Imaging distance of an aquatic image formed by an aquatic display obtained by paraxial approximation

Aquatic display, which forms a floating image in water without a screen, used in behavioral experiments of aquatic organisms as optical stimuli, and accurately determining the imaging position in the water is important for aquatic display. We derive a formula for the imaging distance of an aquatic image by an aquatic display, including both the thickness and refractive index of the water tank by using paraxial approximation. To examine the accuracy of the derived theoretical formula, the imaging distances are estimated at various imaging distances. Our theoretical formula is measured to have 4% error by comparing with experimental results.