Uniform Illumination of Concentrated Sunlight in Photovoltaic Solar Modules With Plane-Focusing Fresnel Lenses

An algorithm has been developed for modeling the parameters of microprismatic specialized plane-focusing Fresnel lenses. Such lenses are more effective for application in photovoltaic modules compare to the traditional point-focusing Fresnel lenses for concentrating the sunlight. The method for manufacturing above lenses by diamond cutting technique and subsequent thermal pressing of silicone blanks is proposed. The optical characteristics of new plane-focusing lenses for photovoltaic solar modules, which are made using our simulation results, have been experimentally investigated with collimated laser beam.

The simplest way to reduce the inhomogeneity is to diminish the focal distance [7]; it is well-known that the longer the focal length, the focal spot is larger, and the more uniformly the solar cell is illuminated. However, with an increase in the focal spot diameter, the solar cell area should be enlarged accordingly. Manuscript [2]; (b) transforming plane-focusing Fresnel lens [20].
The other popular way is the kaleidoscope scheme or Keller integrator [8]. These methods are very effective, but they are too complicated and therefore are less suitable for mass application in solar modules. Many researches [9], [10], [11], [12], [13], [14] have proposed the designs of special 3D lenses, even for square irradiance of photocell surface. However, such methods also are very prominent theoretically, but rather complicated for their practical implementation -the special very complicated machines are necessary [15], [16], [17], [18] to manufacture such polygonal lenses.
However, it is possible to solve completely the problem of non-homogenous irradiation and to diminish the connected high internal ohmic losses in HCPV modules by applying the new transforming plant-focusing Fresnel lens [19], [20], which creates a uniformly illuminated light circle in the lens focal plane ( Fig. 1(b)).
The plane-focusing optics [20] also reduce the focal length f of concentrator lens; minimal f-values for transforming lenses [20] are restricted only by the limit angles α kmax [21] of refractive microprisms of the lens. Accordingly, the diminished f-value reduces the dimensions of concentrator single cell and HCPV modules, it is especially important for space solar photocells applications.
So, the aim of our paper is to deliver an algorithm for creating plane-focusing Fresnel lenses for solar concentration modules, which is adapted for lens mass manufacturing by thermo-pressing method [22]. Other aim is to investigate the main properties of fabricated lenses for photovoltaic modules.

II. ALGORITHM FOR MODELING FRESNEL TRANSFORMING OPTICS
Earlier in our papers [19], [20] an algorithm was proposed for modeling the parameters of transforming microprismatic plane optics. Such Fresnel lenses form in the focal plane a light circle of required radius r V with an almost homogeneous irradiation distribution. The scheme of light beam refraction for this lens is shown in Fig. 2, where f is the lens focal distance; n 0 and n 1 are the refractive indices of the medium and the material of microprism, accordingly; R k is the radius of annular prismatic zones k = 1, 2, 3, …, N of the lens; ΔR k is the width of the refractive k-zone of the lens in the direction of axis X 0 ; r V = R 1 is the outer radius of the light spot in the focal plane; γ k is the angle of observation for k-zone from the focus F; α k is the prism refractive angle and h k is the relief depth for k-zone.
We realized our calculation results for lens manufacturing by the diamond cutting method [22], [23]. Such technique ensures the formation of working facets of exceptional geometrical accuracy and mirror-like surface quality. Recently, we have successfully applied this method for manufacturing the microprism structures for computer eye-glasses [24].
