**4.1.7.2 Lambertian surface**

A Lambertian Surface is an ideal surface whose radiance is the same in all directions of the hemisphere above the surface. This very special surface has many useful properties. It is basically a radiation source that looks equally bright when viewed from any direction. It can be a radiation source itself, such as a black body radiator, or it can be a material that either transmits or reflects the radiation incident upon it in such a manner. Such a material is also called an isotropic diffusing surface. The basic concepts for such a reflecting diffusing surface are shown in Figure 6.

There are no real materials that have the ideal properties of a Lambertian surface, but pressed PTFE (polytetrafluoroethylene) powder or BaSO4 may be used if corrections for their non-ideality are made when required. The reflectance properties of these near-ideal diffusing reflectors are critical to accurate reflectance measurements and their threedimensional reflection properties are extensively studied (Höpe & Hauer, 2010).

Fig. 6. An isotropic diffusing reflecting surface

## **4.1.7.3 Irradiance to radiance**

The Lambertian diffusing reflecting surface is very useful in obtaining a radiance source from an irradiance source. When determining the relationship between radiometric quantities it is very useful to consider the flow of flux and then add the geometric constraints, as we did in Section 4.1.7.1 above. The geometric configuration shown in Figure 6 shows the conversion from the incident radiation from an irradiance source into the output reflected radiation with the properties of radiance. If the known irradiance at the area *A* from our source is *E*, then the incident flux *i* upon the area *A* is:

$$
\Phi\_i = E \cdot A \tag{10}
$$

A Lambertian Surface is an ideal surface whose radiance is the same in all directions of the hemisphere above the surface. This very special surface has many useful properties. It is basically a radiation source that looks equally bright when viewed from any direction. It can be a radiation source itself, such as a black body radiator, or it can be a material that either transmits or reflects the radiation incident upon it in such a manner. Such a material is also called an isotropic diffusing surface. The basic concepts for such a reflecting diffusing

There are no real materials that have the ideal properties of a Lambertian surface, but pressed PTFE (polytetrafluoroethylene) powder or BaSO4 may be used if corrections for their non-ideality are made when required. The reflectance properties of these near-ideal diffusing reflectors are critical to accurate reflectance measurements and their three-

*<sup>i</sup> E A* (10)

Reflected Radiation

The Lambertian diffusing reflecting surface is very useful in obtaining a radiance source from an irradiance source. When determining the relationship between radiometric quantities it is very useful to consider the flow of flux and then add the geometric constraints, as we did in Section 4.1.7.1 above. The geometric configuration shown in Figure 6 shows the conversion from the incident radiation from an irradiance source into the output reflected radiation with the properties of radiance. If the known irradiance at the

area *A* from our source is *E*, then the incident flux *i* upon the area *A* is:

Incident Radiation

dimensional reflection properties are extensively studied (Höpe & Hauer, 2010).

Equation (3) above for a point source.

Diffusing Surface

> Area A

Fig. 6. An isotropic diffusing reflecting surface

**4.1.7.3 Irradiance to radiance** 

**4.1.7.2 Lambertian surface** 

surface are shown in Figure 6.

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 The flux reflected from the area *A* is the incident flux times the reflectance () of *A*:

$$
\Phi\_r = \rho \cdot \Phi\_i = \rho \cdot E \cdot A \tag{11}
$$

We can relate the radiance (*L*) of *A* to the reflected flux *r* using the definition of radiance from Equation (6) and the assumption of a constant radiance in all directions (Lambertian surface):

$$
\Delta d \phi = \mathcal{L} \cdot A \cdot \cos a \cdot d\alpha \tag{12}
$$

where *d* is a small element of flux emitted from *A* into the small element of solid angle *d* . The total flux emitted from *A* by this constant radiance *L* is obtained by integrating *d* over all the output directions from *A*:

$$\Phi\_r = \int L \cdot A \cdot \cos \alpha \cdot d\alpha = L \cdot A \cdot \left[\cos \alpha \cdot d\alpha = L \cdot A \cdot \pi \tag{13}$$

Combining Equations (11) and (13) we obtain:

$$L\_{IDR} = \frac{\rho \cdot E}{\pi} \tag{14}$$

Equation (14) is the basic equation relating the radiance of an isotropic reflecting diffuser due to a known irradiance of the diffuser, where we have introduced the subscript IDR to indicate that this is the radiance from an Isotropic Diffuse Reflector with reflectance .

