miércoles, 3 de febrero de 2010

Optical Properties of Thin Films



Thin films were fixst exploited practically for their optical properties. In the latter part of the nineteenth and first half of the twentieth centuries, the reflecting properties of metal films were utilized in assorted components of precisión optical equipment. A noteworthy example was the Fabry-Perot interferometer developed in 1899, which required mirrors of very high re-flectance or "finesse." This instrument enabled impressive accuracy to be attained in spectroscopy, thereby greatly advancing research in this field. The utility of dielectric films in optical applications was interestingly recognized as a result of observations by early spectroscopists and microscopists, notably Lord Rayleigh and Fraunhofer. They noticed that atmospheric corrosión of the lens surfaces of their instruments actually resulted in an enhanced overall transmission rather than a deterioration of performance. Interference effects of a surface layer was quickly discovered as the cause and it was not long before this damaging effect was capitalized upon in the form of antireflection (AR) coatings. These were first produced commercially by chemical etching, a process which persisted until the 1950s. In the mid-1930s, however, AR coatings were first produced by vacuum evaporation techniques and eventually proved to be more versatile and reliable than those made by etching. Coated lenses then found rapid application in optical imaging equipment such as cameras, telescopes, binoculars, and microscopes. Similar coatings were sub-sequently employed in dielectric mirrors, optical filters, and selective ab sorbere (Ref. 1).

Current interest in optical films still centers around traditional optical components. But, in addition, a variety of new applications has emerged—end mirrors for lasers, antireflection coatings for solar cells, and films for energy conservation systems, to mention a few. The portion of the electromagnetic spectrum involved in virtually all optical film applications falls within the span from the ultraviolet (UV) to the infrared (IR) with particular emphasis focused in the visible región. This chapter will therefore be limited to the effects in this broad spectral domain. A common optical theory governs the phenomena exhibited by the many thin-film coating applications irrespective of operating wavelength range. Specific metal, dielectric, and semiconductor films have been deposited, frequently in layered combinations, to produce the necessary optical characteristics. For these reasons the broad topical outline of the chapter includes:

11.2. Properties of Optical Film Materials
11.3. Thin-Film Optics

11.4. Multilayer Optical Film Applications

Additional discussion of thin films employed in optoelectronic devices and optical communications (Chapter 7), integrated optics (Chapter 14), and optical recording (Chapter 14) can be found in the indicated chapters.
Ampie measure of the overall importance of optical film properties and components is found in the relatively large number of books devoted to the subject (Ref. 2-6). With regard to these the excellent treatment by Anders is recommended for its lucid presentation of the analysis, design, and production of coatings. Similarly the more recent and accessible book by Pulker is recommended for its wealth of useful data on optical coatings, and how they are processed and used. Readers seeking a more specialized treatment of particular topics should consult the widely acclaimed series Physics of Thin Films. Of the 67 review anieles published in 13 volumes until 1987, fully 19 deal with optical properties and applications of metal and dielectric thin films.





Properties of Optical Film Materials
General Considerations

In order to understand the optical behavior of films and film systems, one must become familiar with the optical constants of materials, their origins, magni-tudes, and how they depend on the way films are processed. The purpose of Section 11.2 is to provide a brief survey of these issues. The unifying concept that embraces all optical properties is the interaction of electromagnetic radiation with the electrons of the material. On this basis, optical properties are interpretable from what we know of the electronic structure and how it is affected by atomic structure, bonding, impurities, and defects. Quantum mechanics is required for a detailed description, but this is well beyond the scope intended here. Discussións will stress meanings and implications rather than mathematical rigor.
Electromagnetic radiation propagates differently in materials than in free space because of the presence of charge. As a result, there is a change in the wave velocity and intensity of the radiation described by the complex index of refraction
N=n~ik.                                   (11-1)
Of the total radiation energy incident on an object, a fraction R is reflected from the top surface and a fraction T is transmitted through the bottom surface. The remaining fraction is lost through electronic absorption (A) processes and by scattering (5) at surface and volume imperfections. Surface roughness, internal boundaries, and density fluctuations arising from porosity, pinholes, microcracks, particulate incorporation, and impurities are sources of scattering. Adding the various contributions gives
R + T+ A + S = 1.                         (11-4)
Of the terms, R is of the greatest interest to us. For light passing through a médium of refractive index n0, impinging normally on a transparent film of index w,,