Our algorithm [20] for obtaining the homogeneous focal spot by focusing structure suitable for diamond-cutting method involves, firstly, the setting of central zone of the lens, which in the process of lens manufacturing remains flat. The size R 1 of this zone is determined primarily by technological requirements: during diamond-cutting the speed of rotation of the cutting tool at the point R 1 = 0 is zero; hence the value of R 1 cannot be zero or very small. Then, the algorithm supposes the formation a not-illuminated "dark" areas of variable radius r j in the center of focal spot. It is necessary for compensation of a light beam concentration in the narrow central area of the round focal spot. This fact is investigated in detail in [19] and is explained by the transformation of larger annular light fluxes while focusing into the light rings with the smaller diameters. So, in our algorithm the refractive angles α k of the lens are calculated taking into account these "dark" areas of radii r j with irradiated light rings of nominal widths W j = r V −r j in the focal plane. After setting the radius R 1 of lens central zone the angles γ k of ray inclination by all prismatic zone k are determined at lens radii R = R k. These values γ k determine the refraction angles α k of each circular lens zone k and are calculated by Snell's law [21]: At the next modeling stage [20] the process of light beam narrowing should be taken into account by applying the appropriate correction of the pitch ΔR k and the depth h k of microrelief for a lens with a necessary light diameter D L . We will call this process as zone correction (ZC). For any angles α k and γ k , the necessary correction of the width ΔR k and the depth Δh k of the microrelief for each j-group can be defined as: As the angles α k increase, each k-zone forms a narrower ring in the focal plane with outer radius r k instead r V and with width Δr jk = r k −r j . This process is obviously illustrated in Fig. 2. To expand the refracted annular light beam from radius r k to radius r V , it is necessary to increase proportionally the radii of lens refractive zones R k = W j + ΔR k and, accordingly, the depths of the microrelief h k = W j tgα k + Δh k , where a nominal relief pitch W j = r V −r j is different for different j-groups of prismatic zones (j = 1, 2, …, M).
Then, the optimization of focusing process (FO) is necessary to achieve the uniformity of focal light distribution [20]. This process means that the corresponding prismatic k-zones of the lens with radii R k and widths ΔR k should direct the refracted light beams into certain focal annular light areas with nominal widths W j = r V −r j , where j = 0, 1, 2 … M. The width of annular illuminated areas W j we will consider as the nominal relief pitch of microrelief that is necessary for further calculating the parameters of refractive prismatic zones. The number of diameters W j is equal to M-value. Usually the value of M = 3-4 is enough for obtaining a spot homogeneity.
The next problem is connected with the cutting instrument limitation. Used diamond cutters have a cutting edge that usually is no more 1.2-1.5 mm. Therefore the length of inclined working surfaces of microprisms ΔR k /cosα k or the widths of prismatic zones ΔR k = (R k+1 −R k ) cannot be larger than ∼1.0 mm. So, when uniformly illuminated light spot has rather large radius r V > 5-10 mm, in order to avoid this limitation, it is proposed [19] to form zones k = 1, 2, 3, …, N from several annular constituent microprismatic elements N c with the same refractive angles α k and widths ΔR kc . The total width ΔR kc of prismatic elements for each of this composed k-zones in the sum should be equal to the width ΔR k . The value of ΔR k = ΣΔR kc for each zone k is defined by an appropriate variation of the relief depth h k and the number of constituent elements N c .
At the final stage of simulation, taking into account the stated light focusing optimization scheme (FO) and width zone correction (ZC) the geometrical parameters of necessary transforming Fresnel lens can be calculated. The process of light beam concentration and all stages of our algorithm are considered in detail in our previous papers [19], [20].
Proposed algorithm of light irradiation homogenization [19], [20] allows manufacturing the single lens from any optical plastic; usually we fabricate lenses from polycarbonate. During this process a diamond cutter movement is performed along the axis X 0 (Fig. 2). This movement is responsible for formatting the prism working surfaces of widths ΔR k . This axis X 0 is strictly perpendicular to the plane of rotation of relief forming plate or to the direction Z 0 , responsible for forming the relief depths h k . So, usually we manufacture Fresnel transforming lenses [22] with the angle θ = 0 between the reverse facet of all formed microprisms and the plane orthogonal to the relief forming plate. During manufacturing process the additional angle β θ k = 90 − α k is very important, which sets the necessary angle of inclination ξ k for the cartridge with diamond cutter of angle α G to the axis X 0 : angle ξ k = β θ k − α G . For creating HCPV modules the hundreds of single concentration lenses are necessary. So, our algorithm [19], [20] should be modified to create the lens design, which is suitable for lens mass replication. Our previous experience [22], [25] stated that microprisms in lens structure should be formed with the reverse angle θ ≈ 3.0-4.0 deg. (Fig. 2) to simplify the separation of a primary prism-matrix with a final plastic prism-sample under thermo-pressing process [22]. So, we have to set all microrelief radii ΔR k and depths h k and angles β k in a new coordinate system (X θ , Z θ ) that should be turned to this angle θ relative to system (X 0 , Z 0 ). These new values of ΔR θ k , h θ k and β θ k can be obtained as following: We will start the movement of the cutting tool at the point "0" (see Fig. 2) and our calculations should be connected with this initial point. Because the back angle θ for relief forming plate inclination relative to axis X 0 in coordinate system (X 0 , Z 0 ) is the non-zero one, the distances accumulation takes place along the axis (X θ , Z θ ) and it is necessary to make appropriate correction in values ΔR θ k and h θ k for moving diamond cutter in new coordinate system (X θ , Z θ ): The data obtained by formulas (4) for back angle θ = 3.0 deg. are shown in Fig. 3(a) for lens # 27 with f = 25 mm and D L = 50 mm created from polycarbonate for green spectrum zone with n 1 = 1.585. Fig. 3(b) illustrates the complementary angle β θ k versus radius R θ k in new coordinates. Data obtained for X θ k , Z θ k and β θ k allow organize a movement of cartridge with diamond cutter tool along the axis (X θ , Z θ ).
