## A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length |

Optics Express, Vol. 20, Issue S6, pp. A941-A953 (2012)

http://dx.doi.org/10.1364/OE.20.00A941

Acrobat PDF (974 KB)

### Abstract

We propose a method for calculating the optical response to partially-coherent light based on the coherence length. Using a Fourier transform of a randomly-generated partially-coherent wave, we demonstrate that the reflectance, transmittance, and absorption with the incidence of partially-coherent light can be calculated from the Poynting vector of the incident coherent light. We also demonstrate that the statistical field distribution of partially-coherent light can be obtained from the proposed method using a rigorous coupled wave analysis. The optical characteristics of grating structures in photovoltaic devices are analyzed as a function of coherence length. The method is capable of providing a general procedure for analyzing the incoherent optical characteristics of thick layers or nano particles in photovoltaic devices with the incidence of partially-coherent light.

© 2012 OSA

## 1. Introduction

1. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl. **20**(1), 12–20 (2012). [CrossRef]

2. M. A. Green, “Silicon photovoltaic modules: a brief history of the first 50 years,” Prog. Photovolt. Res. Appl. **13**(5), 447–455 (2005). [CrossRef]

6. S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev. **107**(4), 1324–1338 (2007). [CrossRef] [PubMed]

7. M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett. **12**(6), 2894–2900 (2012). [CrossRef] [PubMed]

10. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**(3), 205–213 (2010). [CrossRef] [PubMed]

11. 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**(19), 3978–3987 (2002). [CrossRef] [PubMed]

11. 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**(19), 3978–3987 (2002). [CrossRef] [PubMed]

16. J. S. C. Prentice, “Optical generation rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D Appl. Phys. **32**(17), 2146–2150 (1999). [CrossRef]

11. 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**(19), 3978–3987 (2002). [CrossRef] [PubMed]

12. E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. **44**(35), 7532–7539 (2005). [CrossRef] [PubMed]

13. 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**(24), 24715–24721 (2010). [CrossRef] [PubMed]

17. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**(5), 1068 (1995). [CrossRef]

21. N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys. **110**(11), 114506 (2011). [CrossRef]

## 2. Numerical model

*f*) and sinusoidally oscillates like a monochromatic coherent wave during phase-maintaining time. The phase-maintaining length, which is not a fixed value, follows a Gaussian distribution in this paper. The coherence length (

*L*), which is the average value of the phase-maintaining length, is given bywhere

_{c}*W*is the spectral width of the partially-coherent source and

_{s}*c*is the speed of light in a vacuum. The variance in the phase-maintaining length distribution is determined for each simulation case. The spectral width of the source can be appropriately changed in order to analyze the optical characteristics of incident light with various coherence lengths. The difference between this pseudo source and the coherent source is that the phase profile of the pseudo source changes abruptly for a random value between 0 and 2π at the end of the phase-maintaining time. This abrupt phase shift results in spectral broadening in the frequency domain. After one phase shift, the source maintains the same temporal frequency during the next phase-maintaining interval, which is also randomly determined from the same distribution in phase-maintaining time. As shown in Fig. 1(a), the

*τ*,

_{c1}*τ*, and

_{c2}*τ*represent the first, second, and third phase-maintaining interval of a partially-coherent wave, respectively.

_{c3}*T*) is sufficient for representing the characteristics of the partially-coherent source. By sampling the pseudo wave with the number of samples (

*N*), the real part of the optical field (

*x*[

_{re}*n*]) then can be discretely expressed aswhere

*f*is the frequency of the source,

*n*is the sampling sequence number,

*N*is an odd number, and

*i*-th phase-maintaining interval. The term

*x*[

_{re}*n*], the pseudo source can be decomposed into the sum of coherent sources in the frequency domain. The

*X*is the weighted value of frequency component, which is defined asSince

_{k}*x*[

_{re}*n*] is the discrete set of purely real input,

*X*is equal to the complex conjugate of

_{k}*X*. Thus, the respective real and imaginary parts (

_{-k}*x*[

_{im}*n*]) of complex partially-coherent source are expressed as The entire expression for a partially-coherent wave (

*x*[

*n*]) then can be expressed asStarting with the real form of a pseudo wave in time domain, the complex form of the pseudo source can be modeled as the sum of coherent sources. The pseudo source in the Fourier domain is illustrated in Fig. 1(b). Since only data within the specific range is sufficient to represent the characteristics of the partially-coherent light and this can increase the speed of the calculation, we find the maximum component in the Fourier domain, using data only within the effective spectral width (

*W*), and data that remains outside of it can be discarded, as shown in Fig. 1(b). Since the process of discarding data reduces the total energy of the wave, the sum of the energy should be normalized to satisfy the conservation of energy law. This modified source is the partially-coherent source, which we are interested in modeling in the Fourier domain. By using an inverse discrete Fourier transform (IDFT), this partial coherent source in the Fourier domain can be converted back into the wave in the time domain as shown in Fig. 1(c). The modified wave in the time domain is not exactly equal to the pseudo source in Fig. 1(a). However, it is sufficient to represent the incoherent optical characteristics of the partially-coherent source itself. Therefore, a randomly-generated partially-coherent source, which has a specific frequency both in the time and frequency domains can be obtained.

