Newton's rings

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Newton's rings observed through a microscope. The smallest increments on the superimposed scale are 100μm.
"Newton’s rings" interference pattern created by a plano-convex lens illuminated by 650nm red laser light, photographed using a low-power microscope.

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 flat surface. It is named after Isaac Newton, who first studied them 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.

How the interference fringes form.

The light rings are caused by constructive interference between the light rays reflected from both surfaces, while the dark rings are caused by destructive interference. Also, the outer rings are spaced more closely than the inner ones. Moving outwards from one dark ring to the next, for example, increases the path difference by the same amount, λ, corresponding to the same increase of thickness of the air layer, λ/2. Since the slope of the convex lens surface increases outwards, separation of the rings gets smaller for the outer rings. For surfaces that are not convex, the fringes will not be rings but will have other shapes.

The radius of the Nth Newton's bright ring is given by

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

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

[Note that the figure at left has the sign of the interference backward. There's a sign change in the fields reflected at the second interface but not at the first interface, reversing the interference pattern from that shown. The limiting case, at the center of the pattern, is equivalent to no gap, and hence like a continuous, non-reflecting medium, consistent with the famous dark reflection spot, as seen in the picture on the right.]

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.

The above formula is applicable only for Newton's rings obtained by reflected light.

Theory

The experimental setup: a convex lens is placed on top of a flat surface.
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.

There is 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, and here the light goes from a higher refractive index (n) value to a lower n value. The light passes through this boundary and suffers no phase change. Also at this boundary, some light is transmitted into the air, and some light is reflected. The light that is transmitted to the air travels a distance, t, before it is reflected at the flat surface below; the air-glass boundary causes a half-cycle phase shift because the air has a lower refractive index than the glass. The two reflected rays now travel in the same direction to be detected. The convex lens touches the flat surface below, and from this point, as one gets farther away, the distance t increases, because the lens is curving away from the surface:

2Rt=t^{2}+x
t\ll x so t^{2}\lll x therefore:
2t={X \over R} and finally, we have:
t={X \over 2R}

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