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Ophthalmic Optics

Handbook of Ophthalmic

Published by Carl Zeiss, 7082 Oberkochen, Germany.

Revised by Dr. Helmut Goersch

ZEISS

Germany All rights observed. Achrostigmat®, Axiophot®, Carl Zeiss T*®, Clarlet®, Clarlet The publication may be repro• Aphal®, Clarlet Bifokal®, Clarlet ET®, Clarlet rose®, Clarlux®, duced provided the source is Diavari ®, Distagon®, Duopal ®, Eldi ®, Elta®, Filter ET®, stated and the permission of Glaukar®, GradalHS®, Hypal®, Neofluar®, OPMI®, the copyright holder ob• Plan-Neojluar®, Polatest ®, Proxar ®, Punktal ®, PunktalSL®, tained. Super ET®, Tital®, Ultrafluar®, Umbral®, Umbramatic®, Umbra-Punktal®, Uropal®, Visulas YAG® are registered trademarks of the Carl-Zeiss-Stiftung

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Printed in Germany HANDBOOK OF OPHTHALMIC OPTICS: Preface 3

Preface

A decade has passed since the appearance of the second edition of the "Handbook of Ophthalmic Optics"; a decade which has seen many innovations not only in the field of ophthalmic optics and instrumentation, but also in standardization and the crea• tion of new terms. This made a complete revision of the hand• book necessary.

The increasing importance of the contact in ophthalmic optics has led to the inclusion of a new chapter on Contact Optics. The information given in this chapter provides a useful aid for the practical work of the ophthalmic and the ophthalmologist.

The "Handbook of Ophthalmic Optics" is intended both as systematic reading material and - due to its extensive listing of optical terms - a reference work. It is not intended as, and cannot take the place of a textbook. Reference literature, a list of specialist terms and tables are contained in the newly arranged appendix.

We would like to extend our gratitude to all those who have contributed to the creation of this third edition of the hand• book. Our special appreciation is due to Dr. Helmut Goersch, who also edited the German edition, Dr. Heinz Baron for the chapter on Contact Optics, and to Mr. M. Jalie, SMS A, FBDO (Hons), Hon CGIA, MBIM, Head of Department of Applied Optics, City and East London College, London, England, with• out whose kind and indefatigable assistance the English edition would not have been possible at all.

Carl Zeiss Oberkochen 4 HANDBOOK OF OPHTHALMIC OPTICS: Contents

Physical Optics See also page 8 Components of tracing. of . of light. . Optical formation with lenses and lens systems. Aberrations.

Wave optics Electromagnetic radiation. Interference and . Polarisation.

Light technology . Material properties.Light sources. Light guides.

Physiological Optics See also page 58 The Structure of eye. . The . Visual performance. Colour vision. Emmetropic eye. Ametropic eye. Monocular correction of eye. Fusion and vergence. Binocular space . Phoria and tropia. Anisometropia and aniseikonia. Binocular correction of eye.

Spectacle Optics See also page 100 Spectacle lenses Terminology. Single-vision lenses with spherical power. Single-vision lenses with astigmatic power. Single-vision lenses with prismatic power. Bifocal, multifocal and progressive lenses. Special types of spectacle lenses. Lens power determination. Image-forming properties. Light-transmission properties.

The lens/eye system Terminology. Monocular centration. Binocular centration. Accommodative effort and amplitude of accommodation. Space perception. Low vision aids. HANDBOOK OF OPHTHALMIC OPTICS: Contents 5

Contact Optics See also page 156 Contact lenses Terminology. Contact lenses with spherical power. Contact lenses with astigmatic power. Bifocal and multifocal contact lenses. Image-forming properties. Light-transmission properties. /eye system Terminology. Hard contact lenses and spherical ametropia. Hard contact lenses and astigmatic ametropia. Soft contact lenses. Optical differences from spectacle lens correction. Reasons for use.

Instrument Optics See also page 188 Optical instruments Lupes. . Telescopes. Photographic lenses. Projection lenses. Endoscopes. Geodetic instruments.

Ophthalmic and Focimeters. Instruments for vision testing. ophthalmological instruments Instruments for subjective vision testing. Equipment for lens fitting. Equipment for contact lens fitting. Other instruments.

Materials See also page 246 Composition and properties. Shaping process. Strengthening techniques.

Plastics Composition and classification. Plastics for spectacle lenses. Plastics for contact lenses. Plastics for spectacle frames.

Metals and other materials Noble metals. Alloys for spectacle frames. Other materials.

Appendix Tables page 277 Specialist terms page 333 Bibliography page 338 Index page 340 8 PHYSICAL OPTICS

Geometrical Optics Components of Light ray 11 Bundles and pencils 1 1 Angles, distances and points 11 Optical image formation 12

Reflection of light Law of reflection 12 Total reflection 13 13

Refraction of light 15 Law of refraction 15 15 Plane parallel plates 17 17

Lenses Lenses with spherical power 20 Surface power 20 Equivalent power and 21 Vertex power and vertex focal length 22 Principal points 23 Lenses with astigmatic power 24

Optical image formation with Optical systems 26 lenses and lens systems Determination of the image 26 formation 30 formation 30 Newton's formula 30 Astigmatic image formation 31 Stops 31

Aberrations Requirements on image formation 33 33 33 of oblique incidence 34 Field curvature 34 34 35 Corrected optical systems 36 PHYSICAL OPTICS 9

Wave Optics Electromagnetic radiation Light 37 Velocity of light 38

Interference and diffraction Interference 38 Newton's rings 39 Reduction of reflections 39 Interference filters 40 Diffraction 40 41

Polarisation Brewster's law 41 Bi-refringence 42 43 Optical activity 43

Light Technology Photometry Terminology 44 Luminous efficacy 46

Material properties Terminology 46 Influence of light path 4S

Light sources Daylight 49 Incandescent lamps 50 Fluorescent lamps 50 Spectral lamps 50 51 Standard illuminants 51 Colour temperature ^2

Light guides Principle 52 Numerical 52 Attenuation 53 Solid and liquid light guides 53 Optical fibres and fibre bundles 54 Image carriers and shape converters 54 Tapered light guides 55 Optical 55 PHYSICAL OPTICS: Geometrical Optics 11

Geometrical Optics Components of ray tracing

Light ray A light ray is an imaginary mathematical line denoting the direction of propagation of light energy; single light rays do not exist in reality. The light rays are perpendicular to the wave fronts of wave optics and in geometrical optics serve to represent changes in light propagation through optical components.

Bundles and pencils Light rays with a common point of intersection form a homo- centric bundle. If the rays emanate from this point of intersec• tion which lies at a finite distance, the bundle is divergent; if they run towards the point of intersection, it is convergent. The point of intersection for a parallel ray bundle lies at infinity. A shows the ray path in one plane and contains the point of intersection of the rays.

Angles, distances and The angles and distances used in geometrical optics to describe points a ray path are given a mathematical sign in accordance with the following rules. Angles in the anti-clockwise direction are taken to be positive, and those in a clockwise direction to be negative. The correct direction of rotation for the angles of incidence, reflection and refraction is obtained by rotating the towards the ray. For the angle of deviation, the direction of the incident ray is rotated towards the refracted ray. For angles between a ray and the optical axis, the ray is rotated towards the axis. Distances are taken to be positive when measured in the direc• tion of the light and negative when measured in the opposite direction; distances perpendicular to the optical axis are taken to be positive when measured upwards, and negative when they are measured downwards. The radius of curvature of an inter• face is measured as the distance from the vertex to the centre of curvature. In graphic representations the direction of light should run from left to right if possible. The parallel displacement of a ray to the right (of the direction of light) is taken to be positive, and negative when to the left. Angles are denoted by small Greek letters, distances by small Roman letters and points by capital Roman letters (Table 1). 12 PHYSICAL OPTICS: Geometrical Optics

Optical image formation Optical image formation involves the creation of a related image point O' for each object point O; in the image formation process the position of the point of intersection changes for the rays of each individual bundle. The angle between two bundle rays (the vergence of the ray bundle) can be retained in optical image formation (e.g. in image formation through a plane parallel plate if the same refractive index is present on both sides of the plate), or it can be changed (change in the vergence of the homocentric ray bundle, e.g. in image formation through a lens). As the ray directions are reversible, the object and the image can be interchanged. Two points are termed optically conjugate if one is the image of the other. Optically conjugate points and the distances and angles used to denote their position are given the same letter; a prime is affixed to the letter for the image-side quantity (Table 1). For paired quantities which are not optically conjugate, a crossbar is added to the letter for the object-side quantity (focal points and focal lengths) Gaussian space is the term given to the paraxial space in which the aberrations present in optical image formation are negligi• ble. The optical construction of a ray path in geometrical optics does not take into account the ever present diffraction of wave optics.

