Reflection off a mirror

Transcription

1 Jurnal f Mdern Optics Vl. 54, N. 1, 10 January 2007, Reflectin ff a mirrr H. F. ARNOLDUS* Department f Physics and Astrnmy, Mississippi State University, P.O. Drawer 5167, Mississippi State, Mississippi, , USA (Received 31 May 2006) Electrmagnetic radiatin incident upn a perfect cnductr induces a current density at the surface f the material. This surface current density is the slutin f an integral equatin, and it is shwn that fr a flat surface this equatin has a very simple slutin. The current density at a pint f the surface is determined by the incident magnetic field at the same pint. The surface current density generates an electrmagnetic field, and the ttal radiatin field is the sum f this field and the incident field. It is shwn explicitly that inside the material the field f the current density cancels exactly the incident field. The reflected field can be expressed as an integral ver the knwn surface current density, and is therefre determined by the incident magnetic field at the surface nly. When the surce f the incident field is knwn, the reflected field can be expressed in terms f a surce functin, which is determined by the current density f the surce. It is shwn that this apprach leads t a simple way f cnstructing the mirrr image f an arbitrary surce, and we illustrate the cncept by determining the mirrr image f an electric quadruple. 1. Intrductin Scattering f electrmagnetic radiatin by an bject r a structured surface f infinite extent has been the subject f numerus studies. The simplest case is the reflectin and transmissin f a plane plarized wave by a flat semi-infinite medium, r a by an infinite layer. The reflected and transmitted waves are again plane plarized waves, and their amplitudes are the Fresnel cefficients [1]. The prblem f scattering f a plane wave by a dielectric r metallic sphere (Mie thery) is anther example f a prblem that can be slved analytically [2, 3]. When the bject r the surface is mre cmplex, ne usually has t resrt t numerical methds. In the differential equatin apprach the incident field, the scattered field, and the field in the material are expanded nt a cmplete set f functins. Then bundary cnditins are impsed, which leads t a set f linear equatins fr the expansin cefficients, and this set is slved numerically [4 6]. In the integral equatin apprach, the scattered field and the field inside the material are a slutin f a linear integral equatin, with the incident field as the inhmgeneus term. The field inside the material and the scattered field are expanded nt a cmplete set f functins, * arnldus@ra.msstate.edu Jurnal f Mdern Optics ISSN print/issn nline ß 2007 Taylr & Francis DOI: /

2 46 H. F. Arnldus which then leads t a set f linear equatins fr the expansin cefficients. This is knwn as the Methd f Mments [7]. There exist many variatins n this apprach [8], because f the existence f a variety f integrals f Maxwell s equatins. Particularly interesting is the case where the bject r the surface is a perfect cnductr. Then there is n field inside the material, s nly the scattered field has t be cnsidered. Scattering f a plane wave by a perfectly cnducting bject r surface has been studied fr a large array f gemetries [9 17]. A prblem f a different nature arises when the incident field is nt a plane wave, but radiatin emitted by a particular surce. Fr example, when an electric diple is lcated near a slab f dielectric r metallic material, then part f the radiatin reflects and part f it is transmitted by the layer. Diple radiatin cntains bth travelling and evanescent waves, and upn transmissin an evanescent wave can be cnverted int a travelling wave, and vice versa. This prblem can be slved in clsed frm [18], and the methd can be generalized t multiple fields f arbitrary rder [19, 20]. The methd f slutin relies n an angular spectrum representatin f the multiple fields. Such a representatin is a superpsitin f plane waves, and the reflectin and transmissin f each plane wave is accunted fr by Fresnel cefficients. Fr a multiple near a perfect cnductr, the reflected field can als be btained by the methd f images [21]. Here we cnsider the reflectin f an arbitrary incident field by a flat perfectly-cnducting surface (mirrr), and we shw that this prblem can be slved in clsed frm. When the surce f the incident field is knwn, this slutin als prvides a methd fr btaining the mirrr image f this surce. 2. Reflectin ff a perfect cnductr Let us first cnsider an electrmagnetic field, incident upn the surface f a perfect cnductr, as shwn in figure 1. Inside the perfect cnductr there is n field and n current density. The nly current appears as a surface current density i(r), which is a vectr in the lcal tangent plane. We assume a harmnic time dependence with angular frequency! s that i(r) is the cmplex amplitude f i(r, t), e.g. i(r, t) ¼ Re[i(r)exp( i!t)], and similarly fr ther time dependent fields. The vectr ptential, r Hertz vectr, fr i(r) is with PðrÞ ¼ ð ds 0 iðr 0 Þgðr r 0 Þ, 4 ð1þ gðr r 0 Þ¼ eik jr r 0 j jr r 0 j, ð2þ the Green s functin fr the scalar Helmhltz equatin, and k ¼!/c. The integral in equatin (1) runs ver the entire surface. The magnetic field generated by the surface

