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2.3. LOCALIZED OPTICAL MODES AND LOW THRESHOLD LASING IN PHOTONIC
SPIRAL MEDIA
Belyakov Vladimir A., Senior Researcher, Professor. Landau Institute for Theoretical Physics, Moscow, Russia
Abstract: The recent original theoretical investigations on the localized opical modes and low threshold lasing in spiral photonic media are presented. Main attention is paid to the analytical approach to the problem. One of the most studied media of this kind are the cholesteric liquid crystals. It is why the localized opical modes and lasing in photonic spiral media are studied for the certainty at the example of chiral liquid crystals (CLCs). The chosen here model (absence of dielectric interfaces in the studied structures) allows one to get rid off the polarization mixing at the surfaces of the CLC layer and the defect structure (DMS) and to reduce the corresponding equations to the equations for the light of diffracting in the CLC polarization only. The dispersion equations determining connection of the EM and DM frequencies with the CLC layer parameters and other parameters of the DMS are obtained. Analytic expressions for the transmission and reflection coefficients of the DMS are presented and analyzed. As specific cases are considered DMS with an active (i.e. transforming the light intensity or polarization) defect layer and CLC layers with locally anisotropic absorption. It is shown that the active layer (excluding an amplifying one) reduces the DM lifetime (and increase the lasing threshold) in comparison with the case of DM at an isotropic defect layer. The case of CLC layers with an anisotropic local absorption is, in particular, analyzed and it is shown that due to the Borrmann effect the EM life-times for the EM frequncies at the opposite stop-bans edges may be signifinately different and concequently the lasing threshold for the EM frequencies at the both stop-band edges are different. The options of experimental observations of the theoretically revealed phenomena in spiral photonic media are briefly discussed.
Index terms: Spiral photonic media, Chiral liquid crystals, edge and defect optical modes, low threshold lasing
1. Introduction
Recently there have been much activities in the field of localized optical modes, in particular, edge modes (EM) and defect modes (DM) in chiral liquid crystals (CLCs) mainly due to the possibilities to reach a low lasing threshold for the mirrorless distributed feedback (DFB) lasing [1-4] in CLCs. The EM and DM existing as a localized electromagnetic eigenstate with its frequency close to the forbidden band gap and in the forbidden band gap, respectively, were investigated initially in the periodic dielectric structures [5]. The corresponding EM and DM in CLCs, and more general in spiral media, are very similar to the EM and DM in one-dimensional scalar periodic structures. They reveal abnormal reflection and transmission [1,2], and allow DFB lasing at a low lasing threshold [3].
Almost all studies of the EM and DM in chiral and scalar periodic media were performed by means of a numerical analysis with the exceptions [6,7] where the known exact analytical expression for the eigenwaves propagating along the helix axis [8,9] were used for a general study of the DM. The approach used in [6,7] looks as fruitful one because it allows to reach easy understanding of the DM and EM physics, and this is why it deserves further implementation in the study of the EM and DM.
In the present work, analytical solutions of the EM and DM (associated with an insertion of a layer in the perfect cholesteric structure and CLCs with local anisotropy of absorption) are presented and some limiting cases simplifying the problem are considered.
2. The Boundary-Value Problem
To investigate EM in a CLC, we have to consider a boundary problem, i.e. transmission and reflection of light incident on a CLC layer along the spiral axis [10-12]. We assume that the CLC is represented by a planar layer with a spiral axis perpendicular to the layer surfaces (Figure 1). We also assume that the average CLC dielectric constant eo coincides with the dielectric constant of the ambient medium. This assumption practically prevents conversion of one circular polarization into another at layer surfaces [11,12], and allows to have only two eigenwaves with diffracting circular polarization taken into account.
In the view of refs. [10-13], we state here only the final expressions for the amplitude transmission T and reflection R coefficients for light incident on a CLC layer of thickness L. These are given as
R(L)= 6sinqL/{(qT/K2)cosqL+i[(T/2K)2+ (q/K)2-l]sinql_} (la) T(L)=exp[iKL](qT/k2)/{(qr/k2)cosqL +I[(t/2k)2+ (q/k)2-l]sinqL}.
(lb)
where,
q= k{1+(t/2k)2 - [(r/r)2 + 62 and,
Here S is the dielectric anisotropy with en and £j. as the local principal values of the CLC dielectric tensor
[10-12], k = oj€o1/2/c with c as the speed of light, and
r = 47:/p with p as the cholesteric pitch.
Figure 2 demonstrates that the values of the reflection coefficient are strongly oscillating functions of the frequency close to the stop-band edges. The same is happening with the eigenwave amplitudes excited in the layer (see Figure 3) [13,14]. At the points of maxima close to the stop-band edges, the eigenwave amplitudes are much larger than the incident wave amplitude. It turns out that the frequencies of the eigenwave amplitude maxima coincide with the frequencies of zero reflection for a nonabsorbing CLC (see Figure 2).
L
CLC
Figurel. Schematic of the boundary problem for edge mode.
