Photographic and geometrical optics
INTRODUCTION
In Chapter 2 we were acquainted with the behaviour of light, mainly in terms of its wave and particle nature, which are the subjects of physical and quantum optics respectively. This chapter is largely concerned with geometrical optics, the part of optics dealing with light’s propagation in terms of rays, i.e. how it travels in straight lines, also known as rectilinear propagation. The concept of light rays is helpful in studying the formation of images by lenses. Further, in many light-related phenomena, such as reflection, refraction and transmission, the properties of light are altered according to different physical properties of the optical media with which they interact. This is the subject of photographic optics in this chapter. Finally, the photometry of image formation will be discussed at the end of the chapter.
PHOTOGRAPHIC OPTICS
Optical materials
Optical elements are made from various transparent and reflective materials, particularly glasses, plastics and crystals. Optical glass is used for photographic lenses, being highly transparent with suitable refractive properties. Early glasses were of the crown and flint varieties, but pioneering work by Abbe, Schott and Zeiss in the 1880s gave many other types by adding elements such as barium, boron, phosphorus, lanthanum and tantalum. Lenses use glasses of both low and high refractive index and low and high dispersion (see Chapter 2). Glasses are characterized by their refractive index (n) and dispersion measured as the Abbe number, V, and location on a ‘glass map’, a graph of n against V. Additional properties are important such as resistance to staining and density. Glass is manufactured by melting the raw materials in platinum crucibles to avoid contamination, and then casting or pressing in moulds to approximate shapes or ‘blanks’. It is essential that glass is homogeneous in its properties. Optical glass should ideally be spectrally non-selective as regards absorption but lower grade glass as may be used for condenser lenses absorbs slightly in the red and blue, giving a greenish tinge to the element. This may not matter in an illumination system. Addition of suitable minerals gives ‘dyed-in-the-mass’ coloured glass suitable for absorption filters.
Plastics (polymers) include transparent types for imaging systems, translucent and coloured types for filters and diffusers, and opaque varieties for constructional purposes. The first plastic lens of 1934 was of thermo-plastic polymethylmethacrylate (PMMA) and shaped by moulding. Other varieties are used to provide chromatic correction, e.g. PMMA and polystyrene. Plastics can be moulded and extruded in complex shapes, including fibres and aspheric surfaces as well as in large sizes. They can be surface coated and used in hybrid glass–plastic lenses. Disadvantages include low refractive index, birefringence, poor scratch resistance and focus shifts due to thermal effects. One material, allyl diglycol carbonate (coded as CR-39), is used as an ‘optical resin’ for accessories such as filters, using dyes to produce uniform or graduated tones and colours. Constructional plastics have properties of low friction and dimensional accuracy suitable for moulded devices such as lens barrels, focusing mounts and camera bodies. Materials such as polycarbonate may be made opaque by addition of graphite. Thin films of gelatin, collodion and nitrocellulose are used for flexible ‘gel’ filters and pellicle beamsplitters.
Crystalline materials have optical uses. Iceland spar and herapathite were used for polarizing optics until replaced by polymeric colloids. Natural mica is used for quarter-wave plates (see Chapter 2). Natural and synthetic quartz are useful for heat-resistant elements such as condensers and for UV optics. Fluorite or calcium fluoride grown as large crystals has useful properties for chromatic correction and UV lenses.
Reflection
Light interacts with materials in various ways, many of which are introduced in Chapter 2. When light encounters an opaque surface, specular and diffuse reflections occur (Figure 6.1). A perfectly diffuse surface reflects the incident light equally in all directions so that its brightness is seen as constant, irrespective of viewpoint. Few surfaces have such properties and there is usually a slight sheen depending on the degree of gloss. Most surfaces have a reflectance, R, in the range 0.02–0.9 (2–90%) represented by matt black paint and a typical white paper respectively. A glossy surface gives little scattering and a specular reflection of the light source is seen. The incident light is reflected at an angle equal to the angle of incidence, both measured from the normal to the surface. Suppression of reflections may be possible using the properties of polarized light (see Chapter 2, Light polarization).
Refraction
In Chapter 2 we saw the refraction of light travelling between transmitting media with different refractive indices. For a specific wavelength, λ, the ratio of velocities in a vacuum and the medium gives the refractive index, nλ, of the medium. A decrease in velocity causes the light ray to be deviated towards the normal and vice versa. The relationship between the angle of incidence, i, angle of refraction, r (Figure 6.1), and the refractive indices n1 and n2 of the two media is given by Snell’s Law (also known as the Law of Refraction):
Figure 6.1 An oblique incident light ray undergoing refraction when passing from air to glass.
Refractive index varies with wavelength, producing chromatic aberrations. The usual value is quoted for the d-line in the helium spectrum at 587 nm, denoted nd. Gases and liquids change in density with variations in temperature or pressure, causing refraction changes.
A plane parallel-sided block of optical glass does not deviate an incident oblique ray but the emergent ray is laterally displaced. This can cause image aberrations or a focus shift in the case of glass filters. Refraction of a ray emerging from a dense medium increases as i increases until a critical value is reached, when the ray does not emerge as it has undergone total internal reflection (TIR). This property is used in mirror prisms and fibre-optic devices.
Transmission
The energy budget of light for a lens is R + A + T = 1, where R, A and T are surface reflectance, absorption by the medium and transmittance respectively. To maximize T, both R and A must be minimized.
