Virtual and remote experiments for radiometric and photometric measurements

For many reasons, experiments should be available in distance learning. When expensive, complex, or challenging experimental set-ups are unavailable or impracticable, virtual and remote labs can serve as an alternative for promoting practical experimental skill development and discovery-based learning. Virtual and remote labs can also serve as a useful supplement to in-class experiments, when...

• the experimental set-up is difficult to implement, time-consuming to arrange, or expensive [1],
• the procedures include dangerous objects, e.g., radioactivity [1] or high-power lasers [2],
• instrumentation must be shared due to limited lab capacities [1, 3], or
• cannot be provided due to logistical reasons [2],
• part-time students cannot attend on-site events,
• grown-ups in case of an initiative for life-long learning must integrate experimental activities in their daily routine,
• disabled students cannot perform an experimental task due to physical limitations [2], or
• the process of the experiment requires distinct scaffolds (scaffolding is of special interest in educational research, see section 2.3).

Experiments can be provided to distance-education learners in different ways. The closest thing to the on-site experiment is a remote experiment, where a real apparatus at a location A is remotely controlled by a user at another location B over the Internet (figure 1). In this case, the measured values can be recorded and, in addition, a live video of the apparatus can be transmitted [4, 5].

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Remote labs enable students to take control of experiments in an active yet harmless way. In this environment, students can watch their peers during the activity and learn from their successes and mistakes. Another advantage of remote labs is the ability to record actual data in real time, along with measuring errors. Thereby, the measuring accuracy is determined by the measuring instruments used.

A remote lab is available from anywhere at any time, but cannot be controlled by multiple users at the same time. Furthermore, the greatest benefit of this methodology for educational research is that the actions of users are immediately registered in full detail and can be evaluated later. All this is done without disturbing the user in any way [6].

Remote labs are typically used...

• to prepare students for hands-on sessions in traditional higher education labs,
• to enrich on-site experiments,
• in distance education,
• for real-time demonstrations in lectures and classrooms, and
• to highlight effects and phenomena (augmented reality).

Instead of carrying out a real experiment, simulations can also be used to acquire experimental competence. In a simulation, the user selects the initial parameters of a given dynamic model. The simulation then determines the temporal development of the dependent variables. Often, the calculated dependent variables are processed medially and presented as dynamic graphics. Neither measurement errors nor measurement accuracy are provided, but they could be simulated or determined from an error propagation of the inaccuracy of the input parameters. Contrary to remote experiments, simulations are always accessible to any number of users at the same time.

The concept of ISE was originally developed with the intention to combine the advantages of real film and simulation [7]. However, an ISE is also a link between remote experiment and simulation. The ISE uses a large number of photographs of the experimental apparatus to provide all possible settings and observable experimental results to be photo-realistic. Since only a limited number of photographs and associated measured values can be stored, measurement errors are usually not provided, but can be derived from stored measurement data or simulated.

Simulations and ISEs can be summarised under the term virtual experiment. One or more virtual experiments are provided in a virtual lab. Analogously, one or more remote experiments are provided in a remote lab. While a remote experiment is always conducted over the Internet, most virtual experiments could technically also be downloaded and be used offline.

Remote labs and virtual labs have different advantages and disadvantages (table 1), the VirtualRemoteLab approach offers most experiments as remote lab implementation along with a similar-looking virtual lab version. This allows teachers to choose the best solution for a certain purpose.

According to Hattie [8], 'Teachers are among the most powerful influences in learning.' When students acquire knowledge on their own this component is missing and has to be replaced by the learning environment. One general advantage of distance labs over on-site labs is the possibility to implement scaffolds.

Scaffolds are support strategies tailored to the needs of a single student. Especially in discovery learning students need scaffolding [9]. Typical strategies to facilitate learning by scaffolding are as follows.

• Interpretative support enables access to complementary knowledge, activates prior knowledge, helps in general problem analysis, supports the generation of suitable hypotheses, and promotes deeper understanding [10].
• Experimental support is intended to elucidate design of scientific investigations, promote a control-of-variables strategy (vary-one-thing-at-a-time) and assist in the formation of predictions, observations, comparison of results and the gathering of reasonable conclusions [10].
• Reflective support strategies, which stimulate self-regulation processes are of great importance for discovery learning [10].
• Besides those strategies listed above, teachers often use enrichment scaffolds, which link the problem in discussion with a related problem area to achieve a higher processing depth [11].
• Implicit scaffolds support student engagement and inquiry by channelling interactions with a learning tool by affordances and constraints [12]; affordances guide the student by possible actions that can be conducted with a tool; constraints guide the learner by what cannot be done in a learning environment.

While scaffolds are easy to implement in virtual labs, the design of scaffolding for remote labs requires deeper considerations due to the large amount of possible user actions.

Comparing the learning effectiveness of on-site laboratories, virtual laboratories and remote labs has been the subject of many studies. Tzafestas et al compared real, virtual, and remote experimentation, and found no significant differences in learning effectiveness in the three dimensions: low-level skills × mid- and high-level skills × understanding [13]. Zacharia and Olympiou compared hands-on experiments with virtual experiments in the case of thermodynamics and found no significant differences in acquisition of declarative knowledge [14]. A blended learning scenario combining hands-on and virtual experiments in case of optics has proven to be more effective compared to either hands-on experiments or virtual experiments alone [15]. This effect was also observed when assembling electrical circuits [16].

