1. (a) Starting from Maxwell's equations for a homogeneous, linear, isotropic, dielectric material:

show that:

[ε(x, y, z) μ₀∂²/∂t² -  ∇²] H (x, y, z, t) = 0.

Be sure to explain any assumptions or approximations made.

(b) In lectures, we showed that for a dielectric material the refractive index "n" is given by:

n_ω = √(ε_ω/ε₀)

Where ω denotes the frequency dependence.

Would you expect the refractive index for a good conductor to be very different to that of a dielectric? In no more than half a page, please explain.

2. Suppose that we wish to design a refractive lens for focusing X-rays by shaping a suitable sample of homogeneous material whose complex index of refraction at a given frequency, ω, is given by nω = 1 - δ_ω + iβ_ω, where δ_ω > 0 and β_ω > 0.

Show that a thickness profile of the form T(r_⊥) = Ar_⊥² etched into a slab of material will produce a lens whose focal length is given by ?? = 1/(2Aδ_ω). You may assume that the thin lens formula is applicable and r_⊥ is the position vector.

3. The amplitude of the exit wave for a 1D reference beam holography experiment is assumed to be composed of a top hat function (width = 1 pixel), and two overlapping Gaussian functions (σ = 1). The 1D profile for this experiment is shown below:

Fig 1: 1D line profile of amplitude of exit wave

a) Study this profile carefully and then either sketch or plot the corresponding autocorrelation function. (Note you may use a computer program if you wish).

b) As the width of the top hat function (which can be thought to represent a small pinhole) increases, the autocorrelation function appears increasingly ‘blurred'. Using Fig 1 estimate the width of the top hat which reduces visibility (V) of features in the autocorrelation function to 50%.

Where: V = (I_MAX - I_MIN)/(I_MAX + I_MIN)

4. Consider a detector that has 1024 × 1024 pixels that are 50 μm wide. For 10 keV x-rays, how far should the detector be placed from the sample to measure information to 2 Å resolution at the detector's edge?

5. In no more than 1 page describe how anomalous scattering can be used to aid in structure determination in macromolecular crystallography. In your answer consider the variation of the form factor with energy as well as the resulting Patterson function.