Total and Frustrated Reflection of a Gaussian Optical Beam

Transcription

1 Problem Total ad Frustrated Reflectio of a Gaussia Optical Beam Kirk T. McDoald Joseph Hery Laboratories, Priceto Uiversity, Priceto, NJ (July 7, 009) Whe a electromagetic wave i a medium with idex of refractio ecouters a iterface with a regio of idex < the wave ca be totally reflected, with oly a evaescet (surface) wave excited i the regio of lower idex. I this case, eergy is (largely) trasported parallel to the iterface i the regio of lower idex. Discuss the flow of eergy whe the icidet wave has limited trasverse extet. I particular, cosider a weakly focused, liearly polarized Gaussia beam that is icidet from the medium with lower idex. Also discuss the flow of eergy whe a Gaussia beam is icidet from a medium of idex oto a regio of idex < of thickess d beyod which the medium has idex. Solutio We will use the time-average Poytig vector, S = Re(E B )/μ (i SI uits), to discuss the flow of eergy i waves with electric field E ad magetic field B.. Weakly Focused, Liearly Polarized Gaussia Optical Beams We use so-called Gaussia beams to describe approximate wave solutios to Maxwell s equatios that have limited trasverse extet. However, eve if the icidet beam is cylidrically symmetric about its axis of propagatio, the refracted beam will i geeral have a elliptical cross sectio. Hece, i Appedix A we make a small geeralizatio of typical presetatios of Gaussia beams (see, for example, sec..4 of []) to cosider elliptical beams. The first-order Gaussia beam of agular frequecy ω that propagates alog the z-axis with See, for example, prob. of

2 y-polarizatio i a medium with permittivity ɛ ad permeability μ has fields E x 0, E y E 0 e ρ /(+z /z0 ) e i{kz[+ρ /k(z +z0 )] ωt ta (z/ )}, () +z /z0 E z iy e i ta z/ E kw0y y, +z /z0 B x = c E y, B y =0, B z = ix kw0x where c is the speed of light i vacuum, c E y k 0 = ω c, k = k 0, = c ɛμ, ρ = e i ta z/ +z /z 0 x w 0x, () + y, (3) w0y w 0j for j = x or y is the characteristic radius of the beam at its waist (focus), θ 0j is the diffractio agle ad is the Rayleigh rage, as show i the figure below, which are related by θ 0j = w 0j, = + kw0x kw0y ( = kw 0 = kθ 0 ) if w 0x = w 0y = w 0. (4) Near the focus (ρ <, z < ), the beam ()-() ca be approximated as the plae wave, E x =0, E y = E 0 e ρ e i(kz ωt), E z = iy E kw0y y, (5) B x = c E y, B y =0 B z = ix kw0x c E y, (6) which obeys E =0= B recallig eq. (4). The equatios E = B/ (ct) ad H = D/ (ct) are satisfied up to terms of order ρ w 0j /. We are iterested i trasverse distaces ρ, so the approximatio (5)-(6) is a good solutio to Maxwell s The forms (5)-(6) could also be deduced quickly by first assumig E y ad B x to be a plae wave with a Gaussia trasverse modulatio, ad the eforcig coditios E =0= B to determie E z ad B z.

3 equatios provided w 0j, i.e., θ 0j. This is the case i the preset problem, where we wish to explore the behavior of very weakly focused optical beams. The flow of eergy i this elliptical beam is described by the (real) Poytig vector, ( ReE ReB ɛ S = = μ μ E 0 e ρ x si[(kz ωt)], y ) si[(kz ωt)], cos (kz ωt). kw0x kw0y (7) The time-average flow of eergy is, of course, oly i the directio of propagatio of the wave. I additio, there is a oscillatory trasverse flow of eergy. 3. Reflectio ad Refractio at a Sigle Iterface The classic formalism for plae waves is reviewed i Appedix B. We first cosider the refractio of a weakly focused Gaussia beam at a sigle iterface betwee liear, isotropic media of idices = c ɛ μ ad = c ɛ μ, usig otatio as show i the figure below. For simplicity, we restrict our attetio to the case that the electric field is perpedicular to the plae of icidece (i.e., tothex-z plae). The icidet, reflected ad trasmitted beams each have the Gaussia form (5)-(6), with respect to axes (x i,y i,z i ), (x r,y r,z r )ad(x t,y t,z t ), where the z-axes are i the directios of propagatio of the various beams. The trasformatio betwee the axes (x i,y i,z i ) of the icidet beam ad the laboratory axes (x, y, z) is ad x i =cosθ x si θ z, y i = y, z i =siθ x +cosθ z, (8) ρ i = x i + y i wix wiy = cos θ x si θ xz +si θ z w ix + y, (9) wiy where θ is the agle betwee the axis of the icidet beam ad the z-axis, ad we assume that axis of the icidet beam passes through the origi. The compoets of a vector A with respect to the laboratory frame are related to those with respect to axes (x i,y i,z i )by A x =cosθ A xi +siθ A zi, A y = A yi, A z = si θ A xi +cosθ A zi. (0) 3 Near its waist, a Gaussia beam is similar to a wave iside a coductig wave guide, which latter case also exhibits steady logitudial, ad oscillatory trasverse, flow of eergy []. 3

