Photographic Lenses Tutorial

By David Jacobson jacobson@hpl.hp.com

Summary:
     This posting contains a summary of optical facts for photographers. It is
     more detailed that a FAQ file, but less so than a text book. It covers
     focusing, apertures, bellows correction, depth of field, hyperfocal
     distance, diffraction, the Modulation Transfer Function and illumination.

Archive-name: rec-photo/lenses/tutorial
Last-modified 1994/3/19
Version: 1.3

Lens Tutorial
by David M. Jacobson
jacobson@hpl.hp.com
Revised March 19, 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 a
straight way from the rear nodal point at exactly the same angle to the lens's
axis as the entering ray had. The nodal points are equivalent to the principal
points when the front and rear media are the same, eg air, so for practical
purposes the terms can be used interchangeably. And again, the more proper
terms are primary nodal point and secondary nodal point.

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 the 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 it 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.

n 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.

n 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)

n 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 field strength (of the electric or magnetic field) at angle
phi from the axis is proportional to

lambda/(phi Pi R) * BesselJ1(2 phi Pi R/lambda),

where R is the radius of the aperture, lambda is the wavelength of the light,
and BesselJ1 is the first order Bessel function. The power (intensity) is
proportional to the square of this.

n 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 diaphragm 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 saggital) and tangential.

The MTF for an an ideal diffraction-free lens is a constant 1 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. Flare would make the MTF of a lens be
less than one even at zero spatial frequency. 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 each step accepts as input solely the intensity of the output of
the previous state, or some function of that intensity. 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, and the OTF induced by a .03mm circle of confusion of a
de-focused but otherwise perfect and diffraction free lens. (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 formulas is

OTF(lambda,N,spf) =
     = 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 * BesselJ1(Pi C spf)/(Pi C spf)

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.

Here is a graph of the OTF of both a .03mm circle of confusion and an f/22
diffraction limit.

  OTF

     #
   1 ***
     #..**
     #  ..**
     #    ..*             ***   .03 mm circle of confusion
 0.8 #      .*            ...   555nm f/22 diffraction
     #        *.
     #         *..
     #          * ..
 0.6 #           *  .
     #            *  .
     #             *  ..
     #              *   ...
 0.4 #               *     ..
     #               *       ..
     #                *        ..
     #                *          ...
 0.2 #                 *            ...
     #                  *              ...
     #                   *                ...
     #                    **                 ..      **********
   #########################*##################.*****..........*****........##
     #                       **              ***                    ********
     #          20         40  ***     60****      80         100         120
     #                           ********
     #
                           spatial frequency (cycles/mm)

Although this graph is linear in both axes, the typical MTF is presented in a
log-log plot.

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 </=
10^-8 Lambert). In between is the mesopic region. The peak response of the eye
shifts from 555 nm to 510 nm as scene luminance is decreased from the photopic
region to the scotopic region. The standard lumen is approximately 1/680 of a
watt of radiant energy at 555 nm. Standard values for other wavelengths are
based on the photopic response curve and are given with two-place accuracy by
the table below. The values are correct no matter what region you're operating
in - they're based only on the photopic region. If you're operating in a
different region, there are corrections to apply to obtain the eye's relative
response, but this doesn't change the standard values given below.

Wavelength, nm   Lumens/watt         Wavelength, nm  Lumens/watt
      400           0.27                600              430
      450          26                   650               73
      500         220                   700                2.8
      550         680

Following are the standard units used in photometry with their definitions and
symbols.

