Grounded Coplanar Waveguide (GCPW)

• GCPW is a planar transmission line featuring a central conductor flanked by ground planes, enabling both symmetric and odd-mode detection.
• It integrates a slot-line bridge that captures odd-mode currents, providing a novel contrast mechanism in near-field scanning microscopy.
• The design ensures accurate impedance control and supports high-resolution spatial measurements through advanced NSMM techniques.

A grounded coplanar waveguide (GCPW) is a planar transmission line structure characterized by a central conductive strip flanked by two conductive ground planes on the same substrate, with a focus on controlled ground potentials and symmetry. The geometry, excitation, and signal pickup mechanisms of GCPW enable it to serve as an advanced sensing element in near-field scanning microwave microscopy (NSMM), particularly when combined with transmission-line resonator (TLR) techniques for high sensitivity and resolution. Integration of a slot-line bridge enables measurement of ground potentials imbalance, introducing a novel contrast mechanism for distinguishing symmetries and inhomogeneities in scanned samples (Gladilovich et al., 2024).

1. Structure and Geometric Considerations

The canonical GCPW for NSMM applications is fabricated on a dielectric substrate, specifically Rogers TMM10 (relative permittivity $\varepsilon_r \approx 9.8$, loss tangent $\tan \delta \approx 0.002$), with a thickness %%%%2%%%% μm and copper cladding thickness $t = 17$ μm. The GCPW assembly consists of several distinct regions:

• Central Conductor: Width $W$ is dimensioned via TXLine software to yield $Z_0 \approx 50$ Ω.
• Gaps: Symmetric gaps $s$ are placed on each side of the central conductor.
• Ground Planes: Two ground planes ($G \gg W + 2s$) provide stable reference potentials.
• Quarter-Wave Feed: Coupling occurs via a capacitive slot to a coplanar feed line, also designed for 50 Ω operation.
• Slot-Line Bridge: A narrow slot line etched between the ground planes picks up odd-mode currents caused by ground potential imbalance, transitioning to a microstrip on the substrate's backside, connected to the $S_{21}$ port.
• Open-End Region: The distal end of the half-wave resonator is left open, allowing direct near-field interaction with the sample.

This architecture enables both direct near-field probing and mode-selective signal measurement, as documented in the schematic and photographs (Fig. 1 (a–c) (Gladilovich et al., 2024)).

2. Ground Potentials Imbalance: Field Distributions and Physical Origin

In a perfectly symmetric CPW, excitation supports only the even quasi-TEM mode, maintaining equal rf potentials on both ground planes $V_{g_1}(x) = V_{g_2}(x)$, so $\Delta V(x) \equiv V_{g_1}(x) - V_{g_2}(x) = 0$. Perturbation occurs when a sample is positioned within the near field at the open end:

• Resonant Frequency Shift: Changes in boundary condition due to sample presence yield shifts sensed in the reflection coefficient ($S_{11}$).
• Capacitance Asymmetry: If the sample is misaligned with respect to the central conductor, the capacitance between the center and ground plane 1 ($C_1$) diverges from that to ground plane 2 ($C_2$). The resulting antisymmetric voltage profile excites the odd (slot) mode, satisfying $V_{g_1} = -V_{g_2}$.

The imbalance magnitude at the open end is captured as: $\Delta V \approx V_0\,\frac{C_1 - C_2}{C_1 + C_2}$ where $V_0$ is the balanced resonator voltage. Odd-mode electromagnetic field lines are concentrated across the slot between ground planes; net odd-mode current is detected by the slot-line bridge. This configuration is validated by modeling and experimental data (Gladilovich et al., 2024).

3. Analytical Framework and Core Equations

3.1 Characteristic Impedances

Closed-form expressions for even and odd mode characteristic impedances in symmetric CPW are: $$Z_{\rm even} = \frac{30\pi}{\sqrt{\varepsilon_{\rm eff,e}}}\frac{K(k')}{K(k)}, \quad Z_{\rm odd} = \frac{30\pi}{\sqrt{\varepsilon_{\rm eff,o}}}\frac{K(k)}{K(k')}$$ where $k = \frac{W}{W + 2s}$, $k' = \sqrt{1 - k^2}$, and $K(\cdot)$ denotes the complete elliptic integral of the first kind; $\varepsilon_{\rm eff,e}$ and $\varepsilon_{\rm eff,o}$ are effective permittivities for respective modes.

3.2 Coupled Line Model

The half-wave resonator is modeled as a distributed line ($\ell = \lambda/2$ in the CPW medium), supporting both modes. Sample-induced perturbations alter per-unit-length shunt capacitance: $$C_e(x) \to C_e + \Delta C_e, \quad C_o(x) \to C_o + \Delta C_o$$ with

$$\Delta C_e \approx \frac{1}{2}(\Delta C_1 + \Delta C_2), \quad \Delta C_o \approx \frac{1}{2}(\Delta C_1 - \Delta C_2)$$

Signal transmission from the excitation port (even mode) into the odd-mode pickup ($S_{21}$) is derived via cascading coupled lines, following standard coupled-line theory.