Optical Properties of Metáis and Mirrors

The large density of empty, closely spaced electrón energy states above the Fermi level in a metal plays an important role in influencing its optical properties. Incident photons over a wide wavelength range are readily ab sorbed by conduction-band electrons. These excited electrons move to higher energy levéis where they undergo collisions with lattice ions, and the extra energy is dissipated in the form of phonons. The lattice is thus heated, and we speak of absorption. Alternatively, if the probability of colliding with an ion is small, the electrón will emit a photon as it drops back to a lower energy level. This results in the strongly reflected beam exhibited by metáis in the visible and infrared región. The characteristic color of some metáis is due to the preferential absorption of some portion of the visible spectrum. In gold, for example, the green portion is absorbed, and the metal assumes the coloration of the reflected red and yellow light. Silver and aluminum reflect all portions of the visible spectrum and therefore appear to have a white color.
A noteworthy feature of the optical response is the decreased reflectivity in the ultraviolet and, in particular, the abrupt absorption edge exhibited by the noble metáis. The absorption is due to interband electrón transitions (e.g., 3d -+ 4s in Cu). In aluminum, however, and rhodium, to a lesser extent, high reflectance extends into the ultraviolet. In the infrared all metáis are very highly reflective. This fact is explained by the Hagen-Rubens relation, which holds when the electrón relaxation time is short compared with the period of the incident wave. Physically, electrons can then respond sufficiently rapidly to prevent the external electric field from penetrating the metal.
The óptica! properties of thin films are somewhat different from those of

bulk metáis. In ultrathin films (< 100 A thick) variations in film continuity make the concept of optical constants problemática!. In thicker continuous films, n and k valúes tend to be slightly smaller and larger, respectively, than comparable bulk valúes. The differences stem largely from variables encoun-tered in evaporation or sputtering, the almost universally employed processes for depositing metal film mirrors. In metáis such as Al, which readily getter gases, the reflectivity is sensitive to deposition conditions. Lower operating pressures and higher deposition rates result in purer films with enhanced reflectivities as shown in Fig. 11-2. The reason is due in part to the absence of surface oxides, nitrides, etc., which serve to increase absorption. More noble metáis like Rh and Pt are, on the other hand, relatively immune to deposition conditions. Other variables affecting reflectivity include incident angle of vapor flux, surface topography, and substrate temperature (Refs. 7, 8). In metáis such as Al, Ag, and Au, high substrate temperaturas result in coarser-grained films with rough surfaces that tend to diffusely scatter light. A contrary behavior is exhibited by Rh whose reflectivity is enhanced by some 2-6% over the spectral range from 0.4-2.2 /¿m, when deposited at 400°C rather than 40°C.
The reader should appreciate that even small reflectance enhancements are very significant in mirror applications. Therefore, much care is exercised during deposition, and steps are frequently taken subsequently to prevent degradation of reflectance due to oxidation (Al), sulfide tarnishing (Ag), and mechanical scratching. Mirrors are therefore frequently coated with hard transparent protective overlayers. Thus films of SiO, Si02, and A1203 are commonly used to protect evaporated Al mirrors, but usually at the cost of increasing absorbance. The superior reflectance of Ag is offset by its poor adhesión and susceptibility to atmospheric attack. Various substrate film glue layers (e.g., A1203, nichrome) and coatings (e.g., A1203, SiO) have been used to improve adhesión, but Ag film use in mirrors remains restricted. Films of Rh are ideally suited as front surface mirrors because they are hard and chemically very durable giving them long-term stability. Adhesión to fused silica based substrates is often poor, however. Despite its relatively low reflectivity, evaporated Rh has found application in telescope mirrors, optical reflectivity standards and in mirrors for medical applications.


