# Lens Tutorial

```by David M. Jacobson
jacobson@hpl.hp.com
Revised December  9, 1995

This note gives a tutorial on lenses and gives some common lens
formulas.  I attempted to make it between an FAQ (just simple facts)
and a textbook.  I generally give the starting point of an idea, and
then skip to the results, leaving out all the algebra.  If any part of
it is too detailed, just skip ahead to the result and go on.

It is in 6 parts.  The first gives formulas relating subject and image
distances and magnification, the second discusses f-stops, the third
discusses depth of field, the fourth part discusses diffraction, the
fifth part discusses the Modulation Transfer Function, and the sixth
illumination.  The sixth part is authored by John Bercovitz.  Sometime
in the future I will edit it to have all parts use consistent notation
and format.

The theory is simplified to that for lenses with the same medium (eg
air) front and rear: the theory for underwater or oil immersion lenses
is a bit more complicated.

Subject distance, image distance, and magnification

In lens formulas it is convenient to measure distances from a set of
points called "principal points".  There are two of them, one for the
front of the lens and one for the rear, more properly called the
primary principal point and the secondary principal point.  While most
lens formulas expect the subject distance to be measured from the
front principal point, most focusing scales are calibrated to read the
distance from the subject to the film plane.  So you can't use the
distance on your focusing scale in most calculations, unless you only
need an approximate distance.  Another interpretation of principal
points is that a (probably virtual) object at the primary principal
point formed by light entering from the front will appear from the
rear to as a (probably virtual) image at the secondary principal point
with magnification exactly one.

"Nodal points" are the two points such that a light ray entering the
front of the lens and headed straight toward the front nodal point
will emerge going straight away from the rear nodal point at exactly
the same angle to the lens's axis as the entering ray had.  The nodal
points are identical to the principal points when the front and rear
media are the same, e.g. air, so for most practical purposes the terms
can be used interchangeably.

In simple double convex lenses the two principal points are somewhere
inside the lens (actually 1/n-th the way from the surface to the
center, where n is the index of refraction), but in a complex lens
they can be almost anywhere, including outside the lens, or with the
rear principal point in front of the front principal point.  In a lens
with elements that are fixed relative to each other, the principal
points are fixed relative to the glass.  In zoom or internal focusing
lenses the principal points may move relative to the glass and each
other when zooming or focusing.

When a camera lens is focused at infinity, the rear principal point is
exactly one focal length in front of the film.  To find the front
principal point, take the lens off the camera and let light from a
distant object pass through it "backwards".  Find the point where the
image is formed, and measure toward the lens one focal length.  With
some lenses, particularly ultra wides, you can't do this, since the
image is not formed in front of the front element.  (This all assumes
that you know the focal length.  I suppose you can trust the
manufacturers numbers enough for educational purposes.)

So      subject (object) to front principal point distance.
Si      rear principal point to image distance
f       focal length
M       magnification

1/So + 1/Si = 1/f
M = Si/So
(So-f)*(Si-f) = f^2
M = f/(So-f) = (Si-f)/f

If we interpret Si-f as the "extension" of the lens beyond infinity
focus, then we see that this extension is inversely proportional to a
similar "extension" of the subject.

For rays close to and nearly parallel to the axis (these are called
"paraxial" rays) we can approximately model most lenses with just two
planes perpendicular to the optic axis and located at the principal
points.  "Nearly parallel" means that for the angles involved, theta
~= sin(theta) ~= tan(theta).  ("~=" means approximately equal.)  These
planes are called principal planes.

The light can be thought of as proceeding to the front principal
plane, then jumping to a point in the rear principal plane exactly the
same displacement from the axis and simultaneously being refracted
(bent).  The angle of refraction is proportional the distance from the
center at which the ray strikes the plane and inversely proportional
to the focal length of the lens.  (The "front principal plane" is the
one associated with the front of the lens.  I could be behind the rear
principal plane.)