3. Edge Mode (Non-Absorbing Liquid Crystal)
In a non-absorbing CLC, y = 0 in the general expression for the dielectric constant e=e0(l+/y)i. The calculations of the reflection and transmission coefficients as functions of the frequency in accordance with eqs. (1) (Figure 2) give the well-known results [8-12], in particular, T+R = 1
Figure 2. Intensity reflection coefficient R calculated versus the frequency for a nonabsorbing CLC layer (cfeO.05, N=L/p=250)
Here and in all figures below 6[2(u-{^/(dcc/n)-1]s plotted at the frequency axis (shown close to one frequency edge of the stop band).
The mentioned relation between the amplitudes of eigenwaves excited in the layer and incident waves at the specific frequencies shows that the energy of radiation in the CLC at the layer thickness for these frequencies is much higher than the corresponding energy of the incident wave at the same thickness (Compare Figure 2 and Figure 3).
Hence, in complete accordance with ref. [13], we conclude that at the corresponding frequencies, the incident wave excites some localized mode in the CLC. To find this localized mode, we have to solve the homogeneous system of linear equations [14].
•O 003 -0.002 -O.OOl O O.OOl 0.002 0.003
FRBQUBNCY
Figure 3. Squared eigenwave amplitude in a nonabsorbing CLC layer calculated versus the frequency ((^0.05, N=L/p=250)■
The solvability condition for this homogeneous system determines the discrete frequencies of these localized modes:
tgqL= ](qi/Kl)/[(T/2K)2+ (qA)2-l] (2)
Generally, solutions to eq. (2) for the EM frequencies
(l»em can to be found only numerically. The EM frequencies Wem turn out to be complex, which can be presented as ^em =w(l+iA) where A is a small parameter in real situations.
Fortunately, an analytical solution can be found for a sufficiently small A ensuring the condition/.lm(c/) «1 . In this case, OJ and A, are determined by the conditions
qL=nn and
A=-1/2<5(n^)7(6/-r/4)3, (3)
where the integer number n is the edge mode number (n = 1 corresponds EM frequency (reflection coefficient minimum) closest to the stop-band edge).
The field distributions corresponding to the EM numbers n = 1, 2, 3 are presented in Figure 4. It shows that the EM field is localized inside the CLC layer, and its energy density experiences oscillations inside the layer with the number of the oscillations equal to the EM number n. However, the total field at each point of the CLC layer is represented by two plane waves propagating in the opposite directions [14].
Thus, the intensities of the waves propagating in the opposite directions can be calculated separately at any point in the layer. These distributions are of a special interest close to the layer surfaces (See Figure 5). It happens that at the layer surface, the intensity of the wave directed inside the layer is strictly zero, but the intensity of the wave propagating outside the layer is non-zero (although small) [14]. This means that the EM energy is leaking from the layer through its surfaces and the EM life-time rm is finite.
For sufficiently thick CLC layers, as the thickness L increases, the em life-time tw (as analytical solution shows [14]) increases as the third power of the thickness, and is inversely proportional to the square of the EM number n. The EM life-time rm is given as
rm = 1 /Iiii(«Jem) = (L/c)(5L/pn)2. (4)
COORDINATE
Figure 4. The calculated EM energy (arbitrary units) distributions inside the CLC layer vs. the coordinate (in the dimension-less units zr) for the three first edge modes (<f= 0.05, N = 16.5, n = 1,2,3).
Figure 5 The calculated EM energy (arbitrary units) distributions close to the CLC layer surface versus the coordinate (in the dimensionless units zr) for the plane wave directed inside (bold line) and outside the layer for the first edge mode (¿=0.05, N=16.5, n=l). 4. Absorbing Liquid Crystal
For the beginning we assume for simplicity that the absorption in the LC is isotropic. We define the ratio of the imaginary part to the real part of the dielectric constant as y i.e. 8=e0(l+/y)i Figure 6 illustrates the 1-R-T, i.e. the absorption in the layer, dependence on the frequency for a positive y where here R and T are now the intensity reflection and transmission coefficients, respectevely. Now R+T < 1. It happens that, for each n, the maximum absorption (i.e. maximal 1-R-T), occurs for
(5)
where a - ôZ.r/4 (in a typical situation, a » 1). From eq.(5), it follows that the maximum absorption occurs for a special relation between ô, y and L As was shown in refs. [11,15], just at the frequency values determined by eq. (5), the effect of anomalously strong absorption reveals itself for an absorbing chiral LC (Figure 6).
FREQUENCY
Figure 6. The absorption 1-R-T calculated vs. the frequency (I = 300, I = L T= 47T N, 6= 0.05) for 7 = 0.001.
5. Amplifying Liquid Crystal
We now assume that y < 0, which means that the CLC is amplifying. If |/| is sufficiently small, the waves emerging from the layer exist only in the presence of at least one external wave incident on the layer. In this case, R+T > 1 (or 1-R-T < 0), which just corresponds to the definition of an amplifying medium.
Figure 7 R calculated versus the frequency (7=300, l=Lr^0.05j (top) close to the threshold gain for the first lasing edge mode (
y = -0.00565), (bottom) close to the threshold gain for the second lasing edge mode ( v =-0.0129).
