Within the framework of this thesis, different optical multilayer structures are investigated concerning their reflectance and transmittance. These, at first glance complicated, systems can be theoretically analyzed by a fast algorithm, the transfer matrix method. This procedure is an often used numerical technique in the modeling of one-dimensional problems. This chapter is a long and rather tedious account of some basic theory which is necessary in order to make calculations of the properties of multilayer thin-film coatings using the transfer matrix method [46, 47, 48, 51,*18].
2.1 Maxwell's equations and plane electromagnetic waves
Thin-film problems can be solved by solving Maxwell's equations together with the appropriate material equations [*16].
In isotropic media these are:
In anisotropic media, equations (2.1) to (2.7) become much more complicated with σ, ε and μ being tensor rather than scalar quantities. Anisotropic media are covered by Yeh  and Hodgkinson and Wu . The International System of Units (SI) is used as far as possible throughout this work. Table 2.1 shows the definitions of the quantities in the equations together with the appropriate SI units.
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Table2.1 the definitions of the quantities in the equations together with the
appropriate SI units
Symbol Physical quantity SI unit Symbol for
E Electric field strength volts per meter V m-1
D Electric displacement coulombs per
square meter C m-2
H Magnetic field strength amperes per meter A m-1
j Electric current density amperes per square
meter A m-2
B Magnetic flux density or
Magnetic induction tesla T
ρ Electric charge density coulombs per C m-3
σ Electric conductivity siemens per meter S m-1
μ Permeability henries per meter H m-1
ε Permittivity farads per meter F m-1
Table2.2 the values of ε0, μ0 and c
Symbol Physical quality Value
c Speed of light in vacuum 2.997925 * 108 m s-1
μ0 permeability of a vacuum 4π * 10-7 H m-1
ε0 permittivity of a vacuum(= μ0-1 c-2 ) 8.8541853 * 10-12 F m-1
Additions to the above equations are:
where ε0 and μ0 are the permittivity and permeability of free space, respectively. εr and μr are the relative permittivity and permeability, and c is a constant that can be identified as the velocity of light in free space. ε0, μ0 and c are important constants, the values of which are given in table 2.2.
The following analysis is brief and incomplete. For a full, rigorous treatment of the electromagnetic field equations it can be referred to Born and Wolf .
By manipulating the above Maxwell's equations and solving for E
A similar expression holds for H.
The solution of equation (2.11) in the form of a plane polarized plane harmonic wave, may be represented by expressions of the form
Where x is the distance along the direction of propagation, E is the electric field, the electric amplitude, ω the angular frequency of this wave and an arbitrary phase. A similar expression holds for H, the magnetic field:
where , and N are not independent. The physical significance is attached to the real parts of the above expressions.
Refractive index (N) is defined as the ratio of the velocity of light in free space c to the velocity of light in the medium υ. When the refractive index is real it is denoted by n but it is frequently complex and then is denoted by N.
N = c/υ = n - ik. (2.14)
N is often called the complex refractive index, n the real refractive index(or often simply as the refractive index because N is real in an ideal dielectric material)
and k is known as the extinction coefficient
N is always a function of x. k is related to the absorption coefficient α by
α = 4n k/x. (2.17)
The phase change suffered by the wave on traversing a distance d of the medium is, therefore,
and the imaginary part can be interpreted as a reduction in amplitude (by substituting in equation (2.12) ).
The change in phase produced by a traversal of distance x in the medium is the same as that produced by a distance nx in a vacuum. Because of this, nx is known as the optical distance, as distinct from the physical or geometrical distance. Generally, in thin-film optics one is more interested in optical distances and optical thicknesses than in geometrical ones.
2.2 The Optical Admittance and irradiance
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The optical admittance is defined as the ratio of the magnetic and electric fields
y = H / E (2.19)
and y is usually complex. In free space, y is real and is denoted by Y:
Y= (ε0/ μ0 )1/2 =2.6544 x 10-3 S (2.20)
The optical admittance of a medium is connected with the refractive index by
y = NY (2.21)
The irradiance of the light, defined as the mean rate of flow of energy per unit area carried by the wave, is given by
I = ½ Re (E H* ) (2.22)
This can also be written
I = ½ n y E E* , (2.23)
Where * denotes the complex conjugate.
2.3 The simple boundary
Thin-film filters usually consist of a number of boundaries between various homogeneous media and the effect which these boundaries will have on an incident wave which is important to calculate. A single boundary is the simplest case.
Figure 2.1 Plane wavefront incident on a single surface
First, it is convenient to consider absorption-free media, i.e. k = 0. The arrangement is sketched in fig. 2.1
Again, a plane polarized plane harmonic wave will be assumed. The incident wave will be split into a reflected wave and a transmitted wave at the boundary, so the objective is the calculation of the parameters of these waves. Without specifying their exact form for the moment, they will certainly consist of an amplitude term and a phase factor. The amplitude terms will not be functions of x, y or r, any variations due to these being included in the phase factors.
