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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 findings 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 interface 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 partially transmitted and partially reflected. The transmitted Hi−HH r = t (3)

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%. At an incidence angle of θi = 90°, all of the incident light is reflected, so the interface acts as a mirror. 2 Er 2 R =2 = r (19) Brewster’s Effect Ei By analyzing Eq. 20 and Fig. 3, we observe that the reflectiv- and thus, ity for the wave polarized in the plane of incidence vanishes

2 when θi + θt = 90°, so the denominator in the right-hand side tan ()θi− θ t of Eq. 20 becomes infinite. At this point, all incident light RP = 2 (20) tan ()θi+ θ t that is polarized parallel to the plane of incidence is transmitted. If the incident wave has both polarization components sin2 (θ− θ ) (or its polarization is random), the reflected wave is com- R = i t (21) S 2 pletely S-polarized. sin (θi+ θ t) The value of the angle of incidence at which this occurs

From these, we obtain intensity transmissivities as follows: is known as Brewster’s angle θB. Writing Snell’s law at Brewster’s angle: 4sinθsin θ cosθcos θ TR1 i t i t (22) PP= − = 2 2 π sin (θi+ θ t )cos ()θi − θ t nsinθ= n sin θ = n sin −θ = n cos θ (25) 1 B 2 t 2 ( 2 B) 2 B

4sinθisin θ t cosθicos θ t TRSS=1 − = 2 (23) we find an explicit expression for that angle: sin (θi+ θ t ) n tanθ = 2 (26) Note that, in contrast to the reflection coefficient, the inten- B n1 sity transmissivity is not simply the square of the amplitude transmissivity, as two additional factors must be taken into which is referred to as Brewster’s law. account. First, one must account for the refractive index of Brewster’s law may be understood by the following the propagation medium, which enters the expression for intuitive argument (Fig. 4). Consider an interface between the intensity (Eq. 18). Second, the intensity is calculated vacuum and glass. The reflected wave is generated by ele- per unit of the wavefront area, and the wavefronts of the mentary molecular dipoles inside the glass that are excited incident and transmitted wave are tilted with respect to the by the transmitted wave. These oscillations are parallel to

interface at different angles θi and θt, respectively. There- the electric field in this wave. But when the transmitted and fore, the intensity transmissivity is given by reflected wave vectors are directed at a right angle to each other, the electric field in the transmitted P-polarized wave, 2 n cosθ E n cosθ and hence the elementary dipoles inside the glass, oscillate T = 2 t t = 2 t t 2 (24) 2 parallel to kr. Hence, the dipoles would have to excite a n1 cosθi E n1 cosθi i wave propagating in the same direction as the direction of A graph of the reflectivities (Eqs. 20, 21) for the vacuum– their oscillation, and this is impossible because the electro- glass interface as a function of the angle of incidence is illus- magnetic wave is transverse. trated in Fig. 3. At normal incidence, the S- and P-polarized Phase of the Reflected Wave For the direction of the incident wave close to normal, we find the amplitude reflectivities (Eqs. 14 and 16) to be nega-

Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 tive if n1 < n2 (if the sign convention of Figs. 1 and 2 is used).

Fig. 3 Intensity reflectivities of the S- and P-polarization components at the interface of vacuum and glass with the index Fig. 4 Reflection and transmission at Brewster’s angle. The of refraction of 1.5. arrows correspond to the electric field vectors. 4 Fresnel Equations

This implies that the phase of the wave shifts by 180° when we rewrite using Eq. 27 as kt⋅ E t = 0. Accordingly, we find reflection from a medium with a higher index of refraction that occurs. For the S-polarization case, the amplitude reflectiv-

