13Thin Films

13.1Introduction

Optical thin films are ubiquitous in optical systems. The majority of refracting surfaces have antireflection coatings to reduce loss. Films are placed over metal mirrors to protect the metal surface and boost their reflectance. Many thin films are used to spectrally filter light, to change its spectral content, such as bandpass filters that block light outside a given spectral range. Beam splitter coatings are used to divide or combine wavefronts; beam splitter coatings are necessary in interferometers and many other applications. Many beam splitter coatings are designed to be non-polarizing, so as to not change the polarization state. Other beam splitter coatings are designed as polarizing beam splitters, to reflect one polarization state and transmit the orthogonal state.

All coatings affect the polarization state of incident light. Amplitude coefficients characterize the polarization properties of the coating; these amplitude coefficients for thin films relate electric fields across interfaces in the same way as the Fresnel coefficients of Chapter 12; they characterize the amplitude and phase changes of the s- and p-polarized components of the light.

This chapter reviews the optics of homogeneous and isotropic thin films. First, the reflectance and transmission of single-layer thin films are derived and studied. Then, the general algorithms for the optical properties of an arbitrary multilayer thin film are presented, and the polarization performance of several important families of coatings is studied from the perspective of the optical engineer and lens designer.

Surfaces with thin film coatings, such as antireflection coatings and beam splitter coatings, have amplitude reflection and transmission coefficients that are calculated by the application of the Fresnel coefficients to multilayer stacks of optical materials. Four types of interfaces are listed below:

• Homogeneous interface—Properties of the interface are constant over the clear aperture.

• Inhomogeneous interface—The composition changes in space or the coating thickness changes over the clear aperture.

• Isotropic interface—Refractive indices of all materials are the same in all directions and for all polarizations.

• Anisotropic interface—Birefringent or optically active materials are used and the refractive index depends on the polarization state. Anisotropy may arise from the use of birefringent materials, from strain, and also from coating microstructure, since many deposited coatings grow as arrays of microscopic pillars, which induces form birefringence.1,2 Also, normally isotropic materials become anisotropic when subjected to strong electric or magnetic fields.

13.2Single-Layer Thin Films

Consider a plane wave of light incident on a thin layer of refractive index n1 and thickness t on a substrate of refractive index n2, as shown in Figure 13.1. The optical film is characterized by the amplitude reflection and transmission coefficients of the s and p eigenpolarizations. The plane wave is represented by a light ray in the figure. As the ray propagates through the film, part of the flux reflects and refracts at each interface, with some of the light undergoing multiple reflections. Each path through the film is referred to as a partial wave. Each partial wave has its own amplitude and phase, and the partial waves will constructively or destructively interfere. Thus, the overall amplitude reflection and transmission are determined by the summation of all the partial reflected or transmitted waves. These transmissions and reflections are wavelength and incident angle dependent.

Figure 13.1A single layer of thin film coating with index n1 on a substrate with index n2. Multiple reflections within the film generate a set of reflected and transmitted partial waves.

The relative amplitude and phase carried by each partial wave is calculated by Fresnel equation and the phase thickness of the film. The individual complex amplitude coefficients of the ray and top/bottom interface interaction are given by Equations 13.4 through 13.11. The ray angles in each medium are obtained from Snell’s law (Equation 13.1). These angles can be rewritten as in Equations 13.2 and 13.3. Figure 13.2 shows the Fresnel coefficients’ subscripts at each interface: air→film (01), film→substrate (12), and film→air (10).

Figure 13.2Fresnel amplitude coefficients of each interface for reflections (left) and transmissions (right).

$$n_0\sin {\theta }_0=n_1\sin {\theta }_1=n_2\sin {\theta }_2,$$13.1

13.1

$$\cos {\theta }_1=\sqrt{1-\frac{n_0^2{\sin }^2{\theta }_0}{n_1^2}},$$13.2

13.2

$$\cos {\theta }_2=\sqrt{1-\frac{n_0^2{\sin }^2{\theta }_0}{n_2^2}},$$13.3

13.3

$$r_{01s}=\frac{n_0\cos {\theta }_0-n_1\cos {\theta }_1}{n_0\cos {\theta }_0+n_1\cos {\theta }_1}=-r_{10s},$$13.4

13.4

$$r_{01p}=\frac{n_1\cos {\theta }_0-n_0\cos {\theta }_1}{n_1\cos {\theta }_0+n_0\cos {\theta }_1}=-r_{10p},$$13.5

13.5

$$r_{12s}=\frac{n_1\cos {\theta }_1-n_2\cos {\theta }_2}{n_1\cos {\theta }_1+n_2\cos {\theta }_2},$$13.6

13.6

$$r_{12p}=\frac{n_2\cos {\theta }_1-n_1\cos {\theta }_2}{n_2\cos {\theta }_1+n_1\cos {\theta }_2},$$13.7

13.7

$$t_{01s}=\frac{2n_0\cos {\theta }_0}{n_0\cos {\theta }_0+n_1\cos {\theta }_1},$$13.8

13.8

$$t_{01p}=\frac{2n_0\cos {\theta }_0}{n_1\cos {\theta }_0+n_0\cos {\theta }_1},$$13.9

13.9

$$t_{12s}=\frac{2n_1\cos {\theta }_1}{n_1\cos {\theta }_1+n_2\cos {\theta }_2},$$13.10

13.10

$$t_{12p}=\frac{2n_1\cos {\theta }_1}{n_2\cos {\theta }_1+n_1\cos {\theta }_2},$$13.11

13.11

The phase thickness 2β is the phase difference between adjacent rays for both transmission and reflection (derived in the Appendix), where

$$\beta (n_0,n_1,{\theta }_0,d)=\frac{2\pi }{\lambda }{n}_{1}d\cos {\theta }_1.$$13.12

13.12

The s and p amplitude reflectance and transmittance coefficients for a single-layer coating calculated by the summation of all partial wave complex amplitudes (derived in the Appendix) are shown in Equations 13.13 through 13.16.

$$\begin{array}{l}{r}_s({\theta }_i,n_0,n_1,n_2,d)=\frac{r_{01s}+r_{12s}{e}^{i2\beta }}{1-r_{12s}{r}_{10s}{e}^{i2\beta }}\\ \phantom{r_s({\theta }_i,n_0,n_1,n_2,d)}=\left|r_s\right|e^{-i{\varphi }_{rs}}\end{array}$$13.13

13.13

$$\begin{array}{l}{r}_p({\theta }_i,n_0,n_1,n_2,d)=\frac{r_{01p}+r_{12p}{e}^{i2\beta }}{1-r_{12p}{r}_{10p}{e}^{i2\beta }}\\ \phantom{r_p({\theta }_i,n_0,n_1,n_2,d)}=\left|r_p\right|e^{-i{\varphi }_{rp}}\end{array}$$13.14

13.14

$$\begin{array}{l}{t}_s({\theta }_i,n_0,n_1,n_2,d)=\frac{t_{01s}{t}_{12s}}{1-r_{10s}{r}_{12s}{e}^{i2\beta }}\\ \phantom{t_s({\theta }_i,n_0,n_1,n_2,d)}=\left|t_s\right|e^{-i{\varphi }_{ts}}\end{array}$$13.15

13.15

$$\begin{array}{l}{t}_p({\theta }_i,n_0,n_1,n_2,d)=\frac{t_{01p}{t}_{12p}}{1-r_{10p}{r}_{12p}{e}^{i2\beta }}\\ \phantom{t_p({\theta }_i,n_0,n_1,n_2,d)}=\left|t_p\right|e^{-i{\varphi }_{tp}}\end{array}$$13.16

13.16

The absolute values indicate the fraction of the incident amplitude that takes each path. The arguments of the amplitude coefficients, ϕ, indicate the phase change between the incident and exiting beam. The phase is measured relative to the incident surface for reflection and the emerging surface for transmission.