Surface reflections are reduced by thin coatings of material applied by vacuum deposition and other methods. The principle for a single anti-reflection layer is that reflections from the air–layer and layer–medium interfaces interfere destructively to enhance transmission. The condition is that the coating has thickness of λ/4 and index √n. In practice, magnesium fluoride is used for a single hard coating, giving a characteristic purple appearance to the surface by reflected light. Reflectance is reduced to under 2%. A double layer gives less than 1% (Figure 6.2), while multiple layer ‘stacks’ of coatings with layers of differing thicknesses and refractive indices can reduce values to much less than 1%. Some 7–15 layers may be used as needed.
Multiple coatings can also be used to enhance reflectance as in mirrors for SLR cameras, or can be spectrally very selective and reflect (or transmit) very narrow spectral bands with sharp cut-off. These are termed interference filters and find use in applications such as multi-spectral photography and colour printing. Multi-layer coatings can also be used to make infrared transmitting or reflecting filters, often called ‘cold’ or ‘hot’ mirrors respectively.
Transmittance, T, of an optical system depends also on the aperture stop, which is important in determining precise exposures. It is usually measured as the f-number, N, which is defined geometrically (see later in the chapter, also Chapter 2), but the true f-number may be significantly different.
Figure 6.2 The changes in surface reflection using various types of anti-reflection coatings as compared with uncoated glass (for a single lens surface at normal incidence).
Dispersion
We saw in Chapter 2 that refraction of light decreases with increase in wavelength (dispersion), but it does so in a non-linear fashion. A variety of glasses are needed in lens design for compensation. Dispersion by simple glass lenses causes two distinct forms of chromatic or colour aberrations. Due to axial or longitudinal chromatic aberration (LCA), the focal length, f, increases with wavelength, λ (fλ), and is corrected by combining two suitable positive and negative lenses of different glasses, to equalize f for two wavelengths such as the Fraunhofer F and C lines for the visible spectrum, giving an achromatic lens. This achromatic doublet has a residual uncorrected secondary spectrum, illustrated in Figure 6.3, further reduced by using three glasses and optimized for three wavelengths, termed apochromatic, although this may just mean a reduced secondary spectrum by the use of special anomalous (extraordinary) dispersion glasses, termed ED glasses. Chromatic correction does not usually include the IR and UV regions, and a focus correction must be made after visual focusing to allow for fλ in these regions. The focusing scale may have an infrared focusing index. A special form of correction, superachromatic, extends correction from 400 to 1000 nm.
By contrast, transverse chromatic aberration (TCA), or lateral colour, illustrated in Figure 6.4, is dispersion of obliquely incident rays and the ‘colour fringing’ effects observed worsen rapidly as the image point moves off-axis. It is a difficult aberration to correct, but for general-purpose lenses, symmetrical construction is effective. Lateral colour sets the performance limits to long focus refracting lenses, but fluorite and ED glass elements give effective correction.
Reflecting elements such as full or partial mirrors do not disperse light and mirror lenses can be used for long-focus lenses as well as for UV and IR work and microscopy. Pure mirror designs are called catoptric and are seldom suitable for photography. Additional refracting elements are needed for full aberration correction, giving catadioptric lenses.
Figure 6.3 Types of chromatic correction for lenses. The letters on the vertical axis represent the wavelengths of the Fraunhofer spectral lines used for standardization purposes. Residual aberration is shown as a percentage change in focal length.
Polarization
In Chapter 2 we explained the physical mechanisms and types of light polarization. In photographic applications linear or plane polarized light can be converted into circularly polarized light by means of a quarter-wave plate, which is a birefringent material of specific thickness that splits the plane polarized light into two equal perpendicular components with a phase difference of λ/4. The emergent beam proceeds with a circular helical motion. Left- and right-handed circular polarization is possible. Circularly polarizing filters have a specialized photographic application with some SLR cameras which use metal film beamsplitters to sample the light from the lens for exposure determination or for an autofocus system, as illustrated in Figure 6.5. The beamsplitter varies its percentage split of an incident beam depending on its amount of linear polarization, which in turn can affect exposure determination or light reaching the autofocus module. Circularly polarized light is unaffected. A linear polarizing filter is very useful in controlling surface reflections that become partly or wholly polarized from dielectric materials – that is, most materials except metals. Colour saturation can be improved. Blue skies may be darkened as scattered light is polarized relative to the sun’s position.
Figure 6.4 Lateral (or transverse) chromatic aberration (TCA) for off-axis points in a lens that has been corrected for longitudinal chromatic aberration (LCA) only. The effect is a variation in image point distance y from the axis for red and blue light.
Figure 6.5 Polarized light and beamsplitters. (a) With natural light (N) a beamsplitter (B) reflects light to the viewfinder (V) and transmits to a photocell or autofocus module (C) in ratio 3:1. (b) With linear polarizing filter (λ) in 45° position, the ratio is 62:38. (c) With linear polarizing filter in 135° position, the ratio is 88:12. (d) With circular polarizing filter made by addition of quarter wave plate Q, the ratio is constant at 3:1 for all positions.
GEOMETRICAL OPTICS
Simple lens
A simple lens consists of a single element with two surfaces. One surface may be flat. A curved surface may be concave, convex or possibly aspheric. The focal length, f, and power, P, of a lens with refractive index nd are given by the lens makers’ formula:
where r1 and r2 are the radii of curvature of the surfaces, being positive measured to the right of the vertex of the surface and negative measured to the left. Depending on the values for r1 and r2, focal length can be positive or negative. In terms of image formation, a positive or converging lens refracts parallel incident light to deviate it from its original path and direct it towards the optical axis and form a real image that can be focused or projected on a screen. A negative or diverging lens deviates parallel light away from the axis and a virtual image is formed, which can be seen or can act as a virtual object for another lens but cannot itself be focused on a screen. The location, size and orientation of the image may be determined by simple calculations. The object and image distances from the lens are termed conjugate distances and commonly denoted by the letters u and v respectively; see Figure 6.6.