As indicated in the discussion of a Lambertian source, there is no perfect isotropic diffuser. This means that the reflectance in Equation (14) will not be the same for all directions of both the input irradiation and the output radiance. For accurate conversions from irradiance to radiance we will need to take account of this effect. We introduce a radiance factor () that is defined (CIE S017, 2011) as: *radiance factor* (at a surface element of a non self-radiating medium, in a given direction, under specified conditions of irradiation): ratio of the radiance of the surface element in the given direction to that of the perfect reflecting or transmitting diffuser identically irradiated and viewed. The *reflected radiance factor* is denoted by *<sup>R</sup>* . For completeness in this description, we also show that the quantities we are discussing are functions of the wavelength , as well as angle.

For a perfect reflecting diffuser (PRD) illuminated by an irradiance *E*( ) , ( 1 ), so that Equation (14) becomes:

$$L\_{\rm PRD}(\mathcal{X}) = \frac{E(\mathcal{X})}{\pi} \tag{15}$$

Since this is a perfect reflecting diffuser, this equation is true for all directions of *LPRD* from the diffuser.

For an imperfect reflecting diffuser (RD) surface,

$$\mathcal{J}\_R(\mathcal{X}, \mathcal{G}) = \frac{L\_{RD}(\mathcal{X}, \mathcal{G})}{L\_{PRD}(\mathcal{X}, \mathcal{G})} \tag{16}$$

where indicates the direction of the radiance *L* from the surface, for some defined irradiation condition. From Equation (16) we obtain:

$$L\_{\rm RD}(\mathcal{X}, \mathcal{G}) = \mathcal{J}\_{\rm R}(\mathcal{X}, \mathcal{G}) \cdot L\_{\rm PRD}(\mathcal{X}, \mathcal{G}) = \mathcal{J}\_{\rm R}(\mathcal{X}, \mathcal{G}) \cdot L\_{\rm PRD}(\mathcal{X}) \tag{17}$$

with the second equality due to the fact that the PRD radiance ( ) *LPRD* is independent of angle.

Combining Equations (15) and (17) we obtain:

$$L\_{\rm RD}(\mathcal{X}, \mathcal{G}) = \mathcal{J}\_{\rm R}(\mathcal{X}, \mathcal{G}) \cdot L\_{\rm PRD}(\mathcal{X}) = \frac{\mathcal{J}\_{\rm R}(\mathcal{X}, \mathcal{G}) \cdot E(\mathcal{X})}{\pi} \tag{18}$$

In our application for converting from irradiance to radiance we usually have the irradiance incident normally upon the diffuser, and the radiance observed at 45°. Therefore, the specific reflectance factor we require is 0/45 ( ) , where the term 0/45 indicates 0° incidence radiation and 45° output radiance. The final equation relating the input irradiance and the output radiance for this 0/45 geometry is therefore:

$$L\_{RD}(\mathcal{X}, 45) = \frac{\beta\_{0/45}(\mathcal{X}) \cdot E(\mathcal{X})}{\pi} \tag{19}$$

The radiance factor 0/45 ( ) will need to be measured (Höpe & Hauer, 2010) for the particular reflecting diffuser that we are using.

#### **4.1.7.4 Radiance to irradiance**

The transfer of radiant energy or flow of radiant flux from a radiating surface to a receiving surface is calculated in a general way for any geometrical configuration using elemental beams of radiance from small areas of the source propagating to small elements of area at the receiver (Grum & Becherer, 1979). The simplest configuration that is often used in calibration laboratories is shown in Figure 7.