These thin film coefficients (Equations 13.13 through 13.16) are the Fresnel-like amplitude coefficients that are used in the polarization ray tracing P matrix (Chapter 9) for each ray intercept during a polarization ray trace as in Equation 9.15,

$$P_t=O_{transmit}\left(\begin{array}{lll}{t}_s& 0& 0\\ 0& t_p& 0\\ 0& 0& 1\end{array}\right)O_{in}^{-1}\text{ and }{P}_r=O_{reflect}\left(\begin{array}{lll}{r}_s& 0& 0\\ 0& r_p& 0\\ 0& 0& 1\end{array}\right)O_{in}^{-1},$$13.17

13.17

for refraction and reflection.

13.2.1Antireflection Coatings

Antireflection coatings are designed to improve the transmission of optical elements. A significant side benefit is that antireflection coatings almost always reduce polarization aberration as well. The most common antireflection coating is a quarter wave thick (optical path length n t) layer of a low-index material. For a quarter wave single-layer coating n1d = λ/4, at normal incidence θ1 = θ2 = θ3 = 0°, the reflectance is

$$R={\left(\frac{n_0{n}_2-n_1^2}{n_0{n}_2+n_1^2}\right)}^2$$13.18

13.18

and

$$R=0\text{ when }{n}_1=\sqrt{n_0{n}_2}.$$13.19

13.19

With a bare glass of index 1.5 in air, zero reflectance is obtained with a quarter wave coating of index $\sqrt{1.5}=1.225$. Thin film coating materials with refractive indices as low as 1.225 are not readily available. The most common low-index coating material is magnesium fluoride, MgF2. A single-layer antireflection coating is commonly made of a quarter wave of MgF2; reflectance R = 1.4% on glass. The s and p intensity coefficients, diattenuation, and retardance are compared with the uncoated surface in the following.

The intensity transmission of MgF2 film on glass as a function of the film thickness, shown in Figure 13.3. Has maxima at quarter and three-quarter wave thicknesses.

Figure 13.3Normal incidence transmission through MgF2 coating on an n = 1.5 substrate as a function of the coating’s optical thickness (t/n).

With a quarter wave MgF2 film, the coated glass has over 98% transmission and less than 2% reflection at normal incidence, as shown in Figure 13.4. The transmission diattenuation decreases and the reflection diattenuation increases slightly, as shown in Figure 13.5, because of this quarter wave coating. The phase change on transmission for the s- and p-components are almost identical and are quadratic for small angles, as shown in Figure 13.6, indicating a very small contribution to the overall power of the lens incorporating this coating (Section 13.4). Both transmission and reflection retardance increases quadratically with angle, as shown in Figure 13.7.

Figure 13.4s and p (left) transmission and (right) reflection intensity coefficients of the λ/4 MgF2 coating on glass at wavelength 0.55 μm.

Figure 13.5(Left) Transmission and (right) reflection diattenuation of the λ/4 MgF2 coating on glass at wavelength 0.55 μm.

Figure 13.6Phase change on transmission for the s- and p-components of the λ/4 MgF2 coating at wavelength 0.55 μm.

Figure 13.7(Left) Transmission and (right) reflection retardance in degrees of the λ/4 MgF2 coating on glass at wavelength 0.55 μm are both negligible.

13.2.2Ideal Single-Layer Antireflection Coating

The ideal single-layer antireflection coating, where the layer index is the geometric mean of the incident and substrate indices (Equations 13.19), has remarkably low diattenuation and retardance. Figure 13.8 shows the performance of a quarter wave of MgF2 on a substrate with a refractive index equal to the MgF2 index squared. On the left, the diattenuation is compared to the uncoated diattenuation as a function of angle of index. The diattenuation remains almost on the x-axis from zero to beyond θ = 30°. As shown by Azzam,3 the variation of diattenuation with angle of incidence is sixth order; thus, there is no quadratic or fourth-order variation in the diattenuation with angle. Similarly on the right, the retardance remains remarkably low with angle, being of fourth order; the retardance has no quadratic variation for this index and thickness.

Figure 13.8(Left) Diattenuation and (right) retardance of the quarter wave MgF2 coating (t = λ/(4n1)) on a substrate with refractive index $n_{sub}=n_{{\text{MgF}}_2}^2=1.9044$, n0 = 1 and n1 = 1.38.

13.2.3Metal Beam Splitters

Beam splitters divide the amplitude of incident light into transmitted and reflected partial waves. Here, the single-layer equations analyze a single film of aluminum on a glass substrate at normal incidence and at a 45° incident angle. The performance of aluminum beam splitters operating at an angle of incidence of 45° will be explored to determine how close the performance can come to ideal non-polarizing behavior.

At normal incidence, reflectance and transmission are approximately equal for a ~4.2-nm-thick aluminum film as seen in Figure 13.9.

Figure 13.9(Left) Intensity transmission (orange) and reflection (blue), and (right) phase transmission and reflection for normal incidence as a function of thickness for an aluminum film (n1 = 1.2 + 7.26i) on glass (n2 = 1.5) at 0.6 μm wavelength.

The intensity and phase for transmission and reflection of an aluminum film at a 45° incident angle are shown in Figure 13.10. The s- and p-transmission coefficients and the s- and p-reflection coefficients never cross, demonstrating that a non-polarizing beam splitter is not possible with a single aluminum layer. However, the four coefficients are very close to each other at a ~3.6-nm-thick aluminum film.

Figure 13.10(Left) Intensity transmission and reflection for aluminum film (n1 = 1.2 + 7.26i) on glass substrate (n2 = 1.5) for 45° incident angle at 0.6 μm. (Right) Phase changes for transmission and reflection for the aluminum film on glass substrate.

13.3Multilayer Thin Films

We now consider the calculation of the reflectance and transmission of a multilayer thin film coating. The objective is the determination of the amplitude coefficients, rs, rp, ts, and tp as functions of wavelength λ and angle of incidence θ. These amplitude coefficients are needed to construct the P matrix for polarization ray tracing.

Consider a homogeneous and isotropic thin film consisting of a set of Q parallel layers. The incident medium is assumed transparent. The coating thicknesses can be formed into a vector of thicknesses T,

$$T=\left(t_1,t_2,\dots ,t_q,\dots ,t_Q\right).$$13.20

13.20

These thicknesses are in unit of length, such as millimeters, and are thus called metric thicknesses to distinguish them from the optical thicknesses specified in waves. The calculations will be performed one wavelength at a time; thus, at each wavelength, the refractive indices of each layer can also be arranged into a vector Λ,

$$\Lambda =(n_1,n_2,\dots ,n_q,\dots ,n_Q).$$13.21

13.21

Specification of the coating includes the thickness of each layer and their complex refractive indices.

