1 Total Internal Reflection and the Related Effects 1 (Translation of the Russian version appeared in the Proceedings of Tartu State University No. 42, p. 94-112, 1956) N.Kristoffel Circle of theoretical physics. Supervisor docent Paul Kard 1. Introduction In 1909 Professor of Moscow University A.A.Eichenwald presented first the theory of total internal reflection basing on the electromagnetic theory of light [1]. Schafer and Gross following the method of Eichenwald have considered the passing of electromagnetic waves through a thin plate at the incident angle exceeding the limiting angle of the total reflection. They have reached accordance with the theory in the corresponding experiments [2]. Wiegrefe has studied the total reflection in the case of an arbitrary linear polarization of the incident light [3]. Recently principally novel experiments with the aim of proving the penetration of light into the second medium on total reflection have been performed by Goos and Hänchen [5,7]. In these investigations the shift of the totally reflected beam has been measured. The theory of this shift has first been developed in papers [6], [8]. The aim of the present contribution is to generalise the theory of total internal reflection for the case of elliptic polarization of the incident light. Until now this general situation has been considered in few cases and the theories of shifts presented hitherto avoid it totally, being restricted to the linear polarization perpendicular or parallel to the plane of incidence. 2 The total internal reflection is present under the condition n sin >Θ n = 2 , (1.1) n1 Θ where is the angle of incidence, n1 and n2 , the absolute refraction indices of the first and the second medium. The field in the second medium can be obtained from the usual Fresnel formulae 3 if one substitutes there sin Θ sin Θ '' = , (1.2) n i cos Θ '' −= sin 2 −Θ n 2 , (1.3) n where Θ '' is the complex refraction angle. 1 Awarded a prize at the contest of Tartu State University (in April, 1954). 2 Here and in what follows the direction of the linear polarization is taken being determined by the electric vector. 3 The magnetic permeability ist taken equal to one. 2 2. The field in the optically sparse medium Let us chose the interface between the media to be the xy plane; the z axis is directed downward, the x axis, to the right, the y axis, to the reader. Let the first medium be above and the second, below (see Fig. 1). Fig.1. The electric field of an infinite incident plane wave can be divided into two components – parallel and perpendicular to the plane of incidence. A distinct phase shift between them is present in the case of elliptic polarization. Therefore we write dd = i(ω t− kr ) E p Ap e dd , (2.1) = i(ω t− kr +δ ) E y Ay e d ω where is the frequency, k is the wave vector, Ap and Ay are the amplitudes of the components mentioned, and δ , the phase shift. For the field in the medium of smaller index one can formally write analogous expressions d d '' = '' i(ω t− k'' r ) E p Ap e d d (2.1) '' = '' i(ω t− k'' r +δ ) E y Ay e d where k '' = nk . From the Fresnel formulae 3 Θ Θ '' = cos2 sin '' Ap Ap sin( Θ+Θ )'' cos( Θ−Θ )'' (2.3) Θ Θ '' = cos2 sin '' Ay Ay sin( Θ+Θ )'' and taking account of (1.2) and (1.3), one finds Θ '' = 2n cos iψ Ap A p e (1−n2 )(sin 2 Θ− n 22 cos Θ ) , (2.4) 2cos Θ φ A'' = A e i y y 2 1 − n where sin 2Θ − n 2 tan ψ = n2 cos Θ 2Θ − 2 (2.5) sin n 2 tanφ= = n tan ψ . cos Θ However, as further ''= '' Θ Ax A p cos '' ''= − '' Θ Az A p sin '' , then Θ2 Θ − 2 '' = 2cos sin n i(ψ− π /2) Ax A p e (1−n2 )(sin 2 Θ− n 22 cos Θ ) Θ '' = 2cos iφ Ay A y e (2.6) 1 − n2 2sinΘ cos Θ φ A'' = − A e i . z p −2 2 Θ− 22 Θ (1n )(sin n cos ) The components of the magnetic vector in the second medium will be calculated from the second Maxwell equation. Introducting the designations κ = k sin 2 −Θ n (2.7) τ = ωt − k sin Θx , (2.8) and going to real quantities, the electromagnetic field in the second medium is obtained in the form '' = Θ ψ −κz τ +ψ E x cos2 sin p eA sin( ) '' = ϕ −κz τ + δ + ϕ Ey cos2 yeA cos( ) 2 −κ E '' −= sin Θcos ψ eA z cos( τ +ψ ) z n2 p 4 '' = − Θϕ−κ z τδφ + + 2.9 Hx2 n1 cos sin Ae y sin( ) ( ) '' = ψ −κz τ +ψ H y 2n1 cos p eA cos( ) '' =Θφ−κ z τδφ ++ Hz2 n1 sin cos Ae y cos( ) . In addition to the periodic alteration in time and along the x axis, the field in the second medium decreases exponentially with moving off from the separation plane. The −κ 2π e z exp−z sin 2 Θ − n 2 exponential factor can be presented in the form λ and if 1 sin 2 −Θ n 2 is not too small, a remarkable weakening of the field in the second medium occurs. Already at the z -s that are comparable to the wavelengths of light, the field in the second medium decreases quite rapidly. Only for the angles of incidence close to the limiting one, the electromagnetic field in the medium of smaller index has noticeable values at relatively large depths. The calculation of the field of the reflected wave shows that in the reflected wave there appears a phase shift with respect to the incident wave, which equals 2ψ for the || - 4 component and 2ϕ for the ⊥ -component. 3. Energy flux curves in the optically sparse medium To investigate the movement of the energy in the second medium we use the Umov-Poynting vector: d e d d S '' = E ''( ×H )'' . (3.1) 4π Substituting here (2.9), we arrive at cn sin Θ cos 2 Θ 2 '' = 1 −2κz 2 2 τ +δ +ϕ + n 2 2 τ +ψ S x e Ay cos ( ) Ap cos ( ) π 1( − n 2 ) sin 2 −Θ n 2 cos 2 Θ cn A A '' = 1 yp Θ 2 Θ ψ δ +ϕ −ψ −2κz S y sin cos sin sin( )e (3.2) π 1− n 2 cn cos 2 sin 2 −Θ n 2 2 '' = 1 −2κz 2 τ +δ +ϕ + n 2 τ +ψ S z e Ay sin (2 ) Ap sin (2 ) π 1(2 − n 2 ) sin 2 −Θ n 2 cos 2 Θ We stress that the y-component of the Umov-Poynting vector is nonzero and does not depend on time. It means that the movement of the energy in the direction perpendicular to the plane of incidence is stationary. 5 Averaging the z-component of the Umov-Poynting vector in time, we obtain zero 4 || - the component in the plane of incidence; ⊥ - the component perpendicular to the plane of incidence. 5 '' = We mentione M.Born’s mistake supposing in [4] that S y 0 (p. 62). 5 d '' = S y 0 (3.3) This means that the amount of energy flowing through the interface into the second medium equals to the one returning from there. In other words, the total internal reflection is really „total“: all the energy entered into the second medium, returns to the first medium with no losses. Especially significant is the circumstance that the light intensity in the second medium depends on the polarization of the incident light (see (3.2)). Eichenwald has stated the opposite. For a linearly polarized light (in the plane of incidence) Eichenwald finds the following components of the Umov-Poynting vector (using the designations of Eichenwald): 4πx ε − z π = 2 a τ − 4 − f x A e 1 cos (t ax ) 8πk 2 τ 4πx ε − z π −= 2 τ 4 − f z A e sin (t ax ) 8πk τ For a light polarized perpendicular to the plane of incidence Eichenwald presents no formulae in his paper, however these one can be easily calculated: 4πx ε − z π = 2 a 2 ϕ − 2 τ − 4 − f x B (sin n )e 1 cos (t ax ) 8πk 2 τ . 4πx ε − z π −= 2 2 ϕ − 2 τ 4 − f z B (sin n )e sin (t ax ) 8πk τ These formulae differ from the preceeding ones firstly, by the factor sin 2 ϕ − n 2 and, secondly, instead of B there stands the amplitude A . Eichenwald states that at A = B both formulae must coincide, but it is not so, which is one side of Eichenwald’s mistake. The other one consists in the fact that A = B does not mean the equality of the amplitudes in the incident wave, which is what Eichenwald apparently supposes. Now we find the energy flux curves in the second medium. These are the curves to which the Umov-Poynting vector is tangent at all time moments. For the projection of the flux curves on the xz plane we obtain the following differential equation: 2 2 2 n A p '' 2 2 A sin2(τδφ+++ ) sin2( τψ + ) dzS sin Θ − n y 2Θ − 2 2 Θ =z = × sinn cos . (3.4) dx S '' 2sin Θ n2 A 2 x A2cos( 2 τδφ+++ )p cos(2 τψ + ) y sin2Θ −n 2 cos 2 Θ Analogously for the projections of the flux curves on the xy plane 6 S'' AA1− n 2 sinψ sin( δ + φψ − ) dy =y = p y .