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Energy Express

  • Editor: Christian Seassal
  • Vol. 22, Iss. S3 — May. 5, 2014
  • pp: A686–A704
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Light diffraction by concentrator Fresnel lenses

Thorsten Hornung and Peter Nitz  »View Author Affiliations


Optics Express, Vol. 22, Issue S3, pp. A686-A704 (2014)
http://dx.doi.org/10.1364/OE.22.00A686


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Abstract

Fresnel lenses are widely used in concentrating photovoltaic (CPV) systems as primary optical elements focusing sunlight onto small solar cells or onto entrance apertures of secondary optical elements attached to the solar cells. Calculations using the Young-Maggi-Rubinowicz theory of diffraction yield analytical expressions for the amount of light spilling outside these target areas due to diffraction at the edges of the concentrator Fresnel lenses. Explicit equations are given for the diffraction loss due to planar Fresnel lenses with small prisms and due to arbitrarily shaped Fresnel lenses. Furthermore, the cases of illumination by monochromatic, polychromatic, totally spatially coherent and partially spatially coherent light (e.g. from the solar disc) are treated, resulting in analytical formulae. Examples using realistic values show losses due to diffraction of up to several percent.

© 2014 Optical Society of America

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

1. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013). [CrossRef]

]. 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

2. G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009). [CrossRef]

]. 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 Δrl=1.5λf as a rule of thumb for a good choice of prism sizes Δrl for Fresnel lenses in general (λ is the wavelength of the incedent light, f is the focal length of the Fresnel lens) [3

3. J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979). [CrossRef]

, 4

4. A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007). [CrossRef]

]. Dürr et al. have performed extensive computer simulations to analyse diffraction effects by very small concentrator Fresnel lenses [5

5. F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010). [CrossRef] [PubMed]

, 6

6. F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010). [CrossRef]

].

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.
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.
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.

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

7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

]. Kirchhoff’s diffraction theory is an example of such a description. It is based on Kirchhoff’s integral theorem [7

7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

9

9. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

]
Y˜(P)=14πS(Y˜(Q)ÑeiksseikssÑY˜(Q))n^dS
(1)
which describes the complex field Y˜ at point P as a function of its values on a closed surface S. The integration is carried out over all points Q of this surface. The unit vector n^ represents the inward directed normal of the surface element dS. As shown in Fig. 2, the length s=|s| is the distance between point Q on the surface and P, the point of observation. The nabla operators act on the spatial coordinates of point Q. Kirchhoff‘s integral theorem (1) assumes a quasi-monochromatic field described by the wave vector k=2π/λ.
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 S. The boundary conditions are selected such that only the regions within the aperture contribute to the integral.

To describe diffraction problems, the surface of integration S 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 S, only that part S located within the aperture contributes to the surface integral in (1). In the following, the field Y˜(Q)=B˜eikq/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

7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

], 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.

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

10. W. Hoßfeld, Tageslichtsteuerung mit prismatischen Mikrostrukturen im Übergangsbereich von diffraktiver und geometrischer Optik (doctoral thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2004).

]. 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

11. T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802). [CrossRef]

] 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

8. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883). [CrossRef]

, 9

9. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

], whereby initially no wave originating at the aperture edge was evident. However, Maggi [12

12. G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).

] 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

13. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924). [CrossRef]

, 14

14. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917). [CrossRef]

] again developed a diffraction theory based on Kirchhoff’s description of diffraction, which is identical to that of Maggi in essential aspects.

