Fundamental limits for path lengths of light in isotropic absorbers are calculated. The method of calculation is based on accounting for occupied states in optical phase space. Light trapping techniques, such as scattering or diffraction, are represented by the way how the available states are occupied. One finding of the presented investigation is that the path length limit is independent of the light trapping mechanism and only depends on the conditions for light incidence to, and escape from the absorber. A further finding is that the maximum path length is obtained for every light trapping mechanisms which results in a complete filling of the available states in phase space. For stationary solar cells, the Yablonovitch limit of 4dn2, with n the refractive index of the absorber, is a very good approximation of this limit.
© 2014 Optical Society of America
Limits for the path length enhancement in absorber layers are of keen interest for different concepts in the field of opto-electronics. They are used as a benchmark for the performance of light trapping structures in solar cells [1,2] and they are of some importance for the out-coupling of light in LEDs .
Approaches have been presented to calculate the limiting path length for the absorption of light in solar cells for different light trapping mechanisms. Most notably is the approach presented by Yablonovitch in 1982  to calculate the maximum absorption enhancement induced by scattering. This work introduced the 4n2 factor, famously referred to as Lambertian- or Yablonovitch limit, since. Another limit, which relates to optimized in- and out-coupling of light into an absorber in combination with scattering was presented in 1990 by Miñano . This limit can be seen as a generalisation of the Yablonovitch limit. The limiting path length for a certain type of scattering, which is symmetric, coherent and conserves étendue , was presented by Kirchhartz in 2009 . This limit is related to diffraction of light. A general approach that discusses the impact of the angular distribution of light was presented by Yu et al. .
In this work we present a method for calculating the path length of light in an absorber based on three basic principles: Liouville’s theorem for optics, also sometimes referred to as conservation of étendue , the reciprocity of light beams  and the density of states in optical media . None of these principles specifies how a certain optical device scatters light and they can therefore be applied generally. A particular scattering system is specified by the way light occupies states inside the absorber. This allows describing any type of scatterer, regardless of the actual scattering mechanism, be it incoherent scattering, coherent scattering  or plasmonic scattering . In fact, it will turn out that the limiting path lengths are independent of the scattering mechanism.
Special attention will, furthermore, be paid to the significance and definition of the condition for light incidence. We will show that the solid angle under which the absorber receives and into which the absorber emits light are playing a key role here. It will be shown in particular that limiting path lengths for stationary solar cells in operation and in the laboratory are different.
A particular example here is the comparison between light trapping induced by diffraction and scattering. A certain discussion about which mechanism is superior has been conducted on this topic . In theoretical investigations and laboratory tests, it has been shown that it is possible to surpass the Yablonovitch limit by using scattering structures [15, 16]. However, other investigations have shown that in practice, there is little difference in the achievable current enhancement [17, 18]. The results of this paper show that under laboratory conditions, diffraction can principally achieve higher absorption enhancements than scattering. However, the results also show that for any stationary solar cell outside, the limits for diffraction and scattering are practically the same and that the limits for optimum in-coupling and out-coupling conditions are equal.
2. Investigated situations
In this work we will consider three different setups that are sketched in Figs. 1(a), 1(b) and 1(c). The quantity that we are interested in is the path length of light inside an absorber layer with thickness d. Note that all considerations use the limit of low absorption; even though the device is referred to as an absorber, absorption is actually not considered. It should be noted that the actual light path in presence of finite absorption is always smaller than the case of zero absorption .
The absorber shall be equipped with a perfect rear side reflector and a scattering surface on the front. A perfect reflector at the rear is equivalent to considering an absorber with thickness 2d and two equal surfaces. The characteristics of the scattering surface shall remain unspecified for now; later we will investigate how different scattering characteristics affect the path length in the cell. The difference between the three setups is in the condition of light incidence and escape. This difference deserves a more detailed discussion:
Figure 1(a) defines a situation in which light incidence is diffuse; i. e. light reaches the absorber from all directions at once. The escape is not affected and all internal light within the escape cone of total internal reflection escapes the system. This situation describes a stationary solar cell that is illuminated by a completely diffuse light source. It also describes the situation of a stationary solar cell that is illuminated by a moving light source that occupies every position on the hemisphere on its course over time. This means that we treat an illuminated solar cell as an ergodic system. A more detailed discussion about ergodic behavior can be found in . Mathematically the solid angle of the ergodic light source Ω is described byFig. 1(a) is therefore a good description for a stationary solar cell.