Reflection of light

Law of reflection The reflected ray AR lies in the determined by the incidence normal LA and the incident ray PA (Fig. 1). The angle of reflection i' is equal but opposite in sign to the angle of incidence i: (1) i=-i'. The angle of reflection is independent of the (col• our) of the light. PHYSICAL OPTICS: Geometrical Optics 13

R

L

P'

Fig. 1 Fig. 2 Reflection of light Critical angle of incidence

Total reflection Total internal reflection (reflection without loss) occurs when light is incident at the boundary between two media with different refractive indices, the light travelling in the denser medium, and the angle of incidence i, is greater than the critical

angle ic. If the refractive index of the denser medium is n, and that of the rarer medium n', then

(2) sin ic = (with n' < n).

For reflection in air (n' = 1) sin ic = 1 /n (Fig. 2 and Table 2). The critical angle of incidence is dependent on the wavelength (colour) of light. experienced close to heated air at ground level, e.g. over roads, are caused by total reflection. Application: reflecting prisms.

Mirrors A plane (Fig.3) produces a virtual point image on the incidence normal (A',B',C) at the same distance behind the mirror as that of the object point (A,B,C) in front. In an angular mirror (Fig.4) the deflection of a ray is twice as large as the angle formed by the two mirror sides. It does not change when the angular mirror is rotated about any axis of rotation parallel to edge K. (Application: marking of right angles with a = 45° in surveying). The rotating mirror (Fig. 5) rotates the reflected ray by twice the amount of the mirror rotation. (Application: rotating mirror galvanometer.) 14 PHYSICAL OPTICS: Geometrical Optics

Fig. 6 Fig. 7 Concave mirror Convex mirror

A concave spherical mirror focuses parallel rays incident in the Gaussian space at one point. This focal point F' lies midway between the centre of curvature and the pole of the mirror: SC — r (Fig. 6). The focal length P is:

(3) f " §•

In the convex spherical mirror the focal point is virtual (Fig. 7). Here (3) also applies. Spherical mirrors display aberrations. Paraboloidal mirrors for searchlights reflect the rays coming from the light source located at the focal point F' as parallel rays, regardless of the size of the mirror's aperture. PHYSICAL OPTICS: Geometrical Optics 15

Refraction of light

Refractive index The refractive index n of a substance is the ratio of the velocity

of light Co in a to the velocity cn in the substance concerned (or the ratio of the corresponding ).

(4) n = ? = ^-

In this definition the refractive index of the vacuum is 1 for any wavelength of light. As n « 1.0003 for air in normal conditions (200 C and 1013 hPa), air is often used as a reference instead of a vacuum. In glass with a refractive index of n = 1.5 light travels only 2/3 as fast as in air; its wavelength is reduced by the same factor.

Law of refraction A light ray which is obliquely incident on the interface between two media is deflected from its original direction (Fig. 8). The refracted ray AB lies in the plane of incidence determined by the incidence normal LL and the incident ray PA. The angle of incidence i and the angle of refraction i' follow Snell's law:

(5) n • sin i = n' • sin i'.

Dispersion The refractive index of a substance is dependent on the frequen• cy (wavelength or colour) of light. This property leads to the breaking down of white light into its monochromatic compo• nents during refraction (dispersion). In order to define the refracting properties of a glass type, the values of n are given for a specific number of spectral lines (Table 3). The refractive index for the light of yellow helium (d) is known as the mean

refractive index nd of the substance. The difference between the refractive indices for the light of the blue (F') and red (C)

spectral lines of hydrogen is the mean dispersion An = nF - nc. Differences in the refractive index for other wavelengths are called partial dispersions. An important quantity for the correction of chromatic aberra•

tions is the ve of a glass type, which is the ratio of Fig. 8 the refractivity in air to the mean dispersion: Refraction of light (n, n', refractive index in front of and behind the inter• face) 16 PHYSICAL OPTICS: Geometrical Optics

(6) VP = nP - nc If two media display two different refractive indices for light of a specific frequency (colour), the one with the larger refractive index is the optically denser, and the other the optically rarer medium. Fig. 9 shows the refractive index of some substances as a function of the wavelength.

1.80-

w Dense flint glas s

Lanthanum crow n yljoi m. 1.70-

Calcit 9 , , ^*'""*''>« -"^^^(ordin ary ray)

Extra dense*^ 1.60-

(extra ordinary ray) Quart i (ordinary ray) ^R39^"*"—- Spectacle crown glass 1.50- " " . (extraordinary ray) Fused Fluorite

1.40-

.^Water^

UV vio et blue green yellow red IR

Wavelength X in air Fig. 9 Refractive indices of some substances as a function of the wavelength X in air PHYSICAL OPTICS: Geometrical Optics 17

Plane parallel plate If the same medium exists on both sides of a plane parallel plate. a light ray incident obliquely on the plate emerges from the glass at the same angle at which it enters (Fig. 10). The amount of parallel displacement v increases with 1. increasing plate thickness d, 2. increasing angle of incidence i 3. increasing quotient of the refractive indices of the plate material and the surrounding medium n. The parallel dis• placement is

(7) v = d.sin(i-i') COS 1 The point of intersection of a ray bundle in the Gaussian space is displaced by the distance x:

(8) x-d-(l-J).

For a plate with n' = 1.5 in air (n = 1) the displacement x is thus one third of the thickness of the plate. If the surrounding medium is optically rarer than the material of the plate (Fig. 11), the displacement takes place in the direction of the light (x is positive); otherwise, it will occur against the direction of light (x is negative).

Fig. 10 Fig. 11 Parallel displacement of a light ray by Passage of a ray bundle through a a plane parallel plate (n' > n) plane parallel plate (n' > n)

Prism A in an optically rare medium deviates a light ray towards the prism base. The angle of deviation d depends upon 1. the prism angle a 2. the angle of incidence i 3. the quotient of the refractive indices of the prism material n' and the surrounding medium n. 18 PHYSICAL OPTICS: Geometrical Optics

If the ray passes symmetrically through the prism, the angle of deviation is at its minimum (Fig. 12). For small prism angles a (wedge) and small angles of incidence i, the following formula provides good results for n = 1: (9) d = (n'-l)a.

Fig. 12 For a prism material with n' = 1.5, the angle of deviation is then Minimum deviation by a prism approximately half the prism angle. In ophthalmic optics the (n' > n) deviation of a ray of light from its original direction is given in cm/m (international denotation: prism A). The devia• tion is 1 cm/m when the lateral deflection is 1 cm on a screen which is placed at a distance of 1 m and is perpendicular to the direction of the original ray. The relationship between the prismatic deviation P and the angle of deviation d is (Fig. 13):

(10) P = 100— tan d. m For prisms made of spectacle crown with n'= 1.525, a prism angle of a = 1 ° leads to a deviation in air of P = 0.916 cm/m in accordance with formulae (9) and (10).