For an abitrary, scalar, monochromatic wave Y˜, which satisfies the Helmholtz equation (2+k2)Y˜=0 with the Laplace operator 2 and the wavenumber k, the divergence of the integrand in Kirchhoff’s integral theorem (1) vanishes according to
(Y˜eiksseikssY˜)=Y˜2eikss+eikssY˜eikss2Y˜eikssY˜=0.
(2)
Accordingly, a vector potential W˜(Q,P) exists with ×W˜(Q,P)=Y˜eiksseikssY˜. This vector potential usually has some singularities. In order that the Kirchhoff integral can be calculated, all singularities of W˜(Q,P) are surrounded by circles of radius ϵi and removed from the surface of integration. The Kirchhoff integral over the remaining area S can be rewritten according to Stoke’s theorem as
Y˜(P)=S(Y˜(Q)eiksseikssY˜(Q))n^dA=S×W˜(Q,P)n^dA=δSW˜(Q,P)l^dl+iδSiW˜(Q,P)l^dl.
(3)
Here, l^ is the unit vector tangential to the edge δS of the aperture or the edge δSi of the circles around the singularities and dl is an element of this edge (Fig. 4).
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 Pis calculated by integrating over all points Q on the edge of the aperture.
By taking the limit, the field
Y˜(P)=δSW˜(Q,P)l^dl+ilimϵi0δSiW˜(Q,P)l^dl=Y˜b(P)+Y˜g(P)
(4)
at point P is obtained. It consists of a ring integral along the aperture edge and contributions from the singularities in the vector potential W˜. In a physical interpretation, Y˜b(P) represents the boundary diffraction wave originating from the aperture edge and the contributions from the singularities can be understood as geometrical-optical waves Y˜g(P).

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

16. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part I,” J. Opt. Soc. Am. 52(6), 615–622 (1962). [CrossRef]

]
W˜(Q,P)=B˜4πeikqqeiksss×qsqsq.
(5)
Here, the nomenclature presented in Fig. 2 and Fig. 4 is used. The spatial vectors for the points P and Q are designated by p and q respectively and their difference as s=qp. The scalar quantities p, q and s represent the magnitudes of the corresponding vector quantities.

The vector field obviously has a singularity where sqsq=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 Pis located in the geometrical-optical shadow, these singularities do not occur. They make the following contributions
Y˜g(P)={B˜eikppinfrontofthefocalpointB˜eikppbehindthefocalpoint0inthegeometrical-opticalshadow
(6)
to the field at point P [17

17. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part II,” J. Opt. Soc. Am. 52(6), 626–636 (1962). [CrossRef]

]. This corresponds exactly to the field according to geometrical optics.

The geometrical-optical wave interferes with the diffraction wave that originates at the edge of the aperture
Y˜b(P)=δSW˜(Q,P)l^dl=δSB˜4πeikqqeiksss×qsqsql^dl.
(7)
The integral is of the form
G˜(x)eikS(x)dx.
(8)
If the phase kS(x) of the integrand changes very rapidly compared to its amplitude G˜(x), most of the contributions average out over one period and vanish. In this case, the stationary phase method is a very good approximation. Only those regions in which the phase is stationary make significant contributions to the integral [13

13. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924). [CrossRef]

]. The integral can be expressed as a series in these regions [7

7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

, 17

17. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part II,” J. Opt. Soc. Am. 52(6), 626–636 (1962). [CrossRef]

]. With the extreme values of S(x) located at xi, the first term of the series is

G˜(x)eikS(x)dxi2π2S(x)x2|x=xiG˜(xi)e14iπeikS(xi)k.
(9)

This approximation diverges if P is located too close to the edge of the geometrical-optical shadow. There, the amplitude 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 2S(x)/x2 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
Y˜b(P)B˜22πkj=01[(1)j(2sΦ2)12sin(s,q)1cos(s,q)e14iπeik(s-q)ssin(θq)]Φ=jπ
(10)
with
s(Φ)=q2+p22qp(cosΦsinθqsinθp+cosθqcosθp),
(11)
s(0)=q2+p22qpcos(θpθq)ands(π)=q2+p22qpcos(θp+θq)
(12)
and thus
2s(Φ)Φ2=qpcosΦsinθqsinθps(Φ)(qpsinΦsinθqsinθp)2s3(Φ),
(13)
2s(Φ)Φ2|Φ=0=qpsinθqsinθps(0)and2s(Φ)Φ2|Φ=π=qpsinθqsinθps(π).
(14)
In addition,
sin(s(0),q)=ps(0)sin(θpθq),sin(s(π),q)=ps(π)sin(θp+θq)
(15)
cos(s(0),q)=qpcos(θpθq)s(0)andcos(s(π),q)=qpcos(θp+θq)s(π).
(16)
Adding the geometrical-optical field (6) results in

Y˜(P)=Y˜b(P)+Y˜g(P)B˜eikq22πkpsinθqqsinθp[sin(θpθq)s(0)q+pcos(θpθq)e14iπeiks(0)s(0)sin(θp+θq)s(π)q+pcos(θp+θq)e14iπeiks(π)s(π)]+Y˜g(P).
(17)