ity has the same sign for all incidence angles; for the EEt= t (, −C0,)− iS (32) P-polarization, it changes sign when the angle of incidence exceeds Brewster’s angle. for the P-polarization case (where we assumed, as previ- Because the amplitude transmission coefficients are ously, that the x-component of the transmitted electric field always positive, the transmitted wave does not experience vector is real and negative) and any phase shift with respect to the incident wave. EEt= t (0, 1, 0) (33) Total Internal Reflection for the S-polarization. ⋅ Another phenomenon that can be derived from examining Now by applying Faraday’s law ∇× E(, r t)(= −μ H r,) t the Fresnel equations is the phase shift of the wave that has to Eq. 27, we find for the electric and magnetic field ampli- undergone total internal reflection. Total internal reflec- tudes of the transmitted wave: tion occurs when n1 > n2 and (n1/n2)sin θi > 1; thus, Snell’s law cannot hold. The result is that the incident wave is k ×EHt = μω t (34) totally reflected and the transmitted wave is of evanescent rather than plane wave character. Since the behavior of the and thus, evanescent wave is largely counterintuitive, it is instruc- n2 tive to briefly summarize its properties before proceeding HEt= t (,0 i, 0) (35) cµ to modify the Fresnel equations for situations involving 2 such waves. for the P-polarization case and The spatiotemporal behavior of the electric field in this wave can be written as n2 HEt= t (, −iC 0,)S (36) cµ2 ikt ⋅r − iωt E(,r t ) =Et e + c.c. (27) for the S-polarization. Note that Eq. 4, which we used for

where ω is the angular frequency, kt is the wave vector, and plane waves, is not applicable to evanescent waves. c.c. refers to the complex conjugate term. The component Equalizing the x- and y-components of the electric and of the wave vector that is parallel to the interface must be magnetic field amplitudes above and below the surface, we the same for the incident and transmitted waves: obtain the Fresnel equations for the case of total internal

()kt x =() ki x = (/ωc) n1 sin θi. Since the evanescent wave must reflection: comply with the wave equation: 2 2 2 2 (/n2 µ2 )cos θi − i(/n1 µ1) n 1 sin θi − n2 2 2 2 rP = − (37) ¨ 2 2 2 2 ∇E(, r t)( = n2 /)c E(,r t) (28) (/n2 µ2 )cos θi + i(/n1 µ1)n 1 sin θi − n2 2 2 2 2 2 we find that ()kt x +() kt z = n2ω / c ; thus, the component of 2 2 2 the transmitted wave vector that is normal to the interface (/n1µ 1)cos θi − (/i µ2)n 1 sin θi − n2 rS = (38) 2 2 2 is imaginary: (/n1µ 1)cos θi + (/i µ2)n 1 siin θi − n2

ω 2 2 2 These results can be interpreted as follows. In the case of ()kt z = i n1 sin θi − n2 (29) c regular refraction, the z-component of the transmitted wave vector equals Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 This is not surprising because substituting an imaginary

(kt)z into Eq. 27, we obtain a wave that decays exponen- tially with the distance from the interface, as expected from 2 n1 2 an evanescent wave. We thus find, ()kt z = n2 (/ωc) cos θt = n2 (/ω c)1 − sin θi (39) n2 ω 2 k = n (,S0,) iC (30) t 2 c For total internal reflection, this component becomes complex, so one can formally write: where S = (n1/n2) sin θi and n2 CS=2 −1 (31) 1 2 cosθt =iC = i 2 sin θi −1 (40) n2 Knowing the wave vector components, we can determine the components of the electric field amplitude vector in the Substituting this expression into Eqs. 7 and 12, one obtains transmitted wave using Gauss’s law ∇ ⋅ E(, r t)= 0, which Eqs. 37 and 38, respectively. Fresnel Equations 5

The absolute values of the numerators and denomina- tors of Eqs. 37 and 38 are equal; thus we find for total inter- i nal reflection, that R = R = 1, in accordance with Eq. 19. P S d On the other hand, the amplitude reflectivity being a complex number implies that the reflected wave experiences an x optical phase shift with respect to the incident wave, which z is given by Fig. 5 Good-Hänchen effect (the totally internally reflected 2 2 2 beam undergoes a spatial dislocation shift by a small distance d δP − π µ2 n1 n 1 sin θi − n2 tan = − 2 (41) with respect to the position of its specular reflection, illustrated 2 µ1 n2 cosθi by dashed lines).