Figure 13.11 shows an eight-layer film stack, and each layer has a thickness and index associated to it. For the coating on curved substrates, the film will be treated as having locally flat layers parallel to the tangent plane to the substrate.

Figure 13.11Schematic of an eight-layer multilayer thin film stack on a substrate. Each layer is specified by its thickness t and refractive index n.

13.3.1Algorithms

The amplitude coefficients for homogeneous and isotropic multilayer coatings can be calculated by a straightforward matrix method first introduced by Abelès.4,5 This algorithm is systematized by the characterized matrix for transparent substrates. This algorithm is in widespread use and is at the core of most commercial thin film simulation software.6

The algorithm uses characteristic matrices specified for each film layer to calculate the transmission and reflections of the multilayer film assembly. For Q layers of films,

$$\left(\begin{array}{l}B\\ C\end{array}\right)=\left[{\displaystyle \prod _{q=1}^Q\left(\begin{array}{ll}\cos {\beta }_q& i\sin {\beta }_q\text{/}{\eta }_q\\ i{\eta }_q\sin {\beta }_q& \cos {\beta }_q\end{array}\right)}\right]\left(\begin{array}{l}1\\ {\eta }_m\end{array}\right),$$13.22

13.22

where the phase thickness of layer q is

$${\beta }_q=\frac{2\pi n_q{d}_q\cos {\theta }_q}{\lambda },$$13.23

13.23

with complex refractive index ƞqfor layer q. ƞm is the characteristic admittance of the substrate. The s- and p-polarizations for the TE and TM components have different characteristic matrices and are calculated separately with different characteristic admittances:

$${\eta }_{s,q}=\sqrt{\frac{{\epsilon }_o}{{\mu }_o}}{n}_q\cos {\theta }_q\text{ and }{\eta }_{p,q}=\sqrt{\frac{{\epsilon }_o}{{\mu }_o}}\frac{n_q}{\cos {\theta }_q}.$$13.24

13.24

The overall reflection and transmission is calculated by the matrix product Equations 13.22 for each layer, which operates on the substrate vector. The amplitude reflection coefficient is

$$r=\frac{{\eta }_0-Y}{{\eta }_0+Y}=\frac{{\eta }_{0}B-C}{{\eta }_{0}B+C}$$13.25

13.25

where surface admittance $Y=\frac{C}{B}$, and the phase change upon reflection is

$${\varphi }_r=-\arctan \left[\frac{\mathrm{Im}\left[{\eta }_m(BC*-CB*)\right]}{{\eta }_m^{2}BB*-CC*}\right].$$13.26

13.26

The complex transmission coefficient is

$$t={\left(\frac{2{\eta }_0}{{\eta }_{0}B+C}\right)}^{*}$$13.27

13.27

and the corresponding phase change of transmission measured relative to the emerging surface is

$${\varphi }_t=-\arctan \left[\frac{-\mathrm{Im}\left[{\eta }_{o}B+C\right]}{\mathrm{Re}\left[{\eta }_{o}B+C\right]}\right].$$13.28

13.28

The intensity coefficient is the ratio of the exiting to the incident flux. The reflection and transmission intensity are

$$R=\left(\frac{{\eta }_{0}B-C}{{\eta }_{0}B+C}\right){\left(\frac{{\eta }_{0}B-C}{{\eta }_{0}B+C}\right)}^{*}\text{ and }T=\frac{4{\eta }_0\mathrm{Re}[{\eta }_m]}{({\eta }_{0}B+C)({\eta }_{0}B+C)*},$$13.29

13.29

respectively. Absorption is zero if all of the film’s refractive indices are real; otherwise, the absorption is

$$A=\frac{4{\eta }_0\mathrm{Re}[BC*-{\eta }_m]}{({\eta }_{0}B+C)({\eta }_{0}B+C)*}.$$13.30

13.30

13.3.2Quarter and Half Wave Films

Some simple relationships can be used in thin film design when the film thickness is a multiple of a quarter wave; these thicknesses have the maximum and minimum effect on reflection and transmission. When the dielectric lossless film is half wave thick with β = mπ/4, where m = 0, 2, 4, …, its characteristic matrix becomes

$$\pm \left(\begin{array}{ll}1& 0\\ 0& 1\end{array}\right).$$13.31

13.31

This absentee layer has no effect on the reflectance and transmittance for light at the wavelength for which the film is half wave. A film is quarter wave thick when m = 1, 3, 5, … and the corresponding characteristic matrix is

$$\pm \left(\begin{array}{ll}0& i\text{/}\eta \\ i\eta & 0\end{array}\right).$$13.32

13.32

The surface admittance Y becomes

$$\frac{{\eta }_1^2{\eta }_3^2{\eta }_5^2\dots }{{\eta }_2^2{\eta }_4^2\dots {\eta }_m^2}\text{ or }\frac{{\eta }_2^2{\eta }_4^2\dots {\eta }_m^2}{{\eta }_1^2{\eta }_3^2{\eta }_5^2\dots }$$13.33

13.33

for an odd or even number of quarter wave films. Many common film designs use quarter wave layers, which are labeled as

• L low-index quarter wave layer

• M medium-index quarter wave layer

• H high-index quarter wave layer

For a low-index quarter wave layer, the partial wave that reflects from the first surface of the layer and the partial wave that reflects from the second surface are 180° out of phase, minimizing reflection. Some example quarter wave designs are the following:

• Air L Glass quarter wave antireflection coating

• Air HLHLHLHL Glass alternating eight-layer coating

13.3.3Reflection-Enhancing Coatings

Coatings with high reflectance can be readily fabricated from alternating high- and low-index layers of quarter wave thickness. These reflective coatings provide an alternative to metal coatings and can provide much higher reflectance; however, the spectral bandwidth and polarization properties can be quite different. Reflection-enhancing coatings provide a good example of how to interpret the results of thin film calculations to interpret the effects of coatings on wavefront and polarization aberrations.

Figure 13.12 (left) shows a schematic of a two-layer dielectric multilayer coating with an alternating high- and low-index layer on an n-BK7 substrate. With our coating shorthand notation, the coating is Air HL Glass, a quarter wave of high index on a quarter wave of low-index material. For this example, hafnium oxide, HfO2 is chosen as the high index material (nH = 1.94) and MgF2 is the low-index material (nL = 1.39).

Figure 13.12(Left) Depiction of the family of partial waves reflecting from an Air HL Glass coating. For the Air HL Glass coating at normal incidence, all the partial waves exit the top surface in phase, enhancing reflectivity. (Right) Reflectance from (HL)N coatings increases rapidly with the number of layer pairs exceeding 99% with nine layer pairs.

In the quarter wave antireflection coating (Air L Glass), the partial waves reflecting from the air-to-MgF2 interface and the partial wave reflecting from the MgF2-to-glass interface are π out of phase after exiting the coating, thus suppressing reflection. For the Air HL Glass coating at normal incidence, the partial waves from the Air–H, H–L, and L–H interfaces, shown in Figure 13.12 (left), are all in phase after reflecting and exiting the top of the coating, thus enhancing reflection, but because the amplitude coefficients are small, only ~25% of the light is reflected from Air HL Glass. This structure is now used as a building block to build high reflection coatings, because as HL layer pairs are added, all the reflected partial waves remain in phase.