Figure 1(b) defines a situation in which light incidence is restricted to a certain solid angle on the hemisphere of incidence Ω. Escape also here is not affected and light within the escape cone is escaping the absorber. Situation 1a can be seen as a special case of this situation with Ω = 1. Another special case of this situation is obtained if Ω is the solid angle of the solar disk (Ω = 0.0047 rad). This special case corresponds to a stationary solar cell at any given moment in time (for sake of simplicity we will only consider the case in which the light source is centred on the surface normal of the absorber). It also corresponds to a solar cell mounted on a tracking system. It is important to note that Fig. 1(b) also corresponds to testing conditions in laboratories, whereas Ω is the solid angle of the solar simulator (Ω = 0.1 rad). This statement will become important later, as it shows that there is a significant difference between stationary solar cells in the laboratory and under testing conditions, where absorption enhancement is concerned.
Figure 1(c) defines a situation similar to Fig. 1(b) with an additional directionally selective filter [20, 21]. This filter shall be designed in a way that it is completely transparent for all direction from which the light source shines light in it and it shall be completely reflective for all other directions. As a consequence, the escape cone for light coming from the absorber is now confined to the solid angle of the light source.
3. Light path length for different scattering conditions
In the following we shall restrict the discussion to isotropic absorbers. Isotropy here means that the 1D optical density of states is given by
Using this density of states and Snell’s law it is trivial, that the number of states inside the absorber is n2 higher than in the surrounding air
3.1 Scattering that increases the phase space volume
Scattering that increases the phase space volume, is typically referred to just as scattering. In solar cells such scattering is realized by randomly textured surfaces such as glass, TCO, or even textures in the absorber material itself.
A Lambertian scatterer is characterized by a cosine angular intensity distribution. This means, that a Lambertian scatterer fills the complete phase space in a single scattering event, i.e. the DOS in Eq. (2) equals the occupied states (see Fig. 2).
To investigate the path length enhancement we start with the situation depicted in Fig. 1(c). It is trivial to show that a fraction of the incident light equal to Ω / n2 is scattered into the escape cone. Here and further throughout the paper we will assume that all light within the escape cone is actually escaping (i.e. we neglect Fresnel reflections). The path length this case is calculated by
The factor 1 / sin θ represents the geometrical path enhancement for light scattered at an angle θ with respect to the surface normal (note that the choice of sine or cosine depends only on the convention, of the coordinate system). The integral is the average path length on a single path through the absorber; calculating the integral results in a factor of 2. The sum takes into account the escape of light on every interaction with the front surface. The factor 2 at the beginning, finally, represents the mirror.
This result can be used to evaluate some special cases. The first special case to consider is the one depicted in Fig. 1(a). In this case we have Ω = 1 and4] already pointed out, the result remains valid for restricted incidence, i.e. the situation shown in Fig. 1(b). It is worth noticing that this means that the absorption enhancement for such a case can be measured in the laboratory; size and position of the light source are of no relevance. As long as the escape cone defines how much light escapes from the system, The Yablonovitch limit is valid for a Lambertian scatterer.
As Eq. (4) indicates, the situation changes if a directionally selective filter is placed in front of the absorber (this corresponds to Fig. 1(c)). As a last special case in this section, we consider the case that absorption and escape are restricted to the solid angle of the sun. In this case we have Ω = sin θs2 with θs the polar angle of the solar disk. Consequently we obtain5].
A quick excursion shall be given, how the path length for non-Lambertian scatterers can be calculated. As an example we shall investigate Phong scattering . Phong scattering is characterized by a cos θ p characteristic, with p the Phong exponent. We will only consider the situation shown in Figs. 1(a) and 1(b) here; extension to the other case is trivial. The fraction of light escaping on each interaction ζ is now given byFig. 3.
