Numerical Aperture and Resolution

What is resolution and what does it have to do with the numerical aperture number of an objective lens (or a condenser lens, for that matter)? Resolution can be defined as the ability of a microscope to allow one to distinguish between small objects. In other words, how crisp and sharp is an image at any given magnification? The numerical aperture number is directly related to the cone of light from the specimen at its vertex which is brought into the lens. Simply put, when light hits an object, it diffracts. A single beam of light will be split into several different diffraction orders bent at increasing angles from the original impinging beam. The easiest way to understand this property of light is to consider what happens when a beam of light is shined through a pinhole onto a dark background. If the image produced on the other side of the pinhole is examined, one finds a light pattern known as an Airy disk. It looks like a negative target with a large central disk of light surrounded by a series of thin concentric circles of light of decreasing brightness the further away from the center they are. An image of an Airy disk is shown on the left.
 
Airy disk 

Light transmission curve for an Airy disk

(Redrawn from Francon)*
What has happened is that the light coming out of the pinhole has been diffracted into several different orders represented by the concentric circles. The same type of thing happens when light hits a microscopic specimen; the diffraction orders spread out. The bigger the cone of light brought into the lens, the more of these diffraction orders which can be collected by it, and the more information it has to form a resultant image and the higher the resolving power of the lens will be. The bigger a cone of light that can be brought into the lens, the higher its numerical aperture is. Therefore the higher the numerical aperture of a lens, the better the resolution of a specimen will be which can be obtained with that lens. If you are interested in learning how the Airy disk is formed and how the light is diffracted, CLICK HERE.

The second advantage of using a higher numerical aperture is that since more orders of diffraction from the object are brought into the lens, more light generally is brought into a higher numerical aperture lens producing brighter images. This becomes a major consideration for darkfield and fluorescence applications where the microscopist is imaging a bright object against a dark background. The diagram to the lower right may be useful for this discussion:
 
Numerical aperture is defined by the formula     N.A. = i sin q

where I is the index of refraction of the medium in which the lens is working, and q is one half of the angular aperture of the lens. All high dry lenses work in air which has a refractive index of 1.0. Immersion oils have a considerably higher refractive index, sometimes even up to 1.56. It can be seen from the diagram and from the formula that using an immersion oil: 

  1. bends more light into the lens capturing more orders of diffraction from the object. (Keep in mind that finer details or more closely spaced objects will give much higher angles of diffraction than will larger objects with less fine details).

    2.  

     
     
     
     
     

  2. will allow a lens to have an N.A. greater than one. It is not possible for a dry lens to have an N.A. greater than one. 
 (Redrawn from Gray)*
Notice the third image in the diagram showing the case of the immersion lens with the N.A. of 1.3. A simple example can be given to understand how the use of an immersion oil can allow the lens to gather those outermost diffraction orders. Have you ever noticed when you look at an aquarium, if you look at the corner you can see the same fish from both the end and the side of the tank. How does this happen? Well, the index of refraction of water is greater than that of air; thus, when the light coming from inside the aquarium hits the air, it is bent at the interface because of the difference in refractive indices allowing you to see the fish from both the end and the side at the same time. The same thing happens when light in the immersion oil hits the end element of the lens and light is bent inward and the end result is that more diffraction orders are collected by the lens. And again, the more diffraction orders (image information) used to form the resultant image, the higher the resolution of the lens will be.

The following diagram shows what happens to the Airy disk with increasing numerical aperture. The diffraction maxima are narrowed and more are brought into the lens to contribute to the final image. These curves can be correlated with the previous diagram. (Redrawn from Francon)*

*Diagrams redrawn from Francon, M. 1961. Progress in Microscopy. Pergamon Press: London  (also Row, Peterson and Co.: Elmsford, NY.) and Gray, P. 1964. Handbook of Basic Microtechnique. McGraw-Hill: New York.

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