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Intelligent Reflecting Surfaces: Physics, Propagation, and Pathloss Modeling
Özgecan Özdogan, Emil Björnson, Erik G. Larsson
TL;DR
Prior work used two incompatible IRS pathloss models without physical derivation. This letter uses physical optics to derive the far-field model, explains coherent beamforming by individually diffuse-scattering elements, and disproves one conjectured model.
Problem
IRS communication analysis lacks consensus on propagation modeling, with two incompatible pathloss models conjectured without derivation from physical principles.
Method
The letter uses physical optics to derive far-field IRS propagation and models the surface as sub-wavelength scattering elements whose phase shifts control the reflected beam.
Results
The received power is proportional to the square of the IRS area and to 1/(d_i r)^2, disproving the conjecture that it scales as 1/(d_i + r)^2 in the studied far-field setup.
Takeaways & Limitations
Individually diffuse-scattering IRS elements can jointly form a coherent beam with beamwidth inversely proportional to the IRS size.
Abstract
from arXiv · showhide
Intelligent reflecting surfaces can improve the communication between a source and a destination. The surface contains metamaterial that is configured to "reflect" the incident wave from the source towards the destination. Two incompatible pathloss models have been used in prior work. In this letter, we derive the far-field pathloss using physical optics techniques and explain why the surface consists of many elements that individually act as diffuse scatterers but can jointly beamform the signal in a desired direction with a certain beamwidth. We disprove one of the previously conjectured pathloss models.
I. INTRODUCTION
Reconfigurable reflecting surfaces are proposed to control wireless propagation, but their communication modeling lacks consensus. This letter addresses two incompatible pathloss conjectures by deriving propagation from physical principles.
- Reconfigurable surfaces can create real-time controllable propagation environments by controlling how incident waves are reflected.
- Prior work conjectured incompatible pathloss models for surfaces composed of many scattering elements and many ideal mirrors.
- The letter analyzes scattering from passive metallic surfaces and derives how an IRS controls the directivity of its scattered wave.
- The resulting pathloss model provides a basis for physically grounded IRS system models.
II. PRELIMINARIES: PASSIVE METALLIC SURFACE
The paper models scattering from a finite perfectly conducting rectangular plate under a far-field plane-wave approximation. Physical-optics analysis characterizes the scattered field and its specular maximum.
- The passive surface is a finite, negligible-thickness rectangular perfectly conducting plate of size a×b in the horizontal plane.
- The source polarization is chosen with the electric field parallel to e_x and the magnetic field in the e_y-e_z plane.
- A distant point source is approximated as producing a plane wave because wavefront curvature across the plate is neglected in the far field.
- Physical-optics techniques derive the far-field scattered-field expression while neglecting edge effects.
- The scattered-field magnitude scales with plate area and incident-field strength, with the incident field from a point source decreasing with distance.
- For the considered polarization, the scattered-field power is maximized at θs=θi, the specular direction.
A. Beamwidth of the Scattered Wave
A finite metallic plate produces a beam-like scattered field whose beamwidth narrows with aperture size and widens with wavelength. The received power follows the far-field plate-area and distance scaling.
- A. Beamwidth of the Scattered Wave: The scattered field forms a beam around θs=θi, with 3-dB beamwidth defined by the angular deviation producing half the peak squared magnitude.
- A. Beamwidth of the Scattered Wave: The 3-dB beamwidth is inversely proportional to plate width b and proportional to wavelength λ.
- A. Beamwidth of the Scattered Wave: A plate with a=b=10λ and θi=30° has approximately 5.7° 3-dB beamwidth, versus approximately 5.2° under the second-order approximation.
- A. Beamwidth of the Scattered Wave: When a and b are no larger than λ, the scattered field is nearly equally strong across observation angles.
- A. Beamwidth of the Scattered Wave: For a receiving antenna in the specified far-field regime, received power is proportional to (ab)^2/(d_i r)^2, with wavelength- and angle-dependent proportionality.
- A. Beamwidth of the Scattered Wave: The plane-wave approximation eventually breaks down as plate dimensions grow without bound, where geometric optics can model an ideal mirror asymptotically.
B. Multiple Adjacent Metallic Surfaces
Multiple adjacent metallic plates can be modeled by superposition when coupling across sufficiently large gaps is negligible. Their fields may interfere constructively or destructively at the receiver.
- Sufficiently separated adjacent plates allow coupling effects to be neglected, so their scattered fields can be superposed.
- Relative phase shifts among the plate fields produce constructive or destructive interference at a given receiving location.