Optical Effects in Dielectrics
Dielectric materials employed in optical coating applications include fluorides (e.g., MgF2, CeF3) oxides (e.g., A1203, Ti02, Si02) sulfides (e.g., ZnS, CdS) and assorted compounds (e.g., ZnSe, ZnTe). Bonding characteristics ranging from ionic to covalent are represented in these materials. An essential common feature of dielectric optical materials in their very low absorption (a < 103/cm) in some relevant portion of the spectrum; in this región they are essentially transparent (e.g., fluorides and oxides in the visible and infrared, chalcogenides in the infrared).
The refractive index n is basically the only optical constant of interest insofar as optical coating design is concerned. In different materials the refracted beam travels at different velocities because incident and forward-scattered beams interact and produce a phase shift. This phase shift can be interpreted in terms of a difference in velocity between incident and refracted beams. As might be expected the magnitude of n depends on the strength of the refracted beam, which in turn depends on the density of electrons. In polar materials such as oxides, glasses, and compound semiconductors the lattice is essentiaJly a collection of permanent electric dipole moments (pe). The magnitude of pe is equal to the product of the effective charge (nuclear, electronic) and the distance between charge centers. Wave retardation occurs because of the interaction between the electromagnetic radiation {i field) and the stretched permanent dipoles. In nonpolar solids like diamond and Si, the incident radiation creates induced electric dipole moments by displacing atomic electrons relative to the nuclear charge. In this case, pe = apé", where ap is known as the polarizability. It is beyond our scope to derive the link between the microscopic polarization and the macroscopic index of refraction known as the Lorenz-Lorentz equation, namely
3     (n2 - 1) M
47rATA (n2 + 2)   p '

where A^ is Avogadro's number, M is the molecular weight, and p is the density (Ref. 9). Thus, high refractive Índices are associated with large ionic polarizabilities, which increase with the size of the ion and with the degree of negative charge on isoelectric ions. In glasses and cubic crystals, n is independent of crystallographic directions. In other crystal systems, n is large in close-packed directions. Similarly more open structures of high-temperature polymorphic phases have lower refractive Índices than crystalline forms of the same composition.  In Si02,  for example,  in order of increasing density
"«i.» = 146> "mdymite = 1-47, ncr¡stobaiite = l-49, and /7quanz = 1.55. Com-pounds with predominant covalent bonding have higher refractive índices than ionically bound solids. For example, in order of increasing degree of covalent bonding «ZnC,2 = 1.68, n?n0 = 2.08, nZnS = 2.37, «ZnSe = 2.57, and nZnTc = 3.56.
The excellent transmission of dielectric materials in the visible región of the spectrum is terminated at short wavelengths with the onset of the ultraviolet absorption edge. The critica! radiation wavelength \ at which this occurs is given by the familiar equation
\c = hc/Eg       or        \(fim) = 1.24/Eg (eV).     (11-8)
Valúes physically correspond to electronic transitions from the filled valence-band levéis across the energy gap Eg to the unfilled conduction-band states. Múltiple peaks near the UV absorption edge indícate the complexity of these processes. At long wavelength the high optical transmission is once again limited, this time by absorption due to the vibration of lattice ions in resonance with the incident radiation. The frequency of máximum absorption is related to the forcé constant and masses of vibrating anions and cations.

Optical Effects in Semiconductors
Lastly, some elementary aspects of the optical properties of semiconductors will be briefly considered. The absorption coefficient behavior of a number of important semiconductor materials was previously noted in Fig. 7-13. The most prominent feature is the rapid decrease in absorption at the critical or cutoff wavelength \c. Here as in dielectrics the magnitude of the energy gap between the valence and conduction band governs the valué of Xc. For wavelengths larger than Xc, semiconductors are essentially transparent because no mechanisms exist to excite electrón transitions. However, for wavelengths shorter than Xc, electrons can be stimulated into the conduction band. In addition, the generated free carriers can now absorb quanta of energy and occupy excited levéis in the continuum of conduction-band states. Semiconduc tors then behave like metáis and are highly reflective.
An interesting phenomenon occurs in this regime of free-carrier light absorption. This effect has relevance to semiconducting In203:Sn films that are widely used because they have the unusual property of being transparent conductors (see Section 11.2.5). In semiconductors (and metáis), valúes of n and k are generally not independent of each other but are connected by electromagnetic theory through so-called dispersión relations (Ref. 10)
n2 - k2 = e, - ncq2/m*e0{o>2 + 72),                  (11-9)
2nk=           \\           5V.                           11-10
m*e0w(w2 + 72)

