Finding the Maximal Contrast of Two Elliptical Gaussian Mode Beams with Aligned Ellipticities

TL;DR

The paper addresses the problem of quantifying the maximal interferometric contrast between two aligned elliptical Gaussian beams by deriving a closed-form expression for $C$ in terms of beam powers, waists $w_{0x}, w_{0y}$, and radii of curvature $R_x, R_y$. The approach models the interference of two elliptical Gaussians, uses Fresnel-type integrals to obtain a rigorous $C$, and validates the theory with a free-space Michelson interferometer, measuring ten beam parameters to compute a theoretical limit $C_{\mathrm{theory}} = 0.968 \pm 0.005$ and an experimental maximum $C_{\mathrm{exp}} = 0.950 \pm 0.007$. The results demonstrate that the analytical formula is a practical tool for modeling and optimizing elliptical-beam interferometers, while also highlighting real-world factors such as misalignment and non-Gaussian features that can reduce achievable contrast. This work provides a concrete framework for predicting and maximizing fringe visibility in high-precision optical measurements.

Abstract

Interferometric contrast is a key factor limiting the sensitivity of precision optical measurements, including the laser interferometers used in gravitational-wave detection. While standard formulas describe the interference of circular Gaussian beams, many real systems use beams with elliptical cross sections, where differing waists and radii of curvature can reduce fringe visibility. This paper derives an analytic expression for the maximum contrast achievable between two aligned elliptical Gaussian beams, written entirely in terms of their geometric and power parameters. We then test the formula using a free-space Michelson interferometer in which all beam parameters are independently measured through beam profiling and nonlinear fitting. In our experiment, the predicted maximum contrast was 0.968 while the experimentally optimized value was 0.950. The small discrepancy is consistent with expected imperfections such as beam rotation, mode mismatch, and non-Gaussian aberrations. This work provides a practical tool for modeling and optimizing elliptical-beam interferometers.

Figures (6)

• Figure 1: Diagram showing the interference of two elliptical coaxial Gaussian beams. Beam 1 is blue, beam 2 is red, and each is described by two characteristic beam waist parameters (one for each transverse direction). The power of the total beam is shown in purple on a screen, whose cross section should be composed of nested circular fringes.
• Figure 2: Simple diagram of our Michelson interferometer, where red lines indicate laser paths and squares with a diagonal line indicate beamsplitters. The positions of the beam collimators $z_{1,2}$ are given with respect to the combining beamsplitter which we defined as $z_0=0 \text{ m}$.
• Figure 3: Photograph of the Michelson interferometer used in our experiment. The two beam collimators, two beamsplitters, and photodetector used to determine contrast are shown and labeled. Extraneous beamsplitters and beam dumps not used in this experiment are partially grayed-out.
• Figure 4: The $\hat{x}$ and $\hat{y}$ profiles of beam 1 in our laser interferometer at the position $z = 0.051 \; \text{m}$. Gray scatter points show the actual ADC values measured by our camera, while the blue curves show the Gaussians-of-best-fit for each profile according to \ref{['eq:gauss']}. The beam widths $w_{x,y}$ of each Gaussian is printed with its uncertainty. The correlation coefficient $R^2$ between each Gaussian and its raw data is also shown.
• Figure 5: The left panel shows the five $\hat{x}$ beam width measurements of each beam, and their curve of best fit as according to \ref{['def']}. The right panel does the same, but for the $\hat{y}$ direction instead. The x-axis of each panel shows the distance between the camera and the beam collimator, denoted as focal displacement $\Delta z$. Finding focal position of the $j$th beam from its focal displacement is simply $z_{0_j} = \Delta z + z_j$ where $z_{1,2}$ are defined above. The four correlation coefficients for the width function of the $j$th beam in the $\hat{u}$ direction, labeled as $R^2_{u, j}$, are printed as well. The eccentricities of beams 1 and 2 are 0.694 and 0.123 respectively.