A.4.1 Optical quantities

In the optical domain two groups of quantities are distinguished: radiometric and photometric quantities. The radiometric quantities are valid for the whole electromagnetic spectrum. Photometric quantities are only valid within the visible part of the spectrum (0.35<λ<0.77 μm). They are related to the mean, standardized sensitivity of the human eye and are applied in photometry. Table A.10 shows the major radiometric and corresponding photometric quantities. Some of them will be explained in more detail.

The radiant intensity I (W/sr) is the emitted radiant power of a point source per unit of solid angle. For a source consisting of a flat radiant surface the emitted power is expressed in terms of W/m², called the radiant emittance E_s. It includes the total emitted energy in all directions. The quantity radiance also accounts for the direction of the emitted light: it is the emitted power per unit area, per unit of solid angle (W/m² sr).

A flat surface can receive radiant energy from its environment, from all directions. The amount of incident power E_d is expressed in terms of W/m² and is called the irradiance.

In radiometry the unit of (emitting or receiving) surface area is often expressed in terms of ‘projected area’. A surface with (real) area S that is irradiated from a direction perpendicular to its surface receives power equal to E_d·S (W). When that surface is rotated over an angle ϕ, relative to the direction of radiation, it receives only an amount of radiant power equal to E_d·S·cos ϕ (Fig. A.2). To make these radiometric quantities independent of the viewing direction, some authors express them in terms of projected area instead of real area, thus avoiding the cosϕ in the expressions.

In Table A.10 the radiance L of a surface is defined following this idea: it is the emitted radiant energy in a certain direction, per solid angle (to account for the directivity) and per unit of projected area, that is per unit area when the emitting surface is projected into that direction. These quantities are used for deriving formulas for sensors that are based on reflecting surfaces (Chapter 7: Optical sensors).

All units have their counterpart in the photometric domain (right-hand side of the table). However, we will not use them further in this book. The radiometric quantities in the table do not account for the frequency or wavelength dependence. All of them can be expressed also in spectral units, e.g., the spectral irradiance E_d(λ), which is the irradiance per unit of wavelength interval (W/m² μm). The total irradiance over a spectral range from λ₁ to λ₂ then becomes:

$E_{\text{d}}={\int }_{{\lambda }_1}^{{\lambda }_2}{E}_{\text{d}}\left(\lambda \right)\text{d}\lambda$ (A.8)

(A.8)

To characterize optical sensors special signal parameters are used, related to the noise behavior of these components (Table A.11).

Note that the signals can be expressed in voltage, current or power. Irrespective of this choice, the S/N ratio is a dimensionless quantity. The parameters D and NEP represent the lowest perceivable optical signal. NEP is, in words, equal to the power of sinusoidally modulated monochromatic radiation that produces an RMS signal at the output of an ideal sensor, equal to the noise signal of the real sensor. This means that, when specifying NEP, the modulation frequency, the sensor bandwidth, the temperature and the sensitive surface area should also be specified, as all these parameters are included in the noise specification. To avoid wavelength and detector area dependency in the specifications, the parameters NEP* and D* are used: the spectral NEP and the spectral D, respectively. Since noise power of a sensor is usually proportional to the sensitive area A and the bandwidth Δf, the noise current and noise voltage are proportional to √(AΔf); hence, NEP*=NEP/√(AΔf) and D*=√(AΔf)/NEP.

A.4.2 Radiant energy from a unit surface with Lambertian emission

If a surface is a perfect diffuser (its radiance is not a function of angle), then I=I₀·cos ϕ (Fig. A.3).

This is known as the cosine law of Lambert. Many surfaces scatter the incident light in all directions equally. A surface that satisfies Lambert’s law is called Lambertian.

All radiant energy emitted from a surface A with radiant emittance E_s has to pass a hemisphere around that surface (Fig. A.4).

The area of surface element dS on the hemisphere, corresponding with a solid angle dΩ equals:

$\text{d}S=r^2\text{dΩ=}{r}^2\sin \varphi \text{d}\varphi \text{dϑ}$ (A.9)

(A.9)

where r is the radius of the hemisphere. According to its definition in Table A.10 the radiant emittance from the emitting surface A in the direction of dS is:

$\frac{\text{d}{P}_{\text{s}}}{\text{d}A}=L\cos \varphi \text{dΩ}$ (A.10)

(A.10)

Since all radiant energy from A passes the hemisphere, the total radiant emittance of A equals the integral of dP_s/dA over the full surface area of the hemisphere. Further, since for a Lambertian surface the radiant emittance is independent of the direction, L is constant. So

$E_s={\iint }_{\mathrm{hemisphere}}L\cos \varphi d\mathrm{\Omega }={\int }_0^{2\pi }{\int }_0^{\pi /2}L\cos \varphi \sin \varphi d\varphi d\vartheta =\pi L$ (A.11)

(A.11)

or the radiant emittance of a Lambertian surface equals π times the radiance of that surface.

A.4.3 Derivation of relations between intensity and distance

The transfer characteristic of an optical displacement sensor (Chapter 7: Optical sensors) depends strongly on the geometry of the configuration. Fig. A.5 depicts the major parameters involved in the direct mode.

