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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 Optical Society of America
1. Introduction
Photovoltaic devices have gained increasing attention, due to their potential usage for solving energy problems. Because recent photovoltaic technologies have low energy conversion efficiencies compared to other technologies [1], replacing existing energy sources such as fossil-fuel and nuclear-fuel is not urgently important. In order to solve this problem, various structures and fabrication methods such as crystalline silicon, thin-films, and organic solar cell technology have been proposed by several research groups [2–6].
Many recent studies for improving light absorption in photovoltaic devices have concluded that metallic nanoparticles or corrugated metallic films, supporting surface plasmon scattering and coupling, can be used to enhance light absorption in the active region [7–10]. Those light trapping structures can significantly increase energy conversion efficiency at a specific frequency but only if the structure is appropriately designed. Optimization of the geometry, with a view to improving the absorption in the active region has been studied by many groups using a variety of numerical methods, which are based on the assumption that the incident light is perfectly coherent, analogous to a monochromatic laser. However, radiated light from the sun is not perfectly coherent because radiated waves from the sun have a finite coherent length and a finite spectral width. Moreover, this incoherent characteristic of incident light can significantly change the optical characteristic of photovoltaic devices [11]. In the absorption power spectrum, narrow oscillation peaks that are numerically calculated under the assumption of coherent light may decrease or not be observable in practical experiments. Thus, improving numerical methods for calculating the relation between the optical responses and incoherent characteristics of a source that has a finite coherent length becomes a necessity [11–16].
One of the representative methods for incoherent light modeling is the generalized transfer-matrix method (GTMM) [11, 12] that usually focuses on the incoherent optical response related to the macroscopic roughness of an interface or impurities in layers. Thus, the method is suitable for calculations related to mixed coherent-incoherent multilayers, but the structure is restricted to a stack of uniform multilayers. Although phase information related to the incident wave is lost because this method uses the square of complex amplitudes of reflection and transmission coefficients. As a result, it can be used to calculate the incoherent optical response of a perfectly incoherent layer. Even though the fundamental principle resulting in incoherence is different, a similar phase discontinuity is generated from both GTMM and our method. Thus, GTMM results for a perfectly incoherent layer can be used as a reference to compare the cases of a perfectly incoherent layer and perfectly incoherent light. Another proposed method is a modified transfer-matrix method, which is based on the random phase shift at the interface of each layer [13]. Although this method can be used to solve optical response due to partially-coherent light, it is necessary to define the degree of incoherence, which is an empirical fitting parameter with no physical meaning. None of the above methods consider the effect of incident light, which has a finite coherent length. A few methods [15, 16] consider the effect of a partially coherent source with normal incidence by fitting a Gaussian function to the entire solar spectrum.
In this report, we present a method for modeling a partially-coherent source with a finite coherence length. The degree of incoherence can be physically modeled by the relation between the coherence length and the spectral width. Thus, the method makes it possible to take into account the characteristics of a partially-coherent source with respect to the coherence length and spectral width. After generating a source with a finite coherence time, we decompose the incoherent source into the sum of coherent sources using a discrete Fourier transform. Consequently, it is possible to obtain the partially-coherent optical response of the structure as the superposition of coherent optical responses. Unlike the previous methods, it is not necessary to calculate the square of the Fresnel coefficients with phase information lost [11, 12] or to define the degree of partial coherency in each layer [13]. Moreover, we expect that it can be combined with the previous method to consider both spatial and temporal incoherent factors. Furthermore, the statistical field distribution with the incidence of partially-coherent light can be calculated based on the proposed method using a rigorous coupled wave analysis (RCWA) [17–20]. Since RCWA has an advantage for calculating optical response with oblique light, our method can be extended to simulate an optical response with partially coherent light as a function of the incident angle [21]. The relation between optical characteristics and partially-coherent light is analyzed in photovoltaic devices with grating structures.