However, if the imaginary part of the dielectric tensor (i.e. ; ) reaches some critical negative value, the quantity (R+Tj diverges and the amplitudes of waves emerging from the layer are nonzero even for zero amplitudes of the incident waves. The corresponding y is a minimum threshold gain at which the lasing occurs. The equation determining the threshold gain (/) coincides with eq. (2). However, it should be solved now not for the frequency but for the imaginary part of the dielectric constant (/).
For a very small negative imaginary part of the dielec-
trie tensor (|y|) and L\\m(q)\ « 1 {L\m(q) « 1},, the threshold values of the gain for the EM can be represented by analytic expression [14]:
y = -6(nn)2/(6L V4)3. (6)
It also follows from Figure7 that the different threshold values of y correspond to the different edge lasing
i? 3
50
100
150
200
modes (divergent R and T) in Figure7. This means that separate lasing modes can be excited by changing the gain (r).
6. Boundary-Value Problem for Defect Mode
The defect mode structure (DMS), which is under consideration here, is shown in Figure 8. The solution to the boundary problem is carried out in the similar way as for a CLC layer above. As such, we give below the final results (all the simplifications assumed above for the CLC layer are implemented for the DMS too).
I
d
I
Figure 8. Schematic of the structure with a defect layer (DMS).
There is an option to obtain formulas determining the optical properties of the structure depicted in Figure 8 via the solutions found for a single CLC layer [14]. If one uses the expressions for the amplitude transmission T(L) and reflection R(L) coefficients for a single cholesteric layer (1a) and (1b) (see also[11,12]), the transmission and reflection intensity coefficients for the whole structure may be presented in the following forms:
T(d,L) = | [TJdexp(ikd)]/[1-exp(2ikd) RdRJ |2, ((7))
R(d,L) = | {Re+Ru TeTuexp(2ikd)/[1-exp(2ikd) RdRJ} |2, (8)
where Re(Te), Ru(Tu) and Rd(Td) (Figure 5) are the amplitude reflection (transmission) coefficients of the CLC layers (Figure 1) for the light incidences on the outer (top) layer surface, the inner top CLC layer surface from the inserted defect layer, and the inner bottom CLC layer surface from the inserted defect layer, respectively. It is assumed in deriving eqs. (7) and (8) that the external beam is incident on the structure (Figure 8) from the above only.
The calculated reflection spectra inside the stop band for the structure of Figure 8 for non-absorbing CLC layers are presented in Figures 9. The figures show the minima of R at some frequencies inside the stop band at positions which depend on the defect layer thickness d. The corresponding minima of R(d,L) and maxima of T(d,L) happen at frequencies corresponding to the DM frequencies [1-3,6,7].
For the layer thickness d = p/4, which is just one-half of the dielectric tensor period in a CLC, these maxima and minima are situated just at the stop band center. In the d/p interval 0 < d/p < 0.5, the value of DM frequency moves from the high frequency stop band edge to the low frequency one. Figures 6 present only R(d,L) because, for a non-absorbing structure 1 -R(d,L) -T(d,L) =0, Figures 6 show that, at some frequency, the reflection coefficient R(d,l) = 0. From eq. (8), one finds that the equation determining the frequencies of the reflection coefficient zeros is presented by the formula:
Re [1-exp(2ikd) RdRu]+Ru TeTuexp(2ikd)=0. (9)
7. Defect Mode
Similarly to the case of EM, the DM frequency ojd is determined by the zeros of the determinant of the system corresponding to the boundary-value solution for the structure depicted in Figure 8 [16, 17]. The determinant can be explicitly given as
{exp(2ikd)sin2qL-exp(-ixL)[(rq/K2)cosqL+i((T/2K)2+(q/K)2-l)sinqL]2/62]}=0.
(10)
It is to be noted that, for a non-absorbing CLC, at a finite length L does not reach zero for a real value of . However, it can reach zero when ¡s complex.
Using eq. (8), the dispersion eq. (10) may be reduced to the expression containing the amplitude reflection coefficients R(L) of the CLC layers. Thus, one can finally have
1- R(L)dR(L)uexp(2ikd) =0 (11)
The field of DM in each CLC layer is a superposition of two CLC eigenmodes [11,12], and can be easily found. In particular, the coordinate dependence of the squared modulus of the whole field is presented in Figure 10. It shows that, for larger dielectric anisotropy à, the DM field presents a more sharp growth toward the place where the defect layer occurs. Similarly to the EM case, at the external surfaces of the DMS, only the amplitude of the wave directed toward the defect layer reduces strictly to zero [16, 30].
The amplitude of the wave directed outwards is small, but does not reduce to zero. This is why there is a leakage of the DM energy outwards through the external surfaces of the DMS. The ratio of the corresponding energy flow to the whole DM energy accumulated in the DMS determines the inverse life-time.
s
b o
0 . 0
1 ° '
0 , 2
Figure 9. R(d,L) vs. the frequency for a non-absorbing CLC
(y = 0) at d/p = 0.1 (top) and d/p = 0.25 (bottom); J = 0.05,1 = 200,1 = Lt = 4,tN, where N is the director half-turn number at the CLC layer thickness L.
mneunwcv
For non-absorbing CLC layers, the only source of decay is the energy leakage through their surfaces. The analysis of the corresponding expressions [16,17] shows that the DM lifetime Tm is dependent on the position of the DM frequency 0 inside the stop band, and reaches a max-
k = r/2
imum just at the middle of the stop band, i.e. ' '.
