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On the Nature of the Evanescent Wave On the Nature of the Evanescent Wave Milan Milosevic MeV Technologies, Westport, CT 06880 USA This is an unusual paper in that it does not address a particular research STANDARD ELECTROMAGNETIC THEORY topic or present a novel experimental method or a new theoretical result. This paper addresses our basic understanding of the nature of the We start with the concept of the electromagnetic wave. evanescent wave, the wave that is the basis of the entire field of Attenuated Maxwell equations in a homogeneous nonferromagnetic Total Reflection (ATR) spectroscopy. I recently had the opportunity to medium, characterized by the dielectric constant e and without reexamine the foundations of ATR spectroscopy and was surprised to have sources, combine into the wave equation for both the electric had to change my own mental picture of the evanescent wave that I have and the magnetic field.1–4 For the electric field of the built over the last 25 years. Over the years I have had numerous electromagnetic wave we have discussions with a large number of workers in the field as well as with my former mentor, and one of the originators and the principal developer of ]2 e ]2 ATR spectroscopy, the late N.J. Harrick. Everything brought up in all 2 À 2 2 Eðx; tÞ¼0 ð1Þ these discussions was perfectly consistent with my old mental picture of ]x c ]t the evanescent wave. Thus, I believe that the picture of the evanescent wave that I had is virtually universally held by workers in the field. This where E(x,t) is the amplitude of the electric field vector at the paper describes the new picture of the evanescent wave that emerged from position x and time t and c is the speed of light in vacuum. The said reexamination process. solution to Eq. 1 is a plane wave: Evanescent wave; ATR spectroscopy; Total internal Index Headings: 2pinkxÀixt reflection. Eðx; tÞ¼E0e ð2Þ pffiffiffi where n = e is the refractive index of the medium. The magnetic field of an electromagnetic wave can be INTRODUCTION expressed in terms of its associated electric field as 2pcn Attenuated total internal reflection (ATR) has become the B ¼ k 3 E ð3Þ most ubiquitous IR spectroscopic technique. The technique is x based on the absorption of the evanescent wave that, in total The wavevector k points in the direction of wave propagation internal reflection, forms on the low refractive index side along 1 a totally reflecting interface. This wave is of great significance and has a magnitude (also known as wavenumber ) k given by to ATR spectroscopy, and in this paper I hope to further clarify x 2pn some characteristics of the evanescent wave. For the sake of 2pnk ¼ ¼ ð4Þ c k specificity and consistent notation, in what follows, some of the well known elementary facts about electromagnetic theory are Here x is the angular frequency and k is the wavelength of explicitly reiterated. light in vacuum. It is first demonstrated that electromagnetic theory demands The phase in the exponential term in Eq. 2 represents the that the electromagnetic wave is a transverse wave and that it phase of oscillations of the electric field. For x in the direction propagates through a medium with speed c/n where c is the of k a constant phase w in the exponent is achieved when speed of light in vacuum and n is the refractive index of the medium. Then it is demonstrated that electromagnetic theory 2pnkx À xt ¼ w predicts both total internal reflection and the existence of the evanescent wave in total internal reflection. This point of constant phase for the wave occurs when What is puzzling is that the evanescent wave as provided by w þ xt w c electromagnetic theory appears not to be a transverse wave and x ¼ ¼ þ t not to propagate through the medium by the speed demanded 2pnk 2pnk n by the theory. In other words, the point of constant phase of the wave travels Furthermore, a conceptual difficulty arises with how the at the speed c/n. This is one of the results of the electromagnetic energy carried by the evanescent wave made it electromagnetic theory that I want to bring up. through the totally reflecting interface in the first place. To solve the mystery of where the electromagnetic energy in the As evident from Eq. 3 the two fields propagate together evanescent wave comes from, the picture of the evanescent and oscillate in phase in every point in space. From Eq. 3 it is wave needs to be adjusted. This adjusted picture then removes also evident that the two fields are mutually perpendicular, all the apparent puzzles posed by the structure of the expression and both are perpendicular to the wave