3.2 Scattering that conserves the phase space volume
Scattering that conserves that phase space volume is basically a redirection of light at the scattering surface. Such scattering in solar cells can be realized by tilted mirrors or diffraction gratings. For path length limits for diffraction also see .
Conservation of the phase space volume requires that the number of states inside the region into which light is redirected is conserved. For sake of simplicity, we will assume closed regions for the redirected light. All given arguments are, however, also valid for arbitrary light distributions in the absorber.
90 degree conical tilt
In the first example we consider a 90 degree tilt of the incident light cone. A tilt by a specific angle is what we would obtain from a diffractive line grating or a tilted mirror. A 90 degree tilt marks the tilt that would result in maximum path length enhancement and can be considered as an optimum case. We will make one further assumption here, which is that the scattering device is symmetrical (i.e. the tilt is divided equally into two tilts into + 90 and −90 degree). Due to the reciprocity of light beams, a symmetrical setup results in a “V” shape of the light in the absorber. In other words: all light is coupled out of the absorber at the second interaction with the front surface. As we will show later, the symmetrical device assumption results in a non-optimum path length. Sketches of the 90 degree conical tilt and the symmetry assumption are shown in Fig. 4.
This example proved to be difficult to calculate and we can only give an approximate solution. As described in the last paragraph Eqs. (9) and (10) need to be solved with a parameterization corresponding to the tilted cone. We obtain
90 degree spherical tilt
The 90 degree conical tilt preserves the shape of the incident light cone. If we do not confine ourselves to this situation, we have another option that results in a slightly longer path length. This option is the 90 degree spherical tilt. In the 90 degree spherical tilt, the light is scattered into the shape of a ring filling up every possible azimuth direction. As the phase space is conserved, this reduces the spread in the polar direction, allowing scattering in a closer range of the 90 degree axis. The reciprocity condition, mentioned in the last example holds strictly in this case; in other words: the described situation can only result in the “V” shaped path. A sketch of this situation is shown in Fig. 5.The equations describing 90 the degree spherical tilt areFig. 1(a) for diffuse incidence (Ω = 1), we obtainFigs. 1(b) and 1(c), Ω = sin θs2) we obtain
It is interesting to note that for both cases 90 degree conical tilt and 90 degree spherical tilt, the path length is proportional to instead of. In fact, the 90 degree spherical tilt completely corresponds to the described scattering cases, apart from this difference.
In the next section we will show that the proportionality to is related to the assumption of a symmetrical device which results in an incomplete filling of the phase space.
Successive filling of the phase-space volume
Neither of the discussed tilt scenarios results in the highest possible path length. A simple gedankenexperiment can help us to construct a more optimal scenario. For this purpose we need to quickly revisit the reasons for using the “V” shape. The “V” shape is a consequence of the reciprocity of light paths for symmetrical scattering devices. Reciprocity, however, is not strictly required. A device that would tilt the cone in only one direction (for example to + 90 degrees and not to −90 degrees) would not need to redirect the light backwards or out of the absorber. Instead, such a device could redirect the light cone into another direction.
The choice of directions is limited only by Liouville’s theorem which tells us that light from different incidence directions has to have different directions after the interaction with the scattering device (otherwise we could turn the light paths around and create a device that would reduce the phase space of light). A consequence of Liouville’s theorem is that we can theoretically construct a scatterer that redirects light into any direction that it hadn’t redirected light to in a previous interaction. Furthermore, we don’t want this scatterer to redirect light into any direction that is within the escape cone. To put it differently: we can construct a scatterer that successively fills the phase space of light inside the absorber until it reaches the escape cone. This scenario is sketched in Fig. 6.
The sequence in which the phase space volume is filled on this path is of no consequence. To calculate the path length that results from a successive filling, we can therefore use a succession of ring shapes, even though, strictly speaking, the reversibility of light paths would not allow us to do this. Proceeding along this line, we know that every ring has to occupy the same phase space.