In these formulas, e, and e0 are the permitivities of the solid and free space, respectively, nc is the carrier concentration in the conduction band, m* is the effective mass of the charge carrier, q is the electronic charge, w is the angular frequency of the incident radiation of wavelength \(<a = 2irc/X), and 7 is the reciprocal of the carrier relaxation time. Basically, 7 is inversely related to the electrical conductivity. When n = k, the optical properties change radically. The critical valué of co for which this happens is called the plasma frequency up, and corresponding to it is the plasma wavelength Xp. At the optical transition the conductivity is high, and y2 may be neglected. It is then easy to show that
2-kc ( £0e{m* \'/2
which only depends on the carrier concentration. When X> \p, the film exhibits metal-like reflection, whereas if X < Xp the high transmittance of a dielectric occurs. This means that In203:Sn films can be prepared to possess the desired, though not independent, admixture of optical reflectivity and electrical resistivity.
In photonic devices the operating wavelength is usually either Xc (light-emit-ting diodes, lasers) or less than Xc (photodetectors, solar cells). It is essential in the latter applications that light be absorbed for the devices to opérate. The opposite is required of semiconductors in optical coating applications where transparency is essential. Another difference relates to structure. In photonic devices either bulk or thin-film single-crystal semiconductors are essential for operation. On the other hand, polycrystalline or amorphous Ge and Si are deposited in optical coatings. The extensión of the spectral regime of trans parency down to \c = 0.82 fim for amorphous relative to polycrystalline Si, is an advantage in optical coatings; in solar cell use however, the constricted range of absorption is a disadvantage, other things being equal. Films possess-ing tailor-made energy band gaps and refractive Índices, prepared from compound semiconductor mixtures, have been discussed previously.
Optical Properties of Thin Dielectric and Semiconductor Films
The optical properties of dielectric and semiconductor films are collected in Table 11-2. Practically all of the data are for evaporated films, but some results for sputtered films are included. Only a portion of the exhaustive information on these films, provided by Pulker (Ref. 5) in his excellent treatment of this subject, is considered here. Among the important issues concerning optical films are the magnitudes of n and k, the spectral range of transmission, the dependence of properties on film structure and deposition processes, and environmental effects. These issues will now be addressed in turn.

Thin-Film Optics
Nonabsorbing Films (One Interface)
In this section we present the theory of the optical behavior of a single transparent (nonabsorbing) film on a nonabsorbing substrate. Transparent films are of great practical importance because applications of all kinds are based on one or, more commonly, múltiple layers of such films. Crucial to understand-ing these applications is how radiation is reflected and transmitted at a single interface between two media of differing index of refraction.
Let us consider Fig. 11-7, which depicts an interface between an optically isotropic médium with an index of refraction ny and a vacuum región of index n0 (= 1). Electromagnetic radiation is incident on the n0/nl interface at an angle 4>Q and refracted in the médium at an angle 0, with respect to the normal. The electric field vector of the incident wave, i¡, is decomposed into two components, <á°/~L and <?,", respectively perpendicular and parallel, to the plañe of incidence. Through the use of the Maxwell equations with appropriate interfacial boundary conditions, the amplitudes of the components of the transmitted (<í',"L , <£",") and reflected ($0X , <f0") waves in the médium and vacuum can be determined. The results of the calculation are the celebrated Fresnel coefficients for reflection (r) and transmission (/) at the  n0/nl.



. Nonabsorbing Film (Two Interfaces)
 We now consider reflection of light at both film interfaces in Fig. 11-8 where radiation may be partially reflected or transmitted. The separation between the boundaries, d, is of the order of magnitude of the wavelength X so that interference effects occur. It is assumed that the film is plañe parallel and homogeneous. The optical properties of the film will now depend on n0, «,, n2, d, and X. If a plañe wave of unit amplitude is normally incident on the n0/nl interface at A, the fraction reflected is rt = (n0 - nx)/(nx + n0), and the wave transmitted has amplitude y\ — r2. At point B of the film-sub-strate (nl /n2) interface, r2 = (w, - n2)/{n2 + nx) so that a wave of ampli tude r2 v 1 - r2 is reflected back to point C on the first interface. Here the wave is reflected again into the film with amplitude —rxr2y\ — r2e'&. Compared with the first reflection at A, the emerging wave at C has twice traversed the optical thickness, nld, of the film. The total path difference, 2nld, implies a phase 6 of amount 8 = (27r/X)(path difference), and there-fore 5 = 4irnid/\. The portion of the wave transmitted into free space at C is attenuated by the factor y\ - r2 so that its amplitude is given by r2(l - r2). Adding the two emergent waves in amplitude and phase yields.



