reeat the receding roblem for H mode (H z only). 7.5 A oint light source is laced at a distance z on the axis of an oaque disk of radius a ( ): Determine the light intensity along the axis behind the disk as a function of distance z: 7.6 A oint light source is laced at a distance z on the axis of a circular oening of radius a in an oaque screen. Determine the light intensity behind the disk as a function of axial distance z: 7.7 how that in Fresnel di raction of incident wave normal to the late, the maximum intensity occurs at y ' : x = and is given by max ' :37 where is the intensity of the incident wave. 7.8 how that near a knife edge, an E-mode axial electric eld and corresonding magnetic eld H = E z (; ) = A sin ; A cos i! e x + sin e y ; satisfy the boundary conditions. The x comonent of the magnetic eld is discontinuous at = and = : nterret this eculiarity. how that the intensity of a wave scattered by a knife edge is insensitive to the observing angle : (A knife edge aears shiny regardless of observing angle.) 7.9 The transmission cross section of a small (ka ) circular aerture in a conducting late is T = 64 7 k4 a 6 ; where a is the hole radius. how that the magnitude of the electric eld in the aerture should be of the order of E a ' kae ; where E is the incident eld. Note: For incident eld olarized in the x direction, Bouwkam found the electric eld comonent in the aerture resonsible for di raction, E y (x; y) = 8k x + y a 3 a x y E : 3

24 (E x (x; y) is an odd function of both x and y:) ntegration over the aerture yields E y (x; y) d = which is consistent with the e ective magnetic diole, E y (r; ) rdrd = 6 3 ka3 E ; m = 8 3 a3 H : At the edge of a thin conductor, the electric eld arallel to the edge vanishes in a manner E k = ; while the normal comonent diverges as E? = = where is the distance from the edge. For examle, the electric eld at the rim of a charged conducting disk behaves as E? _ q = : a (a ) a uch roerties can be exloited to nd electric eld when the boundary involves shar conducting edge.) 7. n low frequency limit, scatterimg by a conducting shere may be analyzed using diole aroximation. Relevant diole moments are = 4" a 3 E ; m = a 3 H ; where E and H are the elds of the incident lane wave. how that the di erential scattering cross section is " d d = k4 a 6 cos sin + cos cos # ; the total scattering cross section is and that the force exerted on the shere is = 3 k4 a 6 ; F z = 4 3 k4 a 6 " E ; (N). 4