1. Introduction

High-concentration photovoltaic systems use optical elements to focus direct sunlight onto multi-junction solar cells that have record efficiencies above 44% [1]. In most systems, the optical concentrator consists of a primary optical element that concentrates the sunlight onto the entrance aperture of a secondary optical element (SOE) which is directly attached to the solar cell. There are also systems that do not use SOEs. The entrance aperture of the SOE or, if there is no SOE, the active area of the solar cell is often circular. The primary optical element has to concentrate as much sunlight as possible onto this target area.

Many high-concentration photovoltaic systems use refractive Fresnel lenses as primary optical elements [2]. Typically, these Fresnel lenses are flat with a smooth surface directed towards the sun and a structured surface directed towards the focal point. This is opposite to the way plano-convex lenses are commonly used, i.e. with the flat surface directed towards the focal point.

Fresnel lenses reduce the volume of the lens compared to a full lens. Smaller prism sizes in the Fresnel lens save more optical material, thus reducing system costs. For manufacturing reasons, the facets of the annular Fresnel lens prisms are often conical surfaces instead of segments of an aspherical lens surface. Reducing the prism size therefore also reduces the beam spread caused by the conical prism surfaces. However, smaller prism sizes also increase diffraction losses due to the Fresnel prisms. Diffracted light may spill outside the target area and be lost for energy conversion. Because these diffraction-induced losses reduce the energy output of the photovoltaic system, even losses of a few percent are highly relevant.

There is little literature on losses due to diffraction by concentrator Fresnel lenses. Several publications give the formula $\Delta r_{\text{l}}=\text{1}\text{.5}\sqrt{\lambda f}$ as a rule of thumb for a good choice of prism sizes $\Delta r_{\text{l}}$ for Fresnel lenses in general ($\lambda$ is the wavelength of the incedent light, f is the focal length of the Fresnel lens) [3, 4]. Dürr et al. have performed extensive computer simulations to analyse diffraction effects by very small concentrator Fresnel lenses [5, 6].

In this paper we take an analytical approach. In many cases, this makes complex and time-consuming wave-optical computer simulations unnecessary. It also provides insight into some aspects concerning the structure of the losses caused by diffraction that cannot be obtained by numerical computations.

After analyzing the applicability of diffraction theories that are based on Kirchhoff’s integral theorem (sections 2 and 3), an approximate formula is derived for the diffraction loss in the simple case of a quasi-monochromatic, spatially coherent wave being diffracted at the prisms of a Fresnel lens with aspherical Fresnel prism surfaces (section 4). Subsequently, the formula is extended to light of arbitrary wavelength and Fresnel lenses with a wide variety of shapes, including conical Fresnel facets (section 5). As a third step, an approximate treatment of partially coherent light from a homogeneous light source, e.g. the solar disc, is added (section 6). The main results are summarized in section 7.

2. Edge radii of mass-manufactured Fresnel lenses

The following discussion based on diffraction theory assumes that the radii of rounded edges of the Fresnel lens prisms are not significantly larger than the wavelength of light. To test whether a typical, industrially mass-produced concentrator Fresnel lens meets this requirement, the cross-section of a resin cast from a silicon-on-glass Fresnel lens was investigated in a scanning electron microscope. Figure 1 shows the cross-section through a typical edge from the peak of a Fresnel structure, where the greatest degree of edge rounding is usually expected. The radius of the rounded edge is less than a micrometre and thus of the same order of magnitude as the wavelength of light. It is thus permissible to treat the edges as being sharp in the calculations.

 

Fig. 1 Scanning electron micrograph of the resin cast from an industrially produced, silicon-on-glass Fresnel lens. It shows the detail of a cross-section through this negative of the lens form. The structure illustrated here originates from the peak of a Fresnel prism. The radius of the rounded edge is less than 1 µm.

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3. Application of Kirchhoff’s Integral Theorem to Fresnel Lenses

For diffracting apertures which are large compared to the wavelength, the diffraction image for unpolarised, quasi-monochromatic light at distances which are also large compared to the wavelength can be described to a very good approximation by the diffraction of a scalar wave [7]. Kirchhoff’s diffraction theory is an example of such a description. It is based on Kirchhoff’s integral theorem [7–9]

 

Fig. 2 Application of Kirchhoff’s integral theorem to the diffraction of an incoming spherical wave by a circular aperture. The field at point $P$ is calculated by integrating over all points $Q$ of the surface ${\mathcal{S}}_{\circ }$. The boundary conditions are selected such that only the regions within the aperture contribute to the integral.

