Field analysis for a highly overmoded iris line with application to THz radiation transport

Vector field analysis is presented for a highly overmoded iris-line structure that can act as a medium of transportation for THz radiation. The axisymmetric structure is capable of supporting hybrid modes with desirable features such as low propagation loss, uniformly linear polarization and approximately Gaussian intensity profile across the iris. A specific application that can benefit from these desirable features is the transportation of THz undulator radiation over hundreds of meters to reach the experimental halls at the Linac Coherent Light Source (LCLS) facility at SLAC, Stanford. Such a structure has been modeled before as a boundary-value problem using Vainstein’s complex-impedance boundary condition and assuming infinitely thin screens. Given that physical realizations of such screens must have finite thickness and that the THz wavelength in the 3–15 THz range is expected to be smaller than convenient screen thicknesses in practice, the question of the impact of finite screen thickness on propagation performance becomes rather pressing. To address this question, we present a mode-matching analysis of the structure as an open resonator with finite screen thickness and perturbatively clustered (localized) field expansions, for computational feasibility. The effect of screen thickness is seen to lower the attenuation constant on the iris line, which is dominated by diffraction loss. Ohmic loss due to the finite conductivity of metallic surfaces at the screen edges is found to be small compared to diffraction loss. The propagation loss predictions based on the Vainstein model are compared with the numerical results from mode matching for infinitely thin screens, where the former method is observed to agree with the numerical results better at higher Fresnel numbers (highly overmoded structures). The properties of the dominant mode fields are formally derived from first principles and a recommended approach is discussed for the inclusion of screen-thickness effects into propagation loss estimations.

Figure 1

(a) The iris-line geometry shown in three dimensions, without the enclosing chamber at the outer radius. (b) A cross section in the iris line, showing dimensions and labeling.

Figure 2

(a) A canonical example of Fresnel diffraction of a plane wave caused by a single knife-edge obstruction (thin and absorptive screen). The field strength of the wave traveling from point $S$ to point $P$ diminishes according to the function $|F(\nu )|$ shown. As the point $P$ moves deeper into the shadow region behind the screen, the attenuation is increased. The Fresnel parameter $\nu =h\sqrt{2(d_1+d_2)/({\lambda }_0{d}_1{d}_2)}$ defining the depth of shadow (or clearance) is found geometrically from the dimensions shown. (b) A simplified illustration of rays lost to diffraction, as they hit an absorptive screen or get diffracted into the shadow region bound by two such screens. (c) A simplified illustration of rays lost to reflection or diffraction in the region between two conductive screens.

Figure 3

Power loss for 150 m (to reach the near experimental hall at LCLS) for the frequency range 3–15 THz, using an iris line with a period of $b=30\mathrm{cm}$ and two iris radii ($a=5.5$ or 10 cm). This prediction is based on the Vainstein model and assumes zero screen thickness. The larger radius is recommended for lower loss, if mechanically feasible.

Figure 4

Regions of analysis for the open-resonator model, shown over the cross section for one period (gap) in the iris-line structure.

Figure 5

A qualitative sketch demonstrating the effect of iris loading on an otherwise-smooth pipe as the depth of iris loading (added admittance) is changed. In this sketch ${\beta }_n$ is assumed to be real (no loss). For a smooth pipe, the continuous parabola (green) is observed, where phase velocity $v\ge c$ and $k_0\ge {\beta }_0$ for propagating modes, asymptotically leading to the limit of TEM propagation at the 45-deg line. As we start loading the pipe with shallow (weak) irises, the propagated modes are allowed in certain passbands, attenuated in stop bands, and the dispersion curves (red) become periodic in ${\beta }_n$, with every side lobe period (index $n$) now corresponding to a Fourier harmonic bin, where ${\beta }_n={\beta }_0+2\pi n/b$. This allows for slow and fast waves to propagate (depending on the used harmonic). If the irises are gradually removed, the passband curves (red) expand and coalesce back into the parabolic curves of the smooth pipe (green) [20]. If the iris loading increases, the passbands (blue) are compressed, and in the limit they become flat spectral lines, with wider stop bands in between. Since our structure is highly overmoded with a paraxial incidence, we effectively operate in the perturbation region (purple shade) in the vicinity of the parabolic curve and make our choice for the $n=0$ harmonic band accordingly; see the simplified example of point $({\beta }_0^{*},k_0^{*})$ on the figure.

