F-number of the imaging optical system
~Understanding through geometrical optics definitions and ray diagrams~
The F-number of an imaging optical system is a crucial parameter that affects image quality and illumination. This page organizes the geometrical optical definition along with ray diagrams and summarizes the relationships between related parameters from a practical perspective.
This page will proceed with the discussion based on the following rules.
Rule A: Sign convention
Regarding dimension notation, in accordance with Appendix: Definitions of Direction and Sign in One-Dimensional Vectors and Angles , the up-down direction is considered positive when pointing upwards, the left-right direction is considered positive when pointing to the right, and the rotational direction is considered positive when turning counterclockwise.
Rule B: Definition of the paraxial region
All rays discussed on this page are assumed to be paraxial rays with sufficiently small angles. That is, when the angle is θ, the following approximation is applied:
sin θ = θ, tan θ = θ, cos θ = 1
・Geometric optical definition of the F-number of an imaging optical system and derivation of the related formulas.
In imaging optical systems, the F-number is a fundamental parameter that influences many performance elements, including illumination, image quality, depth of field, and noise characteristics. However, in imaging systems, the effective F-number is determined not only by the F-number of the lens itself, but also by multiple geometric optical quantities such as the entrance pupil position, exit pupil position, magnification, and image-side distance. This page begins with the definition of the F-number in imaging optical systems and systematically derives the relationships between related parameters using ray diagrams.
Here, if we superimpose the light rays entering the entrance pupil and exiting the exit pupil with the other major light rays, the diagram will look like this.
* Apply Rule A (Symbol Convention)
and Rule B (Definition of Paraxial Region).
however,
f: focal length
z: object distance
z': Image distance
EP: Entrance pupil position
XP: Exit pupil position
HD: Front principal point position
HD': Back principal point position
HH': Distance between principal points
FFinf : Front focal position
BFinf : Back focal position
For the derivation of these optical system parameters, please refer to (PX) .
First, we define the F number as follows:
At this point, the following can be said:
Furthermore, pupil magnification is defined as follows:
At this point, the following can be said:
This can be expressed using the following optical parameters.
Furthermore, the magnification is defined as follows:
At this point, from Newton's equation, we can say the following:
Here, the effective F number is defined as follows:
Applying the formulas shown so far, we can derive the following formula representing the effective F number.
The lateral magnification β is a quantity that changes depending on the position of the object or image, but the entrance pupil diameter and exit pupil diameter do not depend on position. Therefore, pupil magnification can also be treated as a constant that is not affected by the position of the object or image.