OSA's Digital Library

Energy Express

Energy Express

  • Editor: Christian Seassal
  • Vol. 21, Iss. S4 — Jul. 1, 2013
  • pp: A616–A630
« Show journal navigation

First-principle calculation of solar cell efficiency under incoherent illumination

Michaël Sarrazin, Aline Herman, and Olivier Deparis  »View Author Affiliations


Optics Express, Vol. 21, Issue S4, pp. A616-A630 (2013)
http://dx.doi.org/10.1364/OE.21.00A616


View Full Text Article

Acrobat PDF (2201 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

Because of the temporal incoherence of sunlight, solar cells efficiency should depend on the degree of coherence of the incident light. However, numerical computation methods, which are used to optimize these devices, fundamentally consider fully coherent light. Hereafter, we show that the incoherent efficiency of solar cells can be easily analytically calculated. The incoherent efficiency is simply derived from the coherent one thanks to a convolution product with a function characterizing the incoherent light. Our approach is neither heuristic nor empiric but is deduced from first-principle, i.e. Maxwell’s equations. Usually, in order to reproduce the incoherent behavior, statistical methods requiring a high number of numerical simulations are used. With our method, such approaches are not required. Our results are compared with those from previous works and good agreement is found.

© 2013 OSA

1. Introduction

Fig. 1 (a) Sketch of a light trapping diffracting structure (ZI < z < ZIII) with many diffraction orders. Domains I and III are respectively the incident and emergence media. (b) Model of field amplitude Fin(t) for the incoherent monochromatic incident light. m(t) is the corresponding modulation which characterizes the random phase switching at different times τi. τc is the coherence time, equal to the average value <τi>.

It must be noted that, as far as the propagation of the electric field of incident optical radiation is concerned, a solar cell, whatever the complexity of its structure is, behaves as a linear system (thanks to Maxwell equations), which is fully characterized by its scattering matrix (see Appendix A). However, as soon as energy fluxes need to be calculated, linearity does not stand anymore since the intensity (Poynting vector flux) is proportional to the electric field squared, i.e. IE · E* = |E|2. The way the square modulus of the fields enters into the calculation of the absorption (reflectance, transmittance) of the system is far from being trivial (see, for example, the case of RCWA method, in Appendix B). In any case, there exists no transfer function relating linearly the solar cell absorption to the incident Poynting vector flux. The fact that the relevant quantities (absorbed flux, photocurrent,...) do not obey the superposition principle (since I ∝ |E|2) prevents us from applying common principles of the linear response theory [17

17. R. G. Brown and P. Y. C Hwang, Introduction to random signals and applied kalman filtering (John Willey and Sons, 1992).

]. In particular, the incoherent response cannot be treated in the same way as the coherent one, a fact that is known for long time in optics [18

18. M. Born and E. Wolf, Principles of optics (Cambridge University Press, 1999).

], at least for simple cases, such as thin films.

2. Incoherent absorption

It is important to point out that the incoherent absorption Aincoh(ω) is physically different from the coherent one Acoh(ω). Indeed, Acoh(ω) can be considered as an intrinsic property of the solar cell which only depends on its geometry and on its constitutive materials. By contrast, Aincoh(ω) reflects the effective measured response of the solar cell while it interacts with its environment, i.e. for instance when the photocurrent is produced and flows in a closed electrical circuit.

In a numerical computation approach, the absorption coefficient A can be obtained from at least two ways. In FDTD for instance, Eq. (2) can be used as such, since maps of the electromagnetic field are directly computed. In RCWA, reflection R and transmission T are calculated and A is deduced from A = 1 − RT. In the following, we consider a RCWA formalism of the light propagation (see appendix B), and we use it to determine the effective incoherent absorption Aincoh(ω) (see Appendix A). The main results of Appendix A are summarized hereafter.

It can then be shown (see Appendix A) that the incoherent absorption Aincoh(ω) results from the convolution product, noted ★, between the coherent absorption Acoh(ω) and the incoherence function I(ω):
Aincoh(ω)=I(ω)Acoh(ω).
(7)
This extremely simple formula is easy to use in practice. But its demonstration is not obvious and is therefore detailed in Appendix A. Eq. (7) can also be used to compute incoherent reflection and transmission spectra from their coherent counterparts (see appendix A).

