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Numerical Aperture

Numerical aperture is very important in many optical problems. In a microscope the brightness of the image and the resolving power of the instrument depend on the numerical aperture.

The numerical aperture (N.A.) is the name given by Abbe to the product where is the refractive index of the medium through which light enters the objective of the microscope and is the semi-angle of the extreme cone of rays coming from a luminous object towards the microscope objective.

N.A. =

It is independent of any refraction at plane surfaces at right angles to the axis of the instrument because, according to the laws of refraction at plane surfaces

sin = ’ sin = constant.

Thus, the value of N.A. remains constant as the beam travels from one-medium to another if they are separated by plane surfaces.

The object is generally placed on a glass slide embedded in a material such as Canada balsam and protected by a thin cover glass.

If the medium between the cover glass and the front lens of the objective is of refractive index than the cover glass i.e. air

N.A. = sin = sin

for air = 1

= 90˚

for glass = 1.5

N.A. = 1.5 sin = 1 × 90 = 1

∴ = 42˚ i.e. most of the rays are lost by total internal reflection.

If we allow for some working distance between the cover glass and the objective the maximum N.A. is not reached and the maximum possible value = 0.95.

If the space between the cover glass and the objective is filled with oil of the same refractive index as that of cover glass, the aperture is much increased because the rays will proceed in straight lines through the surface of separation.

∴ N.A. = sin 90˚

N.A. = 1.5 × 1 = 1.5

The angle of 90˚ is not reached due to the thickness of cover glass and the working distance. It has been found that the practical value for = 67.5˚.

N.A. = 1.5 × sin 67.5˚ = 1.40

Hence for a dry objective maximum N.A. = 0.95

and for an oil immersion objective N.A. = 1.40