7.1 Introduction

The field of mirrorless distributed feedback (DFB) lasing in photonic structures consisting of many layers of chiral liquid crystals has recently attracted much attention, mainly due to the possibilities of reaching a low lasing threshold for DFB lasing [1,2,3,4,5,6,7,8]. For definiteness, we study photonic liquid crystals with the example of the best-known type of photonic liquid crystals, i.e., cholesteric liquid crystals (CLC) . The related theory is mainly based on numerical calculations [9] whose results are not always interpreted in the framework of a clear physical picture. Several recent papers [10,11,12,13,14,15,16,17] have shown that an analytic theoretical approach to the problem (sometimes limited by the introduction of approximations) can be used to build a clear physical picture of the linear optics and lasing in the relevant structures. In particular, the physics and the role of localized optical modes (edge and defect modes) in the structures under consideration was clearly demonstrated. The most promising results in DFB lasing relate to defect modes (DM) [12,13,14]. The defect modes existing at a structure defect as a localized electromagnetic eigenstate with its frequency in the forbidden band gap were investigated initially in the three-dimensional periodic dielectric structures [18]. The corresponding defect modes in chiral liquid crystals, and more generally, in spiral media, are very similar to the defect modes in one-dimensional scalar periodic structures. They reveal abnormal reflection and transmission inside the forbidden band gap [1, 2] and allow DFB lasing at a low lasing threshold [3]. The qualitative difference from the case of scalar periodic media consists in the polarization properties. The defect mode in chiral liquid crystals is associated with the circular polarization of an electromagnetic field eigenstate whose chirality sense coincides with that of the chiral liquid crystal helix. Two main types of defects have been studied in chiral liquid crystals up to now. One is a plane layer of some substance, differing from the CLC, dividing a perfect cholesteric structure into two parts, and lying perpendicular to the helical axis of the cholesteric structure [1]. The other defect type is a jump in the cholesteric helix phase at some plane perpendicular to the helical axis (without insertion of any substance at the location of this plane) [2]. Recently, many new types of defect layers have been studied [19,20,21,22,23,24,25,26], such as a CLC layer whose pitch differs from the pitch of two layers sandwiched between these layers [8]. Clearly, there are many versions of the dielectric properties of the defect layer, but considerations below are limited to the first type of defect, a layer inserted in a chiral liquid crystal. Our focus is on the active defect layers (absorbing, amplifying, or changing the light polarization). The reason for this is connected with both experimental research on DFB lasing in CLCs where dyes are placed in a defect layer [26] and the general idea that the unusual properties of DMs manifest themselves most clearly just at the middle of the defect structure , i.e., at the defect layer, where the DM field intensity reaches its maximum. We therefore assume that there is no absorption in the CLC layers of the DMS, and that absorption, amplification, or changes in the light polarization occur only in the defect layer. The analytic approach to studying a DMS with an active defect layer is very similar to the DM studies discussed in Chap. 5 [12, 13], and we therefore present below the final results of the present investigation, referring the reader to [12, 13] for full details of the investigation.

In the present chapter, we discuss an analytical solution of the defect mode associated with insertion of an active defect layer in the perfect cholesteric structure, for light propagating along the helical axis, and we consider some limiting cases simplifying the problem [27].

7.2 Defect Mode at Amplifying (Absorbing) Defect Layer

To consider the defect mode associated with the insertion of an isotropic layer in the perfect cholesteric structure, we have to solve Maxwell equations and a boundary problem for an electromagnetic wave propagating along the cholesteric helix for the layered structure depicted in Fig. 6.1. This investigation was performed in [12, 13] under the assumption that the CLC layers can be absorbing or amplifying in Fig. 6.1. It is possible to use the results of [12, 13] in the present case of an amplifying (absorbing) isotropic defect layer and non-absorbing CLC layers, introducing only some physically clear changes in the formulas obtained in [12, 13]. In this section, we retain the assumptions in [12, 13] that the average dielectric constant ε0 of the CLC coincides with the dielectric constant of the defect layer and the external medium, so that polarization conversion is absent at the interfaces and only light with the diffracting circular polarization has to be taken into account. The main notations of the papers [12, 13] are also maintained in this chapter. As is well known [9], much information about the DM is available from the spectral properties of the DMS transmission, T(d, L), and reflection, R(d, L), coefficients.

Formulas for the optical properties of the structure depicted in Fig. 6.1 can be obtained using the expressions for the amplitude transmission, T(L), and reflection, R(L), coefficients for a single cholesteric layer (see Chap. 1 [28, 29]). The transmission, \( | {{T}({d},{L})} |^{2} \), and reflection, \( | {{R}({d},{L})} |^{2} \), intensity coefficients (of light with diffracting circular polarization) for the whole structure may be presented in the form

$$ | {{T}({d},{L})} |^{2} = | {[ {{T}_{\text{e}} {T}_{\text{d}} \exp ({{\text{i}kd}}(1 + {{\text{i}g}}))} ]/[ {1 - \exp (2{\text{i}}{kd} (1 + { {\text{i}g}})){R}_{\text{d}} {R}_{\text{u}} } ]} |^{2} , $$
(7.1)
$$ | {R(d,L)} |^{2} = {\{ {R_{\text{e}} + R_{\text{u}} T_{\text{e}} T_{\text{u}} \exp (2\text{i}kd(1 + \text{i}g))/[ {1 - \exp (2\text{i}kd(1 + \text{i}g))R_{\text{d}} R_{\text{u}} } ]} \}} |^{2} , $$
(7.2)

where Re(Te), Ru(Tu), and Rd(Td) are the amplitudes of the reflection (transmission) coefficients of the individual CLC layers (see Fig. 6.1) for light incident on the outer top layer surface, on the inner top CLC layer surface from the inserted defect layer, and on the inner bottom CLC layer surface from the inserted defect layer, respectively. It is assumed in deriving (7.1) and (7.2) that the external beam is only incident on the structure (Fig. 6.1) from above. The factor (1 + ig) is related to the defect layer alone and corresponds to the dielectric constant of the defect layer having the form ε0(1 + 2ig), with a small g that is positive for an absorbing defect layer and negative for an amplifying one.

The expressions for the amplitude transmission, T(L), and reflection, R(L), coefficients for a single non-absorbing cholesterol layer of thickness L for light with the diffracting circular polarization are given by (6.1) (see also [28,29,30,31,32]).

The defect mode frequency ωD is determined by the dispersion equation [compare with (6.18)]:

$$ \begin{aligned} & \{ {\exp (2{ {\text{i}kd}}(1 + { {\text{i}g}}))\sin^{2} {qL} - \exp ( - {\text{i}}\tau {L})[ {(\tau {q}/\kappa^{2} )\cos {qL}} } \\ & \quad + {{ { {\text{i}}( {(\tau /2\kappa )^{2} + ({q}/ {\kappa})^{2} - 1} )\sin {qL}} ]^{2} /\delta^{2} } ]} \} = 0 \\ \end{aligned} $$
(7.3)

For CLC layers of finite thickness L, the DM frequency ωD is a complex quantity which may be found by solving (7.3) numerically. For very small values of the parameter g, the reflection and transmission spectra of the DMS with an active defect layer are similar to the spectra studied in Chap. 6 [12, 13] (see Fig. 7.1). In particular, the frequency positions of dips in reflection and spikes in transmission inside the stop band just correspond to Re[ωD], and this observation is very useful for solving the dispersion equation numerically. For its part, the DM lifetime is shorter for absorbing defect layers than for a non-absorbing defect layer [12, 13].