We will first consider the cases shown in Figs. 1(a) and (c), in which incidence and escape are equal. For these cases we obtainFig. 1(b) is slightly different; incident light is restricted, the light source is fixed but light can escape into the complete hemisphere. In this case we obtain
There are several conclusions to be drawn from the calculations presented above:
- The maximum possible path length for light in an absorber is independent of the scattering mechanism and only depends on the initial conditions as they are depicted in Fig. 1. We conclude this from the comparison of Eqs. (4) and (19). We also conclude this from the fact that the sequence of the summation in Eq. (19) is of no consequence.
- The maximum path length enhancement is obtained for every scattering process that fills the entire phase space volume. Any scattering process that results in partial filling of the phase space will result in a smaller path length. Both statements are direct consequences of Liouville’s theorem and the reversibility of light paths. The second statement (incomplete filling) can also be proven in the following way: If there is a volume of the phase space unfilled, there are certain angles missing in the sum in Eq. (19). These angles can be considered by summing up over all angles and subtracting the missing angles. The result is consequently and necessarily smaller than 4 d n2/Ω.
The obtained path length enhancement was explicitly shown for several exemplary systems. Two examples were chosen for which the phase space is filled completely: Lambertian scattering and a successive filling of phase space. Lambertian scattering is a random scattering process and can be approximated with random scattering surfaces. Successive filling is directed process and can be approximated by coherent scattering or diffraction. In both cases, however, the significant boundary condition is that phase space is filled completely which results in an equal path length enhancement – the results of Eqs. (4) and (19).
Furthermore, two systems were investigated in which the phase space is not completely filled: Phong scattering and symmetrical conical scattering. The path length enhancement for these cases is given in Eqs. (8), (13) and (15) respectively. In all cases, the result is smaller than for the cases with a complete filling.
- The conclusion that Phong scattering does not fill the entire phase space might not be intuitive at first, because Phong scattering results in a continuous distribution. It becomes clear by looking at the density of occupied states. This density is smaller than the available phase space volume for all cases with p > 1. Hence, the phase space is not completely filled.
- The maximum possible path length enhancement for diffuse light incidence and escape is given by the Yablonovitch limit (Fig. 1(a)). For restricted incidence and emission, the Miñano limit is valid (Fig. 1(c)). A consequence of this conclusion is that for every stationary solar cell, the Yablonovitch limit is a very good approximation for the achievable path length enhancement. This is due to the fact that a stationary solar cell and a moving light source can be considered as an ergodic system. In particular, the situation in a laboratory (Fig. 1(b)) is fundamentally different from that of a stationary solar cell under operation and cannot be directly compared, since the situation of light incidence is different. For a fair comparison, the path length enhancement achieved in the laboratory can be corrected by a factor with the integral in the denominator being close to 1 for most locations in which solar cells are operated.
- One particular conclusion of this investigation is that the limit for absorption enhancement in a solar cell is equal for all scattering mechanisms, be it incoherent scattering (i.e. scattering by randomly textured surfaces), coherent scattering (e.g. by diffractive gratings or holograms), plasmonic scattering or even reflection at titled mirrors, provided the mechanism fills the entire phase space of light if the initial conditions for illumination and escape are equal.
Comparing coherent and incoherent scattering, Eqs. (5), (17) and (20) show that under laboratory conditions, a higher absorption enhancement is possible by diffraction than by scattering, i.e. it is possible to exceed the Yablonovitch limit. However, for a stationary outdoor system, considering ergodic behavior, we have Ω ≈1. In all such cases the limit becomes L = 4 d n2.
An exception is given for all non-stationary devices, for example if a tracking system is used. A solar cell tracking the sun and a solar cell tested in the laboratory receive light only from a restricted range. Under testing conditions, the illumination range depends on the light source and can range from laser light with a solid angle of close to zero to lamp arrays with opening angles of several degrees. For the tracked system, the range is restricted by the diameter of the sun and the tracking accuracy. Tracking systems have been installed for different types of solar cells  and any concept that could exceed the Yablonovitch limit could be of interest here.