3 Reflectin ff a mirrr 47 Inc ˆn(r) + i(r) r S Vacuum Perfect cnductr Figure 1. Surface S is the bundary between vacuum and a perfectly cnducting metal. An incident field induces a current density i(r) n the surface. The current density generates an electrmagnetic field, which is the reflected field in vacuum, and in the material this field cancels exactly the incident field, s that there is n field inside the material. The current density n the surface is a slutin f equatin (8), which has the incident magnetic field as inhmgeneus term. current density is rp(r), and the ttal magnetic field is BðrÞ ¼BðrÞ inc þrpðrþ, ð3þ with B(r) inc the incident magnetic field. Explicitly, BðrÞ ¼BðrÞ inc ð ds 0 iðr 0 Þrgðr r 0 Þ: ð4þ 4 Equatins (3) and (4) hld fr any pint r nt n the bundary surface S. Fr a field pint r in the vacuum, the reflected field is BðrÞ r ¼rPðrÞ: ð5þ On the ther hand, fr a field pint r in the material we have B(r) ¼ 0, and therefre the field generated by i(r) must cancel exactly the incident field. When we mve the field pint r acrss S frm the material t the vacuum, the magnetic field jumps frm zer t sme finite value. This discntinuity cmes frm the integral n the right-hand side f equatin (4). Let us indicate by r þ and r field pints just utside and inside the medium, and near pint r in S. With a limit prcedure it can be shwn using equatin (4) that [22] Bðr Þ¼BðrÞ inc 1 2 iðrþ^nðrþ 4 P ds 0 iðr 0 Þrgðr r 0 Þ, ð6þ where ^nðrþ is the unit nrmal n S at r, directed frm the material t the vacuum. Fr the integral n the right-hand side we take pint r in S. The factr rg(r r 0 ) in the integrand f the integral has a singularity at r 0 ¼ r, and it is understd that we leave ut a small circle arund r, and that in the end we let the radius f this circle shrink t zer. In this sense, the integral is a principal value integral, ð

4 48 H. F. Arnldus which is indicated by P. This integral is the same fr B(r þ ) and B(r ), s fr the difference we btain Bðr þ Þ Bðr Þ¼ iðrþ^nðrþ, and this is the usual bundary cnditin fr the magnetic field at an interface. Therefre, this bundary cnditin is autmatically satisfied by equatin (4), n matter what the surface current density is. Then, inside the perfect cnductr the magnetic field is zer, s we have B(r ) ¼ 0. Frm equatin (6) we then find ð 1 2 iðrþ^nðrþþ 4 P ds 0 iðr 0 Þrgðr r 0 Þ¼BðrÞ inc : ð8þ Given the incident field, this is a linear integral equatin fr i(r), with B(r) inc as the inhmgeneus term. In a numerical apprach, this equatin is slved with the Methd f Mments. After slving, P(r) is cmputed with equatin (1), and then the reflected field fllws by taking the curl, accrding t equatin (5). Finally, the reflected electric field fllws frm a Maxwell equatin: 3. Current density in a mirrr ð7þ EðrÞ r ¼ ic2! rbðrþ r: ð9þ Equatin (8) can nly be slved fr i(r) analytically if the shape f the surface is simple enugh. Fr instance, it can be slved in clsed frm fr a sphere in terms f vectr spherical harmnics. Anther example, which we shall cnsider here, is a flat surface. The gradient f the Green s functin is rgðr r 0 Þ¼ðr r 0 Þ ik jr r 0 j 2 1 jr r 0 j 3 e ik jr r 0j : Let the surface be the xy-plane, and let the material ccupy the regin z50. The integratin in equatin (8) runs ver the xy-plane, s the pint r 0 is in the xy-plane. Fr the principal value integral, pint r is als in the xy-plane, and therefre the vectr r r 0 lies in the xy-plane, and s des rg(r r 0 ). Since the current density i(r 0 ) is als in the xy-plane we see that i(r 0 ) rg(r r 0 ) is alng the z-axis. Cnsequently, the principal value integral in equatin (8) is a vectr alng the z-axis. We nw take the crss prduct f equatin (8) with e z. The cntributin frm the integral vanishes, and e z ½iðrÞ^nðrÞŠ ¼ iðrþ, since ^nðrþ ¼e z. Equatin (8) becmes ð10þ iðrþ ¼ 2 e z BðrÞ inc, ð11þ which is the explicit slutin fr i(r), given the incident magnetic field. Apparently, fr a flat surface the current density at pint r n the surface nly depends n the value f the incident magnetic field at that same pint. In the general case f

5 Reflectin ff a mirrr 49 equatin (8), the current density at r depends n the current density at every ther pint r 0 thrugh the principal value integral. Interesting t nte is that if we set B(r ) ¼ 0 in the bundary cnditin (7), and then take the crss prduct with e z we find iðrþ ¼ 1 e z Bðr þ Þ: ð12þ Here, B(r þ ) is the ttal field just utside the medium, which is the sum f the incident field and the reflected field. The additinal factr f 2 in equatin (11) indicates that the incident field and the reflected field cntribute equally t i(r) at pint r. Frm equatin (6) with B(r ) ¼ 0 we have Bðr þ Þ¼ iðrþe z, ð13þ s B(r þ ) is a vectr in the xy-plane. The incident and the reflected field will bth have a cmpnent alng the z-axis at z ¼ 0 þ, and these cmpnents cancel exactly. 4. The magnetic field With the current density in the mirrr given by equatin (11), the vectr ptential (1) becmes PðrÞ ¼ 1 ð 2 e z ds 0 Bðr 0 Þ inc gðr r 0 Þ: ð14þ Taking the curl and simplifying with a vectr identity then yields fr the magnetic field BðrÞ ¼BðrÞ inc 1 ð ds 0 Bðr Þ 2 gðr r0 Þ þ 1 ð ð15þ 2 e z ds 0 ½Bðr 0 Þ inc rgðr r 0 ÞŠ: Since the integrals run ver the xy-plane, this shws explicitly that the ttal magnetic field ff the xy-plane is determined cmpletely by the given value f the incident field B(r) inc in the xy-plane. Fr z40, the sum f the secnd and the third term n the right-hand side is the reflected field. Fr z50, the three terms n the right-hand side shuld add up t zer, since the magnetic field in the material is zer. We shall shw this in sectin 6. The magnetic field can als be expressed in terms f an angular spectrum, which is particularly useful if the surce f the incident radiatin is knwn. We adpt Weyl s representatin f the Green s functin [23] gðr r 0 Þ¼ i 2 ð d 2 1 eiðr r 0 Þþijz z 0j : ð16þ