2. Numerical model
A simple diagram for generating a partially-coherent wave is shown in Fig. 1 . Figure 1(a) shows the unit-amplitude pseudo source in the time domain, which maintains a constant temporal frequency (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 (Lc), which is the average value of the phase-maintaining length, is given by
where Ws is the spectral width of the partially-coherent source and 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 τc1, τc2, and τc3 represent the first, second, and third phase-maintaining interval of a partially-coherent wave, respectively.
Fig. 1 (a) Temporal behavior of a pseudo source in the time domain, (b) discrete Fourier transform of the pseudo source in Fourier domain, and (c) the modified partially-coherent source that originates from the data within the effective spectral width in time domain. The red diamonds represent data within the effective spectral width and the black diamonds represent the data outside of it.
The same process is repeated until the sampling time interval (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 (xre[n]) then can be discretely expressed as
where f is the frequency of the source, n is the sampling sequence number, N is an odd number, and is the phase shift at the end of the i-th phase-maintaining interval. The term in Eq. (2) is determined from the relation given bywhere is the initial phase and is the random phase that is maintained during the phase-maintaining interval. By using discrete Fourier transform of xre[n], the pseudo source can be decomposed into the sum of coherent sources in the frequency domain. The Xk is the weighted value of frequency component, which is defined asSince xre[n] is the discrete set of purely real input, Xk is equal to the complex conjugate of X-k. Thus, the respective real and imaginary parts (xim[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 (Weff), 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.Moreover, the partially-coherent simulation result can be easily calculated by superposing the simulation results of the coherent sources, which can be obtained by using existing numerical simulation methods such as RCWA, the finite element method (FEM), and the finite-difference time-domain method (FDTD). This superposition principle can be justified as follows:
The electric and magnetic fields in Cartesian coordinates can be expressed as
where Em and Hm represent the electric and magnetic fields with specific frequency, and ωm is the frequency of wave. The z component of instantaneous Poynting vector (Sz) 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 (T) and obtain the time-average Poynting vector (Pz) aswhere Pz,m is the time-average Poynting vector calculated from the wave with a single frequency component like a monochromatic wave. The Pz,m is expressed aswhere Sz,m is the z component of an instantaneous Poynting vector with a monochromatic wave. Since ωn is not an arbitrary number and is chosen from the sampling time interval (ωn = 2πfn = 2πn/T), the sampling time interval (T) is a multiple of each period (Tn = 1/fn = T/n). 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.3. Simulation results
Our method proposed in the previous section was used to calculate the optical characteristics of various structures with the incidence of partially-coherent lights. In the numerical analysis, the length of the sampling time is related to the wavelength of the incident light and the coherence time. The relative scale between the wavelength of the incident light and the length of the sampling time is more important than the length of the sampling time itself. As the wavelength of incident light increases, the sampling time also must increase to represent a partially coherent wave with little randomness, which is caused by not including the sufficient value for the wavelength. Moreover, the sampling time must increase to include a sufficient number of phase-maintaining intervals as the coherence time increases. Furthermore, this increases the number of iterations required to average the results. After selecting a proper sampling time, it is necessary to select the sampling number. In order to maintain an appropriate sampling interval (T/N) to reconstruct with minimal distortion in the discrete Fourier transform (DFT) process as the sampling time (T) increases, the sampling number (N) must increase proportionally. This increases the computation time for the DFT process as well as the computation time for coherent cases using RCWA. Thus, those factors have to be appropriately considered to obtain an efficient computation time. The values of the sampling parameters are then sufficient to reproduce the original wave without distortions when a Fourier transform and an inverse Fourier transform are used. Finally, a proper effective spectral width needs to be selected to reduce the overall computation time. This effective spectral width is also related to the computation time and the distortion in the DFT process since using only data from within the effective spectral width and discarding data that remains outside of it can result in the loss of the original incoherent characteristics. Thus, the modified form of the pseudo source must also be taken into consideration as to whether it is sufficient to represent the characteristics of the partially-coherent source. After satisfying those criteria, a single result is calculated from the method proposed in the previous section. Since simulation results of this method can include random noise, the results of the 100 runs are averaged to minimize noise. This repeating process does not distort the characteristics of the partially-coherent source.
