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Abstract
It is shown how to efficiently convert solar into electrical energy, taking advantage of laser amplification and intra-cavity use of a low-efficiency converter. The latter may consist of a low-efficiency transparent photovoltaic cell or a thermoelectric cell integrated into a metallic laser-cavity mirror, constituting a minor intra-cavity loss for the laser operation. The overall power-conversion efficiency is derived and discussed for a variety of current solid-state laser materials. It is shown that power-conversion efficiencies comparable with commercial silicon photovoltaic cells are obtained with current laser materials.
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1. INTRODUCTION
To obtain electrical power from solar irradiation, single-junction photovoltaic cells are of common use, whose power-conversion efficiency is governed by the Shockley–Queisser limit [1]. Commercial photovoltaic cells are in general first generation crystalline or poly-crystalline silicon cells [2–4]. To reduce material cost, second generation photovoltaic cells use thin-film technology [5], based on materials such as cadmium telluride, copper indium gallium selenide (CIGS), amorphous or nanocrystalline silicon, and gallium arsenide. Apart from this, multijunction thin-film cells have been developed to further increase the power-conversion efficiency [6]. Commonly, the scarcity and toxicity of most of these ingredients represent a mayor disadvantage. Therefore, in third generation thin-film technologies new materials are studied, such as organic materials or Perovskites, suffering, however, from still low efficiencies and stability problems [7]. The highest efficiencies of up to 39.5% are presently obtained with GaAs multijunction cells [8] and up to 47.6% with concentrator multijunction cells [9].
On the other hand, solar-pumped lasers [10] convert a spectral window of the solar irradiation into a single-wavelength coherent laser beam [11–14] and promise a variety of applications, such as a Mg-based energy cycle [15,16]. Whereas the first solar lasers required highly concentrated sunlight to surpass the comparably high lasing threshold [11,12], solar lasers based on Nd:YAG waveguides, sensitized by luminescent nanocrystals, reached lower thresholds, requiring concentrations of only 230 suns [13]. Furthermore, Nd:YAG materials, codoped with, for example, ${{\rm Cr}^{3 +}}$, allow for ${{\rm Nd}^{3 +}}$ emission via nonradiative energy transfer from ${{\rm Cr}^{3 +}}$ [17,18], providing high efficiencies [19,20]. Alternatively, Yb:YAG [21] can be sensitized by ${{\rm Nd}^{3 +}}$ with efficient energy transfer [22], rendering codoped Ce,Cr,Nd,Yb:YAG ceramics promising candidates for high-efficiency solar lasers [20,23–28].
In this contribution, both concepts previously mentioned shall be combined in order to obtain power-conversion efficiencies comparable to commercial Si photovoltaic cells, however, using a low-efficiency photovoltaic cell or, equivalently, a thermoelectric device. The efficiency is shown to be boosted by the laser action and by the fact that the power-converting cell is placed inside the laser cavity. The paper is organized as follows: in Section 2 the generic power-conversion efficiency of the proposed system is derived for a typical four-level laser medium. Section 3 continues with the evaluation of the convertible solar photon flux and a discussion of end and side pumping geometries. In Section 4 the pump geometries are compared with respect to threshold concentrations and maximum power-conversion efficiencies, and the results are compared for different current laser materials. Finally, in Section 5 a summary and conclusions are given.
2. LASER-ASSISTED SOLAR-ENERGY CONVERSION
The system considered for implementing laser-assisted conversion of solar energy consists of a solar-pumped laser with the power conversion being performed at low efficiency inside the laser cavity. This system is mounted on a solar tracker in order to permit high concentration factors for the incident direct solar radiation. We consider a standing-wave laser cavity, comprised of two highly reflective mirrors, M1 and M2, a cylindrically shaped coaxial laser medium, LM, and a power-conversion cell, P, that may be a transparent photovoltaic cell or a thermoelectric device integrated in one of the cavity mirrors; see Fig. 1. The pumping of the laser medium shall be provided by concentrated solar irradiation. The intra-cavity power-conversion cell acts as absorber and contributes to the overall cavity losses with its power-conversion efficiency ${\eta _{\rm p}}$, which we consider to be rather low, i.e., of the order of 5%.