Ronellys Flores---CRF---libro the materials science of thin films






Optical Properties of Thin Films


Thin films were fixst exploited practically for their optical properties. In the latter part of the nineteenth and first half of the twentieth centuries, the reflecting properties of metal films were utilized in assorted components of precisión optical equipment. A noteworthy example was the Fabry-Perot interferometer developed in 1899, which required mirrors of very high re-flectance or "finesse." This instrument enabled impressive accuracy to be attained in spectroscopy, thereby greatly advancing research in this field. The utility of dielectric films in optical applications was interestingly recognized as a result of observations by early spectroscopists and microscopists, notably Lord Rayleigh and Fraunhofer. They noticed that atmospheric corrosión of the lens surfaces of their instruments actually resulted in an enhanced overall transmission rather than a deterioration of performance. Interference effects of a surface layer was quickly discovered as the cause and it was not long before this damaging effect was capitalized upon in the form of antireflection (AR) coatings. These were first produced commercially by chemical etching, a process which persisted until the 1950s. In the mid-1930s, however, AR coatings were first produced by vacuum evaporation techniques and eventually proved to be more versatile and reliable than those made by etching. Coated lenses then found rapid application in optical imaging equipment such as cameras, telescopes, binoculars, and microscopes. Similar coatings were sub-sequently employed in dielectric mirrors, optical filters, and selective ab­sorbere (Ref. 1).
Current interest in optical films still centers around traditional optical components. But, in addition, a variety of new applications has emerged—end mirrors for lasers, antireflection coatings for solar cells, and films for energy conservation systems, to mention a few. The portion of the electromagnetic spectrum involved in virtually all optical film applications falls within the span from the ultraviolet (UV) to the infrared (IR) with particular emphasis focused in the visible región. This chapter will therefore be limited to the effects in this broad spectral domain. A common optical theory governs the phenomena exhibited by the many thin-film coating applications irrespective of operating wavelength range. Specific metal, dielectric, and semiconductor films have been deposited, frequently in layered combinations, to produce the necessary optical characteristics. For these reasons the broad topical outline of the chapter includes:
11.2. Properties of Optical Film Materials
11.3. Thin-Film Optics
11.4. Multilayer Optical Film Applications
Additional discussion of thin films employed in optoelectronic devices and optical communications (Chapter 7), integrated optics (Chapter 14), and optical recording (Chapter 14) can be found in the indicated chapters.
Ampie measure of the overall importance of optical film properties and components is found in the relatively large number of books devoted to the subject (Ref. 2-6). With regard to these the excellent treatment by Anders is recommended for its lucid presentation of the analysis, design, and production of coatings. Similarly the more recent and accessible book by Pulker is recommended for its wealth of useful data on optical coatings, and how they are processed and used. Readers seeking a more specialized treatment of particular topics should consult the widely acclaimed series Physics of Thin Films. Of the 67 review anieles published in 13 volumes until 1987, fully 19 deal with optical properties and applications of metal and dielectric thin films.


Properties of Optical Film Materials
General Considerations

In order to understand the optical behavior of films and film systems, one must become familiar with the optical constants of materials, their origins, magni-tudes, and how they depend on the way films are processed. The purpose of Section 11.2 is to provide a brief survey of these issues. The unifying concept that embraces all optical properties is the interaction of electromagnetic radiation with the electrons of the material. On this basis, optical properties are interpretable from what we know of the electronic structure and how it is affected by atomic structure, bonding, impurities, and defects. Quantum mechanics is required for a detailed description, but this is well beyond the scope intended here. Discussións will stress meanings and implications rather than mathematical rigor.
Electromagnetic radiation propagates differently in materials than in free space because of the presence of charge. As a result, there is a change in the wave velocity and intensity of the radiation described by the complex index of refraction
N=n~ik.                                   (11-1)
Of the total radiation energy incident on an object, a fraction R is reflected from the top surface and a fraction T is transmitted through the bottom surface. The remaining fraction is lost through electronic absorption (A) processes and by scattering (5) at surface and volume imperfections. Surface roughness, internal boundaries, and density fluctuations arising from porosity, pinholes, microcracks, particulate incorporation, and impurities are sources of scattering. Adding the various contributions gives
R + T+ A + S = 1.                         (11-4)
Of the terms, R is of the greatest interest to us. For light passing through a médium of refractive index n0, impinging normally on a transparent film of index w,,