_{eff}*E*and

_{m}*H*represent the electric and magnetic fields with specific frequency, and

_{m}*ω*is the frequency of wave. The

_{m}*z*component of instantaneous Poynting vector (

*S*) is given byThe exponential components in Eq. (8) do not vanish when the frequency of the electric and magnetic fields are different. Due to the oscillation terms in Eq. (8), we integrate the instantaneous Poynting vector over the sampling time interval (

_{z}*T*) and obtain the time-average Poynting vector (

*P*) aswhere

_{z}*P*is the time-average Poynting vector calculated from the wave with a single frequency component like a monochromatic wave. The

_{z,m}*P*is expressed aswhere

_{z,m}*S*is the

_{z,m}*z*component of an instantaneous Poynting vector with a monochromatic wave. Since

*ω*is not an arbitrary number and is chosen from the sampling time interval (

_{n}*ω*), the sampling time interval (

_{n}= 2πf_{n}= 2πn/T*T*) is a multiple of each period (

*T*). Thus, the oscillation terms, which appear in Eq. (8), vanish in Eq. (9) and only non-oscillation terms remain in the time-average Poynting vector. Therefore, it is possible to easily calculate the optical characteristics of reflectance and transmittance from the sum of the time-average Poynting vector with coherent waves.

_{n}= 1/f_{n}= T/n## 3. Simulation results

*T*) and number (

*N*) are 10

^{4}and 2 × 10

^{3}fs, respectively. The sampling time was selected so as to include at least 500 periods of incident wavelength. Moreover, it can include at least about 20 times the coherence packet as the coherence length changes. The effective spectral width is 100 THz, in which most of the energy of the original source is contained. The degree of partial coherence is represented by the coherence time (

*T*), which can be expressed asHere,

_{c}*L*is obtained from Eq. (1) and the spectral width can be properly selected to model the characteristics of the source itself. The solid blue curve shown in Figs. 2(b) and 2(c) represents the coherent case, while the green, red, azure, violet, and brown curves represent the partially-coherent case with coherence times of 95, 41, 20, 10, and 5 fs, respectively. Throughout this paper, the same coherence times and line colors are used to represent the spectral response of partially-coherent wave. The solid black curve represents incoherent limit, which is calculated from GTMM. The GTMM is used for comparison with our method only in simple layer structures since the GTMM cannot calculate complex geometry that contains grating structures or nanoparticles. Those results can be used to compare the difference between incoherent simulation results and partially-coherent results. As the coherence time decreases from 95 to 5 fs, the calculated reflection and absorption spectra based on the proposed method become closer to those based on the GTMM, which shows the accuracy of the proposed numerical method.

_{c}10. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. **9**(3), 205–213 (2010). [CrossRef] [PubMed]

**41**(19), 3978–3987 (2002). [CrossRef] [PubMed]

13. 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**(24), 24715–24721 (2010). [CrossRef] [PubMed]

## 4. Conclusion

## Acknowledgment

## References and links

1. | M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl. |

2. | M. A. Green, “Silicon photovoltaic modules: a brief history of the first 50 years,” Prog. Photovolt. Res. Appl. |

3. | G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO |

4. | I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe |

5. | T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci. |

6. | S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev. |

7. | M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett. |

8. | P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt. |

9. | S. C. Kim and I. Sohn, “Simulation of energy conversion efficiency of a solar cell with gratings,” J. Opt. Soc. Kor. |

10. | H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. |

11. | C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. |

12. | E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt. |

13. | 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 |

14. | S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys. |

15. | J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys. |

16. | J. S. C. Prentice, “Optical generation rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D Appl. Phys. |

17. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

18. | M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A |

19. | H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A |

20. | H. Kim, J. Park, and B. Lee, |

21. | N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys. |

22. | E. D. Palik, ed., |

23. | M. Bass, ed., |

24. | M. Bass, ed., |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(040.5350) Detectors : Photovoltaic