Similar data on X θ k , Z θ k and β θ k can be easily obtained by proposed algorithm for others lens diameters D L and focal lengths f, as well as for another plastic material -silicon or polymetylmetacrylate, from which the lenses usually are made [2], [9] for HCPV solar concentration modules.

III. MASS MANUFACTURING OF PLANE-FOCUSING FRESNEL LENSES
The algorithm allows modeling a lens that forms a uniformly illuminated area in focal plane at any focal distance f, even with the minimum possible focal length. Thus, the lens with optical diameter D L = 56 mm, that corresponds to traditional rectangular lens of size ∼40 × 40 mm [2], [9], can be easily made by our algorithm with diminished focal lengths f -from the usual ones of 75-105 mm to 20-25 mm. However, for a smaller focal length, the prism refractive angles approach to their limiting value α kmax , at which the total internal reflection [21] occurs for transmitted light beams. Therefore, the light transmittance τ R 0 also decreases with decreasing f values. However, the thickness of concentrator module, which can be made from the array of such lenses, also decreases. This will allow minimize the size of concentrator module that is decisive often while such modules construction.
For illustration we have performed the simulation of the lens for solar concentrators for quite small focal distance f = 25 mm with lens light diameter D L = 50 mm and back angle θ = 0 deg., which forms in the focal plane the round uniformly illuminated area with radius of r V = 1.5 mm [25]. Such lens # 25c can ensure the sunlight concentration multiplicity k C ∼ 280. This lens forms in the focal plane the "dark" area of r 0 = 0.5 mm, which should be irradiated by diffuse light beams scattered at the microrelief.
We cannot simulate exactly the process of real diffuse scattering, therefore to optimize the lens design we have simulated and manufactured for future experimental investigation the similar lenses with the same focal distance f = 25 mm, light diameter D L = 50 mm and outer radius r V = 1.5 mm, but with different radii of focal "dark" area r 0 = 0.7 and 0.9 mm. The proposed optimization schemes FO of light beams focusing for all simulated lenses #25c [25], #27 and #28 are illustrated in Fig. 4, where J CR (R) is the intensity of concentrated light.
Light intensities J CR (R) distribution for radii r for focal light spot depending on lens radii R were calculated according to our algorithm proposed in [19]. The simulation has showed that for lens # 27 with r 0 = 0.7 mm the correction procedure realizes 22 prismatic zones corresponding to 3 values W j = 0.8; 0.5 and 0.3 mm, each of W j contributed by 18, 2 and 2 prismatic k-zones of the lens, accordingly. For lens #28 with r 0 = 0.9 mm the correction procedure realizes 29 prismatic zones corresponding to 3 values W j = 0.6; 0.4 and 0.2 mm, each of W j contributed by 25, 2 and 2 prismatic k-zones, accordingly.