Optical Properties of Metáis and Mirrors

The large density of empty, closely spaced electrón energy states above the Fermi level in a metal plays an important role in influencing its optical properties. Incident photons over a wide wavelength range are readily ab­sorbed by conduction-band electrons. These excited electrons move to higher energy levéis where they undergo collisions with lattice ions, and the extra energy is dissipated in the form of phonons. The lattice is thus heated, and we speak of absorption. Alternatively, if the probability of colliding with an ion is small, the electrón will emit a photon as it drops back to a lower energy level. This results in the strongly reflected beam exhibited by metáis in the visible and infrared región. The characteristic color of some metáis is due to the preferential absorption of some portion of the visible spectrum. In gold, for example, the green portion is absorbed, and the metal assumes the coloration of the reflected red and yellow light. Silver and aluminum reflect all portions of the visible spectrum and therefore appear to have a white color.
A noteworthy feature of the optical response is the decreased reflectivity in the ultraviolet and, in particular, the abrupt absorption edge exhibited by the noble metáis. The absorption is due to interband electrón transitions (e.g., 3d -+ 4s in Cu). In aluminum, however, and rhodium, to a lesser extent, high reflectance extends into the ultraviolet. In the infrared all metáis are very highly reflective. This fact is explained by the Hagen-Rubens relation, which holds when the electrón relaxation time is short compared with the period of the incident wave. Physically, electrons can then respond sufficiently rapidly to prevent the external electric field from penetrating the metal.
The óptica! properties of thin films are somewhat different from those of
bulk metáis. In ultrathin films (< 100 A thick) variations in film continuity make the concept of optical constants problemática!. In thicker continuous films, n and k valúes tend to be slightly smaller and larger, respectively, than comparable bulk valúes. The differences stem largely from variables encoun-tered in evaporation or sputtering, the almost universally employed processes for depositing metal film mirrors. In metáis such as Al, which readily getter gases, the reflectivity is sensitive to deposition conditions. Lower operating pressures and higher deposition rates result in purer films with enhanced reflectivities as shown in Fig. 11-2. The reason is due in part to the absence of surface oxides, nitrides, etc., which serve to increase absorption. More noble metáis like Rh and Pt are, on the other hand, relatively immune to deposition conditions. Other variables affecting reflectivity include incident angle of vapor flux, surface topography, and substrate temperature (Refs. 7, 8). In metáis such as Al, Ag, and Au, high substrate temperaturas result in coarser-grained films with rough surfaces that tend to diffusely scatter light. A contrary behavior is exhibited by Rh whose reflectivity is enhanced by some 2-6% over the spectral range from 0.4-2.2 /¿m, when deposited at 400°C rather than 40°C.
The reader should appreciate that even small reflectance enhancements are very significant in mirror applications. Therefore, much care is exercised during deposition, and steps are frequently taken subsequently to prevent degradation of reflectance due to oxidation (Al), sulfide tarnishing (Ag), and mechanical scratching. Mirrors are therefore frequently coated with hard transparent protective overlayers. Thus films of SiO, Si02, and A1203 are commonly used to protect evaporated Al mirrors, but usually at the cost of increasing absorbance. The superior reflectance of Ag is offset by its poor adhesión and susceptibility to atmospheric attack. Various substrate film glue layers (e.g., A1203, nichrome) and coatings (e.g., A1203, SiO) have been used to improve adhesión, but Ag film use in mirrors remains restricted. Films of Rh are ideally suited as front surface mirrors because they are hard and chemically very durable giving them long-term stability. Adhesión to fused silica based substrates is often poor, however. Despite its relatively low reflectivity, evaporated Rh has found application in telescope mirrors, optical reflectivity standards and in mirrors for medical applications.