6 50 H. F. Arnldus Vectr k k is a vectr in the xy-plane, and the integratin is a surface integral ver the k k plane. The parameter is defined by 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < k 2 ¼ k2 jj, < k qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi, ð17þ : i k 2 jj k2, > k with k k the magnitude f k k. Fr k k 5k, is real and the partial wave is travelling, whereas fr k k 4k the partial wave is evanescent. These waves decay expnentially away frm the plane z ¼ z 0. We substitute the expressin (16) int equatin (15), with z 0 ¼ 0 since r 0 is in the xy-plane, and wrk ut the derivatives. We then btain the alternative representatin fr the magnetic field near the mirrr BðrÞ ¼BðrÞ inc þ 1 ð 4 2 d 2 1 sgnðzþ ~Bð Þ inc e z Bðkjj ~ Þ inc þ sgnðzþe z e i rþijzj, ð18þ with ~Bð Þ inc ¼ ð ds 0 Bðr 0 Þ inc e ir 0, ð19þ the spatial Furier transfrm f B(r) inc in the xy-plane. Expressin (18) hlds fr bth z40 and z50. Fr z40, we have sgn(z) ¼ 1, and with the ntatin K ¼ e z, ð20þ the result (18) can be simplified t BðrÞ ¼BðrÞ inc ð d 2 1 eik þr K þ e z ~Bð Þ inc, z > 0: ð21þ The secnd term n the right-hand side is the reflected field. Similarly, fr z50 we can write equatin (18) as BðrÞ ¼BðrÞ inc 1 ð 4 2 d 2 e ik r ~Bð Þ inc 1 ð 4 2 e z d 2 1 eik r K ~Bð Þ inc, z < 0, ð22þ and this shuld be zer. 5. Surce f the incident field Equatins (21) and (22) give the magnetic field fr an arbitrary incident field B(r) inc. Hwever, we have nt used any prperty f this field, and in particular that it has

7 Reflectin ff a mirrr 51 z=b z=a x j(r) r j(x) r r z=0 Figure 2. The surce f the incident radiatin is the current density j(r), which is lcated in the regin a5z5b. Fr a lcalized surce arund sme pint r, it is advantageus t use the psitin vectr n t indicate a pint in the surce distributin. e z t satisfy Maxwell s equatins, when cmbined with the crrespnding electric field. Let us nw cnsider the situatin where the incident field is emitted by a lcalized surce in the regin z40, as shwn in figure 2. The surce has a given current density j(r), and the radiated magnetic field is the curl f the vectr ptential, as in equatin (4), but nw the integral runs ver the vlume f the surce. We have fr the field generated by the surce BðrÞ ¼ 4 ð d 3 r 0 jðr 0 Þrgðr r 0 Þ: In rder t btain an angular spectrum representatin f this expressin, we substitute Weyl s representatin (16) fr the Green s functin. T simplify the appearance f z z 0 in the Green s functin, we shall nly cnsider the slutins in z4b and z5a in the ntatin f figure 2. Fr z4b we have z z 0 ¼ z z 0, since the value f the integratin variable z 0 is restricted t a5z 0 5b, and similarly fr z5a we have z z 0 ¼ (z z 0 ). With equatin (20) we have ð23þ ðr r 0 Þþjz z 0 j¼k ðr r 0 Þ, ð24þ where the upper (lwer) sign refers t the regin z4b (z5a). Then we define the tw surce functins Dð Þ ¼ i ð d 3 rjðrþe ikr, ð25þ! in terms f which the magnetic field f the surce can be represented as BðrÞ ¼ i! ð 8 2 d 2 1 eik r k Dð Þ : ð26þ Here again the upper (lwer) sign refers t the regin z4b (z5a). This is a superpsitin f plane waves, and each partial wave has a wavevectr K ¼ K k e z. The z-cmpnent is. Therefre, the slutin in z4b travels away frm the surce regin twards larger z values fr real, and decays away frm the surce in the psitive z directin fr imaginary. Similarly, the slutin in the

8 52 H. F. Arnldus regin z5a travels r decays twards the xy-plane. The slutin with the lwer sign in equatin (26) is the incident field n the mirrr, s we have explicitly BðrÞ inc ¼ i! ð 8 2 d 2 1 eik r K Dð Þ : ð27þ 6. Magnetic field in terms f the surce functin D(k k ) Equatin (27) expresses the incident field in terms f the surce functin D(k k ), whereas in equatin (21) the reflected field is expressed in terms f the Furier transfrm ~Bð Þ inc f the incident field in the xy-plane. We shall nw derive a relatin between these tw functins f k k. The incident field in the xy-plane fllws frm setting z ¼ 0 in equatin (27), which gives Bðr 0 Þ inc ¼ i! ð 8 2 d 2 1 eir 0 K Dð Þ, xy plane: ð28þ Then we multiply by expð ik 0 jj r0 Þ, as in equatin (19), and we integrate ver the xy-plane. With the spectral representatin f the tw-dimensinal delta functin ð ds 0 e ið k 0 jj Þr0 ¼ 4 2 ð k 0 jjþ, ð29þ we btain the simple relatin ~Bð Þ inc ¼ i! 2 K Dð Þ : ð30þ When we substitute this int equatin (21) we find BðrÞ ¼BðrÞ inc i! ð 8 2 d eik þr K þ e z K Dð Þ, ð31þ fr the magnetic field. The angular spectrum representatin f B(r) inc is given by equatin (27), which hlds fr the regin 05z5a. The secnd term n the righthand side f equatin (31) is the reflected field, and this representatin hlds f curse fr all z40. Next we cnsider the field in z50, given by equatin (22). Frm equatin (30) we have K ~Bð Þ inc ¼ 0, and therefre the last term n the right-hand side f equatin (22) vanishes. When we cmbine equatins (27) and (30) we have BðrÞ inc ¼ 1 ð 4 2 d 2 e ik r ~Bð Þ inc : ð32þ This shws that the incident field B(r) inc fr all r in z5a is determined by knwledge f this field in the xy-plane nly, since the evaluatin f ~Bð Þ inc nly invlves the field in the xy-plane. Frm equatin (32) we see that the first integral n the