Our previous investigation [25] testifies that in practice for above "three-peak" scheme FO the lens, manufactured by diamond cutting, due to diffuse scattering demonstrates the final light distribution that is practically flat. The proposed optimization scheme FO of light beams focusing for lens # 27 is illustrated in Fig. 4(b): created prismatic zones #1-18 reflect light beams to the focal light ring of width W 1 = r V −r 0 = 0.8 mm, zones #19-20 -to the ring of W 2 = r V −r 1 = 0.5 mm, and zones #21-22 reflect light beams to the ring of W 3 = r V −r 2 = 0.3 mm. We will call this scheme of focusing optimization as FO: 0.7 (18)-1.0 (2)-1.2 (2). Under above conditions the focal light distribution has "dark" area of radius r 0 = 0.7 mm and three light peaks (M = 3) for r j = 0.7; 1.0; 1.2 mm with outer radii r V = 1.5 mm.
Similarly, for lens # 28 (Fig. 4(c)) created prismatic zones # 1-25 reflect light beams to the light ring of width W 1 = r V −r 0 = 0.6 mm in the focal plane, zones #26-27 -to the ring of W 2 = r V −r 1 = 0.4 mm, and zones #28-29 reflect light beams to the ring of width W 3 = r V −r 2 = 0.2 mm. We will call this scheme of focusing optimization as FO = 0.9 (25)-1.1 (2)-1.3 (2). Under above conditions the focal light distribution has "dark" area of radius r 0 = 0.9 mm and three light peaks (M = 3) for r j = 0.9; 1.1; 1.3 mm with outer radii r V = 1.5 mm.
Using obtained data on FO and ZC the modeling of lens relief depth h k depending on radius R k can be performed. Calculated structure of the relief for above lens-concentrator # 27 for R k < 25.16 mm and # 28 for R k < 25.05 mm are shown in Fig. 5.
The refractive indices n 1 (λ) were used from the data [26]. The simulation of parameters was performed for polycarbonate (n 1 = 1.585) for the wavelength λ = 0.532 μm, which is most suitable for solar concentrators systems.
The lens-concentrator # 27 with a focus of f = 25 mm has a fairly high total light transmission τ R 0 = 75.42%, which is defined by the Fresnel refraction [21] at the both sides of relief forming plate. Calculated values τ R for each prismatic k-zone of the lens depending to radii R k are shown in Fig. 6(a) (curve τ R ). Similar data are evaluable for lens # 28.
We have modulated lens refractive zones with microprism angles α k ≈ 0.5-38.2 deg. having prism reverse angle θ = 3 deg. So, another reason for light transmission diminishing τ V is the light beam vignetting due to the reverse angle θ ࣔ 0. It is easily to calculate (see Fig. 2) that value τ V = (zone 1-2)/(zone 0-2), thus: The final light beams transmission for each k-zone τ S = τ R τ V , where τ R values are calculated by known Fresnel formulas [21] and values τ V are calculated according to (5). Obtained values τ S versus radii R k are shown also in Fig. 6(a) (curve τ S ). Note, that lens-concentrator # 27 due to above vignetting τ V depending on the prismatic angle α k and on the reverse angle θ = 3 deg. has a total light transmission τ S 0 that is diminished to value τ S 0 = 72.96% compared to the refraction value τ R 0 = 75.42%.
The total light transmittance τ S 0 decreases for lenses with a small focal length f due to enlarging the microprism refractive angles α k , which are approaching the limit values α kmax . Therefore, the optimal aperture numbers for plane-focusing lenses should be f/D L ≈ 1.5-2.0.
The similar lens-concentrator # 4 with a larger focal distance f = 41 mm, made earlier and investigated in [25], has the total light transmission τ R 0 = 88.64% even up to the light diameter D L ≈ 50 mm due to the smaller refractive angles α k and diminished refraction losses τ R for each lens radii R k . For this lens the focusing scheme FO = 0.75 (3)-1.5 (1)-2.5 (1)-3.5 (1). Fig. 6(b) illustrates the transmission values τ S = τ R τ V and τ R for this lens # 4. The diminished due to the vignetting τ V the total transmission τ S 0 for this lens is equal to 83.45%. Data obtained indicate that diminishing the focal distance f markedly diminishes the light transmission τ S = τ R τ V mainly due to the enlarged refraction losses τ R . However, this process can also diminish the total thickness and the weight of single solar concentrator cell, constructed with such lenses, that is more important. This fact should be taken into account during the construction of any solar concentration modules.
