This article was downloaded by: [University of Calgary], [Alexander Lvovsky] On: 22 August 2013, At: 11:24 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Encyclopedia of Optical Engineering Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/doi/book/10.1081/E-EOE Fresnel Equations Alexander I. Lvovsky a a Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada Published online: 27 Feb 2013 To cite this entry: Alexander I. Lvovsky . Fresnel Equations. In Encyclopedia of Optical Engineering. Taylor and Francis: New York, Published online: 27 Feb 2013; 1-6. To link to this chapter: http://dx.doi.org/10.1081/E-EOE-120047133 PLEASE SCROLL DOWN FOR CHAPTER Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Fresnel Equations Alexander I. Lvovsky Department of Physics and Astronomy, University of Calgary, Calgary, Alberta, Canada Abstract The Fresnel equations, which determine the reflection and transmission of light incident on an interface of two media with different indices of refraction, are among the most fundamental find- ings of classical optics. This entry offers a detailed derivation of the equations and discusses some of their major consequences (in particular, Brewster effect, total internal reflection, and the Goos- Hänchen shift), as well as applications both in everyday optics and in specialized equipment. INTRODUCTION wave will propagate at angle θt which is determined by Snell’s law: The Fresnel equations relate the amplitudes, phases, and polarizations of the transmitted and reflected waves that sinθ n i = 2 (1) emerge when light enters an interface between two trans- sinθt n1 parent media with different indices of refraction, to the cor- responding parameters of the incident waves. These where n1 and n2 are the refractive indices of the two media. equations were derived by Augustin-Jean Fresnel in 1823 The angle θr of the reflected wave is equal to θi according as a part of his comprehensive wave theory of light. How- to the law of reflection. We denote the amplitudes of these ever, the Fresnel equations are fully consistent with the rig- two waves as Et and Er, respectively. Our goal is to deter- orous treatment of light in the framework of Maxwell mine these amplitudes. equations. To accomplish this, we apply the boundary conditions The Fresnel equations are among the most fundamental for the electric and magnetic fields at an interface between findings of classical optics. Because they describe the two media with different electromagnetic properties, which behavior of light at optical surfaces, they are relevant to are known from electrostatics. Specifically, the components virtually all fields of optical design: lens design, imaging, of the electric field E and magnetic field H, which are tan- lasers, optical communication, spectroscopy, and hologra- gent to the surface, must be continuous across the boundary. phy. Good understanding of the principles behind Fresnel Because the electromagnetic wave is transverse, the field equations is necessary in designing optical coatings and incident onto the interface can be decomposed into two Fabry-Perot interferometers. polarization components, one P-polarized, i.e., with the elec- This entry begins with a detailed derivation of the tric field vector inside the plane of incidence, and the other Fresnel equations based on Snell’s law and the boundary one S-polarized, i.e., orthogonal to that plane. (Under the relations for the electric and magnetic fields at an interface plane of incidence, we understand the plane that is formed between two media with different electromagnetic proper- by the vector ki and the normal to the interface.) We will ties. We then proceed to discuss the primary consequences derive the Fresnel equations for these two cases separately. of these equations, such as intensity reflectivities and We begin by concentrating on the case when the inci- Brewster’s effect. The final section of the entry is dedicated dent wave is P-polarized (Fig. 1). Due to symmetry, the to numerous applications of the Fresnel equations. transmitted and reflected waves will have the same polari- Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 zation. Because the E, H, and k vectors must form