Newton discovered an example of interference which is known as “Newton’s rings”. In this case a lens L is placed on a sheet of plane glass, L having a lower surface of very large radius of curvature. By means of a sheet of glass G monochromatic light from a sodium flame, for example, is reflected downwards towards L; and when the light reflected upwards is observed through a microscope M focused on H, a series of bright and dark rings is seen. The circles have increasing radius, and are concentric with the point of contact T of L with H.

Consider the air-film PA between A on the plate and P to the microscope, while the remainder of the light passes straight through to A, where it is also reflected to the microscope and brought to the same focus. The two rays of light have thus a net path difference of 2*t, *where *t = *PA. The same path difference is obtained at all points round T which are distant TA from T; and hence if 2*t=mλ, *where *m* is an integer and λ is the wavelength, we might expect a bright *ring *with centre T. Similarly, if 2*t = *(*m + 1/2*) λ, we might expect a dark ring.

When a ray is reflected from an optically *denser *medium, however, a phase change of 180^{o} occurs in the wave, which is equivalent to its acquiring an extra path difference of λ/2. The truth of this statement can be seen by the presence of the dark spot at the centre, T, of the rings. At this point there is no geometrical path difference between the rays reflected from the lower surface of the lens and H, so that they should be in phase when they are brought to a focus and should form a bright spot. The dark spot means, therefore, that one of the rays suffers a phase change of 180^{o}. Taking the phase change into account, it follows that

2*t = m*λ for a *dark* ring . . . . (1)

2*t= (m + ½)*λ for a *bright ring . . . *(2)

and where *m *is an integer. Young verified the phase change by placing oil of sassafras between a crown and a flint glass lens. This liquid had a refractive index greater than that of crown glass and less than that of flint glass, so that light was reflected at an optically denser medium at each lens. A bright spot was then observed in the middle of the Newton’s rings, showing that no net phase change had now occurred.

The grinding of a lens surface can be tested by observing the appearance of the Newton’s rings formed between it and a flat glass plate when monochromatic light is used. If the rings are not perfectly circular, the grinding is imperfect.

**MEASUREMENT OF WAVELENGTH BY NEWTON’S RINGS**

The radius *r *of a ring can be expressed in terms of the thickness, *t, *of the corresponding layer of air by simple geometry. Suppose TO is produced to D to meet the completed circular section of the lower surface PQ of the lens, PO being perpendicular to the diameter TD through T, Fig. 29.10. Then, from the well-known theorem concerning the segments of chords in a circle, TO. OD = QO. OP. But AT = *r = *PO, QO = OP = *r,* AP = *t = *TO, and OD = 2*a* – OT = 2*a *– *t.*

* t (*2*a) – t) = r *x *r = r ^{2}*

* *2*at – t ^{2}=r^{2}*

But *t ^{2}*is very small compared with 2

*at,*as

*a*is large.

2*at = r ^{2}*

2*t = r^{2}*

* a . . . . . *(i)

* *

But 2*t = (m + ^{1}/_{2}) λ *for a bright ring.

__r____ ^{2 }__= (

*m +*(3)

^{1}/_{2})λ . . . .* a *

* *

The first bright ring obviously corresponds to the case of *m = *o in equation (3); the second bright ring corresponds to the case of *m = *0 in equation (3); the second bright ring corresponds to the case of *m = *1. Thus the radius of the 15th bright ring is given from (3) by *r*^{2}/*a = *14½λ, from which λ = 2*r*^{2}/29*a*. Knowing *r *and *a, *therefore, the wavelength λ can be calculated. Experiment shows that the rings become narrower when blue or violet light is used in place of red light, which proves, from equation (3), that the wavelength of violet light is shorter than the wavelength of red light. Similarly it can be proved that the wavelength of yellow light is shorter than that of red light and longer than the wavelength of violet light.

The radius *r *of a particular ring can be found by using a travelling microscope to measure its diameter. The radius of curvature, *a, *of the lower surface of the lens can be measured accurately by using light of known wavelength λ^{1}, such as the green in a mercury-vapour lamp or the yellow of a sodium flame; since *a = r*^{2}/(*m+*½) λ* ^{1}* from (3), the radius of curvature

*a*can be calculated from a knowledge of

*r, m, λ*

^{1}.

**VISIBILITY OF NEWTON’S RINGS **

When white light is used in Newton’s rings experiment the rings are coloured, generally with violet at the inner and red at the outer edge. This can be seen from the formula *r*^{2} = (*m* + ½) λ*a, *(3), as *r*^{2} α λ^{1}. Newton gave the following list of colours from the centre outwards: *First order:* Black, blue, white, yellow, orange, red. *Second order:* Violet, blue, green, yellow, orange, red. *Third order: *Purple, blue, green, yellow, orange, red. *Fourth order*: Green, red. *Fifth order: *Greenish-blue, red.* Six order:* Green-blue, pale-red. *Seventh order:* Greenish-blue, reddish-white. Beyond the seventh order the colours overlap and hence white light is obtained. The list is known generally as “Newton’s scale of colours”. Newton left a detailed description of the colours obtained with difference thicknesses of air.

When Newton’s rings are formed by sodium light, close examination shows that the clarity, or visibility, of the rings gradually diminishes as one moves outwards from the central spot, after which the visibility improves again. The variation in clarity is due to the fact that sodium light is not monochromatic but consists of *two *wavelengths, λ_{2}, λ_{1}, close to one another. These are (i) λ_{2} = 5890 x 10^{-8}cm (*D _{2})*, (ii) λ

_{1}= 5896 x 10

^{-8}cm

*(D*

_{1}). Each wavelength produces its own pattern of rings, and the ring patters gradually separates as

*m,*the number of the ring, increases. When

*m*λ

_{1}= (

*m*+ ½)λ

_{2}, the bright rings of one wavelength fall in the dark spaces of the other visibility is a minimum. In this case

5896*m = 5890 (m +* ½).

*m* = __5890__ = 490 (approx.)

* *12

At a further number of ring *m*_{1, }when *m*_{1}λ_{1} = (*m*_{1} + 1)λ_{2}, the bright (and dark) rings of the two ring patterns coincide again, and the clarity, or visibility, of the interference pattern in restored. In this case.

5896*m*_{1} = 5890 (*m*_{1}+1),

From which *m _{1}* = 980 (approx.). Thus at about the 500th right there is a minimum visibility, and at about the 1000th ring the visibility is a maximum.

It may be noted here that the bands in films of varying thickness, such as Newton’s rings and the air-wedge bands, p.692, appear to be formed in the film itself, and the eye must be focused on the film to see them. We say that the bands are “localized” at the film. With a thin film of uniform thickness, however, bands are formed by parallel rays which enter the eye, and these bands are therefore localized at infinity.