8. Thick CLC Layers
In the case of DMS with thick CLC layers ((1 q L>>1»' some analytical results related to DM can be also obtained similarly as for the EM. In particular, the defect mode life time • reaches a maximum for the defect mode frequency at the stop band center at fixed CLC layers thickness. For the DM frequency at the middle of the stoop band, i.e. at N = ' - the DM life time Tm is given by
rm = [(3/r/cr)(L/p)E0yiexp[2/r&L/p]. (12)
Equation (12) reveals an exponential increase of Tm with increase of the CLC thickness L.
COORDINATE
Figure 10. Coordinate dependence of the squared amplitude of the DM field (arbitrary units) at the DM frequency for various
dielectric anisotropies (from the top to the bottom J - 0.05, 0.04, 0.025) and the defect layer thickness d = p/4 for the cho-lesteric layer thickness L = 50(p/2).
9. Absorbing and Amplifying Liquid Crystals
To take into account the absorption, we again assume e=e0(l+/'y) There are some interesting peculiarities of the optical properties of the structure under consideration (Figure 8). The total absorption at the DM frequency behaves itself unusually. For a small y, the absorption at the DM frequencies (See Figure 11) occurs to be much more than the absorption out of the stop band [30]. It is a manifestation of the so-called anomalously strong absorption effect known for perfect CLC layers at the EM frequency [11,15].
So, at the DM frequency coD, one sees that the effect of anomalously strong absorption (similar to the one for EM [11,15]) exists [16,17]. Moreover, the absorption enhancement for DM for small y is higher than that for EM. In the case of thick CLC layers, the dependence of y on L and other parameters ensuring maximal absorption may
be found analytically. For the position of coD just in the
middle of the stop band, the expression for y ensuring maximal absorption takes the following form y= (4/3-)(p/L) exp[-2-6(L/p)] (13)
The results for the transmission and reflection coeffi-
y < 0
cients at show that, for a small absolute value of the shapes of the transmission and reflection coefficients are qualitatively the same as for zero amplification (- ~ For a growing absolute value of T, a divergence of R(d,L) and T(d,L) happens at some point [16,17] with no signs of noticeable maxima at other frequencies. The corresponding value of y may be considered as close to the threshold value of the gain for the DFB lasing at the DM frequency.
__A
— D . 2 —□ .1 O O . 1
PHBOKBNCy
Figurell The total absorption for an absorbing CLC versus the frequency, 7=0.0003; d/p=0.1, 6=0.05, N=33.
Continuing the increase of the absolute value of y, one finds that the diverging maxima for R(d,L) at the EM frequencies appear (without traces of maximum at the DM frequency) for the gain being approximately four times more than the threshold gain for the DM (See Figure 12).
The observed results show that the DM lasing threshold gain is lower than the corresponding threshold for the EM. Another conclusion revealed from this study is the existence of some interconnection between the LC parameters at the lasing threshold, which for thick CLC layers was found analytically for DM (see [16, 17] and eq. (6) for the EM). A continuous increase of gain results in consequential appearance of lasing at new EM; with the disappearance of lasing at the previous EMs corresponding to lower thresholds (what was experimentally observed [3]).
The mentioned above interconnection between the LC parameters at the lasing threshold in the case of thick CLC layers may be found analytically. If the DM frequency
i^o is located at the stop band center, the corresponding interconnection for the threshold gain (7) is given by the formula in eq. (13) with a negative sign on the right-hand-side of the expression. Equation (13) gives that exponentially small value of for thick CLC layers confirms the statement mentioned above about lower lasing threshold for DM compared to EM.
The defect type considered above is a homogenous layer. The developed approach is applicable also to a defect of phase jump type [2,3,6,7], and therefore, the corresponding results are practically the same as above. Namely, one gets the equation related to the case of a phase jump defect if one performs in the equations presented above a substitution in the factor exp(2ikd), in-
stead of 2kd, the quantity A9 should be inserted, where is the spiral phase jump at the defect plane.
fracting circular polarization) for the whole structure may be presented in the following forms:
Fig.12 | R(d,L) |2 for an amplifying CLC versus the frequency,
(top) y=-0.00117; (bottom) y=-0.0045 d/p=0.1, 6=0.05, N=33.
11. Defect Mode with an Active Defect Layer
The DMs studied above are related to isotropic defect layers. Recently a lot of new types of defect layers have been studied [18-24]. The consideration below will be limited to a birefringent or absorbing (amplifying) layer inserted in a chiral liquid crystal. The reason for that is connected to the experimental [23,29] and theoretical [24,30,31] research on the DFB lasing in CLC where a defect layer is birefringent or absorbing (amplifying) with a general idea that the unusual properties of DM manifest themselves most clearly just at the middle of DMS, i.e. at the defect layer where the intensity of the DM field reaches its maximum. The analytic approach in the study of a DMS with a birefringent or absorbing (amplifying) defect layer is very similar to the previously performed DM studies for isotropic defect layer [16,17]. Therefore, we state here the final results of the present investigation with the suggestion to the readers to go through refs. [16,17] for a detailed approach.
