A novel of Fresnel-type lens for use as a solar collector has been designed which utilizes double total internal reflection (D-TIR) to optimize collection efficiency for high numerical aperture lenses (in the region of 0.3 to 0.6 NA). Results show that, depending on the numerical aperture and the size of the receiver, a collection efficiency theoretical improvement on the order of 20% can be expected with this new design compared with that of a conventional Fresnel lens.
©2012 Optical Society of America
Augustine Fresnel invented the Fresnel lens in France in 1822 as a lightweight equivalent of a conventional lens  which was originally used for controlling the illumination from lighthouses. In recent years Fresnel lenses have been used to concentrate solar energy [2–7].
In such applications the ability of the lens to concentrate the maximum amount of solar power on to the receiver is of paramount importance. There are three principal sources of efficiency loss:
- • “Geometric” losses – that is, losses caused by rays not reaching the desired region at the focus. This may be caused by design factors such as chromatic dispersion or manufacturing factors such as surface roughness or surface form errors.
- • Absorption losses as rays pass through the material thickness. This depends on the choice of material and the optical path length in the lens.
- • Reflection losses at the refracting surfaces. This depends on the choice of material (refractive index) and the design of the lens.
This paper describes a lens design with reduced reflection and geometric losses.
2. Reflection losses
When light reaches the boundary of two materials with different refractive index some of the light is reflected according to the well known Fresnel equations . These reflection losses are dependent on the angle of incidence, θi, and for unpolarised light, are shown graphically in Fig. 1 for light entering and exiting a higher refractive index material. These examples refer to a material of refractive index 1.49 (polymethylmethacrylate, PMMA) in air (nair = 1).
Consider, now, a typical planar Fresnel lens which might be used as a solar collector, such as that shown in Fig. 2 . A ray is shown near the edge of the lens which is refracted and deviated by an angle α. Reflection losses are shown by rays A and B. (Ray A is shown reflecting at a small angle for clarity). In this example the incident light is normal to the first surface and, from Fig. 1, the reflection loss represented by Ray A is approximately 4% of the incident power. The loss represented by Ray B is dependent on the angle of incidence, θi. Figure 1(b) shows that for angles of incidence much greater than about 30° reflection losses increase rapidly.
In general, it is desirable to collect light from as large a collection area as possible. This means the lens numerical aperture (NA) or maximum collection angle, αmax, should be as large as possible. Here, the term “collection angle” or, more specifically, collection half angle refers to the angle between a ray and the optical axis in image space close to the lens focus not the solar subtense angle in object space. For large collection angles reflection losses represented by Ray B increase according to the Fresnel equations . This is shown graphically in Fig. 3 (solid black line) which shows that rays deviated more than about 25° exhibit a rapid fall in transmission efficiency.
3. Geometric losses
It is well known that chromatic dispersion also increases as the refraction angle increases  and this can contribute to what are referred to here as “geometric” losses. This is caused by variation in refractive index, n, with wavelength, λ, and can be calculated by the application of Snell’s law . This is represented schematically in Fig. 4(a) where rays passing through the edge of the lens suffer more chromatic dispersion than those from near the centre of the lens. Depending on the size of the receiver, it is likely that some rays from the edge of the lens will not reach the receiver placed at the lens focus.
Secondly, geometric losses can be caused by angle of incidence variation. Solar illumination subtends 0.5° and hence a solar collector lens operates over an angular input range of +/−0.25°. Again, the application of Snell’s Law shows that losses due to this effect increase with refraction angle and is shown schematically in Fig. 4(b). It can similarly be shown that losses due to surface figure errors and inclination errors of the whole lens also increase with refraction angle.
In summary, a Fresnel lens used for solar power collection should be designed with a large NA but as α (and therefore θi and θt) increase there is a diminishing return of collected power due to an increase in both reflection losses and geometric losses.