Apertures, f-stop, bellows correction factor, pupil magnification

We define more symbols

D       diameter of the entrance pupil, i.e. diameter of the aperture as
seen from the front of the lens
N       f-number (or f-stop)  D = f/N, as in f/5.6
Ne      effective f-number (corrected for "bellows factor",
but not absorption)

Light from a subject point spreads out in a cone whose base is the
entrance pupil.  (The entrance pupil is the virtual image of the
diaphragm formed by the lens elements in front of the diaphragm.)  The
fraction of the total light coming from the point that reaches the
film is proportional to the solid angle subtended by the cone.  If the
entrance pupil is distance y in front of the front nodal point, this
is approximately proportional to D^2/(So-y)^2.  (Usually we can ignore
y.)  If the magnification is M, the light from a tiny subject patch of
unit area gets spread out over an area M^2 on the film, and so the
brightness on the film is inversely proportional to M^2.  With some
algebraic manipulation and assuming y=0 it can be shown that the
relative brightness is

(D/So)^2/M^2 = 1/(N^2 * (1+M)^2).

Thus in the limit as So -> infinity and thus M -> 0, which is the usual
case, the brightness on the film is inversely proportional to the
square of the f-stop, N, and independent of the focal length.

For larger magnifications, M, the intensity on the film in is somewhat
less then what is indicated by just 1/N^2, and the correction is
called bellows factor.  The short answer is that bellows factor when
y=0 is just (1+M)^2.  We will first consider the general case when
y != 0.

Let us go back to the original formula for the relative brightness on
the film.

(D/(So-y))^2/M^2

The distance, y, that the aperture is in front of the front nodal
point, however, is not readily measurable.  It is more convenient to
use "pupil magnification".  Analogous to the entrance pupil is the
exit pupil, which is the virtual image of the diaphragm formed by any
lens elements behind the diaphragm.  The pupil magnification is the
ratio of exit pupil diameter to the entrance pupil diameter.

p       pupil magnification (exit_pupil_diameter/entrance_pupil_diameter)

For all symmetrical lenses and most normal lenses the aperture appears
the same from front and rear, so p~=1.  Wide angle lenses frequently
have p>1, while true telephoto lenses usually have p<1.  It can be
shown that y = f*(1-1/p), and substituting this into the above
equation and carrying out some algebraic manipulation yields that the
relative brightness on the film is proportional to

1/(N^2 ( 1 + M/p)^2)

Let us define Ne, the effective f-number, to be an f-number with the
lens focused at infinity (M=0) that would give the same relative
brightness on the film (ignoring light loss due to absorption and
reflection) as the actual f-number N does with magnification M.

Ne = N*(1+M/p)

An alternate, but less fundamental, explanation of bellows correction
is just the inverse square law applied to the exit pupil to film
distance.  Ne is exit_pupil_to_film_distance/exit_pupil_diameter.

It is convenient to think of the correction in terms of f-stops
(powers of two).  The correction in powers of two (stops) is
2*Log2(1+M/p) = 6.64386 Log10(1+M/p).  Note that for most normal
lenses y=0 and thus p=1, so the M/p can be replaced by just M in the
above equations.

Circle of confusion, depth of field and hyperfocal distance.

The light from a single subject point passing through the aperture is
converged by the lens into a cone with its tip at the film (if the
point is perfectly in focus) or slightly in front of or behind the
film (if the subject point is somewhat out of focus).  In the out of
focus case the point is rendered as a circle where the film cuts the
converging cone or the diverging cone on the other side of the image
point.  This circle is called the circle of confusion.  The farther
the tip of the cone, ie the image point, is away from the film, the
larger the circle of confusion.

Consider the situation of a "main subject" that is perfectly in
focus, and an "alternate subject point" this is in front of or
behind the subject.

Soa     alternate subject point to front principal point distance
Sia     rear principal point to alternate image point distance
h       hyperfocal distance
C       diameter of circle of confusion
c       diameter of largest acceptable circle of confusion
N       f-stop (focal length divided by diameter of entrance pupil)
Ne      effective f-stop Ne = N * (1+M/p)
D       the aperture (entrance pupil) diameter (D=f/N)
M       magnification (M=f/(So-f))

The diameter of the circle of confusion can be computed by similar
triangles, and then solved in terms of the lens parameters and subject
distances.  For a while let us assume unity pupil magnification, i.e. p=1.

