Multilayer Film Applications

Transcription

1 4 8. Multilayer Film Applications 8 Multilayer Film Applications δ i ω n i l i cos θ i π c λ n il i cos θ i π λ l n in i a sin θ a n i 8.. where we used Eq. 8.. to write cos θ i sin θ i n a sin θ a /n i. The transverse reflection coefficients at the M + interfaces are defined as in Eq. 6..: ρ Ti n T,i n Ti n T,i + n Ti, i,,...,m+ 8.. where we set n T n Ta, as in Sec. 6.. and n T,M+ n Tb. The transverse refractive indices are defined in each medium by Eq. 7..: 8. Multilayer Dielectric Structures at Oblique Incidence n Ti n i, cos θ i n i cos θ i, polarization polarization, i a,,,...,m,b 8..4 Using the matching and propagation matrices for transverse fields that we discussed in Sec. 7., we derive here the layer recursions for multiple dielectric slabs at oblique incidence. Fig. 8.. shows such a multilayer structure. The layer recursions relate the various field quantities, such as the electric fields and the reflection responses, at the left of each interface. To obtain the layer recursions for the electric fields, we apply the propagation matrix 7..5 to the fields at the left of interface i + and propagate them to the right of the interface i, and then, apply a matching matrix 7.. to pass to the left of that interface: [ ] [ ][ ][ ] ETi+ ρti e jδ i ET,i+,+ E Ti τ Ti ρ Ti e jδi E T,i+, Multiplying the matrix factors, we obtain: [ ] [ ][ ] ETi+ e jδ i ρ Ti e jδi ET,i+,+, i M, M,..., 8..5 E Ti τ Ti ρ Ti e jδi e jδi E T,i+, This is identical to Eqs. 6.. with the substitutions k i l i δ i and ρ i ρ Ti. The recursion is initialized at the left of the M + st interface by performing an additional matching to pass to the right of that interface: [ ET,M+,+ E T,M+, ] [ τ T,M+ ρ T,M+ ρ T,M+ ][ E T.M+,+ ] 8..6 Fig. 8.. Oblique incidence on multilayer dielectric structure. We assume that there are no incident fields from the right side of the structure. The reflection/refraction angles in each medium are related to each other by Snel s law applied to each of the M + interfaces: n a sin θ a n i sin θ i n b sin θ b, i,,...,m 8.. It is convenient also to define by Eq the propagation phases or phase thicknesses for each of the M layers, that is, the quantities δ i i l i. Using i k n i cos θ i, where k is the free-space wavenumber, k ω/c πf/c π/λ, we have for i,,...,m: It follows now from Eq that the reflection responses, Γ Ti E Ti /E Ti+, will satisfy the identical recursions as Eq. 6..5: Γ Ti ρ Ti + Γ T,i+ e jδi + ρ Ti Γ T,i+ e jδi, i M, M,..., 8..7 and initialized at Γ T,M+ ρ T,M+. Similarly, we obtain the following recursions for the total transverse electric and magnetic fields at each interface they are continuous across each interface: [ ] [ ][ ] ETi cos δi jη Ti sin δ i ET,i+ H Ti jη Ti sin δ i cos δ i H T,i+, i M, M,..., 8..8

2 8.. Lossy Multilayer Structures Multilayer Film Applications where η Ti are the transverse characteristic impedances defined by Eq. 7.. and related to the refractive indices by η Ti η /n Ti. The wave impedances, Z Ti E Ti /H Ti, satisfy the following recursions initialized by Z T,M+ η Tb : Z Ti η Ti Z T,i+ + jη Ti tan δ i η Ti + jz T,i+ tan δ i, i M, M,..., 8..9 The MATLAB function multidiel that was introduced in Sec. 6. can also be used in the oblique case with two extra input arguments: the incidence angle from the left and the polarization type, or. Its full usage is as follows: [Gamma,Z] multidieln,l,lambda,theta,pol; % multilayer dielectric structure where theta is the angle θ θ a and pol is one of the strings te or tm. If the angle and polarization arguments are omitted, the function defaults to normal incidence for which and are the same. The other parameters have the same meaning as in Sec. 6.. In using this function, it is convenient to normalize the wavelength λ and the optical lengths n i l i of the layers to some reference wavelength λ. The frequency f will be normalized to the corresponding reference frequency f c /λ. Defining the normalized thicknesses L i n i l i /λ, so that n i l i L i λ, and noting that λ /λ f/f, we may write the phase thicknesses 8.. in the normalized form: δ i π λ λ L i cos θ i π f f L i cos θ i, i,,...,m 8.. Typically, but not necessarily, the L i are chosen to be quarter-wavelength long at λ, that is, L i /4. This way the same multilayer design can be applied equally well at microwave or at optical frequencies. Once the wavelength scale λ is chosen, the physical lengths of the layers l i can be obtained from l i L i λ /n i. 8. Lossy Multilayer Structures The multidiel function can be revised to handle lossy media. The reflection response of the multilayer structure is still computed from Eq but with some changes. In Sec. 7.7 we discussed the general case when either one or both of the incident and transmitted media are lossy. In the notation of Fig. 8.., we may assume that the incident medium n a is lossless and all the other ones, n i, i,,...,m,b, are lossy and nonmagnetic. To implement multidiel, one needs to know the real and imaginary parts of n i as functions of frequency, that is, n i ω n Ri ω jn Ii ω, or equivalently, the complex dielectric constants of the lossy media: Snel s law given in Eq. 8.. remains valid, except now the angles θ i and θ b are complex valued because n i,n b are. One can still define the transverse refractive indices n Ti through Eq using the complex-valued n i, and cos θ i given by: cos θ i sin θ i n a sin θ a n i, i a,,...,m,b 8.. The reflection coefficients defined in Eq. 8.. are equivalent to those given in Eq for the case of arbitrary incident and transmitted media. The phase thicknesses δ i now become complex-valued and are given by δ i i l i, where i is computed as follows. From Snel s law we have k xi k xa ω μ ɛ n a sin θ a k n a sin θ a, where k ω μ ɛ ω/c is the free-space wave number. Then, i ω μ ɛ i k xi ω n i n a sin θ a, i a,,...,m,b 8.. c Thus, the complex phase thicknesses are given by: δ i i l i ωl i n i c a sin θ a, i,,...,m 8..4 Writing c f λ for some reference frequency and wavelength, we may re-express 8..4 in terms of the normalized frequency and normalized physical lengths: δ i i l i π f l i n i f λ n a sin θ a, i,,...,m 8..5 To summarize, given the complex n i ω as in Eq. 8.. at each desired value of ω, we calculate cos θ i from Eq. 8.., n Ti and ρ Ti from Eqs and 8.., and thicknesses δ i from Eq Then, we use 8..7 to calculate the reflection response. The MATLAB function multidiel implements these steps, with usage: [Gamma,Z] multidieln,l,f,theta,pol; % lossy multilayer structure Once Γ is determined, one may calculate the power entering each layer as well as the power lost within each layer. The time-averaged power per unit area entering the ith layer is the z-component of the Poynting vector, which is given in terms of the transverse E, H fields as follows: P i Re E Ti H Ti, i,,...,m 8..6 The power absorbed within the ith layer is equal to the difference of the power entering the layer and the power leaving it: P loss i P i P i+, i,,...,m 8..7 ɛ i ω ɛ Ri ω jɛ Ii ω, i,,...,m,b ɛ i ω ɛ Ri ω jɛ Ii ω n i ω n Ri ω jn Ii ω ɛ ɛ 8.. The transverse fields can be calculated by inverting the recursion 8..8, that is, [ ] [ ][ ] ET,i+ cos δ i jη Ti sin δ i ETi H T,i+ jη Ti sin δ, i,,...,m 8..8 i cos δ i H Ti

3 8.. Single Dielectric Slab Multilayer Film Applications The recursion is initialized with the fields E T,H T at the first interface. These can be calculated with the help of Γ : E T E T+ + E T E T+ + Γ H T η Ta ET+ E T η Ta E T+ Γ 8..9 where η Ta η /n Ta. The field E T+ is the transverse component of the incident field. If we denote the total incident field by E in, then E T+ will be given by: E in, case E T+ 8.. E in cos θ a, case The total incident power along the direction of the incident wave vector, its z- component, and the power entering the first layer will be given as follows in both the and cases: P in η a E in, P in,z P in cos θ a, P P in,z Γ 8.. where η a η /n a. Thus, one can start with E in η a P in, if the incident power is known. 8. Single Dielectric Slab Many features of oblique incidence on multilayer slabs can be clarified by studying the single-slab case, shown in Fig Assuming that the media to the left and right are the same, n a n b, it follows that θ b θ a and also that ρ T ρ T. Moreover, Snel s law implies n a sin θ a n sin θ. Fig. 8.. Oblique incidence on single dielectric slab. Because there are no incident fields from the right, the reflection response at the left of interface- is: Γ T ρ T ρ T. It follows from Eq that the reflection response at the left of interface- will be: Γ T ρ T + ρ T e jδ ρ T e jδ + ρ T ρ T e jδ ρ 8.. T e jδ These are analogous to Eqs and According to Eq. 8.., the phase thickness can be written in the following normalized form, where L n l /λ : δ π λ λ L cos θ π f f L cos θ π f f 8.. f f L cos θ 8.. At frequencies that are integral multiples of f, f mf, the reflection response vanishes because δ πmf /f πm and e jδ. Similarly, at the halfintegral multiples, f m +.5f, the response is maximum because e jδ. Because f depends inversely on cos θ, then as the angle of incidence θ a increases, cos θ will decrease and f will shift towards higher frequencies. The maximum shift will occur when θ reaches its maximum refraction value θ c asinn a /n assuming n a <n. Similar shifts occur for the -db width of the reflection response notches. By the same calculation that led to Eq , we find for the -db width with respect to the variable δ : Δδ tan ρ T + ρ T Setting Δδ πδf/f, we solve for the -db width in frequency: Δf f π atan ρ T + ρ T 8..4 The left/right bandedge frequencies are f ± Δf/. The dependence of Δf on the incidence angle θ a is more complicated here because ρ T also depends on it. In fact, as θ a tends to its grazing value θ a 9 o, the reflection coefficients for either polarization have the limit ρ T, resulting in zero bandwidth Δf. On the other hand, at the Brewster angle, θ ab atann /n a, the reflection coefficient vanishes, resulting in maximum bandwidth. Indeed, because atan π/4, we have Δf max f atan/π f /. Fig. 8.. illustrates some of these properties. The refractive indices were n a n b and n.5. The optical length of the slab was taken to be half-wavelength at the reference wavelength λ, so that n l.5λ, or, L.5. The graphs show the and reflectances Γ T f as functions of frequency for the angles of incidence θ 75 o and θ a 85 o. The normal incidence case is also included for comparison. The corresponding refracted angles were θ asin n a asinθ a /n 4.9 o and θ 4.6 o. Note that the maximum refracted angle is θ c 4.8 o, and the Brewster angle, θ ab 56. o.

