Newton's Rings — Physica

Equivalent forms

r_m^2 = m\lambda R \;(\text{dark})

r_m = \sqrt{(m+\tfrac12)\lambda R}\;(\text{bright})

Rings whose radii grow like sqrt(m) turn a touch of glass into a precision gauge of curvature and wavelength.

Symbol	Name	SI unit	Dimension	Range
$m$	Ring order Integer ring number from the center	dimensionless	1	1 – 30
$lambda$	Wavelength Wavelength of the illuminating light	nm	L	380 – 750
$R$	Lens radius Radius of curvature of the convex lens	m	L	0.5 – 20
$r_m$	Ring radiusoutput Radius of the m-th dark ring	mm	L	0 – 20

Unit systems

SI:: r_m, R in m; lambda in m
CGS:: in cm
Imperial:: in inches

Where it holds

Near-normal incidence, air gap, monochromatic light. Assumes r << R so the gap thickness

t \sim r^2/(2R)

; breaks down for large rings or non-spherical surfaces.

Discovery

Isaac Newton · 1704

Newton measured the rings precisely in his 'Opticks' and even found their dependence on color, yet — committed to corpuscles — he attributed them to 'fits of easy reflection' rather than waves. Young and Fresnel later explained them as interference.

Try this

Why do concentric colored rings appear where a curved lens touches flat glass?

A thin, wedge-shaped air film between a convex lens and a flat plate makes light interfere. The dark and bright rings have radii that grow as the square root of the ring number — a ruler made of light.

Research status: stable

Common misconceptions

The center is dark, not bright — the half-wave reflection flip from the lower surface, not zero path difference, controls it. Rings are not equally spaced; they crowd outward as sqrt(m).

Derivation

Gap thickness at radius r is

t \sim r^2/(2R)

.

The reflection off the bottom flat adds a half-wave (pi) phase shift, so dark rings require

2t = m \lambda

, giving

r^2/R = m \lambda

, hence

r_m = \sqrt{m \lambda R}

.

Bright rings sit at

2t = (m+1/2) \lambda

.

Limiting cases

m = 0

(contact point)⟶ Dark center — the half-wave phase flip on reflection from the flat

R -> infinity (flat on flat)⟶ Rings spread to infinity; uniform field, no fringes

Thin-Film Interference→Michelson Interferometer Fringe Shift→Young's Double Slit Interference→Lensmaker's Equation→