Within the model of this thesis, different optical multilayer constructions are investigated refering their coefficient of reflection and transmission. These, at first glimpse complicated, systems can be theoretically analyzed by a fast algorithm, the transportation matrix method. This process is an frequently used numerical technique in the mold of unidimensional jobs. This chapter is a long and instead boring history of some basic theory which is necessary in order to do computations of the belongingss of multilayer thin-film coatings utilizing the transportation matrix method [ 46, 47, 48, 51, *18 ] .

## 2.1 Maxwell ‘s equations and plane electromagnetic moving ridges

Thin-film jobs can be solved by work outing Maxwell ‘s equations together with the appropriate stuff equations [ *16 ] .

In isotropous media these are:

In anisotropic media, equations ( 2.1 ) to ( 2.7 ) go much more complicated with ? , ? and ? being tensor instead than scalar measures. Anisotropic media are covered by Yeh [ 47 ] and Hodgkinson and Wu [ 48 ] . The International System of Units ( SI ) is used every bit far as possible throughout this work. Table 2.1 shows the definitions of the measures in the equations together with the appropriate SI units.

Table2.1 the definitions of the measures in the equations together with the

appropriate SI units

## Symbol Physical measure SI unit Symbol for

## SI unit

E Electric field strength Vs per metre V Garand rifle

D Electric supplanting C per

square metre C m-2

H Magnetic field strength amperes per metre A Garand rifle

J Electric current denseness amperes per square

metre A m-2

B Magnetic flux denseness or

Magnetic initiation tesla T

? Electric charge denseness coulombs per C m-3

three-dimensional metre

? Electric conduction siemens per metre S Garand rifle

? Permeability Hs per metre H Garand rifle

? Permittivity Fs per metre F Garand rifle

Table2.2 the values of ?0, ?0 and c

Symbol Physical quality Value

degree Celsiuss Speed of visible radiation in vacuity 2.997925 * 108 m s-1

?0 permeableness of a vacuity 4? * 10-7 H Garand rifle

?0 permittivity of a vacuity ( = ?0-1 c-2 ) 8.8541853 * 10-12 F Garand rifle

Additions to the above equations are:

where ?0 and ?0 are the permittivity and permeableness of free infinite, severally. ?r and ?r are the comparative permittivity and permeableness, and degree Celsius is a changeless that can be identified as the speed of visible radiation in free infinite. ?0, ?0 and degree Celsiuss are of import invariables, the values of which are given in table 2.2.

The undermentioned analysis is brief and uncomplete. For a full, strict intervention of the electromagnetic field equations it can be referred to Born and Wolf [ 8 ] .

By pull stringsing the above Maxwell ‘s equations and work outing for Tocopherol

A similar look holds for H.

The solution of equation ( 2.11 ) in the signifier of a plane polarized plane harmonic moving ridge, may be represented by looks of the signifier

Where ten is the distance along the way of extension, E is the electric field, the electric amplitude, ? the angular frequence of this moving ridge and an arbitrary stage. A similar look holds for H, the magnetic field:

where, and N are non independent. The physical significance is attached to the existent parts of the above looks.

Refractive index ( N ) is defined as the ratio of the speed of visible radiation in free infinite degree Celsius to the speed of visible radiation in the medium ? . When the refractile index is existent it is denoted by Ns but it is often complex and so is denoted by N.

N = c/? = n – ik. ( 2.14 )

N is frequently called the complex refractive index, n the existent refractile index ( or frequently merely as the refractile index because N is existent in an ideal dielectric stuff )

and K is known as the extinction coefficient

N is ever a map of x. K is related to the soaking up coefficient ? by

? = 4n k/x. ( 2.17 )

The stage alteration suffered by the moving ridge on tracking a distance vitamin D of the medium is, hence,

and the fanciful portion can be interpreted as a decrease in amplitude ( by replacing in equation ( 2.12 ) ) .

The alteration in stage produced by a traverse of distance ten in the medium is the same as that produced by a distance nx in a vacuity. Because of this, nx is known as the optical distance, as distinguishable from the physical or geometrical distance. Generally, in thin-film optics one is more interested in optical distances and optical thicknesses than in geometrical 1s.