Figure 6

(a) A clustered expansion in $n$, taking the paraxial dominant harmonic (in red) to be the basic harmonic (${\beta }_n={\beta }_0$) at $n=0$, which corresponds to approximately $N_0$ wavelengths per structure period, where $N_0$ is the nearest integer to $b/{\lambda }_0$. This means that the image of the dominant harmonic is at the index $n=-2N_0$ (in red). Note that the expansion is equivalent to ${\sum }_{-nSteps}^{nSteps}+{\sum }_{-2N_0-nSteps}^{-2N_0+nSteps}$, where we choose $nSteps$. (b) A regular versus clustered expansion in $p$, where the dominant mode in region 2 (excited by the paraxial incidence in region 1) happens at index $p=P_0$, where $P_0$ is the floor integer nearest to $2\mathrm{\Delta }/({\lambda }_0/2)$ from below (i.e., number of half wavelengths in the width $2\mathrm{\Delta }$). The regular expansion here is ${\sum }_{p=0}^{pSteps}$, and the clustered expansion is ${\sum }_{p=P_0-pSteps}^{P_0+pSteps}$, where we choose $pSteps$.

Figure 7

Effect of varying screen thickness $\delta$ from zero to $\sim b/10$ (that is 0.0–30.0 mm) on $\mathrm{Im}[{\beta }_0]$, for the Scale-3 structure with period $b=333.33\mathrm{mm}$, calculated at $pSteps/nSteps=6666/3333$. Selected points on the curve give explicit value of $\delta$ in mm. See data in Table 2 for details.

Figure 8

Effect of varying screen thickness $\delta$ from zero to $\sim b/10$ on $\mathrm{Im}[{\beta }_0]$, for the three scales discussed above, shown together on the same figure for comparison. Selected points on the curve give explicit value of $\delta$ in mm. See data in Table 2 for details.

Figure 9

Propagation loss is shown as a function of frequency for two iris radii ($a=5.5$ or 10 cm), two screen thicknesses (0 or 2 mm) and over a distance of 150 m to reach the near experimental hall in LCLS. The iris-line period is $b=30\mathrm{cm}$ and the effect of thickness is found from the mode-matching analysis and then used to refine Vainstein’s model (assuming that it is the same effect for both cases, namely $\mathrm{\Delta }\mathrm{Im}[{\beta }_0]/\mathrm{Im}[\beta ]\cong -33.9\%$). The larger radius is recommended, if mechanically feasible and does not have a considerable transient at the line’s entrance.

Figure 10

Scale-3 field polarization and amplitude profile plots, calculated at $pSteps/nSteps=6666/3333$ and plotted at $nSteps=1111$: (a)–(c) for $\delta =0\mathrm{mm}$, (d)–(f) for $\delta =2\mathrm{mm}$, and (g)–(i) for $\delta =10\mathrm{mm}$. The polarization and amplitude profiles seem unaffected by the small increase in screen thickness. Subplots (c), (f) and (i) show $E$-field amplitude profile cuts at the $y=0$ plane across the iris, compared to the idealized shape of the $J_0(2.4r/a)$ function. A small sharp spike is observed in the amplitude profile in (f) and (i), which may be a result of low numerical resolution (truncation).

Figure 11

(a) Transverse electric field stream lines and amplitude strength for the dominant hybrid mode ($n=1$, $j=1$) of the iris line at high-frequency operation (overmoded) above cutoff ($k_0\approx \beta$), based on Eqs. (a25) and (a26). This mode is promising for direct coupling with radiation from a THz linear undulator at LCLS. (b) A comparison between the intensity profiles of the Bessel function $J_0^2(r)$ and the Gaussian $e^{-r^2/{\sigma }^2}$, where $\sigma$ is the amplitude rms width.

Figure 12

A comparison between the real part of the field expansion (a37) and the function $J_0(2.4r/a)$. The effect of having a small complex perturbation in the argument of the Bessel function will be to slightly lift the amplitude off zero near the edges, compared to the real function $J_0(2.4r/a)$.

Figure 13

An illustration of the harmonic distribution over the index $n$ under discussion (top axis), where the dashed line depicts the expected decay in harmonic strength as we move away from the band edges (in red) that represent the dominant harmonic and its image. Given the symmetry in the Bessel functions involved in the different bands, it is sufficient to focus our treatment on the middle band, between $-n_0$ and 0, where $k_{tn}$ is real. A convenient positive integer $d$ is introduced in the derivations, for the middle band, and is illustrated in the bottom axis.

Figure 14

(a) A plot of the convex modulation function, $|{\varphi }_d^{-1}|$, responsible for the clustering effect, for the overmoded iris-line example with $a=0.055\mathrm{m}$, $b=0.33\mathrm{m}$, ${ϵ}_1=0.0001$ and $n_0$ of 6666. (b) An example of the effect of the modulation on the oscillatory term $1/J_1^2({\psi }_{1}a)$, plotted using the formula (b30) for the same parameters. The edges of the band (where $d$ tends to 0 or 6666), correspond to the dominant harmonic and its image; clustering is manifested in how the envelope of all harmonics decreases as we move away from band edges.