Cylindrical Multimode Waveguides as Focusing Interferometric Systems

Delivery and focusing of radiation requires a variety of optical elements such as waveguides and mirrors or lenses. Heretofore, they were used separately, the former for radiation delivery, the latter for focusing. Here, we show that cylindrical multimode waveguides can both deliver and simultaneously focus radiation, without any external lenses or parabolic mirrors. We develop an analytical, ray-optical model to describe radiation propagation within and after the end of cylindrical multimode waveguides and demonstrate the focusing effect theoretically and experimentally at terahertz frequencies. In the focused spot, located at a distance of several millimeters to a few centimeters away from the waveguide end, typical for focal lengths in optical setups, we achieve a more than 8.4× higher intensity than the cross-sectional average intensity and compress the half-maximum spot area of the incident beam by a factor of >15. Our results represent the first practical realization of a focusing system consisting of only a single cylindrical multimode waveguide, that delivers radiation from one focused spot into another focused spot in free space, with focal distances that are much larger than both the radiation wavelength and the waveguide radius. The results enable design and optimization of cylindrical waveguide-containing systems and demonstrate a precise optical characterization method for cylindrical structures and objects.


Waveguide power loss methodology
Here we explain how we estimated the transmission of the waveguide, stated as 81.3% in the main text. As shown in Fig. S1 (a), rays entering the waveguide from the QCL have three possible paths: they enter the waveguide (shown as dark green arrows), or they are reected back at the front face of the waveguide, or they miss the waveguide and follow a straight path into free space. The latter two beam paths are illustrated by dark red arrows.
The 4.6 mm inner diameter waveguide used in the experiment has an outer diameter of approximately 6.3 mm. For our power loss analysis, we aim to determine by how much the S2 power of the radiation that has entered the waveguide is attenuated by the time it reaches the other end. This means that only the power contained in the rays indicated by the dark green color in Fig. S1 (a) should be considered as the power that entered the waveguide, used as a reference for the transmission calculation. This allows us to exclude sample-and setupspecic parameters, such as the width of the waveguide walls or the angle of divergence of the QCL emission. We assume that all dark green rays that travel toward the hollow cylindrical inner part of the waveguide enter the waveguide, and neglect wave-optical scattering on the circular edge of the waveguide. Because the diameter of the waveguide, 4.6 mm, is much larger than the radiation wavelength of 159.38 µm, this ray-optical approach is justied.
For this characterisation, we measure the full mode prole at the focus position d 2 = 8.8 mm, as illustrated in Fig. S1 (c), by scanning the Golay cell with a 1 mm aperture in the x and y lateral directions. Then we remove the waveguide and repeat the scanning by placing the Golay cell directly in front of the QCL. The distance between 1 mm aperture and QCL is set to the same distance as d 1 used previously for the measurements with the waveguide.
The 1 mm aperture used for this characterisation is a at, xed-diameter aperture fabricated from a copper sheet, that was not changed or adjusted in any way between the measurements shown in Fig. S2 (a) and (b) to ensure comparability.
The result of the two measurements is shown in Fig. S2. The data in this gure is from the same measurements as presented in Fig. 9 (a), (b) in the main text, except that here the data is not normalised to the integrated power. Instead the original signal is shown in units of millivolts, measured as the output voltage of the Golay cell, captured by the sign-sensitive in-phase component at the output of a lock-in amplier. This allows quantitative comparison of both measurements.
The obtain the full power transmitted through the waveguide, we take a 2D integral of the data shown in Fig. S2 (b), which yields 17.8 mV·mm 2 . To obtain the full power that entered the waveguide, we take a 2D integral of the intensity shown in Fig. S2  line that indicates the radial cut-o for the integration. This gives a value of 21.9 mV·mm 2 .
The obtained power ratio is 81.3 %, which is the estimated transmittance of the radiation power as it travels through the 238 mm-long waveguide.

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2 Comparison of 0.5mm and 1.0mm aperture In Fig. 9 of the main text we have demonstrated the focusing ability of a multimode cylindrical waveguide. We showed a measurement with a 1.0 mm aperture and with a 0.5 mm aperture. These measurements are also shown here in Fig. S3 (a), (b). Here we answer the question, how would the mode prole captured with a 0.5 mm aperture be transformed if it were captured with a 1.0 mm aperture?
To answer this question, we convolve the 0.5 mm aperture measurement, Fig. S3 with a Gaussian function, as described in Eqs. (10)(11) in the main text. We choose the averaging constant a in the Gaussian function e −(x 2 +y 2 )/a 2 /(πa 2 ) such that the peak intensity of the convolved mode prole equals the experimentally measured peak intensity for the 1.0 mm aperture measurement in Fig. S3 (a) of 0.274/mm 2 . This is the case for an averaging constant of a = 0.302 mm, corresponding to an FWHM of 0.503 mm. The result of the convolution is shown in Fig. S3 (c). It shows the convolution of the 0.5 mm aperture measurement in Fig. S3 (b) with a Gaussian with averaging constant a = 0.302 mm, and can be understood as the theoretically expected mode prole that would be observed with a 1.0 mm aperture, based on measurement data of a mode prole with 0.5 mm aperture.
As can be seen, the resulting mode prole is very similar to Fig. S3 (a), the experimental measurement with a 1.0 mm aperture. Even the quantitative area enclosed by a contour line at half the peak value is the same, within 1% accuracy. The observed deviations are very small, and can be observed mainly for two reasons: rstly, the two measurements were carried out at slightly dierent distances d 2 = 8.8 mm in Fig. S3 (a) vs. d 2 = 5.6 mm in Fig. S3