Physical Optics: Newton's Rings

Circular interference formed between a lens and a glass plate with which the lens is in contact. There is a central dark spot around which there are concentric dark fringes.The radius of the nth ring is given by

. Where λ is the wavelength and R is the radius of curvature of the lens.

When a Plano convex lens of long focal length is placed in contact on a plane glass plate (Figure given below), a thin air film is enclosed between the upper surface of the glass plate and the lower surface of the lens. The thickness of the air film is almost zero at the point of contact O and gradually increases as one proceeds towards the periphery of the lens. Thus points where the thickness of air film is constant, will lie on a circle with O as center.

By means of a sheet of glass G, a parallel beam of monochromatic light is reflected towards the lens L. Consider a ray of monochromatic light that strikes the upper surface of the air film nearly along normal. The ray is partly reflected and partly refracted as shown in the figure. The ray refracted in the air film is also reflected partly at the lower surface of the film. The two reflected rays, i.e. produced at the upper and lower surface of the film, are coherent and interfere constructively or destructively. When the light reflected upwards is observed through microscope M which is focused on the glass plate, series of dark and bright rings are seen with center as O. These concentric rings are known as " Newton's Rings ".
At the point of contact of the lens and the glass plate, the thickness of the film is effectively zero but due to reflection at the lower surface of air film from denser medium, an additional path of λ/2 is introduced. Consequently, the center of Newton rings is dark due to destructive interference.

Fig . Experimental arrangement for observing Newton's rings

      Let us consider a system of plano-convex lens of radius of curvature R placed on flat glass plate it is exposed to monochromatic light of wavelength λ normally.
         The incident light is partially reflected from the upper surface of air film between lens and glass and light is partially refracted into the film which again reflects from lower surface with phase change of 180 degree due to higher index of glass plate. Therefore the two parts of light interfere constructively and destructively forming alternate dark and bright rings.
      Now consider a ring of radius r due to thickness t of air film as shown in the figure given below:

According to geometrical theorem, the product of intercepts of intersecting chord is equal to the product of sections of diameter then,

As t is very small then $t^{2}$ will be so small which may be neglected, then,

$r^{2}=2Rt$

${\color{Blue} \Rightarrow (1)}$

Radius for bright ring

The condition for constructive interference in thin film is,

m = 0,1,2,....

From equation (1) putting the value of t in the above equation we get,

$2\left ( \frac{r^{2}}{2R} \right )(1)\; =\: \left ( m+\frac{1}{2} \right )\lambda$

since n = 1 for air film

$r^{2}=\left ( m+\frac{1}{2} \right )\lambda R$

$r=\sqrt{\left ( m+\frac{1}{2} \right )\lambda R}$ ${\color{Blue} \Rightarrow (2)}$

For first bright ring, m = 0

$r_{1}=\sqrt{\frac{1}{2}\lambda R}$

For second bright ring, m = 1

$r_{2}=\sqrt{\frac{3}{2}\lambda R}$

For third bright ring, m = 2

$r_{2}=\sqrt{\frac{5}{2}\lambda R}$

Similarly, for Nth bright ring,

m = N - 1

$r_{N}=\sqrt{\left (N-1+\frac{1}{2 \right )\lambda R}}$

$r_{N}=\sqrt{\left (N-\frac{1}{2 \right )\lambda R}}$ ${\color{Blue} \Rightarrow (3)}$

Radius for dark ring

The condition for destructive interference in thin film is,

2tn = mλ m = 0,1,2,....

By putting the value of t, we get,

$2\left ( \frac{r^{2}}{2R} \right )(1)=m\lambda$

For air n = 1

$r=\sqrt{m\lambda R}$ ${\color{Blue} \Rightarrow (4)}$

For $m=0\Rightarrow r=0$ i.e. point of contact.

Now, if the radius of curvature of plano-convex lens is known and radius of particular dark and bright ring is experimentally measured then the wavelength of light used can be calculated from equation (3) and (4).