How Beam Divergence Affects Collection Loss in Radio and Optical Beams

Geometric Collection Loss

The geometric collection loss quantifies how much of a spreading beam's power a receiver can capture, limited by physical aperture size and beam size at the receiver. Mathematically, it is approximated by:

Ggeom ≈ min ( Ar / Abeam, 1 )

Where:

• A_r: Receiver aperture area (physical size of the collecting surface)
• A_beam: Cross-sectional area of the beam at the receiver’s location

As the beam travels, it spreads and Abeam increases with distance. The receiver collects only the fraction of the beam intercepted by its aperture, represented by the ratio Ar / Abeam. The gain is capped at 1 since the receiver cannot collect more than 100% of the beam.

Intuitively:

• When far away, the beam spreads (large Abeam) and the collected fraction is small.
• When close or with a large receiver, the full beam may be captured (Ggeom = 1).

Applicability to Radio Wave Beamforming

This geometric loss model applies to radio wave beamforming as well, with some important considerations:

• Radio beamforming produces a directional radio wave beam that spreads as it propagates.
• The receiver antenna has a finite effective aperture Ar related to its gain and wavelength.
• The beam area Abeam depends on beamwidth and distance.
• The received power fraction is approximately min(Ar / Abeam, 1), representing beam spreading losses.

Additional formulas:

Ar = (λ² × G) / (4π)
Abeam ≈ π × (r × θ)²
    

Where:

• λ = wavelength
• G = antenna gain
• r = distance from transmitter
• θ = beamwidth in radians

This model mainly applies in the far-field where beam spreading behaves predictably.

Divergence Angle (Beam Divergence)

The divergence angle describes how quickly a beam spreads laterally as it propagates. It’s the angular width of the beam's cone of propagation, typically measured in radians or degrees.

Mathematical description:

At distance r, the beam spot radius w is approximately:

w = r × tan(θ / 2)

For small angles, this approximates to:

w ≈ (θ / 2) × r

Assuming a circular beam spot, the beam area is:

Abeam = π × w² = π × ( (θ / 2) × r )² = π × (θ² × r²) / 4
    

A smaller divergence angle means less spreading and more power captured at the receiver aperture, which is critical in antenna design and optical systems.

Summary

The geometric collection loss formula:

Ggeom ≈ min ( Ar / Abeam, 1 )

quantifies the fraction of beam power collected by a receiver, limited by beam spreading and aperture size. The divergence angle defines how quickly the beam spreads with distance, impacting the beam area and collection efficiency.

This model is widely applicable to radio wave beamforming and optical beams, providing a fundamental understanding of how spatial spreading affects received power. 

 

Further Reading

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