9 Reflectin ff a mirrr 53 right-hand side f equatin (22) is just B(r) inc, and therefre the first tw terms in equatin (22) cancel. This prves that B(r) ¼ 0 fr z Verificatin f the bundary cnditin Our slutin (31) fr the magnetic field in 05z5a was btained by slving the integral equatin (8) fr the current density, after which the field was expressed as the sum f the incident field and the field radiated by i(r). In a mre cnventinal apprach t reflectin f radiatin by an interface, like fr the cmputatin f the Fresnel cefficients fr a layer, ne writes dwn general slutins f Maxwell s equatins at bth sides f the interface, cntaining sme undetermined cnstants. Fr the layer prblem, these are the Fresnel cefficients. These cnstants are then evaluated by impsing the bundary cnditins fr the electric and magnetic fields at the interface. We nw shw that fr ur slutin the bundary cnditin fr the magnetic field is satisfied. The general bundary cnditin fr B(r) is that its nrmal cmpnent must be cntinuus acrss the bundary, which fllws frm Maxwell s equatin rb(r) ¼ 0. Fr a perfect cnductr we have B(r) ¼ 0 inside the material, and therefre it must hld that the nrmal cmpnent f B(r) is zer just utside the medium, as als fllws frm equatin (13). With r þ a pint just utside the material, we must therefre shw that e z B(r þ ) ¼0. Since the z-cmpnent f r þ is zer in the limit where this pint appraches the surface, we have K r þ ¼ k k r þ frm equatin (20). With this bservatin, equatins (27) and (31) can be cmbined as Bðr þ Þ¼ i! ð 8 2 d eir þ K Dð Þ K þ e z K Dð Þ : ð33þ Frm equatin (20) we derive K þ ðe z aþ ¼ðK aþe z þ 2e z ðe z aþ a, ð34þ fr any vectr a. Fr a ¼ K D(k k ), the first term n the right-hand side vanishes, and we have K þ fe z ½K Dð Þ Šg ¼ 2e z fe z ½K Dð Þ Šg K Dð Þ : ð35þ Taking the dt prduct with e z yields e z K þ fe z ½K Dð Þ Šg ¼ ez ½K Dð Þ Š, ð36þ and this gives e z B(r þ ) ¼ 0, which is the bundary cnditin.

10 54 H. F. Arnldus 8. Image surce In the methd f images [24] the mirrr is replaced by an image surce in z50, in such a way that the field radiated by the image surce int the regin z40 is the same as the reflected field in the prblem with the mirrr. We nw cnstruct the image f an arbitrary surce, as in figure 2, and determine the crrespnding surce functin fr the angular spectrum representatin f the image field. Then we prve that the field in z40 f the image surce is equal t the reflected field B(r) r by the mirrr. As usual in the methd f images, we first guess what the image surce is, and then we prve afterwards that it is crrect. Let r be a psitin vectr f a pint in the surce, and we write r ¼ r k þ r?, where the subscripts k and? refer t the parallel and perpendicular cmpnents with respect t the xy-plane, respectively. The mirrr image f this pint is r im ¼ r jj r?, and the mirrr image f a psitive charge is a negative charge, as indicated in figure 3(a). When this psitive charge mves alng the xy-plane, as in figure 3(b), its mirrr image mves in the same directin, but since this is a negative charge, the crrespnding current density is in the ppsite directin. When the psitive charge mves away frm the xy-plane, as in figure 3(c), its negative mirrr image mves in the negative z-directin, crrespnding t a current density in the psitive z-directin. Therefre, if we write j(r) ¼ j(r)? þ j(r) k fr the current density at pint r in the surce, then the current density at the mirrr psitin r im is ð37þ jðr im Þ¼jðrÞ? jðrþ jj : ð38þ r + j(r) + + j(r) r im j(r im ) j(r im ) (a) (b) (c) Figure 3. (a) The mirrr image f a pint with psitin vectr r is the pint with psitin vectr r im, and the mirrr image f a psitive charge is a negative charge, as is illustrated. (b) When the psitive charge mves parallel t the surface, its mirrr image mves in the same directin, but since this is a negative charge, the current density f the image surce at the image psitin is in the ppsite directin. (c) When the psitive charge mves away frm the surface, s des the mirrr image. The crrespnding current density f the image surce at the image psitin is therefre the same as the current density f the surce. These cnsideratins lead t equatin (38) fr the current density f the image surce.