Fig. 1. Solar-pumped laser cavity with two highly reflecting mirrors M1 and M2, a cylindrical laser medium LM (radius $R$, length $L$), and a power-conversion cell P. The laser medium is pumped by concentrated solar irradiation S. The solar irradiation received at the entrance area ${A_{\rm r}}$ is concentrated onto the pump surface ${A_{\rm p}}$ of the laser medium. (a) End pumping: the Fresnel lens FL concentrates the solar light, passing the dichroic mirror M1, highly reflective only for the laser line, onto a base surface of the laser medium. (b) Side pumping: the parabolic mirror PM is used to concentrate the solar light onto the mantle of the laser medium.
A. Generic Power-Conversion Efficiency
In the following, we derive the power-conversion efficiency of the complete system, i.e., the conversion from the total solar irradiation power received to the electric power produced by the intra-cavity power-conversion cell. The laser rate equations [29] determine the stationary dynamics of the laser power ${P_{\rm l}}$ when gain and loss, ${\cal G}$ and ${\cal L}$, respectively, are known for the cavity round-trip time ${\tau _{\rm c}}$,
The gain, on the other hand, is limited by saturation and obeys the dynamicsFrom Eqs. (1) and (2) the stationary laser power is obtained as
The electrical power converted from the laser power, via the power-conversion efficiency ${\eta _{\rm p}}$ of the intra-cavity converter P, is
whereas the received solar irradiation power, ${P_{\rm s}}$, incident on the receiving area, ${A_{\rm r}}$, is with ${{\cal E}_{\rm e}} \approx 1\,{{\rm kWm}^{- {2}}}$ being the solar irradiance at earth surface [32]. Thus, the overall power-conversion efficiency of the system $\eta$, i.e., the relevant parameter that we seek, will beIn order to proceed in obtaining the overall power-conversion efficiency (9), the laser power ${P_{\rm l}}$ has to be obtained. As a first step, the small-signal gain [29], as required in Eq. (5), is defined as
where ${\Phi _{\rm l}}$ is the laser photon flux and $\Delta {\Phi _{\rm l}}$ its increase within one cavity round-trip time, given by The increase of the number of laser photons during a cavity round-trip reads Here the stimulated emission rate on the laser transition is with ${\sigma _{\rm l}}$ being the emission cross section of the laser transition, and where $\Delta {N_{{\rm ss}}}$ is the small-signal inversion of the laser medium. Here, in Eq. (12) $\tau$ is the time the laser light spends in the laser medium during one cavity round-trip. Supposing a cavity length ${L_{\rm c}}$ and a laser medium of length $L$ and index of refraction $n$, this time can be expressed as with the fill factor $\xi$, i.e., the ratio of optical lengths of medium and cavity, beingFig. 2. Four-level laser medium with pumping on the transition $|0\rangle \to |3\rangle$ with pumping rate ${r_{\rm p}}$ and the laser transition $|2\rangle \to |1\rangle$ with stimulated emission rate ${r_{\rm l}}$. The spontaneous decay rates from levels $|i\rangle$ are ${\gamma _i}$ ($i = 1,3$), and the decay rate of the laser transition is $\gamma$.
Returning to Eq. (10) and using Eqs. (11)–(14), the small-signal gain can be written as
Combining Eqs. (5), (6), and (9), the overall power-conversion efficiency of the system becomes then
B. Four-Level Laser Medium
The small-signal inversion can be obtained from the rate equations of the populations of the electronic levels [29]. For a typical four-level laser medium, such as shown in Fig. 2, the stationary population inversion density $\Delta n = {n_2} - {n_1}$ of the laser transition between levels $|2\rangle \leftrightarrow |1\rangle$ is obtained as
For a reasonably good laser medium, typically $\gamma \ll {\gamma _1},{\gamma _3}$, so that the small-signal versions (${r_l} \to 0$) of the inversion (18) and ground-state (20) densities become [29]
The overall small-signal inversion, integrated over the volume $V$ of the laser medium, derives from Eq. (21) and readsWe may assume for simplicity that the laser mode adapts to the small-signal inversion density in the laser medium, which will depend on the pump geometry. In consequence, the mode function $f$, mentioned above, can be identified by the small-signal inversion density $\Delta {n_{{\rm ss}}}$, i.e.,
Given these definitions, now the spatial shape of the small-signal inversion density $\Delta {n_{{\rm ss}}}$, Eq. (21), has to be determined, considering absorption due to the small-signal ground-state population density ${n_{0,{\rm ss}}}$; cf. Eq. (22). This spatial distribution will depend on the geometry of pumping, that is, on whether end or side pumping is considered for the concentrated solar irradiation. This issue will be addressed in the following section.