Luminous flux, F,
     is measured in lumens.
Quantity of light, Q,
     is measured in lumen-hours or lumen-seconds. It is the time integral of
     luminous flux.
Luminous Intensity, I,
     is measured in candles, candlepower, or candela (all the same thing). It
     is a measure of how much flux is flowing through a solid angle. A lumen
     per steradian is a candle. There are 4 pi steradians to a complete solid
     angle. A unit area at unit distance from a point source covers a
     steradian. This follows from the fact that the surface area of a sphere is
     4 pi r^2.
Lamps are measured in MSCP,
     mean spherical candlepower. If you multiply MSCP by 4 pi, you have the
     lumen output of the lamp. In the case of an ordinary lamp which has a
     horizontal filament when it is burning base down, roughly 3 steradians are
     ineffectual: one is wiped out by inter- ference from the base and two more
     are very low intensity since not much light comes off either end of the
     filament. So figure the MSCP should be multiplied by 4/3 to get the
     candles coming off perpendicular to the lamp filament. Incidentally, the
     number of lumens coming from an incandescent lamp varies approximately as
     the 3.6 power of the voltage. This can be really important if you are
     using a lamp of known candlepower to calibrate a photometer.
Illumination (illuminance), E,
     is the _areal density_ of incident luminous flux: how many lumens per unit
     area. A lumen per square foot is a foot-candle; a one square foot area on
     the surface of a sphere of radius one foot and having a one candle point
     source centered in it would therefore have an illumination of one
     foot-candle due to the one lumen falling on it. If you substitute meter
     for foot you have a meter-candle or lux. In this case you still have the
     flux of one steradian but now it's spread out over one square meter.
     Multiply an illumination level in lux by .0929 to convert it to
     foot-candles. (foot/meter)^2= .0929. A centimeter candle is a phot.
     Illumination from a point source falls off as the square of the distance.
     So if you divide the intensity of a point source in candles by the
     distance from it in feet squared, you have the illumination in foot
     candles at that distance.
Luminance, B,
     is the _areal intensity_ of an extended diffuse source or an extended
     diffuse reflector. If a perfectly diffuse, perfectly reflecting surface
     has one foot-candle (one lumen per square foot) of illumination falling on
     it, its luminance is one foot-Lambert or 1/pi candles per square foot. The
     total amount of flux coming off this perfectly diffuse, perfectly
     reflecting surface is, of course, one lumen per square foot. Looking at it
     another way, if you have a one square foot diffuse source that has a
     luminance of one candle per square foot (pi times as much intensity as in
     the previous example), then the total output of this source is pi lumens.
     If you travel out a good distance along the normal to the center of this
     one square foot surface, it will look like a point source with an
     intensity of one candle.
To contrast:
     Intensity in candles is for a point source while luminance in candles per
     square foot is for an extended source - luminance is intensity per unit
     area. If it's a perfectly diffuse but not perfectly reflecting surface,
     you have to multiply by the reflectance, k, to find the luminance.
Also to contrast:
     Illumination, E, is for the incident or incoming flux's areal _density_;
     luminance, B, is for reflected or outgoing flux's areal _intensity_.
     Lambert's law says that an perfectly diffuse surface or extended source
     reflects or emits light according to a cosine law: the amount of flux
     emitted per unit surface area is proportional to the cosine of the angle
     between the direction in which the flux is being emitted and the normal to
     the emitting surface. (Note however, that there is no fundamental physics
     behind Lambert's "law". While assumiming it to be true simplifies the
     theory, it is really only an empirical observation whose accuracy varies
     from surface to surface. Lambert's law can be taken as a definition of a
     perfectly diffuse surface.)
     A consequence of Lambert's law is that no matter from what direction you
     look at a perfectly diffuse surface, the luminance on the basis of
     _projected_ area is the same. So if you have a light meter looking at a
     perfectly diffuse surface, it doesn't matter what the angle between the
     axis of the light meter and the normal to the surface is as long as all
     the light meter can see is the surface: in any case the reading will be
     the same.

There are a number of luminance units, but they are in categories: two of the
categories are those using English units and those using metric units. Another
two categories are those which have the constant 1/pi built into them and those
that do not. The latter stems from the fact that the formula to calculate
luminance (photometric Brightness), B, from illumination (illuminance), E,
contains the factor 1/pi. To illustrate:

B = (k*E)(1/pi)
Bfl = k*E

where:
B = luminance, candles/foot^2
Bfl = luminance, foot-Lamberts
k = reflectivity 0<k<1
E = illuminance in foot-candles (lumens/ foot^2)

Obviously, if you divide a luminance expressed in foot-Lamberts by pi you then
have the luminance expressed in candles /foot^2. (Bfl/pi=B)

Other luminance units are:

     stilb = 1 candle/square centimeter      sb
     apostilb = stilb/(pi X 10^4)=10^-4 L    asb
     nit = 1 candle/ square meter            nt
     Lambert = (1/pi) candle/square cm       L

Below is a table of photometric units with short definitions.

Symbol      Term                 Unit              Unit Definition

  Q      light quantity       lumen-hour          radiant energy
                              lumen-second        as corrected for
                                                  eye's spectral response

  F      luminous flux        lumen               radiant energy flux
                                                  as corrected for
                                                  eye's spectral response

  I      luminous intensity   candle              one lumen per steradian
                              candela             one lumen per steradian
                              candlepower         one lumen per steradian

  E      illumination         foot-candle         lumen/foot^2
                              lux                 lumen/meter^2
                              phot                lumen/centimeter^2

  B      luminance            candle/foot^2       see unit def's. above
                              foot-Lambert   =    (1/pi) candles/foot^2
                              Lambert        =    (1/pi) candles/centimeter^2
                              stilb          =    1 candle/centimeter^2
                              nit            =    1 candle/meter^2

Note: A lumen-second is sometimes known as a Talbot.