Figure 13.12 (right) shows the increasing normal incidence reflection at λ0 = 0.55 μm as the number of HL layers increases. With a four-layer coating (Air HLHL Glass), reflectance increases to 50%. By the time the number of layers increases to 10 alternating layers of HfO2 and MgF2 (Air HLHLHLHLHL Glass), the normal incidence reflection is about 95%, and at 18 alternating layers, reflectivity is over 99%.

This sounds excellent! Why would metal reflectors be used if such high reflectance is available? This high reflectivity is only obtained over a limited range of wavelengths and angles, and outside this range, the reflectivity can be much worse than a metal mirror. Designs with large numbers of layers and high reflectivity have rapid variation of amplitude coefficients with angle and wavelength; hence, this coating family provides a good example of how to interpret the thin film coating calculation output to understand the wavefront aberrations and polarization aberrations that coatings can cause.

For the 10-layer HfO2 and MgF2 reflective coating (Air HLHLHLHLHL Glass) at λ0, the reflectivity is shown in Figure 13.13.

Figure 13.13The reflectivity of a 10-layer reflection-enhancing coating (Air HLHLHLHLHL Glass) at 0.55 μm has a normal incidence reflectivity of 94%, a little bit higher than aluminum (~92%). Some light transmits into the glass substrate (bottom, Ts, Tp).

For 18 alternating layers of HfO2 and MgF2 (Air HLHLHLHLHLHLHLHLHL Glass), the resultant reflection is almost 100% from 0° to ~30° incidence, as shown in Figure 13.14 (left). If the mirror is used beyond 40°, the reflectivity rapidly drops and the coating becomes highly diattenuating. The corresponding phase of the s- and p-light is shown in Figure 13.14 (right). The phases change much more rapidly than a metal film with angle. The difference, ϕp − ϕs, is the retardance, which increases rapidly as the reflectivities change, with the p-component undergoing 180° of phase change in about 50°. The average phase change (ϕp + ϕs)/2 is plotted in magenta, and should be considered as a contribution to the wavefront aberration of any system using this coating. At the edge of the high reflectivity zone, 40° angle of incidence, the average phase has changed by 0.5 radians or ~ λ/12. As is discussed later, since the average phase is changing nearly quadratically, this coating’s quadratic average phase change is a source of defocus and chromatic aberration when the angle of incidence changes across the coating, such as (1) a collimated beam at a spherical mirror or (2) a converging or diverging beam at a flat mirror.

Figure 13.14(Left) Intensity reflectance and (right) phase change on reflection of an 18-layer reflection-enhancing coating versus angle of incidence at 0.55 μm. The s and p light are reflecting from different effective depths.

Figure 13.15 shows the behavior of this 18-layer reflection-enhancing coating design versus wavelength. Note that this film behaves like a bandpass filters with 0.1 μm bandwidth (0.5–0.6 μm) from 0° to 30°. Thus, the excellent reflectance at the design wavelength comes at a cost of reduced spectral bandwidth. As the angle of incidence increases, the width of the p-bandpass reduces faster than the s-bandpass; thus, there is a large diattenuation at the edges of the bandpass. The phases, the coating-induced defocus, and chromatic aberration introduced by this coating are explored further in Section 13.4.

Figure 13.15Intensity reflectance of the 18-layer reflection-enhancing coating versus wavelength for 0° and 30°. Note the “blue shift” of the spectral bandpass to shorter wavelengths as the angle of incidence changes.

For these many-layer reflection coatings, the reflectance is small at each interface and the light penetrates deep into the coating. A large number of partial waves have comparable amplitude, for example, all the singly reflected beams, or all the triply reflected beams, and so on. There is an effective depth where the average light reflects. For the highest reflecting coatings, such as the 18-layer coating, almost no light reaches the bottom of the coating. The result of this effective depth for off-axis rays can be a noticeable offset deff between an incident and reflected ray, as shown in Figure 13.16. The phase difference between s- and p-phases is an indication that the s-polarized component with its higher amplitude reflectance reflects closer to the surface on the average while the p-component reflects from deeper in the coating on the average.

Figure 13.16Light reflecting from a multilayer film can be given an effective depth of zeff based on the mean depth the partial rays reflect from the structure. The exiting ray has a mean offset deff from the incident ray.

13.3.4Polarizing Beam Splitters

An ideal polarizing beam splitter (PBS) divides incident light into its s- and p-components, transmitting the p-component and reflecting the s-component. Real beam splitters approach this ideal but are limited by the laws of multilayer coatings as applied to real materials. Thus, polarizing beam splitters have a limited range of angle of incidence and wavelength over which they are effective, that is, where they exceed an extinction ratio specification such as 1000:1. When illuminated with spherical waves, these behaviors show up as polarization aberrations due to the variation of angle of incidence. The perfect polarizing beam splitter coating could be considered as one that is an ideal antireflection coating for the p-component and an ideal reflector for the s-component. It is difficult to find polarizing beam-splitting coating designs that simultaneously have a high extinction ratio, a broad wavelength range, and a large range of angles of incidence.

The light flux through a polarizing beam splitter can be considered in three classes:7

1. Light that takes the correct path in the correct polarization state

2. Light that takes the incorrect path

3. Light that takes the correct path but ends up in the incorrect polarization state

One approach for PBS coating design is to make a thin film version of the pile-of-plates polarizer, operating near Brewster’s angle. At Brewster’s angle, all of the p-polarized component is transmitted and some of the s-polarized component is reflected. By stacking a group of plates, that is, thin film interfaces near Brewster’s angle, the overall reflectance of the s-component is increased. For the pile-of-plates polarizer, the reflected beam is purely s-polarized and thus has a high degree of polarization. Some of the s-polarized light leaks into the transmitted beam; hence, it has a lower degree of polarization and extinction ratio. Thus, it is generally the case for a PBS that the reflected beam has the higher extinction ratio and diattenuation while the transmitted beam is of lower quality. Thus, when using a PBS as a polarizer, the reflected beam is preferred.

Many optical systems prefer PBS with angles of incidence around 45°, so the transmitted and reflected beams exit orthogonal to each other. The 45° PBS are called polarization beam splitting cubes, shown in Figure 13.17. The polarizing beam-splitting coating is fabricated on the hypotenuse of a right prism of high-index glass, which is then glued to another right prism. The faces are usually antireflection coated to increase overall throughput.

Figure 13.17Schematic of a polarization beam splitting cube (left) showing s-polarized light reflecting and p-polarized light transmitting. (Right) One of the two prisms is coated and cemented to the other prism.