4 8.4. Frustrated Total Internal Reflection 9 8. Multilayer Film Applications θ Δ Γ T f θ a 75 o Δf normal f/f Fig. 8.. θ Γ T f θ a 85 o normal f/f and reflectances of half-wavelength slab. n b, with n a >n b, then there is percent reflection. The transmitted field into the rarer medium n b is evanescent, decaying exponentially with distance. However, if an object or another medium is brought near the interface from the n b side, the evanescent field is frustrated and can couple into a propagating wave. For example, if another semi-infinite medium n a is brought close to the interface, then the evanescent field can tunnel through to the other side, emerging as an attenuated version of the incident wave. This effect is referred to as frustrated total internal reflection. Fig shows how this may be realized with two 45 o prisms separated by a small air gap. With n a.5 and n b, the TIR angle is θ c asinn b /n a 4.8 o, therefore, θ 45 o >θ c. The transmitted fields into the air gap reach the next prism with an attenuated magnitude and get refracted into a propagating wave that emerges at the same angle θ. The notch frequencies were f f /L cos θ.f and f.4f for the angles θ a 75 o and 85 o. At normal incidence we have f f /L f, because L.5. The graphs also show the -db widths of the notches, calculated from Eq The reflection responses were computed with the help of the function multidiel with the typical MATLAB code: na ; nb ; n.5; L.5; f linspace,,4; theta 75; Fig Frustrated total internal reflection between two prisms separated by an air gap. G absmultidiel[na,n,nb], L,./f.^; Ge absmultidiel[na,n,nb], L,./f, theta, te.^; Gm absmultidiel[na,n,nb], L,./f, theta, tm.^; The shifting of the notch frequencies and the narrowing of the notch widths is evident from the graphs. Had we chosen θ a θ ab 56. o, the response would have been identically zero because of the factor ρ T in Eq The single-slab case is essentially a simplified version of a Fabry-Perot interferometer [64], used as a spectrum analyzer. At multiples of f, there are narrow transmittance bands. Because f depends on f / cos θ, the interferometer serves to separate different frequencies f in the input by mapping them onto different angles θ. Next, we look at three further applications of the single-slab case: a frustrated total internal reflection, b surface plasmon resonance, and c the perfect lens property of negative-index media. 8.4 Frustrated Total Internal Reflection As we discussed in Sec. 7.5, when a wave is incident at an angle greater than the total internal reflection TIR angle from an optically denser medium n a onto a rarer medium Fig shows an equivalent problem of two identical semi-infinite media n a, separated by a medium n b of length d. Let ε a n a, ε b n b be the relative dielectric constants. The components of the wavevectors in media n a and n b are: k x k n a sin θ, k ω c a k n a k x k n a cos θ k n b b n a sin θ, if θ θ c jk n a sin θ n b jα zb, if θ θ c 8.4. where sin θ c n b /n a. Because of Snel s law, the k x component is preserved across the interfaces. If θ>θ c, then b is pure imaginary, that is, evanescent. The transverse reflection and transmission responses are: Γ ρ a + ρ b e jkzbd + ρ a ρ b e jkzbd ρ a e jkzbd ρ ae jkzbd T τ aτ b e jkzbd + ρ a ρ b e jkzbd ρ a e jkzbd ρ ae jkzbd 8.4.

5 8.4. Frustrated Total Internal Reflection 8. Multilayer Film Applications Reflectance at θ 45 o Transmittance at θ 45 o.8.8 Γ.6.4 Γ d/λ.5.5 d/λ Fig Frustrated total internal reflection. Fig Reflectance and transmittance versus thickness d. where ρ a,ρ b are the transverse reflection coefficients at the a, b interfaces and τ a + ρ a and τ b + ρ b are the transmission coefficients, and we used the fact that ρ b ρ a because the media to the left and right of the slab are the same. For the two polarizations, ρ a is given in terms of the above wavevector components as follows: ρ a a b a + b, ρ a bε a a ε b b ε a + a ε b 8.4. For θ θ c, the coefficients ρ a are real-valued, and for θ θ c, they are unimodular, ρ a, given explicitly by ρ a n a cos θ + j n a sin θ n b n a cos θ j n a sin θ n b, ρ a jn a n a sin θ n b n b cos θ jn a n a sin θ n b + n b cos θ For all angles, it can be shown that Γ T, which represents the amount of power that enters perpendicularly into interface a and exits from interface b. For the TIR case, Γ, T simplify into: Γ ρ a e αzbd ρ ae, T ρ a e αzbd αzbd ρ ae, α αzbd zb π n a sin θ n b λ where we defined the free-space wavelength through k π/λ. Setting ρ a e jφa, the magnitude responses are given by: Γ sinh α zb d sinh, T sin φ a α zb d+ sin φ a sinh α zb d+ sin φ a For a prism with n a.5 and an air gap n b, Fig shows a plot of Eqs versus the distance d at the incidence angle θ 45 o. The reflectance becomes almost percent for thickness of a few wavelengths. Fig shows the reflectance versus angle over θ 9 o for the thicknesses d.4λ and d.5λ. The reflection response vanishes at the Brewster angle θ B atann b /n a.69 o. The case d.5λ was chosen because the slab becomes a half-wavelength slab at normal incidence, that is, b d π/ atθ o, resulting in the vanishing of Γ as can be seen from Eq The half-wavelength condition, and the corresponding vanishing of Γ, can be required at any desired angle θ <θ c, by demanding that b d π/ at that angle, which fixes the separation d: b d π πd n b λ n a sin θ π d λ n b n a sin θ Fig depicts the case θ o, which fixes the separation to be d.585λ. Γ Reflectance, d/λ.4 θ B θ c θ B.69 o θ c 4.8 o θ degrees Fig Γ Reflectance, d/λ.5 θ B θ c θ degrees Reflectance versus angle of incidence. The fields within the air gap can be determined using the layer recursions Let E a+ be the incident transverse field at the left side of the interface a, and E ± the transverse fields at the right side. Using Eq and 8..6, we find for the TIR case: E + + ρ ae a+ ρ ae αzbd, E ρ ae αzbd + ρ a E a+ ρ ae αzbd 8.4.7

6 8.5. Surface Plasmon Resonance 4 8. Multilayer Film Applications Γ Reflectance, half wavelength at o d/λ.585 However, if the incident plane wave is from a dielectric and from an angle that is greater than the angle of total internal reflection, then the corresponding wavenumber will be greater than its vacuum value and it could excite a plasmon wave along the interface. Fig depicts two possible configurations of how this can be accomplished.. θ θ B θ c θ degrees Fig Reflectance vanishes at θ o. The transverse electric field within the air gap will be then E T z E + e αzbz + E e αzbz, and similarly for the magnetic field. Using we find: [ ] + ρ a [e α E T z zbz ρ ρ ae αzbd a e αzbd e αzbz] E a+ [ H T z ρ a ρ ae αzbd ] [e α zbz + ρ a e αzbd e αzbz] E a+ η at where η at is the transverse impedance of medium n a, that is, with η a η /n a : η a cos θ a,, or parallel polarization η at η a / cos θ a,, or perpendicular polarization Fig Kretschmann-Raether and Otto configurations. In the so-called Kretschmann-Raether configuration [59,594], a thin metal film of thickness of a fraction of a wavelength is sandwiched between a prism and air and the incident wave is from the prism side. In the Otto configuration [59], there is an air gap between the prism and the metal. The two cases are similar, but we will consider in greater detail the Kretschmann-Raether configuration, which is depicted in more detail in Fig It is straightforward to verify that the transfer of power across the gap is independent of the distance z and given by P z z Re[ E T zh T z] Γ E a+ η at Frustrated total internal reflection has several applications [55 588], such as internal reflection spectroscopy, sensors, fingerprint identification, surface plasmon resonance, and high resolution microscopy. In many of these applications, the air gap is replaced by another, possibly lossy, medium. The above formulation remains valid with the replacement ε b n b ε b ε br jε bi, where the imaginary part ε ri characterizes the losses. 8.5 Surface Plasmon Resonance We saw in Sec. 7. that surface plasmons are waves that can exist at an interface between air and metal, and that their wavenumber k x of propagation along the interface is larger that its free-space value at the same frequency. Therefore, such plasmons cannot couple directly to plane waves incident on the interface. Fig Surface plasmon resonance excitation by total internal reflection. The relative dielectric constant ε a and refractive index n a of the prism are related by ε a n a. The air side has ε b n b, but any other lossless dielectric will do as long as it satisfies n b <n a. The TIR angle is sin θ c n b /n a, and the angle of incidence from the prism side is assumed to be θ θ c so that k x k n a sin θ k n b, k ω c 8.5. The geometrical picture in Fig is not valid for θ θ c because the wavevectors are complex-valued.

7 8.5. Surface Plasmon Resonance Multilayer Film Applications Because of Snel s law, the k x component of the wavevector along the interface is preserved across the media. The z-components in the prism and air sides are given by: a k n a k x k n a cos θ 8.5. b jα zb j k x k n b jk n a sin θ n b where b is pure imaginary because of the TIR assumption. Therefore, the transmitted wave into the ε b medium attenuates exponentially like e jkzbz e αzbz. For the metal layer, we assume that its relative dielectric constant is ε ε r jε i, with a negative real part ε r > and a small negative imaginary part <ε i ε r that represents losses. Moreover, in order for a surface plasmon wave to be supported on the ε ε b interface, we must further assume that ε r >ε b. The component within the metal will be complex-valued with a dominant imaginary part: j k x k ε j k x + k ε r + jε i jk n a sin θ + ε r + jε i 8.5. If there is a surface plasmon wave on the ε ε b interface, then as we saw in Sec. 7.7, it will be characterized by the specific values of k x,,b : εεb k x β x jα x k, k ε, b k ε b ε + ε b ε + εb ε + εb Using Eq. 7.., we have approximately to lowest order in ε i : εr ε b εr ε / b ε i β x k, α x k ε r ε b ε r ε b ε r and similarly for, which has a small real part and a dominant imaginary part: β z jα z, α z k ε r εr ε b, β z k ε r ε b ε i ε r ε b / If the incidence angle θ is such that k x is near the real-part of k x, that is, k x k n a sin θ β x, then a resonance takes place exciting the surface plasmon wave. Because of the finite thickness d of the metal layer and the assumed losses ε i, the actual resonance condition is not k x β x, but is modified by a small shift: k x β x + β x,to be determined shortly. At the resonance angle there is a sharp drop of the reflection response measured at the prism side. Let ρ a,ρ b denote the reflection coefficients at the ε a ε and ε ε b interfaces, as shown in Fig The corresponding reflection response of the structure will be given by: Γ ρ a + ρ b e jkzd + ρ a ρ b e jkzd ρ a + ρ b e αzd e jβzd + ρ a ρ b e αzd e jβzd where d is the thickness of the metal layer and β z jα z is given by Eq The reflection coefficients are given by: ρ a ε a a ε ε a + a ε, ρ b bε ε b b ε + ε b where a,b are given by Explicitly, we have for θ θ c : k ρ a ε ε a k x + jε a k x k ε ε k ε ε cos θ + jn a ε a sin θ ε a k x jε a k x k ε ε cos θ jn a ε a sin θ ε k ρ b ε x k ε b ε b k x k ε ε ε k x k ε ε a sin θ ε b ε b ε a sin θ ε b + ε b k x k ε ε ε a sin θ ε b + ε b ε a sin θ ε We note that for the plasmon resonance to be excited through such a configuration, the metal must be assumed to be slightly lossy, that is, ε i. If we assume that it is lossless with a negative real part, ε ε r, then, ρ a becomes a unimodular complex number, ρ a, for all angles θ, while ρ b remains real-valued for θ θ c, and also is pure imaginary, β z. Hence, it follows that: Γ ρ a + Reρ a ρ b e αzd + ρ b e 4αzd + Reρ a ρ b e αzd + ρ a ρ b e 4αzd Thus, it remains flat for θ θ c. For θ θ c, ρ a is still unimodular, and ρ b also becomes unimodular, ρ b. Setting ρ a e jφa and ρ b e jφb, we find for θ θ c : Γ e jφa + e jφb e αzd + e jφa e jφb e αzd + cosφ a φ b e αzd + e 4αzd + cosφ a + φ b e αzd + e 4αzd 8.5. which remains almost flat, exhibiting a slight variation with the angle for θ θ c. As an example, consider a quartz prism with n a.5, coated with a silver film of thickness of d 5 nm, and air on the other side ε b. The relative refractive index of the metal is taken to be ε 6.5j at the free-space wavelength of λ 6 nm. The corresponding free-space wave number is k π/λ 9.94 rad/μm. Fig shows the reflection response versus angle. The TIR angle is θ c asinn b /n a 4.8 o. The plasmon resonance occurs at the angle θ res 4.58 o. The graph on the right shows an expanded view over the angle range 4 o θ 45 o. Both angles θ c and θ res are indicated on the graphs as black dots. The computation can be carried out with the help of the MATLAB function multidiel.m, or alternatively multidiel.m, with the sample code: na.5; ea na^; % prism side er 6; ei.5; ep -er-j*ei; % silver layer nb ; eb nb^; % air side d 5; la 6; % in units of nanometers th linspace,89,89; % incident angle in degrees n sqrteep; % evanescent SQRT, needed if ε i L n*d/la; % complex optical length in units of λ n [na, n, nb]; % input to multidiel for i:lengthth, % reflectance Gai absmultidieln, L,, thi, tm.^; % at λ/λ end plotth,ga;