It should be remarked that, even though the fundamental limits for all scattering devices are equal, there are significant differences for real scattering structures. Whether a scattering mechanism can achieve its maximum potential depends on the scattering efficiencies, which the specific scattering structure provides . The enhancement in generated current furthermore depends on the specific design of the photovoltaic device and factors like parasitic absorption need to be considered.
A final conclusion, that shall be discussed briefly, concerns standard textures applied to silicon wafer solar cells. Such standard textures are pyramidal textures  and isotextures . The presented results indicate that these textures should principally be able to provide very long light paths. Pyramidal textures have a strong loss path at the second interaction with the surface. Subsequently light is trapped efficiently and is quickly scattered. Concepts that reduce the first loss path (e.g. be applying different surface geometries at the front- and rear) will result in close to optimum light trapping. Isotextures feature spherical shapes  that result in close to Lambertian scattering.
In this paper, a method was presented that allows calculating the path length of light in an isotropic absorber for different scattering mechanisms of light. The scattering mechanisms result in different ways of how light occupies the available states in phase space in the absorber. Liouville’s theorem states that the phase space volume of incident light can only be conserved or increased. Based on this theorem, different scattering mechanisms were described, and the corresponding path lengths were calculated.
The calculations show that the path length limit is only achieved for scattering mechanisms that result in a complete filling of the available states in phase space. Vice versa, all mechanisms that result in a complete filling of the available states have the same path length limit. Two examples for such scattering processes are discussed: Lambertian scattering and a successive filling of the phase space. All mechanisms that result in an incomplete filling of the available phase space result in a short path. Examples discussed for such processes are Phong scattering and the 90 degree tilt.
The maximum possible path length depends, furthermore, on the conditions for light incidence onto the absorber and light escape from the absorber. In particular, the conditions in the laboratory and outdoor operation of a stationary solar cell are fundamentally different and cannot directly be compared. A solar cell illuminated by a moving sun can be considered an ergodic system. Consequently, illumination can be treated as being close to hemispherical in most parts of the world. For hemispherical incidence, the path length limit is given by the Yablonovitch limit (L = 4 d n2.) The situation is different for tracked solar cells; here light incidence is restricted to certain directions and the Yablonovitch limit can be exceeded. An even higher path length enhancement is possible, if additionally light escape from the absorber is limited to the same phase space as light incidence. In this case, the highest possible path length is given by the Miñano limit (L = 4 d n2 / Ω)
An additional finding concerns scattering mechanisms that conserve the phase space of light. A special type of such a scattering mechanism is the symmetric scattering mechanism, which results in a V-shaped path inside the absorber. Examples for symmetric device are symmetric gratings and the discussed example is the 90 degree spherical tilt. We find that for symmetric devices, the path length limit is proportional to rather than to .
The path length enhancement factor is a useful tool to rate the light trapping provided by different optical scatterers. However, in practical applications, further considerations need to be taken into account. The impact of scattering depends on the actual absorptivity of the absorber material, including dispersion. Furthermore, in a real device, parasitic absorption processes will occur. As a consequence, different scattering mechanisms will be of different benefit for specific applications.
The results presented in this paper are limited to isotropic absorbers, i.e. absorbers for which the DOS is described by Eq. (2). The results can therefore not be transferred directly to systems with a non-isotropic DOS like photonic crystals and waveguides. It is currently unclear, whether the presented limits also hold for such systems. Considering photonic crystals, non-isotropic systems will probably show spectrally dependent as well as directionally dependent properties. It can be speculated that the presented limits will hold on average but that there are specific conditions (i.e. frequencies) under which other limits are valid (in this respect see also ).
The Solar Energy Research Institute of Singapore (SERIS) is sponsored by the National University of Singapore (NUS) and Singapore’s National Research Foundation (NRF) through the Singapore Economic Development Board (EDB). Special thanks to Liu Zhe for the 3D artwork and to Benedikt Bläsi, Oliver Höhn and Jan Chistoph Goldschmidt from Fraunhofer ISE for helpful discussions.
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