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Landau Institute for Theoretical Physics, Moscow, Russia

- *Corresponding Author:
- Belyakov VA

Landau Institute for Theoretical Physics, Moscow, Russia

**Tel:**+7 499 137-32-44

**E-mail:**[email protected]

**Received Date**: April 25, 2017; **Accepted Date:** April 29, 2017; **Published Date**: May 10, 2017

**Citation: **Belyakov VA (2017) From Liquid Crystals Localized Modes to Localized Modes in Photonic Crystals. J Laser Opt Photonics 4: 153. doi: 10.4172/2469-410X.1000153

**Copyright:** © 2017 Belyakov VA. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.

**Visit for more related articles at** Journal of Lasers, Optics & Photonics

Recently a great attention was paid to the localized optical modes in
photonic liquid crystals (PLC) due to the perspectives of their efficient
application in the linear and nonlinear optics [1,2]. Luckily, an analytic
description of the localized modes is available for PLC due to their spiral
structure [3,4]. The purpose of this paper is to attract attention to the
fact that many results obtained analytically for PLC are also applicable
to the conventional photonic crystals and the corresponding simple
formulas may be useful in the description of the localized modes in
conventional photonic crystals. So, below are given main relationships
for the edge localized modes (EM) in PLC and discussed the options of
their application for describing EM in conventional photonic crystals **(Figure 1)**.

For describing EM in chiral liquid crystals (CLC), for the certainty it
will be assumed that CLC is used for cholesteric liquid crystals, one has
to solve an optical boundary problem for a perfect CLC layer **(Figure
1)**. The solution of the boundary problem [3] results in the following
expressions for the amplitude reflection and transmission coefficients
for a layer of thickness L (it is assumed that the spiral axis of CLC is
normal to the layer surfaces and light of the circular polarization of
the same sense of chirality as the one of CLC spiral propagates along
this axis):

R(L)=δsinqL/{(qτ/κ2)cosqL+i[(τ/2κ)^{2}+ (q/κ)^{2}-1]sinqL} (1a)

T(L)=exp[iκL](qτ/κ2)/{(qt/κ2)cosqL +i[(τ/2κ)^{2}+ (q/κ)^{2}-1]sinqL} (1b)

where

Here δ is the dielectric anisotropy of CLC with ε_{II} and ∈_{⊥} as the
local principal values of the CLC dielectric tensor [3], k=ωϵ_{0}^{1/2}/c with c as the speed of light, and τ=4π/p with p as the cholesteric pitch.

The frequencies of reflection coefficient minima **(Figure 2)** which
may be numerated by integer numbers n [4] determine the real part of
the EM frequency ω_{EM}.

The imaginary part of EM frequency ω_{EM}=ω(1+iΔ) where Δ is a
small parameter can be found from the dispersion equation [4] which
in a general case demands a numerical approach for it solution. The EM
energy is localized inside the layer (with the number of energy density
maxima inside the layer coinciding with the EM number n). For thick
layers the dispersion equation determining via the imaginary part of
EM frequency the EM life-time may be solved analytically:

Δ=- ½ δ(nπ)2/(δLτ/4)^{3} (2)

I.e., the EM life-time for non-absorbing CLC is proportional to the third power of the layer thicknesses.

At the EM frequency effects of anomalously strong absorption or
amplification exist for absorbing or amplifying CLC [5]. It is why, in
particular, the observed lasing threshold at the EM frequency occurs to
be lower than **(Figure 2)** for lasing in the corresponding homogeneous
layer [1,2,5]. The lasing threshold value is determined by a minimal
value of the negative imaginary addition to the dielectric constant
e_{0}(1 -i), (with γ being determined by the population inversion in the
lasing transition) ensuring the lasing. In a general case γ can be found
from the dispersion equation. However, for a thick layer the threshold
values of the gain (γ) for the EM can be presented by the analytic
expression [4]:

γ=- δ(nπ)^{2}/(δLτ/4)^{3} (3)

The equation (3) shows that the lasing threshold at the EM frequency is inversely proportional to the third power of the layer thickness L and formally approaches to zero with a growing L (naturally, the decreasing of γ according (3) is limited by the CLC absorption [6].

The experimentally observed enhancement of some optical effects in CLC at the EM frequency (lowering of the lasing threshold, abnormally strong absorption etc.) described by the presented above analytic theory [4,6] is also observed for conventional photonic crystals [7-11]. However, a theoretical numerical approach is usually applied for interpreting the experiment and as a result rather frequently the physics of studied phenomenon remains not completely clear. To the mentioned above phenomena enhanced at the EM frequency one can add a nonlinear frequency conversion, sum frequency generation, Cherenkov radiations etc. In general, the situation with the theoretical description of localized mode effects in conventional photonic crystals appears as follows. Almost all publications reporting experimental results on the localized modes are accompanied by numerical calculations of the measured quantities showing a good agreement with the measured results. However, a clear physical image of the observed phenomenon is often escaping the theoretical interpretation of the observed results and the most often the explanation is of the type “the effect is enhanced near to the stop-band edge”. It is why due to the common physics of the localized mode effects in CLC and conventional photonic crystals the analytic theory developed for CLC can be applied for conventional photonic crystals as a qualitative guide for a qualitative description of the experiment and clearing the physics of phenomenon.

As examples, below some enhanced, due to the EM, phenomena
experimentally observed in conventional photonic crystals [9-11] are
mentioned and the interpretation related to the localized modes for
one example is proposed. We mention enhancement phenomena in
conventional photonic crystals for the second nonlinear harmonic
generation (SHG) [10], third nonlinear harmonic generation (THG)
[12] and sum frequency generation [11] where the enhancement effects
at the stop-band edge frequency are attributed to the decrease of the light group velocity, increase of the density of states and to the growth
of wave field at this frequency [7-9] **(Figure 3)**.

The enhancement of SHG generation in a ZnS-SrF2 periodic
structure near the photonic band edge [10] was observed at applying
to the sample of femtosecond laser pumping pulses. The SHG was
measured as a function of the pumping wave incidence angle. The SHG
intensity enhancement observation **(Figure 3)** just corresponded to the
pumping wave incidence angle coinciding with the first minimum in
the linear reflection curve for the pumping frequency. As the estimate
shows this angle just corresponds to the first EM frequency coinciding
with the pumping frequency so the enhancement can be interpreted
in the terms of EM as a result of anomalously strong absorption of the
pumping wave.

We considered here the enhancement phenomena related to the localized edge modes. Similar effects are also happening for other types of localized modes, for example, localized defect modes [12]. So, in conclusion should be emphasized that the presented for localized modes in CLC results are of a general nature and are qualitatively applicable for different localized modes in various structures.

The work is supported by the RFBR grants № 16-02-00679_a and № 16-02-00295_a.

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- Chigrinov VG(2013)New Developments in Liquid Crystals and Applications.Choudhury PK editor. NovaPublishers, New York, pp: 199-227.
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__Bowden__CM. (1994) The photonic band edge optical diode. J Appl Phys 75: 1896. - Scalora M, Bloemer MJ, Manka AS, Dowling JP, Bowden CM et al. (1997) Pulsed second-harmonic generation in nonlinear, one-dimensional, periodic structures. Phys Rev A 56: 3166.
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- Dolgov T, Didenko NV, Martem'yanov MG, Aktsipertrov OA, Marowsky G et al. (2002) Third-harmonic generation in silicon photonic crystals and microcavities. : IEEE Xplore 75:17.
- Belyakov VA, Semenov SV (2011) Optical defect modes in chiral liquid crystals. JETP 112: 694.

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