## 2.2 The Optical Admittance and irradiance

The optical entree is defined as the ratio of the magnetic and electric Fieldss

Y = H / E ( 2.19 )

and y is normally complex. In free infinite, Y is existent and is denoted by Yttrium:

Y= ( ?0/ ?0 ) 1/2 =2.6544 ten 10-3 S ( 2.20 )

The optical entree of a medium is connected with the refractile index by

Y = NY ( 2.21 )

The irradiance of the visible radiation, defined as the average rate of flow of energy per unit country carried by the moving ridge, is given by

I = ? Re ( E H* ) ( 2.22 )

This can besides be written

I = ? n Y E E* , ( 2.23 )

Where * denotes the complex conjugate.

## 2.3 The simple boundary

Thin-film filters normally consist of a figure of boundaries between assorted homogenous media and the consequence which these boundaries will hold on an incident moving ridge which is of import to cipher. A individual boundary is the simplest instance.

Figure 2.1 Plane wavefront incident on a individual surface

First, it is convenient to see absorption-free media, i.e. k = 0. The agreement is sketched in fig. 2.1

Again, a plane polarized plane harmonic moving ridge will be assumed. The incident moving ridge will be split into a reflected moving ridge and a familial moving ridge at the boundary, so the aim is the computation of the parametric quantities of these moving ridges. Without stipulating their exact signifier for the minute, they will surely dwell of an amplitude term and a stage factor. The amplitude footings will non be maps of ten, Y or R, any fluctuations due to these being included in the stage factors.

Let the way of extension of the moving ridge be given by unit vector A? where and where I, J and K are unit vectors along the ten, Y and omega axes, severally. ? , ? and ? are the way coefficients.

Let the way cosines of the A? vectors of the transmitted and reflected moving ridges be ( ?t, ?t, ?t ) and ( ?r, ?r, ?r ) severally. The stage factors can be written in the signifier

## .

The comparative stages of these moving ridges are included in the complex amplitudes. For moving ridges with these stage factors to fulfill the boundary conditions for all x, Y, T at z = 0 implies that the coefficients of these variables must be individually identically equal:

? ? ?r ? ?t

that is, there is no alteration of frequence in contemplation or refraction and hence no alteration in free infinite wavelength either. This implies that the free infinite wavelengths are equal:

? ? ?r ? ?t.

Following

0 ? n0?r ? n1?t

that is, the waies of the reflected and transmitted or refracted beams are confined to the plane of incidence. It implies besides that

n0 wickedness ?0 ? n0?r ? n1?t ( 2.24 )

so that if the angles of contemplation and refraction are ?r and ?t, severally, so

?0 = ?r ( 2.25 )

that is, the angle of contemplation peers the angle of incidence, and

n0 wickedness ?0 = n1 transgress ?t.

The consequence appears more symmetrical if we replace ?t by ?1, giving

n0 wickedness ?0 = n1 transgress ?1 ( 2.26 )

which is the familiar relationship known as Snell ‘s jurisprudence. ?r and ?t are so 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 right way.

## 2.3.1 Normal incidence

First, the normal incident of a plane-polarized plane harmonic moving ridge will be examined. The co-ordinate axes are shown in fig. 2.2. The xy plane is the plane of the boundary. The incident moving ridge can be taken as propagating along the omega axis with the positive way of the E vector along the x axis. Then the positive way of the H vector will be the y axis. It is clear that the lone moving ridges which satisfy the boundary conditions are flat polarized in the same plane as the incident moving ridge.

Figure 2.2 Convention specifying positive waies of the electric and magnetic vectors for contemplation and transmittal at an interface at normal incidence.

The mark convention used for the undermentioned subdivisions defines the positive way of E along the x axis for all the beams that are involved. Because of this pick, the positive way of the magnetic vector will be along the Y axis for the incident and familial moving ridges, but along the negative way of the Y axis for the reflected moving ridge. It is of import to specify a mark convention for the electric and magnetic vectors in order that there is a mention for any stage alterations that may happen.

The boundary conditions can now be applied. Since stage factors have already been accounted for merely amplitudes will be considered, and stage alterations will be included in these.