12°

cm - 1° a 1.75 10° m

T3 6° - CO

n

via t 4

o 2" 1 ^ a 0.57° CJ m Fig. 13 c o° *r | i i i Relationship between angular devia• < 0 2 4 6 10 12 14 16 18^20 tion and prismatic deviation in accor• dance with formula (10) Prismatic deviation P

In a rotary prism device (Fig. 14) used as a setting and measur• ing unit for distance measurement, two identical glass prisms are rotated in their planes in opposite directions to each other by the same amount p\ This results in a variable angular devia• tion d of the ray AB corresponding to the parallactic angle for the respective distance setting with a constant deviation plane: PHYSICAL OPTICS: Geometrical Optics 19

(11) d « 2(n' - l)acosp\ Further applications: Herschel's double prism for measuring the fusion capability of a pair of , prism compensator. Rays of different colours are deviated more or less strongly by a prism (Table 4) due to the effect of dispersion. When white light passes through it, the prism therefore produces a , with short-wave light being more strongly refracted than long• Fig. 14 wave. Rotary prism device Application: prisms with large prism angles for spectral units. 20 PHYSICAL OPTICS: Geometrical Optics

Lenses

Lenses with a Spherical lenses are divided into spherical power 1. convex lenses which are thicker at the centre than at the edge 2. concave lenses which are thinner at the centre than at the edge. Fig. 15 shows convex lenses with different shapes but with identical focal lengths. Fig. 16 shows corresponding concave lenses. If surrounded by optically rarer media, convex lenses act as converging lenses (positive or plus lenses), and concave lenses b) c) as diverging lenses (negative or minus lenses). Fig. 15 Parallel rays become convergent, homocentric bundles when Convex lenses they pass through (in the Gaussian space) converging lenses, a) bi-convex and divergent bundles when they pass through diverging lenses. b) plano-convex The optical axis of a lens is the line perpendicular to both c) curved boundary surfaces. It runs through the centres of curvature of m the surfaces. The point of intersection of the optical axis and the boundary surface is known as the vertex of the lens. Any plane containing the optical axis is called a meridian plane. Lenses with a spherical power display the same optical characteristics in all meridian planes. In order to reduce aberrations, lenses with spherical powers are a) c) also designed with surfaces which deviate from the spherical Fig. 16 form, but which display rotational symmetry about the optical Concave lenses axis (aspheric lenses). a) bi-concave b) plano-concave c) curved

Surface power If a spherical surface has a radius of curvature r, the medium in front has the refractive index n and the medium behind the refractive index n', the surface power F of the spherical surface is:

(12) F = n — n

The unit of measure for refractive powers is the dioptre (D). J_ 1 D = 1 m' If in front of a lens with the refractive index n i there is a medium

with the refractive index n1? and behind it a medium with the

refractive index n'2, the surface power of the first surface is: PHYSICAL OPTICS: Geometrical Optics 21

(13) Fl= 51^-21 r, and the surface power of the second surface:

(14) F2 = *LZJ2>. r2 Observed from the optically rarer medium, a convex surface displays a positive, and a concave surface a negative surface power. A spherometer is used to measure the radius of curvature of surfaces; the dioptre scale shows surface powers for a specific refractive index (e.g. n'= 1.523) in air (n =1).

Equivalent power and If d (in m) is the centre thickness of a spherical lens (refractive focal length index n'i), the equivalent power F of this lens is (Gullstrand formula): (15) F - F, + F, - 8 • F, • F, where 8 = d/ n'i is the reduced thickness. For this infinitely , if the centre thickness is negligible, then (16) F = F, + F, For this infinitely thin lens with the refractive index n' in air: 1_ _ 1 (17) F = (n'- !)•(•

If rays are incident upon a lens parallel to the optical axis (in the Fig. 17 Gaussian space), the refracted rays have a common point of Focal points of a converging lens intersection on the optical axis known as the image-side focal point F'. When a bundle of rays emerges from a lens parallel to the optical axis, the incident rays have a common point of intersection on the optical axis known as the object-side focal point F. If the equivalent power is positive, both focal points are real (Fig. 17), i.e. the actual rays intersect. If the equivalent power of the lens is negative, both focal points are virtual (Fig. 18), i.e. the actual rays do not intersect but their imaginary projections do. Related to the focal points are the focal lengths measured from Fig. 18 the corresponding principal points to the focal points (Figs. 17 Focal points of a diverging lens and 18): 22 PHYSICAL OPTICS: Geometrical Optics

image-side focal length f = distance from H' to F', object side focal length f = distance from H to F. The relationship between the equivalent power F and the focal lengths of a lens is

(18) F-f-

If the same medium exists on both sides of the lens (nj = n'2), then (19) f = - f.

Only for a lens in air (ni = n'2 = 1):

(20) F = !=-!.

To obtain the equivalent power in D, the focal length must be substituted in metres (Table 5).

Vertex power and The distances measured from the vertices of a lens along the vertex focal lengths optical axis are called the vertex focal lengths and are desig•

nated fv on the object side and Pv on the focal side. In ophthalmic optics the power of a lens is indicated in terms of

its back vertex power. The back vertex power F'v of an ophthal• mic lens in air is the reciprocal of the image-side (ocular-side)

back vertex focal length Pv of the focal point F' (measured from

the vertex A2) (Figs. 19 and 20)

(21) F0 = ± •v

If the back vertex focal length Pv is substituted in m, the back vertex power Fy is in D (Table 5). The difference between the equivalent power F and the back

vertex power F'v (or the focal length P and the back vertex focal

length Pv is greater, the stronger the curvature of a lens (with constant centre thickness). The relationship of these values for a lens in air is given by the shape S of the lens

F'v = F only for an infinitely thin lens. PHYSICAL OPTICS: Geometrical Optics 23

Principal points The principal points of a lens are the points of intersection of the optical axis and the principal planes perpendicular to this axis. These are conjugate points with magnification m = + 1. The position of the principal points in a lens is determined by the centre thickness, the shape of the lens ("bending"), and the refractive indices of the media. The more a lens deviates from the symmetrical shape (equishape), the further the principal points are shifted in the direction of the more strongly curved surface. For a lens in air the object-side vertex focal length (distance of the principal point H from the vertex of the first surface) is

(23) e = ^-5,

and the image-side vertex focal length (distance of the principal point H' from the vertex of the second surface) is

(24) e'= -§-8. F 24 PHYSICAL OPTICS: Geometrical Optics

The principal points of symmetrical lenses lie inside the lenses. If the lens is not too thick and if n' = 1.5, the principal points divide the lens thickness into three almost equal parts (because

Fi = F2 « F/2). In plano-convex and plano-concave lenses (F)

or F2 = 0) one principal point lies at the vertex of the curved surface, the other one about 1 /3 of the centre thickness away in the lens. The dots in Figs. 15 and 16 indicate the approximate position of the principal points in lenses of different shapes in air. The distance from H to H' is designated as the "interstitium" i.

Lenses with an Lenses with at least one cylindrical, toroidal or atoroidal sur- astigmatic power face are not symmetrical to the optical axis and have a different power in every meridian plane. The rays are united in the two planes of the strongest and weakest refraction only. These two planes are perpendicular to each other and are known as the principal meridians of the lens. Rays which run in one of the other meridian planes in front of the lens are at an angle to each other after refraction (they no longer lie in one plane). For this reason the power of an astigmatic lens can only be measured in the two principal meridians and is given in the form of two

equivalent powers or vertex powers Fv'i and Fv2. The difference between the two principal powers is known as the astigmatic

difference (cylinder): C = Fv2 - Fv'i. The formulae for a lens with a spherical power apply for each of the two principal meridians. The simplest form of an astigmatic lens is a piano-cylinder (Fig. 21). A toroidal surface is produced by rotation of a circular arc about an axis which does not run through the centre of this arc.

