1. Introduction

Concentrating solar light is important for a variety of renewable energy applications. In devising a concentrated photovoltaic (CPV) system [1], for example, one requires a bundle of concentrated light to impinge perpendicularly and uniformly on a planar surface where a multilayered semiconductor solar cell lies. These conditions (concentration, preferred normal direction, uniformity of the light bundle) can be met by optical systems of many kinds. As a first example, we might mention a Cassegrain optical system [2] made of two confocal parabolic mirrors facing each other as in fig. 1. In this system, light coming from infinity is concentrated in a region close to the common focal point F.

Figure 1. Solar light is concentrated by a Cassegrain optical system consisting of two confocal parabolic mirrors facing each other. The primary mirror gathers light rays coming from infinity, projecting them in the vicinity of the common focus F. The secondary mirror sends concentrated light uniformly on the CPV multilayer solar cell. The direction of the light rays is orthogonal to the plane in which the CPV system lies

In this case, however, it is necessary great precision and care in the fabrication of the optical system. One can therefore ask whether a single lens can accomplish the same task of this mirror system. From geometrical optics, under the hypothesis of paraxial rays, we can write, for a spherical thick lens [3]:

(1)

where n₁ and n₂ are the indices of refraction of the medium and of the lens, respectively, p and q are, in the order, the positions of the object and its image, and R is the radius of curvature of the spherical lens. For , we obtain the position of the first focus F in q=f, so that:

(2)

Thus, if we let the second surface to have a focal point coincident with F, we obtain, in a second refraction, light rays emerging parallel to the incoming direction, as shown in fig. 2. In the present work, after having carefully considered spherical and chromatic aberrations, we give details on the optical characteristics of this system, specifying all dimensions of the spherical thick lens able to concentrate light according to a given concentration factor.

Figure 2. Solar light is concentrated by a spherical thick lens. Both sides of the lens are spherically shaped, with different radius of curvature: the upper and lower surfaces have radii of curvature R and r, respectively. The two sides have a common focus in O

2. The Optical System

In this section we give details on the expected optical behavior of the system schematically described in fig. 2. By the thick lens relation corresponding to the bottom side of the lens, we may write a relation similar to Eq. (2) for the quantity f’ shown in fig. 2:

(3)

In this case we have exchanged the indices of refraction in (1) and have required , obtaining the position of the second focus in In order to fix our ideas, let us take n₁=1 and n₂=n, so that:

(4a)

(4b)

We now choose our system to be a solid of revolution obtained by rotating the planar geometry represented in fig. 2 about the optical axis containing the segment OV. By considering the triangles OAB and OA’B’ in fig. 2, we may write the radii and being one half of the upper and lower diameters AB and A’ B’, respectively, as follows:

(5a)

(5b)

where is the angle between the segments CB and CV. From Eqs. (4a-b) and (5a-b) we can derive the following simple proportions:

(6)

The geometrical concentration ratio C_geo can thus be thus defined as the ratio between the projection of the upper and lower spherical areas as follows:

(7)

Therefore, it could appear that we might obtain an arbitrarily high concentration ratio C_geo. In what follows we shall consider this factor to be C_geo400. Nonetheless, problems could arise from spherical and chromatic aberrations. In this way, before considering the total volume and the total mass of the lens, in the next section we shall briefly consider the effects of these types of aberration.

3. Aberrations

In the present section we shall give a brief account of the spherical and chromatic aberrations in the system, noticing that they do not affect the predicted optical behavior described in the previous section.

3.1. Spherical Aberrations

By starting with spherical aberrations, we look for the intersection of a light ray coming from infinity and the optical axis of a spherical thick lens. This intersection depends on the abscissa on which the incident ray impinges on the spherical surface. In this way, rays from infinity are not focused in a single point, so that the phenomenon of spherical aberration arises. In order to compute the maximum deviation from the value of f calculated by means of the paraxial axis hypothesis in (2), we construct the path of the light ray as in fig. 3. The light ray coming from infinity is thus seen to impinge on the spherical interface in P₀=(x₀,y₀). The angles and are comprised between the normal line l_n and the incident and refracted rays, respectively. These two angles are related by Snell’s law, so that:

(8)

Figure 3. Schematic representation of the path of a light ray coming from infinity and refracted inside a thick spherical lens of radius R. The angle of incidence in P0 is , the angle the refracted ray makes with the normal is

As it can be noticed from fig. 3, the refracted line is at an angle from the x-axis, so that the equation of this line is:

(9)

We define y_I = y(0), so that, by setting and and by considering (8), we can write y_I entirely in terms of as follows:

(10)

By a series expansion of the cosine and sine functions to third order in we obtain:

(11)

Therefore, we can estimate the error we make, by assuming paraxial rays, in computing f for an incident ray forming an angle with the normal to P₀, the refracted line is at an angle

(12)

By doing the same with the spherical interface of radius r, by elementary geometry we first notice that the new incident and refracted angles and are related to and by the following simple relations:

(13)

In this way, we find, by the same type of analysis:

(14)

so that:

(15)

For a choice of the maximum angle and of the radius R=24.0 cm, we can compute the maximum deviation from f to be about =3.42 cm for n=1.60 and about =4.51 cm for n=1.49. The corresponding variations of f’ will be Therefore, the excursion of the focuses in this system is rather large.

The question now is, will the calculated spherical aberrations affect the direction of the rays coming out of the lens from the second interface and the uniformity of the bundle, thus hindering the correct functioning of the optical system proposed in the present work? In order to see this, let us assume that, for some reasons, we allow variations of both f and f’ in the system. Because of (4), being only the focal lengths to vary, by taking the derivative of both sides and by approximating the infinitesimal variations of f and f’ to as above, we have:

(16)

In this way, the system will behave in the same way as predicted in section 2, if the ratio of the two variations is such that:

(17)

which, because of (6), is exactly what found in Eq. (15).