Fig. 7.1
figure 1

Reflection \( | {{R}({d},{L})}|^{2} \) versus frequency for a non-absorbing defect and CLC layers (g = 0) at d/p = 0.1 (a) and d/p = 0.25 (b); δ = 0.05, l = 200, l = Lτ = 2πN, where N is the director half-turn number at the CLC layer thickness L. Here and in all further figures in this chapter, the frequency is defined in the same way as in Chaps. 5 and 6 (see Fig. 5.2). Furthermore, δ = 0.05 and the director half-turn number at the CLC layer thickness is N = 33

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7.2.1 Absorbing Defect Layer

As in the case investigated in Chap. 6 [12, 13], i.e., DMs with non-absorbing CLC layers, the effect of anomalously strong absorption also occurs in DMSs with absorbing defect layers . The effect reveals itself at the DM frequency and reaches its maximum, viz. \( 1 - | {{T}({d},{L})} |^{2} - | {{R}({d},{L})} |^{2} \), for a certain value of g that can be found using the expressions (7.1), (7.2) for \( | {{T}({d},{L})} |^{2} \) and \( | {{R}({d},{L})} |^{2} \). Figure 7.2 demonstrates the existence of the anomalously strong absorption effect at the DM frequency, where it can be seen that the maximum value of the anomalous absorption [28, 33] \( (1 - | {{T}({d},{L})} |^{2} - | {{R}({d},{L})} |^{2} ) \) at two differing values of d/p are reached for g = 0.04978 and g = 0.00008891. (Taken with the opposite sign of g, these are the approximate values of the lasing threshold gain for the same DMSs found in the next section.)

Fig. 7.2
figure 2figure 2

Total absorption \( (1 - | {{T}({d},{L})} |^{2} - | {{R}({d},{L})} |^{2} ) \) versus frequency for an absorbing defect layer and non-absorbing CLC layers at g = 0.04978 (a) and g = 0.08 (b) for d/p = 0.1; at g = 0.00008891 (c) and at g = 0.0008891 (d) for d/p = 22.25

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In the case of thick CLC layers \( (| {q} |{L} \gg 1) \) in the DMS, the g value ensuring maximum absorption can be found analytically:

$$ \begin{aligned}{g}_{ \text{t}} &= ({ L}/{ d})| [ 2{\kappa}^{2} /({ q}^{2} {L}\tau ) ]\exp [ - 2| q |L ]\{ 1 + \{ 1/( {2[ {(\tau /\kappa )^{2} + \delta^{2} } ]^{1/2} } ) \\ &\quad - (\tau /2\kappa )^{2} \}/( {1 - [ {(\tau /\kappa )^{2} + \delta^{2} } ]^{1/2} + (\tau /2\kappa )^{2} }] \}^{ - 1} | \end{aligned} $$
(7.4)

For the defect mode frequency ωD in the middle of the stop band, the maximal absorption corresponds to

$$ {g}_{\text{t}} = (2/3 {\pi })({p}/\delta {d})\exp [ - 2{\pi \delta }({L}/{p})]. $$
(7.5)

As the calculations and the formulas (7.4), (7.5) show, the gain g corresponding to the maximal absorption is approximately inversely proportional to the defect layer thickness d.

7.2.2 Amplifying Defect Layer

In the case of DMSs with an amplifying defect layer (g < 0), the reflection and transmission coefficients diverge at some value of \( | {g} | \). The corresponding values of g are the gain lasing thresholds. They can be found by solving the dispersion Equation (7.3) for g or numerically using the expressions (7.1), (7.2) for \( | {{T}({d},{L})} |^{2} \) and \( | {{R}({d},{L})} |^{2} \), or they can be found approximately by plotting \( | {{T}({d},{L})} |^{2} \) and \( | {{R}({d},{L})} |^{2} \) as functions of g. The third option is illustrated in Figs. 7.3, 7.4, and 7.5 where “almost divergent” values of \( | {{T}({d},{L})} |^{2} \), \( | {{R}({d},{L})} |^{2} \), or the absorption \( (1 - | {{T}({d},{L})} |^{2} - | {{R}({d},{L})} |^{2} {)} \) are shown. The values of g used in Figs. 7.3, 7.4, and 7.5 are close to the threshold values ensuring divergence of \( | {{T}({d},{L})} |^{2} \) and \( | {{R}({d},{L})} |^{2} \). The calculation results show that the minimal threshold \( | {g} | \) occurs when ωD lies precisely in the middle of the stop band and \( | {g} | \) is almost inversely proportional to the defect layer thickness. Figures 7.3 and 7.4 actually correspond to the situation where the defect mode frequency ωD is located close to the midpoint of the stop band, and show that there is a decrease in the lasing threshold gain with increasing thickness of the defect layer. Figure 7.6 corresponds to the situation where the defect mode frequency ωD is located close to the stop-band edge and shows that there is an increase in the lasing threshold gain as the defect mode frequency ωD approaches the stop-band edge.

Fig. 7.3
figure 3

Total absorption \( (1 - | {{T}({d},{L})} |^{2} - | {{R}({d},{L})} |^{2} ) \) versus frequency for an amplifying defect layer and non-absorbing CLC layers at g = −0.0065957 for d/p = 0.25

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Fig. 7.4
figure 4

Transmission \( | {{T}({d},{L})} |^{2} \) versus frequency for an amplifying defect layer and non-absorbing CLC layers at g = −0.001000 for d/p = 2.25 (a); at g = −0.00008891 for d/p = 22.25 (b)

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Fig. 7.5
figure 5

Reflection \( | {{R}({d},{L})} |^{2} \) versus frequency for an amplifying defect layer and non-absorbing CLC layers at g = −0.04978 for d/p = 0.1

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Fig. 7.6
figure 6figure 6figure 6

Calculated diffracting polarization intensity transmission coefficient \( | {{T}({d},{L})} |^{2} \) for a low birefringent defect layer versus frequency for diffracting incident polarization and for a birefringent phase shift at the defect layer thickness equal to \( \Delta {\varphi } \) = π/20 (a), π/16 (b), π/12 (c), π/8 (d), π/6 (e), π/4 (f), π/2 (g), and \( \Delta {\varphi } \) = 0 (h) [figure (h) corresponds to the isotropic defect layer] for a non-absorbing CLC with d/p = 0.25

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The analytic approach for thick CLC layers \( (| {q} |{L} \gg 1) \) results in similar predictions, namely the gain threshold value is given by (7.4) “with a negative sign on the right-hand side of this expression.” For thick CLC layers with ωD in the middle of the stop band, the threshold gain is given by the expression

$$ {g}_{\text{t}} = - (2/3 {\pi })({p}/\delta {d})\exp [ { - 2 {\pi \delta }({L}/{p})} ]. $$
(7.6)

Hence, as can be seen from (7.6), the thinner the amplifying defect layer , the higher the threshold gain g.

As mentioned above, the same result is also valid for the absorption enhancement (formulas (7.4) and (7.5)). The thinner the absorbing defect layer , the higher the g value ensuring maximal absorption.

An important result relating to DFB lasing in the DMS with amplifying (absorbing) defect layer can be formulated as follows. The lasing threshold gain in a defect layer decreases as the amplifying layer thickness increases, being almost inversely proportional to the thickness. A similar result holds for the anomalously strong absorption phenomenon, where the value of g in the defect layer ensuring maximal absorption is almost inversely proportional to the defect layer thickness. We note that this decrease in the lasing threshold gain with increasing thickness of the amplifying defect layer cannot be regarded directly as the corresponding reduction in the lasing energy threshold of the pumping wave pulse. The situation depends on the specifics of the pumping arrangement. This question requires separate and more thorough consideration. For example, if we assume that the pumping is arranged such that the product of the gain g and the defect layer thickness d is proportional to the pumping pulse energy, then the threshold pumping pulse energy is almost independent of the defect layer thickness because of the almost inverse proportionality of the threshold gain to the defect layer thickness found above.