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 (JmaxJmin)/(Jmax+Jmin) 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 2N1 edges passing through the points (qi,0,θqi), can then be calculated according to
J(P)prismscϵ0|Y˜bΦ=0(P)|2+prismscϵ0|Y˜bΦ=π(P)|2+prismscϵ0|Y˜g(P)|2=cϵ0p8πksinθp[i=12N1B˜i2sinθqisin2(θpθqi)qisi(0)(si(0)qi+pcos(θpθqi))2+i=12N1B˜i2sinθqisin2(θp+θqi)qisi(π)(si(π)qi+pcos(θp+θqi))2]+j=1N|Y˜g,j(P)|2.
(18)
The amplitudes B˜i differ from edge to edge, because the distance from the focal point varies with qi, but the lens itself is homogeneously illuminated. In addition, the light is refracted by varying angles, i.e. with different reflection losses. If these losses are ignored, the ratio of the amplitudes to each other is equal to the ratio of the distances between the edges and the focal point:

B˜iB˜0=qiq01cosθqi.
(19)

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 (qkp), but still close to the focal point (pq), the boundary diffraction wave in (18) can be approximnated by a Taylor series in s(Φ), so that
Jb(P)cϵ02πkp3sinθpi=12N1B˜i2sinθqi(1sin2(θpθqi)+1sin2(θp+θqi))=cϵ0πkpri=12N1B˜i2rlifiqifipr2+rli2fi2pz2(pr2rli2fi2pz2)2.
(20)
The variables pr and pz represent the radial and axial components respectively of the vector p in cylindrical coordinates.

Often, flat Fresnel lenses are used with a constant pitch, the separation distance between adjacent prisms. This type of lens, with a radius rL and N prisms of constant pitch Δrl=rL/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 rl2j1=rl2j=jrL/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 fi=f. Then applying the approximation of (19) results in

Jb(P)cϵ0B˜02πkpri=12N1rlifqi3f3pr2+rli2f2pz2(pr2rli2f2pz2)2cϵ0B˜02πkpr[2j=1N1rl,2jfq2j3f3pr2+rl,2j2f2pz2(pr2rl,2j2f2pz2)2+rl2N1q2N13pr2+rl,2N12f2pz2(pr2rl,2N12f2pz2)2]=cϵ0B˜02πkpr[2j=1N1jΔrlf(1+j2Δrl2f2)32pr2+j2Δrl2f2pz2(pr2j2Δrl2f2pz2)2+rLf(1+rL2f2)32pr2+rL2f2pz2(pr2rL2f2pz2)2].
(21)

For small pz, (jΔrl/f)2pz2 can be neglected in comparison to pr2 for positions in the shadow sufficiently far away from the shadow edge, so that the irradiance there is described by the expression
Jb(P)cϵ0B˜02πkpr3[2j=1N1jΔrlf(1+j2Δrl2f2)32+rLf(1+rL2f2)32].
(22)
For lenses with many prisms, i.e. for large values of N, the sum in (22) can be approximated by an integral. As a result, we obtain
Jb(P)cϵ0B˜02πkpr3[212N12jΔrlf(1+j2Δrl2f2)32dj+rLf(1+rL2f2)32]cϵ0B˜02πkpr3{[2f5Δrl(1+j2Δrl2f2)52]12N12+rLf(1+rL2f2)32}2cϵ0B˜02f5πkpr3rL[(1+rL2f2)521]N+O(N1).
(23)
For lenses with a constant pitch, the irradiance in the geometrical-optical shadow is approximately proportional to the number of prisms N or inversely proportional to the pitchΔrl=rL/N, as is to be expected.