Cardinal planes
A simple thin lens is normally unsuitable as a photographic lens due to aberrations. A practical lens consists of a number of separated elements or groups of elements, the physical length of which is a significant fraction of its focal length. This is a compound, thick or complex lens. The term equivalent focal length (EFL) is often used to denote the composite focal length of such a system. For two thin lenses of focal lengths f1 and f2, the EFL, f, is given by:
Figure 6.6 Image formation by a positive lens (a, b) and a negative lens (c, d). (a) For a distant subject. F is the rear principal focal plane. (b) For a near subject. Focusing extension E = (v − f); I is an inverted image. (c) For a distant subject ‘O:F’ is the front principal focal plane. ‘I’ is virtual, upright image. (d) For a near subject.
where d is their axial separation.
Thin lens formulae can be used with thick lenses if conjugate measurements are made from two specific planes perpendicular to the optical axis. These are the first and second principal planes that are planes of unit transverse magnification. For thin lenses in air the two planes are coincident. For thick lenses in air the principal planes are separated and coincident with two other planes, the first and second nodal planes (N1 and N2) (Figure 6.7). An oblique ray incident at N1 emerges undeviated on a parallel path from N2. The terms principal and nodal are used interchangeably but when the surrounding optical medium is different for object and image spaces, such as an underwater lens with water in contact with its front element, then the two pairs of planes are not coincident. Focal length is measured from the rear nodal point. For a symmetrical lens, the nodal planes are located approximately one-third of the way in from the front and rear surfaces. Depending on lens design and configuration they may or may not be crossed over or even located in front of or behind the lens. The location of nodal planes can be used to classify major types of lenses (see Figure 6.8).
The third pair of these cardinal or Gaussian planes are the front and rear focal planes through the front and rear principal foci on the axis. Gaussian or paraxial optics are calculations involving subjects located close to the optical axis and small angles of incidence. Detailed information about the image and its aberrations is not given; instead, ray path traces using Snell’s Law at each surface are necessary. The focal ‘plane’ is curved slightly even in highly corrected lenses and is only approximately planar close to the optical axis.
Focal length
A thin, positive lens converges parallel incident light to the rear principal focus, F2, on axis. The distance from this point to the optical centre of the lens is the focal length, f. For a thick lens, focal length is measured from N2. Parallel light from off-axis points is also brought to an off-axis focus at the rear principal focal plane. As light is reversible in an optical system there is also a front principal focus, F1. The closest to a lens an image can be formed is at F2. An object inside F1 gives a virtual image, as formed by a simple microscope, magnifier or loupe. As u decreases from infinity the focus recedes from the lens so, in practical terms, the lens is shifted away from the film plane to focus on close objects. Parameters u, v and f are related by the lens conjugate equation:
Figure 6.7 Image formation by a compound lens. F1, F2, front and rear principal foci; FP, image focal plane; FP1, FP2, first and second principal focal planes; N1, N2, first and second nodal planes (also principal planes).
Figure 6.8 Nodal planes and lens configurations. B, back focal distance; f, focal length; F, photoplane; N1, N2, first and second nodal planes. (a) Simple or long focus lens, B ≈ f. (b) Short focus symmetrical lens, B < f. (c) Quasi-symmetrical wide-angle lens, B << f. (δ) Telephoto lens, B < f. (d) Retrofocus wide-angle lens, B > f.
A Cartesian sign convention is often suitable with the lens taken as the origin so distances measured to the right are positive and to the left are negative. Alternatively, a useful convention is that ‘real is positive and virtual is negative’, and useful derived equations include:
where image magnification, m, is defined as:
For large values of u, v ≈ f, hence m ≈ f/u. So focal length in relation to a film format determines the size of the image.
Focal length can be specified also as dioptric power, P, of the lens, where, for f in millimetres:
Normal sign convention is applicable. A photographic lens of focal length 100 mm has a power of +10 D and one of 50 mm a power of +20 D. This term is used in optometry (measurement of visual performance in humans) and also to specify a close-up lens (see Chapter 10), although other numbering systems may be used. Powers are additive, so two thin lenses of powers +1 and +2 D placed in contact give a combined power of +3 D.
Image magnification
Magnification gives the image size relative to object size. Since in most practical photography the image is very much smaller than the subject, the magnification is less than unity, except in photomacrography (close-up photography – see Chapter 10), where the image is unity or greater. The term ratio of reproduction is often preferred, giving a quantity such as 1:50 instead of magnification 1/50 or 0.02. For a focal systems such as lens attachments and telescopes, either magnifying power or angular magnification is used instead.
From imaging geometry, three types of magnification are definable: lateral or transverse, m, longitudinal, L, and angular, A, with the relationship AL = m. As defined, m = I/O = v/u (see Eqn 6.9) and L = m2. Transverse magnification, m, is mostly used in photography. Note that for m = 0.02, L is 0.0002, so that if depth of field is 3 m, depth of focus will be 1.2 mm – definitions for depth of field and depth of focus are given later in this chapter.
An ideal lens gives a constant magnification in a given perpendicular image plane, but radial variations cause both barrel and pincushion distortion (see Chapter 10), while variation with λ causes lateral colour. Magnification can be determined by direct measurement of object and image dimensions or by use of a scaled focusing screen or a scale on the focusing mount of the lens. Measurements derived from conjugates are more difficult as the exact location of the nodal planes may not be known. Alternatively if m is known, the values of u and v can be calculated from derived equations. For m ≥ 0.1, exposure corrections are usually necessary to the values indicated by hand-held light meters.