- The variables N, a, and b enter the received-power expression through the joint total-area term (Nab)^2.
- For fixed total area, maximum received power is the same whether the area comprises many small plates or a few large plates.
III. SYSTEM MODEL FOR INTELLIGENT METASURFACES
An IRS redirects an incident wave toward a desired angle by shaping the scattered field through a locally tuned surface phase profile. The profile is implemented with discretized sub-wavelength elements, whose resolution creates a trade-off between approximation accuracy and hardware complexity.
- III. SYSTEM MODEL FOR INTELLIGENT METASURFACES: An IRS shapes the scattered field so its main beam points toward a desired reflection angle θr rather than the incident angle θi.This anomalous-reflection operation is needed when the transmitter, surface, and receiver do not share the same angles.
- III. SYSTEM MODEL FOR INTELLIGENT METASURFACES: The required reflection phase profile is obtained by tailoring the surface impedance according to generalized Snell’s law.The local phase φr(y) is selected so the output phase follows the desired reflected-wave phase progression.
- III. SYSTEM MODEL FOR INTELLIGENT METASURFACES: When θr is close to θi, the required phase profile varies more slowly across the surface and is easier to implement.The phase profile is discretized by dividing the surface into sub-λ elements with constant phase shifts.
- III. SYSTEM MODEL FOR INTELLIGENT METASURFACES: Smaller elements approximate the desired local phase more closely, whereas larger elements coarsely quantize it and can mismatch the desired reflection angle.The analysis neglects errors due to phase quantization.
A. Propagation and Pathloss Model
The derived IRS model treats many small elements as contributors whose adjusted currents redirect the scattered field and produce constructive interference toward θr. The resulting far-field pathloss depends on the IRS effective area, transmitter-to-surface and surface-to-receiver distances, and the receiver’s observation angle.
- A. Propagation and Pathloss Model: An IRS requires many small elements to reconfigure local phases and form a main beam at the desired angle θr.The elements individually behave as small scatterers, while their phase alignment determines the collective beam direction.
- A. Propagation and Pathloss Model: The squared scattered-field magnitude is characterized at an arbitrary observation angle θs, with its maximum redirected from θi to θr.The adjusted surface current preserves the intercepted-power relation while shifting the maximum of the scattered field.
- A. Propagation and Pathloss Model: The far-field pathloss expression applies at receiver distance r and depends on the IRS effective area ab cos(θi) as seen from the transmitter.The expression generalizes beyond the stated polarization assumptions while retaining this effective-area dependence.
- A. Propagation and Pathloss Model: The pathloss is maximized at θs = θr, and the main beam narrows as the IRS surface area increases.For dimensions at most λ/2, the IRS nearly acts as a diffuse scatterer.
B. Interpreting an IRS as an Array of Diffuse Scatterers
An IRS can be modeled as many sub-wavelength elements whose individually scattered fields are phase-aligned to produce coherent reflection at the receiver.
- The element-level pathloss expression applies because the element dimensions are no larger than the wavelength.
- For N = NaNb elements, selecting local phases for constructive interference combines their reflected signals coherently at the receiver.
- Each sub-wavelength element acts as a diffuse scatterer, while phase alignment across all elements produces anomalous reflection.
C. System Model for IRS-Supported Communications
The system model represents the direct and IRS-reflected paths jointly, with surface phases selected to align reflected contributions and, when desired, align them with the line-of-sight path.
- The received signal combines the direct source–destination path with the path reflected by the IRS.
- The source–IRS and IRS–receiver line-of-sight channels are represented as normalized phase vectors, while Φ contains the elements’ surface phases.
- Choosing each element phase to align all reflected signal terms maximizes coherent combining through the IRS.
- The selected phases also align the IRS contributions with the line-of-sight path, beyond aligning the N reflected terms with one another.
- For anomalous reflection, the IRS pathloss can be used directly after the element phases are aligned, with the common IRS phase matched to the direct path phase.
IV. SUMMARY AND RELATION TO PRIOR WORK
Physical-optics analysis gives the far-field IRS pathloss and explains practical IRS operation as coherent beamforming by many sub-wavelength scatterers.
- The received signal power is proportional to the square of the IRS area and to 1/(d_i r)^2 in the far-field setup.
- The derived result disproves the conjecture that received power is proportional to 1/(d_i + r)^2 for the studied far-field configuration.
- The system models in,, and are essentially correct when each element’s pathloss is selected according to (20).
- The paper states that practical IRSs use N sub-wavelength elements with distinct phase shifts for coherent beamforming toward a chosen direction.