3. Numerical application

In order to illustrate the usefulness of the convolution formula, i.e. Eq. (7), we use it for the calculation of the incoherent absorption of a 500 nm-thick crystalline silicon (c-Si) slab either planar (see Fig. 2(a)) or corrugated (see Fig. 2(b)).

Fig. 2 Simulation of the absorption spectra of planar and corrugated 500 nm-thick c-Si slabs. The coherent spectra were obtained using RCWA and the incoherent ones using our convolution formula, Eq. (7).(a) Absorption spectra of the planar slab according to various coherence times. (b) Absorption spectra of the corrugated slab according to various coherence times. Inset: corrugated structure ; p = 500 nm, t = 500 nm, h = 300 nm, D = 450 nm, d = 320 nm. The structure follows a super-Gaussian profile with m = 3 (see Ref. [8]).

The first step is the numerical calculation of the coherent absorption using the RCWA method (see Fig. 2, red lines). In the RCWA simulations, we use unpolarized light at normal incidence, where the (x, z) plane is the plane of incidence (polar and azimuthal angles equal to 0) (see Fig. 1(a)). The permittivities of materials are taken from the literature [28

28. E.D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

]. The second step is the convolution of the coherent absorption spectrum with the incoherence function I(ω). This step leads to the incoherent absorption spectra.

In the case of a planar slab, computed spectra are compared with the approximate incoherent absorption spectrum obtained from a standard analytical expression of the absorption using Fresnel equations [18

18. M. Born and E. Wolf, Principles of optics (Cambridge University Press, 1999).

], but in which propagation phases are roughly set to zero to mimic incoherence (orange curve in Fig. 2(a)). We note that this kind of analytical expression must be considered carefully in the present case. Indeed, such an expression considers the effect of incoherence on light propagation only and does not consider the incoherent response of the permittivities of the materials, contrary to our present approach using convolution (see appendix A). In spite of that, both spectra (orange and black curves in Fig. 2(a)) are quite similar, which validates our method (in our method, the incoherence limit is approached by taking τc = 5 fs). The weak discrepancies are due to the absence of incoherent permittivity response in the Fresnel approach.

Fig. 3 Comparison between the coherent spectra obtained with the RCWA method (blue lines) and incoherent spectra with various coherence times, for planar and grating structures, defined in [26]. (a) Reflectance spectrum of an unpatterned c-Si layer (225 nm) deposited on a 75 nm-thick Au film on a glass substrate. (b) Reflectance spectrum of the whole grating structure.

Coming back to Fig. 2, the absorption spectra of a three-dimensional corrugated 500 nm-thick c-Si slab were calculated using the same method as for the planar slab. The corrugation follows a super-Gaussian profile such as the one defined in [8

8. A. Herman, C. Trompoukis, V. Depauw, O. El Daif, and O. Deparis, “Influence of the pattern shape on the efficiency of front-side periodically patterned ultrathin crystalline silicon solar cells,” J. Appl. Phys. 112, 113107 (2012) [CrossRef] .

] (m = 3). The period of the corrugation is 500 nm and its height is 300 nm. The absorption spectra according to various coherence times are shown in Fig. 2(b). We notice that the value of the coherence time affects the absorption spectrum. The highest coherence time (here 95 fs) leads to the strongest oscillations in the spectrum. Physically, these oscillations are due to Fabry-Perot resonances (in the thin-slab) and to guided mode resonances (enabled by the corrugation). With the decrease of τc, these oscillations are expected to be smoothened which is actually observed when τc < 20 fs. This smoothing effect is indeed observed when measuring absorption spectra with an incoherent source.

We finally calculated the photocurent of the corrugated slab, according to Eq. (1) (see Table 1). The aim was to quantify the effect of the coherence time on the value of the photocurrent. The photocurrent fluctuates as the coherence time varies. As expected, the incoherent light behaviour affects both the absorption spectrum and the photocurrent.

