A device that significantly enhances the absorption of incident radiation at visible wavelengths is studied. The device consists of an optical diode based on cholesteric and nematic liquid crystals, as well as a mirror. The diode allows non-symmetric one-way propagation of circularly polarized light around a predetermined region of the spectrum. Via full-wave simulations in both planar and cylindrical geometries, it is shown that combining the proposed device with ordinary absorbing materials results in the doubling of their overall absorption efficiency.
©2012 Optical Society of America
Manipulation of light at visible and optical wavelengths has become an extremely popular field of research in recent years due to the advent of technologies such as metamaterials [1–5], plasmonics [6–8], and photonic crystals [9–11]. One of the most important goals is the ability to design devices that exhibit strong absorption of light over broad bandwidths. Such “perfect absorbers”, inspired by the theoretical concept of the black body , could have significant applications in efficient solar power harvesting, thermal light emitting sources, and cross-talk reduction in optoelectronic devices.
A number of methods have been proposed exhibiting enhanced absorption which is not achievable with conventional materials. One group of concepts are electromagnetic absorbers, in the form of optical black holes and optical attractors [13–17], which are devices designed mainly using transformation optics [18,19] as the electromagnetic analogues of the gravitational black holes predicted by the general theory of relativity, where light is guided and trapped towards a narrow region of space. However, in an optical black hole this is caused by the tuning of the permittivity and permeability of the surrounding region of space, instead of spacetime wrapping. An alternative set of ideas, focused mainly on the improvement of photovoltaic devices, is based on light trapping by plasmonic metallodielectric & metallo-semiconductor structures [20–23]. These structures support surface plasmons [24,25] that concentrate impinging light over very thin semiconductor layers, allowing the same absorption efficiency over much shorter silicon wafer thicknesses, thus significantly decreasing the overall cost.
In this paper we investigate a different method to achieve enhanced absorption of light, utilizing a certain type of photonic crystals, namely cholesteric liquid crystals (CLCs) [26,27]. CLCs consist of liquid crystal molecules that form helical and periodic structures along a certain direction in space. The pitch of a CLC’s helix can be adjusted in order to reflect circularly polarized light propagating along its axis within a certain frequency range. This interesting property of CLCs has been utilized to fabricate tunable optical diodes, i.e. structures that allow non-reciprocal transmission of light [28–30]. In this paper we show how this concept can be expanded to achieve significant enhancement of the absorption efficiency. While we present only a few specific examples here, the absorption enhancement devices described in the following sections can be applied on top of any other absorbing structures and boost their absorption efficiency .
Specifically, in this work, we utilize full-wave electromagnetic simulations to evaluate the optical performance of various structures, based on optical diodes made from cholesteric liquid crystals proposed in . In Section 2, we investigate the performance of a planar optical diode consisting of two layers of cholesteric liquid crystals on either side of an anisotropic nematic layer that acts as a half wave phase shifter. We verify the transmission characteristics of the device as a function of wavelength and demonstrate its non-symmetric behavior, which allows light to be transmitted towards one direction of propagation only. In Section 3, we utilize this concept to create a planar absorption enhancement device by sandwiching an absorbing material between a mirror and the aforementioned diode. We demonstrate that, except for very lossy materials (with loss tangent greater than 0.06), the introduction of the absorption enhancement mechanism doubles the amount of energy that is absorbed by the material. Finally, in Section 4, we examine the absorption enhancing performance of a cylindrical version of the optical diode, which is found to be comparable to the planar scheme.
Two things should be emphasized here. First, enhancing the absorption efficiency of weakly lossy materials offers a double advantage, as not only smaller quantities of absorbing materials can be used, but they can also be of inferior quality, thus in both cases reducing the overall cost of the device. Second, this work is concerned with absorbing materials from a strictly electromagnetic point of view. As such, we model the absorbing materials through their macroscopic dielectric permittivity function only, without modeling microscopically any quantum, atomic, or carrier effects.
2. Planar optical diode operation
In this section, we present full-wave simulations of a planar optical diode illuminated by circularly polarized light . We examine the field distributions inside the diode under circularly polarized plane wave illumination, as well as the transmission of the device as a function of frequency for either forward or backward propagation direction.
