The magnification of a microscope can be determined by using geometrical optics.
- Magnification of a Finite Tube Length Microscope
Let us first consider the simplest case of a microscope as shown in Figure 24. This is a finite tube length microscope that uses a single lens. Only the objective. The sample kept at a distance between 'f' and '2f'. As discussed earlier, for a single lens if an object is placed between f and 2f a real and magnified, but an inverted image is formed on the other side of the lens at a distance between 2f and infinity. Again for the reasons discussed before, we will consider an object shaped like an arrow.
Lets us now look at what magnification is? Simply put, it is the ratio of : (Size of the Image/Size of the Object).
Let us take a closer look at Figure 24.
(Size of the Image/Size of the Object)=(O'A'/OA)
Now let us consider the triangles COA and CO'A'. It is easy to see that these are similar triangles having the same angles.
Hene by property of similar triangles.
Magnification of Finite Tube Length microscope=
(Distance of the image from the center of the Objective/Distance of the object from the center of the Objective)-----(1)
Now let us consider a hypothetical scenario where the inverted image of the arrow is moved or translated across the principal axis to the center of the lens as shown in Figure 25.
A line that is drawn from the top of the object (arrow) and passing through the focal point (F) of the objective will intersect the top of the inverted image of the object.
Lets now look at the similar triangles AOF and FCA''.
Again from the property of similar triangles (CA''/AO) = Magnification= (FC/FO)=(f/r)------------------------------- (2)
Magnification of finite tube length microscope=
(Focal Length of the Objective/Distance of the object from the focal point)-----------------------------------(2)
As we had noted earlier for an object placed at the front of the objective a magnified image is formed at the back of the objective only if the object is kept between 'f' and '2f'.
As the object moves towards f from equation (2) it is obvious that the magnification increases and from equation (1) it turns out that the image will be formed further away from the lens.
It can also be seen from equation (2) that as the object approaches 'f' the magnification approaches infinity which simply means that the beam comes out collimated and the image is formed at infinity. This is also true from equation (1) as the distance of the image from the objective approaches infinity, magnification approaches infinity.
It is now clear that for a finite tube lens microscope the magnification is not a constant, it changes with the distance of the object from the focal point.
Please note that when a manufacturer specifies a magnification for an objective for finite tube length configuration it is matched to a certain tube length. The tube length specifies the distance from the objective where the specified magnification is achieved. If a detector is fixed at this distance, the sample will come into focus only at a fixed distance away from the focal point of the objective.
If the tube length is increased, magnification increases and if the tube length is decreased, magnification decreases.
2.Magnification of an Infinity Corrected Microscope
Now let us consider an infinity corrected microscope as shown in Figure 26. The objective has a focal length of 'fo' and the tube lens has a focal length of 'fi'. Working of the infinity corrected microscope is described in detail in an earlier article.
The object is kept at a focal distance ('fo') away from the objective and the image is formed at a focal distance ('fi') from the tube lens. An infinity path exists between the objective and the tube lens.
For ease of ray tracing, the tube lens is kept focal distance ('fi') away from the objective lens.
From equation (1), magnification=(distance of image from lens/distance of object from the lens)
In an infinity corrected configuration image is formed a focal distance ('fi') away from the tube lens and the object is kept focal distance ('fo') away from the objective.
It then turns out that-
Magnification of Infinity Corrected Microscope=(fi/fo)-----------------------(3)
=(Focal length of tube lens/Focal length of the objective)-------------------(3)
To have a positive magnification the tube lens focal length ('fi') has to be greater than the objective focal length ('fo'). This is mostly the case with most microscopes. They have an objective focal length of few mm and the tube lens focal length typically > 100mm. Please donot confuse the working distance of the objective and focal length of the objective here. The focal length will be much larger than the working distance of an objective.
So, now if magnification depends both on the objective focal length and the tube lens focal length what does it mean to say an objective has a magnification of say 60X. This, in fact, means that the objective will provide a magnification of 60X when coupled with a tube lens of a standard focal length supplied by the manufacturer for microscopes for which that particular 60X objective was designed.
If you mate the objective with a microscope from another manufacturer or a different model by the same manufacturer for which this objective was not designed for you will not get 60X magnification: because the tube lens focal length is different. This may also lead to other problems like with aberration corrections which may not work well as they are matched to a certain tube lens.
In case you are building your own microscope and using one of the commercially available objectives, make sure you know for what tube lens focal length this objective was designed for.