11 Reflectin ff a mirrr 55 The angular spectrum representatin f the field radiated by the image surce is identical in frm as the field generated by j(r), with the explicit frm given by equatin (26). The shape f the image surce is the mirrr image f the surce, since fr every r in the surce there is a crrespnding r im in the image surce. Therefre the regin a5z5b in figure 2 has a mirrr image regin b5z5 a, and fr the field radiated by the image surce in z40 we need the slutin fr z4 a, which is the upper sign in equatin (26). The surce functin fr the image surce is Dð Þ im þ ¼ i ð d 3 r im jðr im Þe ik þr im, ð39þ! as in equatin (25) (we will nt need the surce functin here), and the image field in z4 a is BðrÞ im ¼ i! ð 8 2 d 2 1 eik þr K þ Dð Þ im þ : ð40þ With equatins (20) and (37) we have K þ r im ¼ K r, and fr j(r im ) we substitute the right-hand side f equatin (38) in the integrand f the integral in equatin (39). Fr d 3 r im we write d 3 r, since this is just a symblic ntatin fr the vlume element. Then equatin (39) becmes Dð Þ im þ ¼ i! ð d 3 r½jðrþ? jðrþ jj Še ik r, and the integral nw runs ver the surce, rather than ver the image surce. Cmparisn with equatin (25) then gives ð41þ Dð Þ im þ ¼ DðÞ,? Dð Þ,jj : ð42þ We find that the þ surce functin fr the image surce can be btained frm the surce functin f the surce by reversing the sign f the parallel cmpnent. With D(K k ) ¼ D(k k ),? þ D(k k ),k, and D(K k ),k ¼ e z [e z D(k k ) ], we can als write equatin (42) as Dð Þ im þ ¼ DðÞ þ 2e z ½e z Dð Þ Š: ð43þ The reflected field is given by the secnd term n the right-hand side f equatin (31), and with the identity (35) this can als be written as BðrÞ r ¼ i! ð 8 2 d 2 1 eik þr K Dð Þ 2e z e z K Dð Þ : ð44þ Fr the field by the image surce, equatin (40), we need the crss-prduct f Dð Þ im þ with K þ. T this end we set a ¼ e z D(k k ) in equatin (34). The secnd term n the right-hand side vanishes and with a vectr identity the result can be written as K þ fe z ½e z Dð Þ Šg ¼ e z e z K Dð Þ ez Dð Þ : ð45þ

12 56 H. F. Arnldus Furthermre we have frm equatin (20) K þ Dð Þ ¼ K Dð Þ þ 2e z Dð Þ : Then we crss the representatin (43) f Dð Þ im þ with K þ and use equatins (45) and (46). This yields K þ Dð Þ im þ ¼ K Dð Þ 2e z e z K Dð Þ, ð47þ and when we substitute the right-hand side in equatin (40) fr the field by the image surce, the result is exactly equatin (44) fr the reflected field. This shws that the current density (38) and the surce functin (43) represent an image surce that radiates a field in z40 which is identical t the reflected field. ð46þ 9. Lcalized surce The surce functins D(k k ) frm equatin (25) depend n the rigin f crdinates. When a surce is lcalized arund sme pint r, as in figure 2, then it is mre cnvenient t take this pint as the rigin f crdinates fr the surce functins. We make the change f variables n ¼ r r in equatin (25), which gives Dð Þ ¼ e ik r dð Þ, with the new surce functins d(k k ) defined as dð Þ ¼ i ð d 3 njðnþe ikn :! The field emitted by the surce, equatin (26), then becmes BðrÞ ¼ i! ð 8 2 d 2 1 eik ðr r Þ K dð Þ : ð48þ ð49þ ð50þ The r dependence f the field enters thrugh exp[ik (r r )], and it is nw mre bvius that the field emanates frm the neighburhd f the pint r. The reflected field is given by the secnd term n the right-hand side f equatin (31), and with K r ¼ K þ r im this becmes BðrÞ r ¼ i! ð 8 2 d eik þðr r imþ K þ e z K dð Þ : ð51þ The r dependence f the reflected field enters as exp½ik þ ðr r im ÞŠ, and it appears that this field is radiated by a lcalized surce arund the image psitin r im. Alternatively, the image surce functin frm equatin (43) is with equatin (48) and K r ¼ K þ r im : Dð Þ im þ ¼ e ik þr im dðkjj Þ im þ, ð52þ

13 Reflectin ff a mirrr 57 where dð Þ im þ ¼ dðk jjþ þ 2e z e z d : ð53þ With equatin (40) we then have BðrÞ r ¼ i! ð 8 2 d 2 1 eik þðr r imþ im K þ d þ, ð54þ and this is the alternative t representatin (51). We nw nte that equatins (50) and (54) are identical in frm. The surce radiatin emanates frm the pint r and has d(k k ) as it surce functin. The reflected radiatin, when viewed as the image surce field, appears t cme frm r im, and it has dðþ im þ as its surce functin. 10. Diples When the surce is an atm r mlecule in an excited state, the emitted radiatin during spntaneus emissin is usually electric diple radiatin. The time-dependent electric diple mment f the scillating electrn clud has the frm p e ðtþ ¼Reðp e e i!t Þ, ð55þ with p e the cmplex amplitude, which is referred t as the electric diple mment. When the atm r mlecule is lcated at r, the crrespnding current density is jðrþ ¼ i!p e ðr r Þ, ð56þ and with equatin (49) we find fr the surce functins dð Þ ¼ p e : ð57þ Apparently, fr an electric diple the þ and surce functins are the same. Fr a pint surce, the regin a5z5b in figure 2 becmes z ¼ z, s the angular spectrum f the emitted diple radiatin is given by equatin (50) with d(k k ) ¼ p e, and where the upper (lwer) sign hlds fr z4z (z5z ). The reflected field is given by equatin (51) with d(k k ) ¼ p e. Alternatively, in the image apprach the reflected field is given by representatin (54). Fr the image surce functin, the parallel cmpnent changes sign, s we have dð Þ im þ ¼ p e,? p e,jj p im e : ð58þ When cmpared t equatin (57), we see that the image surce is again an electric diple p im e, but with its parallel cmpnent reversed in sign. This, f curse, is a well-knwn fact and als fllws easily frm cnsideratins as in figure 3.