To best exploit Brewster’s angle, many PBS coating designs operate at angles of incidence around 55°–60°, which provides easier designs with larger angular and wavelength bandwidths. The literature contains many designs for PBS coatings for various wavelengths and angles. An early theory for PBS coatings was developed by MacNeille.8 An example of the MacNeille beam splitter is analyzed on SF5 glass at 550 nm to understand the type of behaviors expected and show how the performance transforms into polarization aberrations. This coating alternates high- and low-index layers of ZnS and MgF2 films with thicknesses:

$$t_{ZnS}=\frac{\lambda }{4n_{ZnS}\sqrt{n_{ZnS}^2+n_{Mg{F}_2}^2}}\text{ and }{t}_{Mg{F}_2}=\frac{\lambda }{4n_{Mg{F}_2}\sqrt{n_{ZnS}^2+n_{Mg{F}_2}^2}}.$$13.34

13.34

Figure 13.18 (left) shows the performance as a function of the number of layers. Tp is weakly dependent on the number of layers since it is the transmitted state at Brewster’s angle. The s-reflectance steadily increases from 1 pair to 10 layer pairs and then oscillates as additional layer pairs are added. With this design, high p-transmission and high s-reflection are obtained between 0.55 and 0.7 μm.

Figure 13.18(Left) Reflected and transmitted intensity of the MacNeille coating as the number of layers increases, optimized at 0.55 μm. (Right) Reflected and transmitted intensity of the 23-layer MacNeille coating. The coating thickness is calculated for wavelength 0.55 μm.

The performance of a 37-layer MacNeille coating optimized for 550 nm is shown in Figure 13.19 for a short wavelength, the optimized wavelength, and a long wavelength. Only the desired s-reflected and p-transmitted beams are shown. At the design wavelength of 550 nm, both p-transmission and s-reflection are high (~100%) and steady between 41° and 47°, providing good polarizing beam-splitting performance over a 6° angular range. At 440 nm, the s-reflection is below 0.2; hence, the majority of both beams are transmitted and there is no polarizing beam-splitting function. At 730 nm, there is a beam-splitting function below 46° and the s-reflectance is high, but the p-transmittance oscillates with wavelength. The remaining p-light is reflected; thus, the quality and the degree of polarization of the reflected beam are poor.

Figure 13.19Intensity and phase of a 37-layer MacNeille coating as a function of incident angle in glass at wavelength 440 nm (top row), 550 nm (middle row), and 730 nm (bottom row). Coating thickness is optimized for 550 nm. (Right) Vertical lines indicate 2π phase discontinuities that occur due to the arctangents in Equations 13.26 and 13.28.

Math Tip 13.1Polynomial Curve Fitting

The equations for multilayer film amplitude coefficients are complicated enough to make analytical manipulation nearly impossible. For the purposes of understanding aberrations and estimating the effects on image formation, treating the coefficients as linear, quadratic, or cubic polynomials provides tremendous insight, facilitating interfacing coating performance with the description of aberrations in a series of orders, second-order wavefront aberrations, fourth order, and so on. Thus, fitting the amplitude coefficients to polynomials provides forms that are easily manipulated and dovetail with aberration theory. The Taylor series approximations (Section 8.5.1) provide an algorithm to calculate polynomials with a close fit in the neighborhood of a point by matching the value, first derivative, second derivative, and so on, through a given order. These Taylor series fits diverge in value, moving away from that point, typically as the next-order polynomial in the series. Thus, a quadratic fit will typically diverge cubically from the function it is fitting, and so forth.

For many purposes, it is better to fit a function over the entire range of interest by using a least square fit instead of matching derivatives with a Taylor series. The square root of the square of the difference (RMS root mean square) between the approximate and exact functions is a common metric for the fit, and the coefficients are calculated to minimize the RMS difference. Let g(x) be an approximation to the exact function f(x). The RMS is calculated on a continuous basis as

$$RMS=\frac{1}{x_2-x_1}\sqrt{{\displaystyle \underset{x_1}{\overset{x_2}{\int }}{\left(f(x)-g(x)\right)}^{2}dx}}.$$13.35

13.35

Similarly, the RMS can be calculated for a discrete set of N point as

$$RMS=\frac{1}{x_N-x_1}\sqrt{{\displaystyle \sum _{n=1}^N{\left(f(x_n)-g(x_n)\right)}^2}}.$$13.36

13.36

The advantage of including the square root in the metric is that the units and scale of the RMS correspond to the function being fit.

Many different functions are used for curve fitting. Sines and cosines are the basis for Fourier series fits. Many forms of polynomials such as Legendre and Chebyshev polynomials provide excellent basis functions for certain problems. Here, our concern is to perform simple polynomial fitting to provide a convenient representation for Fresnel and other thin film functions. It is desired to find the polynomial equation of order N

$$f(x)=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+\dots +a_N{x}^N={\displaystyle \sum _{n=0}^N{a}_n{x}^n},$$13.37

13.37

which passes closest to a set of M + 1 data points dx1, dx2, …, dxm, …, dxM. In general, if M + 1 = N, then a polynomial can be found that passes exactly through the set of data points. If M + 1 > N, the equation that passes closest in the least square sense of minimizing the square of the errors is sought.

$$∆(a_0,a_1,a_2,\dots ,a_N)={\displaystyle \sum _{n=0}^N{\left[f(x_m)-d{x}_m\right]}^2}.$$13.38

13.38

If M + 1 = N, the calculation of polynomial coefficients can be formulated as the matrix equation

$$\left(\begin{array}{lllll}1& x_1& x_1^2& \cdots & x_1^N\\ 1& x_2& x_2^2& & x_2^N\\ 1& x_3& x_3^2& & x_3^N\\ ⋮& & & \ddots & \\ 1& x_N& x_N^2& & x_N^N\end{array}\right)\left(\begin{array}{l}{a}_0\\ a_1\\ a_2\\ ⋮\\ a_N\end{array}\right)=\left(\begin{array}{l}{d}_0\\ d_1\\ d_2\\ ⋮\\ d_N\end{array}\right)=X\stackrel{\rightharpoonup }{a}=d,$$13.39

13.39

where each row vector product evaluates f(x) at one point. For example, for a cubic equation fit to four data points,

$$\left(\begin{array}{llll}1& x_1& x_1^2& x_1^3\\ 1& x_2& x_2^2& x_3^3\\ 1& x_3& x_3^2& x_3^3\\ 1& x_4& x_4^2& x_4^3\end{array}\right)\left(\begin{array}{l}{a}_0\\ a_1\\ a_2\\ a_3\end{array}\right)=\left(\begin{array}{l}{d}_0\\ d_1\\ d_2\\ d_3\end{array}\right)=X\stackrel{\rightharpoonup }{a}.$$13.40

13.40

The solution is found by operating on the data with the matrix inverse of X,

$$X^{-1}\overrightarrow{d}=\stackrel{\rightharpoonup }{a}={(\begin{array}{llll}{a}_0& a_1& a_2& a_3\end{array})}^T,$$13.41

13.41

yielding the polynomial coefficients. Provided all the xn are unique, the solution is exact and unique and passes through all the data. If M + 1 > N, the pseudoinverse$X_p^{-1}$:

$$X_p^{-1}={(X^T\cdot X)}^{-1}\cdot X^T$$13.42

13.42

provides the least squares best-fit solution. Figure 13.20 shows an example of fits of the form

$$f(\theta )=a_0+a_2{\theta }^2+a_4{\theta }^4$$13.43

13.43

Figure 13.20Comparison between Fresnel amplitude coefficients and the fourth-order polynomial fits over the range 0° ≤ θ ≤ 90°.

to the four Fresnel amplitude coefficients. The coefficients a0, a2, and a4 have been adjusted to minimize the area between the amplitude coefficients and the fit curves. All of these fits cross the exact functions three times. Unlike the Taylor series, the values do not match at the origin. Polynomial fits often provide superior approximations over Taylor series, but the method should be chosen to fit the problem.

