A Window Interrupted by a Stick: Two Paths Through Double-Slit Geometry
Year: 2026
Type: preprint
Abstract: A symmetric double aperture can be represented exactly as a transmitting window interrupted by a central opaque region. This note uses that elementary representation to separate two geometrical ingredients normally packaged together as a "double slit": the outer window and the central obstruction. Two controlled paths through aperture geometry are compared, one holding the total open width fixed and the other holding the outer span fixed. At matched obstruction width, the corresponding apertures are nested and differ only by two outer transmitting collars; along constant-shape rays, uniformly illuminated Fraunhofer patterns are related by exact scaling. A known two-interval Fourier phase-retrieval uniqueness result is also specialized to the one-stick geometry. For apertures under a common incident field, standard distinguishability inequalities relate transmitted-field overlap to bounds on the Hellinger and total-variation distances between normalized far-field intensity distributions. For a centered Gaussian beam and fixed central obstruction, the finite-window versus free-standing-stick comparison reduces to an omitted-tail parameter ε, which bounds the change in the normalized diffraction pattern. In the numerically resolved cases studied, the actual total-variation distance is 0.70-0.83 of ε. Thus the outer edges become observationally negligible, to a specified tolerance, when they lie sufficiently far into the tail of the illuminating beam, rather than only in the formal limit of infinite window width. The contribution is organizational and numerical within standard scalar Fourier optics, not a new diffraction law or quantum-foundational claim. Supplementary material: This record also includes the Python scripts and data tables used for the numerical comparisons in the note. They document the aperture geometry conventions, the Fourier-optics calculation of the far-field patterns for a Gaussian beam, and the evaluation of the total-variation and Hellinger distances against the omitted-tail parameter ε. Running the script reproduces the reported ratio of 0.70-0.83 between the actual total-variation distance and ε.
Authors: Yeung, Kacheong, Yeung, Lex
Institution: Bronx High School of Science
Language: English
Subfield: Atomic and Molecular Physics, and Optics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225452241