Refractive index

In case where it was not possible to spot the regions on the surface of the samples with the optical microscope built in the AFM, the larger scan range allowed to inspect an area where the step presumably was located. Scans of various lengths around 10 mm were taken across the region boundaries. Most of the scans taken with the AFM had shown a relatively large curvature of the surface, so in order to prove whether this observation was an effect that only occurred close to the region boundaries or the general nature of the sample, profilometer scans were taken not only across the irradiated samples but also the blank samples to provide for comparison.

Figure 5.1 on the preceding page above shows the approximate position where the line scans were taken. The actual location may have differed to avoid obvious surface structures that would interfere with the measurements.

5.2 Refractive index

5.2.1 Modeling the refractive index

To assess the expected order of magnitude of the change in the refractive index of the irradiated bulk samples, the first task was to find a model that would describe the index changes caused by the radiation. Therefore a first approximation of the refractive index change as a result of densification of the sample due to compaction of the sample is shown in Figure 5.2. Here the change in density is assumed to be uniform throughout the affected volume.

0

t₀

∆t

V₀,ρ₀ V(∆t),ρ(∆t)

Figure 5.2: Schematic of the change in surface level, caused by densification of the material due to the irradiation with X-ray.

The DoseSim simulations show that at a penetration depth of 39.5µm the deposited dose falls down to half of the value at the surface. Be t₀ this maximum depth to which the radiation has any significant effect in the material. With the surface area A and the mass m, thus the density of the blank sample is ρ₀ = m/(At₀). Hence the density of the by the height ∆t compacted sample can be expressed as

ρ(∆t) = m

A·(t_o−∆t) =ρ₀ t₀

t₀−∆t. (5.1)

42 5. Measurements Using the relation between the refractive index and density given in Equation 3.4 on page 18 a relation between the refractive index and ∆t can now be established:

n(∆t) = 1.037 + 0.195·ρ₀ t₀

t0−∆t. (5.2)

5.2.1.1 Refining the model

The first approach to approximate the index profile does not take into account the gradient characteristics of the deposited dose shown in Figure 4.3 on page 30 but assumes a step function for the dose which is obviously a very rough approximation it needs to be refined. In analogy with elasticity theory, the sample is considered a system of n layers of the same thickness ∆z_i as shown in Figure 5.3.

∆z

Figure 5.3: The layer model of the irradiated samples. In an unexposed area all n layers have the same thickness ∆z_i. After irradiation and densification the thickness of a layer iis reduced byδz_i and the layer is shifted to a new position.

When compaction occurs these layers are compressed and the thickness of a single layer decreases to ∆z⁰_i. Additionally they are taken from their initial position z_iz_i+1 to a new position z_i⁰z_i+1⁰ with respect to the bottom line at the depth t = z_n = z_n⁰. This gives rise to the change in width

δz_i = ∆z_i−∆z_i⁰ = (z_i+1−z_i)−(z_i+1⁰ −z_i⁰). (5.3) Together with the relation given in Equation 3.6 on page 19, the compactionκi of a single layer i is thus

κ_i = δz_i

∆zi

=C·D(z_i)^2/3. (5.4)

The total compactionκ is then the sum of all contributions κi

κ=

5.2. Refractive index 43 The sum of all δz_i is the total change of the surface level ∆t. Because the layer thickness can be chosen any small and the dose D(z) is defined continuously, the transition from the discrete ∆z_i to the infinitesimal dz is permitted. The sum becomes an integral to the thicknesst of the sample. In doing so one obtains

∆t =C

Z _t

0 D(z)^2/3dz. (5.7)

From this equation the constantC can be derived:

C = ∆t

·Z _t

0 D(z)^2/3dz

¸₋₁

. (5.8)

In order to link the refractive index to the compaction and thereby to z, the com-paction needs to be expressed through the density. The density ρ(z_i) of a single layeriwith the volume V_i being the product of the mass m_i and the surface areaA (and the densityρ⁰(z_i⁰) of the corresponding compacted layer respectively) is

ρ(z_i) = m_i

V_i = m_i

A_i∆z_i =:ρ_i (5.9)

ρ⁰(z_i⁰) = mi

A∆z⁰_i =:ρ⁰_i. (5.10)

The change in density is thus

δρ_i =ρ_i−ρ⁰_i = mi

The product ∆ziδzi is small and is therefore neglected. With Equation 5.9 one obtains

δρi

ρ_i =−δzi

∆z_i. (5.12)

With the Equations 3.4 on page 18, 5.4 on the preceding page and 5.12 the density ρ(z_i) of a layeri becomes Hereby ρ₀ denotes the density of the unexposed sample. Again the transition from the discrete z_i to a continuous function of z is allowed because the layer thickness can be chosen any small and the doseD(z) is defined continuously.

44 5. Measurements With Equation 5.13 on the previous page there exists now a description of the refractive index as a function of the depth z.

5.2.2 Ellipsometry

Due to the fact that the index change in the samples is not linear but gradient, standard methods based on light transmission through the sample such as interfer-ometry were not applicable to quantify the refractive index of the exposed samples.

Ellipsometry, however, provides a promising tool to measure the refractive index accurately enough while at the same time taking into account the gradient nature of the index profile.

The UVISEL Spectroscopic Ellipsometer³ manufactured by HORIBA Jobin Yvon was used for the measurements. A wavelength range from 400 nm to 800 nm was covered, the integration time (i.e. the time used to measure ∆ and Ψ) was 200 ms and the measurements were repeated five times for each wavelength and the average of these measurements was then calculated. For maximised signal at the detector, the angle of incidence was set to 60^◦ which is close to the Brewster angle of glass at 56^◦.

As illustrated above in Figure 5.1 on page 40, the points of incidence where chosen in a way that they were entirely situated in one of the regions 0-III. At the same time obvious surface features such as cracks were tried to be avoided as they could significantly distort the reflected beam.

5.2.3 Analysis of fibres

The analysis of the tested fibres was carried out by S. A. Wade and is described in the appendix.

3http://www.jobinyvon.de/dedivisions/TFilms/uviselspectro.htm

Chapter 6