3. SOLAR PUMPING
A. Received Photon Flux
The solar irradiation is incident on the receiving area ${A_{\rm r}}$ and by means of an optical system concentrated onto the pumping area ${A_{\rm p}}$, that is defined by the corresponding illuminated surface of the laser medium. Considering a cylindrical medium, coaxial to the laser cavity, the pumping area may be a base of the cylinder (end pumping) or the mantle (side pumping); cf. Fig. 1. In both cases a geometrical concentration factor of the solar irradiation can be defined as
where we assume the system to be mounted on a solar tracking platform in order to allow for high concentration factors. As the concentration optics will cause small losses, described by an overall concentration efficiency ${\eta _{\rm c}}$, we may define the effective concentration factor ${C_{\rm e}} = {\eta _{\rm c}}C$, so that the solar pump rate per emitter at the pumping area of the laser medium is Here ${\sigma _{\rm p}}(\nu)$ is the spectrally dependent absorption cross section of the laser pump transition, and the spectral solar photon direct normal irradiance may be approximated as a Planck spectrumIn order to provide a simplified model of the rather complex spectral dependence of the pump absorption cross section ${\sigma _{\rm p}}(\nu)$, it may be assumed to have a constant value ${\bar \sigma _{\rm p}}$ within a spectral acceptance window between minimum and maximum frequencies, ${\nu _{{\rm min}}}$ and ${\nu _{{\rm max}}}$, respectively, as depicted in Fig. 3(a). In this way the pump rate at the pumping surface of the laser medium becomes
where the solar photon pump flux, received within the spectral acceptance window, is obtained via Eq. (28) asFig. 3. (a) Constant value of the pump absorption cross section, ${\bar \sigma _{\rm p}}$, between a minimum and maximum frequency, ${\nu _{{\rm min}}}$ and ${\nu _{{\rm max}}}$, respectively. Between laser frequency ${\nu _{\rm l}}$ and ${\nu _{{\rm min}}}$ is the quantum defect $h({\nu _{{\rm min}}} - {\nu _{\rm l}})$. (b) Pump photon flux of the solar irradiation for ${\nu _{{\rm min}}} \to {\nu _{\rm l}}$ (vanishing quantum defect) as a function of the minimum wavelength ${\lambda _{{\rm min}}} = c/{\nu _{{\rm max}}}$.
Given a cylindrical laser medium coaxial to the cavity, pumping by concentrated solar irradiation can be performed via a base of the medium, i.e., by end pumping, or by illuminating the mantle of the cylindrical medium, i.e., by side pumping. In both cases, it is well known that the concentration of thermal radiation, such as solar radiation, is limited by fundamental thermodynamic laws [35]. That is, for incident solar irradiation with a spread of the incident angle being $\Delta {\theta _{{\rm in}}} \approx {0.265^{\rm o}}$, and assuming outgoing angles up to $\Delta {\theta _{{\rm out}}}={ 90^{\rm o}}$, the theoretical thermodynamic limit of the concentration factor is
where we assumed absence of a dielectric material in the concentrator optics. This limit is only reached for optimal concentration optics, such as a compound parabolic concentrator (CPC) [35] and is valid for concentrating on a point, which is the case for end pumping. When concentrating on a line, such as for side pumping, the maximum concentration factor isB. End Pumping
In the case of end (longitudinal) pumping, the solar irradiation is incident on a base of the cylindrical laser medium and propagates within the medium in the $z$ direction. Due to absorption of the pump light along the optical axis of the laser, the small-signal inversion density (21) is a function of the propagation depth $z$. In this case the overall small-signal inversion, integrated over the active volume of the laser medium, reads
where ${A_{\rm p}} = \pi {R^2}$ is the base area of the cylindrical laser medium. For obtaining the $z$ dependence we consider the pump rate ${r_{\rm p}}(z)$ obeying Beer’s law, with ${r_{\rm p}}(0) = {r_{{\rm p},0}}$ being defined in Eq. (29). Whereas, strictly speaking, this equation is valid only for each spectral component of the pump rate, each having a different pump absorption cross section, we approximated here using the constant pump absorption cross section, ${\sigma _{\rm p}}(\nu) \approx {\bar \sigma _{\rm p}}$, in accordance with the model shown above.Inserting Eq. (22) and integrating the differential equation (35) from the front face of the medium ($z = 0$), where the pump light is incident, to a position $z $ within the medium, we obtain the solution in terms of the Lambert W function [36],
This density is shown in Fig. 4(a) for two values of the effective concentration factor ${C_{\rm e}}$. It is observed that for the threshold concentration a minimal inversion is obtained around the entrance plane $z \approx 0$. Increasing the effective concentration to the limiting value ${C_{{\rm TL}}}$ accomplishes higher inversion densities at larger depths.