8 8.5. Surface Plasmon Resonance Multilayer Film Applications surface plasmon resonance expanded view The transverse electric and magnetic fields within the metal layer will be given by:.8.8 E T z E + e jkzz + E e jkzz, H T z η T [ E+ e jkzz E e jkzz] Γ θ degrees Fig Γ θ degrees Surface plasmon resonance. Fig shows the reflection response when the metal is assumed to be lossless with ε 6, all the other parameters being the same. As expected, there is no resonance and the reflectance stays flat for θ θ c, with mild variation for θ<θ c. Γ.8.6 reflectance Using the relationship η T /η at + ρ a / ρ a, we have: [ E T z + ρ a ] [e jz + ρ b e jkzd e jkzz] E a+ + ρ a ρ b e jkzd [ ] ρ a [e jk H T z + ρ a ρ b e jkzd ρ b e jkzd e jkzz] E a+ η at 8.5. where η at η a cos θ is the characteristic impedance of the prism. The power flow within the metal strip is described by the z-component of the Poynting vector: Pz Re[ E T zh T z] 8.5. The power entering the conductor at interface a is: P in Γ E a+ Re[ E T zht z] 8.5. z η at Fig shows a plot of the quantity Pz/P in versus distance within the metal, z d, at the resonant angle of incidence θ θ res. Because the fields are evanescent in the right medium n b, the power vanishes at interface b, that is, at z d. The reflectance at the resonance angle is Γ.5, and therefore, the fraction of the incident power that enters the metal layer and is absorbed by it is Γ power flow versus distance Fig θ degrees Absence of resonance when metal is assumed to be lossless. Pz / P in.5 Let E a+,e a be the forward and backward transverse electric fields at the left side of interface a. The fields at the right side of the interface can be obtained by inverting the matching matrix: [ Ea+ E a ] + ρ a [ ρa ρ a ][ E+ E ] [ E+ E ] ρ a Setting E a ΓE a+, with Γ given by Eq , we obtain: E + ρ aγ ρ a E a+ + ρ ae a+ + ρ a ρ b e jkzd E ρ a + Γ E a+ ρ be jkzd + ρ a E a+ ρ a + ρ a ρ b e jkzd [ ρa ρ a ][ Ea+ E a ] 4 5 z nm Fig Power flow within metal layer at the resonance angle θ res 4.58 o. The angle width of the resonance of Fig. 8.5., measured at the -db level Γ /, is very narrow, Δθ.8 o. The width Δθ, as well as the resonance angle θ res, and the optimum metal film thickness d, can be estimated by the following approximate procedure. To understand the resonance property, we look at the behavior of Γ in the neighborhood of the plasmon wavenumber k x k x given by At this value, the

9 8.5. Surface Plasmon Resonance 9 8. Multilayer Film Applications reflection coefficient at the ε ε b interface develops a pole, ρ b, which is equivalent to the condition b ε + ε b, with b, defined by Eq In the neighborhood of this pole, k x k x, ρ b will be given by ρ b K /k x k x, where K is the residue of the pole. It can be determined by: K lim k x k x ρ b lim k x k x bε ε b k x k x k x k x b ε + ε b b ε ε b d b ε + ε b dk x kxk x The derivative d /dk x can be determined by differentiating k z + k x k ε, that is, d + k x dk x, which gives d /dk x k x /, and similarly for db /dk x. It follows that: bε ε b K k x b ε k x ε b Inserting k x,,b from Eq , we obtain: / εεb K k ε b ε ε + ε b The reflection response can then be approximated near k x k x by Γ ρ a + K k x k x e jkzd + ρ a K k x k x e jkzd The quantities ρ a and e jkzd can also be replaced by their values at k x,,b, thus obtaining: k x k x + ρ a K e jkzd Γ ρ a k x k x + ρ a K e jkzd where ρ a ε a a ε ε a + a ε ε a + εεa ε b +ε a ε b ε a εε a ε b +ε a ε b which was obtained using a k ε a k x and Eqs Replacing ε ε r jε i, we may also write: ρ a ε a + j ε r + jε i ε a ε b ε a ε b ε a j b + ja ε r + jε i ε a ε b ε a ε b which serves as the definition of b,a. We also write: ρ a ε a j ε r + jε i ε a ε b ε a ε b ε a + j b + ja ε r + jε i ε a ε b ε a ε b b + a b ja We define also the wavenumber shifts that appear in the denominator and numerator of as follows: Then, Eq becomes, replacing k x β x jα x k x k x Γ ρ k x k x β x a k x k x k ρ β x +jα x ᾱ x a x k x β x β x +jα x + ᾱ x resulting in the reflectance: Γ ρ a k x β x β x +α x ᾱ x k x β x β 8.5. x +α x + ᾱ x The shifted resonance wavenumber is determined from the denominator of 8.5.9, that is, k x,res β x + β x. The resonance angle is determined by the matching condition: k x k n a sin θ res k x,res β x + β x 8.5. The minimum value of Γ at resonance is obtained by setting k x β x + β x : Γ min ρ a β x β x +α x ᾱ x 8.5. α x + ᾱ x We will see below that β x and β x are approximately equal, and so are ᾱ x and ᾱ x. The optimum thickness for the metal layer is obtained by minimizing the numerator of Γ min by imposing the condition α x ᾱ x. This condition can be solved for d. The angle width is obtained by solving for the left and right bandedge wavenumbers, say k x,±, from the -db condition: Γ ρ a k x β x β x +α x ᾱ x k x β x β x +α x + ᾱ x 8.5. and then obtaining the left/right -db angles by solving k n a sin θ ± k x,±. Although Eqs can be easily implemented numerically, they are unnecessarily complicated. A further simplification can be made by replacing the quantities K, ρ a, and by their lossless values obtained by setting ε i. This makes ρ a a unimodular complex number so that ρ a ρ a. We have then the approximations: εr ε / b K k ε r + ε b ε r ε b so that ρ a ε a + j ε r ε a ε b ε a ε b ε a j ε r ε a ε b ε a ε b b + ja, jα z, α z k ε r εr ε b b ε rε a ε b ε a ε a + ε b ε a ε b ε r + ε b The wavenumber shifts then become: ρ a b ja, a ε a εr ε a ε b ε a ε b ε a ε b ε r + ε b k x ρ a K e jkzd b ja K e jkzd β x jᾱ x k x ρ a K e jkzd b + ja K e jkzd β x + jᾱ x k x b ja K e αzd β x jᾱ x k x b + ja K e αzd β x + jᾱ x k x 8.5.6

10 8.5. Surface Plasmon Resonance 8. Multilayer Film Applications with β x b K e αzd, ᾱ x a K e αzd Then, the reflectance becomes in the neighborhood of the resonance: Γ k x β x β x +α x ᾱ x k x β x β x +α x + ᾱ x with a minimum value: Γ min α x ᾱ x α x + ᾱ x In this approximation, the resonance angle is determined from: k n a sin θ res k res β x + β εr ε b x k + b K e αzd 8.5. ε r ε b Since the second term on the right-hand side represents a small correction, a necessary condition that such a resonance angle would exist is obtained by setting θ res 9 o and ignoring the second term: εr ε b n a > n min a 8.5. ε r ε b For example, for the parameters of Fig. 8.5., the minimum acceptable refractive index n a would be n min a.. Thus, using a glass prism with n a.5 is more than adequate. If the right medium is water instead of air with n b., then n min a.4, which comes close to the prism choice. The -db angles are obtained by solving Γ k x k res +α x ᾱ x k x k res +α x + ᾱ x with solution k x,± k res ± 6α x ᾱ x α x ᾱ x,or k n a sin θ ± k n a sin θ res ± 6α x ᾱ x α x ᾱ x 8.5. The angle width shown on Fig was calculated by Δθ θ + θ using The optimum thickness d opt is obtained from the condition α x ᾱ x, which drives Γ min to zero. This condition requires that α x a K e αzd, with solution: d opt α z ln a K α x λ εr ε b 4a ε r ln 4π ε r ε i ε r + ε b 8.5. where we replaced α x from Eq For the same parameters of Fig. 8.5., we calculate the optimum thickness to be d opt 56.8 nm, resulting in the new resonance angle of θ res 4.55 o, and angle-width Δθ.7 o. Fig shows the reflectance in this case. The above approximations for the angle-width are not perfect, but they are adequate. One of the current uses of surface plasmon resonance is the detection of the presence of chemical and biological agents. This application makes use of the fact that the Γ Γ surface plasmon resonance θ degrees Γ expanded view θ degrees Fig Surface plasmon resonance at the optimum thickness d d opt. surface plasmon resonance, n b θ degrees Γ surface plasmon resonance, n b θ degrees Fig Shift of the resonance angle with the refractive index n b. resonance angle θ res is very sensitive to the dielectric constant of the medium n b. For example, Fig shows the shift in the resonance angle for the two cases n b.5 and n b. water. Using the same data as Fig. 8.5., the corresponding angles and widths were θ res o, Δθ.49 o and θ res 7 o, Δθ.5 o, respectively. A number of applications of surface plasmons were mentioned in Sec. 7., such as nanophotonics and biosensors. The reader is referred to [589 67] for further reading. 8.6 Perfect Lens in Negative-Index Media The perfect lens property of negative-index media was originally discussed by Veselago [87], who showed that a slab with ɛ ɛ and μ μ, and hence with refractive index n, can focus perfectly a point-source of light. More recently, Pendry [94] showed that such a slab can also amplify the evanescent waves from an object, and completely restore the object s spatial frequencies on the other side of the slab. The possibility of overcoming the diffraction limit and improving resolution with such a lens has generated a huge interest in the literature [87 469].

11 8.6. Perfect Lens in Negative-Index Media 4 8. Multilayer Film Applications Fig shows the perfect lens property. Consider a ray emanating from an object at distance z to the left of the slab z z. Assuming vacuum on either side of the slab n a n b, Snel s law, implies that the angle of incidence will be equal to the angle of refraction, bending in the same direction of the normal as the incident ray. Indeed, because n a and n, we have: n a sin θ a n sin θ sin θ a sin θ θ a θ corresponding magnetic field is: ωɛ [e jk ŷ E zz Γe jkzz] e jkxx, for z ωɛ [A+ H ŷ k e jk zz A e jk zz ] e jkxx, for z d z ωɛ ŷ E T e jkzz d e jkxx, for z d where k x is preserved across the interfaces, and,k z must satisfy: 8.6. k x + ω μ ɛ, k x + k z ω μɛ 8.6. Thus, ± ω μ ɛ k x and k z ± ω μɛ k x. The choice of square root signs is discussed below. To include evanescent waves, we will define by means of the evanescent square root, setting k ω μ ɛ : sqrte k k k k x, if k x k x j k x k, if k x k Fig Perfect lens property of a negative-index medium with n Moreover, η μ/ɛ μ /ɛ η and the slab is matched to the vacuum. Therefore, there will be no reflected ray at the left and the right interfaces. Indeed, the and reflection coefficients at the left interface vanish at any angle, for example, we have for the case, noting that cos θ cos θ a cos θ a : ρ η cos θ η cos θ a η cos θ + η cos θ a cos θ cos θ a cos θ + cos θ a Assuming that z <d, where d is the slab thickness, it can be seen from the geometry of Fig that the refracted rays will refocus at the point z z within the slab and then continue on to the right interface and refocus again at a distance d z from the slab, that is, at coordinate z d z. Next, we examine the field solutions inside and outside the slab for propagating and for evanescent waves. For the case, the electric field will have the following form within the three regions of z, z d, and z d: E [E ˆx k x [A + ˆx k x ẑ e jkzz + E Γ ˆx + k ] x ẑ e jkzz e jkxx, for z ẑ e jk zz + A ˆx + k ] x ẑ e jk zz e jkxx, for z d k z E T ˆx k x ẑ e jkzz d e jkxx, for z d k z 8.6. where Γ, T denote the overall transverse reflection and transmission coefficients, and A +,A, the transverse fields on the right-side of the left interface i.e., at z +. The We saw in Sec. 7.6 that for a single interface between a positive- and a negativeindex medium, and for propagating waves, we must have > and k z < in order for the power transmitted into the negative-index medium to flow away from the interface. But in the case of a slab within which one could have both forward and backward waves, the choice of the sign of k z is not immediately obvious. In fact, it turns out that the field solution remains invariant under the substitution k z k z, and therefore, one could choose either sign for k z. In particular, we could select it to be given also by its evanescent square root, where n ɛμ/ɛ μ : k z sqrte k n k x k n k x, if k x k n j k x k n, if k x k n By matching the boundary conditions at the two interfaces z and z d, the parameters Γ, A ±,T are obtained from the usual transfer matrices see Sec. 8.: [ ] [ ][ ] E ρ A+ E Γ + ρ ρ A [ ] [ ] [ ][ ] A+ e jk zd ρ E T A e jk zd ρ ρ where, ρ k zɛ ɛ k zɛ + ɛ ζ ζ +, ζ η η k zɛ ɛ where ζ is a normalized characteristic impedance. The solution of Eqs is then, Γ ρ e jk zd ζ e jk zd ρ e jk zd ζ + ζ e jk zd ρ T e jk z d ρ e 4ζ jk zd ζ + e jk zd ζ e jk zd