Fig. 7. Irradiance from a Radiance Source

The source and receiver apertures are parallel and centered upon, and perpendicular to, the optical axis. The source is again assumed to be a Lambertian source with radiance *L*. The apertures are circular with radii RS and RR for the source and receiver respectively. Under

(,) (,) (,) (,) () *LL L RD*

with the second equality due to the fact that the PRD radiance ( ) *LPRD*

*RD R PRD*

 ( ) 

 

*L*

 

(,) () (,) (,) () *<sup>R</sup>*

In our application for converting from irradiance to radiance we usually have the irradiance incident normally upon the diffuser, and the radiance observed at 45°. Therefore, the

radiation and 45° output radiance. The final equation relating the input irradiance and the

0/45() () ( ,45) *RD*

The transfer of radiant energy or flow of radiant flux from a radiating surface to a receiving surface is calculated in a general way for any geometrical configuration using elemental beams of radiance from small areas of the source propagating to small elements of area at the receiver (Grum & Becherer, 1979). The simplest configuration that is often used in

Source Receiver

Rs RR

Distance D

The source and receiver apertures are parallel and centered upon, and perpendicular to, the optical axis. The source is again assumed to be a Lambertian source with radiance *L*. The apertures are circular with radii RS and RR for the source and receiver respectively. Under

  

*<sup>E</sup> L L*

indicates the direction of the radiance *L* from the surface, for some defined

 

*E*

 

will need to be measured (Höpe & Hauer, 2010) for the

Receiver Aperture

> Optical Axis

 

, where the term 0/45 indicates 0° incidence

(19)

(18)

 

is independent of

*<sup>R</sup> PRD <sup>R</sup> PRD* (17)

where

angle.

irradiation condition. From Equation (16) we obtain:

 

Combining Equations (15) and (17) we obtain:

specific reflectance factor we require is 0/45

 ( ) 

calibration laboratories is shown in Figure 7.

(radiance)

Fig. 7. Irradiance from a Radiance Source

Source Aperture

particular reflecting diffuser that we are using.

The radiance factor 0/45

**4.1.7.4 Radiance to irradiance** 

output radiance for this 0/45 geometry is therefore:

these assumptions, the spatial fraction *f* of the total flux emitted by the source radiance aperture that is collected within the receiver aperture is given (Walsh, 1958) by:

$$f = 2R\_R^2 \left[ (R\_S^2 + R\_R^2 + D^2) + \sqrt{(R\_S^2 + R\_R^2 + D^2)^2 - 4R\_S^2 R\_R^2} \right]^{-1} \tag{20}$$

The radiant exitance of a Lambertian source is obtained by the integration of the constant radiance *L* over the hemispherical solid angle to give (Grum & Becherer, 1979):

$$M = \pi \cdot L \tag{21}$$

Therefore the total flux emitted by the source aperture is:

$$
\Phi\_S = (\boldsymbol{\pi} \cdot \boldsymbol{L}) \cdot (\boldsymbol{\pi} \cdot \boldsymbol{R}\_S^2) \tag{22}
$$

and the flux collected within the receiver aperture is:

$$
\Phi\_R = f \cdot \Phi\_S = \pi^2 \cdot R\_S^2 \cdot f \cdot L \tag{23}
$$

The irradiance at the receiver aperture is therefore:

$$E = \frac{\Phi\_R}{\pi \cdot R\_R^2} = \pi \cdot \left(\frac{R\_S}{R\_R}\right)^2 \cdot f \cdot L \tag{24}$$

As we will discuss in Section 7, Equation (23) is the basis for the calibration of blackbody sources, which are then used as absolute radiance sources for the realization of spectral irradiance scales using Equation (24).