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To describe diffraction problems, the surface of integration ${\mathcal{S}}_{\circ }$ is located behind the diffracting aperture and is closed at infinity, as illustrated in Fig. 2. Kirchhoff’s boundary conditions are assumed for the integration, which state that the field within the aperture corresponds to the incident field. The field is set to zero in the shadow at the back of the aperture screen. In addition, the field should vanish at infinity, so that the integral there also vanishes. These assumptions lead to the result that of the area ${\mathcal{S}}_{\circ }$, only that part $\mathcal{S}$ located within the aperture contributes to the surface integral in (1). In the following, the field $\tilde{Y}(Q)=\tilde{B}{e}^{-\text{i}kq}/q$ which is incident on the aperture corresponds to an incoming spherical wave centred on the origin of the coordinate system.

When a concentrator Fresnel lens is described by diffraction theory, Kirchhoff’s diffraction integral is replaced by a sum of diffraction integrals, one for each Fresnel prism. The problem is thus simplified to the diffraction by individual annular Fresnel prisms.

The diffracting aperture is simultaneously the light-refracting surface. In analogy to the usual approach for conventional lenses (see [7], for example), the exterior, exit surface of the Fresnel lens is chosen as the integration surface. This means that diffraction effects are only calculated after light refraction has occurred, creating a spherical wave. This is illustrated in Fig. 3.

 

Fig. 3 Approach to calculate diffraction by a prism of a Fresnel lens. The inner and outer edges of each Fresnel prism are replaced respectively by a circular disc and an aperture surrounded by completely absorbing material. The form of the wave fronts after refraction is considered, which represent an incoming spherical wave if suitably curved slope facets are assumed. The effect of light that is totally internally reflected by the draft facets is ignored.

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The outer edge of the slope facet is replaced by a circular aperture surrounded by completely absorbing material and the inner edge of the slope facet is replaced by a circular disc. The disc and the aperture are in planes which are displaced relative to each other in the direction of the optical axis. This represents a significant simplification compared to the real, totally reflecting dielectric draft facets of the Fresnel lens. Hoßfeld compared the diffraction by a completely absorbing plane to the diffraction by a totally reflecting dielectric wedge [10]. According to his investigations, the diffraction image created by the totally reflecting dielectric wedge is practically indistinguishable from that created by an absorbing screen already only a few wavelengths behind the diffracting structure. Correspondingly, it is assumed that replacing the dielectric draft facets by absorbing aperture screens also represents a good approximation for the following calculations.

Furthermore, it is assumed that the undisturbed, incoming spherical wave is incident on both the aperture and the disc. This assumption is justified if the diffraction by the circular aperture does not significantly affect the electromagnetic wave at the circular disc. For this to apply, the edge of the geometrical-optical shadow of the circular aperture must be many wavelengths away from the edge of the circular disc. Thus, this assumption is justified as long as the light passing through the Fresnel lens does not strike the slope facet at an angle close to the critical angle and therefore exits the slope facet at grazing angle. Typical Fresnel lenses have slope angles much smaller than the critical angle and this assumption holds. .

4. Diffraction loss for idealized Fresnel lenses under coherent illumination

In his qualitative considerations from 1802 on solving the diffraction problem as part of the wave theory of light, Young [11] describes diffraction by an aperture as a process which resembles light scattering by the edge of the aperture. A diffracted wave originates at the edge which draws its energy from the field near the aperture edge. The remaining incident light passes through the aperture without being disturbed.

It was not until much later that the diffraction problem was described by Kirchhoff in a mathematically consistent form [8, 9], whereby initially no wave originating at the aperture edge was evident. However, Maggi [12] demonstrated shortly afterwards that this could be derived mathematically in several cases from Kirchhoff’s integral theorem. Independently of Maggi, at the beginning of the 20th century Rubinowicz [13, 14] again developed a diffraction theory based on Kirchhoff’s description of diffraction, which is identical to that of Maggi in essential aspects.

