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5 Light Sources and Detectors 5.1 INTRODUCTION The most important ‘hardware’ in optical metrology is light sources and detectors. To appreciate the various concepts of these devices, we first introduce the different units and terms for the measurement of electromagnetic radiation. Then the laser is given a relatively comprehensive treatment. The description of detectors involves some understanding of semiconductor technology. Therefore a brief introduction to semiconductors is given in Appendix E. Because of the increasing use of the CCD camera in optical metrology, this device is described separately in Section 5.8. 5.2 RADIOMETRY. PHOTOMETRY To compare light sources we have to make a brief introduction to units and terms for the measurement of electromagnetic radiation (Slater 1980; Klein and Furtak 1986; Longhurst 1967). Below we present the most common radiometric units. Radiant energy, Q, is energy travelling in the form of electromagnetic waves, measured in joules. Radiant flux,  = ∂Q/∂t is the time rate of change, or rate of transfer, of radiant energy, measured in watts. Power is equivalent to, and often used instead of, flux. Radiant flux density at a surface, M = E = ∂/∂A, is the radiant flux at a surface divided by the area of the surface. When referring to the radiant flux emitted from a surface it is called radiant exitance M. When referring to the radiant flux incident on a surface it is called irradiance E. Both are measured in watts per square metre. Note that in the rest of this book, we use the term intensity, which is proportional to irradiance. Radiant intensity, I = ∂/∂, of a source is the radiant flux proceeding from the source per unit solid angle in the direction considered, measured in watts per steradian. Radiance, L = ∂ 2 /∂∂Acos θ, in a given direction, is the radiant flux leaving an element of a surface and propagated in directions defined by an elementary cone containing the given direction, divided by the product of the solid angle of the cone and the area of the projection of the surface element on a plane perpendicular to the given direction. Figure 5.1 illustrates the concept of radiance. It is measured in watts per square metre and steradian. Optical Metrology. Kjell J. G ˚ asvik Copyright  2002 John Wiley & Sons, Ltd. ISBN: 0-470-84300-4 100 LIGHT SOURCES AND DETECTORS Surface normal P d A dΩ q Figure 5.1 The concept of radiance Table 5.1 Symbols, standard units and defining equations for fundamental radiometric and pho- tometric quantities Symbol Radiometric quantity Radiometric units Defining equation Photometric quantity Photometric units Q Radiant energy J Luminous energy lm s  Radiant flux W  = ∂Q/∂t Luminous flux lm M Radiant exitance Wm −2 M = ∂/∂A Luminous exitance lm m −2 E Irradiance W m −2 E = ∂/∂A Illuminance lm m −2 I Radiant intensity Wsr −1 I = ∂/∂ Luminous intensity lm sr −1 LRadiance Wsr −1 m −2 L = ∂ 2 /∂∂A cos θ Luminance lm sr −1 m −2 All of the radiometric terms have their photometric counterparts. They are related to how the (standard) human eye respond to optical radiation and is limited to the visible part of the spectrum. In Table 5.1 we list the radiometric and the corresponding photometric quantities. To distinguish radiometric and photometric symbols they are given subscripts e and v respectively (e.g. L e = radiance, L v = luminance). The radiometric quantities refer to total radiation of all wavelengths. A spectral version for each may be defined by adding the subscript λ (e.g. M eλ or simply M λ ) where for example a spectral flux  λ dλ represents the flux in a wavelength interval between λ and λ + dλ, with units watts per nanometre (Wnm −1 ) or watt per micrometre (W µm −1 ). To represent the response of the human eye, a standard luminosity curve V(λ)has been established, see Figure 5.2. It has a peak value of unity at λ = 555 nm. The conversion RADIOMETRY. PHOTOMETRY 101 200 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 300 400 500 600 700 800 900 Wavelength (nm) V (l) Figure 5.2 The standard luminosity curve