Remote labs have been built in a variety of physics fields. In such environments physical constants can be determined, including the elementary charge with Milikan's oil-drop experiment [17] or the speed of light [18]. Pendulums distributed around the world allow for the measurement and the comparison of the gravitational acceleration at different places [19]. Set-ups containing radioactive sources such as Rutherford scattering [20] and radioactive decay [21] are available as a remote lab variant. There exist remote labs for introductory school level such as the hot wire game or various other platforms involving simple robotics [4]. At an intermediate level—suitable for high school or undergraduate courses—basic physical laws such as Snell's law and Hook's law can be examined [22]. In higher education, topics such as magnetic domains in thin films [23] or Compton x-ray scattering [24] can be considered. Also, medical applications such as the biochemical oxygen demand can be examined [24]. In astrophysics, robotic telescopes are a common tool that are also used for educational purposes [24]. In various projects, control systems are developed with simulations in a virtual remote laboratory and then tested in a remote lab [3, 25]. By comparing the simulation with actual implementation, the limitations of simulations can be denoted [26] (and vice versa).

A remote lab on spectrometry to perform spectrometric, radiometric and photometric measurements has not yet been made available on the Internet. In the following, the physical background, the conceptual development and the educational perspectives of a remote experiment on optical spectrometry with standard lamps is described.

Radiant energy ${Q}_{{\rm{e}}}$ is the energy emitted, transferred, or received in the form of electromagnetic radiation.

Radiant flux, or radiant power, ${{\rm{\Phi }}}_{{\rm{e}}}$ is the power (energy per unit time t) emitted, transferred, or received in the form of electromagnetic radiation:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\displaystyle \frac{{\rm{d}}{Q}_{{\rm{e}}}}{{\rm{d}}t}.\end{eqnarray} \tag{ 1 }$$

     Irradiance ${E}_{{\rm{e}}}$ is the ratio of the radiant power incident on an infinitesimal element of a surface ${S}_{{\rm{d}}}$ to the projected area ${\rm{d}}{A}_{{\rm{d}}}$ of that element ${\rm{d}}{S}_{{\rm{d}}}$,

$$\begin{eqnarray}&&{E}_{{\rm{e}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}}}}{{\rm{d}}{A}_{{\rm{d}}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}}}}{\cos {\theta }_{{\rm{d}}}\,{\rm{d}}{S}_{{\rm{d}}}},\end{eqnarray} \tag{ 2 }$$

where ${\theta }_{{\rm{d}}}$ is the angle between the direction of radiation and the vector normal to ${\rm{d}}{S}_{{\rm{d}}}$.

Exitance ${M}_{{\rm{e}}}$ is the ratio of the radiant power leaving an infinitesimal element of a surface ${S}_{{\rm{s}}}$ to the projected area ${\rm{d}}{A}_{{\rm{s}}}$ of that element ${\rm{d}}{S}_{{\rm{s}}}$,

$$\begin{eqnarray}&&{M}_{{\rm{e}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}}}}{{\rm{d}}{A}_{{\rm{s}}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}}}}{\cos {\theta }_{{\rm{s}}}\,{\rm{d}}{S}_{{\rm{s}}}},\end{eqnarray} \tag{ 3 }$$

where ${\theta }_{{\rm{s}}}$ is the angle between the direction of radiation and the surface element's normal vector.

The solid angle ${\rm{d}}{\rm{\Omega }}$ is the ratio between the area of a portion of a sphere and the square of the corresponding radius r.

$$\begin{eqnarray}&&{\rm{d}}{\rm{\Omega }}=\displaystyle \frac{{\rm{d}}A}{{r}^{2}}.\end{eqnarray} \tag{ 4 }$$

     Radiance ${L}_{{\rm{e}}}$ is the ratio of the radiant power to the infinitesimal elements of both projected area and solid angle when the radiation is incident at an angle of θ (see figure 2). Radiance can be defined at a point on the surface of a source or detector, or at any point on the ray.

$$\begin{eqnarray}&&{L}_{{\rm{e}}}=\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Phi }}}_{{\rm{e}}}}{{\rm{d}}A\,{\rm{d}}{\rm{\Omega }}}=\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Phi }}}_{{\rm{e}}}}{\cos \theta \,{\rm{d}}S\,{\rm{d}}{\rm{\Omega }}}.\end{eqnarray} \tag{ 5 }$$

      Radiant exposure ${H}_{{\rm{e}}}$ is the radiant energy ${Q}_{{\rm{e}}}$ incident on a surface S per unit area

$$\begin{eqnarray}&&{H}_{{\rm{e}}}=\displaystyle \frac{{\rm{d}}{Q}_{{\rm{e}}}}{{\rm{d}}S},\end{eqnarray} \tag{ 6 }$$

or equivalently the irradiance ${E}_{{\rm{e}}}$ on a surface S integrated over time ${\rm{\Delta }}t$:

$$\begin{eqnarray}&&{H}_{{\rm{e}}}={\int }_{\tilde{t}}^{\tilde{t}+{\rm{\Delta }}t}{E}_{{\rm{e}}}\,{\rm{d}}t.\end{eqnarray} \tag{ 7 }$$

      Radiant intensity ${I}_{{\rm{e}}}$ is the ratio of the radiant flux ${{\rm{\Phi }}}_{{\rm{e}}}$ leaving a source to an element of solid angle ${\rm{d}}{\rm{\Omega }}$ propagated in the given direction:

$$\begin{eqnarray}&&{I}_{{\rm{e}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}}}}{{\rm{d}}{\rm{\Omega }}}.\end{eqnarray} \tag{ 8 }$$

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As noted by Zalewski, 'It should be emphasized that the above definitions are precisely accurate only for point sources and point detectors. In practice, a measurement cannot be obtained at a point or an infinitesimal area. Therefore, when the terms defined above are applied to actual measurements it is usually assumed that averages are being discussed' [28].