Mirrors
Plane, spherical, aspherical, faceted and compound mirrors are used as optical elements in various imaging and illumination systems. A plane (flat) mirror gives an image that is virtual, upright, diminished and laterally reversed, located at the same distance behind the mirror as the subject is in front. Plane mirrors deviate the direction of the incident light and have a refractive index of 2 for optical calculations. A series of angled mirrors can be used to invert or laterally correct an image. A mirror can be surface silvered, or be immersed, as with the internal surface of a prism. Rotation of a mirror through an angle K rotates an incident beam through an angle 2K.
Spherical mirrors may be concave or convex and have different imaging properties. Convex mirrors have a negative power and give an erect, minified, virtual image. Paraxial imaging behaviour is given by the mirror formula:
where r is the radius of curvature and f = r/2.
Aspherical mirrors with ellipsoidal or paraboloidal shapes are useful in imaging systems by virtue of the way they focus light. For large r, a spherical mirror approximates to a paraboloid for paraxial imagery.
Figure 6.9 Imaging geometry of a photographic lens. A, object point; A′, conjugate image point; e, focusing extension; EN, entrance pupil; EX, exit pupil; h, w, format height and width; J1, image circle; J2, extra covering power; I, conjugate image plane; N1, N2, nodal planes; O, focused object plane; T, depth of field from planes S′ to R′; θ, semi-field angle of view.
OPTICAL CALCULATIONS
Parameters
The various parameters involved in the geometry of imaging by a photographic lens are shown in Figure 6.9. Information about an image or its optical parameters may be calculated using equations derived from Gaussian or paraxial optics. Special calculators may be used if available. It is assumed that no aberrations are present. Conjugate distances are measured from the first and second principal or nodal planes (points) as appropriate. The symbols used for the optical parameters are defined in Table 6.1.
Equations
The fundamental equation is the lens conjugate equation 6.4 and, assuming real is positive and virtual is negative, various rearrangements are often more convenient to use.
1. Focal length is calculated from:
2. Object conjugate distance is calculated from:
3. Image conjugate distance is calculated from:
4. Magnification and reduction ratio are calculated from:
5. Conjugate sum (u + v) is calculated from:
6. Object size is calculated from:
Table 6.1 Symbols used for optical parameters
SYMBOL |
OPTICAL PARAMETER |
f |
Focal length |
f1 |
Focal length of component 1 in a multi-element system, etc. |
F |
Effective focal length (EFL) of a system with several components |
s |
Axial separation of two optical components in a system |
P |
Optical power of a lens in dioptres |
u |
Object conjugate distance |
up |
Object distance measured from the focal plane |
us |
Object distance for lens focused at distance u, with a supplementary lens |
v |
Image conjugate distance |
D |
Separation between object and image, i.e. sum of conjugates (u + v), ignoring internodal space K |
K |
Internodal or nodal space |
O |
Linear size of object |
I |
Linear size of image |
m |
Optical (image) magnification = I/O = v/u = 1/R |
M |
Print magnification, i.e. degree of enlargement |
R |
Reduction ratio = 1/m = O/I = u/v |
d |
Effective clear diameter of a lens |
N |
Relative aperture or f-number |
N′ |
Effective aperture or effective f-number |
C |
Diameter of circle of confusion |
H |
Hyperfocal distance |
e |
Movement of lens along axis, i.e. focusing extension |
t |
Depth of focus |
T |
Total depth of field = S′ – R′ |
S′ |
Far limit of depth of field |
R′ |
Near limit of depth of field |
7. Image size is calculated from:
8. Effective focal length of two lenses combined:
9. Focal length of one component:
(f1 and f2 are positive when F is positive).
If f1 is known in dioptres (e.g. a supplementary lens), then:
10. Axial separation. Necessary separation for required F is:
11. For near contact to give a small s, approximations are:
12. Object distance with supplementary lenses. When the camera lens is focused on infinity, and a supplementary lens f2 is added, then:
The distance is independent of f1 or F or S. If the camera is focused on nearer objects this does not apply and the combination has to be calculated.
Focusing movements
The forward movement of a lens to focus sharply a near object can be defined in terms of u and f. This distance e is measured from the infinity focus position. Object distance can be determined from f and e. Different formulae apply when object distance is measured from the focal plane and not the front node.
13. For u measured from front node:
14. For u measured from the focal plane:
15. Magnification and camera extension:
16. Focusing movement with supplementary lenses. If F is computed as above, then this can substitute for f in the equations listed. For a camera lens fitted with supplementary lens f2, and focused not on infinity but a near distance u, then the object distance us is given by the approximate formula:
Depth of field and depth of focus
Relevant equations are based on the premise that some degradation of the image is permissible for objects not sharply in focus. The criterion of permissible unsharpness is the diameter, C, of the circle of confusion. Depth of focus, t, is the permissible tolerance in the distance between a lens and the sensitized material:
For near objects, this becomes:
Depth of field, T, is given as the distance between the near and far points in acceptably sharp focus when the lens is focused on a subject distance that is not infinity:
For practical purposes with distant scenes the second term in the denominator may be disregarded, so to a first approximation:
Figure 6.10 shows the effect of focal length, focused distance and f-number on practical depth of field.