The radiative power P(prrZ), 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 Jin=cϵ0B˜02/f2=cϵ0B˜i2/qi2 on the Fresnel lens the incident power is Pin=cϵ0πrL2B˜i2/qi2. Using the more general Eq. (24), the relative loss caused by diffraction thus amounts to
P(prrZ)Pin2πkrL2i=12N1rliqi3fi2rZrZ2rli2fi2pz2=2rZπkrL2i=12N1rlifirZ2rli2fi2pz2(1+rli2fi2)32.
(26)
If a Fresnel lens with constant pitch is assumed, which is thus described by Eq. (25), dividing by the incident radiative power Pin=cϵ0πB˜02tan2ϕ results in
P(prrZ)Pin4Cgeo5πkΔrl(1+tan2ϕ)521tan3ϕ.
(27)
The light loss due to diffraction is proportional to the wavelength and inversely proportional to the pitch of the Fresnel prisms. Equation (27) also shows that when the system is scaled and the Fresnel lens is redesigned afterwards to keep an unchanged Fresnel prism pitch Δrl, the diffraction loss remains constant. The losses increase proportional to the square root of the geometrical concentration factor Cgeo, meaning that the diffraction loss is proportional to rL/rZ. The less obvious dependence on the aperture angle ϕ of the Fresnel lens is plotted in Fig. 5
Fig. 5 Light loss due to diffraction as a function of the f-number f/(2rL)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.
as a function of the f-number f/(2rL)=1/(2tanϕ). For large f-numbers, the diffraction losses according to Eq. (27) are approximately proportional to the f-number and approach the straight line described by

P(prrZ)Pintanϕ12CgeoπkΔrltanϕ.
(28)

5. Diffraction loss for realistic Fresnel lenses under polychromatic illumination

Equation (26) offers the flexibility to treat arbitrary geometrical configurations for Fresnel lenses, including variable pitch Δrl. The focal length of the Fresnel lens fi 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 pz 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, fi 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 zli (Fig. 6).
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.
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 pz of the vector p must be replaced by fizli+pz (Fig. 6). With these changes, Eq. (26) becomes
P(prrZ)Pin2rZπkrL2i=12N1rlifirZ2rli2fi2(fizli+pz)2(1+rli2fi2)32.
(29)
Now, to take account of conical slope facets, the focal length fi of the adjacent slope facet near each diffracting edge is calculated and inserted into the corresponding summand. This calculation method is based on the assumption that Eq. (29), which was derived for an incoming spherical wave, also applies for an incoming conical wave. This is generated by light refraction by a Fresnel prism with a conical slope facet (Fig. 6). Because the incoming wave fronts of the spherical and conical waves are very similar close to the edges, it can be presumed that this assumption is a valid approximation.

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).
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.

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 fi 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].

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

19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

, 20

20. W. H. Carter, “Coherence theory,” in Handbook of Optics M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995, vol. 1).

]. 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

19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

, 21

21. K. K. Sharma, Optics: Principles and Applications (Academic, 2006).

].

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

19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

]). 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
Jb(P)cϵ0πki=12N1[B˜i2rlifi1+rli2fi2τx2+τy2+rli2fi2pz2τx2+τy2(τx2+τy2rli2fi2pz2)2+ϵ1πrSi2Θ(rSi(pxτx)2+(pyτy)2)dτxdτy],
(30)
where Θ(x) is the Heaviside theta function with
Θ(x)={0ifx<01ifx0.
(31)
The normalisation factor, (πrSi2)1, ensures that the intensity incident on the lens does not differ from the coherent case. Because of the poles in the integrand at τx=τy=0 and τx2+τy2(rli2/fi2)pz2=0, the integral would generally not be defined without the term ϵ>0, which has also been inserted. The term is chosen to be so small that it does not significantly affect the integral at other positions than the poles.

The radius rSi 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 ϑin, which describes the apparent size of the light source, and the angle αi 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, rSi is calculated as
rSicosθqi1qitan[arcsin(nsin(αi+arcsin(1nsinϑin)))θqiαi]sinϑin1(fipz)2+rli2cosαi1n2sin2αiϑin.
(32)
This formula represents a lower limit to the estimated radius, because the expansion of the effective target area due to obliquely incident light has been ignored (cosθqi1). This leads to underestimation of the geometrically-optically illuminated area and thus to a conservative estimate of the loss due to diffraction.