Table 1. Computed photocurrents related to the corrugated device of Fig. 2(b) for various coherence times. Photocurrent was integrated over the spectral range: 300 nm – 1200 nm.

table-icon
View This Table

4. Conclusion

We demonstrated that reflection, transmission, and absorption spectra of a photonic structure illuminated with incoherent light can be easily calculated. The only input is the knowledge of their coherent counterparts and of the coherence time of the source (used to determine the incoherence function). The incoherent response is simply the convolution between a function accounting for the incoherent source and the coherent response. This result is theoretically demonstrated from first principle and is confirmed by numerical simulations. In analogy with signal processing theory, it can be interpreted as the consequence, in the frequency domain, of the temporal filtering associated with the intrinsic incoherent modulation. The proposed method allows to significantly simplify the description of incoherent phenomena which are regarded as key problem in solar cells optimization. The inherent simplicity of our method allows to minimize the computational time cost and the algorithm complexity. Reflection, transmission, and absorption spectra, but also the photocurrent, were shown to depend on the coherence time.

A. Appendix: Demonstration of the convolution formula

A.1. Scattering matrix as response function

The following demonstration is based on the formalism of the Rigorous Coupling Wave Analysis (RCWA) [11

11. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]. This analysis takes into account the periodicity of the device and describes the permittivity (ε) through Fourier series. The electromagnetic field is then described by Bloch waves also expanded in Fourier series. In this formalism, the Maxwell’s equations take the form of a matricial first-order differential equation in the z variable. The z axis is perpendicular to the plane (x,y) where the permittivity is periodic (Fig. 1(a)). The essence of the method is to solve this equation but is not the topic of the present article.

Reflected and transmitted field amplitudes are linked to the incident field amplitudes by the use of the scattering matrix (S) which is calculated by solving Maxwell equations using Fourier series (see Appendix B). Let us define Fscat as the scattered field and Fin as the incident field, such that the associated supervectors are:
|Fscat=[N¯III+X¯III+N¯IX¯I],|Fin=[N¯I+X¯I+N¯IIIX¯III].
(A.1)

A.2. R, T, A coefficients for a coherent monochromatic incident wave

Let us first consider a coherent monochromatic incident wave:
|Fin(t)=|Fin(0)eiωct
(A.10)
with an (optical) angular frequency ωc. The response |FX (t)〉 of the device is given by [17

17. R. G. Brown and P. Y. C Hwang, Introduction to random signals and applied kalman filtering (John Willey and Sons, 1992).

]
|FX(t)=SX(t)|Fin(t)
(A.11)
where ★ denotes the convolution product. The response can then be written explicitely as:
|FX(t)=SX(tt)eiωctdt|Fin(0)=eiωctSX(ωc)|Fin(0),
(A.12)
where
SX(ωc)=SX(t)eiωctdt
(A.13)
is the Fourier transform of SX scattering matrix defined in (A.9). We note that SX(ωc)=SX*(ωc) since SX (t) must be real.

From (A.12), we set
|FX(t)=|FX(0)eiωct,
(A.14)
with
|FX(0)=SX(ωc)|Fin(0).
(A.15)
We remind here that X stands either for the reflected (R) or the transmitted (T) wave. Reflection Rcoh and transmission Tcoh coefficients associated with a coherent process can be obtained thanks to:
Rcoh(ωc)=JRJin=FR(0)|FR(0)Jin=Fin(0)|SR(ωc)SR(ωc)|Fin(0)Jin,
(A.16)
and
Tcoh(ωc)=JTJin=FT(0)|FT(0)Jin=Fin(0)|ST(ωc)ST(ωc)|Fin(0)Jin.
(A.17)
The absorption Acoh is then simply given by the energy conservation law: i.e. Acoh = 1 − RcohTcoh. The incident Ponyting flux Jin(t), i.e.
Jin(t)=σ12ε0cεIcosθFin(t)|Fin(t)=σ12ε0cεIcosθFin(0)|Fin(0),
(A.18)
turns out to be time independent (see Eq. (B.13) in Appendix B). Therefore, the time-averaged incident flux Jin is identical to Jin(t)
Jin=1TT/2T/2Jin(t)dt=σ12ε0cεIcosθFin(0)|Fin(0)Jin(t).
(A.19)
This result is only valid as far as the incident wave is coherent.