12 8.6. Perfect Lens in Negative-Index Media Multilayer Film Applications Similarly, the coefficients A ± are found to be: A + ρ Γ [ + ζ + ζ Γ ] E ρ A ρ + Γ [ ζ + + ζ Γ ] E ρ The case is obtained from the case by a duality transformation, that is, by the replacements, E H, H E, ɛ μ, ɛ μ, and ρ ρ, where ρ μ k zμ μ + k zμ ζ ζ +, ζ η μ η k zμ The invariance under the transformation k z k z follows from these solutions. For example, noting that ζ ζ under this transformation, we have: Γ k z ζ e jk zd ζ + ζ e jk zd ζ e jk zd ζ + ζ e jk zd Γk z Similarly, we find T k z Tk z and A ± k z A k z. These imply that the field solutions remain invariant. For example, the electric field inside the slab will be: Ez, k z [ A + k z ˆx k x k z [A + k z ˆx k x k ẑ z ẑ e j k zz + A k z ˆx + e jk zz + A k z Similarly, we have for the magnetic field inside the slab: ωɛ Hz, k z ŷ k z ωɛ ŷ k z ˆx + k x k ẑ z k x k z e jk zz ] ẑ e j k zz e jkxx ] e jkxx Ez, +k z [A+ k z e j k zz A k z ej k zz ] e jkxx [A+ k ze jk zz A k ze jk zz ] e jkxx Hz, +k z Next, we apply these results to the case μ μ and ɛ ɛ, having n. It follows from Eq that k z with given by In this case, ζ k zɛ/ ɛ k z/ ±. Then, Eq implies that Γ for either choice of sign. Similarly, we have T e jkzd, again for either sign of ζ : T e jkzd e jkzd, if k x k, k k x e αzd, if k x 8.6. k, j k x k jα z Thus, the negative-index medium amplifies the transmitted evanescent waves, which was Pendry s observation [94]. The two choices for k z lead to the A ± coefficients: E E ˆx k x ẑ e jkzz e jkxx, for z E ˆx + k x ẑ E ˆx k x ẑ and the corresponding magnetic field: H e jkzz e jkxx, for z d e jkzz d e jkxx, for z d ωɛ ŷ E e jkzz e jkxx, for z ŷ E ωɛ ŷ E ωɛ e jkzz e jkxx, for z d e jkzz d e jkxx, for z d The solution effectively corresponds to the choice k z and is valid for both propagating and evanescent waves with given by In Eq the constant E refers to the value of the transverse electric field at z. Changing the reference point to z z at the left of the slab as shown in Fig. 8.6., amounts to replacing E E e jkzz. Then, 8.6. reads: E E ˆx k x ẑ e jkzz+z e jkxx, for z z E ˆx + k x ẑ E ˆx k x ẑ e jkzz z e jkxx, for z d e jkzz d+z e jkxx, for z d Setting α z as in 8.6.4, we find the evanescent fields: E ˆx k x ẑ e αzz+z e jkxx, for z z jα z E E ˆx + k x ẑ e αzz z e jkxx, for z d jα z E ˆx k x ẑ e αzz d+z e jkxx, for z d jα z The field is amplified inside the slab. The propagation factors along the z-direction agree at the points z z, z z, and z d z, k z ζ + A + E, A k z + ζ A +, A E 8.6. For either choice, the field solutions are the same. Indeed, inserting either set of A +,A into Eqs and 8.6., and using 8.6., we find: e jkzz+z z z e jkzz z zz e jkzz d+z zd z e αzz+z z z e αzz z zz e αzz d+z zd z 8.6.6

13 8.6. Perfect Lens in Negative-Index Media Multilayer Film Applications which imply the complete restoration of the source at the focal points inside and to the right of the slab: E x x, z z z E x x, z zz E x x, z zd z Transmittance ε μ. j ε μ. j ε μ. j 4 Transmittance Fig shows a plot of the evanescent component E x z of Eq versus distance z inside and outside the slab. db db k x /k 4 ε μ..j ε μ.98.j ε μ.5.j ε μ.9.j k x /k Fig Transmittance under non-ideal conditions ɛ, μ are in units of ɛ,μ. Fig Evanesenct wave amplification inside a negative-index medium. Using the plane-wave spectrum representation of Sec. 8.7, a more general singlefrequency solution can be built by superposition of the plane waves and If the field at the image plane z z has the general representation: Ex, z E k x ˆx k x ẑ e jkxx dk x π where the integral over k x includes both propagating and evanescent modes and is given by 8.6.4, then, then field in the three regions to the left of, inside, and to the right of the slab will have the form: Ex, z E k x ˆx k x ẑ e jkzz+z e jkxx dk x, π for z z Ex, z E k x ˆx + k x ẑ e jkzz z e jkxx dk x, π for z d Ex, z E k x π ˆx k x ẑ e jkzz d+z e jkxx dk x, for z d It is evident that Eq is still satisfied, showing the perfect reconstruction of the object field at the two image planes. The perfect lens property is highly sensitive to the deviations from the ideal values of ɛ ɛ and μ μ, and to the presence of losses. Fig plots the transmittance in db, that is, the quantity log Te jkzd versus k x, with T computed from Eq for different values of ɛ, μ and for d.λ.π/k. In the ideal case, because of the result 8.6., we have Te jkzd for both propagating and evanescent values of k x, that is, the transmittance is flat at db for all k x. The left graph shows the effect of losses while keeping the real parts of ɛ, μ at the ideal values ɛ, μ. In the presence of losses, the transmittance acts like a lowpass filter in the spatial frequency k x. The right graph shows the effect of the deviation of the real parts of ɛ, μ from the ideal values. If the real parts deviate, even slightly, from ɛ, μ, the transmittance develops resonance peaks, which are related to the excitation of surface plasmons at the two interfaces of the slab [4,44]. The peaks are due to the poles of the denominator of T in Eq , that is, the roots of ρ zd e jk e jk zd ρ e jk zd ±ρ For evanescent k x, we may replace jα z and k z jα z, where α z kx k and α z k x k n, and obtain the conditions: e α zd ±ρ ± α z ɛ α z ɛ α zɛ + α z ɛ These are equivalent to [4,44]: α tanh z d α zɛ α α, tanh z d α z ɛ 8.6. zɛ α z ɛ For k x k, we may replace α z α z k x in in order to get en estimate of the resonant k x : e kx,resd ± ɛ ɛ e Rekx,resd ɛ + ɛ ɛ ɛ ɛ + ɛ Rek x,res d ln ɛ ɛ ɛ + ɛ 8.6. and for the case, we must replace ɛs byμs. The value k x Rek x,res represents the highest achievable resolution by the slab, with the smallest resolvable transverse distance being of the order of Δx / Rek x,res. If ɛ is real-valued and near ɛ, then, k x,res is real and there will be an infinite resonance peak at k x k x,res. This is seen in the above figure in the first two cases of

14 8.6. Perfect Lens in Negative-Index Media 9 8. Multilayer Film Applications ɛ/ɛ μ/μ. and ɛ/ɛ μ/μ.98 the apparent finite height of these two peaks is due to the finite grid of k x values in the graph. The last two cases have complex-valued ɛ, μ with a small imaginary part, with the resulting peaks being finite. In all cases, the peak locations k x Rek x,res obtained by solving Eqs numerically for k x,res are indicated in the graphs by bullets placed at the peak maxima. The numerical solutions were obtained by the following iterative procedures, initialized at the approximate complex-valued solution of 8.6.: initialize: k x d ln ɛ ɛ ɛ + ɛ for i,,...,n iter, do: α z k x k n α z ɛ α ɛ α z tanh z d k x α z + k or, k x d ln ɛ ɛ, α z k ɛ + ɛ x k n for i,,...,n iter, do: α z k x k α z ɛ α α z tanh z d ɛ k x α z + k n The number of iterations was typically N iter. Both graphs of Fig also show dips at k x k. These are due to the zeros of the transmittance T arising from the numerator factor ρ in At k x k, we have α z and ρ, causing a zero in T. In addition to the zero at k x k, it is possible to also have poles in the vicinity of k, as indicated by the peaks and bullets in the graph. Fig shows an expanded view of the structure of T near k, with the k x restricted in the narrow interval:.99k k x.k. db Transmittance which is itself small. Then, we apply Eq to get α z and from it, the resonant k x,res : α z ɛ α ɛ α z tanh z d k x,res α z + k 8.7 Antireflection Coatings at Oblique Incidence Antireflection coatings are typically designed for normal incidence and then used over a limited range of oblique incidence, such as up to about o. As the angle of incidence increases, the antireflection band shifts towards lower wavelengths or higher frequencies. Any designed reflection zeros at normal incidence are no longer zeros at oblique incidence. If a particular angle of incidence is preferred, it is possible to design the antireflection coating to match that angle. However, like the case of normal design, the effectiveness of this method will be over an angular width of approximately o about the preferred angle. To appreciate the effects of oblique incidence, we look at the angular behavior of our normal-incidence designs presented in Figs. 6.. and 6... The first example was a two-layer design with refractive indices n a air, n.8 magnesium fluoride, n.45 bismuth oxide, and n b.5 glass. The designed normalized optical lengths of the layers were L.94 and L.45 at λ 55 nm. Fig shows the and reflectances Γ T λ as functions of λ, for the incidence angles θ o, o, o, 4 o. Γ T λ percent 4 polarization o o o 4 o Γ T λ percent 4 polarization o o o 4 o.99. k x /k Fig Expanded view of the zero/pole behavior in the vicinity of k x k. For last two cases depicted on this graph that have n ɛμ /ɛ μ, an approximate calculation of the pole locations near k is as follows. Since α z k x k is small, and α z α z + k n, we have to first order in α z, α z k n α z, Fig Two-layer antireflection coating at oblique incidence. We note the shifting of the responses towards lower wavelengths. The responses are fairly acceptable up to about o o. The typical MATLAB code used to generate these graphs was: n [,.8,.45,.5]; L [.94,.45]; la 55; la linspace4,7,; pol te ;

15 8.7. Antireflection Coatings at Oblique Incidence 8. Multilayer Film Applications G absmultidieln, L, la/la.^ * ; G absmultidieln, L, la/la,, pol.^ * ; G absmultidieln, L, la/la,, pol.^ * ; G4 absmultidieln, L, la/la, 4, pol.^ * ; plotla, [G; G; G; G4]; As we mentioned above, the design can be matched at a particular angle of incidence. As an example, we choose θ a o and redesign the two-layer structure. The design equations are still 6.. and 6.., but with the replacement of n i, ρ i by their transverse values n Ti, ρ Ti, and the replacement of k l, k l by the phase thicknesses at λ λ, that is, δ πl cos θ and δ πl cos θ. Moreover, we must choose to match the design either for or polarization. Fig illustrates such a design. The upper left graph shows the reflectance matched at o. The designed optical thicknesses are in this case, L.59 and L.58. The upper right graph shows the corresponding reflectance, which cannot be matched simultaneously with the case. The lower graphs show the same design, but now the reflectance is matched at o. The designed lengths were L.554 and L matched at o 4 unmatched at o The design steps are as follows. First, we calculate the refraction angles in all media from Eq. 8.., θ i asinn a sin θ a /n i, for i a,,,b. Then, assuming polarization, we calculate the refractive indices for all media n Ti n i cos θ i, i a,,,b. Then, we calculate the transverse reflection coefficients ρ Ti from Eq. 8.. and use them to solve Eq. 6.. and 6.. for the phase thicknesses δ,δ. Finally, we calculate the normalized optical lengths from L i δ i /π cos θ i, i,. The following MATLAB code illustrates these steps: n [,.8,.45,.5]; na ; tha ; thi asinna*sinpi*tha/8./n; nt n.*costhi; r nrnt; % for use nt n./costhi c sqrtr^*-r*r^ - r-r^/4*r*r*-r^; de acosc; G r+r*exp-*j*de/ + r*r*exp-*j*de; de angleg - pi - angler/; if de <, de de + *pi; end L [de,de]//pi; L L./costhi:; la 55; la linspace4,7,4; Γ T λ percent Γ T λ percent o o 4 o o matched at o 4 o o 4 o o Γ T λ percent Γ T λ percent o o 4 o o unmatched at o 4 o o 4 o o G absmultidieln, L, la/la,, te.^ * ; G absmultidieln, L, la/la,, te.^ * ; G4 absmultidieln, L, la/la, 4, te.^ * ; G absmultidieln, L, la/la.^ * ; plotla, [G; G; G4; G]; Our second example in Fig. 6.. was a quarter-half-quarter -layer design with refractive indices n air, n.8 magnesium fluoride, n. zirconium oxide, n.6 cerium fluoride, and n b.5 glass. The optical lengths of the layers were L L.5 and L.5. Fig shows the and reflectances Γ T λ as functions of λ, for the incidence angles θ o, o, o, 4 o. The responses are fairly acceptable up to about o o, but are shifted towards lower wavelengths. The typical MATLAB code used to generate these graphs was: n [,.8,.,.6,.5]; L [.5,.5,.5]; la 55; la linspace4,7,4; G absmultidieln, L, la/la.^ * ; G absmultidieln, L, la/la,, te.^ * ; G absmultidieln, L, la/la,, te.^ * ; G4 absmultidieln, L, la/la, 4, te.^ * ; plotla, [G; G; G; G4]; Fig Two-layer antireflection coating matched at degrees.