Photometric quantities characterise visible light as perceived by the human eye. The basic idea is to provide a conversion rule from radiometric to photometric quantities—the luminous efficiency function $V(\lambda )$ (figure 3), empirically based [29] and decided in 1924 by the International Commission on Illumination (CIE) for photopic (daylight-adapted) perception of a 2° point-like light source [30]. Any spectral radiometric quantity ${X}_{{\rm{e}},\lambda }$ can be converted to the corresponding photometric quantity ${X}_{{\rm{v}}}$ by the formula

$$\begin{eqnarray}&&{X}_{{\rm{v}}}={K}_{{\rm{m}}}{\int }_{0}^{\infty }V(\lambda )\,{X}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda ,\end{eqnarray} \tag{ 9 }$$

where ${K}_{{\rm{m}}}=683\,\mathrm{lm}\,{{\rm{W}}}^{-1}$ is the spectral luminous efficacy (the ratio of the luminous flux ${{\rm{\Phi }}}_{{\rm{v}}}$ to the radiant flux ${{\rm{\Phi }}}_{{\rm{e}}}$) for monochromatic radiation of frequency ${\nu }_{0}=540\times {10}^{12}\,\mathrm{Hz}$, $V(\lambda )$ is the luminous efficiency function, and λ is the wavelength. The value of ${K}_{{\rm{m}}}$ derives from the SI definition of the photometric base unit of luminous intensity, the candela [31]:

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The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency $540\times {10}^{12}$ hertz and that has a radiant intensity in that direction of 1/683 watt per steradian.

It is well-known that the $V(\lambda )$ function is erroneous, especially for short wavelengths, and there have been various attempts to redefine the spectral luminous efficiency function [32–34]. However, most measurements in the last 90 years were referenced to the original function. Thus, in order to keep comparability, it has remained the standard for converting radiometric to photometric quantities [35]. However, if another standardised spectral luminous efficiency function is used, a subscript is preferred to reference the function (e.g., ${X}_{{\rm{v}}}^{\prime }$ for the scotopic (low-light adapted) function, or X₁₀ for the 10° photopic observer function [36]).

Luminous energy ${Q}_{{\rm{v}}}$ is the energy emitted, transferred, or received in the form of light as perceived by a standard observer whose sensitivity is defined by the relative luminous efficiency function V_λ:

$$\begin{eqnarray}&&{Q}_{{\rm{v}}}={K}_{{\rm{m}}}{\int }_{0}^{\infty }{V}_{\lambda }\,{Q}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda ,\end{eqnarray} \tag{ 10 }$$

where ${K}_{{\rm{m}}}=683\,\mathrm{lm}\,{{\rm{W}}}^{-1}$, ${Q}_{{\rm{e}},\lambda }$ is the spectral radiant energy and λ is the wavelength.

Luminous flux, or luminous power, ${{\rm{\Phi }}}_{{\rm{v}}}$ is the luminous energy per unit time t emitted, transferred, or received in the form of light as perceived by a standard observer:

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{v}}}=\displaystyle \frac{{\rm{d}}{Q}_{{\rm{v}}}}{{\rm{d}}t}={K}_{{\rm{m}}}{\int }_{0}^{\infty }{V}_{\lambda }\,{{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda .\end{eqnarray} \tag{ 11 }$$

Luminous flux is a measure for the total amount of light sent out by a light source. Hence, it is usually printed on the packaging of lamps and used by customers to compare various lamps. The luminous flux allows for (a) estimation of the perceived brightness, and (b) calculation of the luminous efficacy K:

$$\begin{eqnarray}&&K=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{v}}}}{{{\rm{\Phi }}}_{{\rm{e}}}}=\displaystyle \frac{{\int }_{0}^{\infty }K(\lambda )\,{{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda }{{\int }_{0}^{\infty }{{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda },\end{eqnarray} \tag{ 12 }$$

where ${{\rm{\Phi }}}_{{\rm{v}}}$ is the luminous flux, ${{\rm{\Phi }}}_{{\rm{e}}}$ is the radiant flux, ${{\rm{\Phi }}}_{{\rm{e}},\lambda }$ is the spectral radiant flux, and $K(\lambda )={K}_{{\rm{m}}}{V}_{\lambda }$ is the spectral luminous efficacy. Note that here the luminous efficacy is referenced to the radiant flux, thus, sometimes called luminous efficacy of radiation. More important to protect the environment is the overall luminous efficacy ${K}_{\mathrm{tot}}$, the ratio of luminous flux ${{\rm{\Phi }}}_{{\rm{v}}}$ to total power ${P}_{\mathrm{tot}}$ consumed by the source:

$$\begin{eqnarray}&&{K}_{\mathrm{tot}}=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{v}}}}{{P}_{\mathrm{tot}}}.\end{eqnarray} \tag{ 13 }$$

In order to give an overview, the quantities used to describe the ratio of photometric to radiometric quantities are listed in table 3.

Table 3.  Quantities used to describe the ratio of photometric to radiometric quantities.

