Sony's announcement last week that it successfully manufactured 100 curved digital photographic sensors marks a significant advancement in digital sensor technology -- even if there's not word yet on whether the sensor is ready for mass production.
Joseph Petzval was a 19th century genius who first came up with the formula for field curvature. It basically plays out like this: because lenses are necessarily curved, the image is focused in a curved manner. Thus, a flat photo sensor cannot capture the entire image sharply without various compensation measures.
Looking at the image of the ivy leaves above, the one on the right is uncompensated. Thus when the middle is in focus, the edges are out of focus.
Budding astrophotographers encounter this issue. Starfields that appear fine to their eyes have edges blurred in the photographs. This is because the eye's retinal area, being curved, is not as susceptible to field curvature problems as the flat sensor found in an astrophotography imager. The solution for this is simple: buy a star field flattener.
But if you're a lens manufacturer, it's harder to compensate for this field curvature problem. There are really two options -- tweak the lens or make the imaging surface curved.
It was much easier to curve film and photographic plates. The most famous example would be the Kodak Stretch 35, one of the first disposable cameras ever made.
It had a cheap lens, but since it was made to be a camera to capture wide panoramas, off axis sharpness was important. Kodak ingeniously curved the film plane instead of tweaking the lens to compensate for field curvature, resulting in large cost savings that enabled them to sell this camera as a disposable while still being able to take excellent pictures rivaling the quality of SLR film cameras in 1990. The curved film plane made it possible to get great quality pictures with just a very cheap and simple two element lens.
It's much harder to curve digital photo sensors, and no curved sensor exists in mainstream cameras today. The Kepler Space Telescope, one of the achievements of modern astro imaging technology, uses a curved sensor array in conjunction with tweaks to the lens system to compensate for field curvature and achieve as flat a field of view as possible.
Cameras evolved to have a flat imaging surface, so the lens system had to be tweaked in most mainstream cameras to compensate for field curvature.
This made sense because the lens system also had to compensate for many other defects including distortion, chromatic aberration, coma, and astigmatism. Elements were designed to flatten the field, either alone in the form of field flattener lenses or in conjunction with other correctional elements.
The result was "good enough" but not perfect. In some well-designed lenses, most aberrations were well corrected. But in others, wrong compromises between competing requirements resulted in very poor lenses. In all cases, aberration correction was a huge consideration in lens design and often increased the weight and complexity, which in turn increased the cost and required pretty exotic materials to pull it off.
Sony's curved sensor could simplify lens design substantially, resulting in lighter and cheaper lenses yet maintaining and possibly increasing image quality.
Many hurdles remain, including reducing manufacturing costs and ensuring manufacturing quality, as well as ensuring that the sensors can keep their delicate curvature in the face of thermal expansion and contraction in the field. It's almost certain that the new sensors will not be compatible with current dSLR and mirrorless ILC lenses as these were designed for flat sensors.
However, this new development will likely bring simpler, lighter, and lower-cost lens design at some point in the future and paves the way for better, cheaper, and lighter cell phones, standalone cameras, and telescopes.