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
P(prrZ)cϵ0πki=12N1[B˜i2rlifi1+rli2fi2τx2+τy2+rli2fi2pz2τx2+τy2(τx2+τy2rli2fi2pz2)2+ϵ1πrSi2Θ(rSi(pxτx)2+(pyτy)2)dτxdτy]Θ(px2+py2rZ)dpxdpy.
(33)
Changing the order of integration and summation results in
P(prrZ)cϵ0πki=12N1[B˜i2rlifi1+rli2fi2τx2+τy2+rli2fi2pz2τx2+τy2(τx2+τy2rli2fi2pz2)2+ϵ1πrSi2Θ(rSi(pxτx)2+(pyτy)2)Θ(px2+py2rZ)dpxdpydτxdτy],
(34)
whereby a convolution of the two Heaviside functions with each other can now be found in the second line. This convolution can be solved by taking several geometrical effects into account. The following consideration are limited to the case of rSi<rZ, in which the geometrical image of the sun is smaller than the target area.
Fig. 9 Definition of cases to solve the integral in the second line of Eq. (34). In case a), corresponding to, τr+rs<rz,one of the multiplied theta functions is always equal to zero; in case b), where, τrrs<rz<τr+rsthe integral over the theta functions corresponds to that area of the circle with radius rs which is located outside the circle with radius rz and in case c), where, rz<τrrsthe integral is obtained from the area of the circle with radius rs.
The dark grey regions in Fig. 9 represent the area in which both Heaviside functions are equal to 1. In the light grey regions, only one of the two Heaviside functions is equal to 1. The integral in the second line of Eq. (34) can thus be considered separately for the following three cases.
1πrSi2Θ(rSi(pxτx)2+(pyτy)2)Θ(px2+py2rZ)dpxdpy={0ifτr+rSi<rZ(a)dpxdpyifτrrSi<rZ<τr+rSi(b)1ifrZ<τrrSi(c).
(35)
For the intermediate case shown in Fig. 9(b), the integral can be determined by calculating the geometrical area of the dark grey region. This is done by calculating the areas indicated in Fig. 10.
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 (πφS)rS2 and the difference τrrSsinφSφZrZ between the kite-shaped area bounded by green dashed lines and the circular segment which is bounded by red dots and dashes.
To this purpose, τ is represented in polar coordinates. For τrrSi<rZ<τr+rSi, the following applies:
dpxdpy=1πrSi2((πφSi)rSi2φZirZ2+τrrSisinφSi)=1φSiπφZiπrZ2rSi2+τrsinφSiπrSi.
(36)
The angles
φZi=arccosrZ2+τr2rSi22rZτrandφSi=arccosrSi2+τr2rZ22rSiτr
(37)
are obtained from the cosine rule.

On inserting the three partial solutions a, b, c to the integral into Eq. (34), the expression
P(prrZ)2cϵ0ki=12N1B˜i2rlifi1+rli2fi2[rZrSirZ+rSiτr2+rli2fi2pz2(τr2rli2fi2pz2)2+ϵ(φSiπφZiπrZ2rSi2+τrsinφSiπrSi)dτr+rZrSiτr2+rli2fi2pz2(τr2rli2fi2pz2)2+ϵdτr]
(38)
is obtained. Assuming that the edge of the geometrical shadow is located within the target area, such that rZ>(rli2/fi2)pz2+rSi for all i, the introduced ϵ term can be omitted again and the second integral in the last line of Eq. (38) can be solved. Analogously to Eq. (24), we obtain
P(prrZ)2cϵ0ki=12N1B˜i2rlifi1+rli2fi2rZrSi(rZrSi)2rli2fi2pz2+2cϵ0ki=12N1B˜i2rlifi1+rli2fi2rZrSirZ+rSiτr2+rli2fi2pz2(τr2rli2fi2pz2)2(φSiπφZiπrZ2rSi2+τrsinφSiπrSi)dτr.
(39)
No simple analytical solution can be found for the remaining integral in the second term. However, the integrand can be approximated by a Taylor series of the terms in the last line of Eq. (39), such that the integral can then be solved. This approach results in very long formulae and will not be pursued further here. Instead, it was decided to take the alternative approach of solving the integral numerically for each summand.