Name	Symbol	Unit
Luminous efficacy (of radiation)	$\,K={{\rm{\Phi }}}_{{\rm{v}}}/{{\rm{\Phi }}}_{{\rm{e}}}$	lm W−1
Overall luminous efficacya	${K}_{\mathrm{tot}}={{\rm{\Phi }}}_{{\rm{v}}}/{P}_{\mathrm{tot}}$	lm W−1
Spectral luminous efficacy	$K(\lambda )={{\rm{\Phi }}}_{{\rm{v}},\lambda }/{{\rm{\Phi }}}_{{\rm{e}},\lambda }$	lm W−1
Maximum spectral luminous efficacy	${K}_{{\rm{m}}}$	lm W−1
Luminous efficiency	$\,V=K/{K}_{{\rm{m}}}$
Spectral luminous efficiency	$V(\lambda )=K(\lambda )/{K}_{{\rm{m}}}$

^aOr lighting efficacy.

Illuminance ${E}_{{\rm{e}}}$ is the ratio of the luminous power incident on an infinitesimal element of a surface ${S}_{{\rm{d}}}$ to the projected area ${\rm{d}}{A}_{{\rm{d}}}$ of that element ${\rm{d}}{S}_{{\rm{d}}}$,

$$\begin{eqnarray}&&{E}_{{\rm{v}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{v}}}}{{\rm{d}}{A}_{{\rm{d}}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{v}}}}{\cos {\theta }_{{\rm{d}}}\,{\rm{d}}{S}_{{\rm{d}}}},\end{eqnarray} \tag{ 14 }$$

where ${\theta }_{{\rm{d}}}$ is the angle between the direction of light and the vector normal to ${\rm{d}}{S}_{{\rm{d}}}$.

Luminous exitance ${M}_{{\rm{v}}}$ is the ratio of the luminous power leaving an infinitesimal element of a surface ${S}_{{\rm{s}}}$ to the projected area ${\rm{d}}{A}_{{\rm{s}}}$ of that element ${\rm{d}}{S}_{{\rm{s}}}$,

$$\begin{eqnarray}&&{M}_{{\rm{v}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{v}}}}{{\rm{d}}{A}_{{\rm{s}}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{v}}}}{\cos {\theta }_{{\rm{s}}}\,{\rm{d}}{S}_{{\rm{s}}}},\end{eqnarray} \tag{ 15 }$$

where ${\theta }_{{\rm{s}}}$ is the angle between the direction of light and the surface element's normal vector.

Luminance ${L}_{{\rm{e}}}$ is the ratio of the luminous power to the infinitesimal elements of both projected area and solid angle when the radiation is incident at an angle of θ. Luminance can be defined at a point on the surface of a source or detector, or at any point on the ray:

$$\begin{eqnarray}&&{L}_{{\rm{v}}}=\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Phi }}}_{{\rm{v}}}}{{\rm{d}}A\,{\rm{d}}{\rm{\Omega }}}=\displaystyle \frac{{{\rm{d}}}^{2}{{\rm{\Phi }}}_{{\rm{v}}}}{\cos \theta \,{\rm{d}}S\,{\rm{d}}{\rm{\Omega }}}.\end{eqnarray} \tag{ 16 }$$

Luminous intensity is the luminous flux per solid angle:

$$\begin{eqnarray}&&{I}_{{\rm{v}}}=\displaystyle \frac{{\rm{d}}{{\rm{\Phi }}}_{{\rm{v}}}}{{\rm{d}}{\rm{\Omega }}}=\displaystyle \frac{{\rm{d}}\left[{K}_{{\rm{m}}}{\int }_{0}^{\infty }{V}_{\lambda }\,{{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda \right]}{{\rm{d}}{\rm{\Omega }}}.\end{eqnarray} \tag{ 17 }$$

In order to analyse the radiation field around a particular light source under constant experimental conditions, a probe is used to measure the spectral irradiance.

The irradiance is the total power of electromagnetic radiation incident on a surface per unit area, such as the fibrous cross-section at its tip. Nevertheless, with a bare fibre it is not possible to measure the true irradiance because the coupling of light into the fibre is highly dependent on the angle of incidence. According to Eppeldauer, 'the most important requirement for an irradiance meter is to design its front geometry to satisfy the cosine law of irradiance measurements for the overall angular range of the incident flux' [37]. Furthermore, when the field of irradiation is non-uniform, a probe with a 180° field of view is preferred. The spectrometer's manufacturer provides a so-called cosine corrector: a window made from opaline glass mountable on the tip of the optical fibre that creates a dependency of the spectral irradiance on the angle of incidence that is nearly proportional to the cosine [38].

As Larason, Brown, Eppeldauer and Lykke note: 'Knowledge of the spectral irradiance responsivity of a metre is critical for high accuracy measurements of sources with different spectral power distributions' [39]. Unfortunately, this important calibration measure often does not receive adequate attention.

Figure 4(a) shows the measured spectrum of a standard tungsten halogen lamp (A). For learning purposes, this graph requires special considerations: students may interpret this spectrum as black-body radiation based on the shape of the curve, as well as the fact that thermal radiators such as tungsten lamps behave like black-body radiators. Actually, a tungsten lamp emits only a small part (about 4%) of its radiation in the visible part of the electromagnetic spectrum.

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The important point to keep in mind is that the sensitivity of the detector system is extremely wavelength-dependent. Figure 4(b) shows the measured spectral sensitivity of the detector system used for the experiment with a logarithmic scaling. Note that the spectral sensitivity between 500 and 1000 nm differs by a factor of 1000. Without accurate calibration, no comparison can be made between 'intensities' at different wavelengths, nor can any definitive statements be made about the spectral power distribution of the radiation.

The general procedure for calibration of a radiometric detector using the substitution method is explained in the following section. In section 3.6, the actual implementation we used is described.