The toroidal surface has different radii of curvature (n and r2 in Fig. 22) in the two principal meridians (transverse and equatori• al meridians). In an atoroidal surface the transverse and equato• rial meridians deviate from the circular shape. Application: For the correction of astigmatic ametropia; ana- morphic lens systems for motion pictures. PHYSICAL OPTICS: Geometrical Optics 26 PHYSICAL OPTICS: Geometrical Optics

Optical image formation with lenses and lens systems

Optical systems Optical systems are axially centred lens combinations (objec• tives, , etc.) - or also mirror combinations - which have two focal points, two principal points and an equivalent power. Once the positions of the focal points F and F' and the principal points H and H' have been established in an optical system by computation or measurement, the position and size of an object image can be determined by a drawing or com• putation.

Determination of the image If the refracted rays are convergent in the real part of the image space (space behind the lens), a real image is produced at their point of intersection; if they are divergent, a virtual image is produced at the point of intersection of the backward projec• tions in the virtual part of the image space (space in front of the lens). (Note: The object space is real in front of the lens and virtual behind it. The image space is real behind the lens and virtual in front of it. The space in front of the lens alone is often inaccu• rately described as the object space, and the space behind it alone as the image space.) The distance measured along the optical axis from the object- side principal point H to the object is the object distance /; the corresponding distance from the image-side principal point H' to the image is the image distance /'. Two auxiliary rays serve to construct the image: 1. The ray running parallel to the optical axis in the object space (parallel ray) is refracted at the principal plane H', and the refracted ray runs through the focal point F' (focal ray). 2. The focal ray through F is refracted at principal plane H, and the refracted ray becomes a parallel ray (Figs. 23 to 28). The image position is calculated using the equation:

Inair(n| = n'2 = 1) according to (20) by PHYSICAL OPTICS: Geometrical Optics 27

The construction and computation apply to the Gaussian space only. The size of the image depends on the lateral magnification m, which is the ratio of the image size h' to the object size h

(27) m = ^ ( = yinair).

If m is larger than 1, magnification is then present. If m is positive, the image and the object are in the same 28 PHYSICAL OPTICS: Geometrical Optics

direction and one of them is virtual. If m is negative, then the image and the object are in different directions and both are either real or virtual.

If u is the angle at which an object-side ray intersects the optical axis, and u' is the corresponding angle for the refracted ray, the angle ratio y', or the angular magnification, is the quotient

(28) PHYSICAL OPTICS: Geometrical Optics 29

The optically conjugate axial points with the angular magnifica• tion y = + 1 are the nodal points N and N' of the lens (or the system). As object-side and image-side nodal point rays inter• sect the optical axis at identical angles, they can also be used for image construction.

H' s /7

/ I 1/ A-

>f-—

Fig. 28 Virtual image produced by a negative system with negative m 30 PHYSICAL OPTICS: Geometrical Optics

If the same medium exists on both sides of a lens, H and N as well as H' and N' coincide. If the media in front of and behind the lens are different, the nodal points are displaced towards the denser medium relative to the principal points if the lens has a positive refractive power, and towards the rarer medium if it has a negative refractive power. The distance between the nodal points remains constant and is identical to the interstitium. Principal points, focal points and nodal points are also known collectively as cardinal points.

Real image formation A real image is produced by a positive system if a real object is ' located beyond the object-side focal length (Fig.23) or if there is a virtual object (Fig.24). A negative system produces a real image if a virtual object is located within the object-side focal length (Fig. 25).

Virtual image formation A virtual image is produced by a positive system if a real object is located within the object-side focal length (Fig. 26). A nega• tive system produces a virtual image if a real object is present, or if a virtual object is located beyond the object-side focal length (Fig. 28).

Newton's formulae The distances measured from the focal points F and F' (along the optical axis) to the object and to the image are the extra-focal distances designated x and x'. These distances are used in Newton's equation for image formation: (29) x-x'-f-f. The lateral magnification is thus

(30) p'--!-^£,

and the angular magnification is

(3.)

Astigmatic image Lenses with an astigmatic power image a real object point not as formation an image point, but as two image lines with different image 31

distances. The image lines of an axial object point are perpendi• cular to the optical axis and lie in the principal meridians lying perpendicularly to each other. The ray bundle of an object point which has a circular cross section at any point prior to refrac• tion exhibits an elliptical cross section subsequent to refraction. The size and shape of the varies depending on its posi• tion. The two image lines are the extremes of the cross section. Between them lies the only point where the cross section of the bundle is circular (instead of a point). The cross section forma• tion (Sturm's conoid) in Fig. 29 shows this circle of least confusion (Kr).

Fig. 29 Sturm's conoid to demonstrate the nature of astigmatic imagery

The principal meridian with the mathematically smaller (i.e. weaker positive or stronger negative) refractive power Fp is called the first principal meridian p, while the other is known as the second principal meridian a and has the mathematically

larger refractive power Fa. The image line produced by the first principal meridian lies in the plane of the second principal meridian (yf> in Fig. 29) and vice-versa. For each principal meridian, the formulae for image formation by spherical lenses apply.

The image distance l'c of the circle of least confusion is derived from:

(32)

Stops A bundle of rays is controlled by stops (mechanical stops, lens mounts or "focusing" of rays). These stops are generally cen• tred, i.e. their centres lie on the optical axis of the system and their planes are perpendicular to the optical axis. The stops control image brightness, resolution, aberrations, , and field of view. 32 PHYSICAL OPTICS: Geometrical Optics

The stop which produces the strongest concentration of all ray bundles and which therefore determines the brightness of the image is called the aperture stop or aperture and is often found in the form of an iris diaphragm inside the optical system. Rays which pass through the aperture at the edge of this diaphragm are known as aperture rays. The aperture stop is the viewed from the object, and the viewed from the image. The image of the iris aperture seen when observing an eye is therefore the entrance pupil of the eye. The entrance and exit are conjugate planes and represent the common cross section of all object-side or image-side aper• ture ray cones. When the aperture stop is located in the real object space in the form of the front stop, it is also the entrance pupil. When it is in the real image space in the form of the back stop, it is also the exit pupil. Rays passing through the centre of the entrance pupil are called object-side principal rays, and rays passing through the centre of the exit pupil are known as image-side principal rays. The pupil centres are the centres of perspective. All principal rays are real and pass through the centre of the aperture stop. In a telecentric ray path one of the pupils lies at infinity, this being achieved in a single lens by means of an aperture stop in one focal plane. If an additional stop exists in the plane of the object, in the plane of the image or in the plane of a real intermediate image, the maximum inclination of the principal rays to the optical axis depends on this so-called field stop. Therefore, the field stop alone determines the size of the field of view and sharply defines it. Viewed from the object, the field stop is the entrance port, and when viewed from the image, it is the exit port. If an additional stop exists elsewhere it determines (together with the aperture stop) the size of the field of view. Multiple stops of this type account for decreasing brightness towards the periphery of the image (""). PHYSICAL OPTICS: Geometrical Optics 33

Aberrations

Requirements on image Optical should be sharp, true to scale in the image plane formation and free from colour defects. Deviations from these requirements are called aberrations.

Spherical aberration In a spherical lens, zones concentric with the optical axis have different refractive powers. Only the parallel rays incident in the Gaussian space are collected in the focal point Fo. Rays which are incident parallel to the optical axis and which pass through the lens outside the paraxial space do not intersect at the focal point (Figs. 30 and 31), but form a concentric halo in the focal point plane. This unsharpness is known as the spherical aberra• tion (aperture error), and the surface which envelopes the rays in the image space is called the surface. A ^1 ^

V Fig. 31 / Fig. 30 Spherical aberration in a minus Spherical aberration in a plus lens lens Coma If the object point in Fig. 30 which is assumed to be at infinity moves out of the field centre, the unsharp halo may exhibit a comet-like asymmetry as a result. The asymmetric portion of this unsharpness which is superimposed by spherical aberration is known as coma (Fig. 32). The rays of the middle zone become stigmatic, i.e. they are focused at one point.