Hyperfocal distance, H, is the focus setting, u, which gives S′ as infinity. Use of H simplifies depth of field equations:
Close-up depth of field
When the lens is focused at close distances the depth of field is small and it is sufficient to calculate the total depth of field between the near and far limits rather than each separately:
Figure 6.10 Depth of field (J). Effect of the variables of focal length (f), f-number (N) and focused distance (u) at a constant value for the circle of confusion. (a) Lens aperture varying from f/1.4 to f/16 with a 50 mm lens focused at 5 m. (b) Focused distance varying from 0.5 to 3 m with 50 mm lens at f/5.6. (c) Focal length varying from 28 to 200 mm at f/5.6 focused on 5 m.
THE PHOTOMETRY OF IMAGE FORMATION
The light-transmitting ability of a photographic lens, which principally determines the illuminance of the image at the film plane, is usually referred to as the speed of the lens. Together with the effective emulsion ‘speed’ or photosensor sensitivity and the subject luminance, this determines the exposure duration necessary to give a correctly exposed result.
Stops and pupils
An optical system such as a camera and lens normally has two ‘stops’ located within its configuration. The term stop originates from its original construction of a hole in an opaque plate perpendicular to the optical axis. The field stop, which is the film gate or photosensor array, determines the extent of the image in the focal plane and also the effective field of view of the lens. The aperture stop is located within the lens or close to it, and controls the transmission of light by the lens. It also influences depth of field and lens performance, in terms of the accuracy of drawing and resolution in the image (its ability in reproducing detailed information in the original scene).
In a simple lens the aperture stop may either be fixed, or only have one or two alternative settings. A compound lens is usually fitted with a variable-aperture stop called an iris diaphragm. This is continuously adjustable between its maximum and minimum aperture sizes using five or more movable blades, providing a more or less circular opening. The aperture control ring may have click settings at whole and intermediate f-number values. In photography the maximum aperture as well as the aperture range is important, as aperture choice is a creative control in picture making.
In a compound lens with an iris diaphragm located within its elements, the apparent diameter of the iris diaphragm when viewed from the object point is called the entrance pupil, EN. Similarly, the apparent diameter of the iris diaphragm when viewed from the image point is called the exit pupil, EX. These two are virtual images of the iris diaphragm, and their diameter is seldom equal to its actual diameter. To indicate this difference the term pupil factor or pupil magnification, p, is used. This is defined as the ratio of the diameter of the exit pupil to that of the entrance pupil:
Symmetrical lenses have a pupil factor of approximately 1, but telephoto and retrofocus lenses have values that are respectively less than and greater than unity. Pupil magnification influences image illuminance. The pupils are not usually coincident with the principal planes of a lens; indeed, for telecentric lenses the exit pupil can be at infinity.
Aperture
The light-transmitting ability of a lens, usually referred to as aperture (due to the control exercised by the iris diaphragm) is defined and quantified in various ways. Lenses are usually fitted with iris diaphragms calibrated in units of relative aperture, a number N, which is defined as the equivalent focal length f of the lens divided by the diameter d of the entrance pupil (N = f/d) for infinity focus (see also Chapter 2), so a lens with an entrance pupil 25 mm in diameter and a focal length of 50 mm has a relative aperture of 50/25, i.e. 2. The numerical value of relative aperture is usually prefixed by the italic letter f and an oblique stroke, e.g. f/2, which serves as a reminder of its derivation. The denominator of the expression used alone is usually referred to as the f-number of the lens. Aperture value on many lenses appears as a simple ratio, so the aperture of an f/2 lens is shown as ‘1:2’.
The relative aperture of a lens is commonly referred to simply as its ‘aperture’ or even as the ‘f-stop’. The maximum aperture is the relative aperture corresponding to the largest diaphragm opening that can be used with it.
To simplify exposure calculations, f-numbers are selected from a standard series of numbers, each of which is related to the next in the series by a constant factor calculated so that the amount of light passed by the lens when set to one number is half that passed by the lens when set to the previous number, as the iris diaphragm is progressively closed. As the amount of light passed by a lens is inversely proportional to the square of the f-number, the numbers in the series increase by a factor of √2, i.e. 1.4 (to two figures). The standard series of f-numbers is f/1, 1.4, 2, 2.8, 4, 5.6, 8, 11, 16, 22, 32, 45 and 64. Smaller or larger values are seldom encountered. The maximum aperture of a lens may lie between two of the standard f-numbers above, and will be marked with a number not in the standard series.
Other series of numbers have been used in the past and may be encountered on older lenses and exposure meters. In cameras offering shutter priority and program modes, the aperture setting may be unknown. A change in exposure by a factor of 2 either larger or smaller is referred to as a change of ‘one stop’ or exposure value (EV). Additional exposure by alteration of ‘one-third of a stop’, ‘half a stop’ and ‘two stops’ refer to exposure factors of 1.26, 1.4 and 4 respectively. When the lens opening is made smaller, i.e. the f-number is made numerically larger, the operation is referred to as ‘stopping down’. The converse is called ‘opening up’.
The aperture of a lens is defined in terms of a distant axial object point. However, lenses are used to produce images of extended subjects, so that a point on an object may be well away from the optical axis of the lens, depending on the field of view. Obstruction may occur because of the type and design of the lens, its axial length and position of the aperture stop, and the mechanical construction of the lens barrel. The effect is to reduce the amount of light reaching the focal plane, termed mechanical vignetting or ‘cut-off’ or ‘shading’, which increases as field angle increases causing darkening at the edges of images, and is one factor determining image illuminance as a function of field angle and hence the circle of illumination. Mechanical vignetting must not be confused with the natural fall-off of light due to the geometry of image formation (see later), although both effects occur together. Vignetting may be reduced by designing lenses with oversize front and rear elements, and by careful engineering of the lens barrel. The use of an unsuitable lens hood or two filters in tandem, especially with a wide-angle lens, also causes cut-off and increases peripheral darkening. Vignetting is measured of a decrease in EV with respect to the axial EV and is a particular problem for photosensors.
