Modeling Gravitational Lensing: Analyzing Light Deflection Through a Curved Atmospheric Layer

This article explores light deflection and magnification due to variations in the refractive index, linking laboratory experiments to astrophysical lensing effects. A novel experiment with a curved smoke layer demonstrates how a spatially varying refractive index can bend and subtly magnify a Poisson spot, providing an analogy to astrophysical lensing. This principle extends to refractive index gradients in the solar chromosphere, which differentially bends starlight, causing observable distortions in celestial object’s apparent size and shape. The effects of linear and exponential refractive index profiles on light propagation are analyzed, incorporating effective refractive indices that account for quantum electrodynamic (QED) vacuum effects and frequency-dependent spatial distortions. Unlike gravitational lensing, which results from spacetime curvature, these optical distortions arise purely from refraction within media with refractive index gradients. The findings offer new insights into light propagation and image formation in complex refractive environments, with implications for advanced optical systems and astrophysical observations.

Introduction

Gravitational lensing, a profound consequence of Einstein’s General Theory of Relativity, occurs when massive celestial objects—ranging from stars to galaxy clusters—warp the fabric of spacetime, bending the paths of passing light along geodesics. This effect, analogous to the refraction of light by an optical lens, arises not from changes in refractive index but from spacetime curvature itself. The intervening mass, known as a gravitational lens, distorts and magnifies the images of background sources.

While Newton’s theory of gravity hinted at the possibility of light deflection, Einstein provided a precise theoretical framework. In 1916, he predicted that starlight grazing the Sun would be deflected by 1.75 arcseconds—an estimate famously confirmed by Arthur Eddington’s 1919 solar eclipse observations. The measured shift in the apparent positions of stars near the Sun provided compelling evidence for the curvature of spacetime.

The discovery of the “double quasar” Q0957 + 561 in 1979 by Walsh, Carswell, and Weymann further demonstrated the power of gravitational lensing. Initially appearing as two distinct quasar images separated by approximately 6 arcseconds, both shared nearly identical redshifts (z = 1.41), revealing them to be a single quasar whose light had been split by an intervening galaxy. This discovery ushered in a new era of research, with over 60 lensed quasars now cataloged, each reinforcing Einstein’s predictions [1].

Gravitational Lensing and Rationale for Experimental Simulation

Gravitational Lensing

Cavendish suggested that light deflected by a celestial object follows a hyperbolic path, though he did not formally prove this assertion [2]. Modern measurements of light deflection incorporate the parameter γ, which accounts for spacetime curvature and refines the Newtonian prediction. The total deflection angle $\mathrm{\delta }\mathrm{\theta }$ for a light ray passing the Sun at a distance d is given by [3]:

(1) $$\begin{array}{r}\mathrm{\delta }\mathrm{\theta }=\frac{1}{2}(1+\mathrm{\gamma })\frac{4\text{G}{\text{M}}_{\odot }}{{\text{c}}^2\text{d}}\frac{1+\cos \left(\mathrm{\varphi }\right)}{2}\end{array}$$

where, ${\text{M}}_{\odot }$ is the Sun’s mass, ${\text{R}}_{\odot }$ is its radius, $G$ is Newton’s gravitational constant, c is the speed of light, and $\mathrm{\varphi }$ is the angle between the line connecting Earth to the Sun and the light source. For a light ray passing near the Sun’s surface ($\text{d}\approx {\text{R}}_{\odot }$ and $\mathrm{\varphi }\approx 0$), the deflection simplifies to:

(2) $$\begin{array}{r}\mathrm{\delta }\mathrm{\theta }\approx \frac{1}{2}(1+\mathrm{\gamma })1.‘‘7505\end{array}$$

Gravitational lensing also introduces a time delay between the emission and reception of light, consisting of two components: the geometrical delay ($t_{geom}$) due to the increased path length of deflected rays, and the gravitational delay ($t_{grav}$) caused by the slowing of photons as they traverse the gravitational potential of the lens.

For a lens at redshift $z_L$, the total time delay $t\left(\overrightarrow{s}\right)$ at position $\overrightarrow{s}$ in the gravitational field of the lens is given by:

(3) $$\begin{array}{r}t\left(\overrightarrow{s}\right)=\frac{1+z_L}{c}\frac{D_s{\xi }_0^2}{D_L{D}_{LS}}[\frac{1}{2}{(\overrightarrow{s}-\overrightarrow{y})}^2-\mathrm{\Psi }\left(\overrightarrow{s}\right)]\end{array}$$

where $D_L$ is the distance from the observer to the lens, $D_{LS}$ is the distance from the lens to the source, and $\mathrm{\Psi }\left(\overrightarrow{s}\right)$ is the lensing potential. The deflection angle $\overrightarrow{\hat{\alpha }}$ appears as a two-dimensional vector, and the lens equation is written as:

(4) $$\begin{array}{r}(\overrightarrow{s}-\overrightarrow{y})-\mathrm{\nabla }\mathrm{\Psi }\left(\overrightarrow{s}\right)=\mathrm{\nabla }[\frac{1}{2}{(\overrightarrow{s}-\overrightarrow{y})}^2-\mathrm{\Psi }\left(\overrightarrow{s}\right)=0]\end{array}$$

which follows directly from Fermat’s Principle $\mathrm{\nabla }\text{t}\left(\overrightarrow{s}\right)=0$, stating that light follows paths that minimize travel time [4].