13.4Contributions to Wavefront Aberrations

Thin films contribute to the wavefront aberration, apodization, and polarization aberrations of optical systems. Evaluating these contributions has been one of the main rationales for adding polarization ray tracing capabilities to optical design programs. Consider Figure 13.21, which shows a lens with films drawn in purple on the entering and exiting surfaces, with thicknesses exaggerated for clarity. Note how the films can be considered as meniscus lenses attached to the main lens. Equal radii meniscus lenses have small but non-zero power from the lensmaker’s equation. The equations of paraxial optics could be used to calculate the effect of such thin lenses on the focal length and the longitudinal chromatic aberration of the lens. Would such an algorithm yield the proper correction to the focal length, particularly considering the multiple reflections that occur within the films? A better approach is to consider the lens, the two films, and their paraxial behavior as quadratic phase changes, which can be expressed as

Figure 13.21The thin films (purple) on a glass lens, shown here with exaggerated thickness, act as very weak lenses in combination with the glass lens.

$$\Phi (\theta )\approx {\Phi }_2{\theta }^2.$$13.44

13.44

Φ2 characterizes the quadratic phase change Φ in radians per radian of angle of incidence θ.

Next, consider the defocus and chromatic aberration from a reflecting film, the 18-layer reflection-enhancing film described in Section 13.3.3 and graphed in Figure 13.14. To see the defocus contribution at the reference wavelength of 550 nm, Figure 13.22 shows the s- and p-phases with the average phase shifts (purple) with a quadratic fit to the average phase for small angles (green) where the average phase is seen to be well fit to a quadratic for angles less than 30°. This quadratic contribution indicates the defocus wavefront aberration the coating will contribute when coated on an on-axis mirror and illuminated on-axis. The magnitude of the defocus will depend on the quadratic evaluated for the marginal ray angle of incidence. This defocus varies with wavelength and thus introduces a small longitudinal chromatic aberration. In the regions of the spectrum (Figure 13.15, black) where the reflectance is low or changing rapidly, such as from 475 to 490 nm, the constructive and destructive interference between the partial waves changes rapidly with angle, and the phase changes and thus coating-induced aberrations tend to be larger. Figure 13.23 shows the s, p, and average phase shifts for small angles (0° to 30°) over this spectral range. At 475 nm, the reflectance is very low and the average quadratic phase variation (magenta) is negative. By 480 nm, the average quadratic phase is positive and large, and decreases from 480 to 500 nm, which is in the high reflectance bandpass. The higher-order residual at 475 nm is a contribution to the spherical aberration and higher-order aberrations from the coating. Figure 13.24 plots the average phase for 0, 0.2, and 0.4 radians as a function of wavelength; the closer the curves, the less defocus will be introduced.

Figure 13.22For the 18-layer enhanced reflection coating at its reference wavelength of 550 nm, the s-phase (orange) and p-phase change (red) are approximately quadratic and nearly equal for small angles. The average phase (purple) is well fit by a quadratic (green) for angles less than 30°, indicating that the coating will contribute defocus when illuminated on-axis on a radially symmetric mirror.

Figure 13.23The s (orange), p (red), and average phase change (purple) for the spectral region from 475 to 500 nm, where the reflectivity is changing rapidly. From 480 to 500 nm, the nearly quadratic variation is seen to steadily decrease. This change causes chromatic aberration. At 475 nm where the reflectivity is very low, the phase variation has significant higher-order terms.

Figure 13.24(Left) Intensity reflectance of the 18-layer enhanced reflection coating. (Right) The average phase change on reflection versus wavelength at 0 (purple), 0.2 (blue), and 0.4 (green) radians. The 0 and 0.2 curves are nearly equal. Where the three curves intersect, such as near 520 and 540 nm, the coating’s defocus contribution is zero.

Simple algorithms can evaluate the quadratic phase magnitude coefficient. For example, using finite differences to evaluate the s- and p-phases at 0° and θ0, a small angle of incidence, the quadratic phase change coefficient, Φ2 (Equation 13.44), is approximated as

$${\Phi }_2=\left[\frac{{\varphi }_p({\theta }_0)+{\varphi }_s({\theta }_0)}{2}-\varphi (0)\right]/{\theta }_0^2.$$13.45

13.45

The variation of this defocus contribution is plotted in Figure 13.25 using θ0 = 0.1 radians. The corresponding longitudinal chromatic aberration is seen to be highly wavelength dependent. Thus, Figure 13.24 (right) contains information for the calculation of both the retardance aberration contribution from the thin film and the wavefront aberration contribution.

Figure 13.25Variation of the quadratic phase coefficient, Φ2, for the 18-layer reflection-enhancing coating of Figure 13.15 indicates the relative amount of defocus at different wavelengths, that is, chromatic aberration. Units are radians of defocus wavefront aberration per radian of marginal ray angle at the coating.

Thus, it is seen how contributions of thin films to lens power and wavefront aberration can be calculated from their amplitude coefficients. Usually, the contribution of the thin films to the lens power is very small. There are circumstances where these contributions should be checked, particularly where large numbers of partial waves are significant, or if coatings are used in angular or spectral regions where large oscillations in amplitude occur.

13.5Phase Discontinuities

Abrupt jumps in the phase of amplitude coefficients are another issue with thin film calculations and their interpretation. The phases of the complex amplitude coefficients are generally returned within the range of –π to π by the arctan function of Equation 13.26. Physically, as the angle of incidence or wavelength varies, the phases can vary over many waves. Thus, in phase plots, particularly of thick films, one or more 2π phase discontinuities may occur in the thin film program output but the discontinuities are not real. The phase discontinuities for the quarter wave MgF2 as antireflection coating are plotted in Figure 13.26 where phase discontinuities occur, retardance discontinuities must also occur. Figure 13.24 shows phase discontinuities around 53°.

Figure 13.26The phase changes on transmission through the antireflection coating example show phase discontinuities along a band from 700 nm at grazing incidence to 550 nm at normal incidence. In this case, the s-(left) and p-(right) phase discontinuities are very close together.

These phase discontinuities can prove troublesome for an analysis, such as calculating absolute phase or wavefront aberration; a phase unwrapping operation can be performed separately on the s-phase and p-phase. The image formation calculations for the point spread function and optical transfer function described in Chapter 16 require complex valued amplitude and wavefront maps from the exit pupil to be Fourier transformed. In this Fourier transformation operation, the input is an array of complex values in Cartesian form, z = x + i y. Fortunately, this Fourier transform operation is not affected by 2π phase jumps in the exit pupil function; thus, the unwrapping of phase values is not needed for this important operation.

Interesting phase unwrapping issues regarding thick retarders, similar to the issues of this section, are also a frequent issue, and are treated in Chapter 26.