16 8.8. Omnidirectional Dielectric Mirrors 4 8. Multilayer Film Applications Γ T λ percent 4 polarization o o o 4 o Fig Γ T λ percent 4 polarization o o o 4 o Three-layer antireflection coating at oblique incidence. 8.8 Omnidirectional Dielectric Mirrors Until recently, it was generally thought that it was impossible to have an omnidirectional dielectric mirror, that is, a mirror that is perfectly reflecting at all angles of incidence and for both and polarizations. However, such mirrors are possible and have recently been manufactured [77,774] and the conditions for their existence clarified [77 777]. We consider the same dielectric mirror structure of Sec. 6., consisting of alternating layers of high and low index. Fig shows such a structure under oblique incidence. There are N bilayers and a total of M N + single layers, starting and ending with a high-index layer. δ H π f f L H cos θ H, δ L π f f L L cos θ L 8.8. where L H n H l H /λ, L L n L l L /λ are the optical thicknesses normalized to some λ, and f c /λ. Note also, cos θ i n a sin θ a /n i, i H, L. A necessary but not sufficient condition for omnidirectional reflectivity for both polarizations is that the maximum angle of refraction θ H,max inside the first layer be less than the Brewster angle θ B of the second interface, that is, the high-low interface, so that the Brewster angle can never be accessed by a wave incident on the first interface. If this condition is not satisfied, a wave would not be reflected at the second and all subsequent interfaces and will transmit through the structure. Because sin θ H,max n a /n H and tan θ B n L /n H, or, sin θ B n L / n H + n L, the condition θ H,max <θ B, or the equivalent condition sin θ H,max < sin θ B, can be written as n a /n H <n L / n H + n L,or n a < n Hn L n H + n L 8.8. We note that the exact opposite of this condition is required in the design of multilayer Brewster polarizing beam splitters, discussed in the next section. In addition to condition 8.8., in order to achieve omnidirectional reflectivity we must require that the high-reflectance bands have a common overlapping region for all incidence angles and for both polarizations. To determine these bands, we note that the entire discussion of Sec. 6. carries through unchanged, provided we use the transverse reflection coefficients and transverse refractive indices. For example, the transverse version of the bilayer transition matrix of Eq will be: [ ] e jδ H+δ L ρ T F T ejδh δl jρ T e jδh sin δ L ρ jρ T T e jδh sin δ L e jδh+δl ρ T e jδh δl where ρ T n HT n LT /n HT + n LT and: n HT n H cos θ H n H cos θ H n LT n L cos θ L n L cos θ L polarization polarization Fig Dielectric mirror at oblique incidence. The incidence angles on each interface are related by Snel s law: n a sin θ a n H sin θ H n L sin θ L n b sin θ b 8.8. The phase thicknesses within the high- and low-index layers are in normalized form: Explicitly, we have for the two polarizations: ρ n H cos θ L n L cos θ H, n H cos θ L + n L cos θ H The trace of F T is as in Eq. 6..: a cosδ H + δ L ρ T cosδ H δ L ρ T ρ n H cos θ H n L cos θ L n H cos θ H + n L cos θ L

17 8.8. Omnidirectional Dielectric Mirrors Multilayer Film Applications The eigenvalues of the matrix F T are λ ± e ±jkl, where K acosa/l and l l H +l L. The condition a determines the bandedge frequencies of the high-reflectance bands. As in Eq. 6..6, this condition is equivalent to: cos δ H + δ L ρ T cos δ H δ L Defining the quantities L ± L H cos θ H ± L L cos θ L and the normalized frequency F f/f, we may write: δ H ± δ L π f L H cos θ H ± L L cos θ L πfl ± f Then, taking square roots of Eq , we have: cosπfl + ± ρ T cosπfl The plus sign gives the left bandedge, F f /f, and the minus sign, the right bandedge, F f /f. Thus, F,F are the solutions of the equations: cosπf L + ρ T cosπf L cosπf L + ρ T cosπf L The bandwidth and center frequency of the reflecting band are: Δf f ΔF F F, f c F c F + F f The corresponding bandwidth in wavelengths is defined in terms of the left and right bandedge wavelengths: λ λ F c f, λ λ F c f, Δλ λ λ 8.8. An approximate solution of Eq can be obtained by setting L inthe right-hand sides of Eq. 8.8.: with solutions: cosπf L + ρ T, cosπf L + ρ T 8.8. F acos ρ T, F acos ρ T πl + πl + Using the trigonometric identities acos± ρ T π/ asin ρ T, we obtain the bandwidth and center frequency: Δf f f f asin ρ T, f c f + f f πl + L + It follows that the center wavelength will be λ c c /f c L + λ or, λ c L + λ l H n H cos θ H + l L n L cos θ L At normal incidence, we have λ c l H n H + l L n L. For quarter-wavelength designs at λ at normal incidence, we have L + /4 + /4 /, so that λ c λ. The accuracy of the approximate solution depends on the ratio d L /L +. Even if at normal incidence the layers were quarter-wavelength with L H L L.5, the equality of L H and L L will no longer be true at other angles of incidence. In fact, the quantity d is an increasing function of θ a. For larger values of d, the exact solution of 8.8. can be obtained by the following iteration: initialize with F F, for i,,...,n iter, do: F πl + acos ρ T cosπf L F πl + acos ρ T cosπf L Evidently, the i iteration gives the zeroth-order solution The iteration converges extremely fast, requiring only 4 iterations N iter. The MATLAB function omniband implements this algorithm. It has usage: [F,F] omnibandna,nh,nl,lh,ll,theta,pol,niter % bandedge frequencies [F,F] omnibandna,nh,nl,lh,ll,theta,pol % equivalent to N iter where theta is the incidence angle in degrees, pol is one of the strings te or tm for or polarization, and Niter is the desired number of iterations. If this argument is omitted, only the i iteration is carried out. It is straightforward but tedious to verify the following facts about the above solutions. First, f,f are increasing functions of θ a for both and polarizations. Thus, the center frequency of the band f c f +f / shifts towards higher frequencies with increasing angle θ a. The corresponding wavelength intervals will shift towards lower wavelengths. Second, the bandwidth Δf f f is an increasing function of θ a for, and a decreasing one for polarization. Thus, as θ a increases, the reflecting band for expands and that of shrinks, while their slightly different centers f c shift upwards. In order to achieve omnidirectional reflectivity, the and bands must have a common overlapping intersection for all angles of incidence. Because the band is always narrower than the band, it will determine the final common omnidirectional band. The worst case of overlap is for the band at 9 o angle of incidence, which must overlap with the / band at o. The left bandedge of this band, f, 9 o, must be less than the right bandedge of the o band, f o. This is a sufficient condition for omnidirectional reflectivity. Thus, the minimum band shared by all angles of incidence and both polarizations will be [f, 9 o, f o ], having width:

18 8.8. Omnidirectional Dielectric Mirrors Multilayer Film Applications Frequency Response at 45 Frequency Response at 8 Δf min f o f, 9 o minimum omnidirectional bandwidth In a more restricted sense, the common reflecting band for both polarizations and for angles up to a given θ a will be [f, θ a, f, o ] and the corresponding bandwidth: Δfθ a f o f, θ a In addition to computing the bandwidths of either the or the bands at any angle of incidence, the function omniband can also compute the above common bandwidths. If the parameter pol is equal to tem, then F,F are those of Eqs and Its extended usage is as follows: [F,F] omnibandna,nh,nl,lh,ll,theta, tem % Eq [F,F] omnibandna,nh,nl,lh,ll,9, tem % Eq [F,F] omnibandna,nh,nl,lh,ll % Eq Next, we discuss some simulation examples that will help clarify the above remarks. Example 8.8.: The first example is the angular dependence of Example 6... In order to flatten out and sharpen the edges of the reflecting bands, we use N bilayers. Fig shows the and reflectances Γ T λ as functions of the free-space wavelength λ, for the two angles of incidence θ a 45 o and 8 o. Fig depicts the reflectances as functions of frequency f. The refractive indices were n a, n H., n L.8, n b.5, and the bilayers were quarter-wavelength L H L L.5 at the normalization wavelength λ 5 nm. The necessary condition 8.8. is satisfied and we find for the maximum angle of refraction and the Brewster angle: θ H,max 5.5 o and θ B.75 o Thus, we have θ H,max <θ B. Reflectance at 45 Reflectance at 8 Γ T f percent o f c f c f/f f c o f/f Fig and frequency responses for n H., n L.8. Γ T f percent or lower wavelengths, and the shrinking of the and expanding of the bands, and the shrinking of the common band. At 45 o, there is still sufficient overlap, but at 8 o, the band has shifted almost to the end of the o band, resulting in an extremely narrow common band. The arrows labeled f c and f c represent the band center frequencies at o and 45 o or 8 o. The calculated bandedges corresponding to 9 o incidence were λ λ /F, o 49.7 nm and λ λ /F, 9 o 4.6 nm, with bandwidth Δλ λ λ.4 nm. Thus, this structure does exhibit omnidirectional reflectivity, albeit over a very narrow band. The MATLAB code used to generate these graphs was: na ; nb.5; nh.; nl.8; LH.5; LL.5; la 5; la linspace,8,5; Γ T λ percent λ c λ c o Γ T λ percent λ c o th 45; N ; n [na, nh, repmat[nl,nh],, N, nb]; L [LH, repmat[ll,lh],, N]; Ge *absmultidieln,l,la/la, th, te.^; Gm *absmultidieln,l,la/la, th, tm.^; G *absmultidieln,l,la/la.^; plotla,gm, la,ge, la,g; [F,F] omnibandna,nh,nl,lh,ll,, te ; [Fe,Fe] omnibandna,nh,nl,lh,ll, th, te ; [Fm,Fm] omnibandna,nh,nl,lh,ll, th, tm ; [F,F] omnibandna,nh,nl,lh,ll, th, tem ; Fig and reflectances for n H., n L.8. On each graph, we have indicated the corresponding bandwidth intervals calculated with omniband. The indicated intervals are for o incidence, for and, and for the common band Eq at θ a. We observe the shifting of the bands towards higher frequencies, Because the reflectivity bands shrink with decreasing ratio n H /n L, if we were to slightly decrease n H, then the band could be made to shift beyond the end of the o band and there would be no common overlapping reflecting band for all angles. We can observe this behavior in Fig , which has n H, with all the other parameters kept the same.

