Cone-Shaped Emissions in Cholesteric Liquid Crystal Lasers: The

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Cone-shaped emissions in cholesteric liquid crystal lasers: The role of anomalous scattering in photonic structures César L. Folcia, Josu Ortega, and Jesus Etxebarria ACS Photonics, Just Accepted Manuscript • DOI: 10.1021/acsphotonics.8b01011 • Publication Date (Web): 16 Oct 2018 Downloaded from http://pubs.acs.org on October 22, 2018

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Cone-shaped emissions in cholesteric liquid crystal lasers: The role of anomalous scattering in photonic structures

César L. Folcia1, Josu Ortega2, and Jesús Etxebarria1* *Corresponding Author: E-mail: [email protected] 1Departamento

de Física de la Materia Condensada, Facultad de Ciencia y Tecnología, UPV/EHU, 48080, Bilbao, Spain 2Departamento de Física Aplicada II, Facultad de Ciencia y Tecnología, UPV/EHU, 48080, Bilbao, Spain

The out-of-normal emission in cholesteric liquid-crystal lasers is studied experimental and theoretically. Apart from the well-known dominant laser in the direction of the helix axis, three other types of radiation, with cone-shaped spatial patterns, are identified and characterized. The physical mechanisms responsible for the different emissions are clarified. One of the radiations is a weak colorful lasing emission whose wavelength changes continuously depending on the propagation direction. The wavelengths of the other two radiations take place at the long-wavelength and short-wavelength edges of the photonic bandgap. These emissions are attributed to an anomalous scattering phenomenon that gives rise to energy transfer from the main laser beam to some specific directions where the amount of final photonic states is high. An expression for the scattering cross section, reminiscent of Fermi's golden rule for spontaneous emission in photonic structures, is proposed. Some other phenomena independent of the lasing occurrence but driven by the anomalous scattering are briefly presented.

Keywords: cholesteric liquid crystals, distributed feedback lasers, scattering, photonic structures

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Cholesteric liquid crystals (CLCs) are constituted by chiral molecules that self-assemble forming a helical structure. The structure can be regarded as a one-dimensional photonic crystal when the helical pitch is of the order of the wavelength of visible light. The gap is dependent on light polarization, in such a way that circularly polarized light with the same handedness as that of the helix cannot be transmitted along the helix axis in the spectral region of the gap. The gap region is located between pne and pno, where p is the pitch and ne and no the extraordinary and ordinary refractive indices respectively. The photonic character of CLCs is the basis for their application to build distributed feedback (DFB) lasers.1-16 These lasers are made of CLC materials doped with fluorescent dyes whose emission spectra are significantly altered by the photonic structure.17,18 The lasing characteristics of CLC lasers have received great attention over the last years because of their high versatility and the possibility of giving rise to new low-cost miniaturized devices.19-23 However, further improvement in the performance of these lasers is still needed for potential applications. In this respect it is interesting to point out that some basic features of their working principles are not completely understood yet. In this paper we will study some characteristics of the spectral and angular dependences of the light emission of these devices, with special attention to the analysis of the out-of-normal lasing phenomenon. The detailed emission of CLC lasers is complicated, showing nontrivial spatial and spectral distributions. The main contribution (usually the only considered) is the laser emission in a direction normal to the cell substrates (along the CLC helix axis). If the dye used has its transition dipole moment essentially oriented along the molecular director the emissions


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happen at a wavelength

e  pne (emission at the long-wavelength edge (LWE) of the

gap). This is the usual situation. We will call this kind of radiation type I emission. There are some much less intense contributions around normal emergence at the same e. These contributions have been associated in the literature to diffraction phenomena because their spatial distribution reminds us of Fraunhofer diffraction rings by a circular aperture.3,18,24 These waves will constitute type II emission. At oblique angles there are further lasing emissions at other wavelengths.25-29 In fact there are emissions at all angles. The spectral composition of these waves reveals light of laserlike spectral purity, but their intensities are much smaller than type I emission. This kind of radiation is seldom considered in practice. We will call it type III emission. On some occasions additional emissions can be observed. Several remarkable cases were reported in Refs. [26,30-36] for CLC lasers made of positive uniaxial ( ne  no ) materials. In these works lasing is observed at normal direction with a wavelength o  pno (type I emission at the short-wavelength edge (SWE) of the gap). This is not usual but can be explained by the existence of certain disorder of the dye molecules in the CLC. What is more interesting is the appearance of a rather intense ring at the same wavelength o. This cone laser constitutes what we will call type IV emission and occurs at an internal angle





given by arccos no ne . Apparently, there is a coupling between the light from the central laser and the light from the cone laser, both with the same wavelength. In the first case the SWE mode concentrates the energy in the polarization perpendicular to the director, whereas in the second case (LWE mode) the energy is concentrated in the polarization parallel to the director. Type IV emission phenomena were also reported for laser emission in a 1D dual-periodic photonic crystal based on dichromated gelatin.37


