Faraday Rotation Estimation

An Amateur program used for estimating the loss due to the polarization distortion by Faraday Rotation between to EME stations (DX and Home location)

Overview

In EME communications, signal degradation often occurs due to polarization mismatch. This tool helps operators estimate:

 - Spatial Rotation: Geometric polarization shift caused by the Earth's curvature between two distant stations.
 - Faraday Rotation: The rotation of the plane of polarization as the radio wave passes through the ionosphere, influenced by the Earth's magnetic field ($B$) and Total Electron Content (TEC).

Mathematical Modeling Description

1. Model Overview

The core objective of this project is to model the polarization state evolution in an EME link. We utilize Jones Calculus as the primary mathematical framework to track the complex vector transformations from the transmitter, through the lunar reflection, to the receiver.

2. Coordinate System & Definitions

We define a right-handed orthogonal coordinate system $(\hat{h}, \hat{v}, \hat{k})$ for each station:

• $\hat{k}$: Direction of wave propagation.
• $\hat{h}$: Local horizontal axis (Horizon).
• $\hat{v}$: Local vertical axis (Zenith).

3. Parameterization of Antenna Polarization

Any antenna's polarization state is represented by a normalized polarization vector $\mathbf{P}$. To maintain generality for non-standard polarizations, we use the tilt angle $\psi$ and ellipticity angle $\chi$:

The complex Jones vector $\mathbf{P}(\psi, \chi)$ is formulated as:

$$\mathbf{P} = \begin{pmatrix} P_h \ P_v \end{pmatrix} = \begin{pmatrix} \cos\psi & -\sin\psi \ \sin\psi & \cos\psi \end{pmatrix} \begin{pmatrix} \cos\chi \ i \sin\chi \end{pmatrix}$$

Expanding into its complex form:

$$\mathbf{P} = \begin{pmatrix} \cos\psi \cos\chi - i \sin\psi \sin\chi \ \sin\psi \cos\chi + i \cos\psi \sin\chi \end{pmatrix}$$

Thus, we obtain the source vector $\mathbf{P}{TX}$ and the reference vector $\mathbf{P}{RX}$ for the respective stations.

4. Channel Transfer Matrices

The propagation path involves three major coordinate and phase transformations.

A. Spatial Polarization Angle ($\Theta_{spatial}$)

Due to Earth's curvature, the local "vertical" definitions at two distant points do not align. We compute the Parallactic Angle $\nu$:

$$\nu = \arctan \left( \frac{\sin H \cos \phi}{\sin \phi \cos \delta - \cos \phi \sin \delta \cos H} \right)$$

The net spatial rotation is the sum of both stations' contributions: $\Theta_{spatial} = \nu_D + \nu_H$.

B. Faraday Rotation Integral ($\Omega$)

Faraday rotation is a non-reciprocal effect caused by ionospheric birefringence. The one-way rotation $\Omega$ is given by the integral:

$$\Omega = \frac{e^3}{8\pi^2 \epsilon_0 m^2 c f^2} \int_{path} N_e(s) \mathbf{B}(s) \cdot d\mathbf{s}$$

C. Net Rotation Matrix ($\mathbf{R}_{net}$)

The total rotation angle $\Phi_{net}$ combines both spatial and ionospheric effects:

$$\Phi_{net} = \Theta_{spatial} + (\Omega_{uplink} + \Omega_{downlink})$$

$$\mathbf{R}(\Phi_{net}) = \begin{pmatrix} \cos \Phi_{net} & -\sin \Phi_{net} \ \sin \Phi_{net} & \cos \Phi_{net} \end{pmatrix}$$

D. Lunar Reflection Matrix ($\mathbf{M}_{moon}$)

Reflection off the lunar surface causes a polarization reversal. In the $H/V$ basis, the ideal mirror reflection matrix is:

$$\mathbf{M}_{moon} = \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$$

Note: This reverses the vertical component's phase, causing an RHCP $\leftrightarrow$ LHCP flip.

5. Link Equation & Polarization Loss Factor (PLF)

The signal evolution must follow the chronological path because the reflection matrix $\mathbf{M}$ is non-commutative with the rotation matrices.

Full Signal Evolution

The final incident wave at the receiving end $\mathbf{P}_{final}$ is:

$$\mathbf{P}_{final} = \mathbf{R}(\Phi_{down}) \cdot \mathbf{M}_{moon} \cdot \mathbf{R}(\Phi_{up}) \cdot \mathbf{P}_{TX}$$

Where $\Phi_{up} = \nu_D + \Omega_{D}$ and $\Phi_{down} = \nu_H + \Omega_{H}$.

Efficiency Calculation

The Polarization Loss Factor (PLF), representing the power matching efficiency, is derived from the inner product of the incident wave and the receiving antenna's vector:

$$\text{PLF} = \left| \mathbf{P}_{RX}^\dagger \cdot \mathbf{P}_{final} \right|^2$$

This yields the Final Polarization Transformation Matrix that describes the entire EME link's polarization characteristic.