A.3. R, T, A coefficients for an incoherent monochromatic incident wave

Let us now consider an incoherent quasi-monochromatic incident wave:
|Fin(t)=|Fin(0)m(t)eiωct.
(A.20)
By quasi monochromatic, we mean that the spectral line has a finite though narrow spectral width. As explained in the present article, the function m(t) is a modulation which ensures the incoherent behaviour. On average, m(t) has a coherence time equal to τc (see Fig. 1(b)). Thanks to the Wiener-Khinchine theorem, the autocorrelation function of a random process has a spectral decomposition given by the power spectrum of that process [17

17. R. G. Brown and P. Y. C Hwang, Introduction to random signals and applied kalman filtering (John Willey and Sons, 1992).

]. The function m(t) is then equally characterized by its spectral density:
D(ω)=|m(ω)|2,
(A.21)
where
m(ω)=m(t)eiωtdt
(A.22)
is the Fourier transform of m(t).

The device response is then calculated by the same convolution product as in (A.11):
|FX(t)=SX(t)|Fin(t)=|FX(0)(t)eiωct
(A.23)
where X again stands for the reflected (R) or transmitted (T) wave but now
|FX(0)(t)=UX(t)|Fin(0),
(A.24)
with
UX(t)=m(t)SX(t)eiωct.
(A.25)

A.3.1. Effective response approach

Upon incoherent illumination, the device with its structure and combination of various materials undergoes a set of many incident wave trains randomly dephased with respect to each other. These wave trains have the same pulsation ωc and an average duration equal to the coherence time τc. The coherence time plays a key role while light propagates since it prevents constructive interferences from taking place. But it also changes the way matter responds to light. Indeed, a specific medium mainly interacts with light thanks to electronic and ionic motions of its components. As a consequence, the response of a medium to an incident electromagnetic wave is not instantaneous and occurs according to the dielectric relaxation time. If the incoherence time τc is short enough in comparison with the relaxation time, the medium response is not coherent. In this case, the medium response is not simply given by the value of the permittivity at ω = ωc.

In this way, we must consider that the full response of the medium is a time averaged value of the response recorded during a typical time about Tc such that Tcτc. The time Tc is the sampling time interval [26

26. W. Lee, S.Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length,” Opt. Express 20, A941–A953 (2012) [CrossRef] [PubMed] .

] which reflects the non-instantaneity of any measurement process. For instance, a spectrophotometric measurement is characterized by such a sampling time. Likewise, the photocurrent of a solar cell is a measure of the response of the solar cell to the incident light. In this context, the sampling time corresponds to the recombination/generation time of charged carriers (carrier lifetime). For instance, in silicon, carrier lifetime ranges from 0.1 ns to 1 ms according to the doping density [29

29. B.O. Seraphin, Solar Energy Conversion, Solid-State Physics Aspects (Springer-Verlag, 1979) [CrossRef]

]. These delays are very large compared to the coherence time of sunlight, which is about 3 fs [30

30. E. Hecht, Optics (Pearson Education, 2002)

]. Therefore, the condition Tcτc is well verified in solar cells.

The flux of the Poynting vector for an incoherent incident light JX,incoh is given by [26

26. W. Lee, S.Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length,” Opt. Express 20, A941–A953 (2012) [CrossRef] [PubMed] .

]:
JX,incoh=1Tc0TcJX(t)dt.
(A.26)
The integral can be easily expressed according to the Fourier transform of JX (t), noted JX (ω):
JX,incoh=JX(ω)(12πsin(ωTc/2)ωTc/2)dω.
(A.27)
We note that
limTc+Tc2πsin(ωTc/2)ωTc/2=δ(ω),
(A.28)
where δ is the Dirac distribution. Since Tcτc, JX (ω) should have a frequency distribution spreading over Δω ∼ 1/τc around zero frequency, quite comparable to D(ω), the spectral density of the random process. We can therefore assume that the angular frequencies ω for which JX (ω) is significantly different from zero are such that Tc ≫ 1/ω, i.e. Tc → + roughly speaking. We can then fairly consider the following substitution: (1/2π)sin(ωTc/2)/(ωTc/2) → (1/Tc)δ(ω). As a result, we get:
JX,incoh1TcJX(ω=0).
(A.29)