Fig. 32 The generation of a coma 34

Astigmatism of When the object moves further away from the field centre (e.g. oblique incidence when looking obliquely through a spectacle lens), the stigmatic bundle of rays becomes more and more astigmatic. The two principal section planes of the bundle are the tangential section formed by the system axis and the principal ray, and the sagittal section perpendicular to it containing the principal ray (T and S in Fig. 33).

Fig. 33 Astigmatism of oblique inci- dence

This astigmatism of oblique incidence is dependent on the type of lens and on the positions of the object point and the aperture stop. For the individual points of an object plane perpendicular to the optical axis the astigmatism of oblique incidence results in two curved image surfaces (centres of the image lines) instead of one image plane. For this reason it is also called radial astigmatism.

Field curvature If, with corrected oblique astigmatism, the astigmatic differen• ces are zero for all oblique bundles, a curved image surface remains for point-focal imagery. This deviation from the image plane is called the field curvature.

Distortion The aforementioned aberrations refer to the image formation of an object point and lead to loss in definition in the image plane. In addition, the image of an extended object is no longer true to scale if the aperture stop is not located directly at the lens itself. This distortion depends on the type of lens and on the position of the aperture stop. In the case of a converging lens PHYSICAL OPTICS: Geometrical Optics 35

with a back stop the lateral magnification increases as the distance from the optical axis increases. This leads to pin• cushion distortion. In the case of a converging lens with a front stop, on the other hand, the lateral magnification decreases as the distance from the optical axis increases, resulting in barrel- shaped distortion (Fig. 34). With diverging lenses, the position of the aperture stop has the opposite effect. The lens/eye system has a back stop, with the result that a plus lens produces pin-cushion distortion while a minus lens results in barrel-shaped distortion. In the case of a lens with an astigmatic power the lateral in the two principal meridians are different. This leads to anamorphotic image formation. A circle perpendi• cular to the optical axis, for example, is imaged as an ellipse, a square as a parallelogram or (when principal meridians are parallel with the sides of the square) as a rectangle.

-1 f

a) [ 1 O Fig. 34 Distortion: a centred square (a) perpendicular to the optical axis is imaged in the shape of a pin-cushion (b) or a barrel (c)

Chromatic aberration Owing to dispersion, a lens displays different refractive powers for different wavelengths (colours). This results in chromatic aberrations for image formation in white (polychromatic) light. The focal length is shorter for short-wave (blue) light than for long-wave (red) light; this longitudinal chromatic aberration is shown in Fig. 35. Furthermore, the image-side principal rays of oblique ray bundles are inclined to the optical axis at different angles depending on their colour; this transverse chromatic aberration leads to a different lateral magnification for every colour in the respective image plane (chromatic difference of magnification). 36 PHYSICAL OPTICS: Geometrical Optics

blue

Fig. 35 Longitudinal chromatic aberration

Corrected optical systems Corrected optical systems are combinations of converging and diverging lenses made of different types of glass and with specific shapes and suitably positioned stops. As it is impossible to fully eliminate all aberrations simultaneously, the type of correction depends on the intended application. Achromats are systems for narrow fields of view with longitudinal chromatic aberration corrected for two colours and usually with spherical aberration corrected in one colour. Apochromats are corrected for three colours in the longitudinal chromatic aberration. Anastigmats are systems for large fields of view with corrected astigmatism and corrected field curvature. Aplanatic systems for small fields of view are corrected for spherical aberration, coma and usually also chromatic aberra• tion (achromatic aplanats). PHYSICAL OPTICS: Wave Optics 37

Wave optics Electromagnetic radiation

Light Light is that part of the spectrum of electromagnetic radiation which is perceived by the eye (Fig. 36). This has a waveband of A, = 380 to 780 nm in air (1 nanometre = 10"9 m).

10

cosmic rays

1pm — I -1020 y rays

10-io _ 1nm — X-rays

UV 1015 1um 105 IRand thermal radiation 1 THz 1mm — 1cm — - 1010 UHF - 1GHz

VHF.FM electric short-wave 1 MHz 1 km L- 105 Fig. 36 Wavelength Frequency[Hz] Electromagnetic spectrum X in air [m]

direction of propagation Electromagnetic waves vibrate transversely, i.e. perpendicular• ly to their direction of propagation (Fig.37). The frequency v is the number of vibrations per second and is measured in the unit hertz (1 Hz = 1/s). The frequency of all electromagnetic waves

wavelength X is independent of the medium in which they move. The frequen• Fig. 37 cy (and not the wavelength!) therefore characterises the colour of the light in question (Table 6). 38 PHYSICAL OPTICS: Wave Optics

Velocity of light In a medium with the refractive index n the relationship be•

tween the velocity cn ( velocity) and the wavelength is: medium of vacuum refractive index n: (33) C = n n If the frequency is substituted in hertz and the wavelength in

km, the velocity cn is obtained in km/s. The speed of propaga•

fWVYl I A tion c0 of all electromagnetic waves in a vacuum (space) is one (— Xo-~1 I— XrT-1 of the elementary constants (velocity of light) and equals: -ho

c0 = 299792.46 km/s. Fig. 38 In a medium with the refractive index n, the velocity cn and the Change in velocity and wavelength wavelength X„ of light are less than in a vacuum (Fig. 38), while at the interface of two media the frequency (colour) remains unchanged. The v, with which light are transmitted (simplest example: switching a light source on and off), is decisive for the transmission of information. This group veloci•

ty corresponds to the (cn = c0) in a vacuum; in material, however, its value differs from that of the phase

velocity (c„ = c0/n), namely:

c„ (34) n - Xo dn • dAo In the area of normal dispersion the refractive index decreases with increasing wavelength (dn/dXo < 0), and the group veloci• ty is smaller than the phase velocity.

Interference and diffraction

Interference Interference is the act or process of intervening of two or more waves at the same instant and at the same point in space.In light waves, this phenomenon can be observed in the form of bright (crest plus crest) and dark (crest plus trough) bands. This is, however, only the case with coherent wave trains which have the same source and the same vibration plane and which have a constant phase difference. Two waves of identical wavelength exhibit a phase difference if their crests do not coincide. PHYSICAL OPTICS: Wave Optics 39

Newton's rings When two lens surfaces placed a short distance apart are illuminated, the two reflected wave trains interfere depending on the thickness of the layer of air which separates them. Newton's rings are formed. Adjacent interference fringes indicate a difference in thickness of the layer of half the wavelength of light. If the fit is good, the dark interference fringes are far apart. support Application: inspection of lens surfaces with a test lens Fig. 39 (Fig. 39). Inspection of lens surfaces with the aid of interference phenomena

Reduction of reflections In order to reduce reflections ("anti-reflection coating"), lens surfaces are coated in a high vacuum with a 100 nm-thick, non-absorbing layer of a material with a low refractive index. When this vacuum-deposited coating with the refractive index

n2 borders on air (ni = 1), the two wave trains of the light

reflected at the coating and the lens (n3) neutralise each other by interference if the following two requirements are met (Fig. 40):

1. Phase requirement: the wave crest of one wave train (I) must coincide with the trough of the other (II). This requirement is met if the actual thickness d of the coating is an uneven multiple of a quarter of the wavelength in the coating

2. Amplitude requirement: the amplitudes of the two wave trains must be identical. This requirement is met if the square of the refractive index of the coating equals the refractive index of the lens:

2 (36) n2 = n3. 40 PHYSICAL OPTICS: Wave Optics

Reflections are reduced very effectively by using several layers with different refractive indices, since the interference require• ments can then be met for several wavelengths (Zeiss coatings Super ET, Carl Zeiss T* (T-star), SMC: Super-Multi-Coating). In plastic lenses made of the material CR 39, a layer of quartz is vacuum-deposited on a inhomogeneous primary layer whose refractive index starts at that of CR 39 and gradually increases (Clarlet ET). The antireflection coating increases transmission and enhances .

