Radiators with many layers

 A unit with many boundaries, a radiator, can be achieved using foils placed closely together as shown in fig. 3.3.

  

Figure 3.3: A radiator with many boundaries. The boundaries are numbered from 1 to n - 1 and the media with encircled numbers from 1 to n .

With one boundary two different stationary solutions to the Maxwell equations have to be matched, but with many boundaries a general solution is complicated. As seen in equation (3.4) and fig. 3.2 the radiation is for relativistic particles sharply peaked in the forward direction and from (3.5) with the energy concentrated in the X-ray region. Hence backwards radiation and reflections can be ignored. With this simplification the electric field at the end of the radiator is given as a sum of the fields from the individual boundaries taking interference and absorption into account,

 

$$\vec{E} \omega \vec{\theta} \sum_{j=1}^{n-1} \vec{e}^{j}_{} \omega \vec{\theta} \left( - \sum_{{m \geq j}}^{{n-1}} \sigma_{{m}} + {i}\varphi_{{m}} \right).$$ (53)

$\vec{e}^{j}_{}$ is the single surface amplitude as in (3.1) with the media surrounding boundary j substituting $\xi_{1}^{}$ and $\xi_{2}^{}$ . The absorption coefficient $\sigma_{m}^{}$ is given as

 

$$\sigma_{m}^{} {\frac{l_m}{\lambda_m}}$$ (54)

where $\lambda_{m}^{}$ is the absorption length in the medium and l_m the length of the medium along the particle path. The difference in phase $\varphi_{m}^{}$ for transition radiation from different layers, caused by different times for the particle and the photons to cross the layers, is

 

$$\varphi_{{m}}^{} {\frac{\omega l_{{m}}}{v}} \vec{k}_{{m}}^{} \cdot \vec{l}_{{m}}^{}$$ (55)

where v is the velocity of the particle, $\vec{k}_{{m}}^{}$ the wave vector of the photon with frequency $\omega$ and $\vec{l}_{{m}}^{}$ the vector between the crossing points of the boundaries and the particle as illustrated in fig. 3.4.

  

Figure 3.4: The phase difference of transition radiation from different boundaries is affected by the wave vector in the media, the velocity of the particle and the distance between the boundaries.

The phase difference can be simplified using
   $$\begin{align} \epsilon_{{m}} &= 1 - \xi_{{m}}^2 \\ k_{{m}} &= \sqrt{\epsilon_{... ...} z_{{m}} &= \frac{2}{(\gamma^{-2} + \theta^2 + \xi_{{m}}^2) \omega}\end{align}$$
it is seen that with l_m $\ll$ z_m the two boundaries of the m'th media will have negative interference, i.e. they add up with opposite sign and equal magnitude in (3.8), resulting in no transition radiation. The interpretation is that creation of transition radiation is in fact a macroscopic effect and the effect that there is no transition radiation if the layers have below a certain thickness is called the formation zone effect.

For a single foil placed in a gas with no absorption

 

$$\left(\frac{{d}^2W}{{d}\omega{d}\Omega}\right)_ \sin^{2}_{} \varphi_{1}^{} \left(\frac{{d}^2W}{{d}\omega{d}\Omega}\right)_$$ (56)

using (3.1), (3.4) and (3.8). Since the interference exp(i $\varphi_{m}^{}$) is included in the angular integration of the radiated energy it turns into a complicated integral. In the literature [32] it is expressed in terms of the dimensionless variables $\nu$ and $\Gamma$
  

$$\nu {\frac{2 \omega}{l_{1} \omega_{{P}_1}^2}}$$ = (57)

$$\Gamma {\frac{2 \gamma}{l_{1} \omega_{{P}_1}}}$$ = (58)

The variables $\nu$ and $\Gamma$ are defined such that $\nu$,$\Gamma$ > 1 is the region where the transition radiation is strongly suppressed due to the formation zone effect.

  

Figure: The transition radiation yield from a single foil expressed through the universal function G($\nu$,$\Gamma$) . Note the broad maximum around $\nu$ = 1/$\pi$ .

For a single foil placed in vacuum

 

$$\left(\frac{{d}W}{{d}\omega}\right)_ {\frac{2 \alpha}{\pi}} \nu \Gamma$$ (59)

with G($\nu$,$\Gamma$) plotted in fig. 3.5. The broad maximum around $\nu$ = 1/$\pi$ is important for the design of a detector as will be described in section 3.5. Equation (3.20) only applies to the case with vacuum outside the foils, $\omega_{{P}_2}^{}$ = 0 , but can be extended to the non-vacuum situation through the introduction of $\omega_{{P}_i}^{}$' and $\gamma$' defined as:
 $$\begin{align} \omega_{{P}_1}' &= (\omega_{{P}_1}^2 - \omega_{{P}_2}^2)^{1/2} \... ... 0 \\ \gamma' &= (\gamma^{-2} + \omega_{{P}_2}^2/\omega^2)^{-1/2}.\end{align}$$
In (3.1), (3.4) and (3.8) $\gamma^{-2}_{}$ and $\xi_{i}^{2}$ enters only in the combination $\gamma^{-2}_{}$ + $\xi_{i}^{2}$ which is preserved when substituting $\omega_{{P}_i}^{}$ and $\gamma$ by $\omega_{{P}_i}^{}$' and $\gamma$' . The substitution transforms the general case into the case with vacuum between the foils and (3.20) can be used in the form

 

$$\left(\frac{{d}W}{{d}\omega}\right)_ {\frac{2 \alpha}{\pi}} \nu \Gamma$$ (60)

where $\nu$' and $\Gamma$' are defined using $\omega_{{P}_1}^{}$' and $\gamma$' in (3.18) and (3.19).

From (3.21) it is seen that $\omega_{{P}_1}^{}$' < $\omega_{{P}_1}^{}$ and $\gamma$' < $\gamma$ results in a lower yield from a detector with gas instead of vacuum between the foils. Also the nice feature of a linear response to $\gamma$ as seen in (3.6) disappears. Instead there is a saturation in the transition radiation at

 

$$\gamma \geq \gamma \gamma \infty {\frac{\omega}{\omega_{{P}_2}}}$$ (61)