In the following sections an analytical solution of the DM associated with an insertion of a birefringent or absorbing (amplifying) defect layer in the perfect cholester-ic structure is presented for light propagating along the helical axes.
12. Defect Mode at Birefringent Defect Layer
To consider the DM associated with an insertion of a birefringent layer in perfect cholesteric structure, we have to solve Maxwell equations and a boundary problem for electromagnetic wave propagating along the cholesteric helix for the layered structure, as depicted in Figure 8. Exploiting results obtained in ref. [25] (and using the same simplifications), one easily gets the results related to a birefringent layer. For example, if one neglects the multiple scattering of light of non-diffracting in CLC polarization, the transmission \T(d,L)\2 and reflection \R(d,L)\2 intensity coefficients (of the light of dif-
I T(d,Lf =
\R{d,L)\
1 -M2(k,d,/^n)((Tra-'al) RdRu RjTTM
R..+
1
-M2(k,d,An)(err<C) «A
(14)
(15)
In these equation, Re(Te), Ru(Tu) and Rd(Td) [16,17] are the amplitude reflection (transmission) coefficients of the CLC layer (Figure 8) for the light of diffracting polarization incident at the outer (top) layer surface, the light incidence at the inner top CLC layer surface from the inserted defect layer, the light incidence at the inner bottom CLC layer surface from the inserted defect layer, respectively. Also, ar and are the polarization vectors of light exiting the CLC layer inner surface, reflected at the inner bottom CLC layer surface upon the incidence from the inserted defect layer, and of light whose some polarization vector^transforms to the polarization vector <t. while crossing the birefringent defect layer of thickness d, respectively.
Figure 13a. Diffracting polarization intensity transmission coefficient \T{d,L)\ for a low birefringent defect layer vs. frequency ( 5 = 0.05 and N = 33 is the director half-turn number at the CLC layer thickness L.) for a diffracting incident polarization at the birefringent phase shift at the defect layer thickness A/p = ti/20 at d/p = 0.25.
Figure 13b. Same as Figure 13a at Acp = n16 •
FREQUENCY
Figure 13c. Same as Figure 13a at &<p = nH.
Further, An is the difference of two refractive indices in the birefringent defect layer, and M(k,d,An) is the phase factor related to the light propagating in a birefringent defect layer. The corresponding polarization vectors may be found [11,12,25], and the polarization vector sed may be easily calculated if d and An are known.
In the general case, following eqs. (14) and (15), the calculation of reflection and transmission coefficients is performable analytically. However, it is rather cumbersome. This is why it is studied below in details for the case of a low birefringence.
Following the above mentioned simplification and the assumption that the refractive indices (RIs) of the DMS external media coincide with the average CLC refractive index and the average RIs of defect layer, the RIs of the defect layer may be given by the formulas
An
2 '
An
(16)
where n0 coincides with the average CLC refractive index and An is small i.e. An /n0 < S . The phase difference of two beam components with different eigenwave polarizations at the defect layer thickness is A9 = kd An /n0. Finally, one gets the following expressions for transmission and reflection coefficients of light with a diffracting polarization for the incident beam in the case of low birefringence:
(17)
(18)
Figures 13 illustrate the calculated results for the transmission \t«i,L)\2 coefficients of the light of diffracting polarization for the case of low birefringence corresponding to various values of the birefringent phase factor. Figures 13 show that, at low values of phase shift between eigenwaves at their crossing the defect layer (Acp < nil), the shape of transmission curve is very similar to those for DMS with an isotropic defect layer. How-
for an isotropic defect layer increase of transmission at the DM frequency gradually disappears, and at
A<p = nil (Figure 13 c) does not appear at all.
It is well known that the position of the EM frequency in the stop band is determined by the frequency of transmission (reflection) coefficient maximum (minimum) [13]. Thus, the performed calculations of the transmission spectra (Figures 14) determine a real component of the DM frequency. However, because DM is a quasi-stationary mode, an imaginary component of the DM frequency is not zero [16,17]. A direct way to find the imaginary component of the DM frequency is through solving the dispersion equation. If the multiple scattering of non-diffracting polarization light is neglected, the dispersion equation is presented by the relationship
M2 (k,d, An)sin2(tjrZ)
cxp(-/r£)
^«(^{(¿J.gJ-.W)
(19)
-005 -0.05
FREQUENCY
Figure 14a. Calculated intensity transmission coefficients at a low birefringent defect layer for an amplifying CLC layer vs. the frequency close to their divergence points for diffracting incident polarization at A(p = g = -0.00150; d/p = 2.25.
Figure 14b. Same as Figure 14a at Atp = 7116, g = -0.002355.
ever, approaching A(p to W2 (Figures 138b) the typical
o
J
u
0
Figure 14c. Same as Figure 14a at A<p = /r/2,g = -0.004500.
