8Fresnel Equations
8.1Introduction
The Fresnel equations describe the behavior of light at an interface between two media. The amplitude of the light divides between the reflected and refracted beam based on the refractive indices of the two media. The resulting amplitude coefficients, described by the Fresnel coefficients, are used within our various polarization matrices to include the effects of reflection and refraction in polarization ray tracing calculations. When wavefronts propagate through optical systems, the resulting polarization effects generate diattenuation and retardance aberrations.
One of the most straightforward polarization changes that occur in an optical system are the changes at dielectric to dielectric and dielectric to metal interfaces. When light strikes an interface, it divides into reflected and transmitted beams. The amount of light reflected and transmitted depends on its polarization state and orientation with respect to the interface. The Fresnel equations relate the complex amplitudes of the reflected and transmitted beams to the optical properties of the surface and the angle of incidence. They are used during polarization ray tracing to calculate the changes in polarization as light propagates through optical systems to determine the diattenuation and retardance associated with ray intercepts and ray paths through optical systems.
Optical surfaces with coatings, such as antireflection coatings and beam splitter coatings, have more complex amplitude reflection and transmission coefficients that are calculated by the application of the Fresnel coefficients to multilayer stacks of optical materials as discussed in Chapter 13 (Thin Films). The calculation of the amplitude transmission coefficients for anisotropic materials, such as calcite and sapphire, is derived in Chapter 19.
Our interest is on the effect of interfaces on light propagation. The Fresnel equations can be derived rigorously from Maxwell’s equations applied to a homogeneous and isotropic interface. This derivation is well covered in many other sources. Here, we will focus on the polarization effects, the consequences of Fresnel equations, their magnitudes and forms, and how to integrate these polarization effects into optical design.
This chapter presents the Fresnel equations and then applies the Fresnel equations to describe polarization changes that occur at dielectric interfaces, total internal reflection, and reflection from metals, as in mirrors. The changes in polarization that occur on reflection and refraction are characterized. Then, the resulting polarization aberrations are described for several example optical systems, such as a Cassegrain telescope in Chapter 12 and an astronomical telescope in Chapter 27.
8.2Propagation of Light
8.2.1Plane Waves and Rays
Given a plane wave incident on a plane interface, what are the amplitudes and phases of the reflected and refracted plane waves? Optical design is concerned with plane, spherical, and other wavefronts. In optical ray tracing, light is treated as a light ray. A small patch of a wavefront can be assumed to locally have a plane wavefront. The restriction to plane waves is for the purpose of the derivation. Similarly, an optical interface is plane, spherical, or aspheric surface. But in the vicinity of the light ray, the interface can be considered as a plane surface for the purpose of ray tracing. Thus, the change of the light ray’s polarization upon reflection and refraction can be calculated using the local properties of the interface; this works remarkably well as an approximation.
8.2.2Plane of Incidence
The plane of incidence contains the wavevectorki of an incident plane wave and the surface normalη, as shown in Figure 8.1. The incident medium has refractive index ni. The transmitted medium has refractive index nt. The plane wave is incident with an angle of incidence θi measured from the surface normal. A reflected beam propagates back into the incident medium in the direction kr. A transmitted beam propagates into the transmitted medium in the direction kt at an angle of refraction θt.
Figure 8.1The configuration for reflection and refraction of a plane wave at an interface. The plane of incidence is the plane containing the incident wavevector k and the surface normal η.
8.2.3Homogeneous and Isotropic Interfaces
An interface divides part of an optical system into two volumes with different properties, such as a lens surface or the surface of a mirror. In this chapter, it is assumed that the material is isotropic, homogeneous, non-absorbing, non-magnetic, and has no free charges on the interface.
The following provides the terminology of different types of material and interface:
Homogeneous interface—Interface with uniform optical properties such as constant refractive index n + iκ over the clear aperture, which may be real (transparent medium) or complex (absorbing or metal)
Inhomogeneous interface—Non-uniform interface with varying composition or varying coating thickness
Isotropic material—The same refractive index in all directions and polarizations within the material
Anisotropic material—Birefringent or optically active materials; light experiences varying refractive index as a function of the light’s polarization direction
8.2.4Light Propagation in Media
Light propagates in a vacuum at the speed of light c,
8.1When light propagates through a material, the light’s electric and magnetic field drives the material’s charges, the electrons and protons, which then oscillate at the light’s frequency. In an absorbing material, the atoms and molecules readily make transitions into other electronic and vibrational quantum states of the material, and the light is absorbed into heat or scattered into other directions. Transparent materials, like air, water, and glass, do not have available transitions near the light’s frequency.
Consider light refracting into glass. Monochromatic light drives the charges at the surface into simple harmonic oscillation. The charges are accelerated by the light, and accelerating charges radiate. For a monochromatic plane wave or spherical wave, a whole volume of charge is set oscillating, and the coherent radiation from the volume radiates a beam of light into the forward direction, along the direction of the original beam of light. The charges in the transparent material do not oscillate quite in phase with the incident light; thus, the atoms radiate light that is slightly delayed. The net result of the superposition of the incident light and light reradiated by the atoms is that the light slows down. This is the source of the refractive index of the light.
The refractive index n (also called index of refraction) of transparent material is the ratio of the velocity of light in vacuum to the velocity V of light in a medium,
8.2The angle of refraction θt is related to the angle of incidence θi by Snell’s law
8.3Snell’s law embodies the conservation of momentum. It also describes the phase matching that must occur at an interface between the two media, as shown in Figure 8.2.
Figure 8.2Intersection of an incident set of wavefronts on the interface for a plane wave (left) and a spherical wave (right) matches the intersections of the transmitted set of wavefronts. The incident and exiting wavefronts have different wavelengths and propagation velocities. Since they have the same frequency, they match at the interface.
There is an equilibrium between the light field and the oscillation of the transparent materials charges. The electric field of the light is reduced, while some of the energy is tied up in the charge oscillation. The light’s field is reduced, but its flux is not. When light is incident on an interface, the charges of the atoms and molecules are set into motion. The reflected and transmitted amplitude coefficients of the light’s electric field are calculated by solving Maxwell’s equations at the interface.
Reflection associated with transparent material is divided into two cases, external and internal. External reflection occurs when ni < nt, such as when light reflects from air to glass interface. Internal reflection occurs when nt < ni, such as light reflecting inside glass from an air interface, as shown in Figure 8.3.
Figure 8.3Internal reflection occurs when light reflects from a glass–air interface.
When light reflects from metal surface, the electric field of the reflected light is affected by the complex refractive index of the metal. Metals have free charges that easily move between atoms when an electric field is present or a voltage is applied. Optically, this is the cause of the metal’s complex refractive index, n + iκ. The imaginary part of the refractive index governs how rapidly a wave is absorbed propagating through a material. The incident light’s energy is substantially absorbed within a few tens of nanometers when reflecting from metals with κ greater than 2.
8.3Fresnel Equations
8.3.1s- and p-Polarization Components
The incident light’s electric field is oscillating in a plane transverse to k. It can be divided into a p-polarized component oscillating in the plane of incidence and an s-polarized component oscillating perpendicular to the plane of incidence as shown in Figure 8.4. A basis vector for the s-component 
is constructed as
8.4
Figure 8.4The electric fields for s- and p-polarized components of a beam of light incident on an interface. (Left) s-component perpendicular to the plane of incidence. (Right) p-component in the plane of incidence. The s-component lies on the surface while the p-component has a component into the surface.
which lies on the interface. Since the s-component’s electric field is perpendicular to the plane of incidence, it is also referred to as transverse electric, or the TE mode, as shown in Figure 8.5. The basis vector for the p-component 
is
Figure 8.5(Left) Transverse electric field orthogonal to the plane of incidence. (Right) Transverse magnetic field orthogonal to the plane of incidence.
8.5which lies in the plane of incidence. The p-component is also known as the transverse magnetic, or the TM mode, because the corresponding magnetic field is transverse to the plane of incidence. The set 
forms a right-handed orthonormal basis. The incident light’s polarization vector E is expressed in terms of the s- and p-components as
8.6An arbitrary incident plane wave can be decomposed into s- and p-components that are then reflected and refracted separately. At normal incidence, the definitions of s and p break down, the cross product in Equation 8.4 goes to zero, and 
becomes undefined. Any pair of orthogonal unit vectors on the surface can be chosen since 
and 
have become degenerate.
8.3.2Amplitude Coefficients
Consider a plane wave incident at a homogeneous and isotropic interface. The incident medium is assumed to have no absorption; thus, ni is real. nt can be real or complex. The incident plane wave interacts with the interface, giving rise to reflected and transmitted plane waves. An s-polarized incident plane wave reflects and transmits into an s-polarized reflected wave and an s-polarized transmitted wave. The same is true for an incident p-polarized plane wave. These two incident states, s and p, are the eigenpolarizations for reflection and refraction. Other incident polarization states, combinations of s and p states, are not in general reflected and refracted in the incident state. Therefore, the description of reflection and refraction is simplest in terms of the s- and p-components.
The amplitudes of the plane waves before and after the interface are related by amplitude coefficients rs, ts, rp, and tp. The incident and reflected electric field amplitudes are related as1
8.78.3.3The Fresnel Equations
The amplitude coefficients for an interface associated with two dielectric or an incident dielectric and metal substrate are the Fresnel coefficients. There are four coefficients, an s- and p-coefficient for reflection and an s- and p-coefficient for transmission, given as follows1–4:
8.8
8.9
8.10
8.11Using Snell’s law, the angle of refraction is eliminated, yielding an alternative form for these four coefficients dependent only on angle of incidence,
8.12
8.13
8.14
8.15where n = nt/ni.
Fresnel coefficients are a function of refractive index. Thus, they can appear in two forms: the form for decreasing phase convention calculated by n + iκ, and the form for increasing phase convention calculated with n − iκ.
8.3.4Intensity Coefficients
The Fresnel amplitude coefficients relate the electric field amplitudes of the various wavefronts. The Fresnel intensity coefficients relate the fraction of flux in the reflected and refracted beams to the incident flux. For the reflected beam, the intensity coefficients are just the magnitude squared of the amplitude coefficients,
8.16
8.17For the transmission coefficients, two additional effects must be considered. First, for waves propagating inside materials other than vacuum, the light waves induce charge motions that contribute to the flux. The flux in a material of refractive index n is proportional to n times the flux in vacuum. Thus, a plane wave with amplitude of 1 V/m in silicon, n = 4, conveys four times the energy per unit area as a plane wave in vacuum with the same 1 V/m amplitude. Second, the area of a beam changes on refraction as shown in Figure 8.6. The ratio of the refracted area to incident area is
Figure 8.6(Left) A beam with area Ai incident on an interface (2) refracts into a beam with area At. (1) A circular beam with a unit radius in the incident medium, (2) has a cross section on the interface that is stretched along the plane of incidence. (3) The beam cross section after refraction.
8.18Note that the cross-sectional area of a beam does not change on reflection because θi = θr.
Taking these two factors into account, the Fresnel intensity coefficients become
8.19
8.20where nt/ni accounts for different power conveyed in the different media, and cos θt/cos θi accounts for the change in beam cross section.