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At still larger angles the so-called lasing leaky modes can also appear.38-41 These modes propagate nearly parallel to the substrates due to total internal reflection. They are characterized by a wide spectrum of output light and constitute a serious parasitic effect, which lowers the lasing efficiency of DFB CLC lasers. We will not deal with leaky modes in the present work. In this article, we study experimental and theoretically the four types of emission (I-IV) described above. In Section 2 we present the samples and the experimental method. The experimental results are shown in Section 3. We interpret the results in Section 4, where we propose a theoretical model that accounts for most the experimental facts. Some conclusions are drawn in Section 5.

Materials and Methods

Four CLC samples were studied (CLC1-CLC4). All of them were mixtures based on the nematic material E7 (Synthon) and the chiral twisting agent D* (compound 2 in Ref. [42]). Three of them were doped with a small amount of the dye PM597 (Aldrich). The proportions (in wt. %) together with some of the characteristics of the samples are gathered in Table 1. The chemical structures of E7, D* and PM597 are shown in Figure 1. In all cases the long molecular axes align preferentially along the molecular director. The transition dipole moment of the dye lies also along the long axis of the molecule. Optical textures revealed very good alignment in all samples. See a typical texture in the same Figure.


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Figure 1. Chemical structure of the materials and a typical photomicrograph of one of the samples. Table 1. Compositions, band-gap ranges and thicknesses of the studied samples. The edges of the reflection bands are related to the pitch p through ne and no respectively Sample

E7 wt.%

D* wt.%

CLC1 CLC2 CLC3 CLC4

93.3 94.0 93.3 94.6

5.8 5.1 5.8 5.4

PM597 wt.% 0.9 0.9 0.9 0.0

Reflection band nm

Helical Pitch nm

Thickness mm

343 391 343 370

10 25 28 10

521-592 598-674 521-592 565-638

The band gaps of the materials were characterized by measuring their reflectance spectra with a fiber-based spectrometer (Avantes). The materials were aligned planarly (helix perpendicular to the cell substrates) in cells of several thicknesses.

Figure 2 shows the experimental setup. Cells were optically pumped using a Nd:YAG laser operating at the second-harmonic frequency (wavelength 532 nm). The laser emitted 13 ns-


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long pulses with a repetition rate of 5 Hz. The beam was focused on the samples at normal incidence by using a lens of 20 cm of focal length. Pulsed pump energies were varied close to the lasing threshold to avoid important disturbances of the sample condition. The light polarization was circular right, with opposite sense to that of the CLC helix (left-handed helix) in order to optimize the excitation conditions. The spot on the sample was nearly Gaussian, with a diameter D4σ = 240 μm (FWHM = 144 μm). The light emitted by the sample was collected by a fiber-based spectrometer (AvaSpec 2048) with   0.5 nm of resolution (FWHM). The fiber was situated 30 mm away from the sample, and a notch filter for 532 nm was positioned just before the fiber. Both elements were placed on a rotation stage that had a resolution of 0.5°. For some experiments the fiber spectrometer was replaced by a screen, where the far-field pattern of the light emitted by the sample was projected.

Figure 2. Experimental setup used to measure the light emission spectra at oblique angles.

Experimental results We have observed all kinds of emissions (I-IV) using CLC1 and CLC2. The inset of 2a shows a photograph of the far-field emission pattern of CLC1, where type I and II emissions are visible (a secondary ring can be seen). The wavelength of the radiation is


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e  pne  592 nm , and corresponds to the long-wavelength edge (LWE) of the band gap. The intensity of the light at that wavelength as a function of the external angle of emergence  is represented by the black symbols of Figure 3a.

Figure 3. (a) Measured intensity for CLC1 as a function of the external angle for e  pne  592 nm (black circles) and for type III emission (white circles). The inset shows the image captured on a screen. The photograph was taken at an angle (about 45º) respect to the image plane. Type I (central spot) and type II (ring) emissions can be observed. Type III emission is not appreciated in the picture because of its low intensity. (b) Spectra obtained for some selected angles where type II emission at 592 nm is detected together with type III one, whose wavelength shifts towards shorter wavelengths on increasing external angles.