For absorbing CLC layers in DMS, the anomalously strong absorption effect reveals itself at the value of g ensuring a maximum of the total absorption in the DMS [16,17]. For finite thicknesses of CLC layers, the DM frequency ojd occurs to be a complex quantity, which may be found by a numerical solution of eq. (19). For very small values of the parameter g, the reflection and transmission spectra of MDS with absorbing (amplifying) CLC layers are similar to the studied spectra in refs. [16,17]. In particular, positions of dips in reflection and spikes in transmission inside the stop-band just correspond to Re[ tJp], and this observation is very useful for numerical solution of the dispersion equation. It is of more concern regarding the DM life-time that it reduces for absorbing CLC layers compared to the case of non-absorbing CLC layers [16,17].
14. Absorbing (Amplifying) Isotropic Defect Layer
The study of the DM associated with an insertion of an absorbing (amplifying) isotropic layer (Figure 5) is performed in the same manner as above (see also refs.
[16,17,26). The transmission \W.L)\' and reflection \R(d,L)\2 intensity coefficients (of light of diffracting circular polarization) for the whole structure may be presented as
Figure 14d. Same as Figure 14a at = g = -0.000675.
13. Amplifying and Absorbing CLC Layers
The experiment [3] and the theory [16,17] show that the unusual optical properties of DMS at the DM frequency coD may be effectively used for the enhancement of the DFB lasing. In order to study how the birefringent defect layer influences anomalously strong amplification and absorption effects, we make assumptions [14,17] that the average dielectric constant of CLC has an imaginary addition, i.e. e=eo(l+ry), (note that in real situations g « 1). The value of g may be found from solution of the dispersion eq. (19). Another option (see refs. [16,17]) is to study the transmission and reflection coefficients (of eqs. (17) and (18), respectively) as a function of g.
For amplifying CLC, the value of g corresponding to the divergence of DMS reflection and transmission coefficients just determines the solution to the dispersion eq. (19), and also, the threshold DFB lasing gain in the DMS [16,17]. So, there is an option to find the threshold value of g by calculating the DMS reflection and transmission coefficients for varying g and finding its value at the points of DMS reflection and transmission coefficients divergence.
According to the formulated approach, Figures 14 presenting the values of DMS transmission coefficient close to their divergence points demonstrate the growth of the threshold DFB lasing gain ( | 7 | )with the increase of the birefringent phase factor Acp, and even disappearance of the divergence at DM frequency at Acp =n!2 .
I T(d,L)\2 =
\R{d,L)\2 =
TeTd exp (ikd (1 + /g))
1 - exp (likd (1 + ig)) RjRu RdTeTuexP(2ikd(l+ig))
(20)
R„ +
l-exp(2/fe/(l+/g))JRA
(21)
respectively, where Re(Te), Ru(Tu) and Rd(Td) are determined above. The factor ( 1+ig) is related to the defect layer only, and corresponds to the dielectric constant of the defect layer having its form s=e0(l+2?g) with g being positive for an absorbing defect layer and negative for an amplifying one.
The DM frequency Wo is determined by the following dispersion equation:
cxp(2*M(l + /g))sin2(qI.ycMs*L) (if)cos(?L)+,j(^J -ljsin(ii)
(22)
For finite thicknesses of CLC layers, wD occurs to be a complex quantity, which may be found by a numerical solution of Eq. (22). For very small values of the parameter g, the reflection and transmission spectra of MDS with an active defect layer are similar to the spectra studied in refs. [16,17]. In particular, positions of dips in reflection and spikes in transmission inside the stop-band correspond to Re[wo], Further, the DM life-time is reduced for absorbing defect layers compared to the case of non-absorbing defect layer [16,17].
As in the case of investigated DMs with absorbing CLC layers [16,17], the effect of anomalously strong absorption takes place in DMS with an absorbing defect layer. The effect reveals itself at the DM frequency, and reach-
5
4
3
2
1
Ü
U
es its maximum (maximum of 1-R-T) for a definite value of g, which may be found by using eqs. (20) and (21). Figures 15 demonstrate the existence of the anomalously strong absorption effect. As follows from Figures 15, the maximum values of the anomalous absorption [11,26] are reached for g = 0.04978 and 0.0008891 depending on the defect layer thickness d.
In the case of thick CLC layers (|q|L»l) in the DMS, the value of g ensuring absorption maximum may be found analytically. For the DM frequency wD in the middle of stop-band, the maximal absorption corresponds to
Equation (23) shows that the gain g corresponding to the maximal absorption is approximately inversely proportional to the defect layer thickness d.
In the case of DMS with amplifying defect layer (g < 0), at some value of |g| , divergence of reflection and transmission coefficients occurs. The corresponding values of g are the lasing gain thresholds. Their values may be found numerically using eqs. (20) and (21) for | T(d,L)\1 and | Rtd.L)\-, respectively, or found approximately by plotting | T(d,L>|J and \R(d,L)\- for varying g. The second option is illustrated by Figures 16, 17 and 18, where the almost divergent values of | T(d,L)\2 \R(d,L)\1 0r the absorption (1-R-T) are shown. The used values of g in Figures 16, 17 and 18 are close to the threshold ones ensuring the divergence of and | R(d,L) \2 . The results show that the minimal thr 17WZ)|2eshold |g| corresponds to
the location of coD just in the middle of the stop-band, and |g| is almost inversely proportional to the defect layer thickness. Figures 16 and 17 correspond to the location of the DM frequency <-l>d close to the middle point of the stop-band, and demonstrate the decrease of the lasing threshold gain with the increase in the defect layer thickness.