vector k, which points for the evanescent wave. in the direction of propagation. Since the electric and magnetic fields are both perpendicular to the direction of propagation, the electromagnetic wave is a transverse wave. Received 27 April 2012; accepted 19 September 2012. E-mail: [email protected]. This is another result of the electromagnetic theory that I DOI: 10.1366/12-06707 want to bring up. 0003-7028/13/6702-0126/0 126 Volume 67, Number 2, 2013Ó 2013 Society for Applied Spectroscopy APPLIED SPECTROSCOPY Maxwell equations for empty space free of sources and assume that the wave vector and the electric field amplitude of currents require that the incident wave are known. That means that we know the frequency and the direction of propagation of the incident ] 1 c ½ ðE2 þ B2Þ þ r½ ðE 3 BÞ ¼ 0 ð5Þ wave. The theory then provides the wave vectors and the ]t 8p 4p electric field amplitudes of both the reflected and transmitted The interpretation of expression 5 is that it describes waves. conservation of energy associated with electromagnetic fields. The reflected and transmitted amplitudes are related to the The term incident amplitude through Fresnel equations. The ratios r and t are the Fresnel reflection and transmission amplitude coeffi- 1 cients respectively defined as u ¼ ðE2 þ B2Þð6Þ 8p E r ¼ r ð9Þ is the energy density of the electromagnetic field. Since for Ein electromagnetic waves the electric and magnetic fields have the same magnitudes, i.e., B2 = E2 , we have E t ¼ t 1 Ein u ¼ E2 4p For s-polarized incident light,1–4 The vector quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 N1 cos h À n2 À n1 sin h c 2pc2 rs ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P ¼ ðE 3 BÞ¼ uk ð7Þ 2 2 2 4p x N1 cos h þ n2 À n1 sin h is the so-called Poynting vector, which describes the flow of 2n cos h energy contained in the electromagnetic field. Note that the ts ¼ q1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð10Þ term 2pc/x in Eq. 7 is just the inverse of the wavenumber. 2 2 2 n1 cos h þ n À n sin h Thus the last expression in Eq. 7 is a vector whose magnitude 2 1 is the product of the speed of light and energy density and that and similarly for p-polarized incident light, points in the direction of propagation. The Poynting vector thus qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi describes the flow of energy density u at the speed of light, just 2 2 2 2 as we would expect for energy carried by electromagnetic N2 cos h À n1 n2 À n1 sin h rp ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi waves. Therefore electromagnetic energy flows with the same 2 2 2 2 speed as the wave. This may sound as a tautology, but, as we N2 cos h1 þ n1 n2 À n1 sin h will see, nothing can be taken for granted. The presence of an interface (i.e., discontinuity in the 2 cos h p n1nq2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi medium) means that the solutions of Eq. 1 on one side of the t ¼ ð11Þ 2 2 2 2 interface do not extend to the other side. We have different n2 cos h þ n1 n2 À n1 sin h solutions on the two sides. However, these solutions must come together at the interface. That requirement gives us the For the reflected wave the cross section of the incident beam boundary conditions that then lead to not only Fresnel stays the same. But for the refracted wave it changes as3 equations, but also the laws of reflection and refraction. At c cos u an interface between two media characterized by refractive t ¼ ð12Þ c cos h indices n1 and n2 incident light refracts through and reflects in from the interface. Thus we have incident, transmitted, and reflected waves. The incident and reflected waves are in the where Ct and Cin are the areas of the cross sections of first (incident) medium, while the refracted (transmitted) wave transmitted and incident beams, respectively, and angles h and is in the second medium. All three waves are plane waves u are the angle of incidence and refraction, respectively. described by Eq. 1. Each wave is of the form of Eq. 2 but has Taking Eq. 12 into account the reflectance and transmittance of its own amplitude and wave vector: the interface are 2 2pin1kinxÀixt P E Einðx; tÞ¼E0ine r r 2 R ¼ ¼ 2 ¼jrj ð13Þ Pin Ein 2pin k xÀixt Erðx; tÞ¼E0re 1 r P E2 n C n cos u T ¼ t ¼ t 2 2 ¼jtj2 2 ð14Þ 2pin2ktxÀixt 2 Etðx; tÞ¼E0te ð8Þ Pin Ein n1C1 n1 cos h The three wave vectors kin, kr, kt of the incident, reflected, and In Eqs. 13 and 14, Pin, Pr, Pt, are the power densities carried by transmitted waves, respectively, lie in the same plane (plane of incident, reflected, and transmitted beams, respectively.
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