Peter,

In response to your topic I offer the following thoughts:

(2) how the equations transferred into actual lenses.

I do not believe the Fresnel equations were the primary considerations that influenced the specific design of the Fresnel lens. Rather, I believe it was practical knowledge acquired when developing and testing these theoretical equations.

(1) a simple explanation of the equations

Augustin Fresnel developed mathematical equations in an effort to prove a theory, a theory widely discounted by the traditional physicists at the time. This was the Wave Theory that describes light as waves that spread out from the source and that each color has its own wave length. The majority, the traditionalists supported the Corpuscular Theory, a theory that light was comprised of rapid emissions of tiny particles or energy packets.

In the field of physics, mathematical equations are used to describe behavior of matter and energy. For these equations to be valid and accepted they must describe all behaviors of such matter and energy under any and all conditions.

Fresnel first developed equations to describe the behavior of dispersed light. He then went on to expand these to include reflected, refracted, double refracted and polarized light. In this work he used mirrors, lenses, prism and other objects.

When he published his first work the Corpuscular crowd attempted to use his equations to disprove the Wave Theory. They did this by bench testing using optical devices available at the time. Low and behold they succeeded, succeeded in proving he was right. The tests proved the Wave Theory was correct and that the Corpuscular Theory had big holes in it. One big hole was the Corpuscular Theory could not explain the rainbow effect, which was easily explained by Wave Theory.

This proof of Wave Theory brought about a major paradigm shift in the world of optical physics. It has been said that Fresnel’s work on optical effects caused by the motion of objects was significant in Einstein’s theories of relativity.

The reflection, refraction and transmission characteristics of glass prisms were known before Fresnel first entered the optical physics field. What Fresnel did was to apply prism “technology” to existing convex lens design. This innovative application of prism optics would not have required a great reliance on his previously developed theoretical equations.

The challenge with the lens design was to obtain the correct physical and optical configuration, the placement of the grooves to achieve the optimum prism effect to obtain maximum light transmission and aligned precisely to produce a focused and concentrated beam of light. This would not have required application of the complicated Wave Theory equations. The math for this would involve calculations of angles and distances relative to the focal point of the lens.

Fresnel’s first design was a bull’s eye lens with annular rings surrounding the central convex lens. His next design was the drum lens or cylinder lens. These were dioptric lenses that focus light by refracting or bending. He then devised a system using mirrors to capture the light escaping above and below the lens. This did not perform to his liking because of light loss in reflection from silvered mirrors. Shortly before his death in 1827 he replaced the mirrors with reflecting prisms mounted above and below the Fresnel lens. These prisms served to focus light by both refracting and reflecting light from the illumination source. This final design by Fresnel is the catadioptric configuration of the 1st through 6th Order lenses.

I would suggest that it was Fresnel’s prior hands-on work with lenses, mirrors and prisms and his knowledge of what could be done with these that inspired his thinking. This led to the creation of first Fresnel lens when he was employed by the French Lighthouse Commission in 1822.

As he demonstrated in his earlier the wave theory work, Fresnel was capable of thinking outside the box. He was not constrained by the pre-supposed limits of dioptric lens technology of the time. Unlike many theorists, Fresnel excelled at practical application. He was not a theoretical physicist. He was an experienced civil engineer.