Let the direction of propagation of the wave be given by unit vector Å where and where i, j and k are unit vectors along the x, y and z axes, respectively. α , β and γ are the direction coefficients.
Let the direction cosines of the Å vectors of the transmitted and reflected waves be (αt , βt ,γt ) and (αr , βr ,γr ) respectively. The phase factors can be written in the form
The relative phases of these waves are included in the complex amplitudes. For waves with these phase factors to satisfy the boundary conditions for all x, y, t at z = 0 implies that the coefficients of these variables must be separately identically equal:
ω ≡ ωr ≡ ωt
that is, there is no change of frequency in reflection or refraction and hence no change in free space wavelength either. This implies that the free space wavelengths are equal :
λ ≡ λr ≡ λt.
0 ≡ n0βr ≡ n1βt
that is, the directions of the reflected and transmitted or refracted beams are confined to the plane of incidence. It implies also that
n0 sin ϑ0 ≡ n0αr ≡ n1αt (2.24)
so that if the angles of reflection and refraction are ϑr and ϑt, respectively, then
ϑ0 = ϑr (2.25)
that is, the angle of reflection equals the angle of incidence, and
n0 sin ϑ0 = n1 sin ϑt.
The result appears more symmetrical if we replace ϑt by ϑ1, giving
n0 sin ϑ0 = n1 sin ϑ1 (2.26)
which is the familiar relationship known as Snell's law. γr and γt are then given either by equation (2.24) or by
The negative root of (2.27) must be chosen for the reflected beam so that it can propagate in the correct direction.
2.3.1 Normal incidence
Firstly, the normal incident of a plane-polarized plane harmonic wave will be examined. The coordinate axes are shown in fig. 2.2. The xy plane is the plane of the boundary. The incident wave can be taken as propagating along the z axis with the positive direction of the E vector along the x axis. Then the positive direction of the H vector will be the y axis. It is clear that the only waves which satisfy the boundary conditions are plane polarized in the same plane as the incident wave.
Figure 2.2 Convention defining positive directions of the electric and magnetic vectors for reflection and transmission at an interface at normal incidence.
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The sign convention used for the following sections defines the positive direction of E along the x axis for all the beams that are involved. Because of this choice, the positive direction of the magnetic vector will be along the y axis for the incident and transmitted waves, but along the negative direction of the y axis for the reflected wave. It is important to define a sign convention for the electric and magnetic vectors in order that there is a reference for any phase changes that may occur.
The boundary conditions can now be applied. Since phase factors have already been accounted for only amplitudes will be considered, and phase changes will be included in these.
Electric vector continuous across the boundary:
Magnetic vector continuous across the boundary:
where a minus sign is used because of the convention for positive directions.
The relationship between magnetic and electric field through the characteristic admittance gives
It can eliminate εt to give
the second part of the relationship being correct only because at optical frequencies ,
These quantities are called the amplitude reflection and transmission coefficients and are denoted by ρ and τ respectively. thus
In this particular case, all y real, these two quantities are real. τ is always a positive real number, indicating that according to our phase convention there is no phase shift between the incident and transmitted beams at the interface. The behavior of ρ indicates that there will be no phase shift between the incident and reflected beams at the interface provided n 0 > n1, but that if n0 < n1 there will be a phase change of π because the value of ρ becomes negative.
The energy balance at the boundary has been examined. Since the boundary is of zero thickness, it can neither supply energy to nor extract energy from the various waves. The Poynting vector will therefore be continuous across the boundary, so that we can write:
[Using and equations (2.27) and (2.28) ]
Now, is the irradiance of the incident beam Ii . We can identify
as the irradiance of the reflected beam Ir and as the irradiance of the transmitted beam It .
The reflectance R can be defined as as the ratio of the reflected and incident irradiances and the transmittance T as the ratio of transmitted and incident irradiances. Then
From equation (2.35) we have, using equation (2.36)
(1 - R) = T (2.37)
Equations (2.35), (2.36) and (2.37) are therefore consistent with the ideas of splitting the irradiances into incident, reflected and transmitted irradiances which can be treated as separate waves, the energy flow into the second medium being simply the difference of the incident and reflected irradiances. Remember that all this, so far, assumes that there is no absorption.
