**4. Optical radiation quantities and units**

Having set up this rather formal measurement structure, we can now consider in more detail the measurement of several basic optical radiation quantities used to determine absolute amounts of optical radiation. Human beings have been dealing with light ever since 'the beginning', so they have had a long time to build up a large vocabulary of terms. To understand these terms, we need to consider two important aspects of quantitative optical radiation measurements: the geometrical configuration of the optical radiation that we wish to measure, and the spectral components of this particular geometrical assembly of radiation. Optical radiation is a restricted wavelength range of the electromagnetic spectrum between X-rays and microwave radio waves, from approximately 1 nm to 1 mm (CIE S017, 2011). In this chapter, the focus of our discussions will be upon electromagnetic radiation with wavelengths from approximately 100 nm to 2500 nm, which is an extension from the visible wavelength range, which is approximately from 360 nm to 830 nm.

The basic quantity for the measurement of the amount of electromagnetic radiation is the energy of that radiation, measured with the SI reference unit the joule. This quantity is most

comparison with a reference quantity of the same kind, but by a procedure where the reference is the definition of the measurement unit through its practical realization. In this manner, that primary measurement standard can be said to be "traceable to the SI", rather than to any other laboratory or NMI. For example, the SI unit for luminous intensity, the candela, is defined in terms of a specific amount of radiant intensity (Section 4.2.2). Calibration of sources based upon the realization of the candela is discussed in Section 7.0. In all these measurements, the establishment of the reference and our comparison to the reference will be subject to some errors. It is not possible to establish a reference exactly to definition, as it is not possible to compare two quantities exactly. These errors may be systematic or random, leading to a measurement uncertainty, which is defined (VIM, Section 2.26) as a "non-negative parameter characterizing the dispersion of the quantity values being attributed to a measurand, based on the information used". This definition leaves it open as to the parameter used to describe the dispersion, as well as noting that the value depends upon the information used, implying that our information may not be complete and may include an element of belief rather than absolute knowledge (JCGM 104:2009). As a consequence, a measurement result becomes (VIM, Section 2.9) a "set of quantity values being attributed to

Since most measurements are carried out using references that are quantities of the same kind, and are not a primary measurement standard, these measurements and references must somehow be related to the primary measurement standard. This leads to a sequence of calibrations (calibration hierarchy, VIM Section 2.40) that leads from a reference to the final measuring system, where each calibration depends upon the outcome of the previous calibration. The metrological traceability chain is the sequence of measurement standards and calibrations that were used to relate the measurement result to the reference, and the metrological traceability is the property of the measurement result whereby the result can be related to the reference through a *documented unbroken* chain of calibrations, each contributing to the measurement *uncertainty*. An example of a calibration chain used at an

Having set up this rather formal measurement structure, we can now consider in more detail the measurement of several basic optical radiation quantities used to determine absolute amounts of optical radiation. Human beings have been dealing with light ever since 'the beginning', so they have had a long time to build up a large vocabulary of terms. To understand these terms, we need to consider two important aspects of quantitative optical radiation measurements: the geometrical configuration of the optical radiation that we wish to measure, and the spectral components of this particular geometrical assembly of radiation. Optical radiation is a restricted wavelength range of the electromagnetic spectrum between X-rays and microwave radio waves, from approximately 1 nm to 1 mm (CIE S017, 2011). In this chapter, the focus of our discussions will be upon electromagnetic radiation with wavelengths from approximately 100 nm to 2500 nm, which is an extension from the

The basic quantity for the measurement of the amount of electromagnetic radiation is the energy of that radiation, measured with the SI reference unit the joule. This quantity is most

visible wavelength range, which is approximately from 360 nm to 830 nm.

a measurand together with any other available relevant information."

NMI is discussed in Section 7.0.

**4. Optical radiation quantities and units** 

useful when considering pulses or bursts of radiation that have a time limited duration. In this chapter we will deal with the continuous flow of radiation; hence the (energy per time) quantity flux, or power, will be the appropriate basic quantity for the measurement of the amount of radiation. The SI unit for the quantity flux is the watt (W).

The flux of interest will be the flux that is contained within certain geometrical constraints. There is a set of five geometrical configurations that can be used to form the basis of most radiation measurement quantities: total flux, intensity, radiance, exitance and irradiance. The first four of these quantities are characteristics of radiation emanating from a radiation source, whereas the fifth, irradiance, is characteristic of radiation incident upon a surface.