Figure 2(a) shows a Si thin film on a gold layer. The 225-nm thick Si film is bonded to the 75-nm thick gold layer. A 1-mm thick glass is used as a substrate at the bottom. The refractive index and extinction coefficient of Si and Au were obtained from the literature [22]. The calculated results for the reflectance and absorption spectra with the normal incidence of TM-polarized light are shown in Figs. 2(b) and 2(c), respectively. In all simulations presented in this section, the sampling time (T) and number (N) are 104 and 2 × 103 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 (Tc), which can be expressed as
Here, Lc 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.Fig. 2 (a) Schematic diagram of Si thin film. Calculated (b) reflectance and (c) absorption spectra with coherent and partially-coherent lights.
As shown in Fig. 2(c), absorption peaks can be seen near 430, 480, 565, and, 735 nm whereas absorption dips appear near 415, 455, 520, and 650 nm. The narrow resonance peaks and dips in the solid blue curve originate from Fabry-Perot resonance of the Si layer. The absorption and reflectance spectra are dominantly affected by the thickness of the Si layer in the coherent case.
An interesting feature is that the absorption near the wavelengths of the peak resonance decreases with decreasing coherence length, whereas the absorption near the wavelength of the dip resonance increases with decreasing coherence length. This simulation result indicates that the narrow oscillations, which occur in conventional simulations based on the assumption of coherent light, do not occur or decrease with the incidence of partially-coherent light. Since a partially-coherent wave, having a finite coherence length, contains numerous frequency components, it cannot create precisely the same resonance as the coherent wave. Thus, the degree of resonance behavior in a layer structure decreases as shown in Figs. 2(b) and 2(c). Moreover, the sharpness of the resonance peaks and dips also decrease with decreasing coherence length. The calculated spectra for reflection and absorption approaches the result for the incoherent limit calculated based on GTMM as the coherence length of the partially-coherent light decreases.
Our proposed method can be extended to calculate the optical response with partially coherent light as a function of incident angle. Since a representative advantage of RCWA is that it is capable of calculating an optical response with oblique incidence in nano structures, our method, when combined with RCWA, can easily be used to calculate an optical response with oblique incidence. The same structure shown in Fig. 2(a) was chosen to analyze an optical response with obliquely incident light. Figures 3(a) , 3(b), and 3(c) show absorption spectra for incident angles of 30°, 60°, and 70°. In each case, the optical responses with partially- and perfectly-incoherent incident light are analyzed. As the incident angle increases, multiple reflections or interference effects in the layer decrease. Moreover, the difference between the partially coherent and coherent cases decreases with increasing incident angle. The reason for this is that the magnitude of the interference effect decreases in oblique cases: The incoherent characteristics of the source result in a smooth interference effect.
Figure 4(a) shows a Si thin film with the corrugated back metal contact, which can be used as a backscatter in photovoltaic devices. The 50-nm-height and 400-nm-period gold grating is attached to a 50-nm gold layer. The fill factor of this gold grating is 0.5. The 250-nm thick Si film is placed on the surface of the grating structure. This geometry is well known for serving as a basic structure in photovoltaic devices [10]. The calculated results for the reflectance and absorption spectra in Si layer with the normal incidence of TM-polarized light are shown in Figs. 4(b) and 4(c), respectively. Since the geometry shown in Fig. 4(a) is similar to that shown in Fig. 2(a), similar absorption and reflectance spectra appear in the coherent case, except for one resonance, which appears near 665 nm. This new resonance originates from the effect of the gold grating. Since the gold grating and nanoparticles can create a sharp resonance at a specific frequency, this can enhance the absorption in the active region at a specific frequency. This is the reason why a gold grating is used as a back contactor in this structure. However, the resonance peak near 665 nm contributes not to the
Fig. 4 (a) Schematic diagram of Si thin film with grating structures. Calculated (b) reflectance and (c) absorption spectra with coherent and partially-coherent lights.