Optical Effects in Dielectrics
Dielectric materials employed in optical coating applications include fluorides (e.g., MgF2, CeF3) oxides (e.g., A1203, Ti02, Si02) sulfides (e.g., ZnS, CdS) and assorted compounds (e.g., ZnSe, ZnTe). Bonding characteristics ranging from ionic to covalent are represented in these materials. An essential common feature of dielectric optical materials in their very low absorption (a < 103/cm) in some relevant portion of the spectrum; in this región they are essentially transparent (e.g., fluorides and oxides in the visible and infrared, chalcogenides in the infrared).
The refractive index n is basically the only optical constant of interest insofar as optical coating design is concerned. In different materials the refracted beam travels at different velocities because incident and forward-scattered beams interact and produce a phase shift. This phase shift can be interpreted in terms of a difference in velocity between incident and refracted beams. As might be expected the magnitude of n depends on the strength of the refracted beam, which in turn depends on the density of electrons. In polar materials such as oxides, glasses, and compound semiconductors the lattice is essentiaJly a collection of permanent electric dipole moments (pe). The magnitude of pe is equal to the product of the effective charge (nuclear, electronic) and the distance between charge centers. Wave retardation occurs because of the interaction between the electromagnetic radiation {i field) and the stretched permanent dipoles. In nonpolar solids like diamond and Si, the incident radiation creates induced electric dipole moments by displacing atomic electrons relative to the nuclear charge. In this case, pe = apé", where ap is known as the polarizability. It is beyond our scope to derive the link between the microscopic polarization and the macroscopic index of refraction known as the Lorenz-Lorentz equation, namely
3     (n2 - 1) M
47rATA (n2 + 2)   p '
where A^ is Avogadro's number, M is the molecular weight, and p is the density (Ref. 9). Thus, high refractive Índices are associated with large ionic polarizabilities, which increase with the size of the ion and with the degree of negative charge on isoelectric ions. In glasses and cubic crystals, n is independent of crystallographic directions. In other crystal systems, n is large in close-packed directions. Similarly more open structures of high-temperature polymorphic phases have lower refractive Índices than crystalline forms of the same composition.  In Si02,  for example,  in order of increasing density
"«i.» = 146> "mdymite = 1-47, ncr¡stobaiite = l-49, and /7quanz = 1.55. Com-pounds with predominant covalent bonding have higher refractive índices than ionically bound solids. For example, in order of increasing degree of covalent bonding «ZnC,2 = 1.68, n?n0 = 2.08, nZnS = 2.37, «ZnSe = 2.57, and nZnTc = 3.56.
The excellent transmission of dielectric materials in the visible región of the spectrum is terminated at short wavelengths with the onset of the ultraviolet absorption edge. The critica! radiation wavelength \ at which this occurs is given by the familiar equation
\c = hc/Eg       or        \(fim) = 1.24/Eg (eV).     (11-8)
Valúes physically correspond to electronic transitions from the filled valence-band levéis across the energy gap Eg to the unfilled conduction-band states. Múltiple peaks near the UV absorption edge indícate the complexity of these processes. At long wavelength the high optical transmission is once again limited, this time by absorption due to the vibration of lattice ions in resonance with the incident radiation. The frequency of máximum absorption is related to the forcé constant and masses of vibrating anions and cations.