#### **4.2 Spectral quantities and units**

The field of optical radiation measurements is often divided into three major categories: radiometry, photometry and colorimetry. Our discussion to this point has been primarily centered upon radiometry, the measurement of the quantities associated with optical radiation. However, human beings have been observing optical radiation for quite some time with their built-in detector—the eye. This has resulted in a very large and important field of radiation measurements. The difference between the three categories is basically that between the characterisation of the electromagnetic radiation that is present in our measurements (radiometry, which is essentially a detector that responds to all wavelengths equally), and the evaluation based on what human beings think is present using their visual detection system composed of the eye and the brain (photometry, colorimetry). Photometry evaluates the radiation as a measure of the strength or magnitude of the human visual response, whereas colorimetry characterizes optical radiation in terms of the human ability to distinguish radiation of different wavelengths, which we call colours.

#### **4.2.1 Spectral concepts**

The emission, reflection and absorption properties of the sources, detectors and other materials used for optical radiation measurement are all dependent upon the wavelength

of the radiation under consideration. All the geometrical quantities considered above should be considered to apply at each wavelength we use. The use of the adjective spectral, when applied to any of the quantities *X*, indicates (CIE S017, 2011) that either:


$$X\_{\mathcal{A}} = \frac{\mathrm{d}X(\mathcal{A})}{\mathrm{d}\mathcal{A}}\tag{25}$$

It should be noted that *X* is also a function of wavelength , and to stress this fact, may be written as *X* ( ) without any change in meaning.

As examples, the quantities reflectance () and radiance factor () that we introduced above are functions of wavelength, but not spectral distributions of wavelength. The quantities that involve radiant flux are all spectral distributions. For example, spectral irradiance ( *E* ) is defined (CIE S017, 2011) as the quotient of the radiant power d() in a wavelength interval d incident on an element of a surface, by the area d*A* of that element and by the wavelength interval d:

$$E\_{\lambda} = \frac{\mathrm{d}E(\lambda)}{\mathrm{d}\lambda} = \frac{\mathrm{d}^2 \Phi(\lambda)}{\mathrm{d}A \cdot \mathrm{d}\lambda} \tag{26}$$

The basic concepts are shown in Figure 8. The SI unit for spectral irradiance is W·m-2·m-1, or W·m-3. The unit W·m-2·nm-1 is also used since the wavelength of optical radiation is conveniently measured with the unit nm.

Fig. 8. Spectral distribution components for spectral irradiance

#### **4.2.2 Photometry**

236 Modern Metrology Concerns

of the radiation under consideration. All the geometrical quantities considered above should be considered to apply at each wavelength we use. The use of the adjective spectral, when

2. the quantity referred to is the spectral concentration, or spectral distribution, of *X*, with

d d *X*

As examples, the quantities reflectance () and radiance factor () that we introduced above are functions of wavelength, but not spectral distributions of wavelength. The quantities that involve radiant flux are all spectral distributions. For example, spectral irradiance ( *E*

interval d incident on an element of a surface, by the area d*A* of that element and by the

The basic concepts are shown in Figure 8. The SI unit for spectral irradiance is W·m-2·m-1, or W·m-3. The unit W·m-2·nm-1 is also used since the wavelength of optical radiation is

<sup>2</sup> d() d () d dd

*A*

*X* , or

contained in an elementary range of d of wavelength at the

is also a function of wavelength , and to stress this fact, may be

is defined as the quotient of the radiant

)

in a wavelength

(25)

(26)

applied to any of the quantities *X*, indicates (CIE S017, 2011) that either:

1. *X* is a function of the wavelength , with symbol *X*( )

. This spectral distribution *X*

without any change in meaning.

is defined (CIE S017, 2011) as the quotient of the radiant power d()

*<sup>E</sup> <sup>E</sup>*

d

Fig. 8. Spectral distribution components for spectral irradiance

d

symbol *X*

wavelength , by that range:

quantity d() *X*

It should be noted that *X*

 

wavelength interval d:

conveniently measured with the unit nm.

written as *X* ( )

The retina of the human eye contains two different types of photoreceptors called rods and cones that produce nerve impulses that are passed on to subsequent stages of the human visual system for processing (Ohta & Robertson, 2005). The cones are spread over the entire retina, together with a large concentration within a small central area of our vision called the fovea, which results our high visual acuity at the center of the field of view of the eye. The cones are responsible for our daytime colour vision. The rods are spread over the entire retina except the fovea and are responsible for our night-time, basically black-and-white, vision.