Interference filters Interference filters consist of a glass plate with at least two very thin, partially transparent metallic layers separated by non-me• tallic layers with a thickness of 250 to 500 nm. Narrow spectral transparency maxima are produced in various wavelengths (colours) by interference of the light waves transmitted within these layers. Application: Interference colour filters for different band- widths. Interference filters are also produced with purely mirrors; these consist of X/4-thick layers of non-conductive and non-absorbing materials with alternating large and small refrac• tive indices.

Diffraction Diffraction consists in the propagation of electromagnetic waves laterally to the geometric-optical ray direction (Fig. 41). It occurs when light waves pass through stops, lens mounts, slits, high-contrast microscopic specimen structures propagation (so-called amplitude specimens) or through a fine mosaic of in all low-contrast, light-transmitting specimens with different refrac- plane directions j indices (phase specimens), or when X-rays pass through the wavefront t ve atomic lattice of . An additional interference effect between several adjacent diffraction centres causes strip-like narrow stop propagation. The distance between the strips is increased with Fig. 41 Principle of diffraction increasing wavelength of light. This allows the production of diffraction spectra with diffraction (featuring up to 6000 grooves per mm). Application: The phase contrast method in microscopy is based on diffraction and additional interference with controlled am• plitude and phase. Diffraction is the reason why optical imagery is never 100% sharp; Airy's discs arc produced, reducing the resolution in the PHYSICAL OPTICS: Wave Optics 41

image. This is why a lens should never be stopped down further than f/25 if good resolution is required. Due to diffrac• tion, the upper limit of the useful magnification of telescopes and microscopes is reached when the objective and are so selected that the exit pupil is about 0.5 mm in . The halo around the and moon are due to diffraction at particles in the atmosphere.

Holography While classic only supplies a flat image with an improved , holography involves the recording of the entire three-dimensional structure of a specimen. The specimen is illuminated with coherent light, and the light waves scattered from the object are superimposed with coherent reference light from the same source. The resulting interference pattern is recorded on a high-resolution photosensitive layer (without optics!). A hologram contains in its diffrac• tion pattern the information about the amplitude and phase of the light waves emanating from the specimen and therefore allows these to be reconstructed in suitable monochromatic illumination. This produces a three-dimensional image of the object. For viewing so-called white-light holograms, even day• light or a spotlight are suitable.

Polarisation

Brewster's law Natural light is unpolarised, i.e. for the transverse vibrations of the wave trains there is no preferred direction. When unpolar- ized light meets an interface between air and an electrically non-conductive substance with the refractive index n', one part of the light will enter the medium and, in the case of oblique air (n = 1) incidence, be refracted; the other part is reflected at the inter• face. If the angle of incidence equals the polarising angle /r, all waves of the reflected light vibrate perpendicularly to the plane of incidence (Fig. 42). This is known as plane polarised light. In this case, the reflected ray and the refracted ray are at right angles to each other. For the angle of complete polarisation Brewster's law is

Fig. 42 (37) tan iB = n'. Polarisation by reflection

For crown glass and window glass iB = 56.7°, for water 42 PHYSICAL OPTICS: Wave Optics

surfaces it is 53° and about the same for well-worn roads, ice-rinks, fields covered in snow, glossy coatings of paint, etc. The degree of polarisation denotes what proportion of the light of a bundle is polarised. As the light incident at the Brewster

angle iB is fully polarised subsequent to reflection, the degree of polarisation is 100% in this case. In the vicinity of the Brewster angle the degree of polarisation is reduced, but it is still notice• able. The amount of polarised light gained by reflection is minimal, as glass surfaces reflect only about 8 % of the light incident at the Brewster angle.

Bi-refringence Except along the so-called optical axis, all non-cubic crystals have two different refractive indices for each wavelength (colour) of light. This property is designated bi-refringence, and the two differently refracted partial rays are plane polarised. The polarising directions of ordinary and extraordinary rays are perpendicular. Bi-refringence is used in the design of polarisers (quantity of polarised light up to 45 %) and analysers. Polarising apparatus includes two such elements. In the crossed position the analyser does not transmit any light coming from the polariser (sensitive extinction position). Application: Polarising microscopes for identifying crystals and minerals by utilising their influence on polarised light. Singly-refracting, transparent substances become bi-refringent when they are subjected to mechanical strain. Strain bi-refrin• gence occurs, for example, in lenses which are strained due to insufficient cooling, external pressure or thermal hardening (Fig. 43). Bi-refringent plates produce path differences of the light waves and thus interference colours between crossed polarisers. Application: for strain testing and in polarising interference filters.

Dichroism It is a property of some bi-refringent substances to absorb the polarised light of the ordinary and extraordinary rays different• ly. This so-called dichroism can lead to almost entire absorption of one partial ray. Application: polarising filtersan d membranes. PHYSICAL OPTICS: Wave Optics 43

Fig. 43 Thermally hardened spectacle lens in the strain tester

Optical activity A substance is called optically active if it rotates the vibration plane of linearly polarised light (when looking towards the oncoming light, this optical activity is known as right-handed if the rotation is clockwise, and left-handed if anti-clockwise). The size of the angle of rotation depends amongst other things on the colour of the polarised light: rotary dispersion. Application: polarimeters for determining the concentration of sugar in solution. 44 PHYSICAL OPTICS: Light technology

Light technology Photometry

Terminology As light is electromagnetic radiation, its energy and power can be measured objectively. The sensation of light obtained by the eye is, however, based on physiological processes, making vision a subjective procedure. Physical photometry is the field of objective light measurement using physical receivers, while visual photometry uses the eye as the receiver. In order to measure the power of a light source, it must be placed at the centre of a sphere and the light energy measured per unit time on the surface of the sphere. The solid angle Q is the ratio of a surface area A of a sphere to the square of the radius r of the sphere (Fig. 44):

(38) Q = 4r2 Fig. 44 2 Radiation at the solid angle 0 Q0 is the solid angle which cuts the area A = 1 m from the surface of the standard sphere (radius r = 1 m). This unit for = sr solid angles is called a steradian: Q0 1 (dimension 1). The plane area of the aperture angle of the cone apertaining to QQ, whose point lies in the centre of the sphere, is 65.54°. A solid angle of Q = 4 n sr corresponds to the area of a full sphere (surface of the standard sphere).

The light output perceived by the eye, which is emitted by a light source at a specific solid angle Q, is the luminous flux

(39) I =

The unit of luminous intensity is called candela (cd). Until 1979 this was determined by a standard radiator (black area of A = 1 /60 cm2 at the temperature 2045 K of molten platinum). The present-day definition is based on the luminous efficacy: "The candela is the luminous intensity in a specific direction of a radiation source which emits monochromatic radiation of the PHYSICAL OPTICS: Light technology 45

frequency 540 . 1012 Hz and whose radiant intensity in this direction is 1/683 W/sr." A radiator with the luminous intensity I = 1 cd emits the luminous flux O = 1 lm (1 lm = 1 cd. sr) at the solid angle QQ — 1 sr. A light source with a luminous intensity of I = 1 cd radiating evenly into the space of the full sphere would emit the luminous flux O = 4 n lm. The brightness of a surface which emits or reflects light is characterized by its L. The luminance gives the luminous intensity emitted per unit of the surface which appears to be reduced by the factor cos i at an angle i from the surface normal:

(40) L = A • cos i' Its unit of measurement is cd/m2. (An older unit of measurement is the apostilb: 1 asb = \/n cd/m2). The luminance of the former standard radiator was: L = 6. 105cd/m2. Table 7 shows the luminance values of some radiators and the individual areas of vision. The luminance cannot be increased by optical image formation because the luminous flux fills an even larger solid angle (when the image size is reduced), although it is in fact concentrated on a smaller surface. In practice, the illumination of a specific surface is of particular importance. The illuminance E is the luminous flux O imping• ing on the surface A divided by the size of the surface:

(4.) E-f.