The spectral distribution of radiometric quantities can be specified according to wavelength or frequency interval. For example, the monochromatic radiant power ${{\rm{\Phi }}}_{{\rm{e}},\lambda }$ is the radiant power of an infinitesimal wavelength interval ${\rm{d}}\lambda$, and ${{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda$ is the radiant power of the wavelength interval λ to $\lambda +{\rm{d}}\lambda$. The total radiant power over the entire spectrum is therefore

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}={\int }_{0}^{\infty }{{\rm{\Phi }}}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda .\end{eqnarray} \tag{ 18 }$$

The measured signal from a specific pixel is the integral of the source's power distribution multiplied by the pixel's spectral responsivity,

$$\begin{eqnarray}&&{I}_{n}={\int }_{0}^{\infty }{s}_{n,\lambda }\,{Q}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda ,\end{eqnarray} \tag{ 19 }$$

where I_n is the signal from the nth pixel, ${Q}_{{\rm{e}},\lambda }$ is the spectral radiant energy from the source and ${s}_{n,\lambda }$ is the spectral responsivity of the nth detector pixel (see [39]).

At this point, the detector substitution method is employed, using the known spectral radiometric responsivity of a standard detector under radiance from a calibration standard. The spectral radiometric responsivity of a test detector is then compared to this reference (see [40, 41]). When the standard detector measures the spectral input flux (the total radiant power in the incident beam) ${\text{}}{{\rm{\Phi }}}_{{\rm{e}},\lambda }$, the standard detector's spectral output signal ${I}_{{\rm{S}},\lambda }$ is:

$$\begin{eqnarray}&&{I}_{{\rm{S}},\lambda }={s}_{{\rm{S}},\lambda }\cdot {\text{}}{{\rm{\Phi }}}_{{\rm{e}},\lambda },\end{eqnarray} \tag{ 20 }$$

where ${s}_{{\rm{S}},\lambda }$ is the known spectral responsivity of the standard detector.

The standard detector is then substituted with a test detector to measure the same constant spectral input flux ${\text{}}{{\rm{\Phi }}}_{{\rm{e}},\lambda }$. The test detector's spectral output signal ${I}_{{\rm{T}},\lambda }$ is:

$$\begin{eqnarray}&&{I}_{{\rm{T}},\lambda }={s}_{{\rm{T}},\lambda }\cdot {\text{}}{{\rm{\Phi }}}_{{\rm{e}},\lambda },\end{eqnarray} \tag{ 21 }$$

where ${s}_{{\rm{T}},\lambda }$ is the unknown spectral responsivity of the test detector.

The unknown spectral responsivity ${s}_{{\rm{T}},\lambda }$ of the test detector can then be calculated:

$$\begin{eqnarray}&&{s}_{{\rm{T}},\lambda }={s}_{{\rm{S}},\lambda }\cdot \displaystyle \frac{{I}_{{\rm{T}},\lambda }}{{I}_{{\rm{S}},\lambda }}.\end{eqnarray} \tag{ 22 }$$

To calibrate the radiometric system, the distributor of the spectrometers manufacturer kindly provided us with a NIST-traceable radiometric calibration standard. In this case, the standard was a calibrated tungsten halogen light source with a well-known spectral power distribution. This calibration standard provides known absolute spectral irradiance in $\mu {\rm{W}}\,{{\rm{c}}{\rm{m}}}^{-2}\,{{\rm{n}}{\rm{m}}}^{-1}$ at the fibre port. The spectrometer can be calibrated specifically for a bare fibre or a fibre with a cosine corrector attached.

The spectrum obtained from this measurement was then fitted against the known spectral power distribution of the standard source (see section 3.5). This led to pixel-wise calibration factors s_n that are suitable for the following measurements.

The Ocean Optics Red Tide USB 650 Spectrometer that was used in the experiment contains 651 active pixels and covers a wavelength range of 350–1000 nm. The detector system was calibrated under experimental conditions and the calibration factors were determined as follows.

• 1.  
  At first, the calibration data ${E}_{{\rm{e}},{\rm{S}},\lambda }$ supplied with the radiometric standard was interpolated in order to get the expected spectral irradiance values ${E}_{{\rm{e}},{\rm{S}},n}$ for every detector pixel.
• 2.  
  At this point, a measurement was performed with the entrance slit closed to determine the level of noise ${I}_{\mathrm{noise},n}$.
• 3.  
  Afterwards, the signal ${I}_{\mathrm{signal},n}$ from every detector pixel was recorded under irradiation with the radiometric standard. The previously recorded noise was then subtracted from the signal:

  $$\begin{eqnarray}&&{I}_{n}={I}_{\mathrm{signal},n}-{I}_{\mathrm{noise},n}.\end{eqnarray} \tag{ 23 }$$
• 4.  
  Under the assumption that the irradiance ${E}_{{\rm{e}}}$ on the probe's surface S is constant over the integration time ${\rm{\Delta }}t$, the radiant exposure measured by the nth pixel ${H}_{{\rm{e}},n}$ is

  $$\begin{eqnarray}&&{H}_{{\rm{e}},n}={\int }_{\tilde{t}}^{\tilde{t}+{\rm{\Delta }}t}{E}_{{\rm{e}},n}\,{\rm{d}}t\approx {E}_{{\rm{e}},n}\,{\rm{\Delta }}t\end{eqnarray} \tag{ 24 }$$

  and the radiant energy ${Q}_{{\rm{e}},n}$ measured by the nth pixel is

  $$\begin{eqnarray}&&{Q}_{{\rm{e}},n}=S\,{H}_{{\rm{e}},n}\approx S\,{E}_{{\rm{e}},n}\,{\rm{\Delta }}t.\end{eqnarray} \tag{ 25 }$$
• 5.  
  The calibration factors describe the ratio between the radiation energy ${Q}_{{\rm{e}},n}$ and acquired signal ${I}_{n}$ for each pixel:

  $$\begin{eqnarray}&&{s}_{n}=\displaystyle \frac{S\,{E}_{{\rm{e}},{\rm{S}},n}\,{\rm{\Delta }}t}{{I}_{n}}\approx \displaystyle \frac{{Q}_{{\rm{e}},n}}{{I}_{n}}.\end{eqnarray} \tag{ 26 }$$

