**3.1. Theoretical simulation**

The analysis presented is based on the specular reflection model. There are two important objectives to evaluate with regard to the light quality as a function of the Spectral Power Distribution; these are Correlated Color Temperature (CCT) and the Color Rendering Index (CRI) of the light source, which we evaluate in this work using the guidelines of CIE 13.3.

Color rendering of an illuminant is the effect of the illuminant on the color appearance of objects by conscious or subconscious comparison with their color appearance under a reference illuminant. Colour Rendering serves to describe the effect of an illuminant on the color appearance of objects. The most fundamental natural source is daylight, thus the primary reference for comparing the rendering of a source should be the Standard Illuminant D65.

The Correlated Color Temperature is the temperature of the Planckian radiator whose perceived color most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions (CIE 17.4, 1989). According to this definition, CCT can be calculated using one of the chromaticity diagrams. CIE still recommends to calculate CCT using the 1960 (u, v) chromaticity diagram (now deprecated). On the (u, v) diagram, find the point on the Planckian locus that is at the shortest distance from the given chromaticity point. CCT is the temperature of the Planck's radiation at that point. CCT can be calculated from CIE 1931 chromaticity coordinates x and y in many different ways. Complicated algorithms and simple equations alike have been proposed and used for several decades.

The simulated construction of the prismatic guides is based on a tube internally coated polycarbonate sheet whose outer face is composed of 90º micro-prisms; rays that enter on the guide with an angle suitable undergo total internal reflections. In this model, in order to include the round in corners and the absorption we estimate the spectral reflectance of the inner face of the prism sheet will be estimated in 0.99 throughout the spectrum analyzed. In this section two different types of light guides: circular and rectangular cross sections have been analyzed (Fig. 8).

**Figure 8.** The Coordinate system defines the angles of incidence and observation in the guides. Longitudinal section (a) and transverse section (b).

The photometric profile of the source is an angular distribution adapted to the optimal transmission features of the prismatic film used, and the analysis was restricted to the spectral range between 380 and 780 nm. The CIE Illuminant D65 is the reference light source.

The spectral reflectance of aluminum is estimated as isotropic over the fence and has been experimentally determined using a Hitachi U-3400 spectrophotometer with special accessories for measuring the specular reflectance at 12° incidence. Experimental spectral reflectance measured of aluminum is shown in figure 9.

**Figure 9.** Aluminum spectral reflectance.

162 Dielectric Material

the meridional plane.

Carlo ray tracing.

Illuminant D65.

several decades.

been analyzed (Fig. 8).

**3.1. Theoretical simulation** 

1.5, as is the case of acrylic plastic, then the input light angle must be approximately less than 27.5º from the guide´s axial direction, even though the acceptance cone can be higher in

Two different analysis have been done: Firstly, a theoretical simulation based on the specular reflection model in order to analyse the color characteristics to the output of two guides, rectangular and cylindrical of different lengths, one of them of aluminum material and the other one, internally coated of prismatic film structure this model have been developed by means of a mathematical software as Matlab. Secondly, the prismatic guides proposal evaluates with a ray tracing software through which we show the efficiency for a wavelength evaluated at the end of the guide, this model have been studied with Monte

The analysis presented is based on the specular reflection model. There are two important objectives to evaluate with regard to the light quality as a function of the Spectral Power Distribution; these are Correlated Color Temperature (CCT) and the Color Rendering Index (CRI) of the light source, which we evaluate in this work using the guidelines of CIE 13.3.

Color rendering of an illuminant is the effect of the illuminant on the color appearance of objects by conscious or subconscious comparison with their color appearance under a reference illuminant. Colour Rendering serves to describe the effect of an illuminant on the color appearance of objects. The most fundamental natural source is daylight, thus the primary reference for comparing the rendering of a source should be the Standard

The Correlated Color Temperature is the temperature of the Planckian radiator whose perceived color most closely resembles that of a given stimulus at the same brightness and under specified viewing conditions (CIE 17.4, 1989). According to this definition, CCT can be calculated using one of the chromaticity diagrams. CIE still recommends to calculate CCT using the 1960 (u, v) chromaticity diagram (now deprecated). On the (u, v) diagram, find the point on the Planckian locus that is at the shortest distance from the given chromaticity point. CCT is the temperature of the Planck's radiation at that point. CCT can be calculated from CIE 1931 chromaticity coordinates x and y in many different ways. Complicated algorithms and simple equations alike have been proposed and used for