a right- handed triad for each of the waves, the directions of all DERIVATION field vectors are uniquely defined up to a sign convention, To derive the Fresnel equations, consider two optical which is chosen as illustrated in Fig. 1. The boundary con- media separated by an interface, as shown in Fig. 1. A dition for the electric field then becomes: plane optical wave is propagating toward the interface with Eicosθ i +EEr cos θ i = tcosθ t (2) wave vector ki oriented at angle θi with respect to the inter- face normal. The electric field amplitude of the wave is For the magnetic field, which is collinear in all three waves, given by Ei. this condition takes the form: On incidence onto the interface, this wave will be par- tially transmitted and partially reflected. The transmitted Hi−HH r = t (3) Encyclopedia of Optical Engineering DOI: 10.1081/E-EOE-120047133 Copyright © 2013 by Taylor & Francis. All rights reserved. 1 2 Fresnel Equations Incident wave Reflected wave Incident wave Reflected wave E E Hi r kr i E kr H r H i E i r i r i ki r ki Hr Interface Medium 1 Medium 2 x x t Normal t z Normal z E Ht t Et Ht k kt t Transmitted wave Transmitted wave Fig. 1 Field vectors of the incident, transmitted, and reflected Fig. 2 Field vectors of the incident, transmitted, and reflected waves in case the electric field vectors lie within the plane of waves in case the electric field vectors are perpendicular to the incidence (P polarization). plane of incidence (S polarization). To solve these equations, we need to incorporate the rela- 2(/n µ)cos θ t = 1 1 i (13) tion between the electric and magnetic field amplitudes for S (/n1µ 1)cos θi + (/n2µ 2 )cos θt each wave. We know from Maxwell equations that these amplitudes in any plane electromagnetic wave must satisfy Eqs. 7 and 8, as well as Eqs. 12 and 13 present Fresnel equations in their general form, which is also valid for H = ε E (4) materials with negative indices of refraction (also known as μ metamaterials or left-handed materials). When applying these equations to such materials, absolute values of the where ε and μ are the electric permittivity and magnetic refractive indices must be used.[1–3] permeability, respectively, of the material in which the Most commonly used optical materials are non- wave propagates. Since the index of refraction of a material magnetic, so one can approximate μ1 = μ2 = μ0. Under this is given by n= c εμ, we have: approximation, the permeabilities in Eqs. 7, 8, 12, and 13 cancel, and the Fresnel equations can be further simplified H= n E/µ c and H= n E/µ c (5) i, r 1i 1 t2 t 2 by incorporating Snell’s law: and thus, from Eq. (3), tan(θ− θ ) r = − i t (14) P tan(θ+ θ ) n1() Ei−E r //µ1 = n2Et µ 2 (6) i t Combining Eqs. 2 and 6, we arrive at the Fresnel equations 2sinθtcos θ i tP = (15) for the P-polarized wave: sin(θi+ θ t)cos()θi − θ t (/n1µ 1)cos θt − (/n2µ 2 )cos θi sin(θi− θ t ) rP = (7) rS = − (16) (/n1µ 1)cos θt + (/n2µ 2 )cos θi sin(θi+ θ t) 2(/n1µ 1)cos θi 2sinθtcos θ i tP = (8) tS = (17) (/n1µ 1)cos θt + (/n2µ 2)cos θi sin(θi+ θ t) where we defined the amplitude reflection and transmis- We now proceed toward discussing the main consequences Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 sion coefficients: of the Fresnel equations. E E r = r and t = t (9) Ei Ei CONSEQUENCES AND SPECIAL CaSES In the case of S polarization (Fig. 2), in much the same Intensity Reflectivity and Transmissivity way, we write the boundary conditions as For most practical purposes, the reflection and transmission E +EE = (10) coefficients for the intensity, rather than field amplitudes, i r t are of interest. For a wave of amplitude E propagating in a non-magnetic medium with the refractive index n, we have: −HHicosθ i + r cos θ i = −Htcosθ t (11) 2 from which we derive the second pair of Fresnel equations: I = 2ncε0 E (18) (/n1µ 1)cos θi − (/n2µ 2 )cos θt where c is the speed of light in vacuum and ε is the electric rS = (12) 0 (/n1µ 1)cos θi + (/n2µ 2 )cos θt constant. Because the incident and reflected waves propagate Fresnel Equations 3 in the same medium, we can write for the intensity reflection waves are physically identical and have the same reflectivity coefficient: of about 4%.