4. The use of total internal reflection
Figure 1(b) shows that (for PMMA) for angles of incidence, θi, greater than about 42° all the light is reflected and none is transmitted at the second surface. This phenomenon of total internal reflection (TIR) can be used to contribute to the ray deviation, α, in an efficient way. There are many examples of Fresnel lenses which use a single TIR surface to control light efficiently [10–16]. A design of a Fresnel lens which uses two reflecting surfaces on each Fresnel ring can be shown to increase the collection efficiency further.
Figure 5 shows three ways to deviate a ray by an angle, α. The first is by a conventional Fresnel lens which only uses refraction, the second uses single total internal reflection (at point X) and the third uses double total internal reflection (D-TIR) with reflections at points Y and Z. Comparing the three designs it is evident that the design with the lowest final refraction angle, θt, will be the most efficient because it will have the lowest Fresnel reflection losses and also lowest geometric losses.
By solving Snell’s law for the three geometries and calculating the reflection losses from the Fresnel equations  for the resulting refraction angles, the three geometries can be compared quantitatively. Figure 3 presents this comparison and shows that at small collection angles a conventional Fresnel lens is the most efficient design. For rays on the optical axis (α = 0°) each surface reflects about 4% of the power resulting in a transmission (excluding other losses such as absorption) of 92%. For rays collected at angles greater than about 19° a double TIR design is the most efficient and it remains so until collection angles exceed 38° when a single TIR design becomes more efficient. Above this angle the double TIR design experiences a sharp fall in efficiency and this occurs when rays incident at point Z (Fig. 5) are not total internally reflected; some of the light is transmitted.
Using this information an optimum, hybrid lens can be designed (Fig. 6 ) which uses a minimum refraction angle, θt, at each radial position and, therefore, has an optimum transmission.
5. Example lens design
To illustrate the performance of such a hybrid “Fresnel/D-TIR” lens consider two 0.5NA, PMMA lenses of 100mm diameter (αmax = 30°), one being a conventional Fresnel and the other a hybrid. The thickness of each lens is arbitrarily chosen to be 3mm and the groove height is 1mm (the groove width is variable). For the hybrid lens the transition from Fresnel to hybrid D-TIR occurs at a radius of 30.6mm, corresponding to a collection angle of, α, of 19.5°. The conventional Fresnel lens and its corresponding region on the hybrid lens are designed to focus 550nm (solar peak) perfectly using aspheric groove sections. The D-TIR region is designed with flat (in cross-section) grooves.
The lenses were analysed by Monte Carlo ray tracing using Zemax optical design software . In each case the applied illumination was from the solar spectrum  from 300nm to 2.4µm wavelength and with a solar angular subtense of 0.5°, and the analyses accounted for the absorption and dispersion properties of PMMA. Figure 7 shows the irradiance profiles at a 4mm square region at the focus of each lens. The conventional Fresnel lens produces a lower peak irradiance and a broader profile than the hybrid lens. Figure 8 presents an encircled energy plot of these results showing that the hybrid lens reaches 99% of its maximum efficiency of 86.5% within a receiver diameter of 3mm. At the same diameter a conventional lens has an efficiency of 71.7%, approximately 20% lower than the hybrid design.
It should be stressed that manufacturing constraints have not been included in the analysis. It is expected that such fine D-TIR prisms profiles will be a challenge to manufacture by conventional methods such as moulding. Also, the reflecting surfaces will require higher surface flatness specifications than refracting surfaces for the same ray aberrations. In comparison with single TIR Fresnel lenses, however, it should also be noted that a D-TIR lens has a reduced sensitivity to tracking errors due to the application of two reflecting surfaces.
By appropriate use of the phenomenon of total internal reflection a hybrid Fresnel-type lens has been designed which is efficient even at high numerical aperture. Compared with a similar conventional Fresnel lens a collection efficiency improvement on the order of 20% can be achieved (depending on the size of the receiver).
This work has been funded by the Generalitat Valenciana (IMPIVA) [IMIDTA/2009/1024 and IMIDTA/2010/1140].
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