2.1 Diode simulation setup
The diode consists of three sections. The first section is a 1.8 μm long cholesteric liquid crystal with a left handed helix with pitch (CLC1). The middle section is a half-wave plate (HWP) with length 2.1 μm and ordinary and extraordinary indices and , respectively. The HWP introduces 180° phase retardation between waves polarized linearly along its optical axes, thus inverting the polarization sense for circularly polarized waves. Its thickness is chosen to operate optimally for waves with free space wavelength around 550 nm. The final section is also a 1.8 μm long left-handed cholesteric liquid crystal with pitch . Both CLCs have an ordinary refractive index of and an extraordinary index of .
The cholesteric liquid crystal layers are assumed to always have their optical axes (their local director) in the plane, but their orientation depends on its position along the length of the crystal. The uniaxial local tensor in the laboratory coordinate system can be written as
The total length of the diode is 5.7 μm and is aligned with the z-axis of the simulation domain. All the simulations presented in this paper have been performed in Comsol Multiphysics. In the plane the domain has a square cross section but with periodic boundary conditions in both x and y axes, thus in effect simulating an infinite structure along these directions. While the operation of the diode is in effect one dimensional, Comsol requires a 3D simulation domain when fully anisotropic materials are utilized. The simulation domain also includes 1 μm long sections of air on either end of the diode for improved visualization of the incoming and outgoing waves. Finally, perfectly matched layers (PMLs) are placed at the exit side of the diode (depending on the propagation direction of the incoming wave) in order to absorb any remaining field energy.
The device is illuminated by circularly polarized plane waves of the complex form
Note that in the literature there are two equivalent conventions for defining the polarization sense of a circularly polarized wave. Suppose that the electric field vector rotates clockwise when looking along the direction of propagation of the wave (towards for a wave with , , i.e. from the point of view of the source). This can be defined as a right-handed circularly polarized (RCP) wave if we choose to observe it towards the direction (wave from the point of view of the source, appearing to rotate clockwise), or as a left-handed circularly polarized (LCP) wave if we choose to observe it towards the direction (wave from the point of view of the receiver, appearing to rotate counter-clockwise). Being completely equivalent descriptions, in this paper we choose the former definition of the polarization sense. In any case, however, the underlying physical principle remains that only waves rotating in the same sense as the liquid crystal director are reflected from it.
2.2 Simulation results for the planar diode effect
The diode effect is demonstrated in Fig. 1 , which shows the component of the electric field amplitude distribution () along the diode for four different LCP excitations, after steady state is reached.
In Fig. 1(a) the backward propagating LCP wave at 600 nm is transmitted through the diode and is only absorbed by the PML layer after exiting the structure. Meanwhile, when a LCP wave is propagating backwards, Fig. 1(d), it is not transmitted through the device because it is reflected by the second CLC. The situation reverses itself when the illumination wavelength is 500 nm, as the backward propagating LCP wave is reflected by CLC1 (Fig. 1(b)), while the forward propagating LCP wave is transmitted fine (Fig. 1(c)).
The key to explaining the physics of this behavior is to consider that circularly polarized light is reflected from the CLC if the electric field vector stays parallel to the director of the CLC, rotating in the same sense as it propagates along the CLC. Based on the definition of circular polarization explained in the last paragraph of the previous section, a CLC reflects incident light of the opposite handedness that corresponds to its pitch, while the HWP inverts the handedness of the light. The current choice of CLC pitches are such that CLC1 corresponds to light at 500 nm wavelength, while CLC2 corresponds to light at 600 nm wavelength. For example, in Fig. 1(a), the backward-propagating 600 nm LCP wave is transmitted through CLC2 (since they have the same handedness as viewed from the source), it is subsequently converted to RCP wave from the HWP, and is then also transmitted through CLC1, with its pitch tuned to “operate” at a different wavelength. In Fig. 1(b), however, the 500 nm LCP wave is transmitted through CLC2, converted into an RCP wave in the HWP, and then it is reflected by CLC1, since CLC1 has the opposite handedness (for backward propagation) and its pitch corresponds to the incident wavelength. Similar arguments explain the reflecting and transmitting behavior in Figs. 1(c) and 1(d).
