Visualizing nanometric structures with sub-millimeter waves

The resolution along the propagation direction of far field imagers can be much smaller than the wavelength by exploiting coherent interference phenomena. We demonstrate a height profile precision as low as 31 nm using wavelengths between 0.375 mm and 0.5 mm (corresponding to 0.6 THz–0.8 THz) by evaluating the Fabry-Pérot oscillations within surface-structured samples. We prove the extreme precision by visualizing structures with a height of only 49 nm, corresponding to 1:7500 to 1:10000 vacuum wavelengths, a height difference usually only accessible to near field measurement techniques at this wavelength range. At the same time, the approach can determine thicknesses in the centimeter range, surpassing the dynamic range of any near field measurement system by orders of magnitude. The measurement technique combined with a Hilbert-transform approach yields the (optical) thickness extracted from the relative phase without any extraordinary wavelength stabilization.

of only a single photoconductive pixel and (III) the data evaluation. The single spectrum acquisition time of 24 s (6 ms integration time for 4000 frequency points) can be increased by 1-2 orders of magnitude by replacing the DFB laser diode system with a fast sweep system 1 , where 24 spectra per second with 8750 data points have been recorded. A major bottle neck is the use of a single photoconductive receiver, requiring a 2D scan of the object. Instead, the receiver can be replaced by a second p-i-n-diode-based Terahertz emitter with several tens of microwatt local oscillator power in order to drive a state-of-the-art Terahertz camera, such as a microbolometer 2 or a field effect transistor-based camera 3 . Mixing of the local oscillator signal with the signal transmitted through the object will yield a homodyne mixing term similar to that of a classic Mach Zehnder interferometer or in holographic imaging techniques 4 . The mixing term contains all information required to perform the Hilbert-transform-based algorithm. In this way, the amount of pixels is solely limited by the pixel number of the camera, without the requirement of a mechanical scan. Last but not least, the data evaluation routines can be programmed onto a field programmable gate array (FPGA) which may finally enable real-time image processing.

CORRELATION OF MEASUREMENT PRECISION WITH OPTICAL PATH LENGTH DIFFERENCE
Fig. S1 shows the measured rms height error for various total optical path length differences ∆( ) or, alternatively, homodyne fringe periods ∆ = 0 ∆( ) as shown in Fig. 1b. Since ∆( ) is approximately known from geometry (though with a considerable error), we can estimate the fluctuations of the laser system on the time scale of the frequency sweep (24 s) from Eq. (13) using the slopes shown in Fig. S1. They are estimated to be around ≈209±50 kHz, a value supported by the spectral analysis of the emitted Terahertz signal. This is in the range of the laser linewidth but considerably smaller than the smallest frequency step size used in the measurements, i.e. 40 MHz.

LIMTIATIONS OF THE FABRY-PÉROT CAVITY-ENHANCED IMAGING
For fitting the Fabry-Pérot resonances, the rms noise given by Eq. (11) must be smaller than the visibility of the Fabry-Pérot oscillations of the phase .
is given by the difference between the maximum and minimum phase change caused by the Fabry-Pérot resonances, hence requiring = 2arctan For the concrete example of a silicon wafer with reflectivity = 0.30, = 0.609 rad. For the case of Teflon with = 0.034, = 0.0675 rad. In practice, however, perturbations caused by standing waves in the setup (with phase amplitudes around 0.26 rad) exceed by orders of magnitude, showing that it is difficult to apply the Fabry-Pérot method to Teflon while it works well for silicon.

OTHER NOISE SOURCES
Phase noise by incoherence: The homodyne Terahertz imager uses a coherent detection technique. If the total path length difference of the interferometer arms ∆( ) approaches the coherence length, the visibility of the homodyne fringes (see Fig. 1b)) will be reduced and further phase fluctuations will arise. In the given experiment, the path length difference was below 0.6 m for all measurements. This is much shorter than the self-heterodyne (i.e. short-term) coherence length and still 1-2 orders of magnitude shorter than the long term coherence length.
Amplitude noise caused by the limited dynamic range and the lasers' intensity fluctuations. Its contribution, however, is fairly marginal for the DNRs between 30 and 40 dB used in the experiment.
Phase noise by undesired reflections in the setup: Undesired reflections in the setup, such as the ones arising between the uncoated TPX lenses, the sample and the various optical components in the setup, as well as within the silicon lens of the source or receiver, will also cause Fabry-Pérot-like oscillations that are superimposed on the Fabry-Pérot oscillations caused by the sample. Typically, their oscillation period is much shorter than that caused by the sample, as the distance of the reflecting objects is much larger. These oscillations may partly be reduced with filtering prior to fitting. Still, they severely impact the height resolution.

SAMPLES CONSISTING OF TWO LAYER STRUCTURES
In the manuscript, we only considered the total optical thickness of the structure on the silicon wafer, irrespective of whether the Siemens star was made of the same material as the substrate (Fig. 2) or made of different materials (SiN and SiC on silicon as in Fig. 3). In the following, we evaluate the error caused by this simplification. Fig. S2 shows the physically correct structure. Using the formalism presented in ref. [5], the field transmission coefficient of a two-layer loss-less FP resonator is found to be = 12 23 31 0 ( 2 2 + 3 3 ) 1 + 12 23 2 0 2 2 + 12 31 2 0 ( 2 2 + 3 3 ) + 23 31 2 0 3 3 Where and are the reflection and transmission coefficients when going form interface to interface , and 2 , 3 , 2 and 3 are the refractive indices and the thicknesses of material 2 and material 3, respectively. 0 = 2 / 0 is the vacuum wavenumber.
For 23 ≈ 0 and 23 ≈ 1, the equation reduces to the single-layer FP resonator. This is the case for the Si-SiC interface, where the difference between the phase calculated with the single-layer model and the exact value is less than 0.0005 rad, resulting in an underestimation of the thickness by 1.83 nm (using 2 = 3.416, 3 = 3.1 [6], 2 = 525 m, 3 = 49 nm and a frequency range of 600-800 GHz). We remark that the refractive index of SiC is afflicted by large errors as CVD-deposited material may deviate from its crystalline counterpart or the reported measurements in the literature 6 . The actual comparison between the used approximation and the exact phase is shown in Fig. S3, the error is plotted in Fig. S4.  A similar result is found for the Si-SiN interface, but in this case the difference between the phases increases to a maximum of around 0.002 rad, resulting in an overestimation of the thickness equal to 1.98 nm (using 2 = 3.416, 3 = 2.75 [7], 2 = 508 m, 3 = 250 nm and a frequency range of 600-800 GHz). The errors are summarized in Fig. S5 and S6. The latter also includes the absolute thickness error of SiC on silicon due to the single layer approximation.