Electric vector uninterrupted across the boundary:

Magnetic vector uninterrupted across the boundary:

where a subtraction mark is used because of the convention for positive waies.

The relationship between magnetic and electric field through the characteristic entree gives

It can extinguish ?t to give

i.e.

the 2nd portion of the relationship being right merely because at optical frequences,

Y =nY

Similarly, extinguishing

These measures are called the amplitude contemplation and transmittal coefficients and are denoted by ? and ? severally. therefore

In this peculiar instance, all y existent, these two measures are existent. ? is ever a positive existent figure, bespeaking that harmonizing to our stage convention there is no stage displacement between the incident and familial beams at the interface. The behaviour of ? indicates that there will be no stage displacement between the incident and reflected beams at the interface provided n 0 & A ; gt ; n1, but that if n0 & A ; lt ; n1 there will be a phase alteration of ? because the value of ? becomes negative.

The energy balance at the boundary has been examined. Since the boundary is of nothing thickness, it can neither provide energy to nor extract energy from the assorted moving ridges. The Poynting vector will hence be uninterrupted across the boundary, so that we can compose:

[ Using and equations ( 2.27 ) and ( 2.28 ) ]

Now

i.e.

Now, is the irradiance of the incident beam Ii. We can place

as the irradiance of the reflected beam Ir and as the irradiance of the familial beam It.

The coefficient of reflection R can be defined as as the ratio of the reflected and incident irradiances and the transmission T as the ratio of transmitted and incident irradiances. Then

From equation ( 2.35 ) we have, utilizing equation ( 2.36 )

( 1 – Roentgen ) = T ( 2.37 )

Equations ( 2.35 ) , ( 2.36 ) and ( 2.37 ) are hence consistent with the thoughts of dividing the irradiances into incident, reflected and transmitted irradiances which can be treated as separate moving ridges, the energy flow into the 2nd medium being merely the difference of the incident and reflected irradiances. Remember that all this, so far, assumes that there is no soaking up.

## 2.3.2 Oblique incidence

Oblique incidence will now be considered, still retaining the absorption-free media. There are two orientations of the incident moving ridge which lead to moderately straight forward computations: 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. analogue to the Y axis in Fig. 2.1 ) . In each of these instances, the orientations of the transmitted and reflected vector amplitudes are the same as for the incident moving ridge. Any incident moving ridge of arbitrary polarisation can hence be split into two constituents holding these simple orientations. The familial and reflected constituents can be calculated for each orientation individually and so combined to give the end point. Since, hence, it is necessary to see two orientations merely ; they have been given particular names. A moving ridge with the electric vector in the plane of incidence is known as p-polarized or, sometimes, as TM ( for transverse magnetic ) and a moving ridge 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 analogue and senkrecht ( perpendicular ) . Before proceding the computation of the reflected and transmitted amplitudes, the assorted mention waies of the vectors from which any stage 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 stage factors are surely right, all what is needed is to see the vector amplitudes.

## p-polarized visible radiation

Electric constituent analogue to the boundary, uninterrupted across it:

( B ) Magnetic constituent analogue to the boundary and uninterrupted across it: Here the magnetic vector amplitudes are needed to cipher. Since the magnetic vectors are already parallel to the boundary, Fig. 2.3 can be used and so change over,

Figure 2.3 ( a ) Convention specifying the positive waies of the electric and magnetic vectors for p-polarized visible radiation ( TM waves ) . ( B ) Convention specifying the positive waies of the electric and magnetic vectors for s-polarized visible radiation ( TE waves ) .

Since

At first sight it seems logical merely to extinguish first & A ; deg ; t and so & amp ; deg ; R from equations ( 2.38 ) and ( 2.39 ) to obtain and

And so merely to put

but when the looks are calculated, it is found that R + T ? 1.

this state of affairs can be corrected by modifying the definition of T to include this angular dependance, but an alternate, preferred and by and large adopted attack is to utilize the constituents of the energy flows which are normal to the boundary. The E and H vectors that are involved in these computations are so parallel to the boundary. Since these are those that enter straight into the boundary it seems appropriate to concentrate on them when we are covering with the amplitudes of the moving ridges.

