In [1]:

import radarsimpy

print("`RadarSimPy` used in this example is version: " + str(radarsimpy.__version__))

`RadarSimPy` used in this example is version: 15.2.0

RCS Polarimetry: Co-Pol vs. Cross-Pol¶

Radar polarization describes the orientation of the electric field vector. Common polarization states:

State	Symbol	Electric Field
Vertical	V	Along elevation axis
Horizontal	H	Along azimuth axis
Left circular	LHCP	Rotating CCW (from receiver)
Right circular	RHCP	Rotating CW (from receiver)

In polarimetric radar, the scattering matrix $[S]$ relates incident and scattered fields across all polarization combinations:

$$\begin{bmatrix} E_V^s \\ E_H^s \end{bmatrix} = \begin{bmatrix} S_{VV} & S_{VH} \\ S_{HV} & S_{HH} \end{bmatrix} \begin{bmatrix} E_V^i \\ E_H^i \end{bmatrix}$$

Co-pol ($S_{VV}$, $S_{HH}$): transmit and receive with the same polarization
Cross-pol ($S_{VH}$, $S_{HV}$): transmit one, receive the orthogonal

Corner reflector polarimetric behavior:
A trihedral corner reflector undergoes triple bounce, which preserves the incident polarization — so the co-pol RCS is large, and cross-pol is much smaller. The co-pol to cross-pol ratio (CDR) is a key polarimetric purity metric.

This Example¶

Target: Trihedral corner reflector (STL)
Frequency: 77 GHz
Sweep: 0° to 360° azimuth
VV (Co-pol): transmit V, receive V
VH (Cross-pol): transmit V, receive H

3D Model Loading¶

Import Required Modules¶

In [2]:

import pymeshlab
import numpy as np
import plotly.graph_objs as go
from IPython.display import Image, display

# Set to True for interactive plots; False renders a static JPEG (e.g. for HTML export)
INTERACTIVE = False


def show(fig):
    if INTERACTIVE:
        fig.show()
    else:
        display(Image(fig.to_image(format="jpg", scale=2)))


target = {
    "model": "../models/cr.stl",
    "unit": "m",
    "location": (0, 0, 0),
}

Visualize 3D Model¶

The three mutually perpendicular surfaces of the trihedral corner reflector are visible — these produce the triple-bounce that preserves polarization.

In [3]:

ms = pymeshlab.MeshSet()
ms.load_new_mesh(target["model"])
t_mesh = ms.current_mesh()

v_matrix = np.array(t_mesh.vertex_matrix())
f_matrix = np.array(t_mesh.face_matrix())

fig = go.Figure()
fig.add_trace(
    go.Mesh3d(
        x=v_matrix[:, 0],
        y=v_matrix[:, 1],
        z=v_matrix[:, 2],
        i=f_matrix[:, 0],
        j=f_matrix[:, 1],
        k=f_matrix[:, 2],
        intensity=v_matrix[:, 2],
        colorscale="Viridis",
    )
)
fig["layout"]["scene"]["aspectmode"] = "data"
fig["layout"]["scene"]["camera"] = {"eye": {"x": -1.5, "y": -1.5, "z": 1.25}}
fig["layout"]["height"] = 500
fig["layout"]["margin"] = dict(l=5, r=0, b=10, t=10)

show(fig)

No description has been provided for this image

Polarimetric RCS Simulation¶

Two simulations at 77 GHz, sweeping 0°–360° azimuth, using the same geometry but different receive polarizations.

Parameter	Value
Frequency	77 GHz
Azimuth φ	0°–360°, 1° steps
Elevation θ	90° (fixed)
TX polarization	[0, 0, 1] (vertical, V)
RX polarization (Co-pol)	[0, 0, 1] → $S_{VV}$
RX polarization (Cross-pol)	[0, 1, 0] → $S_{VH}$
Ray density	1 ray/λ²

Expected results:

$S_{VV}$ (Co-pol): large RCS in sectors where the corner faces the radar; symmetric pattern with ~3-fold rotational symmetry from the trihedral geometry
$S_{VH}$ (Cross-pol): substantially lower RCS (15–25 dB below co-pol) — a signature of the triple-bounce polarization-preserving mechanism

In [4]:

# Import time module for performance measurement
import time

# Import RCS simulation function
from radarsimpy.simulator import sim_rcs

# Define observation angles
phi = np.arange(-90, 90, 0.5)    # Azimuth: -90° to +90° in 0.5° steps
theta = np.zeros_like(phi) + 90   # Elevation: fixed at 90° (horizontal plane)

# Radar configuration
freq = 77e9         # Operating frequency: 77 GHz (automotive radar)
pol = [0, 0, 1]     # Incident polarization: Vertical
pol_obs = [0, 1, 0] # Observation polarization for cross-pol: Horizontal
density = 3         # Ray density: 3 rays per wavelength²

# Start timing
tic = time.time()

# Simulate co-polarization RCS (VV: Vertical → Vertical)
# When obs_pol is not specified, it defaults to same as incident pol
co_rcs = 10 * np.log10(
    sim_rcs([target], freq, phi, theta, pol, density=density)
)

# Simulate cross-polarization RCS (VH: Vertical → Horizontal)
# obs_pol parameter specifies different receive polarization
cross_rcs = 10 * np.log10(
    sim_rcs([target], freq, phi, theta, pol, obs_pol=pol_obs, density=density)
)

# End timing
toc = time.time()

# Display execution time
print("Exec time: " + str(toc - tic) + " s")

Exec time: 1.1097238063812256 s

Visualize Polarimetric RCS¶

The co-pol (VV) pattern will show the trihedral's characteristic azimuthal lobes. The cross-pol (VH) response will be significantly lower — the gap between the two curves is the co-pol/cross-pol discrimination ratio.

In [6]:

fig = go.Figure()
fig.add_trace(go.Scatter(x=phi, y=co_rcs, mode='lines', name="Co-Pol (VV)"))
fig.add_trace(go.Scatter(x=phi, y=cross_rcs, mode='lines', name="Cross-Pol (VH)"))
fig.update_layout(
    title="Polarimetric RCS: Co-Pol vs Cross-Pol (77 GHz)",
    yaxis=dict(title="RCS (dBsm)"),
    xaxis=dict(title="Azimuth Angle (Degrees)", dtick=20),
    height=500,
    legend=dict(x=0.02, y=0.98),
)

show(fig)

Summary¶

Trihedral corner reflectors preserve incident polarization (triple bounce) → high co-pol RCS, low cross-pol.
The scattering matrix $[S]$ captures all polarimetric combinations; this example measures $S_{VV}$ and $S_{VH}$.
VH/VV ratio quantifies polarimetric purity — useful for evaluating calibration target quality.

Things to Try¶

Experiment	Parameter to change	Observable effect
HH co-pol	RX pol = `[0, 1, 0]`, TX pol = `[0, 1, 0]`	Should match VV for a symmetric target
HV cross-pol	TX `[0, 1, 0]`, RX `[0, 0, 1]`	Compare with VH (should be similar)
Circular polarization	TX/RX pol = `[1/√2, 1j/√2, 0]`	Cross-pol level changes dramatically
Different frequency	Change freq to 24 GHz	Same qualitative pattern, different magnitude
Dihedral corner	Load a 2-bounce geometry	Dominant cross-pol due to double bounce

Cross-Polarization and Co-Polarization RCS