4 Combiig eqs. (5)-(0), the compoets of the icidet beam i the laboratory frame are E ix = i si θ y E iy, () E iy = E 0i e ρ i e i( si θ k 0 x+ cos θ k 0 z ωt), () E iz = i cos θ y E iy, (3) B ix = [ ] cos θ x si θ z cos θ i si θ E iy, (4) c B iy = 0 (5) B iz = c Similarly, the reflected beam is related by [ si θ + i cos θ cos θ x si θ z ] E iy. (6) x r = cos θ x si θ z, y r = y, z r = si θ x cos θ z, (7) ρ r = x r + y r wrx wrx = cos θ x +siθ xz +si θ z w rx + y,, (8) wrx A x = cos θ A xi +siθ A zi, A y = A yi, A z = si θ A xi cos θ A zi, (9) E rx = i si θ y E ry, (0) E ry = E 0r e ρ r e i( si θ k 0 x cos θ k 0 z ωt), () E rz = i cos θ y E iy, () B rx = [ ] cos θ x +siθ z cos θ + i si θ E ry, c (3) B ry = 0 (4) B rz = c [ si θ + i cos θ cos θ x +siθ z ] E ry, (5) where we have assumed that the reflected beam also makes agle θ with respect to the z-axis, ad that the axis of the reflected beam passes through the origi. Likewise, the trasmitted beam is related by x t =cosθ x si θ z, y t = y, z t =siθ x +cosθ z, (6) ρ t = x t + y t wtx wty = cos θ x si θ xz +si θ z w tx + y, (7) wty A x =cosθ A xt +siθ A zt, A y = A yt, A z = si θ A xt +cosθ A zt, (8) 4

5 E tx = i si θ y E ty, t (9) E ty = E 0t e ρ t e i( si θ k 0 x+ cos θ k 0 z ωt), (30) E tz = i cos θ y E ty, t (3) B tx = c B tz = c [ [ cos θ x si θ z cos θ i si θ t E ty, (3) B ty = 0 (33) si θ + i cos θ cos θ x si θ z t ] ] E ty. (34) The boudary coditios at the iterface z =0arethatE, D z = ɛe z, B z ad H = B /μ are cotiuous. Thus, cotiuity of E y E 0i e ρ i e i( si θ k 0 x ωt) + E 0r e ρ r e i( si θ k 0 x ωt) = E 0t e ρ t e i( si θ k 0 x ωt), (35) for all x ad y, which cofirms the assumptio that θ r = θ i = θ i, verifies Sell s law, tells us that beam waists are related by si θ = si θ, (36) w ix = w rx = cos θ w tx, cos θ w iy = w ry = w ty, (37) ad that the wave amplitudes are related by Similarly, the cotiuity of H x = B x /μ tells us that E 0i + E 0r = E 0t. (38) cos θ μ (E 0i E 0r )= cos θ μ E 0t. (39) Combiig eqs. (38) ad (39) we obtai the usual Fresel relatios 4 for polarizatio perpedicular to the plae of icidece, E 0r = μ cos θ μ cos θ E 0i cos θ + μ, μ cos θ E 0t E 0i = cos θ cos θ + μ. (40) μ cos θ Furthermore, the cotiuity of E x (or D z = ɛe z ) tells us that the Rayleigh rages are related by i = r = si θ t = t. (4) si θ Equatio (4) is ot quite cosistet with eqs. (4) ad (37). 4 See Appedix B 5