Fig. 4. (a) Inversion density as a function of propagation depth $z$ in the case of end pumping for different values of the effective concentration ${C_{\rm e}}$. (b) Dependence of the overall power-conversion efficiency on the effective concentration factor for different lengths $L$ of the laser medium, i.e., different values of ${N_{{\rm ep}}}$. The intra-cavity conversion efficiency is ${\eta _{\rm p}} = 5\%$, the residual cavity loss is ${{\cal L}_{\rm r}} = 1\%$, and the material parameters are chosen for an idealized laser medium; cf. Table 1.
Inserting Eq. (38) into Eq. (34) the overall small-signal inversion is obtained as
In Fig. 4(a) it can be seen that for lower values of ${N_{{\rm ep}}}$, i.e., smaller medium length, the power-conversion efficiency is first increasing with the concentration factor to reach a maximum value where the laser medium becomes saturated. Further increasing the concentration factor leads to a decrease in efficiency as no more solar pump photons can be converted into laser photons and therefore will be lost. Furthermore, increasing the parameter ${N_{{\rm ep}}}$, i.e., the medium length, the maximum efficiency at saturation is increasing. This can be understood, considering the increase of the number of laser emitters available for stimulated emission when increasing the medium length. However, this increase is limited and further increasing ${N_{{\rm ep}}}$ leads to a broadening of the maximum. This can be explained by a progressive saturation depth of the medium.
Below the laser threshold, no laser photons will be generated, but residual pump light can be incident on the intra-cavity converter. This partially absorbed pump light will produce an overall power-conversion efficiency that will be below the efficiency of the intra-cavity converter.
The maximally realizable efficiency of the system results by maximizing the efficiency given in Eq. (41). To obtain this global maximum, we first consider the maximum of Eq. (41) with respect to the number of inverting pump transition ${N_{\rm p}}$. This implies a differential equation for ${{\cal G}_{{\rm ss}}}$, whose solution is given by
where the integration constant $\alpha$ must be determined by comparing Eq. (44) with the known form of ${{\cal G}_{{\rm ss}}}$, as given in Eq. (42). Performing this comparison, the Lambert W function can be expanded with respect to its argument, to obtainAs shown in Fig. 4(b), increasing the length of the medium, i.e., ${N_{{\rm ep}}}$, only contributes to a rise of the range of effective concentration factors, i.e., ${N_{\rm p}}$, where the efficiency approaches its global maximum. This allows us to consider ${N_{{\rm ep}}} \gg 1$ and consequently $\epsilon = \exp (- {N_{{\rm ep}}}) \ll 1$, and to take into account in Eq. (45) only terms up to first order in $\epsilon$. Neglecting the small cavity loss, $\alpha$ results then as
Equation (47) allow us to discuss an optimal case where the cavity and the laser material are ideal, i.e., ${{\cal L}_{\rm r}} = 0$, $\xi = 1$, ${\eta _{\rm c}} = 1$, $\kappa = 1$, and ${N_{{\rm ep}}} \gg 1$. Under these conditions, the maximally realizable efficiency will only depend on the laser wavelength as
It can be observed that due to the absence of residual cavity losses, the conversion efficiency ${\eta _{\rm p}}$ of the intra-cavity converter does not enter the expression for the maximum efficiency, Eq. (48). However, in all cases, increasing ${\eta _{\rm p}}$ enlarges the required threshold concentration factor. In fact, the threshold concentration factor in the idealized case reads
The maximum efficiency, Eq. (48), is shown in Fig. 5, where the peak of the maximum efficiency of 44% is observed at 1150 nm. Increasing the laser wavelength enlarges the absorption window but also introduces an increase of energy loss in the conversion of absorbed photons to the laser photons. In this way increasing the laser wavelength reduces the overall efficiency. It should be noted that the chosen laser wavelength of 1064 nm is close to the optimum wavelength according to Fig. 5. However, when increasing the residual cavity losses the maximum efficiency is reduced.