(240.6680) Optics at surfaces : Surface plasmons

(310.0310) Thin films : Thin films

**ToC Category:**

Photovoltaics

**History**

Original Manuscript: August 27, 2012

Revised Manuscript: October 12, 2012

Manuscript Accepted: October 12, 2012

Published: October 17, 2012

**Citation**

Wooyoung Lee, Seung-Yeol Lee, Jungho Kim, Sung Chul Kim, and Byoungho Lee, "A numerical analysis of the effect of partially-coherent light in photovoltaic devices considering coherence length," Opt. Express **20**, A941-A953 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-S6-A941

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### References

- M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 39),” Prog. Photovolt. Res. Appl.20(1), 12–20 (2012). [CrossRef]
- M. A. Green, “Silicon photovoltaic modules: a brief history of the first 50 years,” Prog. Photovolt. Res. Appl.13(5), 447–455 (2005). [CrossRef]
- G. K. Mor, O. K. Varghese, M. Paulose, K. Shankar, and C. A. Grimes, “A review on highly ordered, vertically oriented TiO2 nanotube arrays: fabrication, material properties, and solar energy applications,” Sol. Energy Mater. Sol. Cells90(14), 2011–2075 (2006). [CrossRef]
- I. Repins, M. A. Contreras, B. Egaas, C. DeHart, J. Scharf, C. L. Perkins, B. To, and R. Noufi, “19.9%-efficient ZnO/CdS/CuInGaSe2 solar cell with 81.2% fill factor,” Prog. Photovolt. Res. Appl.16(3), 235–239 (2008). [CrossRef]
- T. Ameri, G. Dennler, C. Lungenschmied, and C. J. Brabec, “Organic tandem solar cells: a review,” Energy Environ. Sci.2(4), 347–363 (2009). [CrossRef]
- S. Günes, H. Neugebauer, and N. S. Sariciftci, “Conjugated polymer-based organic solar cells,” Chem. Rev.107(4), 1324–1338 (2007). [CrossRef] [PubMed]
- M. G. Deceglie, V. E. Ferry, A. P. Alivisatos, and H. A. Atwater, “Design of nanostructured solar cells using coupled optical and electrical modeling,” Nano Lett.12(6), 2894–2900 (2012). [CrossRef] [PubMed]
- P. Spinelli, V. E. Ferry, J. van de Groep, M. van Lare, M. A. Verschuuren, R. E. I. Schropp, H. A. Atwater, and A. Polman, “Plasmonic light trapping in thin-film Si solar cells,” J. Opt.14(2), 024002 (2012). [CrossRef]
- S. C. Kim and I. Sohn, “Simulation of energy conversion efficiency of a solar cell with gratings,” J. Opt. Soc. Kor.14(2), 142–145 (2010). [CrossRef]
- H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater.9(3), 205–213 (2010). [CrossRef] [PubMed]
- 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(19), 3978–3987 (2002). [CrossRef] [PubMed]
- E. Centurioni, “Generalized matrix method for calculation of internal light energy flux in mixed coherent and incoherent multilayers,” Appl. Opt.44(35), 7532–7539 (2005). [CrossRef] [PubMed]
- 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(24), 24715–24721 (2010). [CrossRef] [PubMed]
- S. Jung, K.-Y. Kim, Y.-I. Lee, J.-H. Youn, H.-T. Moon, J. Jang, and J. Kim, “Optical modeling and analysis of organic solar cells with coherent multilayers and Incoherent glass substrate using generalized transfer matrix method,” Jpn. J. Appl. Phys.50(12), 122301 (2011). [CrossRef]
- J. S. C. Prentice, “Coherent, partially coherent and incoherent light absorption in thin-film multilayer structures,” J. Phys. D Appl. Phys.33(24), 3139–3145 (2000). [CrossRef]
- J. S. C. Prentice, “Optical generation rate of electron-hole pairs in multilayer thin-film photovoltaic cells,” J. Phys. D Appl. Phys.32(17), 2146–2150 (1999). [CrossRef]
- M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A12(5), 1068 (1995). [CrossRef]
- M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A12(5), 1077 (1995). [CrossRef]
- H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A24(8), 2313–2327 (2007). [CrossRef] [PubMed]
- H. Kim, J. Park, and B. Lee, Fourier Modal Method and Its Applications in Computational Nanophotonics (CRC, 2012).
- N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys.110(11), 114506 (2011). [CrossRef]
- E. D. Palik, ed., Handbook of Optical Constants of Solids (Academic, 1985).
- M. Bass, ed., Handbook of Optics, 2nd ed., vol. 2 (McGraw-Hill, 1994).
- M. Bass, ed., Handbook of Optics, 3rd ed., vol. 4 (McGraw-Hill, 2009).

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