The thin-film attack to all this, so, is to utilize the constituents of E and H parallel to the boundary, what are called the digressive constituents, in the looks ? and ? that involve amplitudes.

The digressive constituents of E and H, that is, the constituents parallel to the boundary, have already been calculated for usage in equations ( 2.38 ) and ( 2.39 ) . However, it is convenient to present particular symbols for them: Tocopherol and H.

Then they can be written

The orientations of these vectors are precisely the same as for usually incident visible radiation. Equations ( 2.38 ) and ( 2.39 ) can so be written as follows.

Electric field analogue to the boundary:

Magnetic field analogue to the boundary:

By utilizing a procedure precisely similar to that we have already used for normal incidence,

where y0 = n0Y and y1 = n1Y and the ( R + T = 1 ) regulation is retained. The postfix P has been used in the above looks to denote p-polarization.

It should be noted that the look for ?p is now different from that in equation ( 2.40 ) , the signifier of the Fresnel amplitude transmittal coefficient.

## 2.3.2.2 s-polarised visible radiation

In the instance of s-polarisation the amplitudes of the constituents of the moving ridges parallel to the boundary are

And this consequences in once more an orientation of the digressive constituents precisely as for usually incident visible radiation, and so a similar analysis leads to

where y0 = n0Y and y1 = n1Y and the ( R + T = 1 ) regulation is retained. The postfix s has been used in the above looks to denote s-polarization.

## 2.3.3 The optical entree for oblique incidence

The looks which are derived so far have been in their traditional signifier ( except for the usage of the digressive constituents instead than the full vector amplitudes ) and they involve the characteristic entrees of the assorted media, or their refractile indices together with the entree of free infinite, Y. However, the notation is going progressively cumbrous and will look even more so when the behaviour of thin movies will see.

Since H = y ( A? – Tocopherol ) where Y = NY is the optical entree. It is found that it is convenient to cover with E and H, the constituents of E and H parallel to the boundary, and so a atilt optical entree ? which connects E and H is introduced as

? = H / E. ( 2.52 )

At normal incidence ?= y =nY while at oblique incidence

where the ? and the Y in ( 2.53 ) and ( 2.54 ) are those appropriate to the peculiar medium. In peculiar, Snell ‘s jurisprudence, equation ( 2.26 ) , must be used to cipher ? . Then, in all instances, we can compose

These looks can be used to calculate the fluctuation of coefficient of reflection of simple boundaries between extended media with angle of incidence. In this instance, there is no soaking up in the stuff and it can be seen that the coefficient of reflection for p-polarized visible radiation ( TM ) falls to zero at a definite angle. This peculiar angle is known as the Brewster angle and is of some importance.

The look for the Brewster angle can be derived as follows. For the p-reflectance to be zero, from equation ( 2.46 )

Snell ‘s jurisprudence gives another relationship between:

Extinguishing from these two equations gives an look for

Note that this derivation depends on the relationship Y = New York valid at optical frequences.

## 2.4 The coefficient of reflection of a thin movie

A simple extension of the above analysis occurs in the instance of a thin, plane, parallel movie of stuff covering the surface of a substrate. The presence of two ( or more ) interfaces means that a figure of beams will be produced by consecutive contemplations and the belongingss of the movie will be determined by the summing up of these beams. The movie is called thin when intervention effects can be detected in the reflected or familial visible radiation, that is, when the way difference between the beams is less than the coherency length of the visible radiation, and is called midst when the way difference is greater than the coherency length. The same movie can look thin or thick depending wholly on the light conditions. The thick instance can be shown to be indistinguishable with the thin instance integrated over a sufficiently broad wavelength scope or a sufficiently big scope of angles of incidence.

Normally, the movies on the substrates can be treated as thin while the substrates back uping the movies can be considered thick.

The agreement is illustrated in Fig. 2.4. At this phase it is convenient to present a new notation. The moving ridges in the way of incidence are denoted by the symbol + ( that is, positive-going ) and the moving ridges in the opposite way are denoted by ? ( that is, negative-going ) .

Figure 2.4 Plane wave incident on a thin movie.