The simulated construction of the prismatic guides is based on a tube internally coated polycarbonate sheet whose outer face is composed of 90º micro-prisms; rays that enter on the guide with an angle suitable undergo total internal reflections. In this model, in order to include the round in corners and the absorption we estimate the spectral reflectance of the inner face of the prism sheet will be estimated in 0.99 throughout the spectrum analyzed. In this section two different types of light guides: circular and rectangular cross sections have To determine the spectral distribution of radiation emerging from the rectangular guide, it is used the model developed by Whitehead (Whitehead, L.A., 1982). According to this model, the number of traversals (Δ*n*) per unit length (Δ*z*) of a ray of light undergoes inside the guide is given by the following expression:

$$\frac{\Delta n}{\Delta z} = \frac{\tan \theta}{Dr} \,\,\,\,\,\,\tag{1}$$

Natural Lighting Systems Based on Dielectric Prismatic Film 165

Prismatic guide Aluminum guide

(6)

**Figure 10.** Spectral Power Distribution of the two types of guides for different lengths.

The spectral power distribution is used for the estimation of the spectral ratio resulting. The spectral ratio is used to compare the spectral distribution at the input and output of the

400 450 500 550 600 650 700 750

d10m

Cilindrical Guide Spectrum

Wavelenght (nm)

d15m

'( ) ( ) ( ) *S S* 

Considering *S(λ)* the input spectral power distribution of the source and *S'(λ).* Spectral power distribution ratio emergence from cylindrical guide is greater from rectangular guides with the same dimensions (figure 11). In addition, the ratio decreases significantly in 10 and 15 meters from aluminum guides. The prismatic guide has high transmission across the spectrum and the spectral power is constant. There is a downward trend in short wavelengths in aluminum guide energy due to the spectral reflectance characteristics of the

We define the spectral ratio, *ηλ* as the fraction of the incident spectral power distribution

 

**3.2. Color analysis for circular and rectangular lightguides** 

d15m

d5m

d10m

d5m

guides. The spectral ratio (*ηλ*.) will be:

0

0.2

0.4

U.R.

0.6

0.8

1

material.

transmitted by the guide:

where *Dr* is the average cross-sectional distance travelled by a ray in crossing the guide air space and *θ* is the angle by which any ray deviates from the guide's axial direction.

If the guide air space is rectangular, with dimensions *a* and *b* (fig.8), then

$$Dr = \left(a^{-1} + b^{-1}\right)^{-1},\tag{2}$$

If roughly circular with radius R

$$Dc = \frac{4R}{\pi} \,\prime\,\,\tag{3}$$

In the rectangular case, the dimension *a* is the same as *b* (0.8862 meters) , the cylindrical guide aperture radius is 0.5 meters in order to maintain the same input area than rectangular guide, and the length of the guides (*L*) evaluated are 5, 10 and 15 meters.

The dependence of the radiant flux with the angle to the incident beam is given by equation 4

$$\Phi(\theta) = 2\pi I\_0 \left[ \cos \theta\_2 - \cos \theta\_1 \right] \tag{4}$$

where *I0* is the total intensity power radiated by the source in a specific direction *θ*. Moreover, we obtain the spectral distribution of radiant flux incident *Sλ*, where *Sλ* is the primary illuminant spectral distribution. *θ*1 and *θ*2 are the angular limits and the intensity has *I0* value.

The spectral and angular distribution at the exit of the guide will be:

$$S' = \int\_0^{\pi/2} P(\lambda, \theta) \wp(\lambda, \theta)^n d(\lambda, \theta), \tag{5}$$

*ρ* is the reflectance of the material for each wavelength studied and *n* is considering the number of internal reflections obtained for each incidence angle theta *θ* in the guide.

The prismatic guide has a higher transmission throughout the spectrum (Fig. 10) and the relative spectral power is constant. In the aluminum guide, the energy area of the shorter wavelengths obtained is reduced due to low spectral reflectance of the aluminum.