Type III emission could hardly be observed in the photograph but was easily detected using the fiber spectrometer. Figure 3b shows several examples for different angles, where type II and type III emissions can be observed to coexist. For type III emission, the wavelength  shifts with the angle following the Bragg law,

 ( r )  e cos r 




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where  r is the internal angle of emergence (related to the external angle  by the Snell law). White dots in Figure 3a show the intensity versus the external angle for type III radiation. As can be seen, it is even weaker than type II emission over most of the angular range. Using the CLC2 cell we were able to observe type IV emission (see Figure 4). Both the central spot and ring had a wavelength o  pno  598 nm , which corresponds to the shortwavelength edge (SWE) of the band gap. The pumping power to achieve lasing was much higher than for CLC1. This is normal because the PM597 dye molecules align essentially along the molecular director (the dye order parameter is Sd = 0.4 in E7 16), thus favoring LWE emission. The ring takes place at an external angle of 47º, in accordance with the theory (internal angle  r  arccos(no ne ) ). Preliminary results indicate that the four types of emission are approximately circularly polarized with the same handedness as that of the helix (left).


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Figure 4. Intensity at wavelength o  598 nm vs. external angle  for the CLC2 sample. In the photograph the far-field emission pattern is shown where the type IV light cone is clearly observed. The different color of the central emission is an artifact of the camera due to saturation.

Discussion

General considerations

The theoretical analysis of the laser emission in the CLC materials was carried out using the Berreman matrix method,43 which is an adequate means to treat the propagation of electromagnetic waves in stratified and anisotropic media. The method can be easily adapted to an amplifying medium simply by using a complex dielectric tensor at each layer with a negative imaginary part. This imaginary part is a phenomenological parameter that is actually achieved through the optical pumping. The pumping produces a population inversion in the dye molecules which, in its turn, gives rise to light amplification. In our calculations we have assumed the existence of gain anisotropy, with two principal values

 and   along and perpendicular to the local director. Typically, the ratio    is ||

||

similar to the ratio of absorption coefficients for polarized light parallel and perpendicular to the director. This can be understood, since both the local absorption coefficient and the gain are connected to the orientation of the transition dipole moment of the dye in a similar fashion. Once the emission anisotropy is known, only one parameter is needed to determine the amplifying features of the medium. We used the  || parameter. Under amplifying conditions the sum of the reflectance R and transmittance T verifies R+T >1. As input light,


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we used diffracting eigenstates 44 with intensity value equal to unity, and found the  || parameter that gives an output intensity spectrum characteristic of laser emission. It results that for a critical value  th , the output intensity diverges at a given wavelength. This is the ||

situation for which laser emission occurs. For    th , lasing does not occur anymore. ||

||

Therefore,  th is the parameter to be calculated in order to determine the lasing features of ||

the sample. Usually, the lower the  th parameter the better the laser performance. In the ||

case of normal incidence, approximate analytical expressions for  th have been given in the ||

literature:45

 th  p 3  2 L3 ,

(2)

||

where L is the sample thickness and  is a parameter related to the anisotropy of the refractive indices,



  ne2  no2

 n

2 e



 no2 .

(3)

 th is related with other parameters easier to interpret and more used in practice, such as ||





the photon dwelling time in the cavity  c   2 c th or the threshold for the population ||

inversion of dyes 16,46


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n2th  2 th p e ,

(4)

||

where  e is the cross section of fluorescent emission for the corresponding polarization eigenstate at the laser wavelength and c is the speed of light in vacuum. Essentially n2th is a direct measure of the laser threshold energy. The topic of DOS at normal and oblique incidences in systems with anisotropic gain has been also studied in references [47-50]. The case of localized optical defect modes in CLC samples at normal incidence has also been treated analytically by Belyakov and Semenov.51

Type III emission

The theoretical explanation for laser emission at oblique angles can be accomplished following the Berreman method by using an input wave with the polarization of the diffracting eigenstate at the corresponding angle of incidence. The effect occurs if  ||    . The resulting laser wavelength happens to be given by the Bragg law (see black dots in Figure 5). Moreover, it turns out that the laser at oblique angles would be, in principle, as efficient as the laser along the substrate normal, in the sense that the critical gain  th , shows ||

no important dependence on the angle of incidence (blue dots in Figure 5). Thus similar

n2th and  c values are expected for all angles. This is surprising because, according to Equation 4, the laser efficiency would be determined by the position of the fluorescence band maximum (maximum of  e ). It seems then that we could have the most intense laser


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at an oblique direction, contrarily to the experiments. This could be simply achieved with a dye whose fluorescence was peaked at a wavelength   e cos r  with  r  0 

Figure 5. Calculated threshold gain  th vs. external angle of laser emission  (blue points) ||

and the corresponding wavelength (black points) for the parameters of CLC1. Lines are eye guides.