14 58 H. F. Arnldus Less trivial is the case f a magnetic diple p m at r. The time-dependent magnetic diple mment is given by equatin (55) with p e! p m. The cmplex amplitude f the current density is [25] jðrþ ¼ p m rðr r Þ: ð59þ Frm equatin (49) we btain fr the surce functins dð Þ ¼ 1! p m K, ð60þ and here the þ and surce functins are different. In rder t find the image surce we need t reverse the parallel part f d(k k ). T this end we first write p m ¼ p m,? þ p m,k and use equatin (20) fr K. This gives dð Þ ¼ 1! ðp m,? p m,jj e z þ p m,jj Þ: ð61þ The first tw terms n the right-hand side are parallel t the xy-plane and the third term is perpendicular t the xy-plane. Therefre the image surce functin is dð Þ im þ ¼ 1! ð p m,? þ p m,jj e z þ p m,jj Þ, ð62þ which can als be written in a frm similar t equatin (60) as with dð Þ im þ ¼ 1! pim m K þ, ð63þ p im m ¼ p m,jj p m,?, ð64þ fr the image diple mment. S the mirrr image f a magnetic diple is again a magnetic diple, but it has its perpendicular cmpnent reversed in sign, as cmpared t the riginal diple. 11. Electric quadruple An electric quadruple has a current density [25] jðrþ ¼ i! 6 Q$ rðr r Þ, ð65þ with Q $ a symmetric Cartesian tensr. Frm equatin (49) we find the surce functins t be dð Þ ¼ i 6 Q$ K : ð66þ

15 Reflectin ff a mirrr 59 We nw want t find the image surce functin dð Þ im þ, and express it in the frm dð Þ im þ ¼ i 6 Q$ im Kþ : ð67þ Then equatins (66) and (67) are identical in frm, and we can cnsider Q $ im the image f the electric quadruple. In rder t btain Q $ im we need t find the parallel and perpendicular cmpnents f the right-hand side f equatin (66). It will turn ut t be cnvenient t use spherical basis vectrs e, ¼ 1, 0, 1, rather than Cartesian basis vectrs. This basis is defined as e 1 ¼ p 1 ffiffi ðe x ie y Þ, 2 and e 0 ¼ e z. We then define a basis f unit tensrs as [21] ð68þ E $ ðkþ q ¼ X 0 ð1 1 0 jkqþe e 0, ð69þ with (1 1 0 kq) Clebsch Grdan cefficients. These cefficients are nly nnzer fr k ¼ 0, 1 and 2, and fr a given k the pssible q values are k,..., k. S fr k ¼ 0 we nly have q ¼ 0, and fr k ¼ 1 we have q ¼ 1, 0, 1 and fr k ¼ 2 there are five q values. This gives nine independent tensrs, just like there are nine independent Cartesian tensrs f the frm e e with, ¼ x, y, rz. Then we expand Q $ as Q $ ¼ X kq Q ðkþ q ðkþ E$ q : ð70þ The expansin cefficients Q ðkþ q are cmplex numbers, and fr a symmetric Cartesian tensr Q $ the cefficients with k ¼ 1 vanish. We then btain the expressin dð Þ ¼ i 6 X e X kq 0 Q ðkþ q ð1 1 0 jkqþðe 0 K Þ, ð71þ fr this surce functin, where we have used that the Clebsch Grdan cefficients are real. The terms with ¼1 are in the xy-plane and the term with ¼ 0 is alng the z-axis. Fr the surce functin f the image surce we therefre get a factr ( 1) in equatin (71), with everything else the same. Then we have e 0 K ¼ð 1Þ 0 þ1 e 0 K þ, ð72þ which gives a factr f ( 1) þ0 þ1. The Clebsch Grdan cefficients are nly nnzer fr þ 0 ¼ q, s we can replace this with ( 1) qþ1. We then btain dð Þ im þ ¼ i 6 X kq 0 ð 1Þ qþ1 Q ðkþ q ð1 1 0 jkqþe ðe 0 K þþ, ð73þ