Calculated zone enlargement, corrected according to formulas (2) and (3), is rather large compared to the nominal pitches W j (Fig. 7(a)). Therefore, taking in mind the diamond cutting process for lens manufacturing, for lens # 28 only k-zones #1-16 up to R 16 = 12.123 mm each can be formed by a single prismatic Fig. 7. Calculated pitch enlargement ΔR k (a) and relief pitch ΔR k for each k-zone depending to radii R k (b) for lens # 28 with f = 25 mm: 1-relative enlargement ΔR k /W j (a.u.); 2-absolute value of correction ΔR k (mm). element (see Fig. 6(b)); zones #17-23 for R 23 < 19.021 mm -from two identical microprisms, refractive zones #24-27 for R 27 < 24.297 mm each should be created from three separate prismatic elements, and zones #28-29 for R 29 < 25.883 mm each should be created from two separate prismatic elements. In this case all zone widths ΔR k are less ∼1.15 mm and simulated lens profile can be formed by diamond cutting method [22].
Similarly, for lens #27 only k-zones #1-9 up to R 9 = 9.130 mm each can be formed by a single prismatic element; zones #10-15 up to R 15 = 15.733 mm -from two identical microprisms, refractive zones #16-22 up to R 22 = 25.161 mm each should be created from three separate prismatic elements. This simulated lens design also can be manufactured by diamond microcutting technique [22].

IV. MANUFACTURE AND TESTING THE LENSES FOR SOLAR CONCENTRATION MODULES
A well-established optical production technology used for manufacture of any lenses is ultra-precision machining (UPM). Method UPM using a diamond tool can fabricate lenses with exceptional geometrical accuracy and mirror-like surfaces. Such lenses meet the highest needs of optical industry. The typical set-up of UPM machine [17] is shown in Fig. 8(a). The used technology of direct diamond cutting by computer control being able to guide the tool along the feature as desired. During this process the rake face of the tool is kept orthogonal to the cutting direction at every point [18]. For the relief shaping tool, a single crystal natural diamond tool was used, with a 12 μm nose radius, 20 deg. included angle, 120 deg. opening angle and 15 deg. front clearance angle. To avoid undesired interactions between the other faces of the tool with the facets that are not being machined, the tool is set with a negative 14 deg. rake angle.
We will call this technology as diamond microcutting (DMC) and also will use it to manufacture our simulated microprisms. This technique was effectively developed [22], [23] in the Institute for Information Recording of National academy of Sciences of Ukraine (IIR). The general view of our installation is shown in Fig. 8(b). A diamond cutter tool is hold in special cartridge, which allows moving diamond cutter of cutting angle α G along axis X and Z with calculated inclination (clearance) angle β. Computer control allows set the necessary cutter positions in this coordinate system by few micrometers accuracy (no more ∼5 μm). Diamond cutter for shaping relief manufacture of simulated lenses has almost similar to [18] parameters: 10 μm nose radius R G , 46 deg. front angle α G , clearance angle β changed from 5 to 40 deg. with 86 deg. opening angle ξ G .
It is assumed [9] that the maximum possible optical efficiency of each individual solar module is achieved when using relatively "long-focus" lenses, that can diminish a photocell overheating problem. For the practical design of solar concentrator modules, these authors applied two sizes of Fresnel lenses: 40 × 40 mm with a focus of f = 70 mm and 60 × 60 mm with f = 105 mm. The diameter of photosensitive surface of photocell in the first case is 1.7 mm, in the second -2.3 mm, which corresponds to a concentration multiplicity value k S ∼ 500.
The usage of new plane-focusing lenses [19], [20] can resolve easily the problem of ohmic losses and connected overheating in solar concentration modules, now it is not necessary to enlarge a focal length f.
For instance, our simulated lens # 27.2, made by diamond microcutting technique, ensures the concentration value k C ∼ 280, has focal length only f = 25 mm and makes a uniform focal light distribution, eliminating completely the ohmic losses. The general view of real plane-focusing lens # 27.2 is illustrated in Fig. 9(a). The image of a transformed light beam at the screen for this lens is shown in Fig. 9(b) for nominal observation distance L 0 = 25 mm.