The diffraction theory of Maggi and Rubinowicz, just like Fraunhofer’s und Fresnel’s refraction theory, is based on Kirchhoff’s integral theorem (1) with Kirchhoff’s boundary conditions. In some cases, the surface integral in Kirchhoff’s integral theorem can be split into a coherent sum $\tilde{Y}(P)={\tilde{Y}}_{\text{b}}(P)+{\tilde{Y}}_{\text{g}}(P)$ of a boundary diffraction wave ${\tilde{Y}}_{\text{b}}(P)$ and a geometrical-optical wave ${\tilde{Y}}_{\text{g}}(P)$, whereby the geometrical-optical wave is not subject to any diffraction and the boundary diffraction wave originates only at the edge of the aperture. An overview of further developments related to this theory is given by Sunil Kumar and Ranganath [15]. Miyamoto and Wolf [16, 17] made major extensions to the diffraction theory of Maggi and Rubinowicz. The following derivation is based on their work.

For an abitrary, scalar, monochromatic wave $\tilde{Y}$, which satisfies the Helmholtz equation $({\nabla }^2+k^2)\tilde{Y}=0$ with the Laplace operator ${\nabla }^2$ and the wavenumber $k$, the divergence of the integrand in Kirchhoff’s integral theorem (1) vanishes according to

 

Fig. 4 Nomenclature used in the calculation of the diffraction of an incident spherical wave by a circular aperture according to the Maggi-Rubinowicz diffraction theory. The field at point $P$is calculated by integrating over all points $Q$ on the edge of the aperture.

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This interpretation can be readily understood in the specific calculation of diffraction by a Fresnel prism of a Fresnel lens. For the special case of an incoming spherical wave centred at the origin of the coordinate system, the vector potential is described by the following equation, according to the derivation of Miyamoto and Wolf [16]

The vector field obviously has a singularity where $sq-s\cdot q=0$, i.e. whenever $s$ and $q$ point in the same direction. Expressed in terms of geometrical ray optics, this means that a light ray that passes through the point $Q$ and then the origin of the coordinate system is incident on the observation point $P$. If $P$is located in the geometrical-optical shadow, these singularities do not occur. They make the following contributions

The geometrical-optical wave interferes with the diffraction wave that originates at the edge of the aperture

This approximation diverges if $P$ is located too close to the edge of the geometrical-optical shadow. There, the amplitude $\tilde{G}(x)$ near the singularity in the vector potential (5) no longer changes significantly more slowly than the phase. In addition, the absolute value of the second derivative of the phase with respect to position ${\partial }^{2}S(x)/\partial x^2$ must not become too small at the stationary point. The stationary phase method thus fails for incident light parallel to the lens axis close to the optical axis. The calculation of the fraction of light spilling outside the target area is not hindered by these two divergent cases if the geometrical-optical shadow boundary falls within the target area. This is the most interesting case, because if the shadow boundary lies outside the target area, the resulting geometrical-optical light spillage will dominate over the diffraction loss. The small contribution of diffraction loss will then be of no practical relevance. Since several inaccuracies are involved in a real CPV system, the case of the shadow boundary mathematically falling onto or very close to the edge of the target area is of little practical importance either.

Approximation (9) should now be evaluated for the case of interest here, namely the diffraction of an incoming spherical wave by a circular aperture. The circular aperture should be perpendicular to the optical axis. For a point $P$ not located on the optical axis, the integrand in (7) possesses two stationary points which lie in the plane defined by the optical axis and $P$. By applying (9) to (7), we obtain, with the nomenclature of Fig. 4

In order to describe the diffraction by the Fresnel prism, not only the diffraction by the circular aperture but also the diffraction by the circular disc must be taken into account (see Fig. 3). The boundary diffraction waves from the circular aperture and circular disc are of opposite sign (positive or negative) but are calculated analogously. The diffraction image of a complete Fresnel lens is composed of the superposition of all the boundary diffraction waves and geometrical waves from all prisms of the lens. In customary refractive Fresnel lenses, there is no uniform phase relation between the edges of the prisms. If the lens is illuminated with coherent light, the diffracted waves interfere with each other but they do not display any systematic phase shift with respect to each other. Therefore, it can be assumed that the interference terms essentially cancel out over the complete Fresnel lens. As a rough estimate, the Michelson visibility $(J_{\text{max}}-J_{\text{min}})/(J_{\text{max}}+J_{\text{min}})$ of the interference pattern is approximately equal to one over the number of edges of the lens structure. Accordingly, the visibility of the interference pattern of a Fresnel lens with just 10 prisms is below 5.3%. For a typical Fresnel lens with more than 50 prisms, the diffraction image hardly differs from an incoherent superposition of the intensities. Furthermore, the interference terms modulate the intensity pattern but have little influence on the total light spillage due to diffraction. Altogether, in most cases it is sufficient to calculate the incoherent superposition of the intensities. The intensity of the diffraction image of a Fresnel lens with $N$ prisms, with $2N-1$ edges passing through the points $(q_i,0,{\theta }_{q_i})$, can then be calculated according to