A.3.2. Incoherent response

Since, using (A.24), we have († sign denotes the adjoint matrix, i.e. the conjugate transpose of the matrix)
JX(t)=FX(0)(t)|FX(0)(t)=Fin(0)|UX(t)UX(t)|Fin(0),
(A.30)
we can then deduce from Eqs. (A.25), (A.29) and (A.30):
JX,incoh=1TcFin(0)|IX(ω=0)|Fin(0)
(A.31)
where
IX(ω)=UX(t)UX(t)eiωtdt=12πUX(ω)UX(ω).
(A.32)
From (A.25), we deduce that UX (ω) = m(ω)SX (ω + ωc) and UX(ω)=m(ω)SXt(ωωc), where t sign denotes the transpose of the matrix. Then, (A.32) becomes:
IX(ω)=12πm(ω)SX(ωωc)m(ω)SX(ω+ωc),
(A.33)
from which, by writing explicitly the convolution product, one deduces
IX(ω=0)=12πm(ω)SXt(ωcω)m(ω)SX(ω+ωc)dω.
(A.34)
Since m(−ω′) = m*(ω′) and SXt(ωcω)=SX(ω+ωc), one obtains:
IX(ω=0)=12π|m(ωcω)|2SX(ω)SX(ω)dω.
(A.35)
Then, using Eqs. (A.21) and (A.31), we can deduce
JX,incoh=12πTcD(ωcω)Fin(0)|SX(ω)SX(ω)|Fin(0)dω.
(A.36)

A.3.3. Incident flux

Let us estimate the flux of the incident incoherent wave Jin,incoh. For a non dispersive incident medium, we get:
Jin,incoh(t)=σ12ε0cεIcosθFin(t)|Fin(t)=σ12ε0cεIcosθ|m(t)|2Fin(0)|Fin(0).
(A.37)
Following the same argument as in section A.3.1, at a given angular frequency ωc, we assume that the effective flux impinging on the device Jin,incoh is given by the time average:
Jin,incoh=1Tc0TcJin,incoh(t)dt=Jin1Tc0Tc|m(t)|2dt
(A.38)
where Jin is the incident flux for the coherent wave, see Eq. (A.19). Using the same calculation as in section A.3.1 for estimating the time average, one deduces:
1Tc0Tc|m(t)|2dt=1Tc12πm*(ω)m(ω)|ω=0=12πTcD(ω)dω.
(A.39)
As a consequence, one gets:
Jin,incoh=Jin12πTcD(ω)dω.
(A.40)

A.3.4. R, T, A coefficients

B. Appendix: Coupled waves analysis (RCWA) formalism

The reader will find more details about the present approach in references [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]. We consider as an example a planar dielectric layer with a bidimensional periodic array described by the dielectric function:
ε(ρ,ω)=gεg(ω)eigρ.
(B.1)
In the layer, the dielectric function does not depend on the normal coordinate z, which is used to define the layer thickness, i.e. the layer extends from z = ZI to z = ZIII (Fig. 1(a)). One sets ρ = x1a1 + x2a2, according to the unit cell basis (a1, a2). In such a system, Bloch’s theorem leads to the following electromagnetic field expression in the layer [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
[EH]=g[Eg(z)Hg(z)]ei(k+g)ρeiωt.
(B.2)
It can then be easily shown that Maxwell equations can be recasted in the form of a first-order differential equation system [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

],
ddz[E¯//(z)H¯//(z)]=[0AA˜0][E¯//(z)H¯//(z)],
(B.3)
where Ē// and // are the electric and magnetic field components parallel to the layer surface. In the layer, one can then write [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
[E¯//(zI)H¯//(zI)]=exp{[0AA˜0](zIzIII)}[E¯//(zIII)H¯//(zIII)].
(B.4)

