Hands, Philip J.W.

Mavrogordatos, Themis K.

Morris, Stephen M.

Wilkinson, Timothy D.

This paper aims to give an overview of distributed feedback (DFB) lasing in chiral nematic liquid crystals (CLC) and explain the salient features that render them unique. Particular emphasis is given to the transmission and reflection characteristics of CLCs as well as to the key role of the Density of Photon States (DOS) in the presence of an active medium. Discussions and comparisons with wave propagation in other periodic solid media are also included

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     Edinburgh Research Explorer                                      Laser emission characteristics of dye-doped cholesteric filmsCitation for published version:Mavrogordatos, TK, Morris, SM, Hands, PJW & Wilkinson, TD 2011, Laser emission characteristics of dye-doped cholesteric films. in London Communications Symposium (LCS) and Photonics London. UCL Dept ofElectronic & Electrical Engineering, London Communications Symposium 2011, Photonics London 2011,London, United Kingdom, 8-8 September.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Peer reviewed versionPublished In:London Communications Symposium (LCS) and Photonics LondonGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 28. Apr. 2017Laser Emission Characteristics of Dye-Doped Cholesteric Films Th. K. Mavrogordatos, S. M. Morris, P. J. W. Hands and T. D. Wilkinson Centre of Molecular Materials for Photonics and Electronics (CMMPE), Department of Engineering, University of Cambridge Abstract:  This paper aims to give an overview of distributed feedback (DFB) lasing in chiral nematic liquid crystals (CLC) and explain the salient features that render them unique. Particular emphasis is given to the transmission and reflection characteristics of CLCs as well as to the key role of the Density of Photon States (DOS) in the presence of an active medium. Discussions and comparisons with wave propagation in other periodic solid media are also included.   1.  Introduction Recently, there has been an explosion of interest in band-edge lasing in chiral liquid crystals (LC), which is related to the low-excitation threshold and tuneability of the band-gap when a LC is subjected to a variety of different external stimuli. Distributed Feedback (DFB) lasing occurs at frequencies close to the stop-band edges that are associated with the so called band-edge lasing modes[1,2]. As mentioned in [1], the case of CLCs is unique when compared to other conventional periodic media, as an exact solution of Maxwell’s equations exists owing to their unusual optical properties. Thus, a boundary value problem can be formulated for non-absorbing, absorbing and amplifying Chiral LCs with analytic solutions. In this paper we follow and expand the methodologies presented in [1,2,3] attempting to link our results with experimental data given in [4,5] .  2. Wave propagation in CLCs: transformed frames and ‘reduced indices’ In the chiral nematic phase the director precesses continuously and orthogonally along a single direction (coinciding with the        in our case) giving rise to a helical structure. A full rotation of the director determines the helix pitch  . In [2] de Vries introduces a rotating frame of reference following the precessions of the director, which is schematically shown in [3] alongside transformation relations to the Cartesian frame. In that rotating frame, the eigenwaves have the   dependence                           with                           (2.1)       and                                                                   (2.2) The preceding relation defines the so called ‘reduced refractive indices’ that are a consequence of the introduction of the rotating frame when applying Maxwell’s equations in the CLC cell. The ordinary and extraordinary refractive indices enter the relations through the reduced wavelength            (2.3) with                     (2.4) and                    (2.5) .The eigenwaves have elliptical polarisations in general, as shown by their corresponding unit vectors                               (2.6). The parameter    is the ellipticity factor that reflects changes in the polarisation of the normal modes as the wavelength varies. As we can observe, for specific wavelengths, the reduced index    that corresponds to the first eigenwave is imaginary, setting the limits of the stop band exhibited by the periodic structure (of infinite length). At the edges of the stop band, the ellipticity factor is either zero or tends to infinity giving rise to a linearly polarised wave (in the rotating frame) that is perpendicular or parallel to the director respectively           . In the laboratory frame this corresponds to a circularly polarised wave. As the (finite) number of the director precessions (   decreases, the distance between the two minima of the reflection coefficient either side of the band-gap broadens: this is a result that is also derived in [1] whereby the distance between successive edge modes is inversely proportional to the square of the cell thickness. 