In order to examine in detail the reflection properties of a single CLC layer when light of the opposite handedness is incident upon it, we evaluate the one-dimensional total electric field amplitude distribution along a 2.1 μm long, left-handed CLC with pitch equal to 504 nm (corresponding to 800 nm wavelength), surrounded by air regions (see Fig. 2 ). When the wavelength of the incident RCP wave is 500 nm (propagating along the direction), some reflections are observed in the entrance air region due to impedance mismatch, causing an interference pattern that increases the wave amplitude in that region. However most of the field energy is transmitted on to the other side. When the incident wavelength is 800 nm, the field amplitude is attenuated via reflection inside the CLC region with only 12% of the incident field energy being transmitted (or approximately 1/3 of the field amplitude, as the original amplitude of 1.41 V/m reduces to 0.5 V/m after the diode). The transmitted energy can be further attenuated by increasing the length of the CLC.
In order to characterize the optical performance of the diode, we evaluate its transmission characteristics as a function of frequency (or wavelength) for both directions of propagation. This is achieved by replacing the PML layers of Fig. 1 with a purely absorbing material which is defined by its relative permittivity, equal to . This implies a material with free space properties but with added absorption, which dissipates the incident wave energy. The domains are then terminated along the propagation direction with perfect electric conductors (PEC). The high value of the imaginary part of the permittivity ensures that energy which is not reflected by the diode is being absorbed. The resistive heating power, integrated over the volume of the absorbing regions, is then normalized to the incident power of the circularly polarized plane wave impinging on the diode, yielding the fraction of the transmitted power through the diode.
The transmission characteristics of the optical diode for LCP waves, for both directions of propagation, are reported in Fig. 3 . As expected by theory , the diode reflects incident light in a spectral region between , where is the pitch of each CLC layer and and its refractive indices. This is approximately a range of 70 nm and 90 nm around the central wavelengths of 500 nm and 600 nm for CLC2 and CLC1, respectively.
The response evaluated in Fig. 3 appears less smooth compared to the analytical model prediction using Berreman matrices that was utilized in . In general, the 3D full-wave simulation presented in this paper is expected to provide overall more accurate results compared to a simple analytic approach. However, the continuous rotation of the director in the liquid crystals along the coordinate [Eq. (1)] cannot be perfectly modeled without a prohibitively large number of simulation cells. Still, the region of interest, which is around the bandgap spectral regions of the device, agrees very well with the analytical model of  and is not significantly affected even for lower simulation resolutions.
3. Planar optical absorption enhancement
The diode effect examined in the previous section can be exploited in enhancing the absorption of light by weakly absorbing materials, such as the ones utilized in solar cells. Instead of increasing the thickness of the absorbing material, as required in most scenarios, we place the aforementioned optical diode in front of the absorbing layer of our device. A mirror is also placed behind this absorbing layer. The device is shown schematically in Fig. 4 .
The enhanced absorption device operates as follows. Suppose an LCP wave is incident upon the entrance of the device. We can assume, for example, a wave propagating along the direction at 600 nm, similar to Fig. 1(a). Provided that the diode consists of left-handed liquid crystals, this wave will almost fully transmit through the device, and will appear at its exit with inverted polarization sense (i.e., an RCP wave) due to the presence of the half wave plate. Depending on the absorption strength of the material after the diode, a fraction of the transmitted power will be absorbed, while the rest will reflect off the mirror at the end of the device. The mirror will invert the sense of polarization of the wave, creating an LCP wave that propagates in the opposite direction. Again, a significant fraction of the reflected wave will be absorbed as it propagates towards the diode. However, the LCP wave will now perceive the diode as a reflector, and most of the power will be reflected back into the absorbing layer. Note that, unlike a metallic mirror reflection, the LCP wave will retain its polarization sense again after reflection from the diode . After this second round of absorption, most of the remaining energy exits the device. As it shall be demonstrated, the phenomenon significantly increases the absorbed energy, compared to the device being present in free space without a diode.