at λ = 555 nm is standardized to be K m = 680 lm/W (5.1) This means that 1 W of flux at 555 nm gives the same physical sensation as 680 lm. For other wavelengths the conversion factor is K = K m V(λ)= 680 V(λ)lm/W (5.2) We may use Equation (5.2) to convert any radiometric quantity to the corresponding photometric quantity. For instance, if we have a spectral radiant flux  eλ , the luminous flux is given by  v = 680  ∞ 0 V(λ) eλ dλ(5.3) An average conversion factor is defined by K av =  v / e (5.4) 102 LIGHT SOURCES AND DETECTORS 5.2.1 Lambertian Surface A Lambertian surface is a perfectly diffuse reflecting surface defined as one which the radiance L is constant for any angle of reflection θ to the surface normal. Lambert’s cosine law states that the intensity (flux per unit solid angle) in any direction varies as the cosine of the reflection angle: I = I 0 cos θ(5.5) Since the projected area of the source also varies as cos θ, the radiance becomes inde- pendent of the viewing angle: L = I dA cos θ = I 0 dA (5.6) Assume that an elemental Lambertian surface dA is irradiated by E in W m −2 and that the radiant flux reflected in any direction θ to the surface normal is given by the basic equation d 2  = L d dA cos θ(5.7) The solid angle d in spherical coordinates (see Figure 5.3) is given by d = (r sin θ dθr dφ)/r 2 = sin θ dθ dφ(5.8) The total radiant flux reflected into the hemisphere therefore is given by d h =  2π 0 dϕ  π/2 0 L dA cos θ sin θ dθ = πLdA(5.9) d A dq q r dq r sin q r Figure 5.3 RADIOMETRY. PHOTOMETRY 103 The ratio of the total reflected radiant flux to the incident radiant flux d i = EdA defines the diffuse reflectance of the surface d h d i = ρ = πL E (5.10) The quantity ρE is the radiant flux density reflected from the surface which is equivalent to the radiant exitance M of a self-emitting source, giving M = πL (5.11) for a Lambertian surface. For non-Lambertian surfaces, L is a function of both θ and the azimuthal angle φ and therefore can not be taken outside the integral in Equation (5.9). Many natural surfaces show Lambertian characteristics up to θ = 40 ◦ . In satellite observations, one has found snow and desert to be Lambertian up to about 50 ◦ or 60 ◦ . Most naturally occurring surfaces depart significantly from the Lambertian case for θ greater than about 60 ◦ ,an exception is White Sands, the desert in New Mexico, which is nearly Lambertian for all angles. 5.2.2 Blackbody Radiator A blackbody at a given temperature provides the maximum radiant exitance at any wave- length that any body in thermodynamic equilibrium at that temperature can provide. It follows that a blackbody is a Lambertian source and that it is a perfect absorber as well as a perfect radiator. The spectral radiant exitance M λ from a blackbody is given by Planck’s formula M λ = 2πhc 2 λ 5 [exp(hc/λkT ) − 1] (5.12) where h = Planck’s constant = 6.6256 × 10 −34 Js; c = velocity of light = 2.997925 × 10 8 ms −1 ; k = Boltzmann’s constant = 1.38054 × 10 −23 JK −1 ; T = absolute temperature in kelvin; λ = wavelength in metres. which gives M λ in W m −2 µm −1 . Figure 5.4 shows M λ as a function of wavelength for different temperatures. By integrating over all wavelengths we get the Stefan–Boltzmann law M =  ∞ 0 M λ dλ = σT 4 (5.13) where σ = (2π 5 k 4 )/(15c 2 h 3 ) = 5.672 × 10 −8 Wm −2 K −4 is called the Stefan–Boltz- mann constant. 104 LIGHT SOURCES AND DETECTORS 100 200 400 6001000 10 000 l m T = 2898 µm K 10 000 K 200 K 300 K 500 K 1000 K 2000 K 6000 K 10 −1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 1 Wavelength (nm) Radiant exitance (Wm −2 µm −1 ) Figure 5.4 Spectral radiant exitance from a blackbody at various temperatures according to Planck’s law By differentiating Equation (5.12) we get the wavelength λ m for which M λ is peaked λ m T = 2897.8 µmK (5.14) This relation is called Wien’s displacement law. The blackbody is an idealization. In nature most radiators are selective radiators, i.e. the spectral distribution of the emitted flux is not the same as for a blackbody. Emissitivity is a measure of how a real source compares with