Focus movement amount of finite-corrected optical system
In a finite-corrected optical system (finite conjugate optical system), the amount of movement of the optical system when adjusting the focus differs from the amount of movement of the focus position. Here, we will derive the relational expression.
If the magnification of the finite-corrected optical system is β, then due to the relationship of the longitudinal magnification, the relationship of the change in focus position on the object plane and the image plane with a small change in focus Δ is as shown in the figure below.
Now, let us consider the amount of movement di of the entire optical system required to move the focus position of the image plane by Δi while keeping the object plane fixed . When moving from state (a) to state (d) in the diagram below, equation (1) holds true.
*For dimensional notation, see Appendix: Direction and sign of one-dimensional vectors and angles
From equation (1),
This relational expression is an approximation that holds only when di and Δi are sufficiently small, and it is important to note that the error increases as these values increase.
The relationship between β and di / Δi can be illustrated as follows: As the magnigication approaches unity, the value of di (absolute value) relative to Δi increases rapidly.
Similarly, consider the amount of movement do of the entire optical system required to move the focus position on the object plane by Δo while keeping the image plane fixed. When moving from state (a) to state (d) in the diagram below, equation (2) holds true.
*For dimensional notation, see Appendix: Direction and sign of one-dimensional vectors and angles
From equation (2),
This relational expression is an approximation that holds only when do and Δo are sufficiently small, and it is important to note that the error increases as these values increase.
The relationship between β and do / Δo can be illustrated as follows. As the magnification approaches unity, the value of do (absolute value) relative to Δo increases rapidly.
From the above results, it is important to note that with a finite-corrected optical system with a magnification of 1x, no matter how hard you try, you will not be able to adjust the focus by moving the optical system.