---
title: Grounded Coplanar Waveguide (GCPW)
url: https://www.emergentmind.com/topics/grounded-coplanar-waveguide-gcpw
type: topic
---

# Grounded Coplanar Waveguide (GCPW)

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 [2401.05801].

## 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 $h = 500$ 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) [2401.05801]).

## 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 [2401.05801].

## 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 [2401.05801].

## 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 [2401.05801].

Source: https://www.emergentmind.com/topics/grounded-coplanar-waveguide-gcpw