Purpose of the Experimental Simulation

One of the key challenges in gravitational lensing studies is explaining flux-ratio anomalies observed in lensed quasars. These anomalies arise when the observed brightness ratios of quasar images deviate significantly from theoretical model predictions, presenting a challenge for traditional lensing models. While microlensing—the lensing effect of individual stars within the lensing galaxy—is often considered a primary explanation, it appears to be insufficient.

A study analyzing 44 measurements from 34 image pairs across 23 lens systems found that the mean value of the model anomaly ($⟨\left|\mathrm{\Delta }{m}_{models}\right|⟩=0.74$) is significantly larger than the mean impact of microlensing ($⟨\left|\mathrm{\Delta }{m}_{\text{lines}}\right|⟩=0.33$). This suggests that microlensing alone cannot fully account for these discrepancies. Moreover, the histogram of model anomalies reveals a pronounced tail for $\left|\mathrm{\Delta }{m}_{models}\right|\ge 0.7\text{ mag}$, which is absent in microlensing distributions—further reinforcing the need for alternative explanations.

One possibility is that current lens models are incomplete. While refining these models can reduce some discrepancies ($⟨\left|\mathrm{\Delta }m\right|⟩=0.6\text{mag}$ residual anomalies), astrometric uncertainties contribute minimally to the issue. An alternative approach involves comparing flux ratios from extended emission regions—such as mid-infrared, radio, or broad emission lines—with model predictions. These extended regions are less susceptible to microlensing effects, offering a more robust test of lensing models. However, modeling such regions presents its own challenges, particularly for very large emission regions, which require careful treatment. Integral field spectroscopy (e.g., SINFONI, MUSE, HARMONI) provides a promising method for obtaining more reliable, microlensing-free flux ratios.

Ultimately, a multi-pronged approach is necessary. While continuum flux ratios may be affected by microlensing, broad emission line flux ratios offer a more stable constraint. Even if flux-ratio anomalies remain unexplained, they serve as valuable quality checks, flagging lensed systems with large deviations as potentially unreliable. This is especially critical given the expected influx of new lensed systems from upcoming surveys [5].

Beyond individual lensed quasars, brightness anomalies also appear in the overall distribution of quasar magnitudes. Longo (2012) reported a quasar brightness enhancement of $0.2\text{mag}$ over a broad angular scale (±15°), potentially linked to a gamma-ray structure detected by the Fermi Gamma-ray Space Telescope. One intriguing hypothesis is that this large-scale anomaly results from microlensing by a supergiant molecular cloud (SGMC) in the Galactic halo. Models of such clouds suggest compact clumps ($\sim {10}^{-3}{M}_{⨀},\sim 10AU$) embedded in a fractal structure ($\text{D}=1.64\text{–}2$). An SGMC at $20\text{ kpc}$, with a depth of $8.6\text{ kpc}$ and a total mass of $4.1\times {10}^{10}{M}_{⨀}$ (four times the mass of the Large Magellanic Cloud), could explain the observed anomaly via lensing effects [6].

Investigating Flux-Ratio Anomalies Using Molecular Gas

Flux-ratio anomalies in four-image gravitationally lensed quasars frequently deviate from the predictions of smooth lens models. These discrepancies suggest the presence of small-scale perturbations within the lensing system, potentially caused by dark matter subhalos, massive satellite galaxies, or unaccounted large-scale galactic structures. However, identifying the true cause of these anomalies is complicated by contaminating effects such as stellar microlensing, scintillation, and extinction.

This study introduces a novel approach—the first detection of a flux-ratio anomaly in molecular gas emission (CO 11–10) from a lensed quasar. Unlike optical and radio observations, molecular line emission is immune to extinction and variability, making it a more robust probe of lensing anomalies. This allows for a clearer assessment of whether deviations arise from intrinsic lensing perturbations rather than transient effects.

High-resolution millimeter-wave observations, particularly from ALMA, offer significant advantages. They provide precise lens modeling by resolving lensing structures with high accuracy, reducing degeneracies in source reconstruction, and expanding the range of analyzable systems. Notably, a substantial fraction (~70%) of lensed quasars are FIR-bright, making them ideal targets for such studies.

Analysis of the lensed quasar MG J0414+0534 has revealed a flux-ratio anomaly in its molecular gas emission, allowing researchers to rule out a previously proposed dwarf satellite galaxy as the cause. The consistency of flux ratios across multiple wavebands suggests that the anomaly originates from a perturbation in the lensing potential itself rather than from variability or extinction.

Future studies will integrate both continuum and line emission data to refine mass models and pinpoint the precise origin of the anomaly—whether it stems from a dark matter subhalo or an undetected, more massive structure within the lensing galaxy [7].

Experimental Approach: Modeling Lensing Effects with Smoke

To complement astrophysical observations and theoretical modeling, an experiment was conducted to investigate how inhomogeneous media influence light propagation. This experiment utilized smoke as an analog for the distribution of matter within a gravitational lensing system, such as molecular clouds or dark matter substructures. The diffusion and density variations within the smoke served as a tangible representation of how these structures might distort light paths, providing insight into the origins of flux-ratio anomalies in lensed quasars.

Smoke particles efficiently scatter and absorb light, while aerosols primarily scatter light, with their impact increasing at high relative humidity due to water uptake [8]. This behavior mirrors the way substructures in a gravitational lensing system affect light propagation. Just as smoke and aerosols differentially influence the paths of light rays, so too can molecular clouds or dark matter clumps within a lensing galaxy—leading to flux-ratio anomalies.

