The angular distribution for transition radiation can be obtained by different methods, where the most common ones are the annihilation of the electron with its mirror particle or the conversion of virtual photons of the electromagnetic field of the electron into real photons due to the boundary condition at the conducting surface.
We developed a model, where we derived the radiation from the induced surface current. The emission per frequency interval and solid angle is
where J is the induced surface current and n is a unit vector in the direction of observation. The model has the capability to restrict the emitting surface (limited radiator or diffraction radiation) as well as modeling uneven surface, which is mainly expressed as a position-dependent phase factor exp[i kz z(x,y)] in the integral of W.
The solution for a finite radiator size (circle with radius R) is
The solution converges to the infinite plane solution for a radius larger than the product of gamma and lambda. This is the effective emission size of transition radiation. Otherwise strong diffraction effects become significant and the total emitted radiation power drops down. This implies a significant effect on CTR diagnostic for high energetic beam in the mm range wavelength, because the targets have to be sufficient large to provide enough radiation.
Assuming a 2D periodic structure there are two fundamental limits. For very short wavelength the effective radiation sizes goes down. In this limit the size is smaller than the period length and the radiation pattern becomes that of the flat surface. In the other limits of long wavelength the applied phase modulation exp[i kz z(x,y)] becomes smaller, because kz goes to zero and again approaches the pattern of the flat surface. A change in the radiation pattern is only visible in a bandwidth around the modulation amplitude.
Because the effective radiation size is larger for higher electron energy, it includes more modulation and the emission patterns are more resonant (Fig.1).
Fig.1: Angular distribution of TR at 100 nm for a modulated surface with 100 nm amplitude.
Similar is the emission from a rough surface, with the exception that there is not the limit for very short wavelength, because roughness can be assumed to be fractal, so that on a smaller length scale there is still roughness present. Here a large electron energy means that more bumps are included and the distortion becomes stronger (Fig.2).
Fig.2: Angular distribution of TR at 100 nm for a rough surface
We plan to investigate transition radiation for finite size targets and targets with imprinted roughness or modulation. The set-up will be an electron beam of around 15-20 MeV at the PEGASUS facility. The injector is operated inn thermonic emission mode, which gives a higher charge and thus a better OTR signal.
A digital camera grabs the near or far field (depending on the focal length of the camera) of the OTR image. The set-up is calibrated with the standard image from an uncoated copper mirror with a surface flatness below a quarter wavelength in the visible regime. Early experiments have shown a ring like image in the far zone, with no dependence on the frequency. This is in agreement with the theory. Planned targets are optical gratings and surface with artificially enhanced roughness.
Using a color camera allows us to take three spectral images at the same time. However this comes with the cost of reduced intensity because the captured light is distributed into the three color channels. If the OTR signal is too weak, a monochromatic camera is used instead.