When So is finite
C = D*(Sia-Si)/Sia = f^2*(So/Soa-1)/(N*(So-f))
When So = Infinity,
C = f^2/(N Soa)

Note that in this formula C is positive when the alternate image point
is behind the film (i.e. the alternate subject point is in front of
the main subject) and negative in the opposite case.  In reality, the
circle of confusion is always positive and has a diameter equal to
Abs(C).

If the circle of confusion is small enough, given the magnification in
printing or projection, the optical quality throughout the system,
etc., the image will appear to be sharp.  Although there is no one
diameter that marks the boundary between fuzzy and clear, .03 mm is
generally used in 35mm work as the diameter of the acceptable circle
of confusion.  (I arrived at this by observing the depth of field
scales or charts on/with a number of lenses from Nikon, Pentax, Sigma,
and Zeiss.  All but the Zeiss lens came out around .03mm.  The Zeiss
lens appeared to be based on .025 mm.)  Call this diameter c.

If the lens is focused at infinity (so the rear principal point to film
distance equals the focal length), the distance to closest point that
will be acceptably rendered is called the hyperfocal distance.

h = f^2/(N*c)

If the main subject is at a finite distance, the closest
alternative point that is acceptably rendered is at at distance

Sclose = h So/(h + (So-F))

and the farthest alternative point that is acceptably rendered is at
distance

Sfar = h So/(h - (So - F))

except that if the denominator is zero or negative, Sfar = infinity.

We call Sfar-So the rear depth of field and So-Sclose the front depth
field.

A form that is exact, even when P != 1, is

depth of field = c Ne / (M^2 * (1 +or- (So-f)/h1))
= c N (1+M/p) / (M^2 * (1 +or- (N c)/(f M))

where h1 = f^2/(N c), ie the hyperfocal distance given c, N, and f
and assuming P=1.  Use + for front depth of field and - for rear depth
of field.  If the denominator goes zero or negative, the rear depth of
field is infinity.

This is a very nice equation.  It shows that for distances short with
respect to the hyperfocal distance, the depth of field is very close
to just c*Ne/M^2.  As the distance increases, the rear depth of field
gets larger than the front depth of field.  The rear depth of field is
twice the front depth of field when So-f is one third the hyperfocal
distance.  And when So-f = h1, the rear depth of field extends to
infinity.

If we frame a subject the same way with two different lenses, i.e.
M is the same both both situations, the shorter focal length lens will
have less front depth of field and more rear depth of field at the
same effective f-stop.  (To a first approximation, the depth of field
is the same in both cases.)

Another important consideration when choosing a lens focal length is
how a distant background point will be rendered.  Points at infinity
are rendered as circles of size

C =  f M / N

So at constant subject magnification a distant background point will
be blurred in direct proportion to the focal length.

This is illustrated by the following example, in which lenses of 50mm
and 100 mm focal lengths are both set up to get a magnification of
1/10.  Both lenses are set to f/8.  The graph shows the circle of
confusions for points as a function of the distance behind the
subject.

circle of confusion (mm)
#
#               *** 100mm f/8
#               ... 50mm f/8
0.8 #                                                               *******
#                                                      *********
#                                             *********
#                                         ****
#                                    *****
#                                ****
0.6 #                            ****
#                       *****                                   .......
#                    ***                      ..................
#                  **            .............
0.4 #              ****     .........
#           ***     ....
#         **   .....
#        * ....
#      **..
0.2 #    **.
#  .*.
# **
#*
*######################################################################
0 #
250    500       750     1000     1250    1500     1750     2000
distance behind subject (mm)

The standard .03mm circle of confusion criterion is clear down in the
ascii fuzz.  The slope of both graphs is the same near the origin,
showing that to a first approximation both lenses have the same depth
of field.  However, the limiting size of the circle of confusion as
the distance behind the subject goes to infinity is twice as large for
the 100mm lens as for the 50mm lens.

