F-Mount

Lens mount developed by Nikon, typical used for line scan cameras

F-Theta lenses are a class of Fisheye lenses

F-Theta lenses are also called “equidistant“, “linear scaled“, “äquidistant” or “angular”

F-Theta lenses use an image mapping function of type
$r = f \theta$

Non distortion 180° F-Theta lenses have an image circle or $\pi * FocalLength$ mm

A non distortion F-Theta lens of 1.37mm focal length has an image circle of $d = \pi 1.37$ mm = $4.30$ mm
Therefor a f=3.17mm F-Theata lens with an image diameter for 180° of 4.15mm has a distortion of (4.15/4.3)-1 = 0.965-1 = -3.5%

maintains angular distances

see
Fisheye Types

far point

Most distant point on the optical axis with an image of “acceptable sharpness”

Depth Of Field

$FarPoint= \frac{(FocalLength^2 \cdot ObjectDistance)}{(FocalLength^2 - F\# \cdot CoC \cdot (ObjectDistance-FocalLength))}$

Where CoC is the Circle of Confusion (the largest accepted Airy-disk) in Millimeter.

Alternatively, we can express the FarPoint using the magnification M :

$FarPoint = \frac{(FocalLength^2 \cdot (M + 1))}{(FocalLength \cdot M - F\# \cdot CoC )}$

If we use for example a 1/2.5″ 5 Aptina Megapixel greyscale Sensor mit 2.2 $\mu$ pixel pitch, we can use the pixel diagonal as CoC for crisp images, say $CoC = 2.2\mu \cdot \sqrt{2} = 3.1 \mu = 0.0031mm$
A 5 Mega lens with f=7.2mm focal length and F-stop F2.4, focused to an object distance of 100mm then has a far point of
$FarPoint = \frac{((7.2mm)^2 \cdot 100mm)}{((7.2mm)^2 - 2.4 \cdot 0.0031mm \cdot (100mm-7.2mm))} = 101.35mm$
und einen Nahpunkt von
$NearPoint = \frac{((7.2mm)^2 \cdot 100mm)}{((7.2mm)^2 + 2.4 \cdot 0.0031mm \cdot (100mm-7.2mm))} = 98.69mm$
and thus $DOF = FarPoint - NearPoint = 2.66mm$

If instead we use a 5 Megapixel greyscale Sony Sensor with 3.45 $\mu$ pixel pitch, we can choose as CoC the diagonal of the pixel for crisp images, say $CoC = 3.45 \cdot \sqrt{2} = 4.86 \mu = 0.00486mm$
A 5 Mega lens with f=7.2mm focal length and F-stop F2.4, focussed to 100mm then results in
$FarPoint = \frac{((7.2mm)^2 \cdot 100mm)}{((7.2mm)^2 - 2.4 \cdot 0.00486mm \cdot (100mm-7.2mm))} = 102.13$ mm
und einen Nahpunkt von
$NearPoint = \frac{((7.2mm)^2 \cdot 100mm)}{((7.2mm)^2 + 2.4 \cdot 0.00486 \cdot (100-7.2))} = 97.95$
thus we get $DOF = FarPoint - NearPoint = 4.18mm$

If we use a color sensor instead we can use $CoC=2 \cdot PixelSize$ for crisp images. For the two sensors above we then get:
$DOF_{5 Mega Aptina} = 101.93mm - 98.14mm = 3.78mm$
$DOF_{5 Mega Sony} = 103.06mm - 97.11mm = 5.93mm$

To increase the DOF we can increase the Pixel Size, but we either lose resolution, or (at the same pixel count) the magnification changes)

If you change the focal length of a lens in a way, that (with the same sensor) you get the same FOV (then from a different distance) this results in the same DOF !!!
see also https://www.optowiki.info/blog/can-i-increase-the-dof-by-changing-the-focal-length/

When a lens is focussed to the hyperfocal distance H, the far point is at $\infty$ and the near point is at $\frac{H}{2}$ .
The DOF is $\infty$ then, thus focussing to the hyperfocal distance results in the largest possible DOF.

Fermat’s Principle

If light travels from one point to another, it takes paths of a length that in first approximation is equal to the length of optical paths length closely adjascent to the actual path.

from one point to another light always chooses a path which’s OPL does not change if the path slightly varies.

Light always takes the shortest path in terms of speed but in general not the path of shortest distance.

The actual path taken has either a maximum or a minimum OPL compared to adjascent optical paths or is equal to the OPL of adjascent paths.

Field of View

(usually horizontal) extend of an object that’s visible on a sensor.

Sometimes the HFOV (horizontal field of view), sometimes the VFOV (Vertical Field of View) and sometimes the DFOV (Diagonal Field of View) matters. It’s of utmost importance to state clearly which matters.

first order optics

is the study of perfect optical systems.

Any imperfections and aberrations, also wavelengths, diffraction, interference etc. are ignored.

Parts of first order optics are Gaussian optics ( = paraxial optics).

Results of this analysis are image location and magnification.