As the first step, again this power loss should be expressed as a ratio relative to the incident power. Equation (30) had already been scaled such that the irradiance on the diffracting structure is equal to that in the coherent case, namely Jin=cϵ0B˜i2/qi2, such that the total incident illumination power is equal to Pin=cϵ0πrL2B˜i2/qi2=cϵ0πrL2B˜i2/(fi2+rli2). The relative loss due to diffraction is thus

P(prrZ)Pin2πkrL2i=12N1rlifi(1+rli2fi2)32rZrSi(rZrSi)2rli2fi2pz2+2πkrL2i=12N1rlifi(1+rli2fi2)32rZrSirZ+rSiτr2+rli2fi2pz2(τr2rli2fi2pz2)2(φSiπφZiπrZ2rSi2+τrsinφSiπrSi)dτr.
(40)

In addition, by inserting zli and replacing pz by fizli+pz (Fig. 6), all cases can be included for which Eq. (29) is valid. As a result, the most general form of the formula to calculate the diffraction loss is
P(prrZ)Pin2πkrL2i=12N1rlifi(1+rli2fi2)32[rZrSi(rZrSi)2rli2fi2(fizli+pz)2+rZrSirZ+rSiτr2+rli2fi2(fizli+pz)2(τr2rli2fi2(fizli+pz)2)2(φSiπφZiπrZ2rSi2+τrsinφSiπrSi)dτr].
(41)
The radii rSi must be calculated with the formula
rSi(zlipz)2+rli2cosα1n2sin2αϑin
(42)
adapted from Eq. (32). When rSi0 for all i, Eq. (29) is obtained as a special case. In addition, if zli=fi for all i edges, Eq. (26) results.

The loss according to Eq. (41) for partially coherent illumination has already been plotted for one example in Fig. 7 and Fig. 8, using the spatial coherence of the solar disc (dotted line). As expected, the diffraction loss for partially coherent illumination increases in comparison to the loss for coherent illumination. The closer the edge of the geometrical-optical shadow is to the edge of the target area, the greater is the difference between the effect of coherent and partially coherent illumination.

7. Conclusion

Most concentrating photovoltaic systems with large geometrical concentration factors use Fresnel lenses to focus direct sunlight onto small, high-efficiency solar cells. The segmentation of the lens into many annular prisms introduces diffraction by the edges of these prisms. This causes some of the light to be spilled outside the designated target area. The target area might be the entrance aperture of a secondary optical element or a solar cell surface.

Three equations have been presented that allow calculation of the losses caused by this light spillage under different assumptions. Equation (27) allows calculation of diffraction losses for a Fresnel lens under monochromatic coherent illumination. It neglects any geometrical-optical expansion of the focal spot, e.g. due to chromatic aberration or due to conical Fresnel prism facets. This equation gives a simple and conservative estimate of the diffraction loss under realistic conditions. Equation (29) may be used for Fresnel lenses with a wider variety of surface configurations and correctly includes chromatic aberration in the calculation. Finally, Eq. (41) adds an approximation for the diffraction loss under illumination by partially coherent light, e.g. light that has the same angular divergence as the solar disc.

Calculation examples using realistic values for concentrator Fresnel lenses for concentrating photovoltaic systems show that diffraction losses may be greater than 2% and therefore very significant. Diffraction losses are roughly proportional to the wavelength (Fig. 8), the f-number of the Fresnel lens (Fig. 5) and the square root of the geometrical concentration factor (defined with respect to the target area which might be the entrance aperture of a secondary optical element or a solar cell surface). The diffraction losses are inversely proportional to the pitch of the Fresnel lens (Fig. 7).

A comparison of these results to computer simulations and measurements will be published as a separate paper in near future.

Acknowledgments

The authors would like to thank Andreas Gombert for support with the scanning electron microscope analysis (Fig. 1), Armin Bösch for preparing the analysed sample, and Helen Rose Wilson for translating the draft version of this paper into English.

References and links

1.

M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. 21(1), 827–837 (2013). [CrossRef]

2.

G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev. 13(9), 2645–2652 (2009). [CrossRef]

3.

J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE 193, 63–69 (1979). [CrossRef]

4.

A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik 2(4), 52–55 (2007). [CrossRef]

5.

F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt. 49(12), 2339–2346 (2010). [CrossRef] [PubMed]

6.

F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE 7725, 772509 (2010). [CrossRef]

7.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

8.

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik 254(4), 663–695 (1883). [CrossRef]

9.

G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin 2, 641–669 (1882).

10.

W. Hoßfeld, Tageslichtsteuerung mit prismatischen Mikrostrukturen im Übergangsbereich von diffraktiver und geometrischer Optik (doctoral thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2004).

11.

T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond. 92(0), 12–48 (1802). [CrossRef]

12.

G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math. 16, 21–48 (1888).

13.