Let us define the following unit vectors [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
μI,g=kI,g,zεIωck+g|k+g|,
(B.5)
ηg=k+g|k+g|×ez,
(B.6)
χI,g±=μI,g+|k+g|εIωcez.
(B.7)
One can then expand the electric and magnetic fields (parallel components) in Fourier series [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
EI(ρ,z)=g[NI,g+ηgeikI,g,z(zzI)+NI,gηgeikI,g,z(zzI)+XI,g+χI,g+eikI,g,z(zzI)+XI,gχI,geikI,g,z(zzI)]ei(k+g)ρ
(B.8)
and
HI(ρ,z)=εIcμ0g[NI,g+χI,g+eikI,g,z(zzI)NI,gχI,geikI,g,z(zzI)+XI,g+ηgeikI,g,z(zzI)+XI,gηgeikI,g,z(zzI)]ei(k+g)ρ.
(B.9)
The subscripts I and III stand for the incident medium and emergence medium respectively, and the superscripts + and − denote the positive and negative direction along the z axis for backward (+) and forward (−) field propagation. For each vector g of the reciprocal lattice, NI,g and XI,g are the s and p polarization amplitudes of the reflected field, respectively, and NIII,g+ and XIII,g+, those of the transmitted field. Similarly, NI,0+ and XI,0+ define the s and p polarization amplitudes of the incident field, respectively.

We can then define a transfer matrix T[12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
[N¯I+X¯I+N¯IX¯I]=[T++T+T+T][N¯III+X¯III+N¯IIIX¯III].
(B.10)
Alternatively, we can express the scattered field against the incident field and we define a scattering matrix S[12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
[N¯III+X¯III+N¯IX¯I]=[S++S+S+S][N¯I+X¯I+N¯IIIX¯III].
(B.11)
The flux of the Poynting vector through the unit cell is [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
J=σ12Re(E×H)ezdS.
(B.12)
We get then [12

12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

14

14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

]:
JI+=σ2μ0ωgkI,g,z[|NI,g+|2+|XI,g+|2]Θ(εI(ω)ω2c2|k+g|2)
(B.13)
for the incident flux in medium I,
JIII+=σ2μ0ωgkIII,g,z[|NIII,g+|2+|XIII,g+|2]Θ(εIII(ω)ω2c2|k+g|2)
(B.14)
for the transmitted flux in medium III, and
JI=σ2μ0ωgkI,g,z[|NI,g|2+|XI,g|2]Θ(εI(ω)ω2c2|k+g|2)
(B.15)
for the reflected flux in medium I. In (B.13B.15), Θ(x) is the Heaviside function: i.e. Θ(x) = 1 if x > 0 and Θ(x) = 0 otherwise.

Acknowledgments

M.S. is supported by the Cleanoptic project (Development of super-hydrophobic anti-reflective coatings for solar glass panels / Convention No.1117317) of the Greenomat program of the Walloon Region (Belgium). O.D. acknowledges the support of FP7 EU-project No.309127 PhotoNVoltaics (Nanophotonics for ultra-thin crystalline silicon photovoltaics). This research used resources of the Interuniversity Scientific Computing Facility located at the University of Namur, Belgium, which is supported by the F.R.S.-FNRS under the convention No.2.4617.07.

References and links

1.

L. Tsakalakos, Nanotechnology for Photovoltaics (CRC, 2010) [CrossRef] .

2.

J. Nelson, The Physics of Solar Cells (Imperial College, 2003) [CrossRef] .

3.

M. Zeman, R.A.C.M.M. van Swaaij, J. Metselaar, and R.E.I. Schropp, “Optical modeling of a-Si:H solar cells with rough interfaces: Effect of back contact and interface roughness,” J. Appl. Phys. 88, 6436–6443 (2000) [CrossRef] .

4.

P. Campbell and M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys. 62, 243–249 (1987) [CrossRef] .

5.

E. Yablonovitch and G. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE 29, 300–305 (1982).

6.

O. Deparis, J.P. Vigneron, O. Agustsson, and D. Decroupet, “Optimization of photonics for corrugated thin-film solar cells,” J. Appl. Phys. 106, 094505 (2009) [CrossRef] .

7.

J. Gjessing, A.S. Sudbø, and E.S. Marstein, “Comparison of periodic light-trapping structures in thin crystalline silicon solar cells,” J. Appl. Phys. 110, 033104 (2011) [CrossRef] .

8.