We model the absorbing layer as a 1 μm thick material with a relative permittivity equal to , where the loss tangent can be varied to simulate materials with different absorption strengths. The simulation parameters otherwise remain identical to the ones described in Section 2.1, including the periodic boundary conditions perpendicular to the direction of propagation. We then compare the absorbed power in that layer in two cases. First, without a diode, the absorbing layer (including the end mirror) is placed in free space, and the permittivity varies as . Second, the diode is placed in front of the absorbing layer, which has the same absorbing strength as before, its permittivity varying as . The real parts of the permittivity are different in each case in order to maximize the impedance matching between the different materials and thus minimize unwanted reflections. However the dissipation rates in each case are exactly the same.
The enhanced absorption characteristics are quantified in Fig. 5 . Figure 5(a) reports the absorbed power (as a fraction of the total incident power) as the loss tangent spans three orders of magnitude. It is observed that for loss tangents higher than 0.06, more power is absorbed in the free space scenario than with the diode present. This occurs most likely because the absorption is strong enough, so that a single or double pass through the absorbing material is enough to deposit most of the wave power. An additional mechanism may be attributed to impedance mismatch that appears for large loss tangents between the absorbing region and free space. This mismatch is preventing enough incident power to reach the reflecting mirror in the first place, thus the polarization is not reversed and light is not properly trapped by the diode. On the contrary, for weaker absorption strengths (), the diode mechanism increases the absorption efficiency compared to the free space case. Finally, the reduced absorption at high loss tangents when the diode is present is also partially attributed to the reflection (impedance mismatch) from the diode components, although this effect remains constant for all loss tangents and is inherent in the operation of the diode.
The relative enhancement factor as a function of the loss tangent is reported in Fig. 5(b). The region of interest appears when the loss tangent of the absorbing layer is lower than 10−2. In this region the presence of the diode approximately doubles the absorption efficiency compared to the free space case. It should be noted that although we have modeled the absorbing layer macroscopically as a single isotropic and homogeneous material, this is just to illustrate the operation of the absorption enhancement mechanism in a simple manner. In practice, this mechanism could be applied to more complicated systems  that consist of multiple absorbing layers and utilizing many different materials, as long as the overall effective loss tangent of the absorbing region is less than about 10−2 and the impedance mismatch between the devices remains small. In such cases the absorption enhancement mechanism, when placed immediately after the existing absorption system, would enhance its existing absorption efficiency.
The absorption-enhancing device described here operates properly for one kind of circular polarization only. In order to provide enhancement independent of the polarization of the incident light, a second optical diode (but with inverted orientation compared to the first one) should be placed next to existing one (transversely to the direction of propagation of incident light). Incoming unpolarized light can be first split into left and right circular polarizations before reaching each diode, using a device such as a cycloidal diffractive waveplate (CDW) . This device, which can also be made of cholesteric liquid crystals, has the ability to split unpolarized light into two beams of opposite circular polarizations, which also exit at different locations after the CDW.
4. Cylindrical optical absorption enhancement
In this section we investigate the possibility of applying the optical absorption enhancement mechanism described in section 3 to a cylindrical geometry. This is of interest for geometries where light is collected or focused to a small region of space, using for example a parabolic reflector as shown schematically in Fig. 6(a) . The half cylinder is placed at the focus of the parabolic reflector and consists of the absorbing region which includes the diode. Any cut along the radius of the cylinder yields a one-dimensional optical diode profile, as examined in Section 2. Plane waves propagating perpendicular to the directrix of the parabola are reflected and imping perpendicularly to the cylindrical core for all azimuthal angles.
