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Re: [Rollei] OT: retrofocus & inverse square

> Heres a "curiosity-kills-cats" question for Richard...Do retrofocus 
> wideangle lenses suffer the same corner brightness dropoff as a 
> symmetrical design, or does their increased distance from the film 
> reduce the effect? If not, it would appear there is a free lunch 
> after all. My Nikkor 20mm seems to be reasonably even...no need for a 
> graduated filter. I don't have a Leica Super-Angulon type lens to 
> compare it with. -- John

Hi John. Be happy, the answer is both easy and very hard to find in
the litterature. 

Pardon me for this little tutorial : you ask a delicate question, you
deserve a long, boring, but comprehensive answer ;-);-) "Dura RUG-lex,
sed lex" !!!

Not kidding, there are two issues.

- -1- what is the bellows factor, at the centre of the image field, for
    a given lens design used in close-up ?

- -2- what is the additional drop-off effect with respect to the centre,
    for a given objet-to-image setup ?

Your interest is only in issue#2, but I'll address both for our
beloved RUGgers, so avid of utmost precision in optics.

Retrofocus designs are excellent examples of lenses where the pupils
(entrance and exit) are not located in the principal (or nodal)
planes. For those reasons the usual photometric formulae need to be
adapted with respect to classical formulae, valid only for a single
lens element (nobody uses this in photography !!) and more seriously
valid for all quasi-symmetric lens designs, even compound lenses with
a non-zero distance between the principal (or nodal) planes.

Almost all view camera lenses are quasi-symmetric with pupils located
in the principal planes and therefore behave like a single lens
element as far as photometry is concerned. A good ol' Tessar is not
quite symmetric but the offset of the pupils with respect to the
principal planes is not big enough to yield significant effects when
compared to a symmetric lens. Truly symmetric lenses are Apo-Repro
process lenses like the Apo Ronar®, Apo Artar®, G-claron®,
Makro-Planar® etc.. otherwise enlarging lenses are reasonably
symmetric. Our beloved 7-element 2,8-80 SLR Zeiss Planar® is not quite
symmetric and has a slight amount of retrofocus desing in order to
accommodate for the flipping mirror dimensions.

The degree of asymmetry of the lens as far as photometric effects is
concerned is entirely described by a single parameter : namely the
pupillar magnification ratio (P_M) which is defined as the ratio 

P_M= (exit pupil diameter / entrance pupil diameter). 

P_M is either directly or indirectly (under the form of pupillar
diameters) documented in good manufacturers datasheet. In German, look
for "Eintrittspupille" (entrance pupil) and "Austrittspupille" (exit

In a retrofocus lens design like the brand new Zeiss Distagon® 40mm
CFE-IF, the P_M factor is equal to 2.7. Similar P_M factors apply to
some extreme wide angle view camera lenses like the
Schneider-Kreuznach 28mm Digitar®, an exception to all view camera
lenses including the new "digital" series for which P_M is equal to 1
within a few percent.

Now what are the formulae.

The bellows factor "X" in the general case is given by :

X = (M+P_M)^2/(P_M^2)  

where M is the absolute value of the linear image magnification ratio.

In general, for any thick, asymmmetric, compound lens, M = E/f where E
is the lenght extension with respect to the infinity-focus position.

If the image is ten times smaller that the objet, M = 1/10 = 0.1. In
reality physicists introduce a "minus" sign in the definition of M to
denote the fact that the image is reversed in photographical
conditions. In the above formula there is no sign.

Note that "X" is the multiplicative factor to be applied to exposure
times when departing from the infinity-focus position. If you want to
convert it into f-stops, mathematically you have to compute log2(X).
X=2 : open one f-stop, X=4, open 2 f-stops, etc...

In a purely symmetric lens, or more generally in a lens exhibiting a
unit pupillar magnification ration, the bellows factor is given by the
well-known formula : X_(PM=1) = (M+1)^2 = ((E/f)+1)^2

Example : f=80mm, E = 33mm (bellows of of 33 mm) , M = 0.41
X_(PM=1) = (1.41)^2 = 2 ==> open one f-stop.

OK this solves the question of issue#1. Now what about issue#2.

In the simplest model of a compound lens, there would be no pupillar
distorsion, i.e. the shape and diameter of the exit pupil would be the
same when seen from the corner of the image field. This is valid for
standard lenses and telephotos. However in any modern wide-angle
lense, the exit pupil distorts when seen from the corner. Usually it
looks bigger and seems to turn like an eyeball rotating toward the
line of view. This effects helps minimizing the light fall-off in the
corners and is specific to a given lens design, so no general formaul
can be given.

Now assume that we neglect this effect. Then, the light fall-off is
governed by a fourth-power-cosine law :

X(corner) = X(centre)*cos^4(theta) 

where theta is the angle measured from the **centre of the exit
pupil** and nothing else !!. In a quasi-symmetric lens design, this
angle is exactly the same as the angle of view of the lens in the
object or image space, simply because the exit pupil is located
exactly, or very close to the exit nodal point, for which the rays
exhibit the same angle as in the object space (property of nodal

In a retrofocus design with P_M greater than one, the exit pupil is
located far in front of the lens. The position of the pupils is
governed by the P_M factor because entrance and exit pupils are
conjugated like any objet/image. There is a simple formula giving the
position of the pupils in a asymmetric lens design, but suffice to say
that when P_M is equal to 2.7, the centre of the exit pupil is located
at a longer distance in front of the film than in a symmetrical lens
of same focal length. 

Therefore the "theta" angle is much smaller in the corners, hence the
cos^4(theta) factor is much smaller for a retrofocus than in a
symmetrical design. If you add the effect of pupillar distorsion, all
this explains, that, YES, you are right, a retrofocus lens is less
affected by light fall-off in the corners than a symmetrical design of
same focal lenght because the 'theta' angle is smaller and because
pupillar distorsion counterfights the cos^4(theta) law. In addition a
retrofocus lens is less affected by the bellows factor than a
symmetrical lens. Unfortunately, retrofocus lenses are not good in
macro, symmetrical lenses are preferred near the 1:1 magnification
ratio, so this second advantage is purely theoretical.

To finish with, when the P_M factor become very big, the lens becomes
closer to a so-called "telecentric" lens. Those lenses were used only
in special optical measuring intruments but are very interesting for
digital sensors for which the silicon cell is recessed at the bottom
of a 'well'. Provided that the last lens diameter is as big as the
image field, in theory with a telecentric lens you can get a wide
angle field of view with a very small angle of incidence on the
sensor. Incredible !! The fact that the exit lens diameter has to be
very big has been taken into account in the new 4/3 digital camera and
lens standard. The use of retrofocus lens designs with a P_M value
over 2 is a possible solution to solve the problem of vignetting in
silicon sensors. Athother cheaper solution is to tabulate the
vignetting effect and compensate for it digitally in the image
pre-processing stage. Both solutions are incoprorated in present or
future digital camera designs.

Hope this helps !!!

- -- 
Emmanuel BIGLER
Professor, optics and microtechnology,
National College of Mechanical Engineering and Microtechnology
ENSMM, Besancon, France.