Kitabı oku: «Optical Engineering Science», sayfa 17

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7.7.3 Illumination Levels

Since optical photometry is fundamentally connected to light levels as mediated by the sensitivity of the human eye, levels of illuminance are intimately related to the ability to perform visually based tasks. For the indoor environment, lighting levels may be designed for specific areas. A generally comfortable level of illuminance for a domestic environment is around 100 lx. For an office environment, where moderately demanding visual tasks are to be performed, a level of 300–500 lx is acceptable. For more critical tasks, such as visual inspection, a higher level of 500–2000 lx may be called for. Of course, daylight illumination levels are very much higher, ranging from 1000 lx on an overcast day to 25 000 lx for full sunshine. Table 7.4 sets out some typical illumination levels for different environments:

Another important consideration in illumination sources is their efficiency. The efficiency of domestic and industrial light sources is measured in lumens per watt. From that perspective, the ideal light source is a monochromatic source with a wavelength of 555 nm, giving a maximum efficiency of 683 lm W⁻¹ of optical output. A reasonable approximation to this is the sodium vapour street lamp, providing virtually monochromatic light at 589 nm with an electrical efficiency of 200 lm W⁻¹. However, such a highly coloured source is not acceptable for domestic and industrial applications where broadband or nominally ‘white’ sources are preferred. The least efficient sources are incandescent tungsten sources which are being replaced in domestic and industrial applications due to their poor energy efficiency. Their immediate successors, fluorescent mercury lamps create broadband emission from fluorescent phosphor coatings irradiated by ultraviolet emission from mercury spectral lines. More latterly, these are being replaced by white light LEDs which rely on ultraviolet emission from gallium nitride diodes to create broad band fluorescence from phosphors. Efficiencies of these sources are set out in Table 7.5.

Table 7.5 Luminous efficiencies of different sources.

Figure 7.21 Luminous efficiency vs. blackbody temperature.

In fact, the luminous efficiency of a blackbody source may be calculated directly from the Planck distribution set out in Eq. (7.9) and the luminous efficiency function, V(λ). The plot is shown in Figure 7.21. The peak efficiency occurs around 6000 K and it is, of course, no coincidence that this is close to the solar blackbody temperature of 5800 K. Clearly, the human eye has been ‘designed’ to efficiently harvest light from its primary illumination source.

The brightness of different sources (to the human eye), or luminance, is expressed in candelas per square metre or nits. Representative values range from 80 cdm⁻² for a typical cinema screen to 7 × 10⁶ cdm⁻² for the filament of an incandescent lamp and 1.6 × 10⁹ cdm⁻² for the solar disk. As for the luminous efficiency plot, the luminance of a blackbody source may be derived directly from the Planck distribution and the luminous efficiency curve, V(λ). This plot is shown in Figure 7.22.

7.7.4 Colour

7.7.4.1 Tristimulus Values

The preceding discussion has been wholly concerned with the level of illumination rather than (human) perception of the spectral distribution. This spectral distribution is described by the notion of colour, as perceived by humans. From the perspective of human vision, colour is discerned by the relative stimulation of three types of colour receptors (the cones). To model this process, the CIE, in 1931, proposed a set of colour matching functions, effectively mimicking the relative sensitivity of each type of sensor. The colour matching functions are represented as three separate curves, x(λ), y(λ), and z(λ), and operate, in principle, in a similar manner to the V(λ) curve for photopic efficiency. However, each curve is shifted with respect to the others. The form of these curves is illustrated in Figure 7.23.

Figure 7.22 Luminance vs. blackbody temperature.

Figure 7.23 Colour matching curves.

It must be emphasised that the colour matching curves and the luminous efficiency curve are merely representative of human visual perception. These curves represent the fruits of sustained efforts to find a representative average of human perception. However, not surprisingly, there are considerable variations in spectral sensitivity between individuals.

Quite significantly, the y(λ) curve follows that of the standard V(λ) curve. As for the basic photometric quantities, an input spectral radiance is transformed by integrating across the spectral range using the colour matching functions. However, instead of producing a single luminous flux value, three separate tristimulus values, X, Y, and Z are derived, as below.

