Diving into the focal length of a spherical surface
The world of optics is a beautifully intricate network of ideas, principles, and laws, that helps us understand the universe. As Richard Feynman, a master of explaining complex ideas in accessible ways, once said, “Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry.” One particular piece of this fascinating tapestry is the intriguing concept of the focal length of a spherical surface. This helps me to start to understand why the patterns and colors unfold before me and my Canon camera body and lens. It is not serendipity, there are very well defined laws that could predict the outcome.
Two weeks ago we looked at the reflection and refraction on a flat curved surface. Here we are considering reflection and refraction with a curved surface such as lenses and mirrors. Focal length is fundamental to optics. It’s the distance from the optical center of a lens or mirror to its focus, where parallel rays of light converge after refraction or reflection. But what happens when this concept is applied to a spherical surface? In the photograph “One Million Rising Suns” we see a curved piece of art glass from a vase with a myriad number of bubbles trapped within the glass. The studio and lighting were kept very dark to convey a sense of depth, although the glass was only a few inches across and less than a half inch thick. The white – blue is coming from light reflecting off paper on the studio table surface. The Sun, is we look closely, is one of the studio lights with a red gel filter. The are large, medium and small bubbles each dancing to the these laws of refracting as dictated by nature, the universe or unseen forces beyond. There are bubbles of air in front of others compounding the mirage. The photograph pulled towards membrane theory and how the universe may be described to exist.
Each and every curved or spherical surface is interacting with light precisely according to the rules and doing so in a predictable manner. The course of each light ray cannot waver from the unseen forces that compel its movement. What fascinates me is the fleeting and brief fragment of time that I can see the ephemeral view and capture it with my camera. What a privilege to be in this space, at this time in history and have this equipment available to me.
“The rainbow, “the bridge of the gods,” proved to be the bridge to our understanding of light—much more important” Isaac Asimov
Interestingly, the relationship between the radius of curvature (R) of a spherical surface and its focal length (F) is quite straightforward: F = R/2. This formula holds true when light rays that are close to the axis. They make small angles with the axis. However, as the angle increases, this approximation becomes less accurate. Called spherical aberration, it is the failure of a spherical mirror or lens to bring all incident light to a single point of focus, resulting in a blurred or distorted image.
Our photographic journey gets even more intriguing. As the angle of light increases, the simplicity of focal length calculation and light’s behavior blur, much like a photo out of focus. This blurring, known as spherical aberration, happens because a simple spherical lens fails to bring all incident light to a single focal point. In photography, this translates to a loss of sharpness or a soft glow around the image, particularly when shooting wide open.
Simple spherical surfaces are limited by aberrations, the principles Feynman unpacks in his lectures help to develop more sophisticated optical instruments. For instance, by combining lenses or using aspheric surfaces, we can correct spherical aberration and improve the quality of images produced by telescopes, microscopes, and cameras. These have essentially become stacked glass lenses or mirrors to get to sharper and sharper final images. You can see in the lower left image aberration is a distortion of the light bending into a single focus. For several reasons, light may not focus on on specific spot. One reason would be that light at varying colors (wavelengths) will bend at differing angles based on the refraction index for that wavelength and medium. Camera makers will attempt to solve for these issues by adding different lenses to sharpen the focus of the lens for the camera, telescope or similar optic device.
“I was fascinated with optical illusions” — Temple Grandin
Isn’t it fascinating that the laws that govern light’s path and the design of a camera lens are deeply intertwined? As we click the shutter, we’re not merely capturing a moment, but witnessing the fundamental principles of physics in action. Through this understanding, we can appreciate the true artistry of photography, grounded not just in aesthetics, but in the scientific marvels of optics.
Moving from the theory and back to the practical photography. In the image below we see a sister capture of One Million Setting Suns. The red is originating from the studio light and gel filter, that we see in the air bubbles, being directed more towards the back of the glass and illuminating an otherwise dark studio. The dark studio is only a halo of unknown surrounding the studio light and the gel filter. The blue – white paper is still visible as a hint, suggesting perhaps an atmosphere. The bubbles and bubbles upon bubbles create numerous iterations of a sun rising in alternate universes. It is fascinating that the natural distortions in the air bubbles causes the distortions in the refracted images There is digital manipulation here. This is shot as seen photography.
Tying this all back into the focal length of a sphere and theoretical optics, it is quite amazing to see the laws and formulae on paper have real world and abstract world applications.
So, next time you’re peering through a lens or looking at an IG Reel of mine or photograph, take a moment to appreciate the journey of the light passing through it, following a path dictated by the very fabric of the universe, an elegant principle that existed before science described and will exist forever into the future. In the world of physics, there’s always more to learn, and each discovery is a small step in the grand journey of understanding.
“Photography takes an instant out of time, altering life by holding it still”
Dorothea Lange
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Cheers,
Jim