Image luminance and illumination
A camera lens projects an image formed as the base of a cone of light whose apex is at the centre of the exit pupil (centre of projection). It is possible to derive an expression for the illumination or illuminance at any point in the image of a distant, extended object. The flux F emitted by a small area S of a uniformly diffusing surface (i.e. one that appears equally bright in all directions) of luminance L into a cone of semi-angle u is given by the equation:
Note that the flux emitted is independent of the distance of the source. For an object and an image that are both uniformly diffuse, and whose luminances are L and L′respectively, with an ‘ideal’ lens of transmittance T, then L′ = TL. So the image luminance of an aerial image is the same as the object luminance apart from a small reduction factor due to the transmittance of the lens, within the cone of semi-angle ω. Viewing the aerial image directly gives a bright image, but this cannot easily be used for focusing, except by passive focusing devices or no-parallax methods. Instead, the image has to be formed on a diffuse surface such as a ground glass screen and viewed by scattering of the light. The processes of illumination cause this image to be much less bright than the direct aerial image.
Image illuminance
The projected image formed at the photoplane suffers light losses from various causes, so the image illumination or illuminance, E, is reduced accordingly. It can be shown that the axis is given by:
Equation 6.48 shows the factors influencing image illuminance and exposure. The value of E is independent of u, the subject distance, although the value of v is related to u by the lens equation. The axial value of illuminance is given when θ = 0; then cos θ = 1 and cos4 θ = 1. Hence E = TLA/v2. Now lens area A is given by:
so that:
For the subject at infinity, v = f. Bydefinition, the relative aperture N is given by N = f/d. By substitution we thus have:
This shows that, for a distant subject, on the optical axis in the focal plane, E is inversely proportional to N2. Hence image illuminance is inversely proportional to the square of the f-number. For two different f-numbers N1 and N2, the ratio of corresponding image illuminances is given by:
So it is possible to calculate that the image illuminance at f/4 is one-quarter of the value at f/2.
The exposure H received by a film or a digital sensor during exposure duration t is given by:
and so for a fixed exposure time, as H is proportional to E, then:
Also, as E is inversely proportional to t, so:
hence the exposure times t1 and t2 required to produce equal exposures at f-numbers N1 and N2 respectively are given by:
Also, E is inversely proportional to d2. In other words, image illuminance is proportional to the square of the lens diameter, or effective diameter of the entrance pupil. Thus, by doubling the value of d, image illuminance is increased fourfold. Values may be calculated from:
To give a doubling series of stop numbers, the value of d is altered by a factor of √2, giving the standard f-number series.
An interesting result also follows from Eqn 6.51. By suitable choice of photometric units, such as by taking – in lux (lumen m−2) and L in apostilbs (l/p cd m−2), we have:
So for a lens with perfect transmittance, i.e. T = 1, the maximum possible value of the relative aperture N is f/0.5, so that E = L. Values close to f/0.5 have been achieved in special lens designs.
When the object distance u is not large, i.e. in close-up photography or photomacrography, then the effective aperture N′ = N (1 + m), so:
Usually it is preferable to work with f-number, N, which is marked on the aperture control of the lens, and magni-fication, m, which is often known or set. In addition for photomacrography, when non-symmetrical lenses and, particularly, retrofocus lenses may be used in reverse mode, the pupil factor p = EX/EN must also be taken into account for the effective aperture. The relationship is that N′ = N (1 + m/p) and is due to the non-correspondence of the principal planes from which u and v are measured and the pupil positions from which the image photometry is derived. By substitution:
For image illumination off-axis, θ is not equal to 0. Then cos4 θ has a value less than unity, rapidly tending to zero as θ approaches 90°. Also, a vignetting factor, V, is needed to allow for vignetting effects by the lens with increase in field angle. So the equation for image illuminance, allowing for all factors, is now:
From Eqn 6.61, E is proportional to cos4 q. This is the cos4θ law of illumination, or ‘natural vignetting’ or ‘shading’, due to the combined effects of the geometry of the imaging system, the inverse square law of illumination and Lambert’s cosine law of illumination. The effects of this law are that even a standard lens with a semi-angle of view of 26° has a level of image illuminance at the edge of the image of only two-thirds of the axial value. For an extreme wide-angle lens with a semi-angle of view of 60°, peripheral illuminance is reduced to 0.06 of its axial value.
For wide-angle lens designs, corrective measures are necessary to obtain more uniform illumination over the image area. A mechanical method is a graduated neutral density filter or spot filter, in which density decreases nonlinearly from a maximum value at the optical centre to near zero at the periphery. This can provide a fairly precise match for illumination fall-off. Such filters are widely used with wide-angle lenses of symmetrical configuration. There is a penalty in the form of a +2EV exposure correction factor. Oversize front and rear elements are also used to minimize mechanical vignetting. Illumination fall-off can be reduced if the lens has outermost elements that are negative and of large diameter. Lens designs such as quasi-symmetrical lenses with short back foci, and also retrofocus lenses, both benefit from this technique. The overall effect is to reduce the ‘cos4 θ effect’ to roughly cos3 q. The theoretical consideration of image illuminance applies only to well-corrected lenses that are free from distortion. If distortion correction (which becomes increasingly difficult to achieve as angle of field increases) is abandoned, and the lens design deliberately retains barrel distortion so that the light flux is distributed over proportionally smaller areas towards the periphery, then fairly uniform illuminance is possible even up to angles of view of 180° or more. Fisheye lenses are examples of the application of this principle. The relationship between the distance y of an image point from the optical axis changes from the usual:
of an orthoscopic lens to:
for this type of imagery (θ is given in radians).