The interface between the movie and the substrate, denoted by the symbol B, can be treated in precisely the same manner as the simple boundary already discussed. The digressive constituents of the Fieldss are considered. There is no negative-going moving ridge in the substrate and the moving ridges in the movie can be summed into one attendant positive-going moving ridge and one end point negative-going moving ridge. At this interface, so, the digressive constituents of E and H are

where the common stage factors are neglected and where Eb and Hb represent the end points. Hence

The Fieldss at the other interface a at the same blink of an eye and at a point with indistinguishable ten and y co-ordinates can be determined by changing the stage factors of the moving ridges to let for a displacement in the omega co-ordinate from 0 to ?d. The stage factor of the positive-going moving ridge will be multiplied by exp ( i? ) where

? = 2? N1 vitamin D cos?1 / ?

and ?1 may be complex, while the negative-going stage factor will be multiplied by exp ( ?i? ) . This is a valid process when the movie is thin. The values of E and H at the interface are now, utilizing equations ( 2.58 ) to ( 2.61 ) ,

so that

This can be written in matrix notation as

Since the digressive constituents of E and H are uninterrupted across a boundary, and since there is merely a positive-going moving ridge in the substrate, this relationship connects the digressive constituents of E and H at the incident interface with the digressive constituents of E and H which are transmitted through the concluding interface. The 2 – 2 matrix on the right-hand side of equation ( 2.62 ) is known as the characteristic matrix of the thin movie.

The input optical entree of the assembly is defined by analogy with equation ( 2.52 ) as

Y = Ha /Ea ( 2.63 )

when the job becomes simply that of happening the coefficient of reflection of a simple interface between an incident medium of entree ?0 and a medium of entree Y, i.e.

? = ( ?0 – Yttrium ) / ( ?0 + Y )

Equation ( 2.62 ) can be normalized by spliting through by Eb to give

and B and C, the normalized electric and magnetic Fieldss at the front interface, are the measures from which the belongingss 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 coefficient of reflection can be calculated.

is known as the characteristic matrix of the assembly.

## Matrix Formulation for Isotropic Layered Media: Transportation

## Matrix Method for Isotropic Layered Media

The method described in subdivision 2.4 can be used to cipher the coefficient of reflection of a thin movie. However, when the figure of beds becomes excessively big, the analysis becomes really complicated because of the big figure of equations involved. In this subdivision we will present a matrix method that is a systematic attack to work outing such a job. The matrix method is particularly utile when a computing machine is available that can manage the matrix algebra. It is besides really utile when a big part of the construction is periodic.

## 2.5.1 The analysis of an assembly of thin movies

Let another movie be added to the individual movie of the old subdivision so that the concluding interface is now denoted by degree Celsius, as shown in Fig. 2.5. The characteristic matrix of the movie nearest the substrate is

Figure 2.5 Notation for two movies on a surface.

And from equation ( 2.62 )

Equation ( 2.62 ) can be applied once more to give the parametric quantities at interface a, i.e.

and the characteristic matrix of the assembly, by analogy with equation ( 2.65 ) is,

is, as earlier, C/B, and the amplitude contemplation coefficient and the coefficient of reflection are, from ( 2.64 ) ,

? = ( ?0 – Yttrium ) / ( ?0 + Y )

This consequence can be instantly extended to the general instance of an assembly of Q beds, when the characteristic matrix is merely the merchandise of the single matrices taken in the right order, i.e.

Where

?r = 2? Nr dr cos?r / ?

and where the postfix m is used to denote the substrate or emergent medium

If ?0, the angle of incidence, is given, the values of ?r can be found from Snell ‘s jurisprudence, i.e.

The look ( 2.69 ) is of premier importance in optical thin-film work and forms the footing of about all computations.

The order of generation is of import. If Q is the bed next to the substrate so the order is

M1 indicates the matrix associated with bed 1, and so on. Y and ? are in the same units. If ? is in mhos so so besides is Y, or if ? is in free infinite units ( i.e. units of Y ) so Y will be in free infinite units besides. As in the instance of a individual surface, ?0 must be existent for coefficient of reflection and transmission to hold a valid significance. With that provision, so