For comparison, the focal image of transformed light beam is shown in Fig. 9(c) in the same scale for lens #4.1 with f = 41 mm, r V = 4.5 mm and r 0 = 0.7 mm for observation distance L 0 = 41 mm. The differences in Fig. 9(b) and (c) are in the size of focal light spot: r V = 1.5 (9b) and 4.5 mm (9c), showing the lens simulation capability.
These data were obtained in experimental study of these lenses using a collimated laser beam of the wavelength λ = 0.532 μm with beam diameter d S ≈ 56 mm. The optical scheme of experimental setup is discussed in detail in [20]; the typical profile of collimated laser beam at the screen is shown in Fig. 10(a).

V. DIFFRACTION PHENOMENA FOR TRANSFORMING FRESNEL LENSES
The real profiles of focal light distributions we obtained by Jmage J-1.53 program [25]. To determine the ratio between the geometric dimensions of a light spot in the lens focus at the screen d S and its dimensions d X at the profilogram of this spot for selected scale of scanning by Jmage J-1.53 program, we used the fact that the diameter of the collimated laser beam d S = 56 mm (outer circle in Fig. 10(b), where the typical complete picture at the screen is shown for lens #4.1 [20] (f = 41mm, r V = 4.5 mm, r 0 = 0.7 mm, L 0 = 41mm) with focal light spot of d S = 9.0 mm). The profilogram of this focal spot is illustrated in Fig. 11(a). Simple calculations prove that a distance of 1 cm of the image at the screen corresponds to 17.7 pixels on the profilogram, thus the nominal diameter of the spot d V = 9.0 mm corresponds to a distance in pixels d X = 159.6 pixels (shown in Fig. 11(a) by dashed lines). Similar data for lens # 27 (f = 25 mm, r V = 1.5 mm r 0 = 0.7 mm, L 0 = 25 mm) are shown in Fig. 11(b). The spot diameter d V = 3.0 mm for lens # 27.2 corresponds to the distance in pixels d X = 53.2 pixels (shown in Fig. 11(b) by dashed lines).
The obtained data indicate that the size of light spot at the screen significantly exaggerates the calculated characteristics. Profile enlargement is especially noticeable for lenses with relatively small light spot diameter d V = 3.0 mm; and has a less value for lens #4.1 with larger d V = 9.0 mm. The enlargement scale is the practically the same for all lenses. However, for lenses with a smaller spot diameter d V , the homogeneity is more diminished due to the significantly larger contribution of lateral diffusion zones, which resulting also in spot enlargement comparing to theoretical values.
In our opinion, the reason for these enlargements is the light diffraction on the faces of microprisms, the refraction of light beams inside the microrelief and the diffuse scattering of the light on the mechanical surface defects. For practical consideration of diffraction phenomena during the passage of a light beam through a microprism focusing structure, one can use a simplified model [21], in which the light wave is considered to be plane, and the diffraction is considered to be an amplitude one. In this model, the initial phase difference of the beams Δ D (see Fig. 2) can be taken into account simply by rotating the coordinate system by the refraction angle γ R (λ), which for a certain wavelength λ can be easily determined for each microprism k-zone of the lens using the Snell law [21]. The diffraction angles ϕ D (λ) are counted from the direction of the refraction angle γ R (λ), and to calculate the intensity of diffracted light J D passing through each microprism zone k, one can use the known formulas for the amplitude diffraction grating: where А D = (π . b/λ) sin is usually named as single-slit diffraction factor [24]; ϕ D is the diffractive angle; J 0 is the amplitude of initial light beam from single slit in the direction ϕ D = 0; λ is the wavelength of the considered radiation; b is the width of the luminous beam.
In the direction of beam deflection angle γ k (see Fig. 2), for any microprism lens zone k, the period of the diffraction grating d γ = ΔR γ k = ΔR k cos γ R . The effective aperture b γ , which is the analogous to the quantity b in formula (6), for prism inverse . For a certain wavelength λ, the width of the main diffraction maximum The diffraction angles ϕ D for each microprismatic k-zone of the lens can be easily transformed into a linear beam expansion ΔL D for any focal length f: ΔL D = 2f tg ϕ D . The calculated values of ΔL Dk for lens # 28 with focus f = 25 mm and focal spot r V = 1.5 mm are shown in Fig. 12(a). Diffraction broadening ΔL D for zone width ΔR γk = 1.002 mm (k = 1) and ΔR γk = 0.329 mm (k = 29) for this lens are shown in Fig. 12(b).