**Figure 10.** Spectral Power Distribution of the two types of guides for different lengths.

#### **3.2. Color analysis for circular and rectangular lightguides**

164 Dielectric Material

equation 4

has *I0* value.

guide is given by the following expression:

If roughly circular with radius R

To determine the spectral distribution of radiation emerging from the rectangular guide, it is used the model developed by Whitehead (Whitehead, L.A., 1982). According to this model, the number of traversals (Δ*n*) per unit length (Δ*z*) of a ray of light undergoes inside the

> tan , *<sup>n</sup> z Dr*

where *Dr* is the average cross-sectional distance travelled by a ray in crossing the guide air

<sup>4</sup> , *<sup>R</sup> Dc* 

In the rectangular case, the dimension *a* is the same as *b* (0.8862 meters) , the cylindrical guide aperture radius is 0.5 meters in order to maintain the same input area than

The dependence of the radiant flux with the angle to the incident beam is given by

where *I0* is the total intensity power radiated by the source in a specific direction *θ*. Moreover, we obtain the spectral distribution of radiant flux incident *Sλ*, where *Sλ* is the primary illuminant spectral distribution. *θ*1 and *θ*2 are the angular limits and the intensity

' ( , ) ( , ) ( , ), *<sup>n</sup> SP d*

*ρ* is the reflectance of the material for each wavelength studied and *n* is considering the

The prismatic guide has a higher transmission throughout the spectrum (Fig. 10) and the relative spectral power is constant. In the aluminum guide, the energy area of the shorter

number of internal reflections obtained for each incidence angle theta *θ* in the guide.

wavelengths obtained is reduced due to low spectral reflectance of the aluminum.

The spectral and angular distribution at the exit of the guide will be:

/2

0

rectangular guide, and the length of the guides (*L*) evaluated are 5, 10 and 15 meters.

space and *θ* is the angle by which any ray deviates from the guide's axial direction.

If the guide air space is rectangular, with dimensions *a* and *b* (fig.8), then

(1)

1 11 *Dr a b* ( ), (2)

02 1 ( ) 2 cos cos , *I* (4)

(5)

(3)

The spectral power distribution is used for the estimation of the spectral ratio resulting. The spectral ratio is used to compare the spectral distribution at the input and output of the guides. The spectral ratio (*ηλ*.) will be:

$$\eta(\mathcal{A}) = \frac{S'(\mathcal{A})}{S(\mathcal{A})} \tag{6}$$

Considering *S(λ)* the input spectral power distribution of the source and *S'(λ).* Spectral power distribution ratio emergence from cylindrical guide is greater from rectangular guides with the same dimensions (figure 11). In addition, the ratio decreases significantly in 10 and 15 meters from aluminum guides. The prismatic guide has high transmission across the spectrum and the spectral power is constant. There is a downward trend in short wavelengths in aluminum guide energy due to the spectral reflectance characteristics of the material.

We define the spectral ratio, *ηλ* as the fraction of the incident spectral power distribution transmitted by the guide:

Natural Lighting Systems Based on Dielectric Prismatic Film 167

**Figure 12.** CIE 1931 chromatic coordinates of the studied lightguides.

body which has the same temperature.

lengths than the prismatic guides.

The CCT is obtained from the spectral distribution at the exit of the guide (Table 1). This calculation is carried out for different lengths of the two types of guides considered in this work. To determine the reference light source for a given test source, we must find the CCT of the test source. Once this data is known, the reference light source is a Plankian black

Moreover, the Color Rendering Index (CRI) of the sources at the end of the guides it is calculated (fig.13). From the result obtained (Fig.10), it is possible to determine the significant color change in the aluminum fence, showing less color change difference for all

Iluminant D65 6503 99.97 6503 99.97 Prismatic guide (5m) 6503 99.97 6503 99.97 Prismatic guide (10m) 6503 99.97 6503 99.97 Prismatic guide (15m) 6503 99.97 6503 99.97 Aluminum guide (5m) 4707 89.81 4407 87.74 Aluminum guide (10m) 4126 85.45 3818 82.39 Aluminum guide (15m) 3781 81.98 3478 78.15 **Table 1.** Correlated Color Temperature (CCT) and Color Rendering Index of each spectral power.