Optical Effects in Semiconductors
Lastly, some elementary aspects of the optical properties of semiconductors will be briefly considered. The absorption coefficient behavior of a number of important semiconductor materials was previously noted in Fig. 7-13. The most prominent feature is the rapid decrease in absorption at the critical or cutoff wavelength \c. Here as in dielectrics the magnitude of the energy gap between the valence and conduction band governs the valué of Xc. For wavelengths larger than Xc, semiconductors are essentially transparent because no mechanisms exist to excite electrón transitions. However, for wavelengths shorter than Xc, electrons can be stimulated into the conduction band. In addition, the generated free carriers can now absorb quanta of energy and occupy excited levéis in the continuum of conduction-band states. Semiconduc­tors then behave like metáis and are highly reflective.
An interesting phenomenon occurs in this regime of free-carrier light absorption. This effect has relevance to semiconducting In203:Sn films that are widely used because they have the unusual property of being transparent conductors (see Section 11.2.5). In semiconductors (and metáis), valúes of n and k are generally not independent of each other but are connected by electromagnetic theory through so-called dispersión relations (Ref. 10)
n2 - k2 = e, - ncq2/m*e0{o>2 + 72),                  (11-9)
2nk=           \\           5V.                           11-10
m*e0w(w2 + 72)
In these formulas, e, and e0 are the permitivities of the solid and free space, respectively, nc is the carrier concentration in the conduction band, m* is the effective mass of the charge carrier, q is the electronic charge, w is the angular frequency of the incident radiation of wavelength \(<a = 2irc/X), and 7 is the reciprocal of the carrier relaxation time. Basically, 7 is inversely related to the electrical conductivity. When n = k, the optical properties change radically. The critical valué of co for which this happens is called the plasma frequency up, and corresponding to it is the plasma wavelength Xp. At the optical transition the conductivity is high, and y2 may be neglected. It is then easy to show that
2-kc ( £0e{m* \'/2
which only depends on the carrier concentration. When X> \p, the film exhibits metal-like reflection, whereas if X < Xp the high transmittance of a dielectric occurs. This means that In203:Sn films can be prepared to possess the desired, though not independent, admixture of optical reflectivity and electrical resistivity.
In photonic devices the operating wavelength is usually either Xc (light-emit-ting diodes, lasers) or less than Xc (photodetectors, solar cells). It is essential in the latter applications that light be absorbed for the devices to opérate. The opposite is required of semiconductors in optical coating applications where transparency is essential. Another difference relates to structure. In photonic devices either bulk or thin-film single-crystal semiconductors are essential for operation. On the other hand, polycrystalline or amorphous Ge and Si are deposited in optical coatings. The extensión of the spectral regime of trans­parency down to \c = 0.82 fim for amorphous relative to polycrystalline Si, is an advantage in optical coatings; in solar cell use however, the constricted range of absorption is a disadvantage, other things being equal. Films possess-ing tailor-made energy band gaps and refractive Índices, prepared from compound semiconductor mixtures, have been discussed previously.
Optical Properties of Thin Dielectric and Semiconductor Films
The optical properties of dielectric and semiconductor films are collected in Table 11-2. Practically all of the data are for evaporated films, but some results for sputtered films are included. Only a portion of the exhaustive information on these films, provided by Pulker (Ref. 5) in his excellent treatment of this subject, is considered here. Among the important issues concerning optical films are the magnitudes of n and k, the spectral range of transmission, the dependence of properties on film structure and deposition processes, and environmental effects. These issues will now be addressed in turn.

Thin-Film Optics
Nonabsorbing Films (One Interface)
In this section we present the theory of the optical behavior of a single transparent (nonabsorbing) film on a nonabsorbing substrate. Transparent films are of great practical importance because applications of all kinds are based on one or, more commonly, múltiple layers of such films. Crucial to understand-ing these applications is how radiation is reflected and transmitted at a single interface between two media of differing index of refraction.
Let us consider Fig. 11-7, which depicts an interface between an optically isotropic médium with an index of refraction ny and a vacuum región of index n0 (= 1). Electromagnetic radiation is incident on the n0/nl interface at an angle 4>Q and refracted in the médium at an angle 0, with respect to the normal. The electric field vector of the incident wave, i¡, is decomposed into two components, <á°/~L and <?,", respectively perpendicular and parallel, to the plañe of incidence. Through the use of the Maxwell equations with appropriate interfacial boundary conditions, the amplitudes of the components of the transmitted (<í',"L , <£",") and reflected ($0X , <f0") waves in the médium and vacuum can be determined. The results of the calculation are the celebrated Fresnel coefficients for reflection (r) and transmission (/) at the  n0/nl.

. Nonabsorbing Film (Two Interfaces)
 We now consider reflection of light at both film interfaces in Fig. 11-8 where radiation may be partially reflected or transmitted. The separation between the boundaries, d, is of the order of magnitude of the wavelength X so that interference effects occur. It is assumed that the film is plañe parallel and homogeneous. The optical properties of the film will now depend on n0, «,, n2, d, and X. If a plañe wave of unit amplitude is normally incident on the n0/nl interface at A, the fraction reflected is rt = (n0 - nx)/(nx + n0), and the wave transmitted has amplitude y\ — r2. At point B of the film-sub-strate (nl /n2) interface, r2 = (w, - n2)/{n2 + nx) so that a wave of ampli­tude r2 v 1 - r2 is reflected back to point C on the first interface. Here the wave is reflected again into the film with amplitude —rxr2y\ — r2e'&. Compared with the first reflection at A, the emerging wave at C has twice traversed the optical thickness, nld, of the film. The total path difference, 2nld, implies a phase 6 of amount 8 = (27r/X)(path difference), and there-fore 5 = 4irnid/\. The portion of the wave transmitted into free space at C is attenuated by the factor y\ - r2 so that its amplitude is given by r2(l - r2). Adding the two emergent waves in amplitude and phase yields.

Ronellys Flores---CRF---libro the materials science of thin films