19 8.8. Omnidirectional Dielectric Mirrors Multilayer Film Applications Frequency Response at 45 Frequency Response at 8 range of angles θ a 9 o for both polarizations. The left graph of Fig shows the case n H, n L.8, and the right graph, the case n H, n L.8. Γ T f percent o f c f c Γ T f percent F, F.5 and bandwidths omnidirectional band F, F.5 and bandwidths f/f f c o f/f.5 band band.5 band band Fig and reflectances for n H, n L θ a degrees θ a degrees Γ T f percent At 45 o there is a common overlap, but at 8 o, the band has already moved beyond the o band, while the band still overlaps with the latter. This example has no omnidirectional reflectivity, although the necessary condition 8.8. is still satisfied with θ H,max o and θ B 4.6 o. On the other hand, if we were to increase n H, all the bands will widen, and so will the final common band, resulting in an omnidirectional mirror of wider bandwidth. Fig shows the case of n H, exhibiting a substantial overlap and omnidirectional behavior Frequency Response at 45 f c f c o f/f Γ T f percent Frequency Response at 8 f c o f/f Fig and reflectances for n H, n L.8. The minimum band was [F,F ] [.465,.4] corresponding to the wavelength bandedges λ λ /F 4.84 nm and λ λ /F nm with a width of Δλ λ λ nm, a substantial difference from that of Fig The bandedges were computed with N it in Eq ; with N it, we obtain the more accurate values: [F,F ] [.55,.4]. To illustrate the dependence of the and bandwidths on the incident angle θ a,we have calculated and plotted the normalized bandedge frequencies F θ a, F θ a for the Fig / bandgaps versus angle for n H, n L.8 and n H, n L.8. We note that the band widens with increasing angle, whereas the band narrows. At the same time, the band centers move toward higher frequencies. In the left graph, there is a common band shared by both polarizations and all angles, that is, the band defined by F o, and F, 9 o. For the right graph, the bandedge F, θ a increases beyond F o for angles θ a greater than about 6.8 o, and therefore, there is no omnidirectional band. The calculations of F θ a, F θ a were done with omniband with N iter. Example 8.8.: In Fig , we study the effect of changing the optical lengths of the bilayers from quarter-wavelength to L H. and L L.. The main result is to narrow the bands. This example, also illustrates the use of the iteration The approximate solution and exact solutions for the 8 o bandedge frequencies are obtained from the two MATLAB calls: [F,F] omnibandna,nh,nl,lh,ll,8, tem,; [F,F] omnibandna,nh,nl,lh,ll,8, tem,; with results [F,F ] [.9,.89] and [F,F ] [.5,.66], respectively. Three iterations produce an excellent approximation to the exact solution. Example 8.8.: Here, we revisit Example 6.., whose parameters correspond to the recently constructed omnidirectional infrared mirror [77]. Fig shows the reflectances as functions of wavelength and frequency at θ a 45 o and 8 o for both and polarizations. At both angles of incidence there is a wide overlap, essentially over the desired 5 μm band. The structure consisted of nine alternating layers of Tellurium n H 4.6 and Polystyrene n L.6 on a NaCl substrate n b.48. The physical lengths were l H.8 and l L.6 μm. The normalizing wavelength was λ.5 μm. The optical thicknesses in units of λ were L H.944 and L L.. The bandedges at o were [F,F ] [.6764,.875] with center frequency F c.989, corresponding to wavelength λ c λ /F c.7 μm. Similarly, at 45 o, the band centers

20 8.8. Omnidirectional Dielectric Mirrors Multilayer Film Applications Γ T f percent Γ T λ percent Γ T f percent o Frequency Response at 45 f c f c f/f Γ T f percent Frequency Response at 8 f c o f/f Fig Unequal length layers L H., L L.5. for and polarizations were F c,.7 and F c,., resulting in the wavelengths λ c,.7 and λ c,. μm shown on the graphs are the centers Reflectance at 45 λ c λ c o λ μm Frequency Response at f c f c o f/f Fig Γ T λ percent Γ T f percent Reflectance at 8 λ c λ c o λ μm Frequency Response at f c f c o f/f Nine-layer Te/PS omnidirectional mirror over the infrared. Γ T λ percent only. The final bandedges of the common reflecting band computed from Eq were [F,F ] [.87,.875], resulting in the wavelength bandedges λ λ /F 9.7 and λ λ /F 4.95 μm, with a width of Δλ λ λ 5.4 μm and band center λ + λ /. μm the approximation gives 5.67 and.4 μm. The graphs were generated by the following MATLAB code: la.5; la linspace5,5,4; na ; nb.48; nh 4.6; nl.6; lh.8; ll.65; LH nh*lh/la; LL nl*ll/la; th 45; N 4; n [na, nh, repmat[nl,nh],, N, nb]; L [LH, repmat[ll,lh],, N]; Ge *absmultidieln,l,la/la, th, te.^; Gm *absmultidieln,l,la/la, th, tm.^; G *absmultidieln,l,la/la.^; plotla,gm, la,ge, la,g; Ni 5; [F,F] omnibandna,nh,nl,lh,ll,, te, Ni; % band at o [Fe,Fe] omnibandna,nh,nl,lh,ll, th, te, Ni; %band [Fm,Fm] omnibandna,nh,nl,lh,ll, th, tm, Ni; % band [F,F] omnibandna,nh,nl,lh,ll, th, tem,ni; % Eq [F,F] omnibandna,nh,nl,lh,ll, 9, tem,ni; % Eq Finally, Fig shows the same example with the number of bilayers doubled to N 8. The mirror bands are flatter and sharper, but the widths are the same. Reflectance at 45 λ c λ c o λ μm Γ T λ percent Reflectance at 8 λ c λ c Fig Omnidirectional mirror with N 8. o λ μm Example 8.8.4: The last example has parameters corresponding to the recently constructed omnidirectional reflector over the visible range [774]. The refractive indices were n a, n H.6 ZnSe, n L.4 Na AlF 6 cryolite, and n b.5 glass substrate. The layer

21 8.8. Omnidirectional Dielectric Mirrors 4 lengths were l H l L 9 nm. There were N 9 bilayers or N + 9 layers, starting and ending with n H. With these values, the maximum angle of refraction is θ H,max.7 o and is less than the Brewster angle θ B 7.7 o. The normalizing wavelength was taken to be λ 6 nm. Then, the corresponding optical lengths were L L n L l L /λ.945 and L H n H l H /λ.774. The overall minimum omnidirectional band is [λ,λ ] [65.4, ] nm. It was computed by the MATLAB call to omniband with N i 5 iterations: [F,F] omnibandna,nh,nl,lh,ll,9, tem,ni; la la/f; la la/f; The values of λ,λ do not depend on the choice of λ. Fig shows the reflectance at 45 o and 8 o. The upper panel of graphs has N 9 bilayers as in [774]. The lower panel has N 8 bilayers or 8 layers, and has more well-defined band gaps. The two arrows in the figures correspond to the values of λ,λ of the minimum omnidirectional band. Reflectance at 45 Reflectance at Multilayer Film Applications 8.9 Polarizing Beam Splitters The objective of an omnidirectional mirror is to achieve high reflectivity for both polarizations. However, in polarizers, we are interested in separating the and polarizations. This can be accomplished with a periodic bilayer structure of the type shown in Fig. 8.8., which is highly reflecting only for and highly transmitting for polarizations. This is the principle of the so-called MacNeille polarizers [66,64,64,66,665,68 686]. If the angle of incidence θ a is chosen such that the angle of refraction in the first high-index layer is equal to the Brewster angle of the high-low interface, then the component will not be reflected at the bilayer interfaces and will transmit through. The design condition is θ H θ B,orsinθ H sin θ B, which gives: n a sin θ a n H sin θ H n H sin θ B n Hn L n H + n L 8.9. This condition can be solved either for the angle θ a or for the index n a of the incident medium: n H n L sin θ a n a n H + n L n H n L or, n a sin θ a n H + n L 8.9. Γ T λ percent o Γ T λ percent o In either case, the feasibility of this approach requires the opposite of the condition 8.8., that is, n a > n Hn L n H + n L 8.9. Γ T λ percent Reflectance at o Fig Γ T λ percent Reflectance at o Omnidirectional mirror over visible band. If the angle θ a is set equal to the convenient value of 45 o, then, condition Eq fixes the value of the refractive index n a to be given by: n a nh n L n H + n L Fig depicts such a multilayer structure sandwiched between two glass prisms with 45 o angles. The thin films are deposited along the hypotenuse of each prism and the prisms are then cemented together. The incident, reflected, and transmitted beams are perpendicular to the prism sides. Not many combinations of available materials satisfy condition One possible solution is Banning s [64] with n H. zinc sulfide, n L.5 cryolite, and n a.55. Another solution is given in Clapham, et al, [665], with n H.4 zirconium oxide, n L.85 magnesium fluoride, and n a.65 a form of dense flint glass. Fig shows the and reflectances of the case n H. and n L.5. The incident and output media had n a n b.55. The maximum reflectivity for the component is 99.99%, while that of the component is % note the different vertical scales in the two graphs.

22 8.9. Polarizing Beam Splitters Multilayer Film Applications Reflectance 4 Reflectance Γ λ percent λ c Γ λ percent Fig Polarizing beam splitter Fig Polarizer with n H.4 and n L.85. The number of bilayers was N 5 and the center frequency of the band was chosen to correspond to a wavelength of λ c 5 nm. To achieve this, the normalizing wavelength was required to be λ 78.8 nm. The layer lengths were quarterwavelengths at λ. The bandwidth calculated with omniband is also shown. The Brewster angles inside the high- and low-index layers are θ H 8.5 o and θ L 6.48 o. As expected, they satisfy θ H + θ L 9 o. Γ λ percent Reflectance λ c Γ λ percent Fig Polarizer with n H. and n L.5. 4 Reflectance Fig shows the second case having n H.4, n L.85, n a n b.65. The normalizing wavelength was λ 76.7 nm in order to give λ c 5 nm. This case achieves a maximum reflectivity of 99.89% and reflectivity of only.5%. The typical MATLAB code generating these examples was: nh.; nl.5; LH.5; LL.5; na nh*nl/sqrtnh^+nl^/sinpi/4; nbna; [fe,fe] omnibandna,nh,nl,lh,ll,th, te,5; lac 5; la lac*fe+fe/; la linspace,8,; N 5; n [na, nh, repmat[nl,nh],, N, nb]; L [LH, repmat[ll,lh],, N]; Ge *absmultidieln,l,la/la, th, te.^; Gm *absmultidieln,l,la/la, th, tm.^; plotla,ge; because λ c λ /F c 8. Reflection and Refraction in Birefringent Media Uniform plane wave propagation in biaxial media was discussed in Sec We found that there is an effective refractive index N such that k Nk Nω/c. The index N, given by Eq , depends on the polarization of the fields and the direction of the wave vector. The expressions for the and fields were given in Eqs and Here, we discuss how such fields get reflected and refracted at planar interfaces between biaxial media. Further discussion can be found in [64,57] and [694 75]. Fig. 7.. depicts the and cases, with the understanding that the left and right biaxial media are described by the triplets of principal indices n [n,n,n ] and n [n,n,n ], and that the E-fields are not perpendicular to the corresponding wave vectors in the case. The principal indices are aligned along the xyz axes, the xy-plane is the interface plane, and the xz-plane is the plane of incidence. The boundary conditions require the matching of the electric field components that are tangential to the interface, that is, the components E x in the case or E y in. It proves convenient, therefore, to re-express Eq directly in terms of the E x component and Eq in terms of E y.