6 I the remaider of the body of this ote we suppose that μ = μ = μ 0, where the latter is the permeability of the vacuum. We also suppose that the icidet Gaussia beam is circularly symmetric, ad write the icidet waist as w ix = w iy = w. It follows that the reflected Gaussia beam is also circularly symmetric with the same waist w. The icidet ad reflected beams have Rayleigh rage i = k w/ = k 0 w/, while that of the trasmitted beam is t = i /..3 Total Reflectio at a Sigle Iterface The pheomeo of total reflectio at a iterface betwee optically deser ad lighter media wasdiscussedbynewtoipropositio96ofhispricipia [3], ad i further detail i his Opticks [4]. Newto s model was that particles of light are attracted by the deser medium, ad so follow curved rays that result i a offset betwee the icidet ad reflected path, as show i the figure o the left below (from the Pricipia). Newto s predictio was largely forgotte after the success of the wave theory of light i the early 800 s, but was occasioally discussed i the first half of the 0 th cetury [5, 6, 7, 8]. The, i 947 Goos ad Häche [9] provided experimetal evidece for what is ow called the Goos-Häche shift, d, illustrated i the figure o the above right. Total reflectio occurs whe the idices of the two media obey > ad the agle of icidece, θ, is large eough that si θ =( / )siθ >. I this case, cos θ = si si θ θ = i, (4) is purely imagiary. As a cosequece, the amplitudes (40) of the reflected ad trasmitted waves iclude a phase shift relative to that of the icidet wave, while E 0r = E 0i, ad E 0t E 0i = 4 cos θ. (43) The quatity ρ t of eq. (7) ca ow be writte ρ t = ( si θ ) x i si θ wtx si θ xz + si θ z + y. (44) wty 6

7 Hece, matchig of this form to that of ρ i at z = 0 requires that ρ t = cos θ x w + y w si θ cos θ z ( si θ )w +i si θ cos θ xz, (45) si θ w recallig eq. (9). The factor e Re(ρ t ) is costat o surfaces that are elliptical hyperboloids of sheet about the z-axis, with asymptotic opeig agle θ txz i the x-z plae give by ta θ txz = si θ >. (46) si θ This potetially dramatic behavior is, however, masked by the dampig of the trasmitted wave i z, as follows from the factor e i( si θ k 0 x+ cos θ k 0 z ωt) = e si θ k 0z e i( si θ k 0 x ωt). (47) The time-average flow of eergy i medium is give by S = Re(E t B t ) = Re [E ty Btz μ 0 μ ˆx +(E tzbtx E tx Btz) ŷ E ty Btx ẑ] 0 = E 0t e Re(ρ t ) e si θ k 0z si θ si θ z ˆx μ 0 c t si θ si θ x ẑ. t E 0t μ 0 c e Re(ρ t ) e si θ k 0z si θ ( ˆx x i ) si θ ẑ, (48) otig that t = i /. Eergy flows i the +x directio i medium, but this flow is sigificat oly ear the origi because of expoetial dampig factors. Furthermore, eergy flows i the +z directio for x>0adithe z directio for x>0. That is, eergy flows i curves i medium, as qualitatively predicted by Newto. Lies of S that eter medium from medium at x retur to medium at +x. I particular, the flow of eergy betwee media ad across the plae z =0isdescribed by S z = E 0i 4 cos θ si θ si θ x e (cos θ x +y )/w, (49) μ 0 c i recallig eq. (43). This eergy adds to that i the omial reflected wave at z =0forx>0, S z r = μ 0 Re (E ry (z =0)B rx(z =0))= E 0i μ 0 c cos θ e (cos θ x +y )/w, (50) 7

