Power Normalization in Computational Dosimetry¶

The Problem¶

In computational dosimetry, we compare electromagnetic exposure from:

1. Near-field sources (mobile phones): Antenna fed with measurable input power (e.g., 1 W)
2. Far-field sources (environmental exposure): Plane waves with specified electric field strength (e.g., 1 V/m)

The challenge: How do we normalize results to enable meaningful comparison?

The Core Insight¶

For a plane wave, there is no "source" — the field exists uniformly in space. "1 W" has no direct meaning.

The time-averaged power density (Poynting vector magnitude) is:

Where η₀ = 377 Ω. For E₀ = 1 V/m: S = 1.326 mW/m².

Why Bounding Box Power is Meaningless¶

Some implementations calculate "input power" as P = S × A_bbox (power through the simulation bounding box).

Proof by Contradiction¶

1. SAR depends only on the incident field and phantom — not on the computational domain size.
2. If we define P_input = S × A_bbox, then enlarging the bbox 10× increases P_input 10×.
3. But SAR stays exactly the same.
4. Contradiction: A normalization metric that can be made arbitrarily large without changing SAR is physically meaningless.

Conclusion: The simulation bounding box is a computational convenience with no physical significance for power accounting.

Note: Sim4Life does not provide an "EM Input Power(f)" output for plane wave sources, unlike for antenna (port) sources. This is not an oversight — it reflects the conceptual difficulty of defining "input power" for a plane wave that extends to infinity. The burden of defining a meaningful metric falls on the user.

The Correct Normalization: Power Density (1 W/m²)¶

We define "1 W" for far-field to mean 1 W/m² power density.

Why This Works¶

1. Intrinsic to the field: Power density is a property of the incident wave, not an artifact of computation
2. Direction-independent: Same S regardless of propagation direction
3. Measurable: Exactly what EMF probes (NARDA, etc.) measure
4. Standard in literature: ICNIRP reference levels are defined in terms of power density
5. Reproducible: No dependence on mesh quality, bbox size, or phantom model

The Math¶

At E = 1 V/m, S = 1.326 mW/m². To normalize to 1 W/m²:

Since SAR ∝ E², the scaling factor from 1 V/m simulation to 1 W/m² is:

Comparison with Near-Field¶

Both deliver comparable power to the body — the comparison is physically meaningful.

What About Phantom Cross-Section?¶

We have pre-computed the projected cross-sectional area A(θ,φ) for all phantoms (see data/phantom_skins/README.md). This enables computing the actual power intercepted by the body:

Why Not Use This for Normalization?¶

While physically meaningful, using phantom cross-section for SAR normalization has problems:

1. Poor reproducibility: Different phantoms, meshes, and algorithms give different areas
2. Arbitrary choices: Convex hull vs exact boundary? Which mesh resolution?
3. Not standard: Literature uses 1 W/m², not "1 W per phantom cross-section"
4. Direction-dependent: Same "1 W" would mean different things for different directions

Correct Use of Cross-Section Data¶

The phantom cross-section is valuable for analysis, not normalization:

• Absorption efficiency: η = P_absorbed / P_intercepted (what fraction of intercepted power is absorbed?)
• Worst-case direction: Which angle maximizes/minimizes exposure for a given power density?
• Phantom comparison: Comparing effective target areas across body models

This data is stored in data/phantom_skins/{phantom}/cross_section_pattern.pkl.

Power Balance for Far-Field¶

The Conceptual Issue¶

Power balance = (P_absorbed + P_radiated) / P_input × 100%

For near-field, P_input is unambiguous (power to antenna). For far-field, there's no discrete "input" — the plane wave extends infinitely.

Understanding Sim4Life's RadPower¶

When Sim4Life reports RadPower in the power balance, it computes the total Poynting flux through the simulation boundaries. For a plane wave incident on a phantom:

• Power enters through one face of the bounding box
• Some power is absorbed by the phantom → DielLoss
• The remaining power exits through all faces → RadPower

Crucially, RadPower includes both: 1. Scattered power: Power redirected by the phantom 2. Transmitted power: The portion of the plane wave that missed the phantom and passed straight through

This is why RadPower is closely related to the bounding box size, not just the phantom.

Two Approaches for P_input¶

GOLIAT supports two configurable methods for defining "input power" in far-field power balance. Configure via:

{
    "power_balance": {
        "input_method": "bounding_box"  // or "phantom_cross_section"
    }
}

1. Bounding Box Method (Default)¶

Uses the projected cross-sectional area of the simulation bounding box as seen from the incident direction.

Advantages: - Consistent with Sim4Life: RadPower + DielLoss ≈ S × A_bbox, so balance ≈ 100% - Sanity check: Deviations from 100% indicate numerical issues

Disadvantages: - Domain-dependent: Larger bbox → larger P_input (but same SAR) - No physical meaning: Just verifies FDTD energy conservation

For non-orthogonal directions: The projected bbox area is computed as: \(\(A_{\text{projected}} = |n_x| A_{yz} + |n_y| A_{xz} + |n_z| A_{xy}\)\) where \(\hat{n} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\) is the incident direction.

2. Phantom Cross-Section Method¶

Uses the pre-computed projected cross-sectional area of the phantom from data/phantom_skins/.

Advantages: - Physically meaningful: Represents actual power intercepted by the body - Domain-independent: Same value regardless of simulation bbox size - Enables absorption efficiency: Balance shows what fraction of intercepted power is accounted for

Disadvantages: - Balance >> 100%: Expected! RadPower includes power that bypassed the phantom - Not a sanity check: Can't detect numerical errors from balance alone

Interpreting the Balance¶

Implementation Summary¶

In GOLIAT¶

1. Simulations run at E = 1 V/m (standard Sim4Life plane wave)
2. Analysis scales by 754 (far_field_strategy.get_normalization_factor() returns 754.0)
3. Results reported as "SAR at 1 W/m² incident power density"
4. Power balance uses configurable method (default: bounding_box)

Configuration¶

// In far_field_config.json
{
    "power_balance": {
        "input_method": "bounding_box"  // Options: "bounding_box", "phantom_cross_section"
    }
}

Output Fields¶

The extraction writes these fields to sar_results.json: - input_power_W: Computed input power at 1 V/m - power_balance_input_method: Which method was used - cross_section_m2: The area value used - cross_section_source: Description of the source (e.g., "bbox_x_pos", "phantom_duke")

Conversion Formulas¶

Scaling to arbitrary field E: \(\(\text{SAR}(E) = \text{SAR}_{1\text{W/m}^2} \times S = \text{SAR}_{1\text{V/m}} \times E^2\)\)

Suggested Text for Publications¶

"Far-field exposure simulations were performed using plane wave illumination. Results are normalized to an incident power density of 1 W/m², corresponding to an electric field amplitude of 27.5 V/m in free space. This normalization was chosen because power density is an intrinsic, measurable property of the incident field that does not depend on computational domain size or phantom-specific geometry — ensuring reproducibility across studies. SAR values scale quadratically with field strength (SAR ∝ E²), enabling straightforward translation to any exposure level."

Further Reading¶

• data/phantom_skins/README.md — Pre-computed phantom cross-section data
• docs/technical/skin_mesh_pipeline.md — How phantom outer surfaces were extracted and processed