Compare third order optics

Fisheye Types

Various types of fisheye lenses are available.
Here a short overview:

Fisheye Types:

Type	gnomonic	stereographic	F-Theta	equal area	orthographic
lens class	wide angle	Fisheye	Fisheye	Fisheye	Fisheye
mapping function	$r(\theta) = f \cdot tan \theta$	$r(\theta) = 2 \cdot f \cdot tan \frac{\theta}{2}$	$r(\theta)=f \theta$	$r(\theta) = 2 \cdot f \cdot sin \frac{\theta}{2}$	$r(\theta) = f \cdot sin \theta$
normalized mapping function $h(\theta)=\frac{r(\theta)}{f}$	$h(\theta) = tan \theta$	$h(\theta) = 2 \cdot tan \frac{\theta}{2}$	$h(\theta)=\theta$	$h(\theta) = 2 \cdot sin \frac{\theta}{2}$	$h(\theta) = sin \theta$
meridional scaling $S_{m}={\frac {\mathrm {d} \mathbf {h} (\theta )}{\mathrm {d} \theta }}$	$S_{m}={\frac {1}{\cos ^{2}\theta }}$	$S_{m}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}$	$S_{m}=1$	$S_{m}=\cos {\frac {\theta }{2}}$	$S_{m}=\cos \theta$
sagittal scaling $S_{s}={\frac {\mathbf {h} (\theta )}{\sin \theta }}$	$S_{s}={\frac {1}{\cos \theta }}$	$S_{s}={\frac {1}{\cos ^{2}{\frac {\theta }{2}}}}$	$S_{s}={\frac {\theta }{\sin \theta }}$	$S_{s}={\frac {1}{\cos {\frac {\theta }{2}}}}$	$S_{s}=1$
effective scaling $S=\sqrt{S_{m} \cdot S_{s}}$	$S={\frac {1}{\sqrt {\cos ^{3}\theta }}}$	$S={\frac 1{\cos ^{2}{\frac \theta 2}}}$	$S={\sqrt {\frac {\theta }{\sin \theta }}}$	$S=1$	$S={\sqrt {\cos \theta }}$
N from $S_{m}=S_{s}^{N}$	2	1	0	-1	$\infty$
Balance Deform. vs. Scaling $B={\frac {2\,(N-1)}{N+1}}$	$\frac{2}{3}$	0	-2	$\infty$	2
Curvature $C={\frac {N-2}{3-N}}$	0	$-{\tfrac 12}$	$-{\tfrac 23}$	$-{\tfrac 34}$	-1
Deformation $D={\frac {S_{m}}{S_{s}}}$	$D={\frac 1{\cos \theta }}$	$D=1$	$D={\frac {\sin \theta }\theta }$	$D=\cos ^{2}{\frac \theta 2}$	$D=\cos \theta$
maintains	–	angles	angular distances	areas	planar illuminance
AOV $\alpha_{max}$	<180°	<360°	>= 360°	360°	180°
$\alpha_{weak}$	54°	94°	115°	94°	66°
$\alpha_{medium}$	75°	131°	159°	131°	90°
$\alpha_{strong}$	102°	180°	217°	180°	120°

Source of the various angles and formulae: wikipedia

Here the angle $\theta$ is measured in Radians !

equal area lenses are also called “equisolid angle” or “flächentreu”
F-Theta lenses are also called “equidistant“, “linear scaled“, “äquidistant” or “angular”
sterographic lenses are also called “ conform“or “ winkeltreu”
orthographic lenses are also called” hemispherical“or “orthographisch“

For details see:
equal area
F-Theta
gnomonic
orthographic
stereographic

focal length

The focal length is the distance from the Image side principal plane to the image of objects at infinity.

For single lenses in air that is equal to the distance from the first focal point to the first principal point.
(in each case measured from the left to the right)

Note that this is a positive value for converging lenses and a negative value for the divergent lenses.

The larger the focal length, the smaller the aperture angle of the lens and the smaller the object section that is displayed full-frame on the sensor.
The lens captures less of the object. Extremea are telephoto lenses and finally telescopes.

The smaller the focal length, the larger the aperture angle of the lens and the larger the object section which is displayed full-frame on the sensor.
The lens captures more of the object. Extreme forms are fisheye lenses.

Lenses are typically listed, sorted by focal length. As an approximation, lenses with larger focal length see a smaller portion of the object (in more detail).

There are exceptions! (See: pseudo-knowledge: viewing angle and focal length are equivalent)

The following calculators determine focal lengths. They assume, a pin hole lens model, the Chief Ray Angles are assumed to be the same on object and image side.
Therefore for wide angles a too small focal length is returned .. (as all focal length calculators on the internet do 😉 );

Also keep in mind that the Gauss lens equation and the Newtonian image equation both assume same CRA on object and image side, which is nearly never given for wide angle lenses and for sure not for fisheye lenses.

Therefore when you calculate focal length, object and image distances, they will in general differ from the real world situation you measure!

The following calculator determines focal length from angles.
However, Viewing angles change with the working distance! Also, a Pinhole lens model is assumed. Thus for wide angles a too small focal length is returned .. (as all focal length calculators on the internet do 😉 )

For the next calculator it is very important to correct the distortions before doing the calculation:

OptoWiki Knowledge Base

The Knowledge Base for Technical Optics

Archives

F-Mount

F-Number

F-Stop

F-Theta

far point

Fermat’s Principle

Field of View

first order optics

Fisheye Types

Fisheye Types:

focal length