Fig. 5. Maximally attainable conversion efficiency for ideal laser cavity, ${{\cal L}_{\rm r}} = 0$, $\xi = 1$, and ${\eta _{\rm c}} = 1$, and laser material, $\kappa = 1$, ${N_{{\rm ep}}} \gg 1$. The peak is observed at 1150 nm with an efficiency of 44%.
C. Side Pumping
For side pumping, the pump light is absorbed in the radial direction of the cylindrical laser medium; i.e., the area of pumping is the mantle of the cylindrical medium with area ${A_{\rm p}} = 2\pi RL.$ We consider here an axially symmetric solar pump intensity in order to simplify the following calculations. Of course, the angular dependence of the real pump intensity profile may differ from this symmetry. However, the main features as obtained here will be maintained.
In order to describe the transmission and absorption of a ray of the concentrated solar pump light, we define the radial variable $x $, where $x = 0$ corresponds to the center of the disk-shaped cross section of the medium, $x = - R$ corresponds to the point where the ray enters the medium, and $x = + R$ is the point where the attenuated ray leaves the medium. In this way, Beer’s law for the pump rate becomes
Inserting Eq. (22), and integrating from the surface of the medium, ${-}R$, to the radius $x \in [- R, + R]$, we obtain
Using now the radial coordinate $r$, the small-signal inversion density [cf. Eq. (21)] resulting from Eq. (54) is
Fig. 6. (a) Inversion density as a function of the radial coordinate $x$ for different values of the effective concentration factor ${C_{\rm e}}$. The blue curve corresponds to the concentration at threshold. (b) Power-conversion efficiency for side pumping as a function of effective concentration factor, for different values of radii $R$, i.e., different parameters ${N_{{\rm ep}}}$. The intra-cavity conversion efficiency is ${\eta _{\rm p}} = 5\%$, the residual cavity loss is ${{\cal L}_{\rm r}} = 1\%$, and the material parameters are chosen for the idealized laser medium; cf. Table 1.
The overall small-signal inversion of the medium is obtained, according to Eq. (23), from the small-signal inversion density, Eq. (55), as the integral
This integration must be performed numerically, and in consequence also the overall power-conversion efficiency [see Eq. (17)] is obtained numerically. The result of this numerical evaluation is shown in Fig. 6(b) for various parameters ${N_{{\rm ep}}}$, i.e., radii $R$ of the laser medium. It can be observed that for increasing radius, the overall efficiency rises to increasing maximum values. However, similar to the case of end pumping, a maximum efficiency is observed for further increasing the radius. This can be again explained by progressive saturation of the medium, now, however, in the radial direction. From Fig. 6(b) it is seen that these maximal efficiencies can be reached within the thermodynamic concentration limit.Also here, below the threshold only a residual pump light may be incident on the intra-cavity converter, leading here to an even lower power-conversion efficiency as in end pumping due to the transverse pump direction. It can be shown that for the maximum efficiency the same formula of Eq. (47) applies, with ${N_{{\rm ep}}} = {\bar \sigma _{\rm p}}{n_{\rm e}}R$; in consequence also here the discussion of Fig. 5 is valid.
4. DISCUSSION
As in the proposed setup, the efficiency of the intra-cavity converter is only a minor parameter for determining the overall power-conversion efficiency, and the dominant influence is given by the parameters of the laser material. That is, for maximizing the efficiency of such a system, optimal laser materials must be searched for. As already discussed, side pumping leads in all cases to much lower threshold concentration factors; however, the reachable concentration factors due to the thermodynamic limit are lower than in the case of end pumping. Furthermore, in the side pumping configuration, the laser threshold is decreased by increasing the length $L$ of the cylindrical laser rod, as seen in Fig. 7. It is seen that all laser materials from Table 1 [37] eventually decrease their threshold concentration factor for increasing rod length.