8 ad subtracts from that for x<0. We defie a ceter of eergy x for the reflected wave at z = 0 accordig to x = x S z r dx 0 x S z dx + 0 x S z dx S z r dx = 0 x S z dx S z r dx = 8 cos θ si θ si θ i πw πw 3 8 cos 3 θ = λ 0 ta θ cos θ π si θ, (5) cosθ recallig that w /i =/k = λ 0 /π. Here, λ 0 is the wavelegth i vacuum at agular frequecy ω. Supposig that the reflected beam is cetered o x rather tha x = 0 i the plae z =0, it propagates away from this plae at agle θ with its axis shifted trasversely by d with respect to mirror reflectio, as sketched i the figure o the previous page, where d = x cos θ = λ 0 si θ π si θ. (5) The result (5) is ot that quoted i the literature [0,,, 3, 4, 6, 8, 9, 0,,, 3], d = λ 0 si θ cos θ λ 0 si θ π or. (53) si θ π si θ where differet sets of approximatios are used. While most experimets have bee performed at a sigle agle θ, a recet effort [4] that measured a large rage of agles supports eq. (53). The result (53) is ofte justified by claimig ( that the icidet wave peetrates ito ) medium at agle θ for a distace D λ 0 / π si θ accordigtoeq.(47),at which depth ( it is reflected back ito medium [8]. If so, the it follows that d =Dsi θ = ) λ 0 si θ / π si θ. This result diverges at the critical agle, si θ = /. However, our argumet (5) idicates that the shift of eergy from x<0 to x>0 vaishes at the critical agle... Appedices A Gaussia Beams with Elliptical Cross Sectio May discussios of Gaussia beams emphasize a sigle electric field compoet, such as E y = f(r, z)e i(kz ωt), of a cylidrically symmetric beam of agular frequecy ω ad wave umber k = ω/c propagatig alog the z axis ia medium with idex of refractio. Here, we geeralize to the case of a beam with a elliptical cross sectio. Of course, the electric field must satisfy the free-space Maxwell equatio E =0. Iff(r, z) is ot costat ad E x = 0, the we must have ozero E z. That is, the desired electric field has more tha oe vector compoet. 8

9 To deduce all compoets of the electric ad magetic fields of a Gaussia beam from a sigle scalar wave fuctio, we follow the suggestio of Davis [7] ad seek solutios for a vector potetial A that has oly a sigle Cartesia compoet (such that ( A) j = A j [8]). We work i the Lorez gauge (ad SI uits), so that the electric scalar potetial Φ is related to the vector potetial A by A = c Φ t = ω i c Φ=ik Φ. (54) ω The vector potetial ca therefore have a ozero divergece, which permits solutios havig oly a sigle compoet. Of course, the electric ad magetic fields ca be deduced from the potetials via E = Φ A t = i ω ( A)+iωA, (55) k usig the Lorez coditio (54), ad B = A. (56) The vector potetial satisfies the free-space (Helmholtz) wave equatio, A A c t =( + k )A =0. (57) We seek a solutio i which the vector potetial is described by a sigle Cartesia compoet A j that propagates i the +z directio with the form A j (r) =ψ(r)e i(kz ωt). (58) Isertig trial solutio (58) ito the wave equatio (57) we fid that ψ +ik ψ =0. (59) z I the usual aalysis, oe ow assumes that the beam is cylidrically symmetric about the z axis ad ca be described i terms of three geometric parameters the diffractio agle θ 0,thewaistw 0, ad the depth of focus (Rayleigh rage), which are related by θ 0 = w 0 =, ad = kw 0 kw 0 = kθ. (60) 0 9

10 Here, we cosider the possibility that the beam has a elliptical cross sectio, with major ad mior axes alog the x ad y axes. The waist ad diffractio agle are differet i the x-z ad y-z plaes, but the Rayleigh rage is i commo: θ 0x = w 0x, ad θ 0y = w 0y. (6) We ow covert to the scaled coordiates ξ = x w 0x, υ = y w 0y, ρ = ξ + υ, ad ς = z. (6) Chagig variables ad otig relatios (6), the wave equatio (59) takes the form w 0x ψ ξ + ψ w0y υ + ψ z0 ς + ik ψ ς =0. (63) The paraxial approximatio is that the term i the relatively small quatity /z 0 is eglected, ad the resultig paraxial wave equatio is w 0x ψ ξ + ψ w0y υ + ik ψ ς 0. (64) A educated guess is that the trasverse behavior of the wave fuctio ψ has a Gaussia form, but with a width that varies with z. Also, the amplitude of the wave far from its waist should vary as /z. I the scaled coordiates ρ ad ς a trial solutio is ψ = h(ς)e f(ς)ρ, (65) where the possibly complex fuctios f ad h are defied to obey f(0) = = h(0). Sice the trasverse coordiate ξ ad υ are scaled by the waists w 0x ad w 0y,weseethatRe(f) = w 0j /w j (ς) wherew j(ς) is the beam width i coordiate j = x or y at positio ς. Fromthe geometric parameters (6) we see w j (ς) θ 0j z = w 0j ς for large ς. Hece, we expect that Re(f) /ς for large ς. Also, we expect the amplitude h to obey h /ς for large ς. Pluggig the trial solutio (65) ito the paraxial wave equatio (64) we fid that ( fh w 0x + ) ( ) ξ +f h + υ + ik (h f hρ ) 0, (66) w0y w0x w0y where a idicates differetiatio with respect to ς. We ca defie the Rayleigh rage ad the waists w 0x ad w 0y to be related by ( w 0x + ) = k, (67) w0y so eq. (66) ca be rewritte as [ ( fh+ ih + ρ z0 ξ h kρ w 0x ) ] + υ f if 0. (68) w0y 0