Note that from this point onward the spectral responsivities ${s}_{n,\lambda }$ of all detector pixels are combined in the spectral responsivity s_λ of the detector system, to produce effective calibration factors s_n. Once the calibration factors s_n are estimated, the spectral irradiance can be calculated for every acquired spectrum.

To calculate the spectral irradiance for any acquired intensity spectrum, the following steps are performed by the spectrometer software.

At first, the previously recorded spectral noise ${I}_{\mathrm{noise},n}$ is subtracted from the signal ${I}_{\mathrm{signal},n}$ recorded from every detector pixel under irradiation:

$$\begin{eqnarray}&&{I}_{n}={I}_{\mathrm{signal},n}-{I}_{\mathrm{noise},n}.\end{eqnarray} \tag{ 27 }$$

The energy received by the nth detector pixel is then calculated:

$$\begin{eqnarray}&&{Q}_{{\rm{e}},n}={s}_{n}\,{I}_{n}\end{eqnarray} \tag{ 28 }$$

with the calibration factor s_n for the nth detector pixel.

Dividing out the probe's surface area S leaves the radiant exposure ${H}_{{\rm{e}},n}$ measured by the nth pixel:

$$\begin{eqnarray}&&{H}_{{\rm{e}},n}=\displaystyle \frac{{Q}_{{\rm{e}},n}}{S}.\end{eqnarray} \tag{ 29 }$$

The radiant exposure H_n is the cumulative irradiance ${E}_{{\rm{e}},n}$ measured by the nth pixel during the integration time ${\rm{\Delta }}t$:

$$\begin{eqnarray}&&{H}_{{\rm{e}},n}={\int }_{\tilde{t}}^{\tilde{t}+{\rm{\Delta }}t}{E}_{{\rm{e}},n}\,{\rm{d}}t.\end{eqnarray} \tag{ 30 }$$

Under the simplifying assumption that the irradiance ${E}_{{\rm{e}},n}$ is constant over time, the irradiance ${E}_{{\rm{e}},n}$ measured by the nth pixel may be calculated by dividing the radiant exposure ${H}_{{\rm{e}},n}$ by the integration time ${\rm{\Delta }}t$:

$$\begin{eqnarray}&&{E}_{{\rm{e}},n}=\displaystyle \frac{{H}_{{\rm{e}},n}}{{\rm{\Delta }}t}.\end{eqnarray} \tag{ 31 }$$

Taking into account the wavelength spread ${\rm{\Delta }}{\lambda }_{n}$ of the nth pixel, the spectral irradiance measured by the nth pixel is

$$\begin{eqnarray}&&{E}_{{\rm{e}},\lambda ,n}=\displaystyle \frac{{E}_{{\rm{e}},n}}{{\rm{\Delta }}{\lambda }_{n}}.\end{eqnarray} \tag{ 32 }$$

Since a specific wavelength interval can be assigned to each pixel, the spectral irradiance ${E}_{{\rm{e}},\lambda }$ results from the respective spectral irradiance measured by the individual pixels ${E}_{{\rm{e}},\lambda ,n}$ to a sufficient approximation within the measuring range:

$$\begin{eqnarray}&&{E}_{{\rm{e}},[{\lambda }_{1},{\lambda }_{2}]}={\int }_{{\lambda }_{1}}^{{\lambda }_{2}}{E}_{{\rm{e}},\lambda }\,{\rm{d}}\lambda \approx \displaystyle \sum _{n=n({\lambda }_{1})}^{n({\lambda }_{2})}{\int }_{\lambda (n)-\tfrac{1}{2}{\rm{\Delta }}{\lambda }_{n}}^{\lambda (n)+\tfrac{1}{2}{\rm{\Delta }}{\lambda }_{n}}{E}_{{\rm{e}},\lambda ,n}\,{\rm{d}}\lambda \approx \displaystyle \sum _{n=n({\lambda }_{1})}^{n({\lambda }_{2})}{E}_{{\rm{e}},n}.\end{eqnarray} \tag{ 33 }$$

Figure 5 clearly illustrates how important it is to use appropriate optics, perform the calibration procedure, and account for detector characteristics.

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Using the experimental set-up described in section 4.2, many different objectives can be realised. Table 4 shows the experiments that are accessible by the website http://virtualremotelab.net. Students can compare spectra from different lamps and thus rate the usability of the light sources for various purposes (figure 6). Students can also assess colour temperature and colour error, as well as compare the radiated light with the colour sensitivity of the human eye. As another goal, students can use the remote lab to distinguish between physical and physiological quantities, or they can consider the energy efficiency of the most popular lamp types.

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The flexible positioning of the probe allows for further experimentation. This includes studying the decrease of spectral irradiance as the distance between the source and probe is increased. Expert learners can even examine the spatial and spectral characteristics of radiation from specific light sources. This is of particular interest with light emitting diodes, where radiation is often highly anisotropic.