Here, we examine via simulations the performance of the core of a cylindrical optical absorption enhancement device shown in Fig. 6(b). The device has azimuthal symmetry and is obtained by rotating the one-dimensional device of Fig. 4 360° around its edge (the point where the left edge of the mirror is located). We assume that inward propagating, circularly polarized cylindrical waves are impinging perpendicularly to the surface of the device, i.e. they are described at the impinging surface by the expression
The crystal director in this case lies in the plane of the laboratory frame. The dielectric tensor of each CLC layer is calculated in the laboratory frame after applying two rotations. The first is the rotation along the radial length of the diode by an angle , similar to Section 2 and Eq. (1). The length of each diode is now measured along the direction instead along the direction. The second rotation occurs in the plane by an angle . Thus, the expression for the permittivity tensor for each CLC layer is given by 
First, we examine the performance of the device as a diode, by verifying its selective reflection properties. Figure 7 shows the amplitude of the electric field distribution when inward propagating LCP waves at 500 nm (Fig. 7(a)) and 600 nm (Fig. 7(b)) are impinging on the structure. Moving radially inwards, the computational domain consists of five layers. 1) A 1 μm thick air layer 2) A 5.6 μm thick CLC layer (CLC2) 3) A 2.1 μm thick half wave plate 4) A 5.6 μm thick CLC layer (CLC1) 5) An absorbing material with radial thickness 1 μm and relative permittivity . When compared to the planar diode of Section 2, the thickness of each CLC layer is increased three times in order to achieve comparable attenuation of circularly polarized light along its length (as in Fig. 2). This is necessary to counterbalance the increase in the field intensity that is caused by the focusing of the waves as they propagate towards the center of the device, which would increase as for decreasing in free space. While a full 360° version of the cylindrical diode is modelled in the simulation, a fabricated version of this system would look more like Fig. 6(a), where a semi-circular or a flat mirror would be the innermost component of the device.
It is observed that the cylindrical geometry does indeed preserve the selective reflection characteristics of the diode, as the field inside the device for 500 nm illumination in Fig. 7(a) is much smaller than the field transmitted for 600 nm illumination in Fig. 7(b). For the simulations presented in Fig. 7, the absorbing core of the device was chosen to have in order to effectively absorb all incoming fields.
We also observe that the field distributions in Fig. 7 are not azimuthally symmetric. This may be due to the distribution of the liquid crystal permittivity tensor which, after being transformed to the laboratory frame (which coincides with the axes in Fig. 6(b)), becomes azimuthally asymmetric. Note that this effect is not related to insufficient spatial resolution of the structure in the simulation. It is most likely due to the fact that, near the center of the device, the wavelength is not small enough to resolve the fine details in the rotation of the permittivity tensor, and thus the polarization sense of the wave is not properly rotated as the size of the crystals becomes arbitrarily small. Another relevant reason is that in the cylindrical case the wave scatters off the cylindrical mirror instead of simply reflecting, since the wavelength is comparable to the size of the mirror. The latter mechanism is explained in detail at the end of this section.
Next, we repeat the optical absorption enhancement calculation of Section 3. Instead of a 1 μm thick absorbing region, we have a 0.5 μm thick ring absorber, with a mirror (high conductivity material) with radius 0.5 μm at its core. Then, the fraction of the incident power that is deposited in the absorptive ring around the central mirror of the device is evaluated, when its loss tangent is varied (solid red line), for an illumination wavelength of 600 nm. The result is compared again to the free space case (dashed green line), where no diode is present and only the 0.5 μm-thick ring-shaped absorbing material with is placed in air around the mirror, and is reported in Fig. 8 . It is observed that in the cylindrical case the absorption enhancement mechanism is also present, boosting the efficiency beyond what can be achieved by a simple ring-shaped absorber in air. Similarly to what was observed in the planar scenario, for high values of the loss tangent of the absorption region (larger than 0.02), the absorption efficiency is higher without the optical diode present. However for weakly absorbing materials the presence of the diode enhances again the amount of power deposited, close to 100%.
Figure 8 also reports the absorption efficiency where, at the core of the structure, a plain cylindrical absorber with is placed in air, without a mirror or optical diode around it. For a more fair comparison, the volume of the absorption material is scaled to be the same in this scenario, and thus the cylindrical absorptive core has a radius of 0.86 μm. It is observed that the diode mechanism again provides superior absorption efficiency compared to this case too, especially for weak loss tangents.
Despite the significant absorption enhancement in this scenario, the overall absorption enhancement is slightly lower compared to the planar case. There are a few possible explanations as to why the absorption enhancement is not as strong in the cylindrical case, when it is more than evident in the planar diode case. First, the inwardly propagating cylindrical waves cause an increase in the field amplitude towards the center of the structure. As a result, the field attenuates at a slower rate along the radial direction of the structure compared to the attenuation in the planar case (which is shown in Fig. 2). In order to achieve the same attenuation factor and thus similar diode strength, the length of the CLC layers had to be extended. This causes further unwanted reflections originating from the increased number of liquid crystal molecules. This does not occur at all in the planar case where, excluding the effects of the diode, the field amplitude remains constant along the length of the device.