23 8.. Reflection and Refraction in Birefringent Media Multilayer Film Applications For the case, we write E ˆx E x + ẑ E z E x ˆx ẑ tan θ, for the electric field of the left-incident field, where we used E z E x tan θ. Similarly, for the magnetic field we have from Eq : H N ŷe x cos θ E z sin θ N ŷ E x cos θ E z tan θ η η E x N ŷ E x cos θ + n tan θ N n ŷ E x cos θ cos θ + n sin θ η η n cos θ N η ŷ E x cos θ n n n N n cos θ E x η n N cos θ ŷ where we replaced E z /E x tan θ n /n tan θ and used Eq Thus, Er E x ˆx ẑ n n tan θ j k r e Hr E x n η N cos θ ŷ e j k r E x j k r ŷ e η Similarly, we may rewrite the case of Eq in the form: j k r Er E y ŷ e 8.. Hr E y η n cos θ ˆx + ẑ tan θe j k r E y η ˆx + ẑ tan θe j k r 8.. The propagation phase factors are: e j k r e jkxn sin θ jkzn cos θ and propagation factors 8.. Unlike the isotropic case, the phase factors are different in the and cases because the value of N is different, as given by Eq , or, N n n n sin θ + n cos θ n,, or p-polarization or s-polarization 8..4 In Eqs. 8.. and 8.., the effective transverse impedances are defined by η E x /H y and η E y /H x, and are given as follows: N cos θ η η η n, η n cos θ transverse impedances 8..5 Defining the and effective transverse refractive indices through η η /n and η η /n, we have: n n N cos θ n n n N sin θ n n cos θ where we used Eq for the case, that is, transverse refractive indices 8..6 N cos θ n n N sin θ 8..7 n In the isotropic case, N n n n n, Eqs reduce to Eq Next, we discuss the and reflection and refraction problems of Fig Assuming that the interface is at z, the equality of the total tangential electric fields E x component for and E y for, implies as in Sec. 7. that the propagation phase factors must match at all values of x: e jkx+x e jkx x e jk x+x e jk x x which requires that k x+ k x k x+ k x, or, because k x k sin θ Nk sin θ: N sin θ + N sin θ N sin θ + N sin θ This implies Snel s law of reflection, that is, θ + θ θ and θ + θ θ, and Snel s law of refraction, N sin θ N sin θ Snel s law for birefringent media 8..8 Thus, Snel s law is essentially the same as in the isotropic case, provided one uses the effective refractive index N. Because N depends on the polarization, there will be two different refraction angles for the same angle of incidence. In particular, Eq can be written explicitly in the two polarization cases: n n sin θ n sin θ + n cos θ n n sin θ 8..9a n sin θ + n θ n sin θ n sin θ 8..9b Both expressions reduce to Eq in the isotropic case. The explicit solutions of Eq. 8..9a for sin θ and sin θ are: sin θ n n n sin θ [n n n n n n n n ] sin θ + n n n n sin θ n n sin θ [n n n n n n n n ] sin θ + n n n Hence, the name birefringent. 8..

24 8.. Reflection and Refraction in Birefringent Media Multilayer Film Applications The MATLAB function snel, solves Eqs for θ given the angle of incidence θ and the polarization type. It works for any type of medium, isotropic, uniaxial, or biaxial. It has usage: thb snelna,nb,tha,pol; % refraction angle from Snel s law The refractive index inputs na, nb may be entered as -, -, or -dimensional column or row vectors, for example, n a [n a ] isotropic, n a [n ao,n ae ] uniaxial, or n a [n a,n a,n a ] biaxial. Next, we discuss the propagation and matching of the transverse fields. All the results of Sec. 7. translate verbatim to the birefringent case, provided one uses the proper transverse refractive indices according to Eq In particular, the propagation equations for the transverse fields, for the transverse reflection coefficients Γ T, and for the transverse wave impedances Z T, remain unchanged. The phase thickness δ z for propagating along z by a distance l also remains the same as Eq. 7..8, except that the index N must be used in the optical length, and therefore, δ z depends on the polarization: Using Eq. 8..7, we have explicitly: δ z l kl cos θ Nk l cos θ π λ ln cos θ 8.. δ z π λ l n n n N sin θ, 8..a δ z π λ ln cos θ, 8..b The transverse matching matrix 7.. and Fresnel reflection coefficients 7.. remain the same. Explicitly, we have in the and cases: ρ n n n + n ρ n n n + n n N cos θ n n N cos θ N cos θ + n N cos θ n cos θ n cos θ n cos θ + n cos θ 8.. Using Eq and the and Snel s laws, Eqs. 8..9, we may rewrite the reflection coefficients in terms of the angle θ only: ρ n n n N sin θ n n n N sin θ n n n N sin θ + n n n N sin θ ρ n cos θ n n sin θ n cos θ + n n sin θ 8..4 The quantity N sin θ can be expressed directly in terms of θ and the refractive indices of the incident medium. Using Eq. 8..4, we have: N sin n θ n sin θ n sin θ + n cos θ 8..5 The reflection coefficient behaves like the isotropic case. The coefficient exhibits a much more complicated behavior. If n n but n n, it behaves like the isotropic case. If n n but n n, the square-root factors cancel and it becomes independent of θ: ρ n n n + n 8..6 Another interesting case is when both media are uniaxial and n n and n n, that is, the refractive index vectors are n [n,n,n ] and n [n,n,n ]. It is straightforward to show in this case that ρ ρ at all angles of incidence. Multilayer films made from alternating such materials exhibit similar and optical properties [694]. The MATLAB function fresnel can evaluate Eqs at any range of incident angles θ. The function determines internally whether the media are isotropic, uniaxial, or biaxial. 8. Brewster and Critical Angles in Birefringent Media The maximum angle of refraction, critical angle of incidence, and Brewster angle, have their counterparts in birefringent media. It is straightforward to verify that θ is an increasing function of θ in Eq The maximum angle of refraction θ c is obtained by setting θ 9o in Eq For the case, we obtain sin θ c n /n. As in the isotropic case of Eq. 7.5., this requires that n <n, that is, the incident medium is less dense than the transmitted medium, with respect to the index n. For the case, we obtain from Eq. 8..9a: sin θ c n n n n + n n n maximum refraction angle 8.. This requires that n <n. On the other hand, if n >n, we obtain the critical angle of incidence θ c that corresponds to θ 9 o in Eq. 8..: n n sin θ c n n + n n n critical angle 8.. whereas for the case, we have sin θ c n /n, which requires n >n.

25 8.. Brewster and Critical Angles in Birefringent Media Multilayer Film Applications In the isotropic case, a Brewster angle always exists at which the reflection coefficient vanishes, ρ. In the birefringent case, the Brewster angle does not necessarily exist, as is the case of Eq. 8..6, and it can also have the value zero, or even be imaginary. The Brewster angle condition ρ is equivalent to the equality of the transverse refractive indices n n. Using Eq. 8..6, we obtain: n n n n n N sin θ n n n N sin θ 8.. where N sin θ is given by Eq Solving for θ, we obtain the expression for the Brewster angle from the left medium: tan θ B n n n n n n n Brewster angle 8..4 Working instead with N sin θ N sin θ, we obtain the Brewster angle from the right medium, interchanging the roles of the primed and unprimed quantities: tan θ B n n n n n n n Brewster angle 8..5 Eqs and 8..5 reduce to Eqs and 7.6. in the isotropic case. It is evident from Eq that θ B is a real angle only if the quantity under the square root is non-negative, that is, only if n >n and n >n,orifn <n and n <n. Otherwise, θ B is imaginary. In the special case, n n but n n, the Brewster angle vanishes. If n n, the Brewster angle does not exist, since then ρ is given by Eq and cannot vanish. The MATLAB function brewster computes the Brewster angle θ B, as well as the critical angles θ c and θ c. For birefringent media the critical angles depend on the polarization. Its usage is as follows: ρ θ a n [.6,.6,.5], n [.6,.6,.6] b n [.54,.54,.6], n [.5,.5,.5] c n [.8,.8,.5], n [.5,.5,.5] d n [.8,.8,.5], n [.56,.56,.56] These cases were discussed in [694]. The corresponding materials are: a birefringent polyester and isotropic polyester, b syndiotactic polystyrene and polymethylmethacrylate PMMA, c birefringent polyester and PMMA, and d birefringent polyester and isotropic polyester. Because n n in case a, the Brewster angle will be zero, θ B o. In case b, we calculate θ B 9.4 o. Because n >n and n >n, there will be both and critical angles of reflection: θ c, 76.9 o and θ c, 68. o. In case c, the Brewster angle does not exist because n n, and in fact, the reflection coefficient is independent of the incident angle as in Eq The corresponding critical angles of reflection are: θ c, 56.4 o and θ c, 9 o. Finally, in case d, because n >n but n <n, the Brewster angle will be imaginary, and there will be a critical angle of reflection and a maximum angle of refraction: θ c, 6. o and θ c, 74. o. Fig. 8.. shows the and reflection coefficients ρ θ of Eq versus θ in the range θ 9 o Reflection Coefficients a b c d ρ θ Reflection Coefficients b c d [thb,thc] brewsterna,nb [thb,thc,thc] brewsterna,nb % isotropic case % birefringent case θ θ degrees θ θ degrees In multilayer systems, it is convenient to know if the Brewster angle of an internal interface is accessible from the incident medium. Using Snel s law we have in this case N a sin θ a N sin θ, where θ a is the incident angle and N a the effective index of the incident medium. It is simpler, then, to solve Eq. 8.. directly for θ a : Na sin θ a N sin θ B n n n n n n n n 8..6 Example 8..: To illustrate the variety of possible Brewster angle values, we consider the following birefringent cases: Fig. 8.. and birefringent Fresnel reflection coefficients versus incident angle. The coefficient in case a is not plotted because it is identically zero. In order to expand the vertical scales, Fig. 8.. shows the reflectances normalized by their values at θ o, that is, it plots the quantities ρ θ/ρ o. Because in case a ρ o, we have plotted instead the scaled-up quantity ρ θ. The typical MATLAB code used to compute the critical angles and generate these graphs was: th linspace,9,6; % θ at /4 o intervals na [.6,.6,.5]; nb [.6,.6,.6]; % note the variety of

26 8.. Multilayer Birefringent Structures Multilayer Film Applications Reflectance Fig Normalized Reflectance a b c d θ degrees reflectances normalized at normal incidence. [rte,rtm] fresnelna,nb,th; [thb,thc,thc] brewsterna,nb; na [.54,.6]; nb [.5,.5]; [rte,rtm] fresnelna,nb,th; [thb,thc,thc] brewsterna,nb; % equivalent ways of % entering na and nb % FRESNEL and BREWSR % internally extend % na,nb into -d arrays na [.8,.5]; % same as na[.8,.8,.5] nb.5; % and nb[.5,.5,.5] [rte,rtm] fresnelna,nb,th; [thb,thc,thc] brewsterna,nb; % in this case, θ B [] na [.8,.5]; nb.56; [rte4,rtm4] fresnelna,nb,th; [thb4,thc4,thc4] brewsterna,nb; plotth, abs[rtm; rtm; rtm; rtm4]; We note four striking properties of the birefringent cases that have no counterparts for isotropic materials: i The Brewster angle can be zero, ii the Brewster angle may not exist, iii the Brewster angle may be imaginary with the and reflection coefficients both increasing monotonically with the incident angle, and iv there may be total internal reflection in one polarization but not in the other. 8. Multilayer Birefringent Structures With some redefinitions, all the results of Sec. 8. on multilayer dielectric structures translate essentially unchanged to the birefringent case. We assume the same M-layer configuration shown in Fig. 8.., where now each layer is a biaxial material. The orthogonal optic axes of all the layers are assumed to be aligned with the xyz film axes. The xz-plane is the plane of incidence, the layer interfaces are parallel to the xy-plane, and the layers are arranged along the z-axis. The ith layer is described by the triplet of refractive indices n i [n i,n i,n i ], i,,...,m. The incident and exit media a, b may also be birefringent with n a [n a,n a,n a ] and n b [n b,n b,n b ], although in our examples, we will assume that they are isotropic. The reflection/refraction angles in each layer depend on the assumed polarization and are related to each other by the birefringent version of Snel s law, Eq. 8..8: N a sin θ a N i sin θ i N b sin θ b, i,...,m 8.. where N a,n i,n b are the effective refractive indices given by Eq The phase thickness of the ith layer depends on the polarization: π δ i π λ l N in i a sin θ a n, λ l i in i cos θ i π λ l N in i a sin θ a, where we used Eq and Snel s law to write in the and cases: n i n i n N i sin θ i n N i i sin θ i i n n N i a sin θ a i n i N i cos θ i n i cos θ i n i sin θ i n N i a sin θ a n i n i 8.. To use a unified notation for the and cases, we define the layer optical lengths at normal-incidence, normalized by a fixed free-space wavelength λ : l i n i, λ L i l i n i, λ, i,,..., M 8.. We define also the cosine coefficients c i, which represent cos θ i in the birefringent case and in the isotropic case: N a sin θ a n, i c i N a sin θ a n, i, i,,..., M 8..4 At normal incidence the cosine factors are unity, c i. With these definitions, Eq. 8.. can be written compactly in the form: δ i π λ λ L ic i π f f L i c i, i,,...,m 8..5