Figure15a. Total absorption vs. frequency for an absorbing defect layer and non-absorbing CLC layers at g = 0.04978 for d/p = 0.1.
Figure15b. Same as Figure 15a at g = 0.08.
J V
-0.05_ -004
FRQECY
Figure15c. Same as Figure 15a at g = 0.00008891 for d/p = 22.25.
Figure15d. Same as Figure 15a at g = 0.0008891 for d/p = 22.25.
Figure 18 corresponds to the location of the DM frequency close to the stop-band edge, and demonstrates increase of the lasing threshold gain as toD approaches to the stop-band edge. The analytical approach for thick CLC layers (q>>1) results in the similar predictions, namely, for tJD in the middle of the stop-band, the threshold value of gain is given by eq. (23) with a negative sign on the right-hand-side of the expression.
3
J
o
o
u
0
ü
Figure 16. Total absorption vs. frequency for amplifying defect layer and non-absorbing CLC layers at g = -0.0065957 for d/p = 0.25.
Figure 17a. T(d) vs. frequency for amplifying defect layer and non-absorbing CLC layers at g = -0.001 for d/p = 2.25.
Figure 17b. Same as Figure 17a at g = -0.00008891 for d/p = 22.25.
So, as eq. (23) shows, the thinner defect layer corresponds to the higher threshold gain g. Similar results can also be related to the absorption enhancement (eq. (23)). The thinner defect layer corresponding to the higher g-value ensures maximal absorption.
15. Liquid Crystals with Locally Anisotropic Absorption
The discussed above case of isotropic absorption in CLC does not cover all options happening in CLC. For example, quite common is alignment of dye molecules with clearely presented absorption lines in liquid crystals. If the director distribution in a liquid crystal sample is not homogenious (what is the case of CLC) the local anisot-ropy of absorption in the sample exists and manifests itself in some circumstances [11, 12]. The corresponding effects depend on the value of liquid crystal order parameter and disappeare if the order parameter value is zero, i.e. at the point of liquid crystal phase transition to liquid. The corresponding effects in the transmission and reflection spectra, in particular Borrmann effect, were studied both experimentally [33,34] and theoretically [11,12,33]. In the present section the influence of local anisotropy of absorption on characteristics of localized modes is theoretically studied.
So, let us begin from discussing of the dielectric tensor of a substance with anisotropic absorption. The principal values of corresponding dielectric tensor are complex and have different imaginary parts. In general case all three imaginary parts are different. For simplification of the problem we shall asume that only two principal values of dielectric tensor are complex. Returning to the CLC we assume that the axis corresponding to a real principal value of the dielectric tensor is directed along the spiral axis and two other axes corressponding to complex principal values are rotating around the spiral axis. These rotating a It is why now we have to insert two complex principal values of dielectric tensor in the expressions for dielectric anisotropy after formulas (1). As the result the dielectric anysotropy 5 becomes a complex quantity. Luckelly, the expressions for reflection and transmission coefficients (1) are exact and are applicable to the case of anisotropic absorption which is under concideration here.xes determine the local depending on the coordinate along spiral axis direction of absorption anisotropy.
As we expect the local absorption anisotropy reveal itself in DMS reflection and transmission spectra due to the Borrmann effect [11,12,33]. The Figures 19 and 20 demonstrate Borrmann effect in reflection for a CLC layer and DMS with local absorption anisotropy. Figures 19 and 20 present the calculation results and show that the stop-band edges become nonequivalent in scattering spectra (for no-nabsorbing CLC or CLC with isotropic absorption the reflection is symmetric relative to the stop-band center).
Figure 18. R(d) vs. frequency for amplifying defect layer and non-absorbing CLC layers at g = -0.04978 for d/p = 0.1.
LI
U
u
J
0-
0
j
0.2 -0.1 0 CJ
frequency
Figure.19 Borrmann effect in reflection for EM (see Fig.l) for
locally anisotropic absorption in CLC at 0 = 0.05 +0.015i, N = 300 (Thin curve corresponds to the complete absence of absorption).
The reflection coefficient value close to the frequencies of one stop-band edge is much more large than one close to the frequencies of the other stop-band edge. The Figures 21 and 22 demonstrate Borrmann effect in transmission for a CLC layer and DMS with local absorption anisotropy. Figures 21 and 22 show that the stop-band edges are also nonequivalent in transmission spectra.
Figure 20 Borrmann effect in reflection for DMS (see Fig.8) for locally anisotropic absorption in CLC at J = 0.05 +0.003i, N = 75, d/p=0.1.
The transmission coefficient value close to the frequencies of the same as for the showing an enhancement of reflection stop-band edge is much larger than close to the frequencies of the other stop-band edge.
: "
:. 02
5
7 C . C L&-
Figure 21 Borrmann effect in transmission for EM for locally anisotropic absorption in CLC at S = 0.05 +0.015i, N = 300.