Exposure compensation for closeup photography
The definition of relative aperture, N, assumes that the object is at infinity, so that the image conjugate v can be taken as equal to the focal length f. When the object is closer this assumption no longer applies, and instead of f in the equation N = f/d, we need to use v, the lens extension. Then N′ is defined as equal to v/d, where N′ is the effective f-number or effective aperture.
Camera exposure compensation may be necessary when the object is within about 10 focal lengths from the lens. Various methods are possible, using the values of f and V(if known), or magnification m, if this can be measured. Mathematically, it is easier to use a known magnification in the determination of the correction factor for either the effective f-number N′ or the corrected exposure duration t′. The required relationships are, respectively:
or
and
i.e.
The exposure correction factor increases rapidly as magnification increases. For example, at unit magnification the exposure factor is × 4 (i.e. +2EV), so that the original estimated exposure time must be multiplied by four or the lens aperture opened up by two whole stops.
The use of cameras with through-the-lens (TTL) metering systems is a great convenience in close-up photography, as compensation for change in magnification is automatically taken into account, especially with lenses using internal focusing and for zoom lenses with variable aperture due to mechanical compensation, as the effective aperture may not vary strictly according to theory since focal length changes with alteration of focus too.
Lens flare
Some of the light incident on a lens is lost by reflection at the aireglass interfaces and a little is lost by absorption. The remainder is transmitted to form the image. So the value of transmittance T is always less than unity. Any light losses depend on the number of surfaces and composition of the glasses used. An average figure for the loss due to reflection might be 5% for each aireglass interface. If k (taken as 0.95) is a typical transmittance at such an interface, then as the losses at successive interfaces are multiplied, for n interfaces with identical transmittance, the total transmittance is:
This means that an uncoated four-element lens with eight aireglass interfaces would have reflection losses amounting to some (0.95)8 = 35% of the incident light, i.e. a transmittance of 0.65.
Some of the light reflected at the lens surfaces passes out of the front of the lens and causes no trouble other than loss to the system, but a proportion is re-reflected from other surfaces and may ultimately reach the photoplane. Some of this stray diffuse non-image-forming light is spread uniformly over the surface of the image sensor, and is referred to as lens flare or veiling glare. Its effects are greater in the shadow areas of the image and cause a reduction in the image illuminance range (contrast). The flare light may not be spread uniformly; some may form out-of-focus images of the iris diaphragm (flare spots) or of bright objects in the subject field (ghost images). Such flare effects can be minimized by anti-reflection coatings, baffles inside the lens and use of an efficient lens hood. Light reflected from the inside of the camera body, e.g. from the bellows of a technical camera, and from the surface of the camera sensor, produces what is known as camera flare. This effect can be especially noticeable in a technical camera when the field covered by the lens gives an image circle appreciably greater than the film format. Such flare can often be effectively reduced by use of an efficient lens hood.
The number obtained by dividing the subject luminance range, SLR, by the image illuminance range, IIR, is termed the flare factor, FF, so:
Flare factor is a somewhat indeterminate quantity, since it depends not only on the lens and camera, but also on the distribution of light within and around the subject area. The value for an average lens and camera considered together may vary from about 2 to 10 for average subject matter. The usual value is from 2 to 4 depending on the age of the camera and lens design. A high flare factor is characteristic of subjects having high luminance ratio, such as back-lit subjects.
In the camera, flare affects shadow detail in a negative more than it does highlight detail; in the enlarger (i.e. in photographic printing), flare affects highlight detail more than shadow detail. In practice, provided the negative edges are properly masked in the enlarger, flare is seldom serious. This is partly because the density range of the average negative is lower than the log-luminance range of the average subject, and partly because the negative is not surrounded by bright objects, as may happen in the subject matter. In colour photography, flare is likely to lead to a desaturation of colours, as flare light is usually a mixture approximating to white. Flare may also lead to colour casts caused by coloured objects outside the subject area. Uncorrected flare may be predominantly in the infrared region and can seriously affect photosensor arrays with their extended spectral sensitivity. Suitable suppressive coatings are required on the rear surface of the lens.
T-numbers
Since lens transmission is never 100%, relative aperture or f-number N (as defined by the geometry of the system) does not give the light-transmitting capability of a lens. Two lenses of the same f-number may have different transmittances, and thus different speeds, depending on the type of construction, number of components and type of lens coatings. The use of lens coatings to reduce reflection losses markedly improves transmission, and there is a need in some fields of application for a more accurate measure of the transmittance of a lens. Where such accuracy is necessary, T-numbers, which are photometrically determined values taking into account both imaging geometry and transmittance, may be used instead of f-numbers. The T-number of any aperture of a lens is defined as the f-number of a perfectly transmitting lens which gives the same axial image illuminance as the actual lens at this aperture. For a lens of transmittance T and a circular aperture:
Thus, a T/8 lens is one which passes as much light as a theoretically perfect f/8 lens. The relative aperture of the T/8 lens may be about f/6.3. The concept of T-numbers is of chief interest in cinematography and television, and where exposure latitude is small. It is implicit in the T-number system that every lens should be individually calibrated.
Depth of field calculations still use f-numbers as the equations used have been derived from the geometry of image formation.