This experimental approach provides a conceptual framework for understanding the role of turbulent structures in lensing. By observing how light scatters through the inhomogeneous smoke, the experiment directly visualizes how variations in the medium’s properties create subtle distortions in light propagation. These distortions serve as a proxy for the flux anomalies observed in astrophysical systems, underscoring the importance of accounting for complex, non-uniform mass distributions in gravitational lensing studies.

The controlled laboratory environment enables a systematic exploration of these effects, offering insights that are difficult to obtain through observations alone. This experiment bridges the gap between theory and observation, reinforcing the necessity of considering small-scale structure variations when interpreting gravitational lensing data.

Experimental Setup

To investigate light deflection in a low-density medium, a controlled experiment was designed to analyze the Poisson spot phenomenon within a curved smoke layer. The setup incorporated a coherent light source and a smoke chamber, allowing for a comparative analysis of light deflection in both the chromosphere and a smoke medium.

Equipment

– Light Source: A 532 nm laser diode (Polarization Extinction Ratio: 4 dB) provided a coherent light source. The beam was expanded to 30 mm in diameter using a Galilean beam expander.

– Smoke Chamber and Generation: The experimental chamber consisted of a black matte metal cylinder (radius: 50 mm, height: 50 mm) with two annular slits: an inner radius of 50 mm and an outer radius of 54 mm. The lower slit served as a smoke injection port, while the upper slit facilitated smoke extraction. Charcoal powder was smoldered using a 5 V resistive heater, introducing smoke via the lower slit.

– Smoke Flow Control: A variable-speed axial fan (maximum flow rate: 56 m^³/h) was mounted atop the chamber to extract smoke through the upper slit, ensuring a stable, curved smoke layer (~4 mm thick). To account for fluctuations in smoke density, multiple images were captured and analyzed for consistency. Notably, using smoke at lab temperature in this experiment allows for more precise measurements by minimizing temperature-related variations in the refractive index of the surrounding air, which could affect light deflection.

– Imaging System: A full-frame digital camera equipped with a 150 mm f/2.8 macro lens captured the Poisson spot. Remote operation prevented disturbances from human movement, preserving measurement accuracy.

– Obstacle and Magnification: A 3 mm diameter black matte spherical obstacle was positioned between the chamber and the Galilean beam expander to generate the Poisson spot. A convex mirror, placed 10 m away, provided 80× magnification, with the resulting image projected onto a screen 2 m from the mirror (see Fig. 1a).

Fig. 1. Experimental setup for observing light deflection through a curved smoke layer: (a) Schematic of the experimental setup (b) Location of the Poisson spot in the absence of smoke (c) Displacement of the Poisson spot due to refraction by the curved smoke layer.

Experimental Procedure

In this experiment, the Poisson spot phenomenon served as an analogy for light deflection in gravitational lensing. The experiment was conducted under two conditions: (1) in air and (2) in a controlled smoke-filled environment. The effect of a variable density within a curved smoke layer on light deflection was assessed through multi-photography analysis, with the most stable photographic data selected to account for fluctuations in the smoke layer. It was hypothesized that increased smoke density would enhance light deflection, as the points Q₁, Q₂, and the upper part of the curve collectively form an optical structure resembling a prism. The refractive index gradually decreases in radial distribution from the base of this “prism” to point A (see Fig. 1a), causing a deviation in the light path. This configuration influences light bending in a manner analogous to refraction through a transparent medium with a density gradient, further reinforcing the experimental analogy to gravitational lensing.

A Galilean beam expander generated a 30 mm beam. A 3 mm black ball, positioned 70 mm behind the cylinder, was placed such that the Poisson spot’s trajectory passed 1.5 mm from the cylinder (see Fig. 1a). This configuration prioritizes refraction over diffraction in the Poisson spot’s trajectory analysis.

Observations and Results

This study investigates light deflection using a controlled experiment with a curved smoke layer surrounding a cylindrical region. This setup serves as an analog for how light bends when passing through the Sun’s chromosphere, modeled here as an infinite cylinder for the light beam. Since a star’s surface is vast relative to a single light ray, it can be approximated as an infinite cylinder.

The experiment yielded a measurable deflection of the Poisson spot’s trajectory. The primary deflection observed was 1.779 arcseconds, representing a consistent and representative value within the range of measurements. The maximum observed deflection was 3.2 arcseconds. This data supports the hypothesis that a density gradient within a curved smoke layer can induce a deflection in light, analogous to gravitational lensing. The measured deflection of 1.779 arcseconds is comparable in order of magnitude to the deflections observed by Arthur Eddington in 1919. Further analysis will explore the factors contributing to the variation in observed deflections and potential sources of systematic error.

Analysis of Poisson Spot Deflection

The intensity of the Poisson spot is determined using Fresnel diffraction theory, where the wave amplitude at a point $P$ on the screen is given by the surface integral [9]:

(5) $$\begin{array}{r}U\left(P\right)=U_0\frac{i}{2\lambda }{\int }_s\frac{e^{i\frac{2\pi }{\lambda }{r}_0}}{r_0}\frac{e^{i\frac{2\pi }{\lambda }r}}{r}(1+\cos \mathrm{\chi })ds\end{array}$$

where $r$ is the distance from a point $P_0$ on the circular obstacle to $P$ on the screen, $r_0$ is the distance from the source $O$ to $P_0$, and $\mathrm{\chi }$ is the angle between the extended $O{P}_0$ and the line connecting $P_0$ to $P$.