Figure 22 Borrmann effect in transmission for DMS for locally anisotropic absorption in CLC at l> = 0.05 +0.003i, N = 75, d/p=0.1.
Naturally, that the total absorption (1 - R - T) has to be different at the stop-band edges.
N.. '1.1 0 J _
frequency
Figure 23 Total absorption in CLC layer for locally anisotropic absorption in CLC at (>" = 0.05 +0.015i, N = 300 (see Fig.l).
The Figures 23 and 24 demonstrate that due to existing of the Borrmann effect in the transmission and reflection spectra for a CLC layer and DMS with local absorption anisotropy a suppression of the total absorption occurs at the frequencies close to one of the stop-band edges.
0.15-0.125 -0.1 -0.075-0.OS-0.025 0 0.025 rhiqulifcy
Fig.24 Total absorption in DMS for locally anisotropic absorption in CLC (see Fig.8) at â = 0.05 +0.003i, N = 75, d/p=0.1.
The strength of the discussed effect depends on the order parameter value and the effect disappears for zero order parameter value being at the maximum at its value equal to 1.
The presented reflection and transmission spectra for EM and DM show that the EM and DM life times in the
case of a local absorption anisotropy in CLC depends on the localized mode frequency position relative the stop-band edge frequency growing with approach of the localized mode frequency to the stop-band edge frequency where the manifestation of the Borrmann effect takes place. To find the EM and DM life times in the general case one has to solve numerically the dispersion equations (2) and (10) for EM and DM, respectively. However for sufficiently thick CLC layers an analytical solution may be found.
For example, if the CLC layer thickness L is sufficiently large and the role of inevitably present in real situations isotropic absorption in the determining of the EM lifetime exceeds the role of energy leakage through the layers surfaces the ratio of life-time at the edge frequencies may be estimated by the following expression:
7~r/?"ab =(UE\i(€oy +2eo lm Sj/lCOEuCoy + 1/Tm) (24)
where rB.r^ rm are the life-time at the stop-band edge where the Borrmann effect happens, at the opposite edge, at the edges in the case of non-absorbing CLC (see (4)), respectively, and y is the determining isotropic component of absorption quantity given by the same as above expression s=e0(l+iy),. If y is approaching zero what is happening when the order parameter is approaching 1 (if we neglect all sources of absorption except the dye) T~B is coinciding with rm given by (4), what corresponds suppression of absorption for EM at the stop-band edge frequency in the case of CLC with local anisotropy of absorption. At the opposite stop-band edge frequency the absorption is enhanced and the EM lifetime being proportional to 1/2 Co lm 6. is shorter than
Conclusion
The performed analytical description of the EM and DM (neglecting the polarization mixing) allows one to reveal clear physical pictures of these modes, which is applicable to the EM and DM in general. For example, lower lasing threshold and stronger absorption at the DM frequency compared to the EM frequencies are the features of any periodic media. Note that the experimental studies of the lasing threshold in ref. [3] agree with the corresponding theoretical results [25-28]. Moreover, the experiment ref. [3] confirms also the existence of some interconnection between the gain and other LC parameters at the threshold pumping energy for lasing at the DM and EM frequencies. For a special choice of the parameters in the experiment, the obtained formulas may be directly applied. However, one has to generally take into account a mutual transformation at the boundaries of the two circular polarizations of opposite sense. In the general case, the EM- and the DM-field leakages from the structure are determined as well by the finite CLC layer thickness. Only for sufficiently thin CLC layers, or in the case of the DM frequency being very close to the stop-band frequency edges, the main contribution to the frequency width of the EM and DM is due to the thick-
ness effect, and the above developed model may be directly applied for describing of the experimental data.
The results related to CLCs with local anisotropy of absorption may be useful for optimizing of DFB lasing. Really, the corresponding theoretical predictions show what one of the two stop-band frequencies is preferable for obtaining the most lower lasing threshold.
An important result related to the DFB lasing at DMS with an active defect layer [25-28] may be formulated. The lasing threshold gain in defect layer decreases with the decrease in layer thickness. The similar result relates to the effect of anomalously strong absorption phenomenon where the value of gain in the defect layer (ensuring a maximal absorption) is almost inversely proportional to the defect layer thickness. Note that the obtained results are qualitatively applicable to the corresponding localized electromagnetic modes in any periodic media, and may be regarded as a useful guide in the studies of the localized modes with an active defect layer in general. It should also be mentioned that the localized DM and EM reveal themselves in an enhancement of some inelastic and nonlinear optical processes in photonic LCs. For examples, the experimentally observed effects for the enhancement of nonlinear optical second harmonic generation [35] and lowering of the lasing threshold [36] in photonic LCs have to be mentioned along with the theoretically predicted enhancement of Cerenkov radiation [11,12]. Finally, the results presented here for the EM and DM (see also [16] and [32]) not only clarify the physics of these modes but alsomanifest a complete agreement with the corresponding results of the previous investigations obtained by implementing numerical approach [13].
Acknowledgment
The work is supported by the RFBR grants 15-02-00580_a and 15-02-08757_а.
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