FSO Channel Design with Halo-Inclusive Source

FSO Channel Design with Halo-Inclusive Source A. Chawla REAL Institute, Gurugram November 17, 2025 Abstract -  (FSO)       -   (Airy)     Gaussian          (transmitters)    -       Gaussian—   - Gaussian   - (aperture clipping)   Airy              halo-inclusive     ,                        Gaussian     Airy         , -       -        — (1) - Gaussian-   (2) - - —          inverse-square -                           FSO  ,  ,  -            ,   ,  , MIMO FSO  -     Abstract Free-space optical (FSO) communication models typically assume either a diffraction-limited point source truncated by a circular aperture or an ideal Gaussian beam representing an extended emitter. In practice, real transmitters exhibit both finite source extent and aperture-induced diffraction, producing a characteristic halo that significantly alters far-field intensity distribution and received power. This paper presents a unified halo-inclusive framework that rigorously incorporates both effects within a single scalar field model by multiplying a Gaussian angular envelope with the classical Airy pattern and enforcing energy conservation. Closed-form expressions for total power, on-axis intensity, and channel gain are derived, together with analytical results in two limiting regimes: Gaussian-dominated extended sources and diffraction-dominated point-like sources. The general case is formulated for numerical evaluation. The analysis demonstrates that the conventional inverse-square path-loss law emerges only under restricted geometric conditions and may deviate substantially otherwise. The unified model enables more accurate FSO link design, optimization, and performance prediction. 1 Introduction Science is a universal language, speaking to all who are willing to listen, regardless of race, religion, or nationality. – C. V. Raman It is common to observe LEDs with a “spherical halo” around them, without stopping to wonder about their modeling. Free-space optical (FSO) communication theory frequently assumes an ideal point source truncated by a circular aperture, yielding the familiar Airy diffraction pattern, or a pure Gaussian beam originating from a diffraction-limited laser embedded in a Gaussian aperture [4]. In practice, neither extreme fully describes real transmitters: semiconductor lasers possess finite source size and gain-guided lateral modes, while high-power LEDs have extended emitting areas many times larger than the wavelength. Crucially, no 1 physical source is ever observed without some form of halo—whether a broad Gaussian halo from source extent or narrow diffraction rings from aperture clipping [5]. Despite this, the overwhelming majority of FSO channel models in the literature treat these two contributions separately, or simply ignore the halo altogether. A systematic treatment that retains both effects simultaneously—termed here a “halo-inclusive” model— remains surprisingly sparse. A few works incorporate extended-source Gaussian spreading, and many more treat diffraction-limited Airy patterns, but combined models that conserve total radiated power while preserving the correct functional form of each contribution are rare. This gap matters: the interplay between source size and aperture truncation governs the far-field beam divergence, the fraction of power contained in the central lobe versus rings/halo, and ultimately the received power and link margin in practical systems. The present work closes this gap by starting from the exact scalar field of a uniformly illuminated circular aperture with an additional Gaussian angular envelope, rigorously normalising the radiated power, and deriving closed-form channel gains in the two practical limiting cases. We further provide the general expression for arbitrary source-aperture combinations and highlight why the inverse-square path loss law emerges only under specific (and often misstated) conditions. The remainder of this paper is organized as follows. Section 2 presents the halo-inclusive scalar field model. Section 3 establishes energy conservation and derives the intensity normalization constant. Section 4 and Section 5 develop the two limiting cases—Gaussian-dominated and diffraction-dominated propagation— and their channel gain expressions. Section 6 generalizes the result to arbitrary parameter regimes. Section 7 discusses implications for practical FSO link design and directions for further work. Table of Notation The adjacent Table 1 lists the principal symbols and their meanings as used in the present work. Table 1: Table of Notation Symbol Description LPropagation distance (link range) λWavelength of light k= 2π/λWave number aRadius of the transmitter circular aperture aRX Radius of the receiver circular aperture w0Effective source radius (at transmitter plane) θOff-axis angle from the beam centerline θG=λ/(2πw0)Gaussian 1/eintensity half-angle (far-field divergence half-angle of the extended source) θ2 GAppears in exponents; note the specific factor of 2 used in the paper PTTotal transmitted optical power PRTotal received optical power E(r, θ, t)Far-field scalar electric field E0Electric field amplitude constant (before normalisation) J0(x),J1(x)Bessel functions of the first kind (order 0 and 1) I(L, θ)Far-field intensity distribution (W/m²) I0=cϵ0 2E2 0Unnormalised on-axis intensity constant cSpeed of light in vacuum ϵ0Vacuum permittivity ωAngular frequency of the optical field r⊥Radial coordinate