Example 13.1Phase Discontinuities in Overcoated Gold

Phase discontinuities in amplitude coefficient calculations are a larger problem in thicker coatings, since the optical path length of the light varies more with wavelength and angle of incidence. As an example, consider a thick protective dielectric overcoating on gold. Gold is an excellent reflector in the infrared. However, since gold is so soft, it cannot be easily cleaned. Thus, gold is frequently overcoated with a hard transparent dielectric to provide a protective and cleanable overcoating. Aluminum oxide, as in sapphire, is a preferred choice. Figure 13.27 (top row) plots the s-phase (left) and p-phase (right) as a function of angle of incidence (0° < θ < 90°) and thickness (0 < t < 0.250 μm) and periodic phase changes are seen as the overcoat thickness increases. A phase unwrapping algorithm can unwrap the phase beyond ±π, as shown in Chapter 26.

Figure 13.27(Top row) The phase changes for aluminum oxide Al2O3 overcoated on gold shows periodic phase jump artifacts, phase discontinuities of 2π, as a function of thickness. (Bottom row) s-phase (red) and p-phase (orange) at 30° incident angle. (Bottom left) Phase wrapped between +π and −π. (Bottom right) Phase unwrapped about 0.5 μm thickness.

13.6Conclusion

The equations for single-layer and multilayer thin films have been reviewed and applied to a series of example coatings of particular interest in the polarization analysis of optical systems. The polarization properties are described by s- and p-amplitude reflection and transmission coefficients, in parallel with the Fresnel coefficients.

Single-layer anti-reflection coatings, such as quarter wave of magnesium fluoride on glass, are seen to both boost transmission and greatly reduce diattenuation, solving two problems at once. Beam splitters that divide the flux equally can be made from thin films of metal, a few nanometers thick, but such single-layer coatings have large differences in their s- and p-properties, yielding significant diattenuation and retardance. Making non-polarizing beam splitter coatings requires far more complex coatings. Further, for many vendors and in many catalogs, non-polarizing only means that the s- and p-reflectances and transmittances are equal. It does not always mean that the s- and p-phase changes are equal. Thus, right circularly polarized light will generally reflect and transmit into beams that are not circular but elliptically polarized, unless more complex, phase matching designs are used.

Efficient reflection coatings can be produced from multilayer quarter wave coatings with alternating high- and low-index layers. Such coatings can have far higher reflectivities than bare metal coatings. Because the reflection from each layer is small, a large number of layers are required. Thus, the light goes deep into these coatings and a very large number of partial waves carry significant flux. Since the light goes so deep into the coatings, these high reflectivity designs are much more angle and polarization sensitive than metal coatings, and can have substantial phase variations that contribute to the wavefront aberration. Thus, thick coatings in general can contribute defocus, chromatic aberration, coma, astigmatism, and other aberrations in measurable amounts, and should be analyzed carefully in systems with tight specifications.

Similar coating designs can be used for polarizing beam splitters. Polarizing beam splitters tend to have far more p-polarized light contaminating the transmitted beam than s-polarized light contaminating the reflected beam; thus, when polarizing beam splitters are used as polarizers, the reflected beam is generally preferred. Polarizing beam splitters generally have significant phase variations with angle and can also introduce aberration into converging and diverging beams.

These examples are intended to demonstrate methods for analyzing the plots of the magnitudes and phases of the amplitude coefficients to estimate the wavefront aberrations, apodization, diattenuation, and retardance due to coatings. Then, different coatings can be compared, and the coating aberration information can be integrated with other optical design information. The wavefront aberrations of coatings in general may be small, but they are ignored at the optical designer’s potential peril, especially now that optical analysis software makes it easier to polarization ray trace.

13.7Appendix: Derivation of Single-Layer Equations

Consider an incident beam of light with amplitude Einc. The amplitude reflectance at the air/film, film/substrate, and film/air interfaces are the interface Fresnel coefficients r01, r12, and r10. Similarly, the amplitude transmission Fresnel coefficients through the air/film and film/substrate interfaces are t01, t12, and t10. Figure 13.28 shows the first three reflected partial waves. The amplitudes of the reflected partial waves are as follows:

Figure 13.28(Left) Schematic of a ray refracting and reflecting at a single-layer thin film yielding multiple partial rays. The Fresnel amplitude coefficients of each interface interaction for reflections (middle) and transmissions (right) are shown.

$$\begin{array}{l}{E}_I=r_{01}{E}_{inc}\hfill \\ E_{II}=t_{01}{r}_{12}{t}_{10}{E}_{inc}{e}^{i\alpha }=t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }{E}_{inc}\hfill \\ E_{III}=t_{01}{r}_{12}{r}_{10}{r}_{12}{t}_{10}{E}_{inc}{e}^{i2\alpha }=t_{01}{t}_{10}{r}_{12}^2{r}_{10}{e}^{i2\alpha }{E}_{inc}\hfill \\ ⋮\hfill \\ E_N=t_{01}{t}_{10}{r}_{12}{(r_{12}{r}_{10}{e}^{i\alpha })}^N{E}_{inc}.\hfill \end{array}$$13.46

13.46

Thus, the amplitudes of the partial waves form a geometrical series (a, a x, a x2, a x3, …) where each successive term is multiplied by r12 r10. Let α = 2β be the phase:

$$\begin{array}{l}\alpha =2\beta =\frac{2\pi }{\lambda }∆OP{L}_{I,II},\text{ where}\hfill \\ ∆OP{L}_{I,II}=OP{L}_{ABC}-OP{L}_{AD},\text{ where }OP{L}_{ABC}=2n_1\frac{d}{\cos {\theta }_1}\text{ and }OP{L}_{AD}=2n_{0}d\tan {\theta }_1\sin {\theta }_0\hfill \\ ∆OP{L}_{I,II}=\frac{2n_{1}d-2n_{0}d\sin {\theta }_1\sin {\theta }_0}{\cos {\theta }_1}\text{, where }{n}_0\sin {\theta }_0=n_1\sin {\theta }_1\hfill \\ ∆OP{L}_{I,II}=\frac{2n_{1}d(1-{\sin }^2{\theta }_1)}{\cos {\theta }_1}=2n_{1}d\cos {\theta }_1,\hfill \end{array}$$13.47

13.47

where constructive interference happens when ΔOPLI,II = 2n1d cos θ1 = mλ. The total reflectance is the sum of all E as N → ∞, which is the sum of the geometrical series

$$a+ax+a{x}^2+a{x}^3+\dots =a{\displaystyle \sum _{n=0}^{\infty }{x}^n=\frac{a}{1-x}},$$13.48