#### **4.1 Geometrical quantities and units**

#### **4.1.1 Solid angle**

The geometrical configurations of intensity and radiance make use of the 3-dimensional concept of a solid angle (), which is analogous to the 2-dimensional plane angle (). In Figure 1, *l* is the length of the arc of the circle subtended by the limits of the 2-D object at the center of the circle of radius *r*. The definition of the plane angle is given by

$$a = \frac{l}{r} \tag{1}$$

The SI unit for the plane angle is the radian (rad). A full circle is an angle of 2 rad.

Fig. 1. Two dimensional and three dimensional angles

Similarly, for the 3-D angle, called a solid angle, *A* is the area of the part of the sphere of radius *r* that is subtended by the limits of the 3-D object at the center of the sphere with radius *r*. The definition of the solid angle is given by

$$
\Omega = \frac{A}{r^2} \tag{2}
$$

The SI unit for the solid angle is called the steradian (sr). The complete sphere is a solid angle of 4 sr.

### **4.1.2 Total flux ()**

As indicated above, flux can be measured in many geometrical conditions. However, when radiation sources such as the common incandescent lamps or general lighting service lamps are measured, total flux is used to indicate the total flux output from the lamp into all directions. Specialized equipment, such as integrating spheres or goniophotometers, is required to collect the radiation output into all directions from the lamp (Ohno, 1997). The principles of operation using an integrating sphere are illustrated in Figure 2.

The inside surface of the integrating sphere is coated with a uniformly diffusing material (Lambertian Surface will be discussed in Section 4.1.7.2), such as PTFE (polytetrafluoroethylene) or BaSO4, to provide at the exit port a spatially homogeneous flux that is representative of the total flux output of the lamp. The purpose of the baffle is to prevent radiation directly from the lamp to be incident at the output port, before the sphere has had the opportunity to merge uniformly the radiation output from all directions from the lamp.

Fig. 2. Total flux measurement using an integrating sphere.

#### **4.1.3 Intensity (***I***)**

Intensity is used rather carelessly in daily use, even in physics. However, in optical radiation measurements it has a very specific meaning (Figure 3). First, the direction from the source ( **d** ) in which the intensity is to be defined must be indicated. Then the intensity of the source in this specified direction is defined as the ratio of the flux () leaving the source in that particular direction and propagating into an element of solid angle () containing that specified direction, divided by the size of that element of solid angle. The radiant intensity is defined as

$$I = \frac{\Phi}{\Omega} \tag{3}$$

with SI unit of watt per steradian (W·sr-1).

As indicated above, flux can be measured in many geometrical conditions. However, when radiation sources such as the common incandescent lamps or general lighting service lamps are measured, total flux is used to indicate the total flux output from the lamp into all directions. Specialized equipment, such as integrating spheres or goniophotometers, is required to collect the radiation output into all directions from the lamp (Ohno, 1997). The

The inside surface of the integrating sphere is coated with a uniformly diffusing material (Lambertian Surface will be discussed in Section 4.1.7.2), such as PTFE (polytetrafluoroethylene) or BaSO4, to provide at the exit port a spatially homogeneous flux that is representative of the total flux output of the lamp. The purpose of the baffle is to prevent radiation directly from the lamp to be incident at the output port, before the sphere has had the opportunity to merge uniformly the radiation output from all directions from

Intensity is used rather carelessly in daily use, even in physics. However, in optical radiation measurements it has a very specific meaning (Figure 3). First, the direction from the source

) in which the intensity is to be defined must be indicated. Then the intensity of the source in this specified direction is defined as the ratio of the flux () leaving the source in that particular direction and propagating into an element of solid angle () containing that specified direction, divided by the size of that element of solid angle. The radiant intensity is

e

Lambertian Surfac

*<sup>I</sup>* (3)

principles of operation using an integrating sphere are illustrated in Figure 2.

**4.1.2 Total flux ()** 

the lamp.

t

Detector

Exi Port

Baffl

with SI unit of watt per steradian (W·sr-1).

**4.1.3 Intensity (***I***)** 

( **d** 

defined as

e

Fig. 2. Total flux measurement using an integrating sphere.

Fig. 3. Defining geometry for the quantity Intensity (*I*) and origins of the inverse square law.