The unit of illuminance is lux (lx). 1 lx = 1 lm/m2. The illuminance is dependent on the luminous intensity I of the light source, its distance r from the illuminated surface and the angle of incidence i (inverse square law):

(42) E = i • cos i • f20-

When a light source of the luminous intensity I = 1 cd is positioned in the centre of the standard sphere, the spherical surface is illuminated with the illuminance E = 1 lx. When r = 2 m, E = 1/4 lx, etc. Application: photometer for measuring light sources. PHYSICAL OPTICS: Light technology

In order to darken , not only the illuminance E is important, but also the time of illumination t. The product of both is the H measured in lux seconds (lx • s). (43) H = E t. When a luminous flux

Luminous efficacy In , power is generally given in watts (W). A relationship exists between this unit and the unit lumen. For green light of the frequency 540 • 1012 Hz (wavelength in air: X = 555 nm; maximum light sensitivity of the eye) 1 W = 683 lm. A green light source with the power of 1 W thus has a corresponding light power of 683 lm for the eye, this applying to all conditions of adaptation. For daylight vision, this is the maximum value

for the luminous efficacy Km = 683 lm/W at the same time. Other colours of the spectrum (wavelengths X) have lower values K(X) for the luminous efficacy corresponding to the spectral sensitivity of the eye for daylight vision V(X) (see Fig. 63):

(45) K(X) = V(X)Km. For , the maximum of the spectral sensitivity V'(X) lies at 507 nm and the maximum value for the luminous efficacy

isK'm = 1699 lm/W.

Material properties

Terminology In order to characterise the influence of various media and their interfaces on the distribution of a specific luminous flux, the following units with the dimension 1 (formerly called dimen- sionless) are used (Fig. 45). The reflectance p of an interface between two media is the ratio

of the luminous flux O0 reflected at this surface to the incident luminous flux : PHYSICAL OPTICS: Light technology 4~

(46)

InFig.45D,=^andp2 = |^.

The internal transmittance X; of a body is the ratio of the

luminous flux

Oin entering through the entrance surface:

(47) (f),

Fig. 45 The transmittance x of a body is the ratio of the luminous flux OT Influence of a medium on the lumi• leaving the exit surface to the luminous flux O incident on the nous flux entrance surface: Ox (48) T =

The internal absorptance a* of a body is the ratio of the lumi•

nous flux Oa absorbed between the entrance and exit surfaces to

the luminous flux Oin entering through the entrance surface:

(49) a, = |^=l-xi

where (J) = OIN - OEX in an optically clear (non-) body.

The absorptance a of a body is the ratio of the luminous flux OA absorbed between the entrance and exit surfaces to the lumi• nous flux 0 incident on the entrance surface:

<1> (50) a =

If a medium with the refractive index n is located in front of an interface and a medium with n' is behind it, the reflectance with perpendicularly incident light is:

(51) n' - nV n' + n Fig. 46 shows the reflectance p of some substances in air as a function of the wavelength (colour) of light. In German, is also called "Remission". The light reduction (light attenuation) of a body is the ratio of the difference between the incident luminous flux fl> and the

emerging luminous flux Ot to the incident luminous flux: 48 PHYSICAL OPTICS: Light technology

Fig. 46 Mean values for the spectral reflect• ance of polished metal surfaces (Al — aluminium, Cr = chromium, Au = gold, Cu = copper, Ni = nickel, Ag = silver) in comparison with specta• cle crown glass (K uncoated, T with anti-reflection coating)

Influence of path length For bodies of any thickness consisting of homogeneous, opti• cally clear materials (e.g. coloured glass or clear solutions), the light properties can be calculated if they have been measured on a body made of the same material with a known thickness. If a property is dependent on the wavelength, the calculation must then be performed separately for every wavelength. If, for example, the light transmittance is needed for a specific illumi- nant, it must be calculated from the light transmittance values for the individual wavelengths subsequent to the thickness conversion; a direct calculation using a light transmittance measured at a different thickness is not possible. The following formula applies for optically clear materials:

(53) a + pK + x = 1

where the reflectance pk of a body must be calculated from the internal transmittance Xj of the body and the reflectance q of an interface:

(54) PK = p[l+(l-p)2Xi2]. For the internal transmission and absorption, i.e. neglecting reflection losses, the following equation holds: PHYSICAL OPTICS: Light technology 49

(55) Tj + Oj = 1. The relationship between the transmittance and the internal transmittance is: (56) x = (l-p)2T, If on a body with the thickness di the internal transmittance t\\ is measured or calculated my means of (56) using the measured

transmittance, the internal transmittance xi2 of a body made of

the same material with the thickness d2 is: (57) Tj2 = The internal transmission density A is used to simpify the calculation: (58) A - - logti. Table (8) gives some values using formula (58). The internal transmission density A, which is also known as attenuation (usually called in ), is propor• tional to the path thickness; the factor of proportionality is the extinction coefficient a: (59) A = a • d. In the above formulae multireflections inside the body have been neglected, this always being admissible for reflection on glass surfaces even if they have not been provided with an anti-reflection coating.

Light sources

Daylight The spectral composition and the directional characteristic of daylight are dependent on the position of the sun and weather conditions, ranging from direct sunlight with the blue scattered light of the to totally diffuse illumination through a cloudy sky. Its spectrum is continuous, but the spectral energy distribu• tion is not a smooth curve due to the different thickness and density of the numerous, extremely fine Fraunhofer absorption lines in the solar spectrum and to the atmospheric absorption bands. 50 PHYSICAL OPTICS: Light technology

Incandescent lamps In an incandescent lamp a tungsten filament is heated to a temperature of about 3000 °C. The emitted light displays a continuous spectrum with a very smooth energy distribution curve showing a small blue portion and a maximum in the near . The light output ranges between 8 and 25 lm/W. Standard incandescent lamps have an average life of 1000 operating hours. After this, the luminous flux decreases to 80% of the initial value. If the operating voltage is set higher than the rated voltage, the luminous flux and the light output increase, but the lifetime of the lamp is decreased considerably. With reduced voltage, on the other hand, the opposite occurs: luminous flux and light output decrease and the lifetime of the lamp increases. The brightness of the image produced by an is dependent on the luminance (not the luminous flux !) of the lamp used. For this reason, special incandescent lamps for projection instruments have a high luminance, but a reduced lifetime.

Fluorescent lamps Fluorescent lamps are low-pressure mercury lamps whose in• tense UV radiation (negligibly little of which penetrates to the exterior) excites the fluorescent coating on the inner surface of the lamp. The exciting lines of the mercury vapour (no neon!) are superimposed on the continuous spectrum of the fluores• cent substance. The composition of the fluorescent substance determines the colour of the light with a blue portion increasing from a "warm tone" to "pure white". The luminous flux is dependent on the surrounding temperature and the operating time. The light output and the lifetime of a fluorescent lamp are considerably higher than in incandescent lamps.

Spectral lamps Spectral lamps are metal vapour lamps for the generation of line spectra which are used in combination with suitable colour filters for the production of monochromatic light. In high-pres• sure mercury lamps a weaker continuous spectrum is superim• posed on the line spectrum. The arc discharge of these lamps is fibre-like or punctiform, which increases the luminance to solar brightness. For street , bluish mercury vapour lamps and yellow sodium lamps are also used; these emit a spectrum consisting of only a few lines and result in a pronounced falsification of body colours. PHYSICAL OPTICS: Light technology 51

Lasers " " stands for Light Amplification by of Radiation. Laser light is virtually monochromatic, coherent and parallel, and high light powers are possible. In a laser an exciting light wave is amplified by an active medium. The laser is named after the type of medium used: solid state, liquid (dye) and gas lasers. In solid state lasers, the neodym-YAG laser (k — 1064 nm) has gained considerable significance in ophthalmic surgery (YAG = yttrium-alumini• um-garnet). Liquid lasers contain dissolved dyes as their medi• um. Major gas lasers are the argon laser (k = 488 and 514.5 nm), which is used in ophthalmic surgery for the treatment of retinal detachment, and the helium-neon laser (k = 633 nm).

