Chapter IV
Theoretical Tools
the position of the top surface, see Fig. 14. Supposing that the sample is not homogeneous, but consists of different layers, we can decompose the transfer matrix:
A+(z) A−(z)
=M1M2...Mn
A+(0) A−(0)
(26)
where each layer has a separate transfer matrix denoted by the index, and we have omitted the energy dependence. The number of layers is N. For the procedure determining the transfer matrix for nuclear resonant layers from first principles, we refer the reader to [63]. We give the results here, remarking that the only difference to other methods is the calculation of the index of refraction. The transfer matrix for a single layer is
M1=
1 r01 r01 1
eikzz 0 0 e−ikzz
1 r10 r10 1
1 t01t10
(27) where r01(r10) are the reflection coefficients from the vacuum to the first layer and vice versa, t01(t10)are the respective transmission coefficients, k1z are the wave vector components in the direction of growth (i.e. the direction where the layer changes) andzis the location. k1z=k0zβ where
β= s
1+2(n−1)
k0z (28)
and
rij = βi−βj
βi+βj (29)
tij = 2βi
βi+βj (30)
andnis the index of refraction of the layer in question. All these quantities have to be calculated for each individual layer; the transfer matrix for a stack of layers can be calculated by appending the matrices for all layers. We will call the total transfer matrixMtot, andDis the position where the layer stack ends, see Fig. 14.
A+(D) A−(D)
=Mtot
A+(0) A−(0)
(31)
where
Mtot =
M++ M+−
M−+ M−−
(32)
Now, assuming that the incident wave is incoming from the positive direction, and that there is no field incident from the negative direction, we can setA+(0) = A0andA−(D) =0 where A0is the amplitude of the incoming wave. We then get two equations
1. The transfer matrix algorithm 37
A−(0) =−(MM−+−−(D)(D))A0=RA0
A+(D) = (M++(D)− M+−M(D)M−+(D)
−−(D) )A0=TA0 (33)
whereRandTare the reflectivity coefficients.
For periodic layer structures, we adopt the same formalism. We again decompose the electro-magnetic field illuminating the sample into a forward- and backward propagating component calledA+andA−. The amplitudes of both components at the upper and lower boundaries of a given layer are connected to each other via a transfer matrix
A+(m+1) A−(m+1)
=M(ω)
A+(m) A−(m)
(34)
Here,mis the number of the period of the periodic multilayer. The above can be expressed in vector notation as
A~ =M(ω)A~0. (35)
Mcan be more precisely expressed asei~Fz, where~Fthe scattering matrix of the medium, andzis the propagation distance, i.e. the thickness of the film. Supposing that our multilayer is infinite, we can use an alternative description for the field propagation across one period in the ML structure:
A+(m+1) A−(m+1)
=
eik0za 0 0 e−ik0za
A+(m) A−(m)
(36)
wherek0zis thez-component of the wave vector of the incident light andais the thickness of the period. (zis the so-called surface normal, i.e. the direction in which the multilayer is periodic, see Fig. 14.) The above equation is the Bloch theorem for a two-beam case, which states that in a periodic system, a wave can be described as the superposition of the wave function within a period and a plane wave with a wave vector whose length corresponds to the period length of the system. Upon inserting Eq. (36) into Eq. (34) we can see thateik0za are the eigenvalues ofM.
