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- From: jacobson@cello.hpl.hp.com (David Jacobson)
- Newsgroups: rec.photo.moderated,rec.answers,news.answers
- Subject: Photographic Lenses Tutorial
- Supersedes: <5jhcv5$ng7@cello.hpl.hp.com>
- Followup-To: rec.photo.moderated
- Date: 21 May 1997 11:01:03 -0700
- Organization: Hewlett-Packard Laboratories
- Lines: 850
- Approved: news-answers-request@MIT.EDU
- Expires: 22 June 1997 06:00:00 GMT
- Message-ID: <5lvd8v$r5i@cello.hpl.hp.com>
- NNTP-Posting-Host: cello.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.
- Xref: senator-bedfellow.mit.edu rec.photo.moderated:1201 rec.answers:30897 news.answers:103138
-
- Archive-name: rec-photo/lenses/tutorial
- Posting-Frequency: monthly
- Last-modified 1996/10/26
- Version: 1.10
-
- Lens Tutorial
- by David M. Jacobson
- jacobson@hpl.hp.com
- Revised October 26, 1996
-
- 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 object (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.
-
-
- Object distance, image distance, and magnification
-
- Throughout this article we use the word "object" to mean the thing of
- which an image is being made. It is loosely equivalent to the word
- "subject" as used by photographers.
-
- 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 object distance to be measured from the
- front principal point, most focusing scales are calibrated to read the
- distance from the object 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 form a (probably virtual) erect 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 generally 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
- manufacturer's numbers enough for educational purposes.)
-
-
- So 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 object.
-
- 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. It could be behind the rear
- principal plane.)
-
- Beyond the conditions of the paraxial approximation, the principle
- "planes" are not really planes but surfaces of rotation. For a lens
- that is free of coma (one of the classical aberrations) the principle
- planes, which could more appropraitely be called equivalent refracting
- surfaces, are spherical sections centered around and object and image
- point for which the lens is designed.
-
-
- 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, based on geometric factors, but not absorption
-
- Light from an object point spreads out in a cone whose base is the
- entrance pupil. (The lens elements in front of the diaphragm form a
- virtual image of the physical aperture. This virtual image is called
- the entrance pupil.)
-
- Analogous to the entrance pupil is the exit pupil, which is the
- virtual image of the aperture as seen though the rear elments.
-
- Let us define Ne, the effective f-number, as
-
- Ne = 1/(2 sin(thetaX))
-
- where thetaX is the angle from the axis to the edge of the eXit pupil
- as viewed from the film plane. It can be shown that for any lens free
- of coma the following also holds
-
- Ne = M/(2 sin(thetaE)).
-
- We will ignore the issue of coma throughout the rest of this
- document.
-
- The first equation deals with rays converging to the image point, and
- is the basis for depth of field calculations. The second equation
- deals with light captured by the aperture, and is the basis for
- exposure calculations.
-
- This section will explain the connection between Ne and light striking
- the film, relate this to N, and show to compute Ne for macro
- situations.
-
- If an object radiated or reflected light uniformly in all directions,
- it is clear that the amount of light captured by the aperture would be
- proportional to the solid angle subtended by the aperture from the
- object point. In optical theory, however, it is common assume that
- the light follows Lambert's law, which says that the light intensity
- falls off with cos(theta), where theta is the angle off the normal.
- With this assumption it can be shown that the light entering the
- aperture from a point near the axis is proportional to sin^2(thetaE),
- which is proportional to the aperture area for small thetaE.
-
- If the magnification is M, the light from a tiny object patch of unit
- area gets spread out over an area M^2 on the film, and so the relative
- intensity on the film is inversely proportional to M^2. Thus the
- relative intensity on the film, RI, is given by
-
- RI = sin^2(thetaE)/M^2 = 1/(4 Ne^2)
-
- with the second equality by the defintion of Ne. (Of course the true
- intensity depends on the subject illumination, etc.)