27 8.. Giant Birefringent Optics Multilayer Film Applications where λ is the operating free-space wavelength and f c /λ, f c /λ. This is the birefringent version of Eq A typical design might use quarter-wave layers, L i /4, at λ and at normal incidence. The reflection coefficients ρ Ti at the interfaces are given by Eq. 8.., but now the transverse refractive indices are defined by the birefringent version of Eq. 8..4: n Ti n i N i cos θ i n i cos θ i n i n i n i N a sin θ a, n i N a sin θ a,, i a,,,...,m,b 8..6 With the above redefinitions, the propagation and matching equations remain unchanged. The MATLAB function multidiel can also be used in the birefringent case to compute the frequency reflection response of a multilayer structure. Its usage is still: [Gamma,Z] multidieln,l,lambda,theta,pol; % birefringent multilayer structure where the input n isa M + vector of refractive indices in the isotropic case, or a M + matrix, where each column represents the triplet of birefringent indices of each medium. For uniaxial materials, n may be entered as a M + matrix. 8. Giant Birefringent Optics The results of Sec. 8.8 can be applied almost verbatim to the birefringent case. In Fig. 8.8., we assume that the high and low alternating layers are birefringent, described by the triplet indices n H [n H,n H,n H ] and n L [n L,n L,n L ]. The entry and exit media may also be assumed to be birefringent. Then, Snel s laws give: N a sin θ a N H sin θ H N L sin θ L N b sin θ b 8.. The phase thicknesses δ H and δ L within the high and low index layers are: δ H π f f L H c H, δ L π f f L L c L 8.. ρ n Hn H n L N a sin θ a n L n L n H N a sin θ a n H n H n L N a sin θ a + n L n L n H N a sin θ a n H ρ N a sin θ a n L N a sin θ a n H N a sin θ a + n L N a sin θ a 8.. The multilayer structure will exhibit reflection bands whose bandedges can be calculated from Eqs , with the redefinition L ± L H c H ±L L c L. The MATLAB function omniband calculates the bandedges. It has usage: [F,F] omnibandna,nh,nl,lh,ll,th,pol,n; where pol is one of the strings te or tm for or polarization, and na, nh, nl are -d, -d, or -d row or column vectors of birefringent refractive indices. Next, we discuss some mirror design examples from [694] that illustrate some properties that are specific to birefringent media. The resulting optical effects in such mirror structures are referred to as giant birefringent optics GBO in [694,5]. Example 8..: We consider a GBO mirror consisting of 5-bilayers of high and low index quarter-wave layers with refractive indices n H [.8,.8,.5], n L [.5,.5,.5] birefringent polyester and isotropic PMMA. The surrounding media are air, n a n b. The layers are quarter wavelength at the normalization wavelength λ 7 nm at normal incidence, so that for both polarizations we take L H L L /4. Because the high/low index layers are matched along the z-direction, n H n L, the reflection coefficient at the high/low interface will be constant, independent of the incident angle θ a, as in Eq However, some dependence on θ a is introduced through the cosine factors c H,c L of Eq The left graph of Fig. 8.. shows the reflectance Γ T λ as a function of λ for an angle of incidence θ a 6 o. The and bandedge wavelengths were calculated from omniband to be: [λ,λ ] [54.4, 66.7] and [λ,λ ] [548.55, 644.7] nm. The typical MATLAB code used to generate the left graph and the bandedge wavelengths was as follows: where L H,c H and L L,c L are defined by Eqs. 8.. and 8..4 for i H, L. The effective transverse refractive indices within the high and low index layers are given by Eq. 8..6, again with i H, L. The alternating reflection coefficient ρ T between the high/low interfaces is given by Eq. 8..4, with the quantity N sin θ replaced by Na sin θ a by Snel s law: LH.5; LL.5; na [; ; ]; nh [.8;.8;.5]; nl [.5;.5;.5]; nb [; ; ]; la 7; la linspace4,,6; th 6; N 5; % angle of incidence % number of bilayers

28 8.. Giant Birefringent Optics Multilayer Film Applications Reflectance at o and 6 o 5% thickness gradient Reflectance at o and 45 o 8 and bandwidths Γ T λ percent o Γ T λ percent o Γ T λ percent o λ, λ nm θ a degrees Fig. 8.. Reflectance of birefringent mirror. Fig. 8.. Birefringent mirror with identical and reflection bands. n [na, repmat[nh,nl],, N, nb]; L [repmat[lh,ll],, N]; Ge *absmultidieln, L, la/la, th, te.^; Gm *absmultidieln, L, la/la, th, tm.^; G *absmultidieln, L, la/la.^; plotla,gm, -, la,ge, --, la,g, : ; % N + matrix [F,F]omnibandna,nH,nL,LH,LL,th, tm,; la la/f; la la/f; % bandedge wavelengths The right graph shows the reflectance with a 5% thickness gradient the layer thicknesses L H,L L decrease linearly from quarter-wavelength to 5% less than that at the end. This can be implemented in MATLAB by defining the thickness vector L by: L [repmat[lh,ll],, N]; L L.* - linspace,.5, *N; % 5% thickness gradient The thickness gradient increases the effective bandwidth of the reflecting bands [69]. However, the bandwidth calculation can no longer be done with omniband. The band centers can be shifted to higher wavelengths by choosing λ higher. The reflecting bands can be made flatter by increasing the number of bilayers. Example 8..: In this example, we design a -bilayer GBO mirror with n H [.8,.8,.5] and n L [.5,.5,.8], so that n H n H n L and n H n L n L. As we discussed in Sec. 8., it follows from Eq that ρ ρ for all angles of incidence. As in Ref. [694], the media a, b are taken to be isotropic with n a n b.4. The normalization wavelength at which the high and low index layers are quarter-wavelength is λ 7 nm. The left graph of Fig. 8.. shows the reflectance for a 45 o angle of incidence. Because ρ ρ, the reflection bands for the and cases are essentially the same. The right graph depicts the asymptotic for large number of bilayers bandedges of the reflecting band versus incident angle. They were computed with omniband. Unlike the isotropic case, the and bands are exactly identical. This is a consequence of the following relationships between the cosine factors in this example: c H, c L, and c H, c L,. Then, because we assume quarter-wave layers in both the and cases, L H L L /4, we will have: L +, L H, c H, + L L, c L, 4 c H, + c L, 4 c L, + c H, L +, L, L H, c H, L L, c L, 4 c H, c L, 4 c L, c H, L +, Because the computational algorithm for the bandwidth does not depend on the sign of L, it follows that Eq will have the same solution for the and cases. The typical MATLAB code for this example was: LH.5; LL.5; na [.4;.4;.4]; nb [.4;.4;.4]; nh [.8;.8;.5]; nl [.5;.5;.8]; la 7; la linspace4,,6; tha 45; N ; n [na, repmat[nh,nl],, N, nb]; L [repmat[lh,ll],, N]; Ge *absmultidieln, L, la/la, tha, te.^; Gm *absmultidieln, L, la/la, tha, tm.^; G *absmultidieln, L, la/la.^; plotla,gm, -, la,ge, --, la,g, : ;

29 8.. Giant Birefringent Optics Multilayer Film Applications Γ T λ percent In Fig. 8.., the low-index material is changed slightly to n L [.5,.5,.9]. The main behavior of the structure remains the same, except now the and bands are slightly different. Reflectance at o and 45 o o Fig. 8.. λ, λ nm and bandwidths θ a degrees Birefringent mirror with slightly different and reflection bands. The MATLAB code used to compute the right graph was: theta linspace,9,6; Fe []; Fe []; Fm []; Fm []; Ni ; % incident angles % refinement iterations for i:lengththeta, [fe,fe] omnibandna,nh,nl,lh,ll,thetai, te,ni; [fm,fm] omnibandna,nh,nl,lh,ll,thetai, tm,ni; Fe [Fe,fe]; Fe [Fe,fe]; Fm [Fm,fm]; Fm [Fm,fm]; % frequency bandedges end lae la./ Fe; lae la./ Fe; lam la./ Fm; lam la./ Fm; % wavelength bandedges plottheta,lam, -, theta,lam, -, theta,lae, --, theta,lae, -- ; As the incident angle increases, not only does the band widen but it also becomes wider than the band exactly the opposite behavior from the isotropic case. Example 8..: GBO Reflective Polarizer. By choosing biaxial high/low layers whose refractive indices are mismatched only in the x or the y direction, one can design a mirror structure that reflects only the or only the polarization. Fig shows the reflectance of an 8-bilayer mirror with n H [.86,.57,.57] for the left graph, and n H [.57,.86,.57] for the right one. In both graphs, the low index material is the same, with n L [.57,.57,.57]. The angle of incidence was θ a o. The typical MATLAB code was: Γ T λ percent Polarizer Fig LH.5; LL.5; na [; ; ]; nb [; ; ]; nh [.86;.57;.57]; nl [.57;.57;.57]; la 7; la linspace4,,6; Γ T λ percent N 8; n [na, repmat[nh,nl],, N, nb]; L [repmat[lh,ll],, N]; L L.* linspace,.75,*n; Polarizer and mirror polarizers. Ge *absmultidieln, L, la/la,, te.^; Gm *absmultidieln, L, la/la,, tm.^; plotla,gm, -, la,ge, -- ; % 5% thickness gradient A 5% thickness gradient was assumed in both cases. In the first case, the x-direction indices are different and the structure will act as a mirror for the polarization. The polarization will be reflected only by the air-high interface. In the second case, the materials are matched in their y-direction indices and therefore, the structure becomes a mirror for the polarization, assuming as always that the plane of incidence is still the xz plane. Giant birefringent optics is a new paradigm in the design of multilayer mirrors and polarizers [694], offering increased flexibility in the control of reflected light. The recently manufactured multilayer optical film by M Corp. [5] consists of hundreds to thousands of birefringent polymer layers with individual thicknesses of the order of a wavelength and total thickness of a sheet of paper. The optical working range of such films are between 4 5 nm. Applications include the design of efficient waveguides for transporting visible light over long distances and piping sunlight into interior rooms, reflective polarizers for

30 8.4. Problems 6 improving liquid crystal displays, and other products, such as various optoelectronic components, cosmetics, and hot and cold mirrors for architectural and automotive windows. 8.4 Problems 9 Waveguides 8. Prove the reflectance and transmittance formulas in FTIR. 8. Computer Experiment FTIR. Reproduce the results and graphs of Figures Computer Experiment Surface Plasmon Resonance. Reproduce the results and graphs of Figures Working with the electric and magnetic fields across an negative-index slab given by Eqs and 8.6., derive the reflection and transmission responses of the slab given in Computer Experiment Perfect Lens. Study the sensitivity of the perfect lens property to the deviations from the ideal values of ɛ ɛ and μ μ, and to the presence of losses by reproducing the results and graphs of Figures 8.6. and You will need to implement the computational algorithm listed on page Computer Experiment Antireflection Coatings. Reproduce the results and graphs of Figures Computer Experiment Omnidirectional Dielectric Mirrors. Reproduce the results and graphs of Figures Derive the generalized Snel s laws given in Eq Moreover, derive the Brewster angle expressions given in Eqs and Computer Experiment Brewster angles. Study the variety of possible Brewster angles and reproduce the results and graphs of Example Computer Experiment Multilayer Birefringent Structures. Reproduce the results and graphs of Figures Waveguides are used to transfer electromagnetic power efficiently from one point in space to another. Some common guiding structures are shown in the figure below. These include the typical coaxial cable, the two-wire and mictrostrip transmission lines, hollow conducting waveguides, and optical fibers. In practice, the choice of structure is dictated by: a the desired operating frequency band, b the amount of power to be transferred, and c the amount of transmission losses that can be tolerated. Fig. 9.. Typical waveguiding structures. Coaxial cables are widely used to connect RF components. Their operation is practical for frequencies below GHz. Above that the losses are too excessive. For example, the attenuation might be db per m at MHz, but db/ m at GHz, and 5 db/ m at GHz. Their power rating is typically of the order of one kilowatt at MHz, but only W at GHz, being limited primarily because of the heating of the coaxial conductors and of the dielectric between the conductors dielectric voltage breakdown is usually a secondary factor. However, special short-length coaxial cables do exist that operate in the 4 GHz range. Another issue is the single-mode operation of the line. At higher frequencies, in order to prevent higher modes from being launched, the diameters of the coaxial conductors must be reduced, diminishing the amount of power that can be transmitted. Two-wire lines are not used at microwave frequencies because they are not shielded and can radiate. One typical use is for connecting indoor antennas to TV sets. Microstrip lines are used widely in microwave integrated circuits.