In order to understand the irradiance described by expression (18) better, an approximation for the behaviour of the boundary diffraction wave near the focal point should be examined in more detail. At a location sufficiently far from the optical axis that the stationary phase method (9) can be applied ($q\ll kp$), but still close to the focal point ($p\ll q$), the boundary diffraction wave in (18) can be approximnated by a Taylor series in $s(\Phi )$, so that

Often, flat Fresnel lenses are used with a constant pitch, the separation distance between adjacent prisms. This type of lens, with a radius $r_{\text{L}}$ and $N$ prisms of constant pitch $\Delta r_{\text{l}}=r_{\text{L}}/N$ should thus be analysed as a special case. Both edges of each draft facet are located close to each other between the significantly longer slope facets. The radial length of the draft facet is typically not more than a few percent of the pitch of the lens. To simplify the calculations, the radial length of the draft facet is considered to be negligible and is ignored. Thus, both its inner and outer edges have the same radius $r_{\text{l}2j-1}=r_{\text{l}2j}=j\cdot r_{\text{L}}/N$. The outer edge of the outermost prism of the Fresnel lens must be treated separately, as it does not have a neighbouring draft facet. The structural height of the Fresnel lens is often much less than one percent of the focal length and is also ignored in the following approximation, such that $f_i=f$. Then applying the approximation of (19) results in

For small $p_{\text{z}}$, ${(j\Delta r_{\text{l}}/f)}^2{p}_{\text{z}}^2$ can be neglected in comparison to $p_{\text{r}}^2$ for positions in the shadow sufficiently far away from the shadow edge, so that the irradiance there is described by the expression

The approximations in the Eqs. (20) and (23) are valid for radii $p_{\text{r}}$ which are small compared to the focal length of the lens but still significantly larger than the typical radius $r_{\text{Z}}$of the target area (concentrator solar cell or entrance aperture of secondary optical element). Assuming that the light intensities become negligible for still larger radii, the Eqs. (20) or (23) can be used for an initial estimation of the total loss caused by diffraction effects. This is done by integrating the intensity spilled outside the target area, whereby the integration is extended beyond the valid range for the Taylor series in the direction of infinity. The power which is incident outside a target area with a radius $r_{\text{Z}}$ is derived from Eq. (20) to be

The radiative power $P(p_{\text{r}}\ge r_{\text{Z}})$, which cannot be converted by the solar cell due to diffraction losses, is proportional to the radiative power incident on the Fresnel lens. Thus, not the absolute value but the relative loss is of interest. For a homogeneous irradiance of $J_{\text{in}}=c{ϵ}_0{\tilde{B}}_0^2/f^2=c{ϵ}_0{\tilde{B}}_i^2/q_i^2$ on the Fresnel lens the incident power is $P_{\text{in}}=c{ϵ}_0\pi r_{\text{L}}^2{\tilde{B}}_i^2/q_i^2$. Using the more general Eq. (24), the relative loss caused by diffraction thus amounts to

 

Fig. 5 Light loss due to diffraction as a function of the f-number f/(2r_L)of a Fresnel lens. A Fresnel lens with a pitch of 250 µm and a geometrical concentration factor of 400 was taken as an example for the calculation according to Eq. (27) using light with a wavelength of 589 nm. The approximation (28) for large f-numbers is plotted with the dotted line. Depending on the refractive index of the lens material, f-numbers of less than about 0.5 cannot be achieved with a real lens. Typical concentrator Fresnel lenses have f-numbers close to 1.