However, there is an additional geometrical issue that plays an important role in the determination of the laser intensity. Laser light at a given angle has a small angular tolerance to escape from the sample, i.e., the angle at which laser radiation of a given wavelength can be emitted is not unique but restricted to a narrow angular range. From the Berreman method itself, it is deduced that the amount of emerging light is very sensitive to the angle. For oblique angles the laser light is greatly attenuated as we move a very small angle away from the corresponding incidence. The situation is different around normal incidence, where the margin for non-negligible transmission is much larger, and reasonable


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amount of light can emerge for angles of incidence not exactly zero. This point is essential to calculate the laser intensity at a particular wavelength because, instead of using a unique idealized plane wave (as the Berreman method assumes), we must consider all the incident directions that present amplified transmission at that wavelength. To see this question in more detail we chose a particular wavelength  and computed the



complex transmission coefficient t  of the diffracting eigenstate as a function of the



angle of incidence . The inset of Figure 6 shows t  and

2

for two wavelengths, 1  pne

(laser at normal emergence and at

, respectively). We took a



CLC similar to our CLC1 sample and a gain close to its critical value.[52] Functions t  i

2

attain their maxima at  values corresponding to internal angles given by

 r,i  arccos  i pne  , as expected. On the other hand, though the angular widths of the functions around their maxima are very small (smaller than the angular resolution of our spectrometer), they are very different to each other,

. Consequently,

the integrated transmission, which must be proportional to the laser intensity, presents an abrupt maximum for the normal direction (see Figure 6). The maximum is more pronounced the larger the sample thickness and the closer we approach the critical gain. This behavior explains why the laser in normal direction is the only important radiation in practice. The more perfect the laser the weaker the relative out-of-normal emission. Curiously, however, the laser threshold is essentially independent of the emission angle, because only n2th is involved. We have also checked this point in our experiments: thresholds for type I and type III lasers are essentially the same.


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Figure 6. Angular dependence of laser intensity of type III. Black dots represent the values calculated by means of Berreman’s formalism, whereas white dots are experimental results obtained using the setup of Figure 2 for CLC1. In the inset, the angular profile of the



normalized t  i

2

function is depicted around   0 (red line) and

(black line).

Type IV and II emissions Let us turn now to give an account for the type IV emission. As has been pointed out above, the phenomenon takes place as if there were a coupling between the SWE laser mode at normal emergence and the LWE mode at an oblique angle, both with the same wavelength (see Figure 7a). To explain this phenomenon our proposal is that, physically, this coupling is nothing but light scattering, which gives rise to energy transfer between both modes. We


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will see below that the photonic structure produces a great alteration of the angular distribution of the scattered light. A different approach based on this idea has been also proposed in ref. [53] to analyze the directional spontaneous emission spectra from light sources in photonic crystals. More specifically, we put forward the following expression for the scattering cross section  between initial and final waves of frequency , with wave vectors ki, kf, and polarizations î , fˆ respectively:





 | k i , î, | S | k f , fˆ , |2  k f , fˆ , ,

(5)

where  k i , î, | S | k f , fˆ ,  denotes the matrix element for scattering between the initial





and final states, and  k f , fˆ , is the density of final optical states (DOS). Equation 5 can be considered the counterpart to scattering of Fermi's golden rule for the spontaneous emission in photonic structures. In both cases the key point is the proportionality of the probability for the effect on the amount of available final optical states. The specific form of the matrix element in Equation 5 is not especially important. In general it will be a complicated function involving the elastic constants of the CLC. Here, we simply give an approximate expression for that matrix element, which is valid only in the simpler case of an isotropic liquid:





| k i , î, | S | k i , fˆ , |2  C | î | fˆ |2 |  k f  k i |2 ,



(6)

where   q  is the thermal average of the Fourier transform of the dielectric constant and C is a constant.