16 60 H. F. Arnldus which is with equatin (69) dð Þ im þ ¼ i 6 X kq ð 1Þ qþ1 Q ðkþ ðkþ q ðe$ q K þ Þ: ð74þ Cmparisn with equatin (67) then shws that the image quadruple tensr is Q $ im X ¼ ð 1Þ qþ1 Q ðkþ ðkþ q E$ q, ð75þ and cmparisn with equatin (70) shws that the cmpnent Q ðkþ q ( 1) qþ1 upn reflectin. kq picks up a factr 12. Angular distributin f the emitted pwer When a small surce is lcated near the surface, then the emitted radiatin can be bserved by a detectr in the far field. This radiatin is the sum f the field emitted by the surce, represented by equatin (26) with the upper sign, and the field f the image surce, given by equatin (40). The field in the regin z4b is BðrÞ ¼ i! ð 8 2 d 2 1 h eik þr K þ D þ þdk im jj þ i, ð76þ and the crrespnding electric field fllws frm E(r) ¼ (ic 2 /!)rb(r), which is EðrÞ ¼ i 8 2 " ð d 2 1 eik þr K þ n h K þ D þ þdk im jj þ i : ð77þ Fr a detectr in the far field, lcated at field pint r, we need t cnsider the slutin fr r large. An asympttic expansin f an angular spectrum can be made with the methd f statinary phase [26], and the result takes the general frm ð d 2 1 eik þr Wð Þ 2i eikr r Wð, Þ, where W(k k ) is any functin f k k. At the right-hand side this functin is evaluated at the statinary pint k k, in the k k plane. With (r,, ) the spherical crdinates f the lcatin r f the detectr, this statinary pint is ð78þ, ¼ k sin ðe x cs þ e y sin Þ: ð79þ The value f K þ in the statinary pint is fund t be k ^r, with ^r the unit vectr in the bservatin directin. We btain fr the asympttic apprximatin f

17 Reflectin ff a mirrr 61 the magnetic field BðrÞ! k 4r h e ik r^r D, þ þdk im jj, þ i : ð80þ Fr the electric field a similar expressin is fund, and the result is related t the magnetic field by EðrÞ c^r BðrÞ: The Pynting vectr representing the energy flw in an electrmagnetic field is in general defined as SðrÞ ¼ 1 ReEðrÞBðrÞ : ð82þ 2 Here, E(r) and B(r) are the cmplex amplitudes f the electric and magnetic fields, respectively, but S(r) is in the time dmain. We have drpped terms which scillate with twice the ptical frequency, which has the cnsequence that S(r) is independent f time. With equatin (81), a vectr identity and ^r BðrÞ ¼ 0 this becmes ð81þ SðrÞ ¼ c 2 ½BðrÞBðrÞ Š^r: ð83þ We write an equal sign instead f, which we shall d frm here n. The detected pwer per unit slid angle int the directin ^r is dp d ¼ r2 SðrÞ^r, and with equatins (80) and (83) this becmes dp d ¼! 4 Dh 32 2 " c 3 n h ^r D, D, þ þdk im jj, þ þ þdk im jj, þ i h D, in h ^r D, þ þdk i im jj, þ þ þdk i im jj, þ ð84þ E : ð85þ The result (85) can be simplified further by using equatins (48) and (52) fr a lcalized surce. In the statinary pint we have K þ, ¼ k ^r, s that Dð, Þ þ þ Dð, Þ im þ h ¼ dð,þ þ þ e ik ^rðr im rþ dð, Þ im þ i e ik ^rr : ð86þ Since this expressin is multiplied by its cmplex cnjugate in equatin (85), the phase factr expð ik ^r r Þ cancels. Furthermre, with r im r ¼ 2z e z, and with h ¼ k z as the dimensinless distance between the surce and the surface we have ik ^r ðr im r Þ¼2ih cs. We then intrduce the functin Dð,Þ ¼dð, Þ þ þ e 2ih cs dð, Þ im þ, ð87þ

18 62 H. F. Arnldus q de r Surce q z x xy -plane q Image Figure 4. The directly emitted (de) wave travels directly frm the surce t the detectr, and is bserved under plar angle with the nrmal t the surface. The reflected (r) wave is als detected under angle, and since the angle f incidence equals the angle f reflectin, the angle indicated by? is equal t 2. The wave emanating frm the surce as the incident wave is incident n the surface at angle, and therefre it leaves the surce at angle with the nrmal. The nrmal distance between the surce and the surface is z, and the distance travelled by the incident wave is indicated by x. We see frm the figure that x ¼ z /cs, and the additinal distance travelled by the r-wave, as cmpared t the de-wave, is x þ xcs2, which is equal t 2z cs. This explains the phase factr exp[ik (2z cs)] in equatin (87) fr the reflected wave. Since the reflected wave makes an angle with the nrmal t the surface, it intersects the nrmal line thrugh the surce at an angle. Therefre, the reflected wave appears t cme frm the lcatin f the image surce, a distance z belw the surface. which depends n and thrugh the dependence f the surce functins n k k,, with k k, given by equatin (79). The first term n the right-hand side f equatin (87) cmes frm the radiatin that is emitted by the surce directly twards the detectr, and the secnd term represents radiatin that is emitted twards the surface, and then travels twards the detectr after reflectin. The factr exp(2ih cs) accunts fr the retardatin between the tw signals, as is illustrated in figure 4. The figure als shws that the reflected radiatin appears t cme frm the image surce. The final result fr the detected pwer per unit slid angle then takes the elegant frm dp d ¼! " c 3 Dð,ÞDð,Þ ^r Dð,Þ ^r Dð,Þ, ð88þ which hlds fr any lcalized surce near the surface. The angle dependence enters thrugh the functins D(, ), and thrugh the angle dependence f the unit vectr ^r.