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5. Diffraction loss for realistic Fresnel lenses under polychromatic illumination

The values calculated according to Eq. (27) should be regarded as the theoretical lower limit for the light which spills outside the target area due to diffraction. A perfectly focussing lens and perpendicularly incident, monochromatic and coherent light have been assumed. Due to limitations of the production technology, by contrast, real concentrator Fresnel lenses usually have conical slope facets and thus cannot achieve a perfect focus point. Furthermore, the incident solar radiation is polychromatic and only partially spatially coherent. In total, chromatic aberration, partially coherent light and conical slope facets lead to expansion of the geometrical-optical focus point. As a result, the edge of the geometrical-optical shadow moves closer to the edge of the target area. It is thus expected that more light is diffracted to spill outside the target area than is predicted by Eq. (27). In this section, the assumption of spatially coherent light perpendicularly incident on the Fresnel lens is retained but the light is no longer assumed to be monochromatic. Partially coherent light will be treated in the next section.

Equation (26) offers the flexibility to treat arbitrary geometrical configurations for Fresnel lenses, including variable pitch $\Delta r_{\text{l}}$. The focal length of the Fresnel lens $f_i$ can be chosen separately for each diffracting edge, allowing the height of the Fresnel prism to be taken into account. In addition, the z component $p_{\text{z}}$ of the observation point $P$ allows the diffraction loss to be investigated also outside the focal plane.

If light is incident perpendicularly on a Fresnel lens with conical slope facets, then light which penetrates a Fresnel prism close to the inner edge of the slope facet is incident on the optical axis at a shorter focal length than light which is refracted close to the outer edge of the slope facet. As the origin of the coordinate system has always been located at the focal point of the Fresnel lens up to now, $f_i$ was identical with the z coordinate of the diffracting edge. In order that conical slope facets in Eq. (26) can be taken into account, the z coordinate of the diffracting edge is designated as $z_{\text{l}i}$ (Fig. 6). In addition, because conical slope facets do not lead to a single focal point, an arbitray point on the optical axis is chosen to be the origin of the coordinate system. To do so, the z component $p_{\text{z}}$ of the vector $p$ must be replaced by $f_i-z_{\text{l}i}+{p^{\prime }}_{\text{z}}$ (Fig. 6). With these changes, Eq. (26) becomes

 

Fig. 6 Introduction of the quantity to calculate diffraction for the case that the focal point for the lens region close to the ith edge is not identical to the origin of the coordinate system.

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Figure 7 shows the losses due to diffraction as a function of the pitch of a Fresnel lens which has conical slope facets. The more exact Eq. (29) already predicts noticeably greater diffraction losses than the simple estimation with Eq. (27) for pitch values where the edge of the geometrical-optical shadow is still several millimetres away from the edge of the target area. The dotted line in Fig. 7 results from Eq. (41) which will be derived in section 6.

 

Fig. 7 Relative power loss due to diffraction as a function of the pitch of the Fresnel lens (separation between adjacent Fresnel prisms). The losses according to Eqs. (27) (solid line), (29) (dashed line) and (41) (dotted line) are plotted. The approximations (29) and (41) are plotted only for the range where they are valid, in which the edge of the target area is located in the geometrical-optical shadow region. This example was calculated for a circular Fresnel lens with conical slope facets, 200 mm diameter, 250 mm focal length and a refractive index of 1.45, which is illuminated with light of wavelength 589 nm and concentrates the light onto a target area with a 10 mm diameter (400x geometrical concentration).

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In addition to allowing conical slope facets to be included in the calculation of diffraction effects, Eq. (29) also allows dispersion by the lens material to be taken into account. The focal length $f_i$ of the lens, which is calculated separately for each edge, can be determined for any arbitrary refractive index. If the dispersion function for the lens material is known, the diffraction loss can be determined for any arbitrary wavelength, thereby taking the chromatic aberration of the lens into account. Figure 8 shows the results of calculation examples with and without consideration of chromatic aberration. Results from section 6 are also included in this figure (dotted line).

 

Fig. 8 Relative power loss due to diffraction as a function of the wavelength. The losses are plotted for calculations according to Eqs. (27) (solid line), (29) (dashed line) and (41) (dotted line). The approximations (29) and (41) are plotted only over the range for which they are valid. The calculations were made using a circular PMMA Fresnel lens with the same geometrical configuration as for Fig. 7, assuming a pitch of 250 µm. The underlying dispersion function for PMMA was published by Kasarova et al. [18].