Cylindrical Guide Rectangular Guide CCT (ºK) CRI CCT (ºK) CRI

**Figure 11.** The ratio of spectral power distribution of the two types of guides for different lengths.

For comparing different behavior of light guide regarding color output flux it is used CIE 1931 diagram due to it is more spread on known color representation (CIE 13:3, 1995).

Color coordinates in CIE 1931 chromaticity diagram for studied lengths of aluminum and prismatic guides are show in figure 12. The chromatic coordinates with the light source (D65) chosen for different lengths of guide will be compared. The three consecutive black square points and blue circle points represent the aluminum guide in 5, 10 and 15 meters.

The three consecutive points (d5m, d10m and d15m) represent the aluminum guide. The results of the CIE 1931 chromaticity coordinates of the full spectrum transmitted in the prismatic guides differ from those transmitted in the aluminum guides approaching to the yellow zone. When the length of the aluminum guides increases, there are no changes in the results obtained for the prismatic guide (P), and the result are superposed with the illuminant D65 because the reflectance is maintained for the entire spectrum.

**Figure 12.** CIE 1931 chromatic coordinates of the studied lightguides.

166 Dielectric Material

**Figure 11.** The ratio of spectral power distribution of the two types of guides for different lengths.

For comparing different behavior of light guide regarding color output flux it is used CIE 1931 diagram due to it is more spread on known color representation (CIE 13:3, 1995).

Color coordinates in CIE 1931 chromaticity diagram for studied lengths of aluminum and prismatic guides are show in figure 12. The chromatic coordinates with the light source (D65) chosen for different lengths of guide will be compared. The three consecutive black square points and blue circle points represent the aluminum guide in 5, 10 and 15 meters.

The three consecutive points (d5m, d10m and d15m) represent the aluminum guide. The results of the CIE 1931 chromaticity coordinates of the full spectrum transmitted in the prismatic guides differ from those transmitted in the aluminum guides approaching to the yellow zone. When the length of the aluminum guides increases, there are no changes in the results obtained for the prismatic guide (P), and the result are superposed with the

illuminant D65 because the reflectance is maintained for the entire spectrum.

The CCT is obtained from the spectral distribution at the exit of the guide (Table 1). This calculation is carried out for different lengths of the two types of guides considered in this work. To determine the reference light source for a given test source, we must find the CCT of the test source. Once this data is known, the reference light source is a Plankian black body which has the same temperature.

Moreover, the Color Rendering Index (CRI) of the sources at the end of the guides it is calculated (fig.13). From the result obtained (Fig.10), it is possible to determine the significant color change in the aluminum fence, showing less color change difference for all lengths than the prismatic guides.


**Table 1.** Correlated Color Temperature (CCT) and Color Rendering Index of each spectral power.

Natural Lighting Systems Based on Dielectric Prismatic Film 169

5 m 10 m 15 m

**Figure 14.** Perspective view of Ray trace simulation model of lightguides with a detail of the prismatic

The comparison of efficiency in rectangular and cylindrical prismatic guides is shown in Figure 15; the graphics indicates the influence of the shape in maintaining the transported flux and the efficiency for a wavelength (546 nm). The output flux efficiency of 15 meters cylindrical guide is 2% higher than rectangular guide; in addition, in the first meters the flux

Efficiency (546 nm)

**Figure 15.** Output efficiency results for wavelength of 546 nm in Prismatic Guides.

Compound parabolic concentrator (CPC) is an optical devices used in the solar energy related areas and also in other applications where radiant energy concentration is needed, being defined as one of the first devices that resulted from the practical application of nonimaging optics (Welford and Winston, 1978). Light from a defined range of angles of incidence is reflected by total internal reflection on the parabolic walls of the CPC and

Cylindrical P.G. Rectangular P.G.

structure: cylindrical (a) and rectangular (b).

remains more constant.

**4. Hollow prismatic CPC** 

0.95

0.96

0.97

( U.R.)

0.98

0.99

1

concentrated at the exit of the CPC.

**Figure 13.** CRI of measured lightguides (cylindrical and rectangular).
