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The Effect of Defect Layer on Lasing in Cholesteric Liquid Crystal

The photonic density of states (PDS) of eigen polarizations (EPs) in cholesteric liquid crystal (CLC) cells with a defect layer inside are calculated. The dependences for the PDS and light intensity in the defect layer on the parameters characterizing absorption and gain are obtained. We investigated the possibility of connections between the PDS and the density of the light energy accumulated in the system. The influence of the defect layer and CLC layer on the PDS are investigated. It is shown that the PDS is maximum when the defect is in the centre of the system. We showed also that the subject system can work as a low threshold laser, a multiposition trigger, filter, etc.


INTRODUCTION
Photonic crystals (PCs) consisting of artificial or self-organizing periodic structures have opened possibilities for novel physics and device applications. PCs as laser cavities have been the focus of intense research in recent decades. Lasing in these structures takes place either due to the distributed feedback mechanism or defect modes (in PCs with defects), where spontaneous emission is inhibited in the wavelengths inside the photonic band gaps (PBGs) except at the band edges or defect modes where the photonic density of states (PDS) is large. Especially attractive is the use of cholesteric liquid crystals (CLCs) as band edge lasers which is connected with the fact that the macroscopic properties of CLCs can be manipulated by external stimuli. Lasing in CLC and PCs has been intensely investigated (see, for instance, [1][2][3][4] and the wide literature cited therein). The possibilities of decreasing the lasing threshold in CLC and multilayer systems with CLC layers have been studied both theoretically [5][6][7] and experimentally [8]. In the following three papers [9][10][11] it was reported on the effect of loss and gain on the PDS. Dolganov reported in [12] that a highly sensitive method was used to determine the PDS, as well as a drastic increase in the PDS near the PBG borders and oscillations related to Pendellösung beatings were observed. It is reported in [13] about design and fabrication of a wedge structured CLC film incorporating a spatial gradient of a chiral dopant concentration. A continuous spatial laser tuning in a broad visible spectral range with tuning resolution less than 1 nm is demonstrated, which renders a CLC-based micron-sized laser an important continuously tunable laser device [13]. In review article [14] the ways of employing CLCs in tunable dye lasers were discussed. The investigations in this direction have been intensely going on because the CLC, all the time, exhibit new wonderful and multilateral peculiarities (see, for example, [15][16][17][18][19][20][21][22][23][24]). The performance of light emitting liquid crystal devices may be significantly improved by using more advanced optical structures. To develop new optical architectures one must have at his disposal fundamental understanding of absorption-emission processing and photonic density of states and their other important optical peculiarities, as well as an accurate numerical design tools for their calculation.
In this paper we discussed lasing peculiarities of a CLC layer with an isotropic defect layer inside.

RESULTS AND DISCUSSION
A CLC with an isotropic defect can be treated as a three-layer system: two CLC layers (CLC(1) and CLC(2)) and an isotropic layer (IL) between them (or a Fabry-Perot resonator with diffraction mirrors and an isotropic layer filling). The ordinary and extraordinary refractive indices of the CLC layers are taken to be The defect isotropic layer refractive index was taken to be n =1.8. The problem was solved by Ambartsumian's layer addition modified method [25]. We investigated the peculiarities of absorption, emission and PDS of this system for eigen polarizations (EPs). The EPs are the two polarizations of incident light, which do not change when light transmits through the system.
The presence of a thin defect in the CLC structure is known to initiate a defect mode in the PBG [26,27]. First we investigate some new peculiarities of the PDS for this system. The investigation of the PDS  is important because of the following. For laser emission it has been shown that, for instance, analyzing the case of the Fabry-Perot resonator, the threshold gain g th can be directly related to the maximum PDS max ,  that is, , where n is the refractive index inside the resonator of the length d [28]. Furthermore, according to the spaceindependent rate equations, the slope efficiency of lasers can be shown to be inversely proportional to the threshold energy and, therefore, directly proportional to max  [29]. The PDS, i.e. the number of wave vectors k per unit frequency - is the reverse of the group velocity and can be defined by the expression [29]: where d is One must realize that the concept of the PDS introduced by Eq. (1) is not without controversy, especially in the case of non-periodic systems (more details about this see in [30]).
Let us only note that although it is impossible to ascribe a direct physical meaning to the PDS in Eq. (1) in general case, anyhow, it can be used as a parameter, which can provide some heuristic guidance in experiments for the PDS dispersion-related effects. Moreover, as it had been shown in [9][10][11][12] Eq. (1) is applicable for finite PCs layers, even for the cases with absorption and amplification (naturally, for weak ones).
Below we first show that this equation is applicable (at least qualitatively) to our (aperiodic) system both in the presence of absorption and amplification.
In [10-11, 30,31], it was shown that in the absence of absorption, the PDS was proportional to the integral of the energy stored within the medium. Then in [10,11], it was shown that it takes place both in the presence of absorption and gain ( for more details see  ). n  , that is, if absorption exists in the CLC sublayers and is absent in the defect layer. The CLC layer thickness is: d=44p, the defect layer thickness is: d´=1000nm.
In Fig. 1 the dependences (a, c)  , and (c ,d, g, h) correspond to the case if "" 0 oe nn  and '' 0 n  . The parameters are the same as in Fig.1.
In Fig.2 the evolution (a, c) of the total field intensity 2 E spectra and (b, d) the relative PDS when (a, b, c, d)  For a single CLC layer, as gain increases in the absence of the defect layer, the maximum PDS shifts away from the PBG border, and this does not take place continuously, but in discrete steps [10,11]. Note that when going away from the PBG borders, the critical value of ' x (beyond which the lasing mode is quenched) increases. As it is seen in Fig.2e, if a defect layer is present and gain increases, the PDS increases too, and then quenching takes place. But first it takes place for the defect mode wavelength. It is to be noted that not all modes outside the PBG exist in the case if gain is absent in the CLC layers and it exists in the defect layer. Again, let us note that such changes take place for 2 E too.
Then we considered the influence of the defect layer thickness on the relative PDS. In Fig.3 the evolution of the relative PDS -1,2 / iso spectra when the defect layer thickness increases are presented for (a) the diffracting and (b) non-diffracting EPs. As is well known the defect mode wavelength increases from a minimum to a maximum band gap value if the defect layer optical thickness increases. As we can see in this figure, for this change of the defect layer thickness, the relative PDS 1,2 / iso  on the defect mode does not change smoothly, but with strong oscillations. And this means that there exists a definite value of the defect layer thickness at which the lowest threshold lasing takes place.

3.CONCLUSIONS
Let us note in conclusion that we investigated lasing peculiarities of a CLC layer with a defect inside its structure. We showed that the PDS defined in [29] are applicable (qualitatively, at least) to the subject system, too. In the case of the CLC layer with an isotropic defect layer inside on the defect mode, the 1 / iso  has extremeat the defect mode wavelengthwith a larger height than that at the PBG borders. This provides laser generation with a much lower threshold at the defect modes than that at the PBG borders. Then we investigated the influence of absorption and gain on the PDS and the light field intensity in the defect. Our investigation of the influence of the defect layer thickness on the PDS showed that at the defect layer thickness changes the relative PDS 1,2 / iso  does not change smoothly, but with strong oscillations.
Furthermore, when investigating the influence of the CLC layer thickness on the PDS we showed that though for small CLC layer thicknesses low threshold lasing takes place both for diffracting and non-diffracting EPs, nevertheless, for larger thicknesses it takes place only for the non-diffracting EP. Finally, we investigated the defect location influence on lasing peculiarities of the system.