The eye is sensitive to radiation over a range of approximately 11 orders of magnitude from bright sunlight to a flash of light containing only a few photons. The change in size of the pupil area is only capable of controlling the radiation input to the retina by a factor of 12. The remaining adaptation is provided by the rods and the cones. The high radiation range is mediated by the cones and is known as the photopic range. The low radiation range is mediated by the rods and is known as the scotopic range. The intermediate range is mediated by both the rods and the cones, and is known as the mesopic range.

Each human being will perceive differently the amount and colour of a given beam of radiation. In order to simplify calculations and to provide international standards for the measurement of quantities representing the strength of the human visual response to optical radiation, the CIE has standardized spectral weighting functions to be used for each of the three ranges of photometric measurements. The spectral weighting functions for the photopic and scotopic ranges (Figure 9) are known as the spectral luminous efficiency functions for photopic and scotopic vision, with symbols V() and V'() respectively (CIE Standard S010/E:2004). The mesopic range requires a more complex weighting function based upon a gradual transition between V() and V'() throughout the mesopic region that depends on the visual adaptation conditions (CIE 191:2010).

Fig. 9. CIE spectral luminous efficiency functions

The CIE spectral luminous efficiency functions are relative spectral values with a maximum value of one. Within the SI, photometric quantities and units are obtained by calculation from radiometric quantities and units. This is facilitated with the definition of a quotient K called the luminous efficacy of radiation and defined as (CIE S017, 2011) the quotient of the luminous flux, <sup>v</sup> , by the corresponding radiant flux <sup>e</sup> . (The subscripts v and e are used to indicated photometric and radiometric quantities respectively.)

$$K = \frac{\Phi\_{\text{v}}}{\Phi\_{\text{e}}} \tag{27}$$

The spectral luminous efficiency functions provide the relative spectral values for K(). In order to obtain the absolute scaling, there needs to be a defined scaling factor between radiometry and photometry at one of the wavelengths of the luminous efficiency functions. This is done in the definition (BIPM, 2006) of the SI base unit of luminous intensity, the candela:

"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. It follows that the spectral luminous efficacy for monochromatic radiation of frequency of 540 x 1012 hertz is exactly 683 lumens per watt, K = 683 lm/W= 683 cd sr/W."

This value of K is the scaling factor for all the luminous efficiency functions. The value 683 was chosen to provide the best continuity between the size of the candela historically established using candles and platinum blackbodies before the change in 1979 to link the photometric units to the radiometric units. The definition is given for a specific frequency of radiation, which corresponds to approximately 555 nm in air (555.016 nm in standard air). The definition in terms of frequency avoids wavelength problems caused by the different index of refraction of different media of propagation. This does complicate slightly the determination of the maximum luminous efficacy value *Km* that is used to obtain photometric quantities from radiometric quantities using the CIE spectral luminous efficiency functions that are given as functions of wavelength (CIE S017, 2011). From these definitions we have (CIE S017, 2011): For photopic vision *Km* = 683 *V*(555 nm)/*V*(555.016 nm) lm·W-1=683.002 lm·W-1 ≈ 683 lm·W-1. For scotopic vision *Km* = 683 *V* (507 nm)/ *V* (555.016 nm) lm·W-1=1700.05 lm·W-1 ≈ 1700 lm·W-1. For all other wavelengths *K KV* () () *<sup>m</sup>* and *K KV* () () *<sup>m</sup>* .