-
- For So very large with respect to f, M is approximately f/So and
- sin(thetaE) is approximately (D/2)/So. Substituting these into the
- above formula get that RI = ((D/2)/f)^2 = 1/(4N^2), and thus for So >> f,
-
- Ne = D/f = N.
-
- For closer subjects, we need a more detailed analysis. We will take
- D = f/N as determining D,
-
- Let us go back to the original approximate formula for the relative
- intensity on the film, and substitute more carefully
-
- RI = sin^2(thetaE)/M^2 = ((D/2)^2/((D/2)^2+(So-zE)^2))/M^2
-
- where zE is the distance the entrance pupil is in front of the front
- principal point.
-
- However, zE is not convenient to measure. It is more convenient to
- use "pupil magnification". 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. It can be shown that zE = f*(1-1/p), and substituting this
- into the above equation, carrying out some algebraic manipulations,
- and solving with RI = 1/(4 Ne^2) yields
-
- Ne = Sqrt((M/2)^2 + (N*(1+M/p))^2).
-
- If we further assume thetaE is small enough that
- sin(thetaE) ~= tan(thetaE), the (M/2)^2 term drops out and we get
-
- Ne = N*(1+M/p).
-
- This is the standard equation, and will be used throughout the rest of
- this document. The essence of the approximation is the distinction
- between the axial distance to plane of the entrance pupil and the
- distance along the hypotnuse to the edge of the entrance pupil, which
- is the really correct form. Clearly in typical photographic sitations
- that distinction is insignificant.
-
- An alternative, but less fundamental, derivation is based on the
- relative illumination varying with the inverse square of the distance
- from the exit pupil to the film. This distance is just f*(1+M) - zX,
- where zX is the distance the exit pupil is behind the rear principal
- point. It can be shown that zX = -f*(p-1), so
- Ne/N = (f*(1+M)+f*(p-1))/(f+f*(p-1)) = 1+M/p, and hence Ne = N*(1+M/p).
-
- 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 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 object 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 object 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 object" that is perfectly in
- focus, and an "alternate object point" this is in front of or
- behind the main object.
-
- Soa alternate object 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 object
- 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 object point is in front of
- the main object) 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 object is at a finite distance, the closest
- alternative point that is acceptably rendered is 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. ("!=" means "is not equal to".)
-
- 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 an object the same way with two different lenses,
- i.e. M is the same 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 object 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 circles of
- confusions as a function of the distance behind the object.
-
- 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 object (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 object 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 is 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 sensitivity 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 a dark band receive no more than
- 80% of the light power striking the center of a light band.
- 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
- definitions of the modulation transfer function and of the related
- optical transfer function vary 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 object contrast of an
- object 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, object 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 the 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 formulas for and
- plots of the OTFs three cases:
- 1. Diffraction for an f/22 aperture.
- 2. A .03mm circle of confusion
- 3. The combination of these; more precisely, the OTF of an otherwise
- ideal lens with an f/22 aperture and defocused to produce a .03mm
- circle of confusion.
-
- 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. The OTF goes
- negative at certain frequencies. Physically, this would mean that if
- the test pattern were lighter right on the optical center than nearby,
- the image would be darker right on the optical center than nearby.
- Some authorities use the term "spurious resolution" for spatial
- frequencies beyond the first zero. The MTF is the absolute value of
- the OTF.
-
- Consider the case where there is a combination of diffraction and
- focus error dz, the distance the between the film plane and the plane
- of sharpest focusing. (A focus error of dz by itself would cause a
- circle of confusion of diameter dz/N.) The OTF for this combination
- is given by the following formula, which involves an integration that
- 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
- </= 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
- assuming 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, donl mathis, 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. Thanks to Andy Young for pointing out that
- Lambert's law is only empirical. Thanks to John Bercovitz for
- providing the material on photometry and illumination, for helping me
- improve the presentation, and for nudging me into dusting it off from
- time to time and touching it up.
-
- Copyright (C) 1993, 1994, 1995, 1996 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.
-