Standard illuminants If material properties are to be specified not for monochro• matic radiation, but for "white" light, they are dependent on the spectral composition of this light. The same applies for the translucent or reflected colour of coloured bodies. In order to be able to make clear specifications, so-called standard illumi• nants have been established. The most important of these are (Fig. 47):

- 100

~ i D65/ fC A

y i V

- i / i 1 ~l / -i / i 1 Relative spectral radiance of standard 400 600 nm 800 illuminants A, C and D 65 Wavelength X in air

1. Standard illuminant A represents incandescent lamp light and is obtained by incandescent lamps with the colour temperature 2850 K. 2. Standard illuminant D 65 represents average daylight with the colour temperature 6500 K and is normally obtained by xenon lamps with filters. 52 PHYSICAL OPTICS: Light technology

3. Standard illuminant C is a former (less good as it contains no UV) approximation of daylight and is obtained by incandes• cent lamps with filters.

Colour temperature The so-called colour temperature of a radiator (filament, etc.) is identical to the temperature of the black body which glows in the same chromacity as the radiator. Daylight film for colour photography is adapted to 5500 K, artificial light film to 3200 K. Adaptation between the radiator and the photographic emulsion is achieved by using special colour filters known as conversion filters. Table 9 gives the colour temperatures of some radiators.

Light guides

Principle Light can also be conducted without image formation in a (straight or curved) pipe if, after entering through the entrance aperture, it is reflected without loss each time it strikes the wall of the pipe. In a few special cases and for very short distances pipes which feature an inner metal reflective coating can be used as light guides. As even the best metal mirrors do not achieve the reflectance 1, the losses quickly increase with multi• ple reflection. If the pipe is filled with a light-permeable material whose refractive index is greater than that of its surroundings, however, total reflection occurs for sufficiently large angles of incidence, and the light can even be conducted over large distances virtually without loss.

Numerical aperture If the interior ("core") of a straight cylindrical light guide (Fig.

48) exhibits the refractive index n; and is surrounded by a

medium ("cladding") with the refractive index na, all rays entering through a surface perpendicular to the cylinder axis

from a medium with the refractive index n0 are conducted through the light guide by means of total reflection, provided

their angle u from the cylinder axis is smaller than the angle umax

which results from the AN of the light guide:

Fig. 48 Total reflection in a light guide (60) AN = l/nf^ = n0-sinu, PHYSICAL OPTICS: Light technology 53

AN can easily assume values over 1. If the numerical aperture

n0 • sin u of the light entering the light guide is smaller than the

numerical aperture AN of the light guide, the latter is not fully utilized. This is not necessarily a drawback, as it offers a high degree of safety against light loss caused by bending and form errors. If the incident light emanates from a medium with the

refractive index n0 < AN, it is impossible to fully utilize the numerical aperture of the light guide.

Attenuation Despite total reflection, losses occur in the core (absorption and scatter), in the cladding (absorption) and in the core/cladding interface (surface disturbances) of the light guide. The first two effects can never be fully eliminated. These losses cause the internal transmittance to decrease with increasing length of the light guide in accordance with (57). The internal transmission density A (attenuation) is proportional to the length in accord• ance with (59). The extinction coefficient a (attenuation coefficient) is used to characterise the losses of a light guide and is generally specified in (dB) per length unit. 1 dB corresponds to the linear absorption density 0.1 (Table 8). The normal unit for the characterisation of light guides for illumination purposes is dB/m, for communication fibres dB/km. The extinction coeffi• cient is dependent on the wavelength.

Solid and liquid Rods made of glass, and transparent plastics with light guides polished or fire-polished surfaces can be used without cladding

(outer medium air with na = 1) if they are mounted without supports. The higher their refractive index, the narrower the curvatures of the rods can be. In constructions with cladding, contact from the exterior causes no disturbance. In liquid light guides the liquid must have a higher refractive index than the tube material. The interior of the tubes must be smooth, and they exhibit flexibility with similar cross sections to the rods. Some of the technical problems involved are imper• meability, freedom from bubbles and durability (including mi• crobe infestation). PHYSICAL OPTICS: Light technology

Optical fibres and In optical fibres the fibre core is surrounded by cladding with a fibre bundles lower refractive index; the core and the cladding are made of glass or fused quartz. They are covered by a protective coating which is usually made of plastic. As the cladding insulates the fibres optically from their surroundings, they can be guided in ducts or tubes without any light leaking at the points of contact. By combining many fibres into one bundle, the light-conducting cross section is increased without any decrease in flexibility. The entire bundle cross section is not available for the light conduction, however, due to the cladding and the unused area between the fibres. In fibre bundles which serve only to conduct light without image formation the individual fibres are random• ly arranged.

Image carriers and Fibre bundles which contain fibres in exactly the same configu- shape converters ration at both end-faces allow the transmission of an image projected onto one end-face to the other end-face, although with a raster corresponding to the fibre spacing (Fig. 49). If the fibres are fixed in position at the two end cross sections only and are freely movable in between, a flexible image guide results such as that frequently used for endoscopes. Fibre bundles fixed in position along the entire length are rigid and are mostly used in the form of so-called fibre optic faceplates for adapting flat entrance and exit faces to curved cathodes and the display screens of electro-optical image intensifiers and image con• verters. The fibre length is the same as the plate thickness which, in turn, is smaller than the plate diameter.

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Fig. 49 Image transmission by a light- conducting fibre bundle: a) entrance b) exit a) PHYSICAL OPTICS: Light technology 55

The fibres of a bundle can have different cross-sectional confi• gurations at the two faces of a light guide, but the same surface area (shape converters). One face, for example, can be in the shape of a circle and the other a long, narrow rectangle for adapting a round beam cross section to the slit of a spectro• graph. The fibres of a bundle can also be split up into several bundles (multiple-end light guides); the fibres belonging to the partial bundles can then be arranged in the common cross section in specific zones, or they can be evenly mixed or arranged randomly.

Tapered light guides Solid light guides and even fibre bundles may also be conical in shape. With a "light funnel" of this type the light incident through the larger face can be concentrated within certain limits on a smaller surface. However, the product of the diameter and the numerical aperture of the light entering the light guide is retained, and the largest possible numerical aperture at the exit sets a limit to the reduction in the diameter. If this limit is exceeded (without the numerical aperture of the light guide itself being exceeded), some of the light rays turn back and leave the light guide through the large cross section (Fig. 50). Lumi• nance can therefore be increased just as little by this method as by optical image formation. Fig. 50 shows the multiple mirror images of the limiting sur• Fig. 50 faces a and b. Only rays such as 1 whose extension (marked with Exceeding the aperture in a tapered a broken line) strikes the inner area of the sector figure can light guide emerge through the small face. Rays such as 2 whose extension does not strike the inner area leave the conical light guide again through the large entrance face after multiple reflection.

Optical waveguides Depending on the properties of the material of which they are made, light guides can be used to conduct optical radiation outside the visible spectral region (UV, IR). They are called optical waveguides if the wave properties of light which have not been taken into account until now play a role. Their effect consists in allowing waves, including those within the angular range defined by the numerical aperture, to travel in certain individual directions only. These types of are called modes. While the light guides described so far effect linear conduction of light, planar waveguides restrict propagation to a single plane. They normally consist of a substrate, a thin wave-con- 56 PHYSICAL OPTICS: Light technology

ducting film, whose refractive index must be greater than that of the substrate, and a coating whose refractive index is no greater than that of the substrate and which can also be air. The radiation is conducted by total reflection in the direction per• pendicular to the film plane.