Different situations call for for different light sources (see figure 6). The present experiment allows for the observation of standard lamps with E 27 lamp sockets. Irradiation at a certain position is collected with an optical fibre and analysed with a USB compact spectrometer (see figure 7). The spectrometer that was used projects light through a 25 μm entrance slit onto a linear silicon CCD array with 651 active pixels offering an optical resolution of approximately 2.0 nm full width at half maximum. Radiation within a wavelength range of 350–1000 nm is recorded with 12 bit A/D resolution. With this, 'intensities' are measured within a range of 0–4095 arbitrary units.

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Several parameters in the experiment can be changed by the user. Some adjustable acquisition parameters, for example, include the integration time, the boxcar width (which characterises averaging over neighbouring pixels), and the number of samples to average over (averaging over two or more subsequent spectra). Furthermore, users can choose from six standard light bulbs that are mounted on a carousel: tungsten incandescent light bulbs, halogen incandescent lamps, cool white and soft white compact fluorescent lamps, light-emitting diode lamps and speciality bulbs. The cosine-corrected probe can be moved and positioned in front of the bulb in an area of 114 × 70 square centimetres. The probe can also be rotated. All of these adjustable parameters provide versatility and ensure a realistic experimental experience. With this versatility, the user-interface can be optimised for a distinct user group or a specific experiment (see figure 8). This implicit scaffolding technique (see section 2.3) helps students to identify the important parts of the control section.

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The single modules used to acquire, process, distribute, and display the spectral data, along with their communication paths, are shown in figure 9. On the remote lab spectrometer server, the acquisition of spectra is done by a Windows service. The same programme then computes the spectral irradiance. The acquired and computed data are then sent to a client for display. A notable challenge was ensuring that mobile devices such as smartphones and tablets were able to display the spectra and control the experiment. For this reason, the client programme is built for HTML5 web browsers, which are commonly available nowadays. To avoid problems due to network restrictions and firewalls on the side of the user, spectral data and control data are delivered via a commercial WebSocket service. Authentication is necessary before a user may control the experiment. This is done by a PHP script running on a typical web server that also provides the website for the experiment. An IP camera watches the experiment and sends the captured video stream directly to the client where the video stream is displayed in a browser. Another advantage of using the WebSocket service is that no static IP is needed for the remote-lab server. Only the IP camera needs direct Internet access. This prevents the remote-lab server from risks that would be associated with direct access from the Internet.

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For many experiments it is important that the measured irradiance ${E}_{{\rm{e}}}$ is linearly dependent on the cosine of the incident angle θ (see section 3.3). Hence, students should verify this dependency. When examining the visual part of the spectrum it is suitable to measure the illuminance ${E}_{{\rm{v}}}$ instead. Figure 10(a) shows the illuminance in lux plotted against the cosine of the incident angle θ at x-positions 25 and 45 cm under illumination with a green LED lamp for incident angles $-90^\circ \leqslant \theta \leqslant 90^\circ$. A nonlinear fit to ${E}_{{\rm{v}}}={L}_{{\rm{v}}}\,\cos \theta +{E}_{{\rm{v}},0}$ shows very good correspondence. A linear fit reproduces the data well for $\cos \theta \geqslant 0.4$ ($-66^\circ \leqslant \theta \leqslant 66^\circ$) (see figure 10(b)). It can be concluded that the cosine-corrected probe works well for educational purposes.

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As we saw earlier, equation (5) provides the definition of radiance. Consider a surface S₁ with radiance ${L}_{{\rm{e}},12}$ incident on a second surface S₂. Likewise, S₂ projects radiance ${L}_{{\rm{e}},21}$ onto S₁, and the two surfaces are separated by a light ray of length s₁₂. The net radiant power exchange between elemental areas on each surface is then given by (see [28, p 24.14, equation (19)])

$$\begin{eqnarray}&&{\rm{\Delta }}{{\rm{\Phi }}}_{{\rm{e}}}={\rm{d}}{{\rm{\Phi }}}_{{\rm{e}},12}-{\rm{d}}{{\rm{\Phi }}}_{{\rm{e}},21}=\displaystyle \frac{({L}_{{\rm{e}},12}-{L}_{{\rm{e}},21})\,\cos {\theta }_{1}\,\cos {\theta }_{2}\,{\rm{d}}{S}_{1}\,{\rm{d}}{S}_{2}}{{s}_{12}^{2}},\end{eqnarray} \tag{ 34 }$$

where ${\theta }_{1}$ and ${\theta }_{2}$ are the angles between the ray s₁₂ and the normals to the surfaces S₁ and S₂, respectively (see figure 11(a)).

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The total amount of radiation transferred between the two surfaces is given by the integral over both areas as follows (see [28, p 24.14, equation (20)]):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\int \int \displaystyle \frac{({L}_{{\rm{e}},12}-{L}_{{\rm{e}},21})\,\cos {\theta }_{1}\,\cos {\theta }_{2}}{{s}_{12}^{2}}\,{\rm{d}}{S}_{1}\,{\rm{d}}{S}_{2}.\end{eqnarray} \tag{ 35 }$$

In the simplified case of a source and a receiver, the radiant power emitted by a receiver is zero by definition. In this case, the term L₂₁ is zero and it follows (see [28, p 24.15, equation (23)])

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\int \int \displaystyle \frac{{L}_{{\rm{e}},12}\,\cos {\theta }_{1}\,\cos {\theta }_{2}}{{s}_{12}^{2}}\,{\rm{d}}{S}_{1}\,{\rm{d}}{S}_{2}.\end{eqnarray} \tag{ 36 }$$

For the specialised case of a source and a cosine-corrected detector the radiant power transfer equation becomes

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\int \int \displaystyle \frac{{L}_{{\rm{e}}}\,\cos {\theta }_{{\rm{s}}}\,\cos {\theta }_{{\rm{d}}}}{{s}_{\mathrm{sd}}^{2}}\,{\rm{d}}{S}_{{\rm{s}}}\,{\rm{d}}{S}_{{\rm{d}}},\end{eqnarray} \tag{ 37 }$$

where the subscripts s and d denote the source and the detector, respectively.