To review:

Quantity of light, Q, is akin to a quantity of photons except that here the
number of photons is pro-rated according to how bright they appear to the eye.
Luminous flux, F, is akin to the time rate of flow of photons except that the
photons are pro-rated according to how bright they appear to the eye.
Luminous intensity, I, is the solid-angular density of luminous flux. Applies
primarily to point sources.
Illumination, E, is the areal density of incident luminous flux.
Luminance, B, is the areal intensity of an extended source.

Photometry with a Photographic Light Meter

The first caveat to keep in mind is that the average unfiltered light meter
doesn't have the same spectral sensitivity curve that the human eye does. Each
type of sensor used has its own curve. Silicon blue cells aren't too bad. The
overall sensitivity of a cell is usually measured with a 2856K or 2870K
incandescent lamp. Less commonly it is measured with 6000K sunlight.

The basis of using a light meter is the fact that a light meter uses the
Additive Photographic Exposure System, the system which uses Exposure Values:

Ev = Av + Tv = Sv + Bv

where:
Ev = Exposure Value
Av = Aperture Value = lg2 N^2, where N = f-number
Tv = Time Value = lg2 (1/t), where t = time in sec.s
Sv = Speed Value = lg2 (0.3 S), where S = ASA speed
Bv = Brightness Value = lg2 Bfl

lg2 is logarithm base 2

from which, for example:
Av(N=f/1) = 0
Tv(t=1 sec) = 0
Sv(S=ASA 3.125) E
Bv( Bfl = 1 foot-Lambert) = 0

and therefore:
Bfl = 2^Bv
Ev (Sv = 0) = Bv

From the preceeding two equations you can see that if you set the meter dial to
an ASA speed of approximately 3.1 (same as Sv = 0), when you read a scene
luminance level the Ev reading will be Bv from which you can calculate Bfl. If
you don't have an ASA setting of 3.1 on your dial, just use ASA 100 and
subtract 5 from the Ev reading to get Bv. (Sv@ASA100=5)

Image Illumination

If you know the object luminance (photometric brightness), the f-number of the
lens, and the image magnification, you can calculate the image illumination.
The image magnification is the quotient of any linear dimension in the image
divided by the corresponding linear dimension on the object. It is, in the
usual photographic case, a number less than one. The f-number is the f-number
for the lens when focussed at infinity - this is what's written on the lens.
The formula that relates these quantities is given below:

Eimage = (t pi B)/[4 N^2 (1+m)^2]

or:
Eimage = (t Bfl)/[4 N^2 (1+m)^2]

where:

Eimage
     is in foot-candles (divide by .0929 to get lux)
t
     is the transmittance of the lens (usually .9 to .95 but lower for more
     surfaces in the lens or lack of anti-reflection coatings)
B
     is the object luminance in candles/square foot
Bfl
     is the object luminance in foot-Lamberts
N
     is the f-number of the lens
m
     is the image magnification

References:

G.E. Miniature Lamp Catalog
Gilway Technical Lamp Catalog
"Lenses in Photography" Rudolph Kingslake Rev.Ed.c1963 A.S.Barnes
"Applied Optics & Optical Engr." Ed. by Kingslake c1965 Academic Press
"The Lighting Primer" Bernard Boylan c1987 Iowa State Univ.
"University Physics" Sears & Zemansky c1955 Addison-Wesley

Acknowledgements

Thanks to John Bercovitz for providing the material on photometry and
illumination. Thanks to Andy Young for pointing out that Lambert's law is only
empirical. Thanks to John Bercovitz, donl, and Bill Tyler for reviewing an
earlier version of this file. I've made extensive changes since their review,
so any remaining bugs are mine, not a result of their oversight. All of them
told me it was too detailed. I probably should have listened.

Copyright (C) 1993, 1994, 1995 David M. Jacobson

Rec.photo.* readers are granted permission to make a reasonable number
electronic or paper copies for their themselves, their friends and colleagues.
Other publication, or commercial or for-profit use is prohibited.
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Last modified: Mon Mar 27 18:10:03 1995