Diffraction

When a beam of parallel light passes through a circular aperture it
spreads out a little, a phenomenon known as diffraction.  The smaller
the aperture, the more the spreading.  The normalized field strength
(of the electric or magnetic field) at angle phi from the axis is
given by

2 J1(x)/x, where x = 2 phi Pi R/lambda

and where R is the radius of the aperture, lambda is the wavelength of
the light, and J1 is the first order Bessel function. The
normalization is relative to the field strength at the center.  The
power (intensity) is proportional to the square of this function.

The field strength function forms a bell-shaped curve, but unlike the
classic E^(-x^2) one, it eventually oscillates about zero.  Its first
zero at 1.21967 lambda/(2 R).  There are actually an infinite number
of lobes after this, but about 86% of the power is in the circle
bounded by the first zero.

Relative field strength

***
1 #  ****
#      **
0.8 #        *
#         **
#           *
#            **
#              *
0.6 #               *
#                *
#                 *
0.4 #                  *
#                   *
#                    **
0.2 #                      **
#                        **
#                          **                         *****************
###############################*###################*****###################
#                             *****        ******
#          0.5         1          1.5******    2         2.5          3

Angle from axis (relative to lambda/diameter_of_aperture)

Approximating the aperture-to-film distance as f and making use of
the fact that the aperture has diameter f/N, it follows directly that
the diameter of the first zero of the diffraction pattern is
2.43934*N*lambda.  Applying this in a normal photographic situation is
difficult, since the light contains a whole spectrum of colors.  We
really need to integrate over the visible spectrum.  The eye has
maximum sensitive around 555 nm, in the yellow green.  If, for
simplicity, we take 555 nm as the wavelength, the diameter of the
first zero, in mm, comes out to be 0.00135383 N.

As was mentioned above, the normally accepted circle of confusion for
depth of field is .03 mm, but .03/0.00135383 = 22.1594, so we can
see that at f/22 the diameter of the first zero of the diffraction
pattern is as large is the acceptable circle of confusion.

A common way of rating the resolution of a lens is in line pairs per
mm. It is hard to say when lines are resolvable, but suppose that we
use a criterion that the center of the dark area receive no more than
80% of the light power striking the center of the lightest areas.
Then the resolution is 0.823 /(lambda*N) lpmm.  If we again assume 555
nm, this comes out to 1482/N lpmm, which is in close agreement with
the widely used rule of thumb that the resolution is diffraction
limited to 1500/N lpmm.  However, note that the MTF, discussed below,
provides another view of this subject.

Modulation Transfer Function

The modulation transfer function is a measure of the extent to which a
lens, film, etc., can reproduce detail in an image.  It is the spatial
analog of frequency response in an electrical system.  The exact
definition of the modulation transfer function and the related
optical transfer function varies slightly amongst different
authorities.

The 2-dimensional Fourier transform of the point spread function is
known as the optical transfer function (OTF).  The value of this
function along any radius is the fourier transform of the line spread
function in the same direction.  The modulation transfer function is
the absolute value of the fourier transform of the line spread
function.

Equivalently, the modulation transfer function of a lens is the ratio
of relative image contrast divided by relative subject contrast of a
subject with sinusoidally varying brightness as a function of spatial
frequency (e.g. cycles per mm).  Relative contrast is defined as
(Imax-Imin)/(Imax+Imin).  MTF can also be used for film, but since
film has a non-linear characteristic curve, the density is first
transformed back to the equivalent intensity by applying the inverse of
the characteristic curve.

For a lens, the MTF can vary with almost every conceivable parameter,
including f-stop, subject distance, distance of the point from the
center, direction of modulation, and spectral distribution of the
light.  The two standard directions are radial (also known as
sagittal) and tangential.

The MTF for an an ideal lens (ignoring unavoidable effect of
diffraction) is a constant 1 for spatial frequencies from 0 to
infinity at every point and direction.  For a practical lens it starts
out near 1, and falls off with increasing spatial frequency, with the
falloff being worse at the edges than at the center.  Adjacency
effects in film can make the MTF of film be greater than 1 in certain
frequency ranges.