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Because of the  dependence of the scattering cross section, the scattering can be either enhanced or inhibited respect to the same property in vacuum. If there are no optical modes available as final states the process will be forbidden. If a large  occurs only at a given angle (as in Figure 7b), then scattering will mainly take place at that angle. This is precisely the case of the experiment described in Figure 4.

Figure 7. (a) Condition for the occurrence of type IV emission. In the figure the reflectances corresponding to normal (black) and oblique (red) incidences are depicted. The cone emission occurs at the angle for which the LWE of the oblique reflectance coincides with the SWE at normal incidence (blue arrow). (b) DOS calculated as a function of the external angle for wavelength 598 nm using the parameters of CLC2. A sharp maximum results at 47º, i.e., the external angle where the ring in Figure 4 was observed.

The anomalous scattering can also explain the presence of diffraction-like rings around the laser at normal emergence (type II emission). If the wavelength of the central laser is at the LWE of the gap, this light will undergo important scattering at internal angles given by

 r,i  arccos  pne i 


(7)

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where i are the wavelengths of the secondary minima of the reflectance at normal incidence for the left-handed eigenwaves (see Figure 8a). At those angles the DOS presents secondary maxima with decreasing importance as we move away from the central maximum. Figure 8b shows the computed DOS as a function of the external angle for

  pne using the data of our CLC1 sample. Evidently the emitted radiation will keep its original wavelength (pne) and its intensity will be roughly proportional to that curve. The photograph in Figure 8b is the far-field pattern of type II radiation for CLC1 under rather high pumping power, in order to observe as many rings as possible. A good angular correlation between the photograph and the profile of DOS was observed (see also black dots in Figure 3a).

Figure 8. (a) Condition for the occurrence of type II emission. Black, blue and red lines represent, respectively, the reflectance spectra at normal incidence and at internal angles  r,1 and  r,2 . These angles are defined in Equation 7. 1 and 2 are the wavelengths of two secondary minima of the reflectance at normal incidence. The second and third minima of the blue and red lines coincide with the first minimum at normal incidence (red arrow). (b) DOS versus emission angle for type II radiation at the wavelength indicated in Figure 8a.


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The calculations were carried out using the parameters of CLC1. In the photograph the farfield pattern of type II emission for that sample can be seen. The different color of the central emission is an artifact of the camera due to saturation. A different alternative has been proposed in the literature to explain type II emission.3,18 According to that scheme the structure of concentric rings would be due to a Fraunhoferlike diffraction due to the existence in the sample of small spatially coherent circular areas. In this approach, the diffraction patterns from these circular emitting disks would produce the observed ring structure. In response to this idea we have two important objections: -The ratios between the intensities of the central maximum and secondary maxima are not correctly explained. For example, according to Airy's pattern the intensities of the central and first secondary maximum are in a relation of about 50, much larger than the experimental value (about 5, see black symbols in Figure 3a). - The fit between Airy's pattern and the angular positions for the observed maxima and minima requires the specification of a definite coherence length lc. For CLC1 the fit is reasonable if we take lc ~ 5  m . However, the ring pattern presents very important modifications if cells of different thickness are used. Figure 9 shows the situation for a cell of 28 m (CLC3 sample). As can be seen, now we have many more rings and would need a much larger lc to fit the new pattern. It seems difficult to explain such an important change in lc by a mere thickness variation in the same material. In contrast to this approach, the model of anomalous scattering can explain the new angular positions in a natural way (see Figure 9).


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Figure 9. DOS for the CLC3 sample calculated as a function of external angle for   pne  592 nm. The inset shows an image captured on a screen situated at about 5 cm from the sample. A ring pattern, with many more rings than in Figure 8, is observed. From the diameters of the minima, external angles were estimated with a good correspondence with the minima of the curve. Other details observed for type IV emission are also satisfactorily explained within the scattering model. For example, it is interesting to point out that the central maximum of a laser at the SWE of the gap is not surrounded by diffraction-like concentric rings, contrarily to what happens if the laser emits at the LWE. This feature is very clear not only in our photographs (Figure 4) but in all images published up to now by other authors.26,33,36 However it had never even been mentioned in the literature. The explanation of the effect is evident, since the secondary maxima in transmittance near the SWE occur at wavelengths smaller than pno, ruling out any possibility to form rings. Another remarkable characteristic observed in the pattern of type IV emission has to do with an asymmetry in the location of some weaker rings associated to the main one. As can be noticed in Figure 3b of Ref. [36]


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the secondary rings always occur at angles larger than the most intense ring, and there is no emission at the other side of the ring. The explanation of the effect is again straightforward because the secondary maxima of  around the ring angle (47º in Figure 7b) only take place at larger external angles.

