# Newton's rings

Newton's rings observed through a microscope. The smallest increments on the superimposed scale are 100μm. The illumination is from below, leading to a bright central region.
Newton’s rings interference pattern created by a plano-convex lens illuminated by 650nm red laser light, photographed using a low-power microscope. The illumination is from above, leading to a dark central region.
The experimental setup: a convex lens is placed on top of a flat surface.

Newton's rings is a phenomenon in which an interference pattern is created by the reflection of light between two surfaces—a spherical surface and an adjacent touching flat surface. It is named for Isaac Newton, who first studied the effect in 1717. When viewed with monochromatic light, Newton's rings appear as a series of concentric, alternating bright and dark rings centered at the point of contact between the two surfaces. When viewed with white light, it forms a concentric ring pattern of rainbow colors, because the different wavelengths of light interfere at different thicknesses of the air layer between the surfaces.

## Phenomenon

The phenomenon was first described by Robert Hooke in his 1664 book Micrographia, although its name derives from the physicist Isaac Newton, who was the first to analyze it.

## Theory

Wave interference leading to bright and dark fringes. Note that this figure has the sign of the interference reversed. There is a sign change in the fields reflected at the second interface but not at the first interface, reversing the interference pattern from what is shown. The limiting case, at the center of the pattern, is equivalent to no gap, and hence a continuous non-reflecting medium, consistent with the central dark reflection spot, as seen in the picture on the right.
Newton's rings seen in two plano-convex lenses with their flat surfaces in contact. One surface is slightly convex, creating the rings. In white light, the rings are rainbow-colored, because the different wavelengths of each color interfere at different locations.

The bright rings are caused by constructive interference between the light rays reflected from both surfaces, while the dark rings are caused by destructive interference. Moving outwards from one bright ring to the next, the path difference of interfering rays at the given radius is one wavelength, λ, corresponding to an increase of thickness of the air layer between the glass surfaces by λ/2. For glass surfaces that are not spherical, the fringes will not be rings but will have other shapes.

For illumination from above, with a dark center, the radius of the Nth bright ring is given by

${\displaystyle r_{N}=\left[\lambda R\left(N-{1 \over 2}\right)\right]^{1/2},}$

where N is the bright-ring number, R is the radius of curvature of the glass lens the light is passing through, and λ is the wavelength of the light.

The above formula is also applicable for dark rings for the ring pattern obtained by transmitted light.

Consider light incident on the flat plane of the convex lens that is situated on the optically flat glass surface below. The light passes through the glass lens until it comes to the glass-air boundary, where the transmitted light goes from a higher refractive index (n) value to a lower n value. The transmitted light passes through this boundary with no phase change. The reflected light (about 4% of the total) also has no phase change. The light that is transmitted into the air travels a distance, t, before it is reflected at the flat surface below; reflection at the air-glass boundary causes a half-cycle phase shift because the air has a lower refractive index than the glass. The reflected light at the lower surface returns a distance of (again) t and passes back into the lens. The two reflected rays will interfere according to the total phase change caused by the extra path length 2t and by the half-cycle phase change induced in reflection at the lower surface. When the distance 2t is less than a wavelength, the waves interfere destructively, hence the central region of the pattern is dark.

A similar analysis for illumination of the device from below instead of from above shows that in that case the central portion of the pattern is bright, not dark. (Compare the given example pictures to see this difference.)

Given the radial distance of a bright ring, r, and a radius of curvature of the lens, R, the air gap between the glass surfaces, t, is given to a good approximation by

${\displaystyle t={r^{2} \over 2R}}$,

where the effect of viewing the pattern at an angle oblique to the incident rays is ignored.

## Thin-film interference

The phenomenon of Newton's rings is explained on the same basis as thin-film interference, including effects such as "rainbows" seen in thin films of oil on water or in soap bubbles. The difference is that here the "thin film" is a thin layer of air.