Build & Environment

Requirements

• Compiler: Visual Studio 2026 (or VS2019+ with C++20 support)
• Platform: Windows (x64), compatible with WDK 10.0.26100.0 environments
• Standards: C++20 (/std:c++20)

Compile

• MSVC

  cl /std:c++20 /EHsc /Fe:FaradayRotation.exe `
     FaradayRotation.cpp `
     main_interactive.cpp
     MoonCalendarReader.cpp
     IonexReader.cpp
     InosphereDataProvider.cpp
     WMMModel.cpp

• GCC

  g++ -std=c++20 -O2 -o FaradayRotation \
      FaradayRotation.cpp \
      main_interactive.cpp \
      MoonCalendarReader.cpp \
      IonexReader.cpp \
      InosphereDataProvider.cpp \
      WMMModel.cpp

• Clang

  clang++ -std=c++20 -O2 -o FaradayRotation \
      FaradayRotation.cpp \
      main_interactive.cpp \
      MoonCalendarReader.cpp \
      IonexReader.cpp \
      InosphereDataProvider.cpp \
      WMMModel.cpp

Data Source

In the latest version, we introduced three key files for accurate calculation: TEC Data data.txt , Moon Calendar calendar.dat and WMM Coefficient File WMMHR.COF

For latest data, you can download from websites below:

TEC Data(Sign in needed): https://cddis.nasa.gov/archive/gnss/products/ionosphere/2026/

WMM model(Short survey on data usage needed): https://www.ncei.noaa.gov/magnetic-model-hr-survey-page

Moon Calendar: https://eme.radio/dl7apv-eme-calendar-2026

TEC data is compressed in .Z file and you may need to decompress and rename the file to data.txt

Moon Calendar raw data is presented in HTML Sheets, you may need to convert it to .dat file, the convert tool will be published in few days as convert_calendar.exe. The calendar contained in the repo could be used up to Dec. 2026.

WMM model is available for 2025-2029, you needn't to upgrade it.

Example Calculation

Scenario: EME between BI6DX (OM81ks) and UA3PTW (KO93bs) at 432 MHz.

Parameter	BI6DX (OM81ks)	UA3PTW (KO93bs)
Coordinates	31.79°N, 116.87°E	53.77°N, 38.13°E
Antenna $\psi$	90° (Vertical)	0° (Horizontal)
Moon Elevation	~27°	~27°

Results:

===========================================================================
  EME Faraday Rotation Calculator - Interactive Mode
===========================================================================

This program calculates polarization loss due to Faraday rotation
in Earth-Moon-Earth (EME) communications.

Enter operating frequency (MHz): 432.065

--- DX Station Configuration ---
Enter DX station grid locator (e.g., FN20xa): KO93bs
Enter DX antenna orientation angle psi (degrees, 0=horizontal): 0
Enter DX antenna ellipticity chi (degrees, 0=linear, 45=RHCP, -45=LHCP): 0

--- Home Station Configuration ---
Enter Home station grid locator (e.g., PM95vr): OM81ks
Enter Home antenna orientation angle psi (degrees, 0=horizontal): 90
Enter Home antenna ellipticity chi (degrees, 0=linear, 45=RHCP, -45=LHCP): 0

--- Ionosphere Parameters ---
Data source options:
  1. Load from IONEX file (data.txt)
  2. Use default values (vTEC=25 TECU, B=50uT, inclination=60deg)
  3. Manual input
Select option (1/2/3): 1
Loading IONEX file (data.txt)...
IONEX file loaded successfully!
Loading WMM model (WMMHR.COF)...
WMM model loaded successfully!

Enter observation date and time (UTC):
Year (e.g., 2026): 2026
Month (1-12): 2
Day (1-31): 9
Hour (0-23): 0
Minute (0-59): 37

Ionosphere data retrieved:
  DX vTEC: 4.428 TECU
  Home vTEC: 25.286 TECU
  DX Magnetic Field: 52874.339 nT
  DX Inclination: 70.817 deg
  Home Magnetic Field: 50616.592 nT
  Home Inclination: 49.309 deg

--- Moon Ephemeris ---
Do you have moon elevation/azimuth data? (y/n): y
Enter DX station moon elevation (degrees above horizon): 9.6
Enter DX station moon azimuth (degrees, 0=North, 90=East): 147.6
Enter Home station moon elevation (degrees above horizon): 22.4
Enter Home station moon azimuth (degrees, 0=North, 90=East): 224.9

Moon declination options:
  1. Load from calendar.dat (automatic)
  2. Manual input
Select option (1/2): 1
Moon declination from calendar: -19.592 deg
Enter Earth-Moon distance (km, typical: 356500-406700, default=384400): 403500

Calculating...

Debug - Calculated Moon Elevations:
  DX Elevation: 9.600 deg
  Home Elevation: 22.400 deg
===========================================================================
  Calculation Results
===========================================================================

--- Station Information ---
DX Grid: KO93bs
Home Grid: OM81ks
Ground Distance: 6503.0 km
Frequency: 432.1 MHz (70cm band)

--- Rotation Components ---
Spatial Rotation: -19.523 deg
DX Faraday Rotation: -18.656 deg
Home Faraday Rotation: -129.115 deg
Total Rotation: -167.294 deg

--- Ionosphere Parameters (Precise Model) ---
DX vTEC: 4.43 TECU
DX hmF2: 350.00 km
DX Mapping Factor: 2.8124
Home vTEC: 25.29 TECU
Home hmF2: 350.00 km
Home Mapping Factor: 2.0765

--- Link Parameters ---
Path Length: 807000.0 km
Propagation Delay: 2691.862 ms

--- POLARIZATION LOSS ---
PLF (Polarization Loss Factor): 0.012622
Loss: -18.989 dB
Efficiency: 1.26 %

--- Interpretation ---
Poor: Significant loss. Consider using circular polarization.
===========================================================================

GL on your EME activities! 73s from Izumi@BI6DX