Simple algebraic manipulations give
cos(k0za) = Tr(M(ω))
2 . (37)
This is the dispersion relation of the light in the multilayer system. Conventionally, a dispersion relation is resolved forω; this can only be performed in special cases for periodic systems. It can, however, be calculated numerically. To go in this direction, we have to calculateM. We have already pointed out its relation to the scattering matrix and elaborate on that now. In a nuclear resonant layer which is very thin,Mcan be given as
M=1+iFnd1 (38)
whered1is the layer thickness,Fn the nuclear scattering matrix and1the unity matrix.Fnis
Fn =
fn+k0z fn
−fn −fn−k0z
(39)
where fn = fn(ω) is the frequency-dependent nuclear scattering amplitude. We will assume that the non-resonant parts, that is the electronic scattering amplitude of the layer is included in the nuclear one. This comes down to adding a constant term, since the electronic scattering is basically constant over the energy range of the nuclear scattering amplitude in almost all Mössbauer systems. In an isotopic multilayer, which consists of alternating layers of non-resonant
56Fe and resonant57Fe, the reflection and transmission coefficients at the interfaces are merely due to resonant scattering, which allows us to ignore electronic contributions in the following. For the non-resonant layer then, only changes incurred in the propagation through the layer appear;
the corresponding transfer matrix is
Mnr=
eik0znd2 0 0 e−ik0znd2
(40)
whered2is the thickness of the non-resonant layer. Multiplying Eqs. (38) and (40) and inserting the product into Eq. (37), we get
cos(k0za)
2 =cos(k0znd2) +i(fn+k0z)d1sin(k0znd2) (41) This is the dispersion relation for an infinite ML; it can be calculated numerically. Calculating the dispersion relation for a finite ML is not possible analytically, but we can calculate the transfer matrix for one withNlayers:
MN(ω) = sin(Nk0zd)
sin(k0zd) M(ω)−sin((N−1)k0zd)
sin(k0zd) 1 (42)
From this, we can calculate the reflectivity, transmittivity, and absorption as a function ofωfor this structure
R= MMN(1,2)(ω)
N(2,2)(ω)
T= MMN(2,1)(ω)
N(2,2)(ω)
A=1−R−T.
(43)
This method yields quantitative predictions against which experimental results can be tested.
However, the theoretical predictions refer to the case of an ideal experiment. This includes a
1. The transfer matrix algorithm 39 sample/multilayer of infinite lateral length (or an infinitely small beam), a beam without any divergence and other assumptions that are not warranted in reality, but often fulfilled to a good approximation. We now introduce the methods required to calculate the reflectivity correctly even in imperfect experimental conditions.
For the divergence, the procedure is straightforward [110]. The reflectivity curve is simply convoluted with a Gaussian whose FWHM corresponds to the divergence:
Rcorr(Θ) =
R R(Θ−φ)D(φ)dφ
R D(φ)dφ (44)
where D(φ) is the Gaussian describing the divergence, R is the reflectivity, and Rcorr is the reflectivity taking into account the beam divergence. The next issue is the problem of beam width and sample length. At zero degrees incidence angle (as mentioned we define the incidence angle as the angle between the surface and the beam), half the beam passes the sample and is transmitted into the detector. As the angle of the sample is changed, some of the radiation that was transmitted before is reflected now, and is detected at double the incidence angle. Only part of the beam actually is reflected; the size of this part is referred to as the footprint. As the angle is increased further and further, larger parts of the beam are reflected. This has the curious effect of increasing the reflected intensity of radiation until the tilted sample blocks the whole beam, even though the actual reflectivity decreases as the angle is increased. This effect can be incorporated into the transfer matrix method (TMM) used to calculate the reflectivity by multiplying the reflectivity with an angle-dependent prefactor. The functional form of the prefactor depends on the beam shape; the relevant cases are a square and a Gaussian beam shape. For both, the prefactors can be derived by simple geometrical considerations, given in [130]. We give the results here for the square beam:
f = d
lsinα (45)
wheredis the diameter of the square beam,lis the sample length in the direction of propagation, andαis the angle of incidence. Naturally, this prefactor has to be applied only for those angles for which the footprint is larger than the sample length, that is fromα=0 toα=arcsindl.
For a Gaussian beam, the formula is more complicated:
fg=
RLsin(Θ)/2 g(t)dt Rtm
0 g(t)dt (46)
where gis the Gaussian describing the beam, tis the variable describing the distance from its center,tmis the position at which we assume the Gaussian to be zero (we have chosen six FWHMs for this),Lis the size of the sample, andΘthe angle of incidence.