sharp enhancement around 665 nm but to broadband enhancement between 600 and 700 nm as the coherence length decreases. However, the resonance peak near 665 nm contributes not to the sharp enhancement around 665 nm but to broadband enhancement between 600 and 700 nm as the coherence length decreases. The grating structure functions to improve overall absorption efficiency around the broadband in the partially-coherent case, as shown in Fig. 4(c). In addition, the optical resonances, which originate from the grating structure, decrease more rapidly as the coherence length decreases. The multiple oscillations originating from the Fabry-Perot resonance decrease significantly as the coherence time drops below 10 fs. On the other hand, both the 665-nm resonance induced by the grating and the 597-nm resonance affected by the grating structure disappear when the coherence time drops below 20 fs. Since the grating period is longer than the thickness of the Si layer, the oscillations related to the grating structures need a longer optical path to produce a resonance than oscillations related to Fabry-Perot resonance.
Moreover, the incident wave requires a longer coherence time and length to create a strong resonance without phase mismatch in the case of a long optical path. This indicates that the decay of oscillation with respect to the decrease of the coherence is related to the effective optical path for resonance. The simulation results shown in Figs. 4(b) and 4(c) indicate that the narrow oscillation peak, which is designed to enhance absorption on the assumption of coherent light, shows different optical characteristics in the case of partially-coherent light. Thus, it would be expected that our method is capable of providing a more realistic optical design of optical components such as photovoltaic devices and incoherent holography by considering the partially-coherent nature of sunlight.
Another competitive advantage of the proposed method is that it can be used to calculate the spatial field distribution of partially-coherent light without the loss of phase information, which occurs in previous methods [11–13]. Instantaneous field distribution can be easily calculated from the sum of the coherent field distributions. Moreover, the statistical amplitude distribution with a partially-coherent field can be calculated from the sum of the amplitude distributions with the coherent field and does not require a time-average process. It can be directly calculated from the sum of the amplitude distributions with the coherent light. The same geometry shown in Fig. 4(a) was used to compare the spatial distribution of an electric field between the coherent and partially-coherent cases.
The statistical amplitude distribution of the electric field is illustrated in Fig. 5 . Figures 5(a) and 5(b) are calculated at a wavelength of 525 nm, which corresponds to the resonance peak of the absorption spectra in Fig. 4(c). On the other hand, Figs. 5(c) and 5(d) are obtained at a wavelength of 560 nm, which is matched with the resonance dip of the absorption spectra. Figures 5(a) and 5(c) represent the case where the incident wave is coherent. Figures 5(b) and 5(d) correspond to the case where the incident wave is partially-coherent and the wave has a coherence time of 5 fs. As shown in Figs. 5(a) and 5(b), the overall intensity of the electric field in the absorption layer is higher in the case of a coherent wave than for a partially-coherent wave. In contrast, the overall intensity in the absorption layer is higher in the partially-coherent wave than in the coherent wave when the wavelength is located near the resonance dip, as shown in Figs. 5(c) and 5(d).
Fig. 5 The amplitude distributions of the electric fields (a) with 525 nm (coherent), (b) with 525 nm (partially-coherent), (c) with 560 nm (coherent), and (d) with 560 nm (partially-coherent). The coherence time of partially-coherent wave is 5 fs.
Our method was extended to the analysis of the optical characteristics of photovoltaic devices as a function of coherence length. The typical geometry of a copper indium gallium (di)selenide (CIGS) photovoltaic device is shown in Fig. 6(a) . Since RCWA has an advantage in calculating thick layers, it is an appropriate example to show how our method can be extended to real photovoltaic devices that include a layer of thick glass. A 150-nm CIGS is attached on a 400-nm layer of molybdenum (Mo). The 50-nm thick zinc oxide (ZnO) and 20-nm thick zinc sulfide (ZnS) layers are attached to the CIGS layer. The 1000-nm thick boron doped zinc oxide (BZO), which functions as a transparent conducting layer, is attached on the top of the ZnO. A 1-mm thick glass is used as a substrate at the bottom. The refractive index and extinction coefficients of Mo, ZnO, and ZnS are taken from the literature [22–24]. The refractive index and extinction coefficients of BZO and CIGS are taken from experiment data. The absorption spectra in the CIGS active layer with the normal incidence of TM-polarized light are shown in Fig. 6(b). Since the calculated absorption spectra have many dips or peaks, as shown in Fig. 6(b), the absorption spectra and corresponding total absorption vary as the coherence length changes. This is a quite different result compared with the simulation based on coherent assumption.