(7.25)

From the preceding arguments, the Y tristimulus value is a measure of the luminance of the source. Normalisation of the tristimulus values provides a two dimensional description of colour.

(7.26)

Only the x and y ordinates are used in the standard CIE chromaticity diagram which provides a standardised quantification of the human perception of colour. The chromaticity diagram provides a plot in these two dimensions, with the third degree of freedom effectively corresponding to the luminous flux or intensity. Although it is perhaps obvious from the preceding discussion, this tripartite description of colour is purely an artefact of human vision and in no sense related to any property of light. Indeed, in recording any manifestly complex or subtle spectral distribution, the human eye can only, in effect, describe these by three independent parameters. It is clear that it is very possible for different spectral distributions to produce the same X, Y, Z stimulus values. This effect is known as metamerism. This highlights the limited spectral information that is provided by the three different sensor types. Indeed, two surface coatings (e.g. painted) can appear to be the same colour under one illumination (e.g. fluorescent) but different under another illumination source (e.g. tungsten) because of this effect.

7.7.4.2 RGB Colour

In many instances in describing colour we are interested in the effect of adding or blending colours. As a result of the three sensor types it is clear, in principle, that a linear combination of three different colours may be used to create a wide range colour sensations. The three colours themselves are described by a linear combination of the tristimulus values, X, Y, and Z and are known as primary colours. Definition of the suite of primary colours is arbitrary and established by virtue of convention. The guiding principle is that these three colours must be capable of being admixed to create as wide as possible a range of colours without recourse to negative coefficients in the linear combination. This range is referred to as a gamut. The original standard (1931 CIE) colour representation is the so-called RGB system (Red – Green – Blue) of primary colours using three standard monochromatic stimuli at 700 nm (R), 546.1 nm (G), and 435.8 nm (B). In this scheme, definition of the RGB primary colours from the tristimulus values is given by the following (X, Y, Z) vectorial representation:

(7.27)

The inverse transformation between the two representations is effected by the following matrix:

(7.28)

Presentation of this RGB colour convention is intended for illustrative purposes only. This simple scheme has been largely superseded. In reality, there are a plethora of different primary colour conventions designed with specific applications, such computer screen rendition and so on, in mind. Some conventions take account of the non-linearity of the eye's response. That is to say, we abandon the linear convention hitherto prescribed. Other conventions ensure that uniform movements across colour space correspond to uniform changes in human perception of colour. These are called perceptually uniform colour spaces.

In principle, an equal admixture of primary colour components leads some form of standard white colouration or ‘white point’. The concept of whiteness as a chromatic descriptor is purely associated with the human perception of colour, rather than a fundamental property of a source. However, definition of whiteness, is convention dependent. Rather than defining a colour sensation by virtue of the admixture of RGB, it may also be defined by another three parameter set, HSL or hue, saturation, and luminosity. Hue is a measure of the undiluted colour, loosely corresponding to the equivalent monochromatic wavelength of stimulation. Saturation describes the purity of the colour, or the extent to which white must be admixed with a pure monochromatic colour to achieve the desired colour. The final degree of freedom is provided by luminosity which correlates to the brightness of the sensation, effectively the sum of the RGB components.

Colour difference, ΔE, is a measure of the absolute difference in colour between two different colours. It is generally expressed as the root sum of squares of the difference in each of the three colour ordinates and dependent upon the convention adopted. With this in mind, the concept of colour temperature describes the temperature of the blackbody radiator that most closely matches the colour of interest, i.e. with the smallest colour difference. This is particularly associated with the characterisation of light sources. Somewhat ironically, the term ‘cool’ describes a source with a bluer spectral distribution whereas a ‘warm’ light source refers to illumination with a larger red contribution. This is rather based on human perception and psychology; a ‘cool’, bluer blackbody source is, of course, hotter than a redder ‘warm’ source.

Much of the preceding discussion introduces the topic of colour with treatment of just one, antecedent, colour convention. As such, this provides a useful description of the basic underlying principles. However, the topic in itself is much too broad to provide any comprehensive treatment here and the reader is referred to specialist texts for further study. Some guidance is provided in the short bibliography and the end of the chapter.