An advantage of the MTF as a measure of performance is that under some
circumstances the MTF of the system is the product (frequency by
frequency) of the (properly scaled) MTFs of its components.  Such
multiplication is always allowed when the phase of the waves is lost
at each step.  Thus it is legitimate to multiply lens and film MTFs or
the MTFs of a two lens system with a diffuser in the middle.  However,
the MTFs of cascaded ordinary lenses can legitimately be multiplied
only when a set of quite restrictive and technical conditions is
satisfied.

As an example of some OTF/MTF functions, below are the OTFs of pure
diffraction for an f/22 aperture, the OTF induced by a .03mm circle of
confusion of a de-focused but otherwise perfect and diffraction free
lens, and the combination of these.  (Note that these cannot be
multiplied.)

Let lambda be the wavelength of the light, and spf the spatial
frequency in cycles per mm.

For diffraction the formula is

OTF(lambda,N,spf) = 2/Pi (ArcCos(lambda N spf) -
lambda N spf Sqrt(1-(lambda N spf)^2))   if lambda N spf <=1
= 0                            if lambda N spf >=1

Note that for lambda = 555 nm, the OTF is zero at spatial frequencies
of 1801/N cycles per mm and beyond.

For a circle of confusion of diameter C,

OTF(C,spf) = 2 J1(Pi C spf)/(Pi C spf)

where, again J1(x) is the first order Bessel function.
This goes negative at certain frequencies.  Physically, this would
mean that if the test pattern were lighter right on the optical center
then nearby, the image would be darker right on the optical center
than nearby.  The MTF is the absolute value of this function.  Some
authorities use the term "spurious resolution" for spatial frequencies
beyond the first zero.

When there is a combination of diffraction and focus error dz (which
by itself would cause a circle of confusion of diameter dz/N), the OTF
is given by the following.  It involves an integration which must be
done numerically.  Let s = lambda N spf, and a = Pi spf dz / N.
Then the OTF is given by

OTF = 4/(Pi a) integral y=0 to sqrt(1-s^2) of sin(a(sqrt(1-y^2)-s)) dy
for s < 1
0 for s >= 1

This formula is an approximation that is best at small apertures.

Here is a graph of the OTF of the f/22 diffraction limit, a .03mm
circle of confusion, and the combined effect.

OTF
*
1 *****
# +\$\$*
#   +\$*
#    + *\$             \$\$\$\$ Diffraction
0.8 #     + **\$           **** Circle of confusion
#      ++ *\$\$         ++++ Combined diffraction and circle of confusion
#        + * \$\$
#         + *  \$
0.6 #          ++*  \$\$
#            +*   \$\$
#              *    \$\$
#               *     \$\$
0.4 #                *      \$\$
#                *++++    \$\$
#                 *   +++++ \$\$\$\$
#                  *       +++++\$\$\$\$
0.2 #                   *           ++++\$\$
#                    *               +\$\$\$
#                     *                  \$\$\$\$*****************
#######################**##################**\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$\$******\$\$\$
0 #                        **           *****                         ***
#          20         40   ***** *****        80          100        120
#                              **

Spatial Frequency (cycles/mm)

Note how the combination is not the product of each of the effects
taken separately.

Some authorities present MTF in a log-log plot.

The classic paper on the MTF for the combination of diffraction and
focus error is H.H. Hopkins, "The frequency response of a defocused
optical system," Proceedings of the Royal Society A, v. 231, London
(1955), pp 91-103.  Reprinted in Lionel Baker (ed), _Optical Transfer
Function: Foundation and Theory_, SPIE Optical Engineering Press,
1992, pp 143-153.

Illumination
(by John Bercovitz)

The Photometric System

Light flux, for the purposes of illumination engineering, is
measured in lumens.  A lumen of light, no matter what its wavelength
(color), appears equally bright to the human eye.  The human eye has a
stronger response to some wavelengths of light than to other
wavelengths.  The strongest response for the light-adapted eye (when
scene luminance >= .001 Lambert) comes at a wavelength of 555 nm.  A
light-adapted eye is said to be operating in the photopic region.  A
dark-adapted eye is operating in the scotopic region (scene luminance
```