Fig. 7. Threshold concentration factor ${C_{\rm t}}$ for side pumping as a function of the length $L$ of the laser material. The radius is fixed to $R = 8\;{\rm cm}$. The intra-cavity conversion efficiency is ${\eta _{\rm p}} = 5\%$, the residual cavity loss is ${{\cal L}_{\rm r}} = 1\%$, and the material parameters are shown in Table 1.
Table 1. Parameters of Four Potential Laser Materials for Solar-Energy Conversiona
In Table 1 the relevant parameters for various prominent laser materials are shown, together with estimated threshold concentrations and maximally reachable power-conversion efficiencies for end and side pumping. It can be seen that for both pump geometries, as discussed above, various materials would require concentration factors beyond the thermodynamical limit. These materials are marked with red color. ${{\rm NdYVO}_4}$ shows a considerably lower threshold concentration factor, with a total power-conversion efficiency of the order of 22%. ${{\rm NdYVO}_4}$ supersedes all other materials with an extremely low threshold concentration factor for side pumping of only 21. This is due to its rather high values of the pump and laser emission cross sections ${\bar \sigma _{\rm p}}$ and ${\sigma _{\rm l}}$, respectively. However, it should be noted that these values are optimistic, taken at typical pump wavelengths of the laser materials. A more precise treatment would require the integration of the pump cross section over the spectral solar irradiance [cf. Eq. (27)] using precise data of the spectrally dependent pump cross section of each laser material.
Apart from the proper choice of the laser material, also the intra-cavity converter may show detrimental effects when implemented as a photovoltaic cell. Due to the rather high laser photon flux, the photovoltaic cell would operate at unusually high intensities. In such an operational regime additional effects can occur, such as an increased power loss due to increased photocurrents and stimulated emission [38,39].
To include these effects goes beyond the scope of the present paper but may be the subject of a future publication. On the other hand, an alternative for the intra-cavity converter could be the use of a thermoelectric cell mounted on the backside of one of the cavity mirrors in order to absorb and convert the low fraction of power transmitted through the mirror; it should be noted that these losses within the intra-cavity converter are phenomenologically described by the residual losses ${{\cal L}_{\rm r}}$.
5. SUMMARY AND OUTLOOK
In summary, a novel device for laser-assisted intra-cavity solar-energy conversion has been introduced and analyzed. Instead of receiving the solar irradiation directly on a photovoltaic or thermoelectric cell, the solar irradiation is used to pump a laser medium, so that the power conversion can be performed inside the cavity. This allows for rather low power-conversion efficiencies of the intra-cavity converter without affecting the overall efficiency of the setup. In fact, the system power-conversion efficiency is now governed by the properties of the laser material. The reachable power-conversion efficiencies depend on the spectral acceptance window of the laser material and on its pump absorption cross section, the emission cross section of the lasing transition, and its lifetime. In the present paper, four-level laser media are considered, such as prominent Nd:YAG materials doped with Cr or other elements for increasing the spectral window where solar radiation can be absorbed.
Furthermore, the type of pumping, i.e., end versus side pumping, does influence the shape of the inversion density within the laser medium. As a consequence, side pumping typically will lead to much lower solar concentration factors required to reach the laser threshold.
To increase the power-conversion efficiency, and decrease the required threshold concentration factor, novel laser materials should be investigated, principally for broadening their spectral acceptance window within the visible part of solar irradiation and also for increasing their spectral pump absorption cross section. A promising new material could be represented by the vast possibilities of metal organic frameworks (MOF), which recently have been considered as novel laser materials [40–44]. The presented mechanism may serve as an alternative route to photovoltaics, in order to obtain high efficiencies with novel laser materials.
Funding
Pontificia Universidad Católica de Chile; Agencia Nacional de Investigación y Desarrollo; Comisión Nacional de Investigación Científica y Tecnológica (ACT 1409).
Acknowledgment
IJ, SW, and UV acknowledge funding by Instituto de Física, Pontificia Universidad Católica de Chile. BS acknowledges funding by Millennium Institute for Research in Optics (MIRO). SW, BS, UV, DD, and AC acknowledge prior funding by CONICYT.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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