in the receiver plane NNormalisation integral ensuring total radiated power equals PT H(L)General halo-inclusive channel gain (PR/PT) HG(L)Channel gain in the Gaussian-dominated (extended-source) limit HD(L)Channel gain in the diffraction-dominated (point-like source) limit ξDummy integration variable (often ξ=kaθ or ξ=kar⊥/L) 2 2 Electric Field of a Halo-Inclusive Source We begin with the far-field electric field of a circular transmitter aperture of radius awhose exit-plane illumination is limited both by hard-edged clipping and by the finite angular spread of the source itself. The resulting field in the scalar approximation is the product of a Gaussian envelope and the classic Airy pattern [1]: E(r, θ, t) = E0 rexp (−θ2 θ2 G)[2J1(ka sin θ) ka sin θ]ei(kr−ωt),(1) where θG=λ/(2πw0)is the 1/eintensity half-angle of the Gaussian component determined by the effective source radius w0,k= 2π/λ, and the remaining symbols have their usual meaning [2]. The time-averaged intensity then follows immediately: I(L, θ) = cϵ0 2|E(L, θ)|2=I0 1 L2exp (−2θ2 θ2 G)[2J1(ka sin θ) ka sin θ]2 ,(2) with I0=cϵ0 2E2 0. Having established the intensity distribution, we now enforce energy conservation to relate I0to the total transmitted power PT. 3 Energy Conservation and Intensity Normalisation The total transmitted power is recovered by integrating the intensity over the forward hemisphere: PT=∫2π 0∫π/2 0 I(L, θ)L2sin θ dθ dϕ. (3) Under the paraxial approximation (sin θ≈θ, valid for practically relevant FSO links), this reduces to PT= 2πI0∫∞ 0 exp (−2θ2 θ2 G)[2J1(kaθ) kaθ ]2 θ dθ. (4) Defining the normalisation integral N ≡ ∫∞ 0 exp (−2θ2 θ2 G)[2J1(kaθ) kaθ ]2 θ dθ, (5) the correctly normalised intensity becomes I(L, θ) = PT 2πNL2exp (−2θ2 θ2 G)[2J1(kaθ) kaθ ]2 .(6) The value of Ndepends on the ratio kaθG; analytical solution is possible only in limiting cases, to which we now turn. 4 Limiting Case 1: Gaussian-Dominated (Extended Source) When the source is large compared to the wavelength (w0≫λ), the diffraction pattern collapses to a near-δ-function, and N → 1/(2θ2 G). The intensity simplifies to the familiar extended-source form: I(L, θ) = PT π(LθG)2exp (−θ2 θ2 G).(7) Received power collected by a circular aperture of radius aRX at distance Lis PR=PT[1−exp (−2a2 RX L2θ2 G)].(8) 3 Thus the channel gain in the Gaussian-dominated regime is HG(L) = 1 −exp (−2a2 RX L2θ2 G).(9) 5 Limiting Case 2: Diffraction-Dominated (Effectively Point-Like Source) When the Gaussian spread is negligible (θG→0or equivalently a very large aperture relative to source size), the pattern becomes a pure Airy disc plus rings. Substituting the known integral ∫∞ 0[2J1(x)/x]2x dx = 1 yields the correctly normalised on-axis intensity I(L, θ) = PTk2a2 πL2[2J1(kaθ) kaθ ]2 .(10) The power incident on the receiver aperture is then PR=PT∫kaaRX /L 0[2J1(ξ) ξ]2 ξ dξ =PT[1−J2 0(kaaRX L)−J2 1(kaaRX L)],(11) where we have used the standard encircled-energy identity for the Airy pattern. The diffraction-dominated channel gain is therefore HD(L) = 1 −J2 0(2πaaRX λL )−J2 1(2πaaRX λL ).(12) Both limiting cases recover the expected physical behaviour: H→1for arbitrarily large receiver apertures and the classic 1/L2far-field scaling when aRX is small compared to the beam footprint. 6 General Halo-Inclusive Channel Gain For arbitrary ratios of source size and aperture truncation, the full channel gain retains the form H(L) = 1 N∫aRX 0 exp(−2r2 ⊥ L2θ2 G)[2J1(kar⊥/L) kar⊥/L]2 2πr⊥dr⊥,(13) with Ndefined in Eq. (5). This expression must generally be evaluated numerically but poses no conceptual difficulty. 7 Discussion and Path Forward A key insight emerging from both limiting cases is that the inverse-square path loss law PR∝1/L2holds only when the receiver aperture subtends a small solid angle relative to the beam—exactly the regime where most simplified FSO models are applied. In all other cases, the channel gain deviates significantly from pure geometric loss. Far from being the final word, the halo-inclusive framework presented here serves as a rigorous starting point for a vast array of extensions that were previously ambiguous or inconsistent: • Atmospheric turbulence effects on beams with realistic halo structure [3], • Pointing error statistics when the halo power distribution is non-Gaussian [6], • MIMO FSO with aperture arrays and partial halo capture, 4 • Non-line-of-sight UV scattering channels dominated by broad halos, • Optimisation of transmitter aperture radius agiven fixed source divergence θG. Virtually every existing FSO investigation can be revisited and strengthened by replacing idealised point/Gaussian assumptions with the combined model derived above. The halo is not a nuisance to be approximated away—it is an inherent feature of all real optical sources, and treating it explicitly opens a rich, physically accurate landscape for future theoretical and experimental work. Acknowledgments This work was produced with the assistance of language models. References [1] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999). [2] A. E. Siegman, Lasers (University Science Books, 1986). [3] L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE Press, 2005). [4] A. K. Majumdar and J. C. Ricklin, Free-Space Laser Communications: Principles and Advances (Springer, 2008). [5] A. Belmonte, “Laser propagation through turbulence: An extended source approach,” J. Opt. Soc. Am. A26, 31–40 (2009). [6] A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). 5