δ µ n2sin 2 θ − n2 tan S = − 1 1 i 2 (42) 2 µ n cosθ +∞ 2 1 i i[] kxx+δ () k x Er ()x = ∫E() kx ed kx with the zero phase corresponding to the vector orienta- −∞ tions defined in Figs. 1 and 2. We see that these phase +∞ iδ () kx0 ikx (x+dδ //)dkx shifts are different for the S- and P-polarized waves. In ≈ e∫ E()kx e dkx other words, a linearly polarized wave will generally be −∞ elliptically polarized after it has experienced total internal iδ () kx0 ⎛ dδ ⎞ reflection. =e Ei x + (45) ⎜ dk ⎟ If medium 2 is a metamaterial, the associated phase ⎝ x ⎠ shifts are opposite with respect to those obtained in reflec- Neglecting the constant phase factor, we find that the tion from a right-handed material with the same magni- reflected wave is spatially displaced with respect to the tudes of n and μ .[2,3] 2 2 incident one. The lateral displacement of the reflected beam is then obtained as (Fig. 5): Goos-Hänchen shift d= −dδ/ d k cos θ (46) An important consequence of the phase shift associated x i with the total internal reflection is the spatial displacement Substituting Eqs. 41 and 42 into the foregoing result and experienced by an optical beam undergoing such reflec- keeping in mind that k = k sin , we find the expressions for [4] x i θi tion, known as the Goos-Hänchen shift (Fig. 5). This phe- the Goos-Hänchen shift in the two polarizations.[2,3] nomenon can be understood by analyzing the spatial distribution of the incident field amplitude in the interface 2 2 n2µ 1 1 plane, E() x and its Fourier transform over x, given by dP = 2 i k µ 2 1n1 2 n E()kx , such that sin2 θ − 2 i n2 +∞ 1 E()x= E()k eikx xdk (43) ∫ x x 2 −∞ n2  1− 2  sinθi y n1  (where we neglect the dependence of the field on , which × (47) plays no role in this argument). In other words, we consider 4 2 n  µ  n2  the incident field as a sum of infinitely many plane waves,  2 1 2 2 2      cosθi+ sin θ i − 2  n µ2 n  Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 each having a slightly different x component of its wave 1    1  vector, and hence, a slightly different angle of propagation θ (k ) = arcsin(k /k ). Accordingly, in total internal reflec- n2  i x x i 12 sin tion, each of these plane waves experiences a different − 2  θi 2 µ n1  phase shift, which can be decomposed into the first-order d 2 (48) S = 2 k1µ 1  2  2 Taylor series as µ2  2 2 n2 2 n2   cosθi+ sin θ i − 2  sin θi − 2 µ n n 1  1  1 dδ () kx δ()kx≈ δ () k x0 + ()kx− k x0 (44) dkx where k = ω n /c. As seen from the foregoing equations, at kx0 1 1 incidence angles that are significantly larger than the criti-

where kx0 corresponds to the direction of the incident cal angle, the Goos-Hänchen shift is on a scale of the opti- beam axis. Eq. 44 is valid if the beam diameter greatly cal wavelength. For right-handed materials, it is always in exceeds the wavelength; thus, the relevant range of val- the positive x direction. This can be visualized using the

ues of Δkx is small. For the reflected wave, we then ray picture of light: in total internal reflection, the incident obtain: rays bounce not off the interface, but slightly below the 6 Fresnel Equations

interface, accounting for the existence of the evanescent wave. However, if medium 2 is a metamaterial, the Goos- Hänchen shift is in the negative x direction due to counterintuitive direction of refraction in metamaterials.[2,3]