A. Herman, C. Trompoukis, V. Depauw, O. El Daif, and O. Deparis, “Influence of the pattern shape on the efficiency of front-side periodically patterned ultrathin crystalline silicon solar cells,” J. Appl. Phys. 112, 113107 (2012) [CrossRef] .

9.

K.S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, 1993).

10.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

11.

M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981) [CrossRef] .

12.

M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B 67, 085415 (2003) [CrossRef] .

13.

J.P. Vigneron and V. Lousse, “Variation of a photonic crystal color with the Miller indices of the exposed surface,” Proc. SPIE 6128, 61281G (2006) [CrossRef] .

14.

J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy 61, 21–27 (1995) [CrossRef] .

15.

A. Jin and J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells 92, 1689–1696 (2008) [CrossRef] .

16.

L. Zhao, Y. Zuo, C. Zhou, H. Li, W. Diao, and W. Wang, “A highly efficient light-trapping structure for thin-film silicon solar cells,” Sol. Energy 84, 110–115 (2010) [CrossRef] .

17.

R. G. Brown and P. Y. C Hwang, Introduction to random signals and applied kalman filtering (John Willey and Sons, 1992).

18.

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

19.

J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D 33, 3139–3145 (2000) [CrossRef] .

20.

C.L. Mitsas and D.I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt. 34, 1678–1683 (1995) [CrossRef] [PubMed] .

21.

M.C. Troparevsky, A.S. Sabau, A.R. Lupini, and Z. Zhang, “Transfer-matrix formalism for the calculation of optical response in multilayer systems: from coherent to incoherent interference,” Opt. Express 18, 24715–24721 (2010) [CrossRef] [PubMed] .

22.

A. Niv, M. Gharghi, C. Gladden, O. D. Miller, and X. Zhang, “Near-Field Electromagnetic Theory for Thin Solar Cells,” Phys. Rev. Lett. 109, 138701 (2012) [CrossRef] [PubMed] .

23.

C.C. Katsidis and D.I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41, 3978–3987 (2002) [CrossRef] [PubMed] .

24.

J.S.C. Prentice, “Optical genration rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D 32, 2146–2150 (1999) [CrossRef] .

25.

E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. 44, 7532–7539 (2005) [CrossRef] [PubMed] .

26.

W. Lee, S.Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length,” Opt. Express 20, A941–A953 (2012) [CrossRef] [PubMed] .

27.

R. Santbergen, A. H.M. Smets, and M. Zeman, “Optical model for multilayer structures with coherent, partly coherent and incoherent layers,” Opt. Express 21, A262–A267 (2013) [CrossRef] [PubMed] .

28.

E.D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

29.

B.O. Seraphin, Solar Energy Conversion, Solid-State Physics Aspects (Springer-Verlag, 1979) [CrossRef]

30.

E. Hecht, Optics (Pearson Education, 2002)

OCIS Codes
(030.1640) Coherence and statistical optics : Coherence
(040.5350) Detectors : Photovoltaic
(300.6170) Spectroscopy : Spectra
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:
Photovoltaics

History
Original Manuscript: March 14, 2013
Revised Manuscript: May 8, 2013
Manuscript Accepted: May 9, 2013
Published: May 23, 2013

Citation
Michaël Sarrazin, Aline Herman, and Olivier Deparis, "First-principle calculation of solar cell efficiency under incoherent illumination," Opt. Express 21, A616-A630 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-S4-A616