Diattenuation of the transmitted and reflected light at isotropic surface is defined by the s- and p-Fresnel intensity transmission and reflection coefficients,
8.21Example 8.1Reflection and Transmission at Glass
Consider a beam incident in air ni = 1 reflecting and transmitting from a glass interface with refractive index nt = 1.5. Figure 8.7 (left) shows the Fresnel amplitude transmission coefficients as a function of θi. Near normal incidence, the refracted E into the glass is 0.8 of the incident E. This amplitude monotonically decreases as θi increases and reaches zero at grazing incidence. Figure 8.7 (right) shows the Fresnel amplitude reflection coefficients. These have magnitude 0.2 at normal incidence; rs(0) and rp(0) are equal in magnitude, but because of the sign convention for Fresnel coefficients, discussed below, they are opposite in sign, with rs(0) < 0. rs(θi) monotonically decreases to −1 at θi = 90°; all the light reflects at grazing incidence. rp(θi) decreases, crossing zero at about θi = 57°, which is Brewster’s angle discussed in Section 8.3.6.
Figure 8.7(Left) Amplitude transmission coefficients ts (green) and tp (blue) as a function of incident angle at an interface with incident refractive index ni = 1 and transmitted refractive index nt = 1.5. (Right) Amplitude reflection coefficients rs (red) and rp (orange) as a function of incident angle.
After accounting for the additional factors in the computation of the intensity transmittance, Equations 8.19 and 8.20, the Fresnel intensity transmission coefficients have a different form compared to the amplitude coefficients. Figure 8.8 shows the intensity coefficients. The s intensity reflectance Rs monotonically increases from 0.04 to 1. The p intensity reflectance Rp decreases slowly from 0.04, reaching 0 at Brewster’s angle. At normal incidence, a fraction of the flux (T = 0.96) of both the s- and p-components are refracted into the glass. The flux transmittance of the p-component Tp increases, initially quadratically to 1 at Brewster’s angle, before decreasing rapidly to 0 at grazing incidence. The intensity transmittance of the s-component TS decreases monotonically from 0.96 to 0.
Figure 8.8Intensity transmission coefficients Ts (green) and Tp (blue) and intensity reflection coefficients Rs (red) and Rp (orange) as a function of incident angle at an interface with incident refractive index ni = 1 and transmitted refractive index nt = 1.5.
Figure 8.9 shows the corresponding diattenuation for refraction and reflection. Several features are worth repeating. First, at Brewster’s angle, the reflected light is fully polarized; the diattenuation is 1, only a small fraction of the flux, ~0.17 is reflected. Also at Brewster’s angle, 100% transmission is attained for the p-component; this is very useful in situations such as inside laser cavities, where loss must be minimized. Near normal incidence, the diattenuation, and thus the polarization change, is small. Further, since both refractive indices are real, the Fresnel coefficients are real and there is no retardance, only diattenuation. Note that as the reflected p-component passes through Brewster’s angle, it undergoes a 180° phase change.
Figure 8.9At an interface with incident refractive index ni = 1 and transmitted refractive index nt = 1.5, (left) diattenuation upon transmission as a function of incident angle and (right) diattenuation upon reflection as a function of incident angle.
8.3.5Normal Incidence
At normal incidence, the incident light lies along the normal of the interface and the difference between s and p disappears. The electric field for any incident polarization state lies along the interface, the definition of s-polarization. At normal incidence, the plane of incidence is undefined, so s and p become degenerate. Taking the limit of the Fresnel equations as θi approaches zero, the s and p amplitude transmission coefficients become equal,
8.22Similarly, the two amplitude reflection coefficients become
8.23Thus, the normal incidence intensity transmission and reflection coefficients are
8.24
8.25The derivations of Fresnel amplitude coefficients use right-handed coordinates for both the incident and reflected beams and 
in almost all references. Since the 
basis vector reverses direction, this condition requires a change of sign in one of the transverse basis vectors (s or p) at the interface after reflection, as shown in Figure 8.10. This change of direction requires an extra minus sign in one of the reflection amplitude coefficients to conform to this convention. Since our polarization ray tracing algorithms use global coordinates, this minus sign is not needed. One must be very careful in choosing the sign of amplitude coefficient and stay consistent throughout the ray tracing calculation.
Figure 8.10Examples of right-handed coordinate systems for light at normal incidence. (Left) Incident light basis vectors and (right) reflected light basis vectors. Because the propagation vector k changes direction, one of the coordinates, x or y, must also change sign to retain the right-handed convention; y is shown changing here.
Consider linearly polarized light at or near normal incidence with its polarization oriented at 45° from the incident right-handed coordinates as shown in Figure 8.11. The light causes electrons to oscillate along this 45° and the reflected light is polarized in this same plane. Globally, the plane of polarization of the incident and reflected light are in the same plane. Expressed in right-handed coordinates, this incident light is 45° polarized and the reflected light is 135° polarized! The Fresnel s-coefficient in Equations 8.8 and 8.12 incorporate this minus sign.2
Figure 8.11In a right-handed coordinate system containing k, 45° linear polarization externally reflects into 135° linear polarization.
Figure 8.12 shows left circularly polarized light incident on an interface and the state following reflection. The light’s field moves the reflector’s charges in a clockwise direction rotating from (+x → +y) to (−x → +y). The field of the reflected light continues to rotate in the same direction as a function of time (small orange arrows), but now the direction of propagation is reversed. The helicity of the wave has changed from left to right circularly polarized.
Figure 8.12In a right-handed coordinate system containing k, left circular polarization always reflects into right circular polarization at normal incidence from an isotropic interface.
8.3.6Brewster’s Angle
A special angle related to the Fresnel equations is the Brewster’s angle. Brewster’s angle θB is defined by the angle of incidence θi where the p-reflectance is zero, rp = 0, which occurs at
8.26Figure 8.13 (left) shows Brewster’s angle at varying nt/ni. Note that at nt/ni = 1, Brewster’s angle is 45°.
Figure 8.13(Left) Brewster’s angle θB and (right) the critical angle θC as a function of the ratio of refractive indices nt/ni.