Further effects driven by anomalous scattering

The existence of anomalous scattering in photonic structures also gives rise to peculiar phenomenology independent of the presence of DFB laser emission. Two examples are presented below. We reproduced type IV emission without having DFB lasing by simply illuminating a sample at normal incidence with left-handed circularly polarized light of wavelength

  pno . For the experiment we used the passive sample (CLC4). The light that gets into the sample from the outside behaves as if it were an internally generated SWE laser beam that propagates along the helix axis. In this situation, strong scattering is expected at an









internal angle  r  arccos  pne  arccos no ne which is associated to the highest  , i.e., the LWE of the gap. We carried out the experiment by using a supercontinuum laser followed by a polarizer, an achromatic quarter wave plate, and a high-pass filter of 550 nm. The filter practically eliminated all light with wavelengths   pno . A green ring at an external angle of 47º was observed on a screen (Figure 10a), together with successive rings at larger angles, which correspond to the secondary maxima of  near the LWE (see Figure 7b). We also checked that the effect vanishes for circularly polarized light of the opposite chirality.


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Another remarkable phenomenon can be observed by using input light as in the previous case but with wavelength   pno . It is well known that in this case some light can enter the material. According to Equation 5, this light will be scattered along cones with internal





half openings  r,i  arccos  i , where i are the wavelengths that correspond to a high DOS (   i ). The most important of these wavelengths is precisely the SWE, i.e.,

i  pno . There, we will have a bright ring. Other less important rings will also appear for smaller  i . These angles correspond to the positions of the secondary maxima of  near





the SWE region. For  r  arccos  pno we will have a broad angular range without any





scattering, until we reach the internal angle of the LWE of the gap, arccos  pne . We verified experimentally all these phenomena again with the CLC4 sample (Figure 10b). We used an illumination wavelength   532 nm , and the most intense ring was observed at an external angle of 32º ( pno  565 nm ). Three somewhat weaker rings were also visible at angles 29º, 25º and 21º ( i  560 , 553 and 547 nm). No scattering could be seen between 32º and 59º, the latter corresponding to the LWE of the gap (638 nm). At large angles the scattering was rather diffuse because the overlapping between î and fˆ polarizations decreases (see Equation 6). All these anomalous effects disappeared if the incident light had the handedness opposite to that of the helix.


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Figure 10. (a) Far-field emission of sample CLC4 illuminated with circularly polarized light with the same handedness as that of the helix and wavelength   pno . Cone-





scattering emission can be clearly observed at angles  r  arccos no ne . Three rings (1, 2, 3) are visible. (b) Reflectance spectrum of the CLC4 sample at normal incidence. The inset shows the image captured on a screen when illuminating the sample perpendicularly with left-handed circularly polarized light of wavelength 532 nm (green arrow). Four rings (1, 2, 3, 4) are clearly seen. They correspond to i for minimum reflectance (blue arrows). In both cases the rings disappear when the sample is illuminated with right-handed circularly polarized light. Conclusions We have clarified the origin of out-of-normal emissions in CLC lasers. Apart from the usual laser propagating along the helical axis, only the very weak multi-wavelength type III emission is genuine laser light, in the sense that corresponds to a light amplification phenomenon. In contrast, the comparatively stronger ring structures of type II and type IV emissions are due to energy leakage from the central laser light. The physical mechanism behind this effect is simply light scattering, whose angular dependence is greatly altered in


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photonic structures, being particularly strong where the amount of final photonic states is high.

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When



||

is exactly equal to the critical value

 th

the laser intensity presents a divergence, and its spectral

||

width tends to zero. In practice, there are always imperfections in the sample that limit these exceptional characteristics. These imperfections have been modeled using a gain slightly different from

 th . ||

53

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from light sources in photonic crystals. Phys. Rev. A 2005, 71, 053813.


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For Table of Contents Use Only.

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Synopsis: Cone-shaped emissions in cholesteric liquid crystal lasers


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Title Cone-shaped emissions in cholesteric liquid crystal lasers: The role of anomalous scattering in photonic structures Authors César L. Folcia, Josu Ortega, and Jesús Etxebarria


Cone-Shaped Emissions in Cholesteric Liquid Crystal Lasers: The

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