3.3 Permittivity Estimation via Imbalance

Modification of local permittivity $\Delta \varepsilon_r$ by the sample generates: $$\Delta C_{1,2} \approx \frac{\partial C}{\partial\varepsilon_r}\Delta\varepsilon_r \Rightarrow \Delta V \propto \Delta\varepsilon_r$$ Consequently,

$$S_{21}|_{\rm odd} \approx j\,\kappa\,( \Delta C_1 - \Delta C_2 )\exp(-j\beta\ell)$$

where $\kappa$ is the coupling constant, $\beta$ is the propagation constant. Amplitude and phase are interpreted as: $$|S_{21}| \propto |\Delta\varepsilon_r|,\quad \angle S_{21} \approx \beta\ell \pm \pi/2$$ Phase sign reflects which capacitance ($C_1$ or $C_2$) dominates.

4. NSMM Measurement Strategy and Calibration Protocols

The experimental NSMM setup operates as follows:

1. Excitation: Vector Network Analyzer (VNA) applies probe via the quarter-wave feed (Port 1).
2. Resonator Reflection: The CPW half-wave resonator interacts with the sample, with $S_{11}$ measured at Port 1 for even-mode response.
3. Odd-Mode Pickup: The slot-line bridge intercepts odd-mode current between ground planes, directed to VNA Port 2 for measuring $S_{21}$.

Spatial scanning is achieved with XYZ piezo-stages (1 μm resolution), under computer control. In-situ electronic calibration (ECal) of coaxial feeds ensures accuracy. The resonant frequency is fixed near 6.9–7.0 GHz, set by the CPW geometry (Gladilovich et al., 2024).

5. Electromagnetic Simulation and Empirical Assessment

AWR Design Environment, using TXLine for initial dimensioning and a full-wave 3-D solver (Method of Moments / Finite Element Analysis, adaptive mesh $\sim$10–20 elements per guided wavelength), underpins both modeling and empirical design.

Test conditions include:

• Sample: 500 × 250 μm square recess in copper plate, placed 100 μm beneath the CPW open end.
• Scanning Positions: Lateral positions A, B (centered), and C (shifted ±100 μm).

Key observations:

• $S_{11}$ amplitude shift: $\approx$7 dB at resonance (best resolved at symmetric position B).
• $S_{21}$ amplitude contrast: $\approx$10 dB between B and A/C for a $-42$ dBm excitation.
• $S_{21}$ phase difference: $\approx \pi$ between positions A and C, indicating reversal of odd-mode polarity.
• Spatial discrimination limited by open-end aperture ($\sim$100 μm).

6. Experimental Thin-Film Scanning Performance

6.1 Copper-Recess Three-Position Test

Sample: Rogers TMM10 with 1000 μm slot. GCPW is scanned across three positions (A, B, C); $S_{11}$ and $S_{21}$ measured:

• $S_{11}$: Amplitude/phase overlap for positions A/C, unable to distinguish left/right displacement.
• $S_{21}$: Amplitude shows some asymmetry (attributable to fabrication/alignment variations); experimental $S_{21}$ phase differs by $\sim$60° between A and C, confirming odd-mode pickup capability.

6.2 Granular Aluminum–Sapphire Line Scan

Sample: Two $20$ nm thick granular aluminum (grAl) stripes (1000 μm long, $R \approx 30$ kΩ/□) on sapphire.

• Line scan in X with $6$ μm steps; Z lift-off $30$ μm; $f_0 \approx 6.93$ GHz.
• $S_{21}$ amplitude reveals two clear dips; extracted stripe widths from half-width measurements:
  • First stripe: $987$ μm ($98.7$ % of actual, $1000$ μm).
  • Second stripe: $960$ μm ($96.0$ %).
• $S_{11}$: amplitude barely resolves the stripes (width error $\gtrsim 30$ %).

7. Advantages and Constraints in NSMM Applications

Advantages:

• Enables dual contrast mechanisms: odd-mode ($S_{21}$) in addition to traditional resonant frequency/|$S_{11}$| shift.
• Facilitates discrimination of bilateral symmetry in inhomogeneities via $S_{21}$ phase sign.
• Quantitative measurement of low-contrast structures with sub-percent precision (e.g., $98.7$ % accuracy in grAl test).

Limitations:

• Odd-mode $S_{21}$ signal is weak ($-30$ to $-40$ dB); might require low-noise amplification.
• Contingent upon precise mechanical alignment; parallelism between open-end and sample must be maintained to $\lesssim 1$ μm.
• Intrinsic spatial resolution governed by open-end aperture size ($\mathcal{O}(100$ μm)), unless tip is localized/sharpened further.
• Equivalent-circuit model necessitates treatment of coupled even/odd lines, complicating calibration.

In summary, the integration of a slot-line bridge into a standard CPW half-wave resonator grants direct sensitivity to the odd mode, excited exclusively by asymmetric sample perturbations. The resulting ground potentials imbalance signal ($S_{21}$), operating as a distinct contrast channel, extends the capabilities of NSMM for high-accuracy measurement and symmetry discrimination (Gladilovich et al., 2024).