Second, the permittivity tensors always remain perpendicular to the direction of propagation of the waves. In the planar case, this is also imposed by the periodic boundary conditions. The planar waves always sample the same permittivity tensor at a given length along the diode. In the cylindrical case, however, the nearby permittivity tensors (perpendicular to the direction of propagation) have slightly different values. In a ray optics limit this would not be an issue, as the rays would only “see” a single value of the tensor in the crystal. However the wavelength in our scenario is large enough so that the wave samples an extended region of space as it propagates towards the center of the structure, essentially averaging over the nearby permittivity tensors. In essence, the spatial variation in the permittivity increases dramatically towards the center of the structure, without a corresponding increase in the resolving capability of the incoming waves. The same may hold true for the effect of the half wave plate. This phenomenon is not affected significantly by the spatial resolution of the simulation, and it becomes stronger as the waves approach the center of the structure.
Finally, a third possible reason for the slight deterioration of the absorption in the cylindrical case compared to the planar one is the finite size of the wavelength which causes scattering off the cylindrical core, instead of a simple reflection. It is true that in the ray-optics limit where the wavelength becomes arbitrarily small, the incident wave would be reflected perfectly from the cylinder with its polarization sense inverted, as is the case with the planar reflection. However, in the cylindrical case examined here the wavelength is comparable to the size of the cylindrical mirror. As a result, instead of an ideal reflection (which occurs only when the cylinder is much larger than the wavelength), a more complicated, polarization-dependent scattering from the cylinder occurs. This does not occur in the planar case where, due to the periodic conditions imposed, an effectively infinite mirror plane is simulated, and the reflection is not wavelength dependent.
5. Conclusions and discussion
We have investigated the absorption enhancement of light at visible frequencies, using an optical diode made from cholesteric and nematic liquid crystals. Based on full-wave electromagnetic simulations, we find that, in a planar geometry, the selective reflection properties of the liquid crystal layers result in asymmetric transmission through the device for forward and backward propagation of circularly polarized plane waves over certain frequency ranges. Subsequently, we demonstrate an absorption enhancement mechanism where an absorbing region is sandwiched between a mirror and the optical diode, which is found to double the absorption efficiency when the overall absorption is initially relatively weak (loss tangent <0.06). Similar enhancement was found to extend straightforwardly to cylindrical geometries too.
Regarding practical materials that could be used in the fabrication of the devices examined here, demonstrated examples have utilized polymeric CLCs  and low-molecular-weight CLCs [35,36]. The CLC layers in the optical diode fabricated in , for instance, consisted of a mixture of two commercially available nematic liquid crystal aromatic polyester polymers which contain chiral units. The resulting helical pitches could be controlled by tuning the ratio of the polymers in the final mixture. In the same design, the anisotropic half wave plate was fabricated using dye-doped nematic liquid crystal films.
For solar cell applications, the helical pitch should not vary strongly over a significant temperature range in order for the reflective properties of the optical diode to be maintained. While no direct measurements of the pitch dependence on temperature seem to be available for polymeric CLCs, normally most pure CLCs exhibit a slight decrease of the pitch with increased temperature, in analogy with thermal expansion in crystals . The case is usually different for mixtures, where the rate of change for the pitch can be positive or negative with increasing temperature, depending on the exact material ratio, the solvent, and the temperature range . As an example, a mixture of 85% cholesteric liquid crystal and 15% nematic liquid crystal in  showed a drop in the reflection band from 575 nm to 560 nm as the temperature was increased from 20°C to 60°C.
In all aforementioned works, while the pitch may be varying with temperature, it retains its periodic nature. As a result, optical absorption enhancement is expected to be exhibited with high absorption efficiency, albeit centered at slightly different frequency range. However, due to the broadband absorption occurring in solar cells, this is not expected to impact the overall performance of the devices.
The research leading to these results has received funding from the European Union's Seven Framework Programme (FP7/2007-2013) under Grant Agreement No.228455-NANOGOLD (Self-organized nanomaterials for tailored optical and electrical properties).
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