7.3 Defect Mode at Birefringent Defect Layer

In this section, we focus on a birefringent defect layer and, in particular, the case of low birefringence . As mentioned above, the reason for this is connected with both experimental research on DFB lasing in CLCs where the defect layer is birefringent [26] and the general idea that the unusual properties of DMs manifest themselves most clearly just at the middle of the defect structure , i.e., at the defect layer, where the DM field intensity reaches its maximum. We also assume at the outset that there is no absorption in the CLC and the birefringent defect layer. The analytic approach used to study a DMS with a birefringent defect layer is very similar to the DM studies carried out for an isotropic defect layer in Chap. 6 [12, 13], and we therefore present the final results, referring the reader to Chap. 6 for the details.

7.3.1 Non-absorbing CLC Layers

In this section , we present an analytic solution for the DM associated with the insertion of a birefringent defect layer in the perfect cholesteric structure for light propagating along the helical axes and we consider some limiting cases simplifying the problem. To consider the DM associated with the insertion of a birefringent layer in the perfect cholesteric structure, we have to solve Maxwell’s equations and a boundary problem for the electromagnetic wave propagating along the cholesteric helix for the layered structure with a birefringent defect layer depicted in Fig. 6.1. This investigation has already been performed in Chap. 6 [12, 13] under the assumption that the defect layer in Fig. 6.1 is isotropic. We can therefore use the results of [12, 13] for the present case of a birefringent defect layer and non-absorbing and amplifying (absorbing) CLC layers (keeping the main notation of Chap. 6 here), but introducing just a few physically clear changes in the formulas obtained in Chap. 6. We can no longer assume, as we did in [12, 13], that polarization conversion is absent and only light with the diffracting circular polarization need be taken into account (due to the assumption that the average CLC dielectric constant ε0 coincides with the dielectric constant of the defect layer and the external medium). In fact, due to the birefringence of the defect layer, the light polarization changes in the course of its propagation through the defect layer, from one of its surfaces to the other, and hence, generally speaking, the polarization of the light after crossing the defect layer will differ from the polarization at the first defect layer surface. This is why the polarization component differs from the diffracting polarization, generally speaking, and there is leakage of the correspondingly polarized light from the DMS. The obvious consequence of this leakage is a reduction in the DM lifetime for the case of a birefringent defect layer.

Formulas for the optical properties of the structure with a birefringent defect layer depicted in Fig. 6.1 can be obtained using the expressions for the amplitude transmission, T(L), and reflection, R(L), coefficients for a single cholesteric layer in the presence of dielectric interfaces (see Chap. 3 and [28, 33]). If we neglect multiple scattering of light with the non-diffracting polarization, the transmission, \( | {{T}({d},{L})} |^{2} \), and reflection, \( | {{R}({d},{L})} |^{2} \), intensity coefficients (of light with the diffracting circular polarization) for the whole structure may be presented in the following form:

$$ | {{T}({d},{L})} |^{2} = | [ {T}_{\text{e}} {T}_{\text{d}} {M}({k},{d},\Delta {n})( {{\varvec{\upsigma}}_{{\mathbf{e}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{*} } ) ][ 1 - {M}^{2} ({k},{d},\Delta {n})( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{*} } )^{2} ( {R}_{ \text{d}} {R}_{ \text{u}}] |^{2}, $$
(7.7)
$$ | {{R}( {{d},{L}} )} |^{2} = | {\{ {{R}_{{\text {e}}} + {R}_{\text{d}} {T}_{{\text {e}}} {T}_{\text{u}} {M}^{2} ( {{k},{d},\Delta {n}} )| {( {{\varvec{\upsigma}}_{{\mathbf{e}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )} |^{2} /[ {1 - {M}^{2} ( {{k},{d},\Delta {n}} )|( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )^{2} {R}_{\text{d}} {R}_{\text{u}} } ]} \}} |^{2} , $$
(7.8)

where the meaning of Re(Te), Ru(Tu), and Rd(Td) is the same as in (7.1) and (7.2) and σe, σr, and σed are the polarization vectors of light exiting the CLC layer inner surface, light reflected at the inner bottom CLC layer surface after incidence from the inserted defect layer, and light whose polarization vector σed transforms to the polarization vector σe upon crossing the birefringent defect layer of thickness d, respectively. Finally, Δn is the difference between the two refractive indices in the birefringent defect layer and M(k, d, Δn) is the phase factor for single propagation of the light through a birefringent defect layer. It is assumed in deriving (7.7)–(7.8) that the external beam is only incident on the structure (Fig. 6.1) from above. In the presence of dielectric interfaces, there is light polarization conversion at the inner surfaces of the CLC layers in the DMS under reflection and transmission of light through a CLC layer, and the light field inside CLC layers cannot just be expressed as a superposition of two diffracting eigenmodes of the CLC (generally speaking, two non-diffracting eigenmodes are also present). The corresponding polarization vector inside the defect layer (after the light has crossed the interface between the CLC and defect layers), denoted by σe, can be found (see [28, 29]), and the polarization vector σed can be easily calculated if d and Δn are known. “The same can be said about finding the polarization of light exciting only the diffracting eigenmodes in a CLC layer when incident at the external CLC layer surface of the DMS.” The corresponding polarization in the presence of dielectric interfaces is referred to as the diffracting polarization here. Polarization orthogonal to the diffracting polarization is referred to as a non-diffracting polarization. When incident on the DMS, light with a non-diffracting polarization excites only non-diffracting CLC eigenmodes in the CLC layers of the DMS. The polarization vectors σe, σr, and σed can be presented in the form

$$ {\varvec{\upsigma}}_{{\mathbf{i}}} = ( {\cos {\alpha }_{\text{i}} {\mathbf{e}}_{x} + { \text{e}}^{{{\text{i}} {\beta }{\text{i}}}} \sin {\alpha }_{\text{i}} {\mathbf{e}}_{y} } ), $$
(7.9)

where ex and ey are the unit vectors along the x and y axes and αi, βi are the parameters determining the polarization. For example, αi = π/4 and βi = π/2(− π/2) correspond to right and left circular polarizations.

In the general case, for a DMS with a birefringent defect layer, the transmitted and reflected beams do not correspond to the diffracting circular polarization, and therefore there is reflection and transmission of light with the non-diffracting polarization, even for incident light with the diffracting polarization. Neglecting multiple scattering of light with the non-diffracting polarization, we obtain the reflection, R(d, L)−, and transmission, T(d, L)−, coefficients of light with the non-diffracting circular polarization (for incident light with the diffracting circular polarization):

$$ \begin{aligned} | {{T}({d},{L})^{ - } } |^{2} & = [ {{T}_{\text{e}} {T}_{\text{d}}^{ - } \{ {{M}( {{k},{d},\Delta {n}} )( {{\varvec{\upsigma}}_{{\mathbf{e}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{ \bot} *} ) + ( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )( {{\varvec{\upsigma}}_{{\mathbf{e}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )} } \\ & \quad \times\,{ {( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{\bot} *} ){M}^{2} ( {{k},{d},\Delta {n}} )/[1 - {M}^{2} ( {{k},{d},\Delta {n}} )( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )^{2} ( {{R}_{\text{d}} {R}_{\text{u}} } ]} \}} |^{2} , \\ \end{aligned} $$
(7.10)
$$ \begin{aligned} | {{R}( {{d},{L}} )^{ - } } |^{2} &= | \{ {R}_{\text{e}}^{ - } + {R}_{\text{d}} {T}_{{\text {e}}} {T}_{\text{u}}^{ - } {M}^{2} ({k},{d},\Delta {n})\\ & \quad \times\, ( {{\varvec{\upsigma}}_{{\mathbf{e}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{\bot} *} )/[ {1 - {M}^{2} ( {{k},{d},\Delta {n}} )( {{\varvec{\upsigma}}_{{\mathbf{r}}} {\varvec{\upsigma}}_{{{\mathbf{ed}}}} *} )^{2} {R}_{{\text {d}}} {R}_{\text{u}} } ] \} |^{2} ,\end{aligned} $$
(7.11)

where \( {R}_{{\text {e}}}^{ - } \) is the reflection coefficient of the CLC layer for light with the non-diffracting circular polarization, taking into account dielectric interfaces, for incident light with the diffracting polarization, and T− is the transmission coefficient of the CLC layer for light with the non-diffracting circular polarization, taking into account dielectric interfaces, for incident light with the non-diffracting polarization. Further, \( {\varvec{\upsigma}}_{{{\mathbf{ed}}}}^{\bot} \) is the polarization vector orthogonal to \( {\varvec{\upsigma}}_{{{\mathbf{ed}}}} \). We note that the amplitude transmission coefficients \( {T}_{\text{d}}^{ - } \) and \( {T}_{\text{u}}^{ - } \) are approximately equal to exp[ikLn−/n0], where n− is the refractive index of light with the non-diffracting circular polarization in the CLC layer.