2.3.2 Oblique incidence
Oblique incidence will now be considered, still retaining the absorption-free media. There are two orientations of the incident wave which lead to reasonably straight forward calculations: the vector electrical amplitudes aligned in the plane of incidence (i.e. the xy plane of Fig. 2.1) and the vector electrical amplitudes aligned normal to the plane of incidence (i.e. parallel to the y axis in Fig. 2.1). In each of these cases, the orientations of the transmitted and reflected vector amplitudes are the same as for the incident wave. Any incident wave of arbitrary polarization can therefore be split into two components having these simple orientations. The transmitted and reflected components can be calculated for each orientation separately and then combined to yield the resultant. Since, therefore, it is necessary to consider two orientations only; they have been given special names. A wave with the electric vector in the plane of incidence is known as p-polarized or, sometimes, as TM (for transverse magnetic) and a wave with the electric vector normal to the plane of incidence as s-polarized or, sometimes, TE (for transverse electric). The p and s are derived from the German parallel and senkrecht (perpendicular). Before proceding the calculation of the reflected and transmitted amplitudes, the various reference directions of the vectors from which any phase differences will be calculated must be chosen. The conventions which will be used are illustrated in Fig. 2.3. They have been chosen to be compatible with those for normal incidence already established.
Now, the boundary conditions can be applied. Since the phase factors are certainly correct, all what is needed is to consider the vector amplitudes.
Electric component parallel to the boundary, continuous across it:
(b) Magnetic component parallel to the boundary and continuous across it: Here the magnetic vector amplitudes are needed to calculate. Since the magnetic vectors are already parallel to the boundary, Fig. 2.3 can be used and then convert,
Figure 2.3 (a) Convention defining the positive directions of the electric and magnetic vectors for p-polarized light (TM waves). (b) Convention defining the positive directions of the electric and magnetic vectors for s-polarized light (TE waves).
At first sight it seems logical just to eliminate first °t and then °r from equations (2.38) and (2.39) to obtain and
And then simply to set
but when the expressions are calculated, it is found that R + T ≠ 1.
this situation can be corrected by modifying the definition of T to include this angular dependence, but an alternative, preferable and generally adopted approach is to use the components of the energy flows which are normal to the boundary. The E and H vectors that are involved in these calculations are then parallel to the boundary. Since these are those that enter directly into the boundary it seems appropriate to concentrate on them when we are dealing with the amplitudes of the waves.
The thin-film approach to all this, then, is to use the components of E and H parallel to the boundary, what are called the tangential components, in the expressions ρ and τ that involve amplitudes.
The tangential components of E and H, that is, the components parallel to the boundary, have already been calculated for use in equations (2.38) and (2.39). However, it is convenient to introduce special symbols for them: E and H.
Then they can be written
The orientations of these vectors are exactly the same as for normally incident light. Equations (2.38) and (2.39) can then be written as follows.
Electric field parallel to the boundary:
Magnetic field parallel to the boundary:
By using a process exactly similar to that we have already used for normal incidence,
where y0 = n0Y and y1 = n1Y and the (R + T = 1) rule is retained. The suffix p has been used in the above expressions to denote p-polarization.
It should be noted that the expression for τp is now different from that in equation (2.40), the form of the Fresnel amplitude transmission coefficient.
126.96.36.199 s-polarised light
In the case of s-polarisation the amplitudes of the components of the waves parallel to the boundary are
And this results in again an orientation of the tangential components exactly as for normally incident light, and so a similar analysis leads to
where y0 = n0Y and y1 = n1Y and the (R + T = 1) rule is retained. The suffix s has been used in the above expressions to denote s-polarization.
2.3.3 The optical admittance for oblique incidence
The expressions which are derived so far have been in their traditional form (except for the use of the tangential components rather than the full vector amplitudes) and they involve the characteristic admittances of the various media, or their refractive indices together with the admittance of free space, Y. However, the notation is becoming increasingly cumbersome and will appear even more so when the behavior of thin films will consider.
Since H = y( Å - E) where y = NY is the optical admittance. It is found that it is convenient to deal with E and H, the components of E and H parallel to the boundary, and so a tilted optical admittance η which connects E and H is introduced as
η = H / E . (2.52)
At normal incidence η= y =nY while at oblique incidence
where the ϑ and the y in (2.53) and (2.54) are those appropriate to the particular medium. In particular, Snell's law, equation (2.26), must be used to calculate ϑ. Then, in all cases, we can write
These expressions can be used to compute the variation of reflectance of simple boundaries between extended media with angle of incidence. In this case, there is no absorption in the material and it can be seen that the reflectance for p-polarized light (TM) falls to zero at a definite angle. This particular angle is known as the Brewster angle and is of some importance.
The expression for the Brewster angle can be derived as follows. For the p-reflectance to be zero, from equation (2.46)
Snell's law gives another relationship between :
Eliminating from these two equations gives an expression for
Note that this derivation depends on the relationship y = nY valid at optical frequencies.