A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik 378(5-6), 339–364 (1924). [CrossRef]

14.

A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik 358(12), 257–278 (1917). [CrossRef]

15.

P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana 37(6), 457–488 (1991). [CrossRef]

16.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part I,” J. Opt. Soc. Am. 52(6), 615–622 (1962). [CrossRef]

17.

K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part II,” J. Opt. Soc. Am. 52(6), 626–636 (1962). [CrossRef]

18.

S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater. 29(11), 1481–1490 (2007). [CrossRef]

19.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).

20.

W. H. Carter, “Coherence theory,” in Handbook of Optics M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995, vol. 1).

21.

K. K. Sharma, Optics: Principles and Applications (Academic, 2006).

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(220.1770) Optical design and fabrication : Concentrators

ToC Category:
Solar Concentrators

History
Original Manuscript: January 13, 2014
Revised Manuscript: February 14, 2014
Manuscript Accepted: February 19, 2014
Published: March 26, 2014

Citation
Thorsten Hornung and Peter Nitz, "Light diffraction by concentrator Fresnel lenses," Opt. Express 22, A686-A704 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-S3-A686


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References

  1. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl.21(1), 827–837 (2013). [CrossRef]
  2. G. Zubi, J. L. Bernal-Agustin, and G. V. Fracastoro, “High concentration photovoltaic systems applying IIIV cells,” Renew. Sustain. Energy Rev.13(9), 2645–2652 (2009). [CrossRef]
  3. J. R. Egger, “Use Of Fresnel Lenses In Optical Systems: Some Advantages And Limitations,” Proc. SPIE193, 63–69 (1979). [CrossRef]
  4. A. Davis and F. Kühnlenz, “Optical design using Fresnel lenses,” Optik Photonik2(4), 52–55 (2007). [CrossRef]
  5. F. Duerr, Y. Meuret, and H. Thienpont, “Miniaturization of Fresnel lenses for solar concentration: a quantitative investigation,” Appl. Opt.49(12), 2339–2346 (2010). [CrossRef] [PubMed]
  6. F. Duerr, Y. Meuret, and H. Thienpont, “Down scaling of micro-structured Fresnel lenses for solar concentration: a quantitative investigation,” Proc. SPIE7725, 772509 (2010). [CrossRef]
  7. M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).
  8. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Annalen der Physik254(4), 663–695 (1883). [CrossRef]
  9. G. Kirchhoff, “Zur Theorie der Lichtstrahlen,” Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin2, 641–669 (1882).
  10. W. Hoßfeld, Tageslichtsteuerung mit prismatischen Mikrostrukturen im Übergangsbereich von diffraktiver und geometrischer Optik (doctoral thesis, Albert-Ludwigs-Universität Freiburg im Breisgau, 2004).
  11. T. Young, “On the theory of light and colours,” Philos. Trans. R. Soc. Lond.92(0), 12–48 (1802). [CrossRef]
  12. G. Maggi, “Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo,” Ann. Math.16, 21–48 (1888).
  13. A. Rubinowicz, “Zur Kirchhoffschen Beugungstheorie,” Annalen der Physik378(5-6), 339–364 (1924). [CrossRef]
  14. A. Rubinowicz, “Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen,” Annalen der Physik358(12), 257–278 (1917). [CrossRef]
  15. P. B. S. Kumar and G. S. Ranganath, “Geometrical theory of diffraction,” Pramana37(6), 457–488 (1991). [CrossRef]
  16. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part I,” J. Opt. Soc. Am.52(6), 615–622 (1962). [CrossRef]
  17. K. Miyamoto and E. Wolf, “Generalization of the Maggi-Rubinowicz Theory of the Boundary Diffraction Wave - Part II,” J. Opt. Soc. Am.52(6), 626–636 (1962). [CrossRef]
  18. S. N. Kasarova, N. G. Sultanova, C. D. Ivanov, and I. D. Nikolov, “Analysis of the dispersion of optical plastic materials,” Opt. Mater.29(11), 1481–1490 (2007). [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  20. W. H. Carter, “Coherence theory,” in Handbook of Optics M. Bass, E. W. van Stryland, D. R. Williams, and W. L. Wolfe, eds. (McGraw-Hill, 1995, vol. 1).
  21. K. K. Sharma, Optics: Principles and Applications (Academic, 2006).

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