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6. Diffraction loss under partially coherent illumination

A monochromatic point light source emits a spherical electromagnetic wave which is completely spatially coherent. By contrast, as observed from the earth, the solar disc represents a homogeneous, incoherently radiating light source subtending a half angle of 4.65 mrad. The individual points of the sun’s surface can be treated as independent sources of radiation which do not exhibit any fixed phase-shift relations. For a mathematically correct description, only coherence must be assumed over distances which are shorter than one wavelength, because in other cases the radiated waves mutually extinguish each other [19, 20]. The diffraction pattern created by an aperture can be calculated separately for each point of an incoherent light source. The diffraction patterns calculated in this way are superimposed incoherently, i.e. by addition of the intensities, to obtain the complete diffraction image of the light from the incoherent light source [19, 21].

Because no extinction by destructive interference can occur when the intensities are added, the fraction of light which is spilled outside the target area cannot be less for partially coherent illumination by an incoherent light source than for coherent illumination (assuming that the geometrically-optically illuminated area is located completely within the target area). This conclusion initially appears to contradict the widely held view that diffraction effects diminish with decreasing coherence of the incident light. This opinion is in fact correct with respect to the contrast between the maxima and minima of the diffraction pattern but does not apply to the total fraction of diffracted light.

A commonly applied approach to calculate diffraction effects with partially coherent light consists of determining the complex degree of coherence of the light and taking this into account when the diffraction is calculated (e.g. see [19]). However, the complexity of the diffraction problem considered here makes it improbable that similarly simple solutions to the Eqs. (27) and (29) will be found in this way. Thus, in the following section, the diffraction pattern will be derived directly from the assumption of an incoherently radiating light source.

As a starting point, it is assumed that a slight change in the incident angle of the light changes essentially only the angle at which the individual boundary diffraction waves are emitted. This leads to a shift in the diffraction pattern in the observation plane. Apart from this shift, the diffraction pattern remains unchanged in this approximation. The intensity distribution of the diffracted light can then be approximately described by convolution of the diffraction pattern for normal incidence with the angular distribution of the incident radiation. Applying Eq. (20), the following expression is obtained for the diffraction pattern for a circular, incoherent light source

The radius $r_{\text{S}i}$ specifies the radius of the convolution circle for light which passes through the Fresnel lens close to the ith edge. It is determined by the half-angle ${\vartheta }_{\text{in}}$, which describes the apparent size of the light source, and the angle ${\alpha }_i$ of the light-refracting surface close to the ith diffracting edge (Fig. 6). For a flat concentrator Fresnel lens with a planar entrance surface, the light is refracted on entry to and exit from the lens. For small angles of incidence and n as the refractive index, $r_{\text{S}i}$ is calculated as

Again, the total fraction of illumination power which spills outside the target area due to diffraction is of interest. Thus, the integral is evaluated over the region outside the target area. From Eq. (30), it follows that

 

Fig. 9 Definition of cases to solve the integral in the second line of Eq. (34). In case a), corresponding to, ${\text{τ}}_{\text{r}}+{\text{r}}_{\text{s}}<{\text{r}}_{\text{z}},$one of the multiplied theta functions is always equal to zero; in case b), where, ${\text{τ}}_{\text{r}}-{\text{r}}_{\text{s}}<{\text{r}}_{\text{z}}<{\text{τ}}_{\text{r}}+{\text{r}}_{\text{s}}$the integral over the theta functions corresponds to that area of the circle with radius r_s which is located outside the circle with radius r_z and in case c), where, ${\text{r}}_z<{\text{τ}}_r-{\text{r}}_s$the integral is obtained from the area of the circle with radius r_s.

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Fig. 10 Calculation of the area for case b) in Fig. 9. The area consists of the circular segment bounded by blue dots with an area of $(\pi -{\phi }_S)r_S^2$ and the difference ${\tau }_r{r}_S\sin {\phi }_S-{\phi }_Z{r}_Z$ between the kite-shaped area bounded by green dashed lines and the circular segment which is bounded by red dots and dashes.

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On inserting the three partial solutions a, b, c to the integral into Eq. (34), the expression