11 If w 0x = w 0y, eq. (68) reduces to the form fh+ ih + ρ h(f if ) 0, (69) recallig eq. (60). The key approximatio of this ote is that eq. (68) ca be writte as eq. (69) eve whe w 0x w 0y. Acceptig this approximatio, we see that for eq. (69) to be true at all values of ρ implies that f f = i, ad h = i. (70) fh Thus, f = h is a solutio, despite the differet physical origi of these two fuctios as the trasverse width ad amplitude of the wave. We itegrate the first of eq. (70) to obtai = C + iς. (7) f Our defiitio f(0) = determies that C =. Thatis, f = +iς = iς +ς = e i ta ς +ς. (7) Note that Re(f) = /( + ς ) = w 0j/w j (ς), while f = / +ς, so that f = h is cosistet with the asymptotic expectatios discussed above. The logitudial depedeces of the widths of the Gaussia beam are ow see to be The lowest-order wave fuctio is w j (ς) =w 0j +ς. (73) ψ 0 = fe fρ = e i ta ς +ς e ρ /(+ς ) e iςρ /(+ς ). (74) The factor e i ta ς i ψ 0 is the so-called Gouy phase shift [9], which chages from 0 to π/ as z varies from 0 to, with the most rapid chage ear the. For large z the phase factor e iςρ /(+ς ) ca be writte as e i(/z)(x /w 0x +y /w 0y ) e ikr /(z), recallig eqs. (6) ad (67). Whe this is combied with the travelig wave factor e i(kz ωt) we have e i[kz(+r /z) ωt] e i(kr ωt), (75) where r = z + r. Thus, the wave fuctio ψ 0 is a modulated spherical wave for large z, but is a modulated plae wave ear its waist. To obtai the electric ad magetic fields of a Gaussia beam that is polarized i the y directio we take the vector potetial to be The, A x =0, A y = E 0 iω ψ 0e i(kz ωt) = E 0 iω fe fρ e i(kz ωt), A z =0. (76) A = fy A w0y y. (77)

12 ad the electric field follows from eq. (55) as E x 0, E y E 0 fe fρ e i(kz ωt), E z iyf E kw0y y, (78) where we eglect terms of order /z 0. Similarly, the magetic field follows from eq. (56) as B x = ɛμe y = c E y, B y =0, B z = ixf kw0x c E y. (79) B Fresel Relatios for Plae Waves For referece we preset a derivatio of the well-kow Fresel relatios for reflectio ad refractio of a plae wave at a plaar iterface. The geometry ad otatio are agai as show i the figure below, with the plae of icidece beig the x-z plae. The icidet, reflected ad trasmitted waves ca be writte E i = E 0i e i(k ixx+k iz z ωt), E r = E 0r e i(krxx+krzz ωt), E t = E 0t e i(ktxx+ktzz ωt), (80) B i = k i ω E 0i e i(k ixx+k iz z ωt), B r = k r ω E 0r e i(krxx+k rzz ωt), B t = k t ω E 0t e i(ktxx+k tzz ωt), (8) where we have used the plae-wave Maxwell equatio ik E = E = B/ t = iωb, ad we ote that the wave equatio E = E/ t requires that k i = k ix + k iz = ω c, k r = k rx + k rz = ω c, k t = k tx + k tz = ω c, (8) so that k i = k r = k t. (83) Cotiuity of the tagetial compoet of E at the iterface requires that the argumet of the expoetial factors all be equal there, ad hece k ix = k rx = k tx, (84)