The reflection and transmission coefficients can be calculated in the general case using (7.7), (7.8), (7.10), and (7.11), but these calculations are rather cumbersome. This is why we study the case of low birefringence in detail below and present expressions for \( | {{T}({d},{L})} |^{2} \) and \( | {{R}({d},{L})} |^{2} \) taking into account only the polarization transformation in the defect layer and neglecting transformations of the polarizations at the interfaces and small deviations in the diffracting and non-diffracting polarizations from the circular ones, since this allows simple analytical calculations.

With this simplification and under the assumption that the refractive indices of the DMS external media coincide with the average CLC refractive indices, the refractive indices of the defect layer can be expressed by the formulas

$$ {n}_{\hbox{max} } = {n}_{0} + \Delta {n}/2,\quad {n}_{\hbox{min} } = {n}_{0} - \Delta {n}/2, $$
(7.12)

where n0 coincides with the average CLC refractive index and ∆n is small. The phase factor M(k, d, Δn) is given by

$$ {M}( {{k},{d},\Delta {n}} ) = \exp [{ {\text{i}kd}}]\cos (\Delta {\varphi }/2), $$
(7.13)

where the phase difference between the two beam components with different eigenpolarizations at the defect layer thickness is \( \Delta {\varphi } \) = Δnkd/n0, k = ωn0/c = ωε 1/20 /c.

Finally, in the case of low birefringence , inserting (7.13) into (7.7) and (7.8), we obtain explicit expressions for the reflection and transmission coefficients of light with a circular diffracting polarization for the incident beam with a circular diffracting polarization:

$$ | {{T}({d},{L})} |^{2} = | {[{T}_{\text{e}} {T}_{\text{d}} \exp [{ {\text{i}kd}}]\cos (\Delta {\varphi }/2)]/[ {1 - \exp [ {{\text{i}}2{kd}} ]\cos^{2} (\Delta {\varphi }/2){R}_{\text{d}} {R}_{\text{u}} } ]} |^{2} , $$
(7.14)
$$ \begin{aligned}| {{R}({d},{L})} |^{2} &= | \{ {R}_{\text{e}} + {R}_{{\text {d}}} {T}_{\text{e}} {T}_{\text{u}} \exp [ {{\text{i}}2{kd}} ]\cos^{2} (\Delta {\varphi }/2)/[ 1 - \exp [{\text{i}}2{kd}]\\ & \quad \times\,\cos^{2} (\Delta {\varphi }/2){R}_{\text{d}} {R}_{\text{u}} ] \} |^{2} \end{aligned} $$
(7.15)

If \( \Delta {\varphi }/2 {\pi } \) is an integer, (7.14) and (7.15) are identical to the corresponding equations for the DMS with an isotropic defect layer (see Chap. 6 [12, 13]) and there is no conversion of the diffracting polarization into a non-diffracting one; but if \( \Delta {\varphi }/2 {\pi } \) is not an integer, there is conversion of the diffracting polarization into the non-diffracting one, so light leaks from the DMS, and in particular, the DM lifetime is less than for the case of the corresponding DMS with an isotropic defect layer. This dependence of the DM properties on the phase shift between the eigenwaves when they cross the defect layer opens up ways to control the DM properties. The simplest such possibility involves varying the thickness of the defect layer.

The polarization conversion has the effect of adding the non-diffracting components to the transmitted and reflected beams. For low birefringence , which corresponds to the condition Δn/n0 < δ, the amplitude transmission and reflection coefficients for light with the non-diffracting polarization (and incident light with the diffracting polarization) are given by

$$\begin{aligned} {T}( {{d},{L}} )^{ - } &= [ {{T}_{\text{e}} \exp { {[\text{i}kLn}}_{ - } /{n}_{0} ]\exp [ {\text{i}kd}}]\sin (\Delta {\varphi }/2)]/[1 - \exp [ {{\text{i}}2{kd}} ] \\ & \quad \times\,\cos^{2} (\Delta {\varphi }/2){R}_{\text{d}} {R}_{\text{u}} ],\end{aligned} $$
(7.16)
$$ \begin{aligned}{R}({d},{L})^{ - } & = 1/2{R}_{\text{u}} {T}_{{\text {e}}} \exp [ {{ {\text{i}kLn}}_{ - } /{n}_{0} } ]\exp [\text{i}2{kd}]\sin (\Delta {\varphi })/[1 - \exp [{\text{i}}2{kd}]\\ & \quad\times\,\cos^{2} (\Delta {nkd}/2{n}_{0} ){R}_{\text{d}} {R}_{\text{u}} ],\end{aligned} $$
(7.17)

where n- is the refractive index of light with the non-diffracting circular polarization in the CLC layer.

The calculation results for the transmission coefficients \( | {{T}({d},{L})} |^{2} \) of light with the diffracting polarization for the case of low birefringence are presented in Fig. 7.6 for various values of the birefringent phase factor \( \Delta {\varphi } \) related to single propagation of light in the birefringent defect layer. Figure 7.6 shows that at low values of the phase shift between eigenwaves when they cross the defect layer \( (\Delta {\varphi } < {\pi }/2) \), the shape of the transmission curve is very similar to the one for a DMS with an isotropic defect layer (for \( \Delta {\varphi } \) equal to an integer multiple of 2π or zero, it coincides with the shape of the corresponding curve for the case of an isotropic defect layer). But as \( \Delta {\varphi } \) approaches π/2 (see Fig. 7.6e–g), the increased transmission at the defect mode frequency, typical for an isotropic defect layer, gradually disappears and does not appear at all at \( \Delta {\varphi } = {\pi }/2 \) (Fig. 7.6g). This may be regarded, in particular, as a hint that the DM lifetime decreases with increasing shift between the eigenwaves when they cross the defect layer and that the DM does not exist at all at some value of the shift.

Taking into account the partial conversion of a circular non-diffracting incident polarization into a diffracting one, the picture of the transmission spectrum does not change radically. In Fig. 7.7, the transmission spectra for the total light intensity crossing the DMS (the sum of the intensities for both circular polarizations) calculated using (7.16) and (7.17) show a general decrease in transmission at the DM frequency ωd as \( \Delta {\varphi } \) increases, but it is much slower than for the diffracting polarization, and only for \( \Delta {\varphi } \) close to π/2 does the transmission practically vanish (which demonstrates the polarization conversion in the birefringent layer).