3.2. Chromatic Aberrations

We next consider chromatic aberrations due to dispersion of light inside a dielectric. According to Cauchy’s equation, the index of refraction depends on the wavelength of the electromagnetic radiation in the visible range roughly as follows:

(18)

where A and B are opportune coefficients, which depend on the particular material used. However, we are not particularly interested in the exact way n changes with for reasons which will appear clear later. We only need to know that, if n varies in some way, Eqs. (4a) and (4b) predict variations and which may be written as follows:

(19a)

(19b)

Once again we obtain that the ratio between the variations and is equal to the ratio of the radii R and r, thus verifying (17) as in the previous case of spherical aberrations. Therefore, even though chromatic aberrations modify the focal lengths for rays of different wavelength, these variations do not affect the way the system concentrates light on the CPV cell for the same reasons explained at the end of the previous subsection.

4. Dimensions and Mass

Under the hypothesis that C_geo in the present section we specify the dimensions of the optical system in fig. 2. In this case, the ratios in Eq. (6) are In order to intercept a not negligible solar radiation power and to illuminate a CPV cell of about 1cm², we choose cm so that =0.60 cm. We can thus estimate the radiating power harvested by this optical system by considering the incoming power to be , where E_S is the irradiance of the direct solar radiation, whose peak value is about 0.9 kW/m². By making the assumptions that nearly normal incidence occurs at the upper and lower glass interfaces, so that the transmittances [4] can be set equal to T(n)=4n/(n+1)² and , respectively, and by considering the power dissipated in the glass as negligible, we can write the transmitted power as:

(20)

By now assuming a 35% efficiency of the CPV cell in converting radiating energy in electrical energy, we end up with an estimated value of the electrical power generated by a single element (optical system + CPV cell) of 0. 32 P_t.

We still need to completely specify other geometrical quantities in the optical device. In order not to incur into too much discrepancy with the assumption of quasi-normal incidence, we might take the angle In this way, from (5a) and (5b) we obtain R=24.0 cm and r=1.2 cm, respectively. We shall keep these geometrical quantities as fixed and determine, on the basis of different choices of the index of refraction of the lens, the focal lengths, f and f’.

Next, consider the density of the material used for the thick lens to be denoted by d_L, so that the mass M will be given by M = d_L V, where V is the total volume of the lens itself. In order to calculate V, let us first calculate the volume V_TC of the truncated cone by means of the difference between the cone with base radius and height and the cone with radius and height so that and:

(21)

Let us now define the heights of the upper and lower spherical domes in the optical system, respectively, as follows: so that h/h’=f/f’. The missing volume to V_TC is thus given by the difference between the volume of the upper and lower spherical domes defined as follows

(22)

In this way, the total volume of the optical system can be written as follows:

(23)

Consider first a glass with index of refraction n=1.6. From (4a) and (4b) we find f =64.0 cm and f’ =3.2 cm, respectively. By now taking specifying the density of the glass to be d_L = 2.40 g/cm³, we calculate the volume of the system studied and, successively, its mass. By evaluating V from (23), we have 11590 cm³, so that M=27.8 Kg. Finally, from (20) we obtain at most a radiating power P_t=36.5 Watt, and an electric power of about 12.8 Watt.

One could also wish to look for lighter materials with roughly the same index of refraction, in order to construct an optical device for practical applications. For example, we might choose to use transparent thermoplastic (plexiglas), with an index of refraction n=1.49 and a density of about 1.18 g/cm³. By going through the same calculations, for cm and we obtain f =73.0 cm and f’ =3.65 cm. The volume of this system can thus be calculated to be roughly 12940 cm³, so that the final mass is reduced to about 15.3 Kg. Therefore, even though the space occupied by plexiglas is higher than glass with n=1.6 and with a higher density, we have a conspicuous reduction in the weight of the system. Finally, we can see that the maximum transmitted power in this case is P_t=37.6 Watt, for a peak electric power generated by the cell of about 13.2 Watt.

5. Conclusions

We have studied the optical behavior of a spherical thick lens for light concentration on a concentrated photovoltaic (CPV) cell. The proposed optical system looks like a transparent glass cone to which we take off the inferior part, whose interface is spherically shaped, and to which we add a superior spherical dome. We have noticed that the system can gather incoming light from infinity concentrating it uniformly and with normal incidence on a plane perpendicular to the incoming rays. Spherical and chromatic aberrations are seen not to affect the optical behavior predicted through the simplifying approximation that rays are monochromatic and paraxial. This happens because the system is constructed in such a way that the variation of the focal length f for the first convex portion of the optical device is compensated by a proportional variation of the focal length f’ of the second concave interface. In this respect, the system can be used as a concentrator of solar light for a concentrated photovoltaic (CPV) cell.

In the present work the transmitted radiating power to the CPV cell is calculated under the assumption of normal incidence of light on the first and second interfaces and of negligible power dissipation inside the glass. Under these hypothesis, a device made of transparent thermoplastic (plexiglas), with mass of about 15.3 Kg, a radial aperture cm, and total length of about 70.0 cm, is seen to concentrate solar light with a concentration factor C_geo400 into a circular CPV cell, of approximately 1.1 cm² of area and with 35% efficiency, giving out an electric power output of about 13.2 Watt. These studies, essentially based on geometrical optics, and the specific quantities described in the present work may trigger experimental and analytical research on the interesting topic of optical devices for CPV cells illumination.