Light energy radiates from the source with emitting surface area S₁ and radiance L_e (W/sr m²). The sensor surface area is S₂, the distance from source to sensor is r. Further, the light beam as received by the sensor makes an angle ϑ₁ with the normal on the emitting surface, and an angle ϑ₂ with the normal on the receiving surface. We calculate the radiant power that arrives at the sensor. Assume the source dimensions being small compared to r. Then the light beam that arrives at the sensor has a solid angle equal to

$\text{dΩ=}\frac{S_2\cos {\vartheta }_2}{r^2}\left(\mathrm{sr}\right)$ (A.12)

(A.12)

where S₂ cos ϑ₂ is the projected surface area of the sensor. The emitter emits L_e·S₁ W per steradian; hence, the part ΔP_e of the emitted light falling on the sensor is:

$\Delta P_{\text{e}}=L_{\text{e}}{S}_1\cos {\vartheta }_1\text{dΩ}\text{(}\text{W}\text{)}$ (A.13)

(A.13)

where S₁ cos ϑ₁ is the projected surface area of the source. By substituting Eq. (A.13) in Eq. (A.12) we find for the radiant power arriving at the sensor:

$P_{\text{o}}=L_{\text{e}}\frac{S_1{S}_2\cos {\vartheta }_1{\cos \vartheta }_2}{r^2}(\text{W}\text{)}$ (A.14)

(A.14)

from which follows that the output signal, indeed, is inversely proportional to the square of the distance.

Next we consider the indirect mode, where we distinguish between two different situations. In the first situation the reflecting target is only partly illuminated by the source; all the light from the source is intercepted by the object. In the second situation the whole object falls within the light cone of the source. Fig. A.6 displays the first situation and defines the various parameters. For a better understanding, the object (target) is presented twice, once as the receiving surface and once as the (secondary) emitting surface.

The target intercepts a light beam that makes an angle ϑ_s with the normal vector n_t on the target’s surface. The source can be characterized by an intensity I_s (W/sr) and a beam solid angle Ω_s. The surface area S₁ of the light spot on the object satisfies the equation:

$S_1\cos {\vartheta }_{\mathrm{s}}={\mathrm{\Omega }}_{\mathrm{s}}{r}_{\mathrm{s}}^2({\mathrm{m}}^2)$ (A.15)

(A.15)

where r_s is the distance between the source and the target. The irradiance of the object (actually the lighted part of it) is:

$E_1=\frac{I_{\text{s}}{\mathrm{\Omega }}_{\text{s}}}{S_1}(\text{W/}{\text{m}}^2)$ (A.16)

(A.16)

Since the object surface is assumed to be Lambertian it behaves as a radiant source with radiance L₁=E₁/π (see Eq. (A.11)). Part of the scattered light is received by the detector with sensitive area S_d on a distance r_d from the target. Similar to Eq. (A.12) this sensor receives light over a solid angle Ω_d equal to

${\mathrm{\Omega }}_{\mathrm{d}}=\frac{S_{\mathrm{d}}\cos {\vartheta }_2}{r_{\mathrm{d}}^2}(\text{sr})$ (A.17)

(A.17)

The radiant power received by the sensor is:

$P_{\mathrm{d}}=L_1{S}_1{\mathrm{\Omega }}_{\mathrm{d}}\cos {\vartheta }_1=\frac{E_1{S}_1{\mathrm{\Omega }}_{\mathrm{d}}\cos {\vartheta }_1}{\pi }(\mathrm{W})$ (A.18)

(A.18)

Finally, when substituting Eqs (A.16) and (A.17) into Eq. (A.18) we find for the optical power received by the sensor:

$P_{\mathrm{d}}=\frac{S_{\mathrm{d}}\cos {\vartheta }_1\cos {\vartheta }_2}{\pi r_{\mathrm{d}}^2}\cdot I_{\mathrm{s}}{\mathrm{\Omega }}_{\mathrm{s}}(\mathrm{W})$ (A.19)

(A.19)

So the sensor output is inversely proportional to the square of the distance to the sensor, as long as the object intercepts the whole light beam. Note that P_d does not depend on the angle ϑ_s, because all the light from the source falls on the object, whose surface is assumed to be Lambertian.

Next we consider the situation where the whole object is illuminated by the source, as can happen at large distances and with small objects (Fig. A.7).

The solid angle of the beam falling on the object is:

$d{\mathrm{\Omega }}_{\mathrm{s}}=\frac{S_{\mathrm{t}}\cos {\vartheta }_{\mathrm{s}}}{r_{\mathrm{s}}^2}(\text{sr})$ (A.20)

(A.20)

where S_t is the area of the illuminated part of the object. The radiant emittance now becomes:

$E_1=\frac{I_{\mathrm{s}}d{\mathrm{\Omega }}_{\mathrm{s}}}{S_{\mathrm{t}}}=\frac{I_{\mathrm{s}}\cos {\vartheta }_{\mathrm{s}}}{r_{\mathrm{s}}^2}{(\mathrm{W}/\mathrm{m}}^2)$ (A.21)

(A.21)

Following the same steps as before, we find for the radiant power received by the sensor:

$P_{\mathrm{d}}=L_1{S}_{\mathrm{t}}{\mathrm{\Omega }}_{\mathrm{d}}\cos {\vartheta }_1=\frac{E_1}{\pi }\cdot S_{\mathrm{t}}\cos {\vartheta }_1\cdot \frac{S_{\mathrm{d}}\cos {\vartheta }_2}{r_{\mathrm{d}}^2}=I_{\mathrm{s}}\cdot \frac{S_{\mathrm{t}}{S}_{\mathrm{d}}\cos {\vartheta }_1\cos {\vartheta }_2\cos {\vartheta }_{\mathrm{s}}}{\pi r_{\mathrm{s}}^2{r}_{\mathrm{d}}^2}(\mathrm{W})$ (A.22)

(A.22)

Clearly the sensor output is inversely proportional to the fourth power of the distance between source/sensor and the target.