13.48

where −1 < x < 1. This must be repeated once for the s- and once for the p-amplitude reflection coefficients,

$$\begin{array}{l}r=r_{01}+t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }\underset{N\to \infty }{\lim }\left(1+r_{12}{r}_{10}{e}^{i\alpha }+r_{12}^2{r}_{10}^2{e}^{i2\alpha }\dots +r_{12}^{N-2}{r}_{10}^{N-2}{e}^{i(N-2)\alpha }\right)\\ \phantom{r}=r_{01}+t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }{\displaystyle \sum _{n=0}^{\infty }{\left(r_{12}{r}_{10}{e}^{i\alpha }\right)}^n}=r_{01}+t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }{\displaystyle \sum _{n=0}^{\infty }{\left(-r_{12}{r}_{01}{e}^{i\alpha }\right)}^n}\\ \phantom{r}=r_{01}+\frac{t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }}{1-r_{12}{r}_{10}{e}^{i\alpha }}=\frac{r_{01}-r_{01}{r}_{12}{r}_{10}{e}^{i\alpha }+t_{01}{t}_{10}{r}_{12}{e}^{i\alpha }}{1-r_{12}{r}_{10}{e}^{i\alpha }}\\ \phantom{r}=\frac{r_{01}-r_{12}{e}^{i\alpha }(r_{01}{r}_{10}-t_{01}{t}_{10})}{1-r_{12}{r}_{10}{e}^{i\alpha }}=\frac{r_{01}+r_{12}{e}^{i\alpha }(-r_{01}{r}_{10}+t_{01}{t}_{10})}{1-r_{12}{r}_{10}{e}^{i\alpha }}\\ \phantom{r}=\frac{r_{01}+r_{12}{e}^{i\alpha }(r_{01}{r}_{01}+t_{01}{t}_{10})}{1-r_{12}{r}_{10}{e}^{i\alpha }}=\frac{r_{01}+r_{12}{e}^{i\alpha }}{1-r_{12}{r}_{10}{e}^{i\alpha }}=\frac{r_{01}+r_{12}{e}^{i\alpha }}{1+r_{12}{r}_{01}{e}^{i\alpha }}.\end{array}$$13.49

13.49

For non-absorbing film, the overall reflected and transmitted flux shown in Figure 13.29 must sum to the incident flux, and the following relations hold:

Figure 13.29(Left) An incident lightray (black arrow) with amplitude 1 reflects (blue arrow) and refracts (green arrow) at the interface with amplitude coefficients r01 and t01. (Middle) Light incident into the interface at the opposite direction of the reflected light, with the same amplitude coefficient r01. It refracts (purple) and reflects (black) into the opposite direction as the incident light of the left figure. The two exiting beam amplitude coefficients are altered as the two exiting rays in the left figure. (Right) Similar to the middle figure, A ray is incident in the opposite direction of the refracted light.

$$\{\begin{array}{l}1=r_{01}{r}_{01}+t_{01}{t}_{10}\hfill \\ 0=r_{01}{t}_{01}+t_{01}{r}_{10}\hfill \end{array},\text{ then }\{\begin{array}{l}1=r_{01}^2+t_{01}{t}_{10}\hfill \\ r_{01}=-r_{10}\hfill \end{array}.$$13.50

13.50

For a film surrounded by two identical media, n0 = n2, so r12 = r10 and the reflection coefficient becomes

$$r=\frac{r_{01}+r_{10}{e}^{i\alpha }}{1-r_{10}{r}_{10}{e}^{i\alpha }}=\frac{r_{01}-r_{01}{e}^{i\alpha }}{1-r_{01}^2{e}^{i\alpha }}=\frac{r_{01}(1-e^{i\beta })}{1-r_{01}^2{e}^{i\beta }}.$$13.51

13.51

Similarly, for the amplitude transmission coefficients, ts and tp, the following equation is evaluated twice for s and p:

$$\begin{array}{l}t=t_{01}{t}_{12}\left(1+r_{12}{r}_{10}{e}^{i\alpha }+r_{12}^2{r}_{10}^2{e}^{i2\alpha }+\dots \right)=t_{01}{t}_{12}{\displaystyle \sum _{n=0}^{\infty }{(r_{12}{r}_{10}{e}^{i\alpha })}^n}\\ \phantom{t}=\frac{t_{01}{t}_{12}}{1-r_{10}{r}_{12}{e}^{i\alpha }}=\frac{t_{01}{t}_{12}}{1+r_{01}{r}_{12}{e}^{i\alpha }}.\end{array}$$13.52

13.52

13.8Problem Sets

1. Write the characteristic matrix for a quarter wave layer of MgF2 at 60° and calculate the s- and p-amplitude coefficients.

2. Figure 13.6 shows the s- and p-phase change on transmission for a MgF2 antireflection coating. The optical path lengths and phase changes for the partial waves for each of the two polarizations are equal. Thus, where does the phase change and retardance arise?

3. Consider a high-index (n1 = 2.895) film with quarter wave optical thickness (i.e., half the film thickness period) on a low-index (n2 = 1.386) substrate at wavelength λ = 488 nm. When using these two materials to construct a beam-splitting coating (produce equal power in transmission and reflection), what is the operating angle?

4. Find the condition for zero reflectance from a single-layer thin film at normal incidence. Consider Equations 13.13 and 13.14. Set the amplitude reflectivity coefficients equal to 0, as(θ = 0, n0, n1, n2, t1) = ap(θ = 0, n0, n1, n2, t1) = 0 and solve for the thickness t1 and refractive index n1 that yield zero reflection at normal incidence.

5. For a set of n points, an order n − 1 polynomial that passes exactly through all the points can be found. For example, consider five points (x0, f(x0)), (x1, f(x1)), …(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

   $$\left(\begin{array}{lllll}{m}_{00}& m_{01}& m_{02}& m_{03}& m_{04}\\ m_{10}& m_{11}& m_{12}& m_{13}& m_{14}\\ m_{20}& m_{21}& m_{22}& m_{23}& m_{24}\\ m_{30}& m_{31}& m_{32}& m_{33}& m_{34}\\ m_{40}& m_{41}& m_{42}& m_{43}& m_{44}\end{array}\right)\left(\begin{array}{l}{c}_0\\ c_1\\ c_2\\ c_3\\ c_4\end{array}\right)=\left(\begin{array}{l}f(x_0)\\ f(x_1)\\ f(x_2)\\ f(x_3)\\ f(x_4)\end{array}\right).$$

   

   1. Find the matrix coefficients mij in this equation to calculate the c’s from the set of f(x).

   2. Show the matrix equation that calculates the polynomial coefficients, c0,….

   3. Provide the equation for fitting three points to a quadratic equation yielding c0, c1, and c2. Provide all nine matrix elements as functions of (x1, f(x1)), (x2, f(x2)), and (x3, f(x3)).

6. Continue on the previous problem.

   1. 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)).

   2. Fit the amplitude transmission coefficients ts and tp for an air–silicon interface for the angles.

   3. For which component, ts and tp, is the fourth-order term more significant?

References

1I. J. Hodgkinson, F. Horowitz, H. A. Macleod, M. Sikkens, and J. J. Wharton, Measurement of the principal refractive indices of thin films deposited at oblique incidence, JOSA A 2(10) (1985): 1693–1697.

2I. J. Hodgkinson and Q. H. Wu, Birefringent Thin Films and Polarizing Elements, Singapore: World Scientific (1997).

3R. M. A. Azzam and M. M. K. Howlader, Fourth- and sixth-order polarization aberrations of antireflection-coated optical surfaces, Opt. Lett. 26 (2001): 1607–1608.

4F. Abelès, Researches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieus stratifies. Applications aux couches minces, Ann. Phys. Paris, 12ième Series 5 (1950): 596–640.

5F. Abelès, Researches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieus stratifies. Applications aux couches minces, Ann. Phys. Paris, 12ième Series 5 (1950): 706–784.

6H. A. Macleod, Thin-Film Optical Filters, 3rd edition, Taylor & Francis (2001).

7J. L. Pezzaniti and R. A. Chipman, Angular dependence of polarizing beam-splitter cubes, Appl. Opt. 33.10 (1994): 1916–1929.

8S. M. MacNeille, Beam splitter, U.S. Patent 2,403,731 (1946).