7.7.5 Astronomical Photometry

Astronomical photometry is concerned with the measurement of the magnitude of electromagnetic flux from stellar objects, stars galaxies, and so on. For modern observations, these measurements are almost exclusively dependent upon semi-conductor detectors. For a given stellar object, the ideal measurement might involve the high resolution capture of its spectral irradiance at the earth across a wide range of wavelengths. That is to say, a detailed spectrum of each object should be obtained that can be related to absolute spectral irradiance. However, as stellar objects of interest are almost invariably faint, the amount of flux that is captured in any given wavelength band is necessarily very small. For the majority of measurements, therefore, this approach is impractical. A more practical solution is to use a number of spectrally filtered detectors to monitor flux from the star (via a telescope). Each filter has a relatively broad passband, e.g. 100 nm and centred on some specific wavelength, e.g. 555 nm. Using a small number of these filtered detectors across the ultraviolet, visible, and infrared spectral ranges, provides what amounts to low resolution spectral information for the source. However, since interpretation of the spectral quality of a stellar object is based on a limited number (e.g. 3) of spectrally filtered measurements, there is a clear correlation with visual photometry.

As with visual photometry, conventions for stellar photometry are varied. By way of illustration, we invoke here one set of standard filter response curves which follow the same broad purpose as the colour matching functions in visible photometry. Standard filters include U (ultraviolet), B (blue), V (visible), R (red), I (infrared) as well as Z, Y, J, H, and K bands further into the infrared. The visible filter function follows the standard visible photopic curve with a maximum response at about 555 nm. Standard filter response curves are illustrated in Figure 7.24.

Figure 7.24 Standard astronomical filter response curves.

As for visible photometry, the spectral irradiance of the source, E(λ) is mediated by the filter transmission function, S(λ), to give the effective illuminance, Ep.

(7.29)

Hitherto, we have not introduced any absolute scale into this discussion. In fact, it will be appreciated, as part of a strand running through this chapter, that absolute radiometric and photometric measurements are extremely challenging. That is to say, it is difficult (not impossible) to relate the brightness of stellar objects to fundamental units of flux such as watts. Therefore, in most practical applications, stellar photometry relates the brightness of specific stellar objects to a small number of reference stellar objects. Pre-eminent amongst these reference objects is the star Vega or α Lyrae. The brightness of other stellar objects is expressed as a ratio of their effective illuminance, Ep to that of Vega, Ep₀. However, a simple ratio does not provide a useful impression of the brightness of a source. This is because the human eye is effectively a logarithmic detector, with geometric ratios in flux appearing as a uniform progression in brightness. By the same token, the sensitivity of the human eye covers many orders of magnitude of flux. For this reason, the brightness of a star is described by its apparent magnitude, M, which is based on the logarithm of the illuminance ratio (to the standard). For historical reasons, the base of the logarithm is the fifth root of one hundred.

(7.30)

Note that the reference star illuminance forms the numerator and the measured illuminance the denominator. As a consequence, in this convention, brighter stars have a lower magnitude. Nominally Vega (a bright star) has a magnitude of zero. However, attempts to reconcile these measurements with absolute and other photometric scales have required a small adjustment. For example, in the Johnson convention, the magnitude of Vega is 0.03. So far, the term magnitude, M, has been presented as a simple one parameter description of stellar brightness. However, as indicated in Figure 7.23, this brightness is mediated by a range of standard filters. In presenting stellar magnitudes, the filter band is always indicated by a subscript, e.g. M_U, M_B, M_V, M_R, and so on. The most frequently used is M_V, reflecting the visible brightness of the object. Difference between two of the magnitudes, e.g. M_V and M_B, is an indication of the object's colour. Visual magnitudes of a number of astronomical objects are sketched out in Table 7.6.

Table 7.6 Visual magnitudes of several astronomical objects.