a blackbody and is defined as ε = M  /M (5.15) where M  is the radiant exitance of the source of interest and M is the radiant exitance of a blackbody at the same temperature. ε is a number between 0 and 1 and is in general both wavelength and temperature dependent. When ε is independent of wavelength the source is called a greybody. A more general form of Equation (5.15) can be written to take into account the spectrally varying quantities, thus ε, the emissivity for a selective radiator, as an average over all wavelengths is ε = M  M =  ∞ 0 ε(λ)M λ dλ  ∞ 0 M λ dλ = 1 σT 4  ∞ 0 ε(λ)M λ dλ(5.16) Consider two slabs of different materials A and B, and that each is of semi-infinite thickness and infinite area, forming a cavity as shown in Figure 5.5. Assume that A is a RADIOMETRY. PHOTOMETRY 105 A B M M bb r M bb Figure 5.5 Radiant exitances between a blackbody A and another material B blackbody and that B is a material with emissivity ε,reflectanceρ and absorbtance α,and that the materials and cavity are in thermal equilibrium. Because of the last assumption, the flux onto B must equal the flux leaving B toward A. Thus M bb = ρM bb + M(5.17) where M bb and M are the radiant exitance of A (the blackbody) and B respectively. From the definition of emissivity we have M/M bb = ε = 1 − ρ(5.18) which is referred to as Kirchhoff’s law. Because of conservation of energy, the reflectance, transmittance and absorbance at a surface add up to unity. Since we have assumed semi-infinitely thick materials, the transmittance is zero and we have M/M bb = ε = 1 − ρ = α(5.19) where α is the absorptance of material B. Equation (5.19) states that good emitters and absorbers are poor reflectors and vice versa. We can anticipate that Equation (5.19) holds for any given spectral interval which gives the more general form M λ /M λbb = ε(λ) = 1 − ρ(λ)= α(λ) (5.20) 5.2.3 Examples Let us compare the light from a typical He–Ne laser and a blackbody with the same area as the output aperture of the laser. Assume this area to be 1 mm 2 and the blackbody temperature to be 3000 K, close to the temperature of the filament of an incandescent lamp. From Equation (5.13) we find the blackbody exitance to be 4.6 × 10 6 Wm −2 which gives a radiant flux of 4.6 W. An ordinary He–Ne laser has an output of about 1/1000th of this, not very impressive even if we take into account that most of the radiation from the blackbody is outside the visible region. From Equation (5.11) we find the radiance from the blackbody to be L = M/π = 1.46 × 10 6 Wm −2 sr −1 106 LIGHT SOURCES AND DETECTORS The light beam from the laser has a diverging angle of about λ/d where λ is the wave- length and d is the output aperture diameter. This gives a solid angle of about λ 2 /A where A is the aperture area. The radiance at the centre of the beam is therefore (cos θ = 1) L =  A =  λ 2 (5.21) With a radiant power (flux)  = 5 mW and a wavelength λ = 0.6328 µm, this gives L = 1.2 × 10 10 Wm −2 sr −1 , a number clearly in favour of the laser. Note that the radiance of the blackbody is independent of its area. By decreasing the power of the laser by reducing its output aperture, the radiance decreases accordingly. Figure 5.6 illustrates the imaging of an object of elemental area dA o by a lens system with the entrance and exit pupils as sketched. We assume that the object is a Lambertian surface of radiance L o . The flux incident over an annular element of the entrance pupil is given by d 2  = L o dA o cos θ d(5.22) where d = 2π sin θ dθ(5.23) If θ m is the angle of the marginal ray passing through the entrance pupil, the flux incident over the entrance pupil is d o = 2πL o dA o  θ m 0 sin θ cos θ dθ = πL o dA o sin 2 θ m (5.24) Equation (5.24) is not the product of radiance, area and solid angle, or 2πL o dA o (1 − cos θ m ), as we might at first expect, because the cosine factor, which accounts for the projected area in any direction in the solid angle, has to be included in the integration. We can write a similar expression for the flux d i incident over the exit pupil from a fictitious Lambertian source L i , in the plane of the image. Then, evoking the principle of q q m d A o dΩ′ dΩ d A i Entrance pupil Exit pupil q′ q′ m Figure 5.6 Geometry for determining the radiometry of an optical system RADIOMETRY. PHOTOMETRY 107 the reversibility of light, we can say that this flux, leaving the exit pupil in the direction of the image, gives rise to an image plane radiance L i according to d i = πL i dA i sin 2 θ  m (5.25) where dA i is the image area and θ  m is the inclination of the marginal ray in image space. For a perfect lossless system (that is, one without reflection, absorption and scattering losses), d o = d i ,so L i sin 2 θ  m dA i = L o sin 2 θ m dA o (5.26) Now dA i dA o = m 2 (5.27) the square of the lateral magnification. We assume that the lens is aplanatic, obeying the Abbe sine condition, that is, it exhibits zero spherical abberation and coma for objects near the axis, no matter how low the F -number. (Spherical abberation is illustrated in Figure 2.10). Thus, n sin θ m = mn  sin θ  m (5.28) where n and n  are the refractive indices in object and image space which we set equal to unity. Then sin θ m sin θ  m = m(5.29) and we get L i = L o which shows that the radiance is conserved in a lossless imaging system. Equation (5.25) then gives for the image irradiance E i = d i /dA i = πL o sin 2 θ  m (5.30) As usual in paraxial optics, we approximate sin θ  m by tan θ  m , giving sin θ  m ≈ tan θ  m = D i 2b (5.31) where D i is the diameter of the exit pupil and b is the image distance. By introducing the aperture number F = f/D i where f is the focal length, Equation (5.30) becomes E i = πL o  D i 2b  2 = πL o 4F 2 (1 + m) 2 (5.32) With the object at infinity, b = f and we get E i = πL o  D i 2f  2 = πL o 4F 2 (5.33) 108 LIGHT SOURCES AND DETECTORS Equation (5.33) or a similar form of it, is generally referred to as the ‘camera equation’. It indicates that image irradiance is inversely proportional to the square of the F -(aperture) number. Therefore the diaphragm or stop openings for a lens are marked in a geometrical ratio of 2 1/2 . Recall that we have assumed the object to be a Lambertian surface. For example, for a point source of radiant intensity I as the object, the flux intercepted by the entrance pupil is d = I d = IS a 2 = Iπ  D 2a  2 (5.34) where S is the area and D is the diameter of the entrance pupil, a is the object dis- tance and where we for simplicity assume the entrance and exit pupils to have equal area S. If we take the image area dA i to be equal to the area of the Airy disc (see Section 4.6, Equation (4.33)) dA i = π(r i ) 2 = 1.5π  λb D  2 (5.35) we get for the image irradiance E i = d dA i = 8 3 I λ 2  D 2a  2  D 2b  2 = I 6λ 2  m F 2 (1 + m) 2  2 (5.36) From this expression we see that the image irradiance is dependent on both the object and image distances. In conclusion we might say that for a Lambertian surface we can not increase the image irradiance by placing the lens closer to the object, but for a point source we can, the maximum occurring at unit magnification, i.e. when a = b = 2f . 5.3 INCOHERENT LIGHT SOURCES Most light sources are incoherent, from the candle light to the Sun. They all radiate light due to spontaneous emission (see Section 5.4.1). Here we will consider some sources often used in scientific applications. These are incandescent sources, low-pressure gas discharge lamps and high-pressure gas discharge-arc lamps. They are commonly rated, not according to their radiant flux, but according to their electric power consumption. Tungsten halogen lamps Quartz tungsten halogen lamps (QTH) produce a bright, stable, visible and infrared output and is the most commonly applied incandescent source in radiometric and photometric studies. It emits radiation due to the thermal excitation of source atoms or molecules. The spectrum of the emitted radiation is continuous and approximates a blackbody. Spectral distribution and total radiated flux depend on the temperature, area and emissivity. For a QTH lamp, the temperature lies above 3000 K and the emissivity varies around 0.4 in the visible region.