Newton's rings

Newton's rings is an interference pattern that results when light reflects between two surfaces, most often a curved spherical surface and a flat surface it touches. The effect is named for Isaac Newton, who examined it in 1666. When monochromatic light is used, the pattern appears as a series of concentric rings alternating between bright and dark, centered where the two surfaces meet. With white light, the rings display a rainbow of colors because each wavelength interferes at a different thickness of the air layer between the surfaces.

The phenomenon was first recorded by Robert Hooke in his 1665 book *Micrographia*. Newton studied it in 1666 while isolated at home in Lincolnshire during the Great Plague, which had closed Trinity College, Cambridge. He wrote his observations in an essay titled "Of Colours." The subject sparked a dispute between Newton, who believed light was corpuscular, and Hooke, who argued for a wave nature. Newton did not publish his analysis until after Hooke's death, including it in his 1704 treatise *Opticks*.

The pattern is produced by placing a slightly convex glass lens on an optical flat. The two pieces touch only at the center; elsewhere, a thin air gap exists that grows larger with distance from the center. When monochromatic light shines from above, it reflects from both the lower surface of the lens and the upper surface of the flat. Light passes through the glass to the glass-air boundary, where the transmitted light moves from a higher refractive index to a lower one and undergoes no phase change. The reflected light at this internal boundary also has no phase change. The light that enters the air travels a distance *t* before reflecting off the flat surface below. This reflection at the air-glass boundary causes a half-cycle (180°) phase shift because air has a lower refractive index than glass. The reflected light then travels back through the air gap, returning a distance *t* into the lens. The total extra path length is twice the gap (2*t*). The two reflected rays interfere based on this path length and the half-cycle phase shift from the lower reflection. When 2*t* is zero—where the lens touches the flat—the waves interfere destructively, making the center of the pattern dark. If the device is illuminated from below instead, the center appears bright.

Where the path length difference between the two rays equals an odd multiple of half a wavelength (λ/2), the reflected waves are in phase, leading to constructive interference and a bright area. Where the difference equals an even multiple of half a wavelength, the waves are 180° out of phase, causing destructive interference and a dark area. The 180° phase reversal at the lower reflection ensures the center is dark. This interference creates bright and dark bands called interference fringes, which resemble contour lines on a map, each showing a constant air-gap thickness. The path length difference between two adjacent bright or dark fringes is one wavelength λ, so the gap difference between them is half a wavelength. Because light wavelengths are tiny, this technique can measure very small deviations from flatness. For example, red light has a wavelength of about 700 nm, so the height difference between two fringes is 350 nm—roughly 1/100 the diameter of a human hair. Since the gap increases radially from the center, the fringes form concentric rings. If the glass surfaces are not axially symmetric, the fringes take other shapes.

For illumination from above with a dark center, the radius of the *N*th bright ring is given by *r*_N = [λ*R*(*N* − 1/2)]^(1/2), where *N* is the bright-ring number and *R* is the radius of curvature of the lens.