For distances greater than 1 mm from the occulting circular obstacle, the beam exhibits a linear growth pattern. As intensity decreases, the beam size stabilizes at a finite value, where the zeroth-order Bessel function drops to $e^{-2}$ of its maximum, corresponding to the square of the Bessel function with an argument of 1.75. The Arago–Poisson spot undergoes slight divergence along propagation with an angle:

(6) $$\begin{array}{r}\alpha =\frac{1.75\lambda }{\pi d}\end{array}$$

where $\text{d}$ is the diameter of the circular obstacle [10]. The Poisson spot diameter, ${\text{d}}_{\text{spot}}$, is approximated as [11]:

(7) $$\begin{array}{r}{d}_{spot}\approx \frac{\lambda }{2r_b}\frac{ab}{a+b}\end{array}$$

where $a$ and b are the distances from the light source to the circular obstacle (ball in this experiment) and from the ball to the observation screen, respectively, and $r_b$ is the ball’s radius. Equation (7) estimates a $12.32\mu m$ Poisson spot diameter at point $X_3$ (see Fig. 1a). Given an $80\times$ magnification by a convex mirror (−25 mm focal length) onto a screen at 2000 mm, the measured spot diameter is 103.133 mm (see Fig. 1b). This suggests an actual Poisson spot size of $9\mu m$ at $X_3$, with minor deviation likely due to the Galilean beam expander (see Fig. 1a). However, in both theoretical and measured estimations, using the Poisson spot instead of a laser beam is advantageous, as it eliminates Diffraction-affected distortions, ensuring more precise measurements of light deflection through the curved smoke layer.

Ray Trajectories in a Medium with a Variable Refractive Index

In a medium with a constant refractive index (such as a ball lens), light travels in a straight line. However, in graded-index (GRIN) media—where the refractive index varies with spatial coordinates—photons follow curved trajectories governed by the local refractive index, in accordance with Fermat’s principle. These variations arise due to changes in the medium’s chemical and physical properties. The radiative transfer equation for a medium with a variable refractive index is given by [12]:

(8) $$\begin{array}{r}n(s{)}^2\frac{d}{ds}\left(\frac{I(r_H,s)}{n(s{)}^2}\right)=-\beta I(r_H,s)+n(s{)}^2\kappa I_b\left(r_H\right)+\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(\mathit{r},s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }\end{array}$$

where $I(\mathit{r},s)$ is the specific intensity at position $r$ in direction $s$, and $\varphi (s^{\prime },s)$ is the scattering phase function. The term $\kappa I_b\left(\mathit{r}\right)$ represents thermal emission at intensity $I_b$, while $\kappa$, $\beta$, and ${\sigma }_s$ are the absorption, extinction, and scattering coefficients, respectively. As a result, light propagating through a graded-index medium follows a curved trajectory.

To assess whether the Poisson spot’s trajectory through the curved smoke layer could be approximated as a straight line, a 1:1 digital model was developed. In this model, a light beam—representing the Poisson spot’s trajectory—initially travels parallel to the Z-axis, 1.5 mm from the cylinder’s surface, toward a convex mirror 10,000 mm away. Without interaction with the cylinder, the beam follows a straight path. However, introducing a 4 mm-thick curved smoke layer around the cylinder alters its trajectory. The incident light, entering the smoke at $Q_1$, undergoes a slight bending and exits at $Q_2$ deviating by an angle between $\mathrm{\delta }{\mathrm{\theta }}_{min}\approx 1.779\text{ arcseconds}$ and $\mathrm{\delta }{\mathrm{\theta }}_{max}\approx 1.781\text{ arcseconds}$ (calculated in 9 and 10).

For minimum and maximum deflection distances of $d_s^{min}=6.9mm$ and $d_s^{max}=6.908mm$ (refer to Figs. 1a and 1b), we obtain:

(9) $$\begin{array}{r}\mathrm{\delta }{\mathrm{\theta }}_{min}=\frac{S}{r}\times \frac{3600\times 180}{\pi }=\frac{\left(\frac{6.9}{80}\right)}{10000}\times \frac{3600\times 180}{\pi }\approx 1.779\text{arcsecond }\end{array}$$

(10) $$\begin{array}{r}\mathrm{\delta }{\mathrm{\theta }}_{max}=\frac{\left(\frac{6.908}{80}\right)}{10000}\times \frac{3600\times 180}{\pi }\approx 1.781\text{arcsecond }\end{array}$$

The digital model calculates the curved path length as $\widehat{Q_1{Q}_2}=32.481131\text{ mm}$, compared to the straight-line approximation $\overline{Q_1{Q}_2}=32.4811\text{ mm}$ (see Fig. 2). The difference is only $\mathrm{\Delta }{Q}_1{Q}_2=0.000031\text{ mm}$, with a maximum transverse deviation of just 0.0001 mm. These minuscule differences confirm that the light’s trajectory through the smoke can be effectively approximated as a straight line, simplifying further calculations.

Fig. 2. Light declination by a curved medium layer.

Notably, this result mirrors—at a scaled level—Arthur Eddington’s famous 1919 observation that confirmed Einstein’s prediction of gravitational light bending. However, in this experiment, the observed deflection arises from variations in the refractive index within the curved smoke layer rather than gravitational warping.

Effective Refractive Index of the Curved Smoke Layer

Fig. 2 presents a schematic diagram (not to scale), as a true-to-scale representation would be impractical due to the large focal length relative to the other dimensions. Measurements from the 3D model, with the Poisson spot trajectory passing by a cylinder ($R_c=50\text{ mm}$) surrounded by smoke layer ($R_s=54\text{ mm}$), yielded the following results: $\overline{O{H}_4}=51.5mm$, $\overline{H_1{H}_2}=3.1\times {10}^{-5}\text{mm}$, $\overline{H_2{H}_3}=3\times {10}^{-5}\text{mm}$, and $\overline{H_3{H}_4}=0.0002\text{ mm}$. Furthermore, measurements in the 3D model show that the effective focal length ($\overline{OF}$), $F_{eff}$, is approximately $5,971,128.44\text{ mm}$.

Given that the curved trajectory closely approximates a straight line for these small deflections, (11), which relates the focal length of a ball lens to its refractive index [13], can be applied.

(11) $$\begin{array}{r}F\approx \frac{nD}{4(n-n_{medium})}\end{array}$$

where $D=Q_1{Q}_1^{\prime }=2\overline{O{H}_4}$. However, using (8), we can calculate the effective refractive index of the curved smoke layer along the trajectory ($\widehat{Q_1{Q}_2}$), denoted as $n_{eff(s)}\left(s\right)$. This refractive index transitions from the refractive index of air (1.000293) at the incident point $Q_1$, reaches a maximum near $H_2$, and then decreases back to the refractive index of air at the exit point $Q_2$. Therefore, (11) can be rewritten as:

(12) $$\begin{array}{rl}{n}_{eff\left(s\right)}\left(s\right)& \approx \frac{F_{eff}{n}_{air}}{F_{eff}-\frac{D}{4}}\approx \frac{F_{eff}{n}_{air}}{F_{eff}-\frac{1}{2}\overline{O{H}_4}}\\ & \approx \frac{\frac{51.5mm}{8,6248354\times {10}^{-6}Rad}\times 1.000293}{\frac{51.5mm}{8,6248354\times {10}^{-6}Rad}-\frac{51.5mm}{2}}\approx 1.00029731369\end{array}$$

Ray Trajectory Crossing the Chromosphere

In 2015, the International Astronomical Union (IAU) adopted Resolution B3, redefining the nominal solar radius as 695,700 km, replacing the previously used value of 695,990 km. The PICARD mission measures the solar photosphere in the continuum at heights between 300 and 400 km above the solar limb and also observes the chromosphere—an irregular layer extending from 500 to 2000 km above the photosphere. The chromosphere, approximately 1500 km thick, features the Ca II K spectral line, which is divided into three regions: K1 (450–650 km), K2 (700–1450 km), and K3 (1800–2000 km, core at 393.37 nm in air), where prominences are visible, as described by Vernazza et al. (1981) [14].

The solar atmosphere is stratified into the photosphere and chromosphere. The photosphere, the optically visible layer, is the dominant source of solar irradiance in the visible spectrum. Its surface exhibits a granular morphology, a manifestation of convective heat transfer whereby buoyant, hotter plasma ascends, cools, and subsequently descends. This dynamic process results in transient, bright granular structures with characteristic lifetimes on the order of minutes. Despite being the primary emitter of visible light, the photosphere is relatively cool, with a mean temperature of approximately 6,000 K. The density profile of the solar atmosphere, as illustrated in Fig. 4, provides further insight into its structural characteristics [15].

Discrepancies exist between the solar radius definition used in early space-based observations, such as those from the Skylab mission (1979), and contemporary measurements. Skylab-era data designated the termination of the photosphere as the 0 km reference level, effectively coinciding with the solar limb. However, modern high-resolution solar imaging and spectroscopic techniques place the photospheric boundary at an altitude of approximately 300–400 km above the conventionally defined solar limb. To reconcile these differing reference frames, a systematic offset is required. Given the IAU standard solar radius of 695,700 km, an additive correction of approximately 290 km—representing the average discrepancy in photospheric boundary determination—can be applied. This adjustment aligns the Skylab-era radial coordinate system with modern solar atmospheric models, facilitating comparative analysis of historical and contemporary datasets.

A 1:1,000,000 scaled digital model, based on the smoke experiment’s approach, was developed to analyze the 1.75 arcsecond deflection of starlight by the Sun, a phenomenon observed in 1919.

An incident light ray, parallel to the Z-axis and reaching the chromosphere at $Q_1$, proceeds toward $H_2$, at 200 km above the photosphere at a radial distance of 696,200 km from the Sun’s center (assuming a 696,000 km photospheric radius). The ray then exits at $Q_2$, deviated by 1.75 arcseconds relative to its original trajectory, mirroring the smoke experiment’s design.

Light Path Deviation within the Chromosphere

Assuming the upper boundary of the chromosphere has a radius of 697,700 km, the 3D model measurements show the deviation between the curved trajectory and the straight-line approximation within the chromosphere: $\overline{H_1{H}_2}=0.097km$, $\overline{H_2{H}_3}=0.0971km$, and $\overline{H_3{H}_4}=2.954km$. These small deviations confirm that the deflected light path within the chromosphere ($\widehat{Q_1{Q}_2}$), measuring $91,466.764131\text{ km}$, is virtually indistinguishable from a straight line ($\overline{Q_1{Q}_2}=91,466.764130\text{ km}$) for these small-scale calculations. Furthermore, ${\delta }_L$, the deflection of the light ray due to a $1.75\text{ arcsecond}$ deflection at Earth’s distance (~150 million km), is approximately $1,275.1594\text{ km}$. This represents the star’s apparent positional shift as observed from Earth. Notably, this displacement (${\delta }_L$) is significantly smaller than ${\delta }_y$ (approximately 694,927 km), the distance between the observer on Earth and the Z-axis (the axis connecting the Sun’s center and focal point), as measured in the digital model (see Fig. 2).

Effective Refractive Index of the Sun’s Chromosphere

To determine the effective refractive index along the deflected trajectory, $n_{eff\left(p\right)}\left(s\right)$ the incident ray is assumed to be parallel to the Z-axis. The intersection of the deflected ray with this axis defines the effective focal length, measured in the 3D model given by:

(13) $$\begin{array}{rl}{n}_{eff\left(p\right)}\left(s\right)& \approx \frac{F_{eff}{n}_{ISM}}{F_{eff}-\frac{1}{2}\overline{O{H}_4}}\\ & \approx \frac{\left(\frac{696200km}{8.484239\times {10}^{-6}Rad}\right)\times n_{ISM}}{\left(\frac{696200km}{8.484239\times {10}^{-6}Rad}\right)-\left(\frac{696200km}{2}\right)}\approx 1.000004242137\times n_{ISM}\end{array}$$

where $D=Q_1{Q}_1^{\prime }=696,200km$ and $n_{ISM}$ represents the refractive index of the interstellar medium, which varies based on local chromospheric density and ionization state. A commonly assumed value for a fully ionized medium is $n_{ISM}\approx 1.00000001$though variations may occur in different astrophysical environments.

This result confirms that although the chromosphere’s refractive index is slightly greater than one, it is sufficiently close to that of a vacuum that dispersion effects are negligible over the measured light path. The difference between the curved trajectory’s length and a straight-line approximation within the chromosphere is only 0.00001 km—a negligible difference, especially on the scale of astronomical distances. Therefore, the deflected ray’s image remains essentially undistorted, appearing as if the deflection were solely due to gravitational effects.

Magnification and Distortion: From Smoke Layers to Stellar Chromospheres

Just as a lens induces magnification and distortion, variations in the refractive index within a medium can similarly alter light propagation. This effect is evident in both controlled experimental conditions and astrophysical observations.

Magnification and Distortion in Astronomical Observations

The deflection of two adjacent light rays may vary due to differences in the refractive index or gravitational influence. Specifically, when a pair of rays—one originating from either side of a source—passes near a massive object acting as a gravitational lens, the ray passing closer to the mass experiences greater deflection. This differential bending leads to a distortion, making the source appear stretched.

Astronomical observations provide robust evidence of such magnification and distortion effects. Gravitational lensing can cause celestial sources to appear either enlarged or diminished relative to their intrinsic size. A well-known example is the identification of the “double quasar” Q0957+561, a classic demonstration of strong gravitational lensing effects.

Deflection and Magnification in the Curved Smoke Layer

As shown in Figs. 1a and 1b, the diameter of the Poisson spot ($D_{spot}$) is slightly enlarged—effectively magnified—when smoke is present compared to when it is absent. This occurs because the smoke modifies the trajectory of light by altering its effective focal properties. This phenomenon mirrors the behavior of light in astronomical lensing scenarios, where rays from opposite sides of a source experience unequal deflection when passing through a gravitational field.

In the smoke experiment as shown in Figs. 1b and 1c, the Poisson spot diameter increased from 103.133 mm (without smoke) to 103.141 mm (with smoke), indicating a slight magnification with a corresponding deflection of approximately 1.781 arcseconds due to the refractive properties of the medium (refer to(9) and (10)). This demonstrates a direct analogy to the magnification effects observed in astrophysical systems.

Furthermore, Fig. 3 illustrates the observed magnification structure in the experiment, showing that the image deformation is not uniform. Instead of simple enlargement, the pattern follows stretching and compression, analogous to gravitational lensing effects. However, in this case, the distortion arises not from gravitational curvature but from the refractive index gradient in the curved smoke layer.

Fig. 3. Magnification mechanism in a medium with a variable refractive index.

Specifically, the lower ray, traversing a region of higher refractive index, undergoes greater deflection, whereas the upper ray, passing through a lower refractive index region, experiences less bending. Even with a linear refractive index gradient, the lower ray follows a longer optical path, resulting in more pronounced curvature. This effect is further amplified when the refractive index follows an exponential or more complex, non-uniform distribution, leading to even greater variations in deflection.

As a result, the image in the region of higher refractive index undergoes greater stretching, contributing to magnification. The effective focal length ($F_{eff}$) is longer where deflection is minimal and the refractive index is lowest. Conversely, where deflection is strongest and the refractive index is highest, $F_{eff}$ is shorter.

Understanding the distribution of the effective refractive index is crucial, as it dictates the degree of light bending and the resulting magnification and distortion. Both linear and exponential models provide insight into how refractive index gradients influence optical phenomena across different media.

Deflection and Magnification in the Curved Atmospheric Layer

Magnification in a curved medium layer, where light deflection is on the order of a few arcseconds, corresponds to two key factors. As discussed previously, the light’s path within the medium can be approximated as a straight line. However, the light traverses a variable refractive index, which governs the deflection upon exiting the medium. Furthermore, light rays bend differentially within the medium depending on the refractive index encountered—greater bending occurs in regions of higher refractive index, and less bending in regions of lower refractive index.

Equation (8) describes the attenuation, emission, and scattering of light as it propagates through a non-uniform medium. Since $n\left(r\right)$ appears within the differential operator, it directly influences the evolution of intensity $I(r,s)$ with distance. The refractive index depends on optical frequency, gas pressure, temperature, and other physical properties.

However, as discussed earlier, the deviation between the deflected light path—both within the chromosphere and in the curved smoke layer— ($\widehat{Q_1{Q}_2}$) and the straight-line segment connecting $Q_1$ to $Q_2$ is negligible. This allows us to approximate the path as effectively straight. Consequently, this behavior resembles a ball lens with a variable refractive index along its radius, where an effective refractive index ($n_{eff}$) governs the light’s trajectory. Thus, unlike the standard ray equations for light traveling along curved paths in graded-index media, we can apply (8) without invoking $d/ds\left(ndr/ds\right)=\mathrm{\nabla }n$. Therefore, for a refractive index $n(s)$ varying with spatial position ($s$), we obtain:

$$\begin{array}{r}n{\left(s\right)}^2\left[\frac{n{\left(s\right)}^2\frac{dI(r_H,s)}{ds}-I(r_H,s)\frac{dn{\left(s\right)}^2}{ds}}{n{\left(s\right)}^4}\right]=-\beta I(r_H,s)+n{\left(s\right)}^2\kappa I_b\left(r_H\right)+\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(r_H,s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }\end{array}$$

$$\begin{array}{r}\frac{I(r_H,s)}{n{\left(s\right)}^2}\frac{dn{\left(s\right)}^2}{ds}=\frac{dI(r_H,s)}{ds}+\beta I(r_H,s)-n{\left(s\right)}^2\kappa I_b\left(r_H\right)-\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(r_H,s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }\end{array}$$

$$\begin{array}{r}\frac{2}{n\left(s\right)}=\frac{1}{I(r_H,s)}[\frac{dI(r_H,s)}{ds}+\beta I(r_H,s)-n{\left(s\right)}^2\kappa I_b\left(r_H\right)-\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(r_H,s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }]\end{array}$$

$$\begin{array}{r}n\left(s\right)=2I(r_H,s){[\frac{dI(r_H,s)}{ds}+\beta I(r_H,s)-n{\left(s\right)}^2\kappa I_b\left(r_{H}r\right)-\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(r_H,s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }]}^{-1}\end{array}$$

where $I(r,s)$ is the radiation intensity. If we ignore the thermal emission at intensity $I_b$, which is negligible for visible light observations in mediums like the solar chromosphere (but not necessarily in other materials or conditions), we obtain:

(14) $$\begin{array}{r}n(s)=2I(r_H,s){[\frac{dI(r_H,s)}{ds}+\beta I(r_H,s)-\frac{{\sigma }_s}{4\pi }{\int }_{4\pi }I(r_H,s^{\prime })\varphi (s^{\prime },s)d{\mathrm{\Omega }}^{\prime }]}^{-1}\end{array}$$

In addition, $n(s)$ varies between $n_{ISM}$ at the incident point, $Q_1$and $n_{max}$ at $H_2$, the midpoint of the path, where a photon crosses the solar chromosphere before decreasing symmetrically back to $n_{ISM}$ at $Q_2$ (see Fig. 3). Thus, for $\overline{Q_1{H}_2}$ where $\overline{O{H}_2}\le r_H\le R_{ch}$ we have:

(15) $$\begin{array}{r}{r}_H=\overline{O{H}_2}=\sqrt{{R_{ch}}^2-{(\frac{\left|\overline{Q_1{Q}_2}\right|}{2}-s)}^2}\end{array}$$

Therefore, $I(r_H,s)$ in (14) may be rewritten in terms of $s$, representing the position of a photon from $Q_1$ as it crosses the solar chromosphere. The variation of the refractive index along the chromospheric radius ($R_{ch}$) can be modeled either linearly or exponentially, reflecting the bending of starlight in the Sun’s chromosphere. This phenomenon is analogous to the experimental simulation using a curved smoke layer, where variations in refractive indices induce differential deflection and magnification.

These variations in refractive index affect the total time delay $t\left(\overrightarrow{s}\right)$, analogous to the gravitational time delay discussed in (3). Differential bending alters the effective focal length, shortening it in regions of greater bending and lengthening it in regions of lesser bending. As a result, changes in apparent size and shape introduce magnification and distortion, whose extent depends on the refractive index gradient.

Alternatively, considering the Lorentz-Lorenz equation and the effective refractive index along $\overline{Q_1{Q}_2}$, we obtain:

(16) $$\begin{array}{r}\frac{n_{eff\left(p\right)}{\left(s\right)}^2-1}{n_{eff\left(p\right)}{\left(s\right)}^2+2}=\frac{4\pi }{3}N{\alpha }_m\end{array}$$

where $n_{eff\left(p\right)}\left(s\right)=1.000004242137\times n_{ISM}$, ${\alpha }_m$ is the mean polarizability of the molecule, and $n_{ISM}=1.00000001$. The number density $N$ is given by [16] as:

(17) $$\begin{array}{r}N=\frac{\rho N_{av}}{M}\end{array}$$

where $N_{av}=6.022\times {10}^{23}mo{l}^{-1}$ Avogadro’s Number, and $M$ is molecular weight (mean molecular mass, $\mu$). Substituting into (16), we obtain:

(18) $$\begin{array}{r}\rho =\frac{3\mu (n_{eff\left(p\right)}{\left(s\right)}^2-1)}{4\pi {\alpha }_m{N}_A(n_{eff\left(p\right)}{\left(s\right)}^2+2)}\end{array}$$

Equation (18) yields a density of approximately ${10}^{-6}g.c{m}^{-3}$. According to Fig. 4, this corresponds to a radial distance of 696,000 km, which marks the upper boundary of the photosphere. This suggests that the maximum observed deflection (1.75 arcseconds) during the 1919 solar eclipse was primarily due to the refractive index gradient in the Sun at the transition between the photosphere and chromosphere.

Fig. 4. Temperature and density of the solar atmosphere. (Adapted from NASA, “A New Sun: The Solar Results from Skylab,” John A. Eddy, 1979, p. 2.) This figure has been modified to reflect updated data from the International Astronomical Union (IAU) 2015, utilizing a new nominal solar radius of 695,700 km. Distances are measured from the center of the Sun. The outer boundary of the photosphere is indicated by two dotted lines, representing a height of 300–400 km measured from the solar limb level.

Complex Media and Multi-Distortion Environments

Magnification in a homogeneous medium follows predictable refraction laws. However, the universe isn’t homogeneous. Complex environments like turbulent atmospheres and layered astrophysical media present multiple refractive index variations, leading to nonlinear distortions far beyond simple magnification. Imagine a cosmic “funhouse” of distorted images: stretching, compression, multiple image formation, and asymmetric distortions that mimic gravitational lensing effects. These phenomena can be explored through the lens of scalar quantum electrodynamics (QED), which considers the vacuum polarization effect on photon propagation in curved spacetime. The refractive index, dependent on frequency, is then given by [17]:

(19) $$\begin{array}{r}n(u;\omega )=1-\frac{\alpha }{2\pi \omega }{\int }_0^{1}d\xi \xi (1-\xi )ℱ(u;\frac{m^2}{2\omega \xi (1-\xi )})\end{array}$$

where:

(20) $$\begin{array}{r}ℱ(u;z)={\int }_0^{\mathrm{\infty }-iϵ}\frac{dt}{t^2}i{e}^{-izt}[1-\mathrm{\Delta }(u,u-t)\sqrt{det\mathrm{\Delta }(u,u-t)}]\end{array}$$

Furthermore, considering the effective refractive index as a function of frequency within a fractal space, incorporating an exponential chirp function [18], offers deeper insight into astrophysical optical complexities. The key takeaway is this: while simple refraction explains magnification in controlled settings, astrophysical environments exhibit a dynamic interplay of layered distortions. Variations in density distribution directly influence the refractive index, leading to non-uniform stretching and magnification (see Fig. 4). This, in turn, results in asymmetric image distortions, where different regions experience varying degrees of warping, elongation, and intensity shifts. Images are not just magnified; they are dynamically reshaped, creating a complex observational challenge for astronomers.

Discussion

The total deflection angle $\mathrm{\delta }\mathrm{\theta }$ of a light ray traversing the solar gravitational field at a distance d, as approximated by (1), suggests a reciprocal function with distance, indicative of uniform decay. However, a more nuanced analysis, incorporating (14) and (18), reveals a departure from this simplified $d^{-1}$ dependence.

The spatially varying density profile of the solar atmosphere, as illustrated in Fig. 4, induces fluctuations in the coefficients within these equations, leading to a non-uniform refractive index and correspondingly a decay of $\mathrm{\delta }\mathrm{\theta }$. While both approaches yield similar deflection magnitudes at the photosphere, the variable refraction model captures a more intricate deflection behavior, underscoring the necessity of accounting for the complexity of the solar atmosphere.

This refined understanding of light deflection has profound implications for both theoretical frameworks and future observational endeavors in variable atmospheric regimes. Specifically, this understanding could inform the design of future observational campaigns aimed at measuring light deflection more accurately in varied atmospheric conditions, potentially enhancing our knowledge of solar physics. Additionally, it may provide insights into explaining flux-ratio anomalies observed in earlier sections, linking these phenomena to the complex interplay of light and the solar atmosphere.

While the variable refraction model provides a more detailed depiction of light deflection, further investigations are needed to quantify the precise effects of atmospheric turbulence on light paths. Future studies should also explore the impact of solar flares and other dynamic atmospheric phenomena on light deflection, as these factors may further complicate our understanding.

Conclusion

General Relativity attributes light deflection by massive objects to gravity, explaining image distortions and negligible dispersion as gravitational lensing effects. However, flux-ratio anomalies in lensed quasars expose limitations in current gravitational models, as microlensing alone cannot fully explain observed deviations. This study demonstrates that a variable refractive index in a curved medium—such as the Sun’s atmosphere—can produce similar light-bending effects, independent of gravitational influence. The experiment with a curved smoke layer further supports this concept, illustrating how refractive gradients alone can alter light paths.

This raises an important question: Could the light bending predicted by General Relativity be significantly influenced by atmospheric refraction rather than occurring purely in vacuum? For instance, objects like Mercury, which lack an atmosphere, may exhibit different deflection behavior than previously assumed. Because refractive index variations depend on density, temperature, and pressure, observed starlight bending and magnification may be governed in part by Snell’s Law rather than solely by gravitational curvature.

This perspective challenges our fundamental assumptions about celestial mass estimation. If a larger celestial body deflects light less than a smaller one due to a lower effective refractive index in its solar atmosphere, this contradicts expectations based on gravitational lensing and highlights the overlooked role of optical refraction. Moreover, unexpected brightness enhancements in quasars further complicate conventional models, raising alternative explanations such as dark matter subhalos or supergiant molecular clouds. Notably, the first detection of a flux-ratio anomaly in molecular gas emission—immune to extinction and variability—suggests that intrinsic refractive effects may play a far greater role in shaping astronomical observations than previously acknowledged.

If these refractive effects are more significant than assumed, the very foundation of gravitational lensing as a cosmological tool may require serious reconsideration. Future research must carefully disentangle gravitational and refractive contributions to observed lensing phenomena, ensuring that mass estimates and cosmological inferences remain robust.