From these definitions, all the geometrical photometric quantities may be determined from the corresponding radiometric quantities discussed in Section 4.1 above. For example,

$$\begin{aligned} \clubsuit \spadesuit &= \mathsf{K}\_{m} \int \spadesuit \spadesuit \left( \clubsuit \end{aligned} \begin{aligned} \spadesuit \spadesuit \left( \clubsuit \end{aligned} \right) \cdot \mathbf{V} \left( \clubsuit \right) \cdot \mathbf{d} \mathscr{X} \\ \clubsuit \spadesuit \overleftarrow{\mathbf{Q}\_{\text{v}}^{\prime}} &= \mathsf{K}\_{m}^{\prime} \int \spadesuit \spadesuit \left( \clubsuit \end{aligned} \begin{aligned} \mathbf{d} \mathscr{X} \\ \mathbf{d} \mathscr{X} \end{aligned} \tag{28}$$

where v is the photopic luminous flux in units of lumens, v is the scotopic luminous flux, also in units of lumens, e ( ) is the spectral distribution of radiant flux in units of watts per nanometre and d is the wavelength in units of nanometre.

The CIE spectral luminous efficiency functions are relative spectral values with a maximum value of one. Within the SI, photometric quantities and units are obtained by calculation from radiometric quantities and units. This is facilitated with the definition of a quotient K called the luminous efficacy of radiation and defined as (CIE S017, 2011) the quotient of the luminous flux, <sup>v</sup> , by the corresponding radiant flux <sup>e</sup> . (The subscripts v and e are used

> v e

The spectral luminous efficiency functions provide the relative spectral values for K(). In order to obtain the absolute scaling, there needs to be a defined scaling factor between radiometry and photometry at one of the wavelengths of the luminous efficiency functions. This is done in the definition (BIPM, 2006) of the SI base unit of luminous intensity, the

"The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540 x 1012 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. It follows that the spectral luminous efficacy for monochromatic radiation of frequency of 540 x 1012 hertz is exactly 683 lumens per watt, K =

This value of K is the scaling factor for all the luminous efficiency functions. The value 683 was chosen to provide the best continuity between the size of the candela historically established using candles and platinum blackbodies before the change in 1979 to link the photometric units to the radiometric units. The definition is given for a specific frequency of radiation, which corresponds to approximately 555 nm in air (555.016 nm in standard air). The definition in terms of frequency avoids wavelength problems caused by the different index of refraction of different media of propagation. This does complicate slightly the determination of the maximum luminous efficacy value *Km* that is used to obtain photometric quantities from radiometric quantities using the CIE spectral luminous efficiency functions that are given as functions of wavelength (CIE S017, 2011). From these definitions we have (CIE S017, 2011): For photopic vision *Km* = 683 *V*(555 nm)/*V*(555.016

*V* (555.016 nm) lm·W-1=1700.05 lm·W-1 ≈ 1700 lm·W-1. For all other wavelengths

From these definitions, all the geometrical photometric quantities may be determined from the corresponding radiometric quantities discussed in Section 4.1 above. For

> *K V K V*

 

where v is the photopic luminous flux in units of lumens, v is the scotopic luminous

( ) ( )d ( ) ( )d

 

 

(28)

is the spectral distribution of radiant flux in units of

v e v e

*m m* *<sup>K</sup>* (27)

= 683 *V* (507 nm)/

to indicated photometric and radiometric quantities respectively.)

nm) lm·W-1=683.002 lm·W-1 ≈ 683 lm·W-1. For scotopic vision *Km*

watts per nanometre and d is the wavelength in units of nanometre.

 and *K KV* () () *<sup>m</sup>* .

candela:

683 lm/W= 683 cd sr/W."

*K KV* () () *<sup>m</sup>*

example,

flux, also in units of lumens, e ( )

The relationships between the radiometric and photometric quantities, units and symbols are shown in Table 1. Although luminous exitance has the same unit (lumen per square metre) as illuminance, the unit lux is reserved for illuminance only.


Table 1. Radiometric and Photometric quantities, units and symbols.
