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