Calculations showed that for transforming plane-focusing lenses with small focal spot d V ∼2-3 mm, the microprism pitches ΔR γ k even for microprism maximum k-values are over 200-250 μm (Fig. 7(b)); therefore, the diffraction expansion of transformed light fluxes ΔL Dk (Fig. 12(a)) cannot be significant compared to values of ΔR γ k . Thus, the role of diffraction is very small for focal spot enlargement. For a focal light spot with d V > 8-10 mm, the effective widths of the microprism zones ΔR γ k significantly exceed the values of the diffraction widths ΔL Dk ; therefore, for example, for lens # 4.1 (d V = 9.0 mm) the almost homogeneous focal irradiation is observed for focal areas with r k < 4.5 mm, but for larger radii the diffusion lateral zones exist, which expand the spot. We assume that a spot size enlargement, which is obtained experimentally, can be caused mainly by the reflection of light beams inside the relief (4-5%) and by the diffuse light scattering on the mechanical defects of the relief. Therefore, the diamond tool during the microcutting process should be of high quality; precise tool positioning is necessary; the high accuracy of forming the angles α k and the depths h k of the relief are decisive for obtaining the high-quality homogeneous irradiation.
We cannot investigate the process of diffuse scattering theoretically due to unknown data on the influence of relief defects on the light spot enlargement. To optimize the design of our new solar lenses with small f-value we have investigated experimentally using collimated laser beam the manufactured lenses #25c, #27.2 and #28 created for solar concentration modules and having the same focal distances f = 25 mm and light diameters D L = 50 mm.
The spot profile for little focal distances is strongly dependent on the observation distance L 0 because diaphragm number F for our lenses is very small (F = f/D L = 0.5) and the field depth is small also. Fig. 13(a) shows the spot profiles for lens # 25c with f = 25 mm for different observation length L 0 = 24 mm; 25 mm and 26 mm. For all cases real enlarged profiles exceed the calculated value d V = 3.0 mm (dotted lines in pixels profile).
The profile enlargements for lenses #27 and #28 having the focal light spot r V = 1.5 mm are similar, they slightly dependent on optimization scheme FO (Fig. 13(b)). For further lens creation any lens design cab be used; we have recommended the transforming lens #27.2 with "dark" area of r 0 = 0.7 mm, as the more simple one for its practical implementation.
In contrast, the profile dependence on optimization scheme FO is evident for lenses with large enough focal spots with d V > 8-10 mm (Fig. 14). Optimal lens design corresponds to "dark" area of r 0 ≈ 0.6 mm; such scheme will be realized during the mass manufacturing such lenses for four-plane detector control systems for tracking the moving objects [27].

VI. CONCLUSION
Computational and experimental data confirmed that for creating the solar concentration modules with minimal thermal and ohmic losses the best variant of focusing solar radiation is the usage of new plane-focusing lenses, which transform the refracted light beams into a uniform light circle in the lens focal plane. Proposed specialized lens-concentrators can be used effectively in optical concentration modules for diminishing the ohmic currents and the dimensions of photocells.
The algorithm of mathematical modeling of plane-focusing micro-prismatic lenses was developed to create optimal Fresnel lenses for photovoltaic module with highly concentrated solar radiation. The proposed algorithm is specially adapted for mass manufacturing such lenses by thermo-pressing method.
The calculations of geometric parameters for such specialized microprism lens-concentrators have been carried out. The samples of specialized lens-concentrators were manufactured from the optical polycarbonate by the diamond microcutting method according to our simulation results. The experimental study of the optical and lighting characteristics of these lenses by collimated laser beam was performed. Obtained data indicate that the role of diffraction is very small for focal light spot enlargement, so the main reason can be the diffuse scattering of the light on the mechanical defects of microrelief.
Thus, the high-quality cutting tools only can be used, while the precise tool positioning in the process of diamond microcutting should be realized. The optimal lens design was proposed for mass manufacturing the plane-focusing lenses for solar concentration modules. Obtained data showed the complete compliance of experimental data with theoretical characteristics.