Fig. 7.7
figure 7figure 7figure 7

Calculated total intensity transmission coefficient for a low birefringence defect layer versus frequency for diffracting incident polarization and for a birefringent phase shift at the defect layer equal to \( \Delta {\varphi } \) = π/20 (a), π/16 (b), π/12 (c), π/8 (d), π/6 (e), π/4 (f), π/2 (g), for a non-absorbing CLC with d/p = 0.25

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It is well known [9] that the position of the defect mode frequency in the stop band is determined by the frequency of the transmission (reflection) coefficient maximum (minimum), and therefore the calculations of the transmission spectra carried out here (Figs. 7.6 and 7.7) determine the real component of the DM frequency. But because the DM is a quasi-stationary mode, the imaginary component of the DM frequency is not zero [12, 13]. A direct way to find the imaginary component of the DM frequency is to solve the dispersion equation. In the case of a birefringent defect layer, this equation can be found similarly to the case of an isotropic defect layer [12, 13], and if multiple scattering of light with the non-diffracting polarization is neglected, it can be written

$$ \begin{aligned}& \{ {M}^{2} ( {{k},{d},\Delta {n}} ) \sin^{2} {qL} - \exp( - {\text{i}}\tau {L})[ ( {\tau {q}/\kappa^{2} } )\cos {qL}\\ & \quad + {\text{i}}( {(\tau /2\kappa )^{2} + ({q}/\kappa )^{2} - 1} )\sin {qL} ]^{2} /\delta^{2} \} = 0. \end{aligned} $$
(7.18)

In the general case, the solution of (7.18) has to be found numerically, and in the case of an isotropic defect layer, a detailed discussion of this can be found in [12, 13]. Some simplification of (7.18) occurs in the case of low birefringence , when the phase factor in (7.18) is given by (7.13).

7.3.2 Amplifying and Absorbing CLC Layers

As experiment [3] and theory [12, 13] show, the unusual optical properties of DMSs at the DM frequency ωD (abnormally strong absorption for an absorbing CLC and abnormally strong amplification for an amplifying CLC [12, 13, 28, 33]) can be effectively used to enhance DFB lasing. It is quite natural to study the way the birefringent defect layer influences the abnormally strong amplification and abnormally strong absorption effects. To do so, we assume, as in [12, 13], that the average dielectric constant of the CLC contains an imaginary term, i.e., ε = ε0(1 + 2iγ), where positive γ corresponds to an absorbing medium and negative γ to an amplifying medium. (We note that in real situations \( | {\gamma } | \ll 1 \).) As mentioned above, the value of γ can be found by solving the dispersion Equation (7.18). Another option (see [12, 13]) is to study the reflection and transmission coefficients (7.7)–(7.8), (7.14), and (7.15) as functions of γ close to R(d, L) and T(d, L) at the DM frequency.

For an amplifying CLC, the value of γ corresponding to a divergence in the DMS reflection and transmission coefficients determines the solution of the dispersion Equation (7.18) and also determines the threshold DFB lasing gain in the DMS (see [12, 13]). Therefore, the threshold value of γ can be found by calculating the DMS reflection and transmission coefficients at various values of γ and finding its value at the points where the DMS reflection and transmission coefficients diverge.

This procedure, performed here for a birefringent defect layer at various values of the birefringent phase factor \( \Delta {\varphi } \) related to single propagation of the light in a birefringent defect layer, can be used to establish the dependence of the threshold lasing gain (γ) on the birefringent phase factor \( \Delta {\varphi } \). Figure 7.8 presents values of the DMS transmission coefficient close to their divergence points, showing the increase in the threshold DFB lasing gain \( (| {\gamma } |) \) with increasing birefringent phase factor \( \Delta {\varphi } \), and even disappearance of the divergence at the defect mode frequency when \( \Delta {\varphi } = {\pi }/2 \). This is in good agreement with the transmission spectra calculated in Figs. 7.6 and 7.7. In particular, at \( \Delta {\varphi } = {\pi }/2 \), there is no trace of the typical DM peculiarities in the transmission spectra.

Fig. 7.8
figure 8figure 8figure 8

Calculated transmission intensity coefficients of a low birefringence defect layer for an amplifying CLC layer versus frequency close to their divergence points as a function of γ for diffracting incident polarization with birefringent phase shift at the defect layer equal to \( \Delta {\varphi } \) = π/20, γ = −0.00075 (a), π/16, γ = −0.00085 (b), π/12, γ = −0.00100 (c), π/8, γ = −0.00150 (d), π/6, γ = −0.002355 (e), π/4, γ = −0.003555 (f), π/2, γ = −0.004500 (g), and \( \Delta {\varphi } \) = 0, γ = −0.000675 (h) corresponding to an isotropic defect layer; d/p = 2.25

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For absorbing CLC layers in the DMS, the abnormally strong absorption effect reveals itself at the value of γ ensuring a maximum of the total absorption in the DMS (see [12, 13]). For a finite thickness L of the CLC layers, the DM frequency ωD is a complex quantity, which can be found by numerical solution of (7.18). As in the case of absorbing and amplifying defect layers , the positions of the dips in reflection and the spikes in transmission inside the stop band just correspond to Re[ωD], and this observation turns out to be useful for numerical solution of the dispersion equation for a birefringent defect layer and absorbing CLC layers.

We note that the results obtained here for DMs with a birefringent defect layer open up new options for varying the DM characteristics. An important result relating to DFB lasing at the DMS with a birefringent defect layer may be formulated as follows. The lasing threshold gain increases with an increase in the optical path difference of two eigenwaves at the defect layer thickness. A similar result relates to the effect of the anomalously strong absorption phenomenon, where the value of maximal absorption is dependent on the optical path difference at the defect layer thickness.

7.4 Defect Structure with Dielectric Jump

An isotropic defect layer whose dielectric constant differs from the average dielectric constant ε0 of the CLC layers can also be effectively related to the case of an active defect layer. This is due to polarization conversion at the surfaces, which makes this case similar to the case of a birefringent defect layer. If the dielectric constant of the medium external to the DMS is different from the average dielectric constant ε0 of the CLC layers, polarization conversion also occurs at the external DMS surfaces, but as we shall see below, the polarization conversion at the external DMS surfaces does not affect the DM properties as much as the polarization conversion at the defect layer surfaces. There are no major difficulties in obtaining the DM dispersion equation from the boundary conditions in the general case of dielectric jumps at all interfaces of the DMS. But the DM dispersion equation is rather complicated in the general case (it is connected with a system of 12 linear equations). Therefore, we first demonstrate the role of dielectric jumps for a localized mode for the simplest case of an edge mode (EM) which is related to a CLC layer with dielectric jumps at its surfaces.

7.4.1 Dielectric Jumps at Single CLC Layer

In accordance with the foregoing, we study the transmission and reflection of light by a CLC layer surrounded by media with dielectric constants differing from the average CLC dielectric constant ε0 for light propagation along the helical axis (see the schematic of the boundary problem in Fig. 7.9). Following the approach in Chap. 3 [28, 29, 34], from the boundary conditions, we obtain [slightly modifying the notation in (1.20)] the system of equations for the amplitudes \( {E}_{j}^{ + } \) of eigenwaves in the layer excited by an external wave incident at the layer:

$$ \begin{aligned} & \sum\limits_{{j = {1}}}^{4} {( {{1} + K_{j}^{ + } /\kappa_{{\text{e} ,1}} } )} E_{j}^{ + } = {2}E_{\text{e}}^{ + } ,\quad \sum\limits_{{j = {1}}}^{4} {\text{e}^{{\text{i}K_{j}^{ + } L}} ( {{1} - K_{j}^{ + } /\kappa_{{\text{e},2}} } )} E_{j}^{ + } = {0} \\ & \sum\limits_{{j = {1}}}^{4} {\xi_{j} ( {{1} + K_{j}^{ - } /\kappa_{{\text{e},1}} } )} E_{j}^{ + } = {2}E_{\text{e}}^{ - } ,\quad \sum\limits_{{j = {1}}}^{4} {\xi_{j} \text{e}^{{\text{i}K_{j}^{ - } L}} ( {{1} - K_{j}^{ - } /\kappa_{{\text{e},2}} } )} E_{j}^{ + } = {0} \\ \end{aligned} $$
(7.19)

where the incident, reflected, and transmitted waves and the wave inside the CLC have been written as follows:

$$ \begin{aligned} \overrightarrow {E}^{\text{e}} & = \text{e}^{{\text{i}( {\kappa_{{\text{e},1}} z - \omega t} )}} ( {E_{\text{e}}^{ + } \overrightarrow {n}_{ + } + E_{\text{e}}^{ - } \overrightarrow {n}_{ - } } ) \\ \overrightarrow {E}^{\text{r} } & = \text{e}^{{ -\text{i}( {\kappa_{{\text{e},1}} z - \omega t} )}} ( {E_{r}^{ + } \overrightarrow {n}_{ - } + E_{\text{r}}^{ - } \overrightarrow {n}_{ - } } ) \\ \overrightarrow {E}^{\text{t}} & = \text{e}^{{\text{i}( {\kappa_{{\text{e},2}} z - \omega t} )}} ( {E_{t}^{ + } \overrightarrow {n}_{ + } + E_{\text{t}}^{ - } \overrightarrow {n}_{ - } } ) \\ \overrightarrow {E} & = \text{e}^{{ - \text{i}\omega t}} \sum\limits_{{j = {1}}}^{4} {E_{j}^{ + } (\text{e}^{{\text{i}K_{j}^{ + } z}} \overrightarrow {n}_{ + } + \xi_{j} \text{e}^{{\text{i}K_{j}^{ - } z}} \overrightarrow {n}_{ - } )} \\ \end{aligned} $$

Here, n± are the left and right circular polarization vectors [see (7.9)] and we use the labeling of CLC eigenwaves proposed in Chap. 1. (The subscripts “1” and “4” correspond to the non-diffracting eigenwaves propagating in opposite directions and the subscripts “2” and “3” correspond to the diffracting eigenwaves.)

Fig. 7.9
figure 9

Schematic of the CLC edge mode structure with dielectric jumps at the interfaces

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The wave vectors inside the CLC layer are \( {K}_{1}^{ + } =\tau /2 + {q}_{ + } ,{K}_{4}^{ + } =\tau /2{-}{q}_{ + } \), \( {q}_{ + } = \kappa \{ {1 + (\tau 2\kappa )^{2} + [(\tau /\kappa )^{2} + \delta^{2} ]^{1/2} } \}^{1/2} \), \( {K}_{2}^{ + } =\tau /2 + {q},{K}_{3}^{ + } =\tau /2{-}{q} \) and q is determined by (5.11), \( {K_{j}^{ - } = K_{j}^{ + } - \tau } \), \( {\kappa_{{\text{e},1}} = \frac{{\omega \sqrt {\varepsilon_{1} } }}{c}} \), \( \kappa = {\omega }\,{\varepsilon }_{0}^{1/2} /{c} \), \( {\kappa_{{\text{e},2}} = \frac{{\omega \sqrt {\varepsilon_{2} } }}{{c^{ - } }}} \) and \( {\xi_{\text{i}} = \frac{\delta }{{(K_{\text{i}}^{ + } /\kappa - \tau /\kappa )^{2} - 1}}} \), and the dielectric constants ε1 and ε2 are determined in Fig. 7.9.

The amplitudes of the reflected and transmitted waves are expressed in terms of \( {E}_{j}^{ \pm } \) by

$$ \begin{aligned} E_{r}^{ + } & = \frac{1}{2}\sum\limits_{j = 1}^{4} {\xi_{j} \left( {1 - \frac{{K_{j}^{ - } }}{{\kappa_{{\text{e},1}} }}} \right)E_{j}^{ + } } ,\quad E_{r}^{ + } = \frac{1}{2}\sum\limits_{j = 1}^{4} {\text{e}^{{\text{i}( {K_{j}^{ + } - \kappa_{{\text{e},2}} } )}} \left( {1 + \frac{{K_{j}^{ + } }}{{\kappa_{{\text{e},2}} }}} \right)E_{j}^{ + } } \\ E_{r}^{ - } & = \frac{1}{2}\sum\limits_{j = 1}^{4} {\left( {1 - \frac{{K_{j}^{ + } }}{{\kappa_{{\text{e},1}} }}} \right)E_{j}^{ + } } ,\quad E_{r}^{ - } = \frac{1}{2}\sum\limits_{j = 1}^{4} {\xi j\text{e}^{{\text{i}( {K_{j}^{ - } - \kappa_{{\text{e},2}} } )}} \left( {1 + \frac{{K_{j}^{ - } }}{{\kappa_{{\text{e},2}} }}} \right)E_{j}^{ + } } \\ \end{aligned} $$
(7.20)

It is convenient to introduce the parameters \( {r}_{1} = {\varepsilon }_{0}^{1/2} /{\varepsilon }_{1}^{1/2} = {k}/{k}_{{{ \text{e}},1}} \), \( {r}_{2} = {\varepsilon }_{0}^{1/2} /{\varepsilon }_{2}^{1/2} = {k}/{k}_{{{ \text{e}},2}} \) reducing the ratios \( {K}_{j}^{ \pm } /{k}_{{{\text {e}},{\text{i}}}} \) in (7.19), (7.20) to \( {r}_{\text{i}} {K}_{j}^{ \pm } /{k} \). For the sake of generality, the case of different dielectric constants in the media surrounding the CLC layer is shown in Fig. 7.9, and accordingly, (7.19) and (7.20) relate to the case of different media at either side of the CLC layer.

Examples of calculations, performed using (7.19) and (7.20), which demonstrate the influence of dielectric jumps at the layer surfaces on the transmission and reflection coefficients are presented in [35]. Here, we shall not discuss the transmission and reflection of light by a layer, but concentrate on the influence of dielectric jumps at the layer surfaces on the EM properties. The EMs are determined by the homogeneous system corresponding to (7.19) and the EM dispersion equation for the EM frequency follows from the solvability condition for this homogeneous system [36]. It is known [36] that the real part of the EM frequency coincides approximately with the frequency positions of the minima of the reflection coefficient, and hence the solution of the above-mentioned homogeneous system at the frequency of such a minimum gives the amplitudes of all four eigenwaves in the layer composing the EM in the case where the dielectric constants of the media surrounding the CLC layer are different from ε0. We recall that the EM in the absence of jumps in the dielectric constant at the interfaces is composed only of two diffracting eigenwaves [36]. Owing to the rather cumbersome form of the solution of the homogeneous system, we first use the consecutive approximation approach to solve the system. If the layer is thick enough, the known solution [36] in the absence of dielectric constant jumps can be used as the zeroth approximation. In this approximation, the homogeneous system under consideration reduces to a system of two equations for the amplitudes of two non-diffracting eigenwaves \( {E}_{1}^{ + } \) and \( {E}_{4}^{ + } \). The solution of the homogeneous system found using this method at the EM frequency shows that the amplitudes of the two non-diffracting eigenwaves in the solution for the EM decrease in inversely proportion to the layer thickness L. This result shows that, if the CLC layer is thick enough, the influence of the dielectric constant jumps at the layer surfaces is small, and in the limit of an infinitely thick CLC layer, the EM properties are the same as in the absence of the jumps in the dielectric constant. In Fig. 7.10a, the calculated variations in the EM lifetime versus the layer thickness L are presented for the case where dielectric jumps are absent, and for two values of the dielectric jump (Fig. 7.10b zooms in on a small part of the curve in Fig. 7.10a). The calculations of the EM lifetime versus the layer thickness L presented in Fig. 7.10 confirm the above statement that, as L increases, the EM lifetime (Imω) approaches, with decaying oscillations, the value corresponding to the absence of dielectric jumps.

Fig. 7.10
figure 10

a Calculated EM lifetime versus the CLC layer thickness normalized by the CLC layer flight time ε 1/20 L/c for several values of the dielectric jump at the CLC layer surface. b Zoom on part of (a)

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7.4.2 Dielectric Jumps at Defect Layer

We return to the case of a DMS with an isotropic defect layer and a dielectric constant that differs from the average dielectric constant ε0 of the CLC layers. In the general case of dielectric jumps at all interfaces in the DMS (see Fig. 6.1), we have to determine the 12 amplitudes of the eigenwaves propagating in the DMS (four amplitudes in each CLC layer and four amplitudes for waves propagating in the isotropic defect layer in both directions and with opposite circular polarizations). To simplify the problem, we assume that there are no dielectric jumps at the external DMS surfaces. As we have seen, the dielectric jumps at external DMS surfaces for thick CLC layers do not significantly affect polarization conversion. We therefore take the dielectric jumps into account only at the interfaces with the defect layer. Taking into account the form of the DM solution in the absence of dielectric jumps [12, 13], we have to determine only eight amplitudes of eigenwaves propagating in the DMS (two amplitudes in each CLC layer and four amplitudes for waves propagating in the isotropic defect layer).

If we accept the labeling of the eigenwaves used in Chap. 1 and specify them by superscripts “u” and “d” for the top and bottom CLC layers, respectively, in Fig. 6.1, then the corresponding system includes E u2 , E u4 , E d1 , E d2 , the amplitudes of the eigenwaves in the CLC, and C ±R , C ±L , the amplitudes of the right (left) polarized waves in the defect layer with two (±) possible propagation directions. We assume for definiteness that the diffracting circular polarization is the right-hand one. If we accept the ordering E u2 , E u4 , C +R , C −R , C +L , C −L , E d1 , and E d2 of the amplitudes mentioned above in the equations obtained from the boundary conditions, then the elements of the matrix aik of the corresponding system of equations are as follows:

$$ \begin{aligned} & {a}_{{{ {\text{i}k}}}} = 0\quad{ \text{for}}\,{i} = 5,6,7,8\quad{\text{and}}\quad{k} = 1,2;\quad{a}_{{{ {\text{i}k}}}} = 0\quad{ \text{for}}\,{i} = 1,2,3,4\quad{ \text{and}}\quad{k} = 7,8; \\ & {a}_{11} = \exp ( {{ {\text{i}K}}_{2}^{ + } {L}_{ - } } ) - \exp ( {{ {\text{i}K}}_{3}^{ + } {L}_{ - } } ),\quad {a}_{12} = \exp ( {{ {\text{i}K}}_{4}^{ + } {L}_{ - } } ),\quad {a}_{13} = \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ - } } ), \\ & {a}_{{1{k}}} = 0\quad{ \text{for}}\,{k} = 4,5;\quad{a}_{16} = \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ - } } );\quad{a}_{21} = {\zeta }_{2} \exp ( {{ {\text{i}K}}_{2}^{ - } {L}_{ - } } ) - {\zeta }_{3} \exp ( {{ {\text{i}K}}_{3}^{ - } {L}_{ - } } ), \\ & {a}_{22} = {\zeta }_{4} \exp ( { - { {\text{i}K}}_{4}^{ - } {L}_{ - } } ),\quad{a}_{{2{k}}} = 0\quad{\text{for}}\,{k} = 3,6;\quad{a}_{24} = \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ - } } ), \\ & {a}_{25} = \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ - } } ),\quad{a}_{25} = \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ - } } );\quad{a}_{31} = - {K}_{2}^{ + } \exp ( {{ {\text{i}K}}_{2}^{ + } {L}_{ - } } ) - {K}_{3}^{ + } \exp ( {{ {\text{i}K}}_{3}^{ + } {L}_{ - } } ), \\ & {a}_{32} = - {K}_{4}^{ + } \exp ( {{ {\text{i}K}}_{4}^{ + } {L}_{ - } } ),\quad{a}_{33} = - {k}_{\text{d}} \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ - } } ),\quad{a}_{{3{k}}} = 0\quad{ \text{for}}\,{k} = 4,5; \\ & {a}_{35} = {k}_{\text{d}} \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ - } } );\quad {a}_{41} = {K}_{2}^{ - } {\zeta }_{2} \exp ( {{ {\text{i}K}}_{2}^{ - } {L}_{ - } } ) - {\zeta }_{3} {K}_{3}^{ - } \exp ( {{ {\text{i}K}}_{3}^{ - } {L}_{ - } } ), \\ & {a}_{42} = - {\zeta }_{4} {K}_{4}^{ - } \exp ( { - { {\text{i}K}}_{4}^{ - } {L}_{ - } } ),\quad{a}_{{4 {k} }} = 0\quad{ \text{for}}\,{k} = 3,6;\quad {a}_{44} = - {k}_{\text{d}} \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ - } } ), \\ & {a}_{45} = {k}_{\text{d}} \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ - } } );\quad{a}_{53} = \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{{5{k}}} = 0\quad{ \text{for}}\,{k} = 4,5;\quad{a}_{56} = \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ + } } ), \\ & {a}_{57} = \exp ( {{ {\text{i}K}}_{1}^{ + } {L}_{ + } } ),\quad {a}_{58} = \exp ( {{ {\text{i}K}}_{2}^{ + } {L}_{ + } } ) - {r}_{32} \exp ( {{ {\text{i}K}}_{3}^{ + } {L}_{ + } } );\quad {a}_{{6 {k}}} = 0\quad{ \text{for}}\,{k} = 3,6; \\ & {a}_{64} = \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{67} = {\zeta }_{1} \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{68} = {\zeta }_{2} \exp ( {{ {\text{i}K}}_{2}^{ - } {L}_{ + } } ) - {r}_{32} {\zeta }_{3} \exp ( {{ {\text{i}K}}_{3}^{ - } {L}_{ + } } ), \\ & {a}_{74} = - {k}_{\text{d}} \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{{7{k}}} = 0\quad{ \text{for}}\,{k} = 4,5;\quad{a}_{77} = {K}_{1}^{ + } \exp ( {{ {\text{i}K}}_{1}^{ + } {L}_{ + } } ), \\ & {a}_{78} = - {K}_{2}^{ + } \exp ( {{ {\text{i}K}}_{2}^{ + } {L}_{ + } } ) + {r}_{32} {K}_{3}^{ + } \exp ( {{ {\text{i}K}}_{3}^{ + } {L}_{ + } } );\quad{a}_{{8{k}}} = 0\quad{ \text{for}}\,{k} = 3,6; \\ & {a}_{84} = - {k}_{\text{d}} \exp ( { - { {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{85} = {k}_{\text{d}} \exp ( {{ {\text{i}k}}_{\text{d}} {L}_{ + } } ),\quad{a}_{87} = {\zeta }_{1} {K}_{1}^{ - } \exp ( { - { {\text{i}K}}_{1}^{ - } {L}_{ + } } ), \\ & {a}_{88} = {K}_{2}^{ - } {\zeta }_{2} \exp ( {{ {\text{i}K}}_{2}^{ - } {L}_{ + } } ) - {r}_{32} {\zeta }_{3} {K}_{3}^{ - } \exp ( {{ {\text{i}K}}_{3}^{ - } {L}_{ + } } ),\quad{ \text{where}}\;{r}_{32} = ( { {\zeta }_{2} / {\zeta }_{3} } )\exp ( {4{ {\text{i}qL}}} ). \\ \end{aligned} $$

The dispersion equation for the DM frequency ωD, and in particular, the DM lifetime (ImωD), for a DMS with dielectric jumps only at the interfaces with the defect layer, is determined by the equation obtained by setting the determinant of the above matrix equal to zero. This equation must be solved numerically. However, a simple estimate of the DM lifetime can be obtained.

As is well known, the DM lifetime for a DMS with no dielectric jumps at the interfaces is determined by energy leakage through the external DMS surfaces [13], and the lifetime increases with increasing thickness of the CLC layer, becoming infinite for an infinite thickness. The changes in the DM lifetime for a DMS with dielectric jumps at the interfaces compared to the case without such jumps are mainly due to the conversion of the diffracting polarization into the non-diffracting one and the free escape of light with the non-diffracting polarization from the DMS. If the CLC layer is thick enough, this mechanism predominates over the leakage of light with the diffracting polarization through the external DMS surfaces. This is why, if the CLC layer in the DMS is thick enough, the DM lifetime is mainly determined by polarization conversion at the interfaces with the defect layer. Hence, to estimate the DM lifetime for a DMS with dielectric jumps at the interfaces, we can use the formula for the DM lifetime due to energy leakage through the external DMS surfaces for the case of no dielectric jumps at the interfaces (formula (7.22) in [13]), with the amplitude of the wave with converted polarization at the defect layer surface inserted instead of the diffracting wave amplitude leaking through the external DMS surface. The amplitude of the wave with converted polarization can be found approximately if, when solving the given homogeneous system, we assume that the field in the CLC layers is the same as for the DMS without dielectric jumps. This means that the amplitudes E u2 , C +R , C −R , and E d2 are the same as for the DM in a DMS without dielectric jumps and E d1  = E u4  = 0. However, we must still find C +L and C −L . This is straightforward using the expressions for the DM field in [13]. The next step is to express the nonzero E d1 and E u4 , which determine the field with non-diffracting circular polarization escaping from the DMS through the external surfaces, in terms of the known E u2 , C +R , C −R , E d2 , and the values found for C +L , C −L . A rather more crude estimate may be obtained without finding E d1 and E u4 , by calculating the direct polarization conversion at the interface with the defect layer for light with the diffracting polarization (for the DM field in the DMS without dielectric jumps). To estimate the polarization conversion, we can apply the formulas for polarization conversion at the interface of the CLC and an isotropic medium presented in Chap. 1 [28, 29, 34]. The reflection coefficient for light with the diffracting circular polarization into light with the non-diffracting circular polarization at a semi-infinite CLC layer, denoted R+−, and the transmission coefficient for light with the non-diffracting circular polarization for incident light with the diffracting circular polarization, denoted T+−, are given to zeroth order in δ by

$$ R^{ + - } = (1 - {r})^{2} /(1 + {r})^{2} ,\quad {T}^{ + - } = 4{r}(1 - {r})^{2} /(1 + {r})^{4} , $$
(7.21)

where \( {r} = {\varepsilon }_{d}^{1/2} /{\varepsilon }_{0}^{1/2} \) and εd is the dielectric constant of the defect layer.

Because the circular polarization conversion at the interface of the CLC and an isotropic medium is proportional to the square of the small parameter δ, even in the absence of dielectric jumps [28, 29, 34], polarization conversion at the interfaces should be taken into account if the dielectric jump is sufficiently large \( ( {| {{r} - 1} | > \delta } ) \). Therefore, the expressions (7.21) are accurate enough under these conditions to estimate the influence of the dielectric jumps on the DM lifetime in this case. The results of the corresponding analysis are as follows. The DM lifetime for a DMS with dielectric jumps at the interfaces increases as the thickness of the CLC layers increases to the value for which energy leakage through the external surfaces and leakage due to conversion of light with the diffracting polarization into light with the non-diffracting polarization become approximately equal. With a further increase in the thickness of the CLC layers, the DM lifetime is determined almost exclusively by the polarization conversion at the defect layer surfaces, and becomes practically independent of the CLC layer thickness L or, more correctly, becomes a very slowly increasing function of L. If, following [13], we represent the DM lifetime τdr for the DMS with dielectric jumps at the interfaces as the ratio of the optical field energy in the DMS to the energy flow of light of converted polarization through the defect layer surfaces, then the relation between τdr and the DM lifetime for a DMS without dielectric jumps at the interfaces τd can be estimated as

$$\begin{aligned} \tau_{\text{dr}} &= ({\varepsilon }_{0}^{1/2} /{c}) \int { | {{E}( {\omega }_{\text{D}} ,{z},{t})} |^{2} { {\text{d}z}}/[ {2{r}(1 - {r})^{2} | {{E}_{\text{dr}} }|^{2} /( {1 + {r}} )^{4} } ]}\\ &=\tau_{d} | {{E}^{\text{out}} } |^{2} /[ {2{r}(1 - {r})^{2} | {{E}_{\text{dr}}^{\text{u}} } |^{2} /(1 + {r})^{4} } ], \end{aligned}$$
(7.22)

where r is determined in (7.21) and all other quantities in (7.22) are related to the DM at the DMS without dielectric jumps: E(ωD, z, t) is the EM field in the CLC layer, Edr is the DM field at the defect layer surface of light propagating toward the CLC layer as a function of the z coordinate along the layer normal and the time t, Eout is the EM field of the light propagating out of the CLC layer at the external CLC layer surface, ωD is the DM frequency, and the integration over z is carried out over the thickness L of CLC layer. Equation (7.22) shows that the DM lifetime τdr for a DMS with thick CLC layers and dielectric jumps at the interfaces, in contrast to the lifetime τd of a DM in the DMS without dielectric jumps at the interfaces, does not increase exponentially with L. The exponential increase in τd is compensated in (7.22) by the exponential increase in \( | {{E}_{\text{dr}}^{\text{u}} } |^{2} \) (see [13]). In order to restore the exponential increase in τdr with L, the sharp jumps in the dielectric constant should be replaced by a smooth variation of the dielectric constant at the defect layer surfaces. We note that sharp jumps at the interfaces have a negative effect on the possibilities for lowering the lasing threshold, so smoothing of the dielectric jumps opens up options for lowering the lasing threshold compared with the case of DMSs with jump-like variations in the dielectric parameters.

In general, the localized optical modes in chiral liquid crystals studied theoretically in this chapter for a structure with jumps in the dielectric properties at their interfaces reveal a significant influence of the dielectric jumps on the EM properties, and especially on the DM properties, in particular, its lifetime. The effects studied here pave the way to optimizing the DM parameters by means of a proper choice of the dielectric properties of the defect layer.

7.5 Conclusion

As we have seen, isotropic defect layers with dielectric properties differing from those of the CLC layers in the DMS can effectively be regarded as active defect layers. The analytic description of the defect modes at active defect layers (amplifying (absorbing), birefringent, with dielectric jumps) allow one to obtain a clear physical picture of these modes which applies to defect modes in general (see [16]). For example, a lower lasing threshold and stronger absorption (under the conditions of the anomalously strong absorption effect) when the defect mode frequency lies at the middle of the stop band, compared to the situation when the defect mode frequency lies close to the stop-band edge, are features of any periodic medium. The results obtained suggest numerous ways to influence the DM properties by varying the dielectric characteristics of the defect layer. For a special choice of the parameters in the experiment, the resulting formulas can be applied directly to experiment. Some results can provide a qualitative explanation of the observed effects. This relates, for example, to the circular polarization sense of the wave emitted from the defect structure above the lasing threshold observed in experiment [3], which is opposite to the polarization sense responsible for the existence of the defect mode. An obvious explanation for the “lasing” at the opposite (non-diffracting) circular polarization is as follows. Due to the polarization conversion of the generated wave into a wave with the opposite circular polarization, the converted wave with non-diffracting polarization freely escapes from the structure. As mentioned above, this polarization conversion phenomenon, due to both birefringence and dielectric jumps , also makes a contribution to the frequency width of the defect mode. However, in the general case, a quantitative description of the measurements involves taking into account all possible “active properties” of the defect layer using the above formulas.

We note that the results obtained for the DM in the DMS consisting of CLC layers are qualitatively applicable to the corresponding localized electromagnetic modes in any periodic medium, and can be regarded as a useful guide in any study of localized modes with an active defect layer.