Apparent magnitude describes the relative brightness or illuminance of a stellar object as it appears at the Earth's surface. By virtue of the inverse square law, differences in apparent magnitude might arise from differences in distance or the absolute luminous intensity of the source. The absolute magnitude of a source is equivalent to the relative magnitude of that source if it were placed at a standard reference distance from the Earth (10 pc or 32.6 light years). In this scheme, our own sun is a relatively non-descript source with an absolute magnitude of 4.83.

The relative photometry of stellar sources may, with some difficulty be related to absolute radiometric units. In stellar photometry a standard unit of spectral irradiance is used, the Jansky. However, instead of expressing the spectral irradiance in irradiance per unit wavelength it is expressed as irradiance per unit frequency (Hertz). The Jansky unit is 10⁻²⁶ W m⁻² Hz⁻¹. In the so-called AB Magnitude convention the zero magnitude brightness, for the visible passband, is defined for an imaginary source whose spectral irradiance is flat in the frequency domain and amounts to 3631 Jansky units. Converting from spectral radiance in the frequency domain, Ef(f) to spectral radiance in the wavelength domain, Eλ(λ) is straightforward:

(7.31)

To gain an appreciation of how stellar magnitudes relate to absolute irradiance it would be useful to compute and effective irradiance of a zero magnitude star in the AB system. We may compute an effective irradiance using Eq. (7.29) and the visible transmission function SV(λ) illustrated in Figure 7.23. It is also assumed that over the passband of the filter, the spectral irradiance is flat at 3631 Jansky units. This gives the effective irradiance of a zero magnitude star as about 3.2 × 10⁻⁹ W m⁻².

8
Polarisation and Birefringence

8.1 Introduction

In our treatment of electromagnetic wave propagation we have maintained the convenient fiction that the amplitude of the wave disturbance is a scalar quantity. However, the physical quantities that underlie the amplitude are the electric and magnetic fields. These are unambiguously vector quantities. Indeed, the set of equations (Maxwell's equations) defining electromagnetic propagation are a set of differential equations that establish the relationship between vector quantities. However, in the scalar theory that we have presented to this point, analytical convenience dictates that we imagine an electromagnetic wave to be described by a scalar quantity. This has the benefit of making the analysis somewhat more tractable. We applied this to the analysis of optical diffraction. It is inherently a useful approximation that is applicable under certain constraints. Most particularly, if light is constrained to some optical axis, then the scalar approximation is largely valid provided that all propagation angles with respect to this axis are relatively small. In effect, this is a ‘paraxial approximation’.

Notwithstanding this, we must ultimately accept that the quantities describing the amplitude of electromagnetic radiation are in fact vector quantities. It is the direction of the electric field, E, associated with this radiation that, by convention defines the direction of polarisation of light. Of course, there is also the magnetic field, H, and the two are inextricably linked via Maxwell's equations. As will be seen later, in plane polarised light, the magnetic field vector is orthogonal to the electric field vector and both are orthogonal to the direction of propagation. In reality all light is polarised and what is described as unpolarised light is, in fact, randomly polarised. That is to say, there is a complete lack of correlation or coherence between different vector components of polarisation producing random shifts in the direction of polarisation over an optical cycle.

In this chapter we will also look in a little more detail at the underlying structure of optical materials that contribute to refractive properties. Previously we had understood the refractive property of a material to be associated only with its modification of the local speed of light. The refractive index of a material is, of course, defined as the ratio of the speed of light in vacuo to that in the medium. Refractive effects in almost all materials are produced by the interaction of the electric field of the radiation with the internal atomic structure of the material. This local electric field causes charge separation at the atomic level, leading to the production of electric dipoles. In effect, these dipoles produce an additional electric field which interacts with the imposed electric field. It is this effect that leads ultimately to refraction. In the simplest scenario, in an isotropic material, where ‘all directions are equal’, we might imagine that these dipoles will simply be aligned to the imposed electric field. However, in anisotropic materials, such as crystals, not all directions are equivalent, and these internal dipoles are more readily created where the imposed electric field is in certain specific orientations. The effect of this is that the refractive index of the material varies with the direction of the imposed electric field, or polarisation. This effect is known as birefringence.

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