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. L. Tsakalakos, Nanotechnology for Photovoltaics (CRC, 2010). [CrossRef]
  2. J. Nelson, The Physics of Solar Cells (Imperial College, 2003). [CrossRef]
  3. M. Zeman, R.A.C.M.M. van Swaaij, J. Metselaar, and R.E.I. Schropp, “Optical modeling of a-Si:H solar cells with rough interfaces: Effect of back contact and interface roughness,” J. Appl. Phys.88, 6436–6443 (2000). [CrossRef]
  4. P. Campbell and M. Green, “Light trapping properties of pyramidally textured surfaces,” J. Appl. Phys.62, 243–249 (1987). [CrossRef]
  5. E. Yablonovitch and G. Cody, “Intensity enhancement in textured optical sheets for solar cells,” IEEE29, 300–305 (1982).
  6. O. Deparis, J.P. Vigneron, O. Agustsson, and D. Decroupet, “Optimization of photonics for corrugated thin-film solar cells,” J. Appl. Phys.106, 094505 (2009). [CrossRef]
  7. J. Gjessing, A.S. Sudbø, and E.S. Marstein, “Comparison of periodic light-trapping structures in thin crystalline silicon solar cells,” J. Appl. Phys.110, 033104 (2011). [CrossRef]
  8. A. Herman, C. Trompoukis, V. Depauw, O. El Daif, and O. Deparis, “Influence of the pattern shape on the efficiency of front-side periodically patterned ultrathin crystalline silicon solar cells,” J. Appl. Phys.112, 113107 (2012). [CrossRef]
  9. K.S. Kunz and R.J. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC, 1993).
  10. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  11. M. Moharam and T. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am.71, 811–818 (1981). [CrossRef]
  12. M. Sarrazin, J. P. Vigneron, and J. M. Vigoureux, “Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes,” Phys. Rev. B67, 085415 (2003). [CrossRef]
  13. J.P. Vigneron and V. Lousse, “Variation of a photonic crystal color with the Miller indices of the exposed surface,” Proc. SPIE6128, 61281G (2006). [CrossRef]
  14. J. P. Vigneron, F. Forati, D. André, A. Castiaux, I. Derycke, and A. Dereux, “Theory of electromagnetic energy transfer in three-dimensional structures,” Ultramicroscopy61, 21–27 (1995). [CrossRef]
  15. A. Jin and J. Phillips, “Optimization of random diffraction gratings in thin-film solar cells using genetic algorithms,” Sol. Energy Mater. Sol. Cells92, 1689–1696 (2008). [CrossRef]
  16. L. Zhao, Y. Zuo, C. Zhou, H. Li, W. Diao, and W. Wang, “A highly efficient light-trapping structure for thin-film silicon solar cells,” Sol. Energy84, 110–115 (2010). [CrossRef]
  17. R. G. Brown and P. Y. C Hwang, Introduction to random signals and applied kalman filtering (John Willey and Sons, 1992).
  18. M. Born and E. Wolf, Principles of optics (Cambridge University Press, 1999).
  19. J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D33, 3139–3145 (2000). [CrossRef]
  20. C.L. Mitsas and D.I. Siapkas, “Generalized matrix method for analysis of coherent and incoherent reflectance and transmittance of multilayer structures with rough surfaces, interfaces, and finite substrates,” Appl. Opt.34, 1678–1683 (1995). [CrossRef] [PubMed]
  21. M.C. Troparevsky, A.S. Sabau, A.R. Lupini, and Z. Zhang, “Transfer-matrix formalism for the calculation of optical response in multilayer systems: from coherent to incoherent interference,” Opt. Express18, 24715–24721 (2010). [CrossRef] [PubMed]
  22. A. Niv, M. Gharghi, C. Gladden, O. D. Miller, and X. Zhang, “Near-Field Electromagnetic Theory for Thin Solar Cells,” Phys. Rev. Lett.109, 138701 (2012). [CrossRef] [PubMed]
  23. C.C. Katsidis and D.I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt.41, 3978–3987 (2002). [CrossRef] [PubMed]
  24. J.S.C. Prentice, “Optical genration rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D32, 2146–2150 (1999). [CrossRef]
  25. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt.44, 7532–7539 (2005). [CrossRef] [PubMed]
  26. W. Lee, S.Y. Lee, J. Kim, S. C. Kim, and B. Lee, “A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length,” Opt. Express20, A941–A953 (2012). [CrossRef] [PubMed]
  27. R. Santbergen, A. H.M. Smets, and M. Zeman, “Optical model for multilayer structures with coherent, partly coherent and incoherent layers,” Opt. Express21, A262–A267 (2013). [CrossRef] [PubMed]
  28. E.D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).
  29. B.O. Seraphin, Solar Energy Conversion, Solid-State Physics Aspects (Springer-Verlag, 1979) [CrossRef]
  30. E. Hecht, Optics (Pearson Education, 2002)

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1 Fig. 2 Fig. 3
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited