Simulated Gravitational Lensing in the Undergraduate Lab

Technical Summary: Simulated Gravitational Lensing in the Undergraduate Laboratory

Problem Statement
Gravitational lensing, a prediction of General Relativity where massive bodies distort spacetime and alter the trajectory of light, is a cornerstone of modern astrophysics. However, incorporating this phenomenon into undergraduate physics laboratories is challenging. Traditional observations rely on distant celestial objects, requiring specialized telescopes and complex image-stacking software, making direct student engagement difficult. While previous works have successfully fabricated plastic lenses to simulate the visual effects of gravitational lensing (producing rings and arcs), these studies have not quantitatively analyzed the resulting images to determine the simulated mass of the lens. There is a gap in undergraduate pedagogy for a self-contained experiment that not only visualizes strong, weak, and microlensing regimes but also allows students to recover the lens mass quantitatively using standard laboratory equipment.

Methodology
The paper presents a method for fabricating and utilizing acrylic lenses to simulate point-mass gravitational lenses across three regimes: strong, weak, and microlensing.

• Lens Fabrication: The lenses were designed using Snell's Law to relate the deflection angle ($\alpha$) to the lens thickness ($T$) as a function of radius ($\xi$), based on the deflection formula for a point mass. The resulting profile follows a logarithmic function: $T(\xi) = T(R) - \frac{4GM}{c^2(n-1)} \ln(\frac{\xi}{R})$. Lenses simulating masses of $7.5 \times 10^{22}$ kg, $2.5 \times 10^{23}$ kg, and $7.5 \times 10^{23}$ kg were machined via CNC lathe, sanded, and polished to optical clarity.
• Apparatus: The experimental setup consists of an optical rail holding a webcam (observer), the acrylic lens, and a source object (a black dot or pattern of circles on paper). Distances ($D_L$, $D_S$, $D_{LS}$) are measured directly.
• Data Acquisition and Analysis:
  • Strong Lensing: A single black dot source is viewed through the lens. When aligned, it forms an Einstein ring. The ring's radius is measured using ImageJ, converted to an angular Einstein radius ($\theta_E$), and used with the lens equation to calculate the simulated mass.
  • Weak Lensing: A field of randomly distributed circles is used. The lens induces a tangential shear, distorting circles into ellipses. ImageJ measures the ellipticity ($\epsilon$) of these shapes as a function of radius. The data is fit to the shear profile $\gamma(\theta) = (\theta_E/\theta)^2$ to extract $\theta_E$ and subsequently the mass.
  • Microlensing: A linear actuator moves the lens perpendicular to the line of sight between the observer and a point source. As the lens passes, the source brightness increases. A Python script using OpenCV counts black pixels in webcam frames to proxy brightness. The resulting light curve is fitted to the microlensing magnification equation to determine the Einstein timescale ($t_E$), which yields $\theta_E$ and the mass.

Key Contributions

1. Quantitative Mass Recovery: Unlike previous simulations that focused solely on visual demonstration, this work demonstrates that the simulated mass can be quantitatively recovered from the lensed images in all three regimes.
2. Unified Experimental Platform: The paper shows that a single apparatus, accessible to most undergraduate labs, can simulate and analyze strong, weak, and microlensing phenomena.
3. Pedagogical Integration: The study provides a framework for integrating astrophysical data analysis (model fitting, image processing, error analysis) into second- and third-year undergraduate physics curricula.

Results
The experiments yielded consistent results across all three regimes:

• Strong Lensing: Measurements of the Einstein ring radius for lenses simulating masses of $7.5 \times 10^{22}$ kg, $2.5 \times 10^{23}$ kg, and $7.5 \times 10^{23}$ kg resulted in weighted average masses of $7.5(2) \times 10^{22}$ kg, $2.5(2) \times 10^{23}$ kg, and $7.6(5) \times 10^{23}$ kg, respectively. These values agree with the expected masses derived from the machined curvature.
• Weak Lensing: Analysis of tangential shear yielded a weighted average mass of $7.5(3) \times 10^{22}$ kg for the corresponding lens, consistent with the expected value.
• Microlensing: Fitting the light curves of moving lenses produced a weighted average mass of $7.1(4) \times 10^{22}$ kg for the $7.5 \times 10^{22}$ kg lens. Measurements for heavier lenses also showed reasonable agreement with expected values, though uncertainty increased for the heaviest lens due to the inability to measure the unlensed baseline brightness directly.
• Uncertainty: The primary source of uncertainty in strong and weak lensing was the measurement of the Einstein ring radius or ellipticity fit. In microlensing, uncertainty was dominated by the correlation between the curve width and the unlensed brightness. Systematic errors (small angle approximation, thin lens approximation, camera distortion) were found to be negligible.

Significance and Claims
The paper claims that these experiments provide a practical, accessible introduction to modern observational astronomy for undergraduate students. By allowing students to control parameters (distances, masses) and analyze real-time data, the simulation bridges the gap between theoretical general relativity and observational astrophysics.

The authors position the work as a teaching tool rather than a discovery of new astrophysical phenomena. They note that while the simulation models a point-mass lens (approx. $10^{23}$ kg), which does not correspond to confirmed astrophysical objects like galaxies or clusters, the techniques used (image analysis, model fitting, shear measurement) are directly analogous to those used in professional research. The paper concludes that these experiments successfully simulate the three regimes of gravitational lensing with sufficient accuracy to allow students to independently measure the simulated mass, validating the use of machined acrylic lenses as a robust educational tool for upper-level physics laboratories.