2.4 The reflectance of a thin film
A simple extension of the above analysis occurs in the case of a thin, plane, parallel film of material covering the surface of a substrate. The presence of two (or more) interfaces means that a number of beams will be produced by successive reflections and the properties of the film will be determined by the summation of these beams. The film is called thin when interference effects can be detected in the reflected or transmitted light, that is, when the path difference between the beams is less than the coherence length of the light, and is called thick when the path difference is greater than the coherence length. The same film can appear thin or thick depending entirely on the illumination conditions. The thick case can be shown to be identical with the thin case integrated over a sufficiently wide wavelength range or a sufficiently large range of angles of incidence.
Normally, the films on the substrates can be treated as thin while the substrates supporting the films can be considered thick.
The arrangement is illustrated in Fig. 2.4. At this stage it is convenient to introduce a new notation. The waves in the direction of incidence are denoted by the symbol + (that is, positive-going) and the waves in the opposite direction are denoted by − (that is, negative-going).
Figure 2.4 Plane wave incident on a thin film.
The interface between the film and the substrate, denoted by the symbol b, can be treated in exactly the same way as the simple boundary already discussed. The tangential components of the fields are considered. There is no negative-going wave in the substrate and the waves in the film can be summed into one resultant positive-going wave and one resultant negative-going wave. At this interface, then, the tangential components of E and H are
where the common phase factors are neglected and where Eb and Hb represent the resultants. Hence
The fields at the other interface a at the same instant and at a point with identical x and y coordinates can be determined by altering the phase factors of the waves to allow for a shift in the z coordinate from 0 to −d. The phase factor of the positive-going wave will be multiplied by exp(iδ) where
δ = 2π N1 d cosϑ1 / λ
and ϑ1 may be complex, while the negative-going phase factor will be multiplied by exp(−iδ). This is a valid procedure when the film is thin. The values of E and H at the interface are now, using equations (2.58) to (2.61) ,
This can be written in matrix notation as
Since the tangential components of E and H are continuous across a boundary, and since there is only a positive-going wave in the substrate, this relationship connects the tangential components of E and H at the incident interface with the tangential components of E and H which are transmitted through the final interface. The 2 - 2 matrix on the right-hand side of equation (2.62) is known as the characteristic matrix of the thin film.
The input optical admittance of the assembly is defined by analogy with equation (2.52) as
Y = Ha /Ea (2.63)
when the problem becomes merely that of finding the reflectance of a simple interface between an incident medium of admittance η0 and a medium of admittance Y , i.e.
ρ = (η0 - Y) / (η0 + Y)
Equation (2.62) can be normalized by dividing through by Eb to give
and B and C, the normalized electric and magnetic fields at the front interface, are the quantities from which the properties of the thin-film system will be extracted. Clearly, from (2.63) and (2.65), it can be written
and from (2.66) and (2.64) the reflectance can be calculated.
is known as the characteristic matrix of the assembly.
Matrix Formulation for Isotropic Layered Media :Transfer
Matrix Method for Isotropic Layered Media
The method described in section 2.4 can be used to calculate the reflectance of a thin film. However, when the number of layers becomes too large, the analysis becomes very complicated because of the large number of equations involved. In this section we will introduce a matrix method that is a systematic approach to solving such a problem. The matrix method is especially useful when a computer is available that can handle the matrix algebra. It is also very useful when a large portion of the structure is periodic.
2.5.1 The analysis of an assembly of thin films
Let another film be added to the single film of the previous section so that the final interface is now denoted by c, as shown in Fig. 2.5. The characteristic matrix of the film nearest the substrate is
Figure 2.5 Notation for two films on a surface.
And from equation (2.62)
Equation (2.62) can be applied again to give the parameters at interface a, i.e.
and the characteristic matrix of the assembly, by analogy with equation (2.65) is,
is, as before, C/B, and the amplitude reflection coefficient and the reflectance are, from (2.64) ,
ρ = (η0 - Y) / (η0 + Y)
This result can be immediately extended to the general case of an assembly of q layers, when the characteristic matrix is simply the product of the individual matrices taken in the correct order, i.e.
δr = 2π Nr dr cosϑr / λ
and where the suffix m is used to denote the substrate or emergent medium
If ϑ0, the angle of incidence, is given, the values of ϑr can be found from Snell's law, i.e.
The expression (2.69) is of prime importance in optical thin-film work and forms the basis of almost all calculations.
The order of multiplication is important. If q is the layer next to the substrate then the order is
M1 indicates the matrix associated with layer 1, and so on. Y and η are in the same units. If η is in siemens then so also is Y, or if η is in free space units (i.e. units of Y) then Y will be in free space units also. As in the case of a single surface, η0 must be real for reflectance and transmittance to have a valid meaning. With that proviso, then