Fig. 6 (a) Device structure of the CIGS solar cell. (b) Calculated absorption spectra with coherent and partially-coherent lights.
The CIGS solar cell with a grating is shown in Fig. 7(a) . The period of the grating is 500 nm and the fill factor is 0.5. Absorption spectra in the CIGS active region are shown in Fig. 7(b). The dashed black curve represents the incoherent limit of the structure without a grating, which was calculated based on the GTMM. The new resonance, which is related to the Mo grating, appears at near 650 nm. Moreover, the absorption near 750 nm is enhanced by the grating structure. The calculated spectra of the absorption approach the absorption spectra for the case of a geometry without a grating when the wavelength is below 650 nm. In addition, the grating structure creates a broad absorption band between 650 and 800 nm compared with the structure without a grating. The results show that the enhancement effects of the grating structure change as the coherence length decreases.
Fig. 7 (a) Device structure of the CIGS solar cell with grating structure. (b) Calculated absorption spectra with the coherent and partially-coherent lights.
The spectral current densities of a CIGS solar cell and a CIGS solar cell with a grating structure are shown in Figs. 8(a) and (b) . The spectral current density between 450 and 700 nm is calculated when the normal incident light is TM polarized. We assume that the solar spectrum has AM 1.5 conditions, all absorbed energy contributes to generating electrons, and all of the generated electrons are converted into current without any loss. The results show that the spectral current density variation in a photovoltaic device with varying coherence length is significantly different from the coherent result.
Fig. 8 (a) Spectral current density of CIGS solar cell. (b) Spectral current density of CIGS solar cell with grating structure.
The total calculated current density between 450 and 800 nm is shown in Table 1 . The CIGS with grating structures show a lower efficiency than CIGS without a grating when the coherence time exceeds 49 fs. However, a CIGS with grating structures show a higher efficiency when the coherence time is smaller than 49 fs. The results also demonstrate that our method can provide a more accurate simulation for designing photovoltaic devices with various coherence lengths.
Table 1. Total Current Density (mA/cm2) in the Absorption Layer of CIGS Solar Cells
4. Conclusion
We present a method for calculating the optical response of partially-coherent light. In order to model a partially-coherent wave, we use the random generation method in the time domain considering the spectral width and the coherence length. By changing the coherence length, the characteristics of source itself can be considered. Unlike previous methods, the proposed method uses the physical parameter of coherence length to define the degree of incoherence. Thus, the simulation results based on this method have more physical meaning than that of previous methods. The accuracy of the proposed method was verified by comparison with simulation results based on GTMM. Moreover, the proposed method, when combined with various numerical methods such as RCWA, FEM, and FDTD, can be applied to the analysis of complex geometries, which contain grating structures or nano particles. Since a complex geometry with a partially-coherent source can be easily analyzed using the method, it would be expected that the proposed method is capable of producing more precise modeling results compared to previous methods with the assumption of coherent light. Since there are few ways for calculating partially-coherent characteristics based on RCWA, we expect that the proposed method represents a unique route to calculating the optical response in a partially-coherent wave when RCWA is used. Furthermore, it can be combined with various numerical methods such as RCWA, FEM, and the FDTD method. In addition, instantaneous and statistical field distributions can be calculated using the method. We demonstrate that this method can be used to obtain more accurate simulation results in photovoltaic devices by considering the partially-coherent nature of sun light. We also expect that this method can be applied not only to photovoltaic devices but can also be used in various applications such as incoherent holography and organic light emitting diodes.
Acknowledgment
The authors acknowledge the support of the National Research Foundation and the Ministry of Education, Science and Technology of Korea through the Creative Research Initiative Program (Active Plasmonics Application Systems).
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