Since no p-polarized light is reflected at Brewster’s angle and only s-polarized light is reflected, the reflected light is polarized. When unpolarized light is incident on a surface at Brewster’s angle, the reflected light is linearly polarized transverse to the plane of incidence; the reflected light is s-polarized. Therefore, the degree of polarization becomes one at Brewster’s angle, as shown in Figure 8.14.
Figure 8.14Degree of polarization of unpolarized incident light reflected at an air–glass interface for nt/ni = 1.5.
This characteristic provides a convenient method to produce a polarizer; reflecting a nearly collimated beam from a dielectric medium at Brewster’s angle can work as a polarizer. Although the reflected flux is not so great, about 0.17 for nt/ni = 1.5, the polarizer is extremely easy to construct.
The Brewster’s angle phenomena occurs at the special incidence angle where the angle between the reflected and refracted beams is 90°, θr + θt = 90°. The direction of motion of the charges in the refracting medium is 90° from the angle of refraction. Dipole oscillators do not radiate along that axis. Thus, at Brewster’s angle, the charges in the refracting material cannot radiate (i.e., reflect) light back into the incident medium, and 100% of the p-polarized light is refracted into the refracting medium. This can be extremely useful. For example, many elements are placed into laser cavities at Brewster’s angle and operated with p-polarized light to minimize losses and maximize gain.
8.3.7Critical Angle
Another special angle related to the Fresnel equations is the critical angle. A distinction is made between external reflection where ni < nt and internal reflection where ni > nt. Internal reflection occurs inside glass for example. During internal reflection, only light incident within a range of θi less than the critical angle can escape into the lower index medium. The critical angle occurs when the angle of refraction from Snell’s law is equal to 90°; the refracted light exits the interface at grazing angle. Solving Snell’s law for θt = 90° yields an incident angle, the critical angle θC of
8.27For θi greater than the θC, all incident energy is internally reflected, a condition called total internal reflection or TIR. TIR is extremely useful for designing low loss optics. Many optical systems such as binoculars use TIR prisms to minimize loss. Fold mirrors are often replaced with prisms. The entering and exiting faces are antireflection coated to minimize reflection loss, and the TIR at the hypotenuse is 100%, yielding far less loss than reflection from a metal mirror. A comprehensive list of TIR prism designs is found in Wolfe.5Figure 8.13 (right) plots the critical angle at varying nt/ni. For glasses with 1.5 < ni < 2, the θC falls in the range of 40° to 30°; thus, higher index glasses undergo TIR over a larger range of angles, and such higher indices are often necessary for TIR devices. Much of the sparkle of diamond and cubic zirconia gemstones comes from TIR associated with their high refractive indices.
Below the critical angle, dielectric materials with real refractive index have real Fresnel coefficients and no phase change occurs on internal reflection. The phase changes beyond the critical angle are
8.28Retardance is defined in Section 5.3.2 as the optical path difference between eigenpolarizations, here the s- and p-polarized light. Calculating the phase difference using the phases of the Fresnel coefficients
8.29produces a number, Δ, which contains an extra π phase due to geometrical rotation upon reflection, as described in Figures 8.10 through 8.12. Here, in this normal incidence example, Δ = π, due to the change of the sign of the s-basis vector. s- and p-light should have the same phase change. This extra π phase must be removed to obtain the optical path difference, the physical phase difference,
8.30The issues of this additional π for polarization ray tracing are discussed in Section 17.6.1.
8.3.8Intensity and Phase Change with Incident Angle
Reflections as a function of incident angle are treated in this section. Figure 8.15 plots the Fresnel s and p reflection coefficients for external and internal reflections for an air–glass (n = 1.5) interface. The Fresnel coefficients6 are defined in the incident and exiting local coordinates using the sign convention in broadest use,7 which are both right handed. At normal incidence, s and p reflection coefficients have the opposite sign due to this choice of local coordinates.
Figure 8.15Fresnel external (left) and internal (right) amplitude reflection coefficients for s-polarization (red) and p-polarization (orange) versus the angle of incidence.
Figure 8.16 shows relative phase shifts between s- and p-polarizations for external and internal reflections. A residual Fresnel π phase difference exists as the consequence of coordinate change during reflection. At Brewster’s angle θB, the p Fresnel reflection coefficient passes through 0 and changes sign, causing an additional π phase shift upon reflection for the incident angles greater than θB. The transmitted light is partially polarized while the reflected beam is completely polarized. At critical angle θC, the s- and p-polarized light has equal amplitude but different phase change and results in non-zero retardance.
Figure 8.16The relative phase shift for external (left) and internal (right) reflection calculated from the Fresnel coefficients as a function of the angle of incidence.
8.3.9Jones Matrices with Fresnel Coefficients
The Jones matrices for rays reflecting from or refracting into isotropic media incorporate the Fresnel amplitude coefficients. First, consider a Jones matrix defined in s–p coordinates. In this basis, the reflection and refraction Jones matrices are diagonal matrices,
8.31for a single isotropic surface, because s-polarized light couples entirely to s-polarized light and p-polarized light couples entirely to p-polarized light. Since Jones matrices are expressed in right-handed coordinate systems, the phase shifts are the Fresnel coefficients phase shifts. For angles of incidence smaller than θB, the Jones matrix for reflection shows π phase shift due to the right-handed local coordinate choice before and after the reflection, as shown in Figure 8.10. This value is not the proper retardation because it includes the π geometric transformation of the local coordinates as described in Section 17.6.1. The construction of the polarization ray tracing P matrices for reflection and refraction incorporating the Fresnel coefficients is covered in Section 10.6.
8.4Fresnel Refraction and Reflection
8.4.1Dielectric Refraction
In the Fresnel equations, only the ratio of the refractive indices matters; the coefficients only depend on the ratio. Figure 8.17 shows the effect of the refractive index ratio, n = nt /ni on the Fresnel intensity transmission coefficients. As the refractive index ratio increases, the normal incidence transmission decreases and Brewster’s angle increases. The separation between the s- and p-coefficients occurs more rapidly at higher index. Again, no retardance is present with these purely real refractive indices.
Figure 8.17Intensity transmission coefficients ts (green) and tp (blue) as a function of incident angle for an interface as the ratio of refractive indices n2/n1 changes: 1.5, 2, and 4.
The diattenuation of a dielectric interface is plotted versus refractive index ratio and angle of incidence in the colored contour plot of Figure 8.18. Diattenuation is zero at normal incidence, along the bottom of the graph, at grazing incidence, along the top of the graph, and for total internal reflection, the red region along the upper left side of the contour plot. Along vertical cross sections, diattenuation is seen to rise monotonically to a peak of one, at Brewster’s angle, and then falls monotonically back to zero, at the critical angle for n1/n2 < 1, or monotonically back to zero at grazing incidence for n1/n2 > 1.
Figure 8.18(Left) Transmission diattenuation for dielectric interface with varying refractive index ratio n at varying incident angles θin. Brewster’s angle occurs down the middle of the violet band. (Right) Reflection diattenuation for dielectric interface with varying index n at varying incident angle θin.
8.4.2External Reflection
External reflection occurs when light reflects from a less dense medium to a denser medium, such as from an air–glass interface. Figure 8.19 depicts external reflection of the electric field for three different polarizations: s-polarization states, an equal combination of s- and p-polarization states, and p-polarization states. Since the Fresnel sign conventions can be confusing, it is helpful to visualize the fields just before and just after the interface in three dimensions.
Figure 8.19External reflection for the incident s-polarized states (left), 45° polarization states (center), and p-polarized states (right) as angles of incidence change from 7° to 88° near grazing incidence. Since the p-polarization figures would overlap for different angles of incidence, each incident and reflected pair has been spatially translated for better visualization. The blue arrow indicates the reflected electric field at Brewster’s angle. Each incident electric field vector (black arrowhead) and corresponding reflected electric field vector (colored arrowhead) pair is plotted in the same color. The arrows represent the oscillations of the electric field vector in time. The length of the arrow represents the amplitude of the electric field vector.
Care is needed to properly interpret the signs of the Fresnel coefficients. When calculated properly, polarization ray tracing matrix will yield zero retardance upon reflection at normal incidence and π phase shift for external reflection in the range, θB < θi < θC. The retardance for internal reflection changes rapidly for the angle of incidence greater than θC as shown in Figure 8.16. The reflected electric field vector has smaller amplitude than the corresponding incident electric field. The s-component of the electric field steadily increases from 0.2 to 1.0 as the angle of incidence increases. The p-component of the electric field changes its sign after Brewster’s angle and has zero reflectance at the Brewster’s angle. This is the origin of π retardance for reflections in the range θB < θi < θC.
Figure 8.20 shows the external reflection of circularly polarized light. Since both s- and p-components have a π phase shift upon reflection when θi < θB, left circularly polarized light reflects as right circularly polarized light at normal incidence. Note that “left” and “right” are defined locally relative to the propagation direction. The incident and reflected light circulate in the same direction when viewed in 3D global coordinate for θi < θB. As θi approaches θB, the amplitude of the reflected p-component is lower than the reflected s-component. Thus, it reflects as right elliptically polarized light. When θi = θB, the reflected light becomes purely s-polarized. For θi > θB, both reflected s- and p-components have increasing amplitude toward grazing incidence, while the p-component has an extra π shift before Brewster’s angle. Thus, the left circularly polarized light reflects as left elliptically polarized light. When θi ≈ 90°, left circularly polarized light reflects as left circularly polarized light.
Figure 8.20Left circularly incident light at various incident angles externally reflect from an air–glass interface. The reflected light around normal incident is right elliptically polarized. It becomes linearly polarized at Brewster’s angle (θB = 56.31°) and switches to left elliptically polarized above θB.
8.4.3Internal Reflection
Internal reflection is reflection at an interface from a larger to a smaller refractive index, such as reflection inside glass at a glass–air interface. When light is incident above the critical angle, the intensity reflectance for both s- and p-polarizations are 1; the reflection is lossless. This highly desirable property is responsible for the widespread use of right angle prisms to reflect beams through 90°, the popularity of prisms in binoculars, and many other applications. Since the s- and p-intensity reflectances are equal above the critical angle, the diattenuation is zero, but large retardances are present, as shown in the following internal reflection example.
Figure 8.21 shows the intensity reflection coefficients Rs and Rp and intensity transmission coefficients Ts and Tp for a ray propagating from glass with index n1 = 1.5 to air with index n2 = 1. At normal incidence, 4% of the flux reflects. Rs increases monotonically to 1 at the critical angle = 41.81°. Above θC, the reflectance remains 1 until grazing incidence, θi = 90°. Rp decreases to 0 at Brewster’s angle, θB = 33.69°. Beyond Brewster’s angle, the intensity reflectance rises rapidly to 1.
Figure 8.21(Left) Intensity reflection coefficients Rs (red) and Rp (orange) and (right) intensity transmission coefficients Ts (green) and Tp (blue) as a function of incident angle.
The amplitude reflection coefficients, rs and rp, become complex when θi > θC. Figure 8.22 shows that the magnitude of rs and rp are both 1 above critical angle with changing phase. The Fresnel phase of rs and rp are shown in Figure 8.23. The π Fresnel phase in rp at normal incidence is due to the right-handed Fresnel coordinate. Hence, both s- and p-components have zero phase shift upon internal reflection. The retardance is the phase difference between s- and p-components. Notice that the retardance has infinite slope at θC.
Figure 8.22(Left) Fresnel amplitude reflection coefficient Re(rs) (red), Re(rp) (orange), Im(rs) (purple), and Im(rp) (cyan) as a function of incident angle. (Right) Absolute magnitude of Fresnel amplitude reflection coefficient of rs (red) and rp (orange) as a function of incident angle.
Figure 8.23(Left) Fresnel phase shift of rs (red) and rp (orange) as a function of incident angle. (Right) Retardance upon internal reflection as a function of incident angle.
Figure 8.24 depicts the internal reflected electric field for three different polarizations: s-polarization states, the combination of s- and p-polarization states, and p-polarization states. The reflected p-polarized light starts to have non-zero phase shift at θi = θB, while the s-polarized light starts to have phase shift at θi = θC. The different phase shift between these two components produces retardance. Therefore, the electric field with both components reflects to elliptically polarized light, as shown in Figure 8.24 (middle).
Figure 8.24(Left) The incident and reflected states for internal reflection for s-polarized incident light from 7° to 88°. (Center) Internal reflection for 45° polarized incident light from 0.6° to 86°. The phase shift between s and p beyond Brewster’s angle produces elliptically polarized reflected light. (Right) Internal reflection for p-polarized incident light. Reflected s- and p-polarized components are in phase from normal incidence to Brewster’s angle, and then the phase changes (the arrowheads shift from the end of the line segments).
Figure 8.25 shows the internal reflection of circularly polarized light. At normal incidence, both s- and p-components have zero phase shift upon reflection with a reversed propagation direction; hence, left circularly polarized light reflects to right circularly polarized light. As θi approaches θB, the amplitude of the reflected p-component is lower than the reflected s-component. Thus, it reflects as right elliptically polarized light. When θi = θB, the reflected light becomes purely s-polarized. For θi > θB, both reflected s- and p-components have increasing amplitude toward grazing incidence, while the p-component has an extra π shift. Thus, the left circularly polarized light reflects as left elliptically polarized light. When θi ≈ 90°, left circularly polarized light reflects as left circularly polarized light.
Figure 8.25Internal reflection for left circularly polarized light. The reflected light around normal incident is right elliptically polarized. It becomes linearly polarized at Brewster’s angle (θB = 33.69°) and switches to left elliptically polarized above θB. Its ellipticity increases to left circularly polarized light at critical angle (θC = 41.81°).
8.4.4Metal Reflection
Light reflecting from a smooth metal surface is affected by its complex refractive index, n + iκ, as opposed to the real refractive index of dielectric surface. Whereas transparent dielectrics like glasses and optical plastics have refractive indices greater than one with very small imaginary parts, the refractive indices of metals have a substantial imaginary component. Figure 8.26 lists the refractive indices of some common metals at 633 nm and locates these refractive indices on the (n, κ) complex plane.
Figure 8.26Complex refractive indices of common metals at 633 nm contain a real part n and an imaginary part κ.
The change of amplitude and phase of reflection from a smooth metal surface is calculated using the same Fresnel equations that apply to dielectrics, except that the complex value of the metal’s refractive index is used. The resulting Fresnel coefficients are complex valued. When rs and rp are expressed in polar form, 
, the magnitudes ρs and ρp describe a change in amplitude of the electric field components and ϕs and ϕp describe changes in phase. Since the transmitted part of the wave is rapidly absorbed near the surface of the metal, the transmission coefficients do not apply, but the reflection coefficients work fine.
Example 8.2Fresnel Coefficients for Aluminum
The most common metal reflector is aluminum because of its high reflectivity in the visible and near infrared, low cost, and ease of machining. Aluminum doesn’t corrode, is resistant to humidity, and doesn’t tarnish due to trace atmospheric sulfur like silver does. The spectral variation of aluminum’s refractive index is shown in Figure 8.27.
Figure 8.27(Left) The real part of the refractive index of aluminum versus wavelength, (center) the imaginary part, and (right) a parametric plot of real versus imaginary.
Consider reflection from a smooth aluminum surface at 633 nm where the complex refractive index is 1.374 + i7.620. The amplitude reflection coefficients, shown in Figure 8.28 for s and p, are about 0.955 at normal incidence. The s-component has the greater amplitude reflectance
Figure 8.28(Left) Amplitude reflection coefficients rs (red), rp (orange) and (right) diattenuation for reflection from an Al surface as a function of the angle of incidence.
and increases monotonically to one at grazing incidence, θi = 90°. The p-coefficient decreases, quadratically at first, and then reaches a minimum around θi = 82° of about 0.83, before rapidly increasing to one at grazing incidence (Figure 8.29). This difference in amplitude reflectance causes the diattenuation for reflection (Figure 8.28, right). Figure 8.29 (left) shows the phase shift from the amplitude reflection coefficients for s-polarized (red) and p-polarized (orange) light. The difference in phase shift results in the retardance for reflection (right). Measurement techniques for the phase change at metallic reflection are discussed by Medicus et al.8
Figure 8.29(Left) Phase shift, ϕs (red) and ϕp (orange), upon reflection from Al, and (right) retardance as a function of incident angle.
Light transmitted across the front surface of a metal propagates only a small distance into the metal being absorbed within the first few nanometers. The fraction of this absorbed light is calculated as 1 − R, where R is the intensity reflection coefficient. Values for the absorption of aluminum as a function of angle of incidence for the s- and p-components are plotted in Figure 8.30.
Figure 8.30Reflection (left) and absorption (right) of s (red) and p (orange) polarizations during reflection from Al as a function of incident angle.
8.4.4.1Normal Incidence Reflectance
The normal incidence reflectance of metals from air with n = 1 is shown in Figure 8.31 as a function of complex refractive index. The reflectance always increases with increasing κ. It also increases with increasing n when n is larger than 1, or decreasing n when n is less than 1.
Figure 8.31Reflectance of metals at normal incidence as a function of complex refractive index.
The corresponding phase shift is shown in Figure 8.32. These phase shifts are mostly just below π, except when n is less than the air index 1 and κ is less than 2. When κ = 0, this phase change becomes the same as external reflection at dielectric surface, which is always π for n > 1.
Figure 8.32Phase change upon reflection for metals for normal incidence.
8.4.4.2Retardance and Diattenuation of Metal at Non-Normal Incidence
The retardance of metals at 30° incident angle is shown in Figure 8.33. The reflection retardance decreases with increasing κ. For n > 1 and κ > 1, retardance decreases with both n and κ.
Figure 8.33Retardance (in radians) for reflection from Al at a 30° incident angle. (Left) Shown in false color and 3D plots (left two) for 0 < n < 0.8 and 0 < κ < 0.8. The right two figures are for 1 < n < 4 and 1 < κ < 4.
The diattenuation reflecting off metals at 30° incident angle is shown in Figure 8.34. The diattenuation decreases to 0 as κ increases. When κ = 0, diattenuation decreases with increasing n.
Figure 8.34Diattenuation for reflection from metals at 30° incident angle with varying refractive index.
Figure 8.35 shows the polarization change for light incident on a metal polarized at 45° to the plane of incidence, reflecting from air to Al at various angles of incidence. An unambiguous discussion of the handedness of the reflectance from a metal surface is discussed by Swindell.9
Figure 8.3545° linearly polarized incident light reflects from an air–metal surface at various angles. Black arrowheads indicate incident light, and colored arrowheads indicate reflected light. The arrow pair at each incident angle has the same color.
8.5Approximate Representations of Fresnel Coefficients
The Fresnel coefficients are the first and simplest example of amplitude coefficients; later amplitude coefficients for multilayer films, anisotropic interfaces, and diffractive optical structures will be introduced. The functional forms of Equations 8.8 through 8.11 are complex and the coefficients are difficult to manipulate. Their behavior is not obvious from inspection characteristics of the Fresnel coefficients.
When applying the Fresnel equations, the goal is frequently to understand optical path length, amplitude, phase, diattenuation, and retardance and apply the equations to find suitable configurations of optical elements and coatings to achieve optical system specifications. This understanding is enhanced when the optical system properties can be expressed in terms of simply defined functions. It is frequently helpful to take complex equations, like the Fresnel equations, and replace them with approximate but simpler functions, usually polynomials, which, although approximate, maintain a high degree of accuracy. Then, by reasoning with these approximate functions, the source of aberration can be more clearly described, and methods for aberration compensation can be more easily constructed. Often, approximate functions can improve communication. Within optical design and analysis software, the exact equations will still be used, but for gaining an understanding of sources of polarization aberration, approximate functions can be enabling.
Two types of approximate functions will be used depending on the circumstances, (1) Taylor series and (2) function fits. A Taylor series represents a function in the neighborhood of a point as a series of polynomial terms calculated from the derivatives to the function. In contrast, a fit describes a function over a range as a sum of basis functions with weights adjusted to minimize the difference between the fit and the function.
8.5.1Taylor Series for the Fresnel Coefficients
The Taylor series of a function f(x) in the neighborhood of a point x0 is defined as
8.32The first two terms describe the tangent line through the point. The third term adds a quadratic component, and so on. For example, Figure 8.36 shows the convergence of linear, quadratic, and cubic fits to an rs Fresnel coefficient, each extending the range of accurate representation.
Figure 8.36Linear, quadratic, and cubic Taylor series fits about an angle of incidence of 45° to the rs Fresnel coefficient from air into n = 1.5, showing the increasing range of accurate representation with increasing order.
The Fresnel equations are even functions, symmetric about zero angle of incidence. Thus, the Fresnel coefficient Taylor series about normal incidence have only even terms. Relatively simple and useful closed form expressions for the quadratic coefficients are calculated as follows:
8.33
8.34Figure 8.37 compares the second-order Taylor series fits about normal incidence to the Fresnel amplitude coefficients. The deviations are of fourth order and the fits are quite accurate to 30°.
Figure 8.37Comparison of Fresnel amplitude coefficients for reflection and refraction with quadratic Taylor series fits. The fits are excellent to beyond θ = 30°.
8.6Conclusion
The Fresnel equations describe the polarization state changes on reflection and refraction. In optical systems, each beam of light is typically an aberrated spherical wave, and when it is incident on a surface, the angle of incidence and plane of incidence varies across the beam. Thus, because of the Fresnel equations, the magnitude and orientation of the diattenuation and retardance also vary across the beam. These variations are referred to as polarization aberrations, just as wavefront aberrations are variations of the optical path length and phase across beams and images. The polarization aberrations are divided into diattenuation aberrations and retardance aberrations. These polarization aberrations are generally not large and frequently have a minor or negligible effect on the optical system performance. But it is important to understand their genesis and effect.
In Chapter 12, several optical systems and their polarization aberrations will be analyzed. In each case, the optical system is described and a beam of light is chosen for analysis. The angles of incidence are calculated and used to produce maps of the diattenuation aberration and retardance aberration. The point spread functions are shown as Jones matrices for coherent light or Mueller matrices for incoherent light and the distribution of polarization is shown for sample input polarizations.
8.7Problem Sets
Find the relationship between refractive indices for which the intensity reflection coefficients at normal incidence equal the intensity transmission coefficients.
Show that if a beam is incident on a plane parallel plate from air at Brewster’s angle, it also strikes the rear surface at Brewster’s angle.
Show that for reflection in air from a dielectric medium, both Rs(θ) and Rp(θ) approach the same value as θ→90°. Find the slopes of Rs(θ) and Rp(θ) as θ→90°.
Verify the following relation between the Fresnel s and p amplitude coefficients for the angle of incidence
.Find the real refractive index n where at normal incidence from air tp = rp. What is the ratio of reflection and transmitted fluxes, Rp/Tp?
Find the real refractive index n where at normal incidence from air tp = κ rp, where κ is a real constant. What is the ratio of reflection and transmitted fluxes, Rp/Tp, when κ = 3?
Light refracts from air into a glass interface with refractive index
.Find Brewster’s angle.
At Brewster’s angle, what fraction of the p-polarized flux refracts?
At Brewster’s angle, is the reflected beam or the refracted beam 100% polarized?
What is the angle between the reflected and refracted beam at Brewster’s angle?
For light exiting this interface, glass into air, what is the value of the critical angle?
Design a variable diattenuator using a tilted parallel plate piece of glass with n = 2. A collimated beam is incident on a tilted glass plate and refracts into and then out of the plate. By tilting the glass plate, a variable diattenuation D is introduced. If unpolarized light is incident, the degree of polarization of the exiting light is equal to the plate’s diattenuation. Provide a lookup table for the inverse function θ(D), the angle of incidence as a function of the desired diattenuation. Ignore multiple reflections. Comment on possible implementation issues when this device is constructed.
Consider uncoated dielectric interface without absorption, that is, with real refractive index. Define weakly diattenuating as a diattenuation less than 0.05 and strongly diattenuating as a diattenuation greater than 0.5.
Over what range of refractive index ratios n = n2/n1 with 1/3 < n < 3, and what range of angles of incidence 0° < θ < 90° are dielectric interfaces weakly diattenuating in transmission?
Strongly diattenuating in reflection?
Weakly diattenuating in reflection?
Strongly reflecting in reflection?
Does an uncoated dielectric interface described by Fresnel equations have diattenuation?
Does an uncoated dielectric interface described by Fresnel equations have retardance?
What is the form of the diattenuation and retardance: linear, elliptical, or circular?
The complex refractive index of silver at three wavelengths is as follows:
400 nm
0.138 + 2.005i
710 nm
0.168 + 4.286i
1460 nm
0.374 + 9.449i
At 710 nm, plot the s and p intensity reflection coefficients for angle of incidence between 0° and 90°. At about what angle of incidence is the diattenuation greatest? Does the diattenuation increase toward longer wavelength or shorter wavelength?
At 710 nm, plot the s and p absolute phase changes on reflection for angle of incidence between 0° and 90°. Plot the retardance of angle of incidence between 0° and 90°. For small angles of incidence, the retardance is approximately δ(θ) = δ0 + δ2(θ). Determine δ0 and δ2.
Does the retardance at small angles increase toward longer wavelength or shorter wavelength?
How would one change the refractive index to maximize reflectance and minimize retardance (close to π). What types of materials have complex indices like this?
For a set of n points, an order n − 1 polynomial can be found that passes exactly through all the points. For example, consider five points (x0, f(x0)), (x1, f(x1)), (x2, f(x2)), (x3, f(x3)), and (x4, f(x4)), to be fit to the fourth-order polynomial f(x) = c0 + x c1 + x2c2 + x3c3 + x4c4. The points and coefficients can be related by the matrix equation:
Find the matrix coefficients mij in this equation to calculate the c’s from the set of f(x).
Show the matrix equation that calculates the polynomial coefficients c0, …c4.
Provide the equation for fitting three points to a quadratic equation yielding c0, c1, and c2. Provide all nine matrix elements as a function of (x1, f(x1)), (x2, f(x2)), and (x3, f(x3)).
Provide the equation for fitting three points to an even order fourth-order equation yielding c0, c2, and c4. Provide all nine matrix elements as functions of (x0, f(x0)), (x2, f(x2)), and (x4, f(x4)).
Fit the amplitude transmission coefficients ts and tp of an air/silicon (n = 4) interface for the angles 0°, 30°, and 45°.
For which component, ts or tp, is the fourth-order term more significant?
Perform the least squares polynomial fit with f(θ) = a0 + a2θ2 + a4θ4 to the amplitude coefficients for n1 = 1.0 and n2 = 1.5, as shown in Figure 8.7.
Find the Jones matrix for a real half wave retarder, fast axis at 45°, of sapphire at λ = 589 nm where nO = 1.76817 and nE = 1.76009. Chapter 5 presented the Jones matrices for ideal retarders; now consider a real retarder. Find the thickness t, then the absolute phases for the ordinary and extraordinary rays. Because of these optical path lengths, the retarder’s Jones matrix will not be in symmetric, fast axis unchanged, or slow axis unchanged form (Table 5.4). Evaluate the Fresnel equations at normal incidence for the ordinary and extraordinary modes and include the small resulting diattenuation in the Jones matrix.
References
1R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light, North-Holland, Elsevier Science (1987).
2M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition (1999), pp. 790–852.
3D. H. Goldstein, Polarized Light, 3rd edition, Boca Raton, FL: CRC Press (2010).
4J. A. Stratton, Electromagnetic Theory, New York: John Wiley & Sons (2007).
5W. B. Wolfe, Nondispersive prisms, The Handbook of Optics 2, Handbook of Optics, eds. M. Bass, E. Van Stryland, D. Williams, and W. Wolfe, New York: McGraw-Hill, 1996, 4–1 (1994).
6E. Hecht, Optics, Addison-Wesley (2002), pp. 113–122.
7M. Born and E. Wolf, Principles of Optics, Cambridge University Press (2003).
8K. M. Medicus, A. Fricke, J. E. Brodziak Jr., and A. D. Davies, The effect of phase change on reflection on optical measurements, in Proc. SPIE 5879, Recent Developments in Traceable Dimensional Measurements III, 587906 (2005).
9W. Swindell, Handedness of polarization after metallic reflection of linearly polarized light, J. Opt. Soc. Am. 61 (1971): 212–215.
1 These equations are presented in the decreasing phase convention used throughout the book, as described in Sections 2.17 and 2.18. In the increasing phase convention, the complex amplitude coefficient is the complex conjugate or ρeiϕ. Of course, these two conventions produce the equivalent electric field E.
2 This is the most common Fresnel equation sign convention.