13 θ i = θ r = θ, k rz = k iz, (85) ad si θ = si θ (= si θ t ). (86) The, the cotiuity of the x- ady-compoets of E = μωh k/k at the iterface implies H 0iy H 0ry = μ k tz H μ 0ty = μ cos θ H 0ty, ad E 0iy + E 0ry = E 0ty, (87) k iz μ cos θ otig that E x μk z H y /. Similarly, cotiuity of the tagetial compoet of H = k E/μω at the iterface implies E 0iy E 0ry = μ μ k tz k iz E 0ty = μ μ cos θ cos θ E 0ty, ad H 0iy + H 0ry = H 0ty. (88) otig that H x k z H y /μ. Combiig eqs. (87) ad (88) we fid E 0ry = μ cos θ μ cos θ E 0iy cos θ + μ, μ cos θ E 0ty E 0iy = cos θ cos θ + μ, (89) μ cos θ H 0ry = μ cos θ μ cos θ H 0ty cos θ H 0iy cos θ + μ, = μ cos θ H 0iy cos θ + μ. (90) μ cos θ For the special case of the electric field polarized perpedicular to the plae of icidece, H has o y-compoet, ad we write E 0r = μ cos θ μ cos θ E 0t cos θ E 0i cos θ + μ, = μ cos θ E 0i cos θ + μ, (9) μ cos θ Similarly, for the special case of the electric field polarized parallel to the plae of icidece, E has o y-compoet, ad for each of the three waves, H y E /μ, sowecawrite E 0r E 0i = μ μ cos θ cos θ μ, μ cos θ + cos θ E 0t E 0i = cos θ μ, (9) μ cos θ + cos θ As usual, we ote that the case of iteral reflectio whe si θ =( / )siθ > ca be accommodated i the above formalism by writig cos θ = si θ = si θ = i si θ. (93) I this case, we also write si θ k tz = ik 0, (94) such that the trasmitted wave is damped i z accordig to ( ) e iktzz = e si θ / k 0 z. (95) 3

14 Refereces [] K.T. McDoald, Gaussia Laser Beams with Radial Polarizatio (Mar. 4, 000), [] M. Moricoi ad K.T. McDoald, Eergy Flow i a Waveguide below Cutoff (Mar. 4, 000), [3] I. Newto, Pricipa, Vol. (Lodo, 76; reprit of Cajori s Eglish traslatio, U. Calif. Press, 96), Propositio 96. [4] I. Newto, Opticks, 4th ed. (Lodo, 730; Dover, New York, 95). Book Two, Part, Observatio : Compressig two Prisms hard together that their sides (which by chace were a very little covex) might somewhere touch oe aother: I foud the place i which they touched to become absolutely trasparet, as if they had there bee oe cotiued piece of Glass. For whe the Light fell so obliquely o the Air, which i other places was betwee them, as to be all reflected; it seemed i that place of cotact to be wholly trasmitted,... Book Three, Part, Questio 9: Are ot the Rays of Light very small Bodies emitted from shiig Substaces? For such Bodies will pass through uiform Media i Right Lies without bedig ito the Shadow, which is the Nature of Rays of Light.... The Rays of Light i goig out of Glass ito a Vacuum, are bet towards the Glass; ad if they fall too obliquely o the Vacuum, ther are bet backwards ito the Glass, ad totally reflected; ad this Reflexio caot be ascribed to the Resistace of a absolute Vacuum, but must be caused the the Power of the Glass attractig the Rays at their goig out of it ito the Vacuum, ad brigig them back. [5] S. Boguslawski, Das Feld des Poytigsche Vektors bei der Iterferez vo zwei ebee Lichtwelle i eiem absorbierede Medium, Phys. Z. 3, 393 (9), [6]A.Wiegrefe,Neue Lichtströmuge bei Totalreflexio. Beiträge zur Diskussio des Poytigsche Satzes, A. Phys. 45, 30 (94), Neue Lichtströmuge bei Totalreflexio. Beiträge zurketis des PoytigscheVektors, A. Phys. 50, 77 (94), H. Rose ad A. Wiegrefe, Versuche über die Sichtbarmachug vo Lichtströmuge durch die Eifallsebee im isotrope Medium bei Totalreflexio, A. Phys. 50, 8 (96), [7] J. Picht, Beitrag zur Theorie der Totalreflexio, A. Phys. 3, 433 (99), Die Eergieströmug bei der Totalreflexio, Phys. Z. 30, 905 (99), 4

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