19 Reflectin ff a mirrr 63 Fr an electric diple with diple mment p e the functins d(k k ) þ f the surce and dð Þ im þ f the image are independent f k k, accrding t equatins (57) and (58), s we find immediately Dð,Þ ¼p e þ e 2ih cs p im e : ð89þ Fr a magnetic diple these functins d depend n k k, as fllws frm equatins (60) and (63). They depend n k k thrugh K þ, which becmes k ^r in the statinary pint, and we find Dð,Þ ¼ 1 c ^r ðp m þ e 2ih cs p im m Þ: ð90þ Similarly, we btain frm equatins (66) and (67) Dð,Þ ¼ ik 6 ^r þ e 2ih cs Q $ im Þ, ð91þ ðq$ fr an electric quadruple. Equatins (89) (91) clearly shw that the field by the image surce is accunted fr by adding the image diple mment (image quadruple tensr) t the diple mment (quadruple tensr). Fr the image term the retardatin factr exp(2ih cs) appears, as a result f the fact that this radiatin first travels t the surface where it subsequently reflects, as illustrated in figure 4. It is interesting t see that in this asympttic limit the explicit dependence n the lcatin r f the multiple drps ut. 13. Pwer distributin f an electric quadruple As a nntrivial example we cnsider an electric quadruple lcated near the surface. Let the quadruple tensr be given by Q $ ¼ QE $ ð2þ q, ð92þ with Q a cmplex number and q ¼ 2,..., 2. Frm equatins (70) and (75) we find that the image quadruple mment is given by Q $ im ¼ Qð 1Þ qþ1 E $ ð2þ q ¼ð 1Þ qþ1 Q $ : ð93þ The functin D(, ) becmes Dð,Þ ¼ ik 6 ^r Q$ ½1 þð 1Þ qþ1 e 2ih cs Š, ð94þ and frm equatin (88) we find fr the emitted pwer per unit slid angle dp ck 6 d ¼jQj j1 þð 1Þ qþ1 e 2ih cs j 2 ^r E $ð2þ q ^r E $ ð2þ q " j^r E $ ð2þ q ^rj 2 : ð95þ

20 64 H. F. Arnldus As an illustratin, let q ¼ 0. Frm equatin (69) and the tabulated values f the Clebsch Grdan cefficients we find explicitly 0 ¼ p 1 ffiffi ðe 1 e 1 þ 2e 0 e 0 þ e 1 e 1 Þ, 6 E $ ð2þ ð96þ in spherical unit vectrs, and with equatin (68) this becmes 0 ¼ p 1 ffiffi ð2e z e z 2e x e x e y e y Þ, 6 E $ ð2þ ð97þ in Cartesian unit vectrs. We furthermre have ^r ¼ sin ðe x cs þ e y sin Þþe z cs, which gives ^r E $ ð2þ 0 ^r E $ ð2þ 0 j^r E $ ð2þ 0 ^rj 2 ¼ 3 8 ðsin 2Þ2, ð98þ and this yields fr the angular distributin f the emitted pwer dp ck 6 d ¼jQj ½1 csð2h cs ÞŠðsin 2Þ 2 : ð99þ " This intensity pattern is shwn in figure 5, tgether with the angular distributin f the radiatin fr this quadruple in free space. The lbe structure is a result f interference between the radiatin emitted directly by the surce twards the detectr and the radiatin that is first reflected ff the mirrr. 14. Cnclusins When electrmagnetic radiatin is incident upn a perfectly-cnducting medium, a current density is induced n the surface f the material. This surface current density i(r) is the slutin f the integral equatin (8), which has the incident magnetic field as inhmgeneus term. We have shwn that fr a flat surface f infinite extent the slutin f this equatin can be btained in clsed frm, and is given by equatin (11). This remarkably simple expressin shws that the current density at the pint r f the surface is determined by the value f the incident magnetic field at that same pint. With the current density knwn, the ttal magnetic field is the sum f the incident field and the field generated by i(r), and the result is given by equatin (15), invlving the Green s functin fr the scalar freespace Helmhltz equatin. The ttal magnetic field is determined entirely by the value f the incident magnetic field at the surface f the cnductr, irrespective the surce f the radiatin. With Weyl s representatin (16) f the Green s functin, the magnetic field in z40 can als be written as an angular spectrum, as given by equatin (21). The angular spectrum representatin f the field in z50 is given by equatin (22), and it was shwn that this field vanishes. The current density

21 Reflectin ff a mirrr xy -plane Figure 5. The thin line is a plar diagram f the angular intensity distributin f the radiatin emitted by a q ¼ 0 electric quadruple near the surface f a perfect cnductr, and the thick line is the emissin pattern fr the same quadruple in free space. The radiatin pattern is rtatin symmetric arund the nrmal t the surface. The dimensinless nrmal distance between the quadruple and the surface is h ¼ 6, and bth curves are scaled with the same verall cnstant. generates a magnetic field, and inside the material this field cancels exactly the incident field. When the incident field is radiatin frm a surce with knwn current density j(r), the reflected field can be expressed in terms f a surce functin D(k k ), defined by equatin (25), and the result is given by the secnd term n the right-hand side f equatin (31). In sectin 8 we cnstructed an image surce, and equatin (42) r (43) shws that the surce functin f the image surce can be btained in a simple way frm the surce functin f the surce itself. It was shwn explicitly that the image surce radiates an electrmagnetic field in z40 which is identical t the reflected field. When the surce is lcalized arund sme pint r it is advantageus, but nt necessary, t make the change f crdinates shwn in figure 2. The dependence n the lcatin r f the surce then factrs ut as in equatin (48), leading t the new surce functins d(k k ). We illustrated in sectins 10 and 11 that the apprach with the surce functins leads t a simple methd fr btaining the mirrr image f multiples with, fr example, the mirrr image f an electric quadruple given by equatin (75). Fr a lcalized surce, the pwer emitted per unit slid angle, including the directly emitted radiatin by the surce and the reflected field, was fund t be given by equatin (88). This expressin hlds fr any surce f radiatin near the surface, and its evaluatin nly invlves the surce functins d(k k ).

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