The exact solution for radiation transfer between a circular source and detector is (see [28, p 24.17, equation (32)]):

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\displaystyle \frac{2\,{L}_{{\rm{e}}}\,{(\pi {r}_{{\rm{s}}}{r}_{{\rm{d}}})}^{2}}{{r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2}+{[{({r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2})}^{2}-4{r}_{{\rm{s}}}^{2}{r}_{{\rm{d}}}^{2}]}^{1/2}}.\end{eqnarray} \tag{ 38 }$$

Under the experimental conditions presented here, the result can be approximated by assuming that the sum of the square of the distance and radii is large compared to the product of the radii, that is, $({r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2})\gg 2{r}_{{\rm{s}}}{r}_{{\rm{d}}}$. In this case, the equation reduces to

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\rm{e}}}=\displaystyle \frac{{L}_{{\rm{e}}}{(\pi {r}_{{\rm{s}}}{r}_{{\rm{d}}})}^{2}}{{r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2}}\end{eqnarray} \tag{ 39 }$$

with a relative error here being always lower than $2\times {10}^{-5}$ and therefore negligible compared to measurement uncertainties.

From equation (39), the irradiance at the detector can be obtained

$$\begin{eqnarray}&&{E}_{{\rm{e}}}=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{e}}}}{{A}_{{\rm{d}}}}=\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{e}}}}{\pi \,{r}_{{\rm{d}}}^{2}}=\displaystyle \frac{{L}_{{\rm{e}}}\,\pi \,{r}_{{\rm{s}}}^{2}}{{r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2}}.\end{eqnarray} \tag{ 40 }$$

Since the illuminance is of greater interest than the irradiance for lighting purposes, and since with a well-calibrated spectrometer the transfer from irradiance to illuminance is only a pixel-wise linear conversion, it is useful to consider the decrease of illuminance with growing distance:

$$\begin{eqnarray}&&{E}_{{\rm{v}}}=\displaystyle \frac{{L}_{{\rm{v}}}\,\pi \,{r}_{{\rm{s}}}^{2}}{{r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{s}_{\mathrm{sd}}^{2}}.\end{eqnarray} \tag{ 41 }$$

Figure 11(b) shows the very good nonlinear fit using ${E}_{{\rm{v}}}={L}_{{\rm{v}}}\,\pi \,{r}_{{\rm{s}}}^{2}/[{r}_{{\rm{s}}}^{2}+{r}_{{\rm{d}}}^{2}+{(x-{x}_{0})}^{2}]$, with ${L}_{{\rm{v}}}=(192,516\pm 1,113)\,\mathrm{cd}\,{\mathrm{sr}}^{-1}\,{\mathrm{cm}}^{-2}$, ${r}_{{\rm{s}}}=0.435\,\mathrm{mm}$, ${r}_{{\rm{d}}}=0.195\,\mathrm{cm}$, and ${x}_{0}=(-7.46\,\pm 0.03)\,\mathrm{mm};$ ${R}^{2}=0.9997$.

It is important for students to realise that the inverse-square law is an approximation for point sources of light. Extended light sources can also be approximated as projections of point light sources, leading to effective origins of the approximated point light sources far beyond the measured light source. Although a fit to ${E}_{{\rm{v}}}={I}_{{\rm{v}}}/{(x-{x}_{0})}^{2}$ where ${I}_{{\rm{v}}}$ is the luminous intensity shows good fit parameters, it is the responsibility of the experimenter to choose the best approximation for a given purpose. To find the best approximation it is very important to consider the spatial distribution of luminance of the lamp being analysed.

When choosing the appropriate light source for a specific lighting situation, it is very important to consider the spatial characteristics of a lamp. In former times, reflector lamps were used to direct light. With LED lamps, knowledge of the spatial distribution is much more important because light-emitting diodes radiate highly anisotropic. Hence, students should assess the spatial distribution of luminance for different LED lamps. Furthermore, the following experiment is vicarious for other devices which show anisotropic directional characteristics of radiation or sensitivity (eg., loudspeakers and microphones).

The spatial distribution of luminance can be estimated by measuring illuminance in a plane around the light source. To measure the spatial distribution of irradiance the following assumptions have to be made: first, the position of the lamp ${x}_{\mathrm{lamp}}$ (e.g., the centre of the lamp) has to be defined by the user. The coordinates and probe angles for the following measurements can then be calculated as illustrated in figure 12. However, a configuration of the graphical user interface with predefined offset positions is available. Then, only polar coordinates have to be chosen.

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Figure 13(a) shows a photograph of four typical LED lamps. The measured illuminance in a circle around each lamp is plotted in figure 13(b). The plot clearly shows the differences in spatial distribution of luminance for the four lamps. From plots such as this, students learn how important it is to know the spatial characteristics of a lamp when choosing appropriate light sources for specific lighting situations.

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