#### **4.1.4 Irradiance (***E***)**

The quantity irradiance is a measure of the amount of radiation incident upon a surface, as illustrated in Figure 4.

Fig. 4. Defining geometries for the quantities irradiance (*E*) and exitance (*M*).

The irradiance at the surface is defined as:

$$E = \frac{\Phi}{A} \tag{4}$$

where is the total flux incident upon the surface element of area *A*. Irradiance is measured in terms of the SI unit of watt per square metre (W·m-2).

#### **4.1.5 Exitance (***M***)**

Exitance is very similar to irradiance except that the direction of flow of the flux is reversed—it is leaving the surface, as shown in Figure 4. The radiant exitance from the surface is defined as:

$$M = \frac{\Phi}{A} \tag{5}$$

where is the total flux leaving the surface element of area *A*. Radiant exitance is measured in terms of the SI unit of watt per square metre (W·m-2).

#### **4.1.6 Radiance (***L***)**

This quantity is the most detailed of the five geometrical quantities we are considering. It takes into account that a radiation source is not a point, but an extended surface, and that a radiation source does not emit the same flux into all directions. Therefore we have a quantity (Figure 5) that specifies the flux () emitted by a radiation surface from a specified area (A) on the surface and in a specified direction ( **d** ) from the surface, and into a specified solid angle () containing the given direction **d** . The vector direction **n** is the perpendicular to the radiation source element of area A, and is the angle between **n** and **d** .

Fig. 5. Defining geometries for the quantity radiance and related projected area. The radiance of the element of area is defined as

$$L = \frac{\Phi}{A\_p \cdot \Omega} \tag{6}$$

The projected area Ap is the size of the area A as seen in the specified direction **d** :

$$A\_p = A \cdot \cos a \tag{7}$$

*A*

where is the total flux leaving the surface element of area *A*. Radiant exitance is measured

This quantity is the most detailed of the five geometrical quantities we are considering. It takes into account that a radiation source is not a point, but an extended surface, and that a radiation source does not emit the same flux into all directions. Therefore we have a quantity (Figure 5) that specifies the flux () emitted by a radiation surface from a specified

perpendicular to the radiation source element of area A, and is the angle between **n**

Fig. 5. Defining geometries for the quantity radiance and related projected area.

*L A*

The projected area Ap is the size of the area A as seen in the specified direction **d**

*p*

cos *A A <sup>p</sup>*

**Surface of radiation Source**

**Projected Area**

(6)

(7)

 :

(5)

) from the surface, and into a specified

is the

and

. The vector direction **n**

*M*

in terms of the SI unit of watt per square metre (W·m-2).

area (A) on the surface and in a specified direction ( **d**

**Radiance**

**Element of area A**

The radiance of the element of area is defined as

solid angle () containing the given direction **d**

**4.1.6 Radiance (***L***)** 

**d** .

#### **4.1.7 Quantity relationships**

Several useful relationships between the quantities discussed above may be obtained by adding specific geometric constructions and/or source constraints to the measurement configurations that use these quantities. These are particularly useful to change from a radiation source of one quantity to provide a radiation source of another quantity, such as using an irradiance source to provide a source of radiance.

#### **4.1.7.1 The inverse square law**

This is the most common and useful relationship. In its basic form, it relates the irradiance (*E*) of a surface due to the output of a radiant intensity source (*I*) placed at a known distance (r) from the surface. The geometry is shown in Figure 3.

The same flux passes through each of the areas A1 and A2. Since the sizes of these areas are increasing with distance r from the source, it is evident that there will be less flux per unit area on surfaces farther from the source. From the definitions of intensity (*I*), solid angle (), and irradiance (*E*), we may derive the relation between *E*, *I*, and r:

$$
\Delta \Phi = I \cdot \Omega = I \cdot \frac{A}{r^2} \tag{8}
$$

The irradiance on any of the surfaces is then obtained as

$$E = \frac{\Phi}{A} = \frac{I}{r^2} \tag{9}$$

This is a very useful concept, but some restrictions should be noted:


the intensity of these devices (CIE 127, 2007). The recommended geometries prescribe exact solid angles and distances from the LED, together with specific considerations for the alignment of the LED with the measurement direction. Since the radiant intensity from LEDs is not constant with distance or solid angle, the resulting quantities measured are called Averaged LED Intensities, rather than true intensities as defined in Equation (3) above for a point source.
