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Effective F-Number and relative Signal amplitude

Effective F-Number and relative Signal amplitude

Effecive F-Number

For small magnifications β< 0.1 when calculating signal amplitude or the limit of lens resolution caused by diffraction, the F-number K (=focal length/aperture diameter) is replaced by an effective F-number K’(=image distance/aperture diameter).

With a nominal F-number K and small magnification β the effective F-number K’ is calculated from

{!{!{K'=K\cdot (1+\beta)}!}!}

Example

Nominal F-number      K = 4
Magnification     β = 1
Effective F-number   K ’ = 2 · K = 8

Relative signal amplitude

The relative signal amplitude of two effective F-numbers is given by
{!{!{R=\left(\frac{{K'}_2}{{K'}_1}\right)^2}!}!}

For konstant β, the effect of stopping down a K=1.4 lens is as follows
F-number KRelative signal amplitude R
1.41
20.5
2.80.25
40.125
5.60.06
80.016

Imaging Parameters

Schematic depiction of the imaging system and definition of variables used.

f = Lens focal length (mm)
S = Sensor length (mm)
L = Length of Region of Interest (ROI) of object (mm)
a = Object range (mm)
a’ = Image distance: Distance from sensor to HH’ (mm)
β Magnification
w = Field angle
OO’ = Distance from sensor to measured zone (mm)
s’A = Flange focal length (mm)
∆s’  = Lens extension (mm)
LT = Tube length 
A = Working distance (mm)
HH’ = Principal point distance (mm) (can lengthen or shorten OO’)
s’K = Camera flange length consisting of focus adapter series FA22 and extension rings series ZR (mm)
LO = Lens length (mm)