Anti-reflection coatings
An effective practical method of increasing the transmission is by applying thin coatings of refractive material to the aireglass interfaces or lens surfaces. The effect of single anti-reflection coating is to increase transmittance from about 0.95 to 0.99 or more. For a lens with, say, eight such interfaces of average transmittance 0.95, the lens total transmittance increases from (0.95)8 to (0.99)8, representing an increase in transmittance from 0.65 to 0.92, or approximately one-third of a stop, for a given f-number. In the case of a zoom lens, which may have 20 such surfaces, the transmittance may be increased from 0.36 to 0.82, i.e. more than doubled. Lens flare is reduced, giving an image of improved contrast.
The effect of a surface coating depends on two principles. First, the surface reflectance R, the ratio of reflected flux to incident flux, depends on the refractive indices n1 and n2 of the two media forming the interface. In simplified form (from Fresnel’s equations) this is given by:
In the case of a lens surface, n1 is the refractive index of air and is approximately equal to 1, and n2 is the refractive index of the glass, typically approximately equal to 1.51. Reflectance increases rapidly with increase in the value of n2. In modern lenses, using glasses of high refractive index (typically 1.7–1.9), such losses would be severe without coating.
Secondly, in a thin coating there is interference between the light wavefronts reflected from the first and second surfaces of the coating. With a coating of thickness t and refractive index n3 applied to a lens surface Figure 6.11), the interaction is between the two reflected beams R1 and R2 from the surface of the lens and from the surface of the coating respectively. The condition for R1 and R2 to interfere destructively and cancel out is given by:
where r is the angle of refraction and λ the wavelength of the light. Note that the light energy lost to reflection is transmitted instead. For light at normal incidence this expression simplifies to:
To satisfy this condition the ‘optical thickness’ of the coating, which is the product of refractive index and thickness, must be λ/4, i.e. one-quarter of the wavelength of the incident light within the coating. This type of coating is termed ‘quarter-wave coating’. As the thickness of such a coating can be correct for only one wavelength it is usually optimized for the middle of the visible spectrum (green light) and hence looks magenta (white light minus green) in appearance. By applying similar coatings on other lens surfaces, but matched to other wavelengths, it is possible to balance lens transmission for the whole visible spectrum and ensure that the range of lenses available for a given camera produce similar colour renderings on colour reversal film, irrespective of their type of construction.
Figure 6.11 An anti-reflection coating on glass using the principle of destructive interference of light between reflections R1 and R2.
The optimum value of n3 for the coating is also obtained from the conditions for the two reflected wavefronts to interfere destructively and cancel (see Chapter 2). For this to happen the magnitudes of R1 and R2 need to be the same. From Eqn 6.71 we can obtain expressions for R1 and R2:
By equating R1 = R2 and taking n1 = 1, then n3 = On2.
So the optimum refractive index of the coating should have a value corresponding to the square root of the refractive index of the glass used.
For a glass of refractive index 1.51, the coating should ideally have a value of about 1.23. In practice the material nearest to meeting the requirements is magnesium fluoride, which has a refractive index of 1.38. A quarter-wave coating of this material results in an increase in transmittance at an aireglass interface from about 0.95 to about 0.98 as the light energy involved in the destructive interference process is not lost but is transmitted.
Types of coating
• Evaporation. The original method of applying a coating to a lens surface is by placing the lens in a vacuum chamber in which is a small container of the coating material. This is electrically heated and evaporates, being deposited on the lens surface. The deposition is continued until the coating thickness is the required value. This technique is limited to materials that will evaporate at sufficiently low temperatures.
• Electron beam coating. An alternative technique to evaporative coating is to direct an electron beam at the coating substance in a vacuum chamber. This high-intensity beam evaporates materials even with high melting points which are unsuitable for the evaporation technique. Typical materials used in this manner are silicon dioxide (n = 1.46) and aluminium oxide (n = 1.62). The chief merit of the materials used in electron beam coating is their extreme hardness. They are also used to protect aluminized and soft optical glass surfaces.
• Multiple coatings. Controlled surface treatment is routinely applied to a range of other optical products. With the advent of improved coating machinery and a wider range of coating materials, together with the aid of digital computers to carry out the necessary calculations, it is economically feasible to extend coating techniques by using several separate coatings on each aireglass interface. A stack of 25 or more coatings may be used to give the necessary spectral transmittance properties to interference filters, as used in colour enlargers and specialized applications such as spectroscopy and microscopy. By suitable choice of the number, order, thicknesses and refractive indices of individual coatings the spectral transmittance of an optical component may be selectively enhanced to a value greater than 0.99 for any selected portion of the visible spectrum (see Figure 6.2).
The use of double and triple coatings in modern photographic lenses is now almost universal and even greater numbers are routinely used, making available a number of advanced lens designs that would otherwise have been unusable because of flare and low transmittance.
BIBLIOGRAPHY
Freeman, M., 1990. Optics, tenth ed. Butterworths, London, UK.
Hecht, E., 2002. Optics, fourth ed. Addison-Wesley, San Francisco, CA, USA.
Jacobson, R.E., Ray, S.F., Attridge, G.G., Axford, N.R., 2000. The Manual of Photography, ninth ed. Focal Press, Oxford, UK.
Jenkins, F., White, H., 1981. Fundamentals of Optics, fourth ed. McGraw-Hill, London, UK.
Kingslake, R., 1992. Optics in Photography. SPIE, Bellingham, WA, USA.
Ray, S., 2002. Applied Photographic Optics, third ed. Focal Press, Oxford, UK.