APS-C and full frame conversion and comparison; crop factor, focal length, depth of view, depth of field; Canon, Nikon, Sony.

Advanced Photo System type-C (APS-C) is an image sensor format approximately equivalent in size to the Advanced Photo System "classic" negatives of "25.1×16.7 mm". All APS-C variants are considerably smaller than 35 mm standard film which measures 36×24 mm (Full Frame FF sensor)

Sensor sizes range from 20.7×13.8 mm to 28.7×19.1 mm, but are typically about 22.5×15 mm for Canon for example (crop factor=1.62× ).

For the Sony APS HD CMOS sensor of Alpha series (from alpha 100 to alpha 77 (II)), it's 23.5 x 15.6mm (also for Nikon); sensor surface=366.60mm^2, and Sensor Pixel Area = 15.28µm^2 (for 24.3MPix), and crop factor=1.55×. Then a 35mm (24x36mm)-->"35*1.55=54.25mm". 

A crop factor (sometimes referred to as a "focal length multiplier", even though the actual focal length is the same) can be used to calculate the field of view in 35 mm terms from the actual focal length.

Many companies manufacture a range of lenses "optimised" for APS-C sensors. For full frame lenses, the sensor is only in the center.

The full-frame lens is bigger, heavier (than APS lens), and uses a larger filter because it has to cover a larger sensor area.

Moreover they are also some differents designs Sony A-Mount and Sony E-Mount camera .

Shortening the flange focal distance to 18 mm (E-Mount) compared with earlier offerings from Sony which used 44.5 mm (A-mount).

http://stephane-mottin.blogspot.fr/2017/06/future-of-sony-minolta-mount-dead-or.html

Sony also produces E-mount lenses designated as "FE", which cover the entire full-frame image circle.

optics

basis : the 5 parts

http://demonstrations.wolfram.com/AnatomyOfTheEye/

http://demonstrations.wolfram.com/OpticalModelOfTheHumanEye/

The start is the pinhole camera

https://en.wikipedia.org/wiki/Pinhole_camera

No lens, only a pin-hole.

The main problem is the quantity of light due to the fact of the very small surface of the hole.

The second problem is an "infinite" depth of view" (all parts are not blurred).

Then with a lens, we have a large "hole" (diaphragm).

With the aperture, we control the depth of view".

The only problem with FF-->APS-C is the size of the sensor and then the angle of view.

Angle of view from lenses at the same focal length

The angle of view (AOV) describes the angular extent of a given scene that is imaged by a camera (~ also this more general term field of view).
See also

http://masochismtango.com/2010/02/05/angles-of-view-and-crop-factors/

 Aperture

http://demonstrations.wolfram.com/DiaphragmShutter/

The f-number

The f-number f/#  of an optical system such as a camera lens is the ratio of 2 lengths, of the system's focal length to the diameter of the entrance pupil (effective aperture). 

A 100 mm focal length f/4 lens has an entrance pupil diameter of 25 mm. 

The standard f-stop scale which is an approximately geometric sequence of numbers that corresponds to the sequence of the powers of the square root of 2:   

 f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32, f/45, f/64,...

f/√2, f/(√2)^2, f/(√2)^3, ...  (2.8 is 2.828...)

Then the surface increase of 2 when you change the aperture f-number, for example 2.8->4.

When you change a full frame to an APS-C, the aperture of a lens never changes.... Only the focal length and then the f-number. It lets the same amount of light in no matter what. A 2.8 is a 2.8 ever and ever. If you have a smaller sensor doesn't use the whole image coming in.

Fundamentally, f/# is the ratio of the effective focal length (EFL) of the lens to the effective aperture diameter (DEP).

Diameter of the "first entrance lens" or pupils

Moreover remember that the diameter of the first entrance lens  (or single lens "diaphragm-like") is very important for the quantity of light and also for the resolution.

and with some effects:

http://slideplayer.com/slide/4593133/

https://www.edmundoptics.com/resources/application-notes/imaging/limitations-on-resolution-and-contrast-the-airy-disk/

This figure  shows the difference in spot sizes between a lens set at f/2.8 and a lens set at f/8.

As pixels continue to reduce in size, this effect becomes more of an issue and eventually is very difficult to overcome. The Airy Disk, or minimum spot size can be calculated using the f/# and wavelength in μm:

You can see the interest of the dimensionless number f/# .

The lambda is the wavelength of light (blue ~ 0.4µm).

f/#	Airy Disk Diameter (μm) at a Wavelength of 520nm
1.4	1.78
2	2.54
2.8	3.55
4	5.08
5.6	7.11
8	10.15
11	13.96
16	20.30

The definition of f/# (see below) is limited in the sense that it is defined at an infinite working distance where the magnification is effectively zero. Most often, the object is located much closer to the lens than an infinite distance away, and f/# is more accurately represented by the working (f/#)w:

In the equation for working f/#,   m represents the paraxial magnification (ratio of image to object height) of the objective. Note that as m approaches zero (as the object approaches infinity), the working f/# is equal to the infinite f/#. It is especially important to keep (working f/#w) in mind at smaller working distances. For example, an f/2.8, 25mm focal length lens operating with a magnification of -0.5X will have an effective working f/# of f/4.2. This impacts image quality as well as the lens’s ability to collect light.

f/# and Numerical Aperture (NA)

It can often be easier to talk about overall light throughput in a lens in terms of the cone angle, or the numerical aperture (NA), of a lens. 

The numerical aperture of a lens is defined as the sine of the marginal ray angle in image space, and is shown in:

Visual Representation of f/#, both for a Simple Lens (a) and a Real-World System (b)

It is important to remember that f/# and NA are inversely related:

Also we must consider some aberrations for example:

Optical Depth of Field

Optical Depth of Field (DoF) Calculator for Cameras

http://demonstrations.wolfram.com/OpticalDepthOfFieldCalculatorForCameras/

depth of field (DOF), also called focus range or effective focus range is the distance between the nearest and farthest objects in a scene that appear "acceptably sharp" in an image. 

In fact focus is possible at only one distance and  at that distance, a point ("subpixel") object will produce a point image.

The diameter of the circle increases with distance from the point of focus; the largest circle that is indistinguishable from a point is known  as the circle of confusion. 

Doubling the f-number will approximately double the depth of field...

others

An other point is the quality of lenses:

lens transmittances of 60%–90% are typical. 

and many many bad effects as tilt, flare...

http://demonstrations.wolfram.com/LensFlares/

Macro

You’re admiring your work when you see an insect on the flower which you hadn’t noticed before.You have a macro lens and now you want to fill the photo with that insect.At this point, the APS-C camera might be more helpful because you won’t have to stand as close to the insect to fill the frame. While that might be nice for people who don’t like getting close to critters, being at a greater distance also means you’re less likely to scare the insect. And in macro photography, trying not to frighten off your skittish subjects is a big deal and you’ll welcome the advantage you have from being able to work at a greater distance.

But that’s not all.

Because getting enough depth of field is notoriously difficult in macro photography, you’ll probably also appreciate the slight increase in depth of field that comes from photographing your insect at the greater distance.
http://www.mdavid.com.au/photography/apscversusfullframe.shtml

Depth of field increases as you move further away from your subject. 

This can give an APS-C camera an advantage when taking macro photos 

because they fill the frame with your subject from a greater distance. 

Note that this graphic is not drawn precisely to scale.