Applications

One of the primary consequences of the Fresnel equations Fig. 6 Fresnel rhomb. is that any interface between transparent optical media results in a significant fraction of the light being reflected. This is particularly important for complex lens systems Conclusion such as microscope, telescope, and camera objectives. We have derived the Fresnel equations from the first princi- Given that the spurious reflectivity at a single glass–air ples of wave optics. Subsequently, we discussed the conse- interface is 4%, a system of 8 optical elements will suffer quences of these equations, such as the Brewster effect and from about 50% loss due to Fresnel reflections. the optical phase shift in partial and total internal reflection. To avoid these losses, antireflection coatings are com- Finally, we discussed a few applications of the Fresnel monly used in lens systems. In fiber optics, an alternative equations and the related effects in optical design. solution is offered by index-matching materials: liquid or gel substances whose index of refraction approximates that of the fiber core. Placing an index-matching fluid in fiber Historical notes connectors and mechanical splices greatly reduces Fresnel Augustin-Jean Fresnel (1788–1827) is one of the found- reflection at the surfaces and thus decreases the power loss. ing fathers of the wave theory of light. In response to an Brewster’s effect is extensively used in photography. 1818 competition held by the French Academy of Sci- Unpolarized light, incident on a building window or water ences, Fresnel wrote a memoir describing diffraction as a surface, becomes largely S-polarized after reflection. wave phenomenon. Although the corpuscular (Newto- Dependent on the orientation of a polarizing filter in front nian) concept of light was universally accepted at that of the camera, the amount of the reflected light can be reg- time, Fresnel’s theory received immediate experimental ulated. In particular, aligning this filter to transmit only the confirmation, thus revolutionizing contemporary optical P polarization permits taking pictures of objects beneath science. In 1823, Fresnel was unanimously elected a the surface or behind the window. member of the Academy, and in 1825 he became a mem- Polarizing sunglasses provide another example of prac- ber of the Royal Society of London. At that time, Fresnel tical application of Brewster’s effect. These sunglasses are developed his theory based on the theory of elastic ether. designed to block horizontal polarization, which helps In 1827, the Royal Society of London awarded him the reducing glare from horizontal objects such as water or Rumford Medal. road surfaces. Sir David Brewster (1781–1868) is mostly remembered A further application of Brewster’s effect is found in laser for his invention of the kaleidoscope and optical improve- physics, specifically in gas laser design. The end windows of ments of the microscope. However, his main experiments laser tubes are routinely manufactured to be oriented at the were on the theory of light and its uses. His first paper, Brewster angle with respect to the cavity mode, with an aim “Some Properties of Light,” was published in 1813. to eliminate reflection losses in the P-polarization. In this Brewster’s Law was named after him in 1814 when he way, a stronger gain per cavity roundtrip can be achieved for made measurements on the angle of maximum polarization one of the polarization components while reducing the gain using biaxial crystals. He was awarded all three of the prin- for the other. This helps in obtaining strong emission in a

Downloaded by [University of Calgary], [Alexander Lvovsky] at 11:24 22 August 2013 cipal medals of the Royal Society for his optical research single polarization mode. (Copley medal, 1815; Rumford medal, 1818; Royal medal, An interesting application of Fresnel equations was pro- 1830). He was also knighted in 1831. posed by Fresnel himself. As mentioned earlier, total internal reflection causes different phase shifts to the S- and P-polarized components of the incident wave. Fresnel used References this phenomenon to design an optical element that converts 1. Veselago, V.; Braginsky, L.; Shklover, V.; Hafner, C. Nega- light polarization from linear into circular. This is accom- tive refractive index materials. J. Comp. Theor. Nanosci. plished by means of two total internal reflections in a paral- 2006, 3 (1), 1–30. lelepiped prism, as illustrated in Fig. 6. For a prism made 2. Berman, P.R. Goos-Hänchen shift in negatively refractive of glass with a refractive index of 1.5, an internal reflection media. Phys. Rev. E 2002, 66 (6). angle of incidence of 54.6° can be used. It should be noted 3. Berman, P.R. Goos-Hänchen shift in negatively refractive that at present, polarization transformations in free space media. Phys. Rev. E 2005, 71 (3). are typically performed by birefringent waveplates rather 4. Goos, F.; Hänchen, H. Ein neuer und fundamentaler versuch than the Fresnel rhomb. This is because waveplates are zur totalreflexion. Ann. Phys. 1947, 436 (7–8), 333–346. more compact and do not distort the beam position.