3. Density of (photon) states (DOS): definition and importance  By invoking Fermi’s Golden Rule, in the semi-classical approach, the photon emission rate of light with a certain polarisation and wave vector    from an excited fluorescent molecule is                                  ,          (3.1) where    is the density of states for the normal mode  ,    is the electric dipole moment,     is the electromagnetic vector potential corresponding to the plane monochromatic electromagnetic wave with wave vector    ,that represents the perturbation (related to the eigenfield of the normal modes supported by the structure,     and     ) and                 denote the wave functions of the initial and final state between which the transition occurs.  The above expression can be written as                        (3.2) denoting the projection of the optical field onto the direction of the dipole moment    . The DOS is defined as (the absolute value of) the inverse slope of the dispersion relation [3]. As for any structure, given the transmission coefficient                                     (3.3), the DOS for a CLC reads                                                         (3.4) for                                                                                           (3.5) In the figures that follow, we depict the DOS for the two normal light modes that propagate parallel to the helical axis as well as their relative intensity contribution. These results are obtained after performing the orientational average of the squared projections of the transition dipole moment with respect to the rotating frame planes. The degree of order is indicated by an order parameter                     (3.6), where   is the angle between the director and the dipole moment (for perfect alignment      while for poor alignment     ).   Figure 3.1: Normalised DOS of the eigenwaves    (a) and    (b) for a CLC cell with thickness       and relative dielectric anisotropy       (left). Relative intensity contributions of the eigenwaves    (c) and     (d) for a CLC cell with the same thickness and relative dielectric anisotropy and order parameter         (right).The yellow area marks the stop band. As we can observe, the DOS of the eigenwave     peaks at either side of the stop band (denoted as short and long-wavelength edge) and the same trend is also followed by the emission intensity (which is dependent upon the order parameter   ). The same result can be obtained from the analysis carried out in [1], where the transmission coefficient is expressed as a ratio of the (complex) amplitude of the diffracted eigenwaves and the transmitted wave, upon the formulation of a boundary value problem. For an absorbing or an amplifying CLC structure, now, losses can be incorporated in our analysis by introducing a small (wavelength independent) imaginary part   in the expression for the average dielectric constant                   , (3.7) where a negative   corresponds to gain and a positive   corresponds to losses, as we have assumed an           dependence for the normal modes [3]. The introduction of an imaginary part in the dielectric constant affects substantially both the reflectivity of the structure (as well as the transmittance) and the DOS.   Figure 3.2:Intensity reflection of a 30-period CLC(a) with a small negative imaginary part of the mean dielectric constant,γ                     (gain)(b)with real mean dielectric constant      (c) with a small positive imaginary part of the dielectric constant                     (losses) . In all,       .    Figure 3.3: DOS (algebraic value)of the eigenwave     for a 30-period CLC cell (a) with a small negative imaginary part of the mean dielectric constant          (gain)(b) with real mean dielectric constant,     (c) with a small positive imaginary part of the mean dielectric constant        (losses).In all cases       . 0.7 0.8 0.9 1 1.1 1.2 1.3024681012 Density of States (DOS) of the eigenmode E1reduced wavelength || = |/(np)|normalised density of states0.7 0.8 0.9 1 1.1 1.2 1.31.00051.0011.00151.0021.00251.0031.0035 Density of States (DOS) of the eigenmode E2reduced wavelength || = |/(np)|normalised density of statesshort wavelength edgelong wavelength edge0.7 0.8 0.9 1 1.1 1.2 1.3024681012141618 Relative Intensity Contribution of the eigenmode E1reduced wavelength || = |/(np)|normalised intensity0.7 0.8 0.9 1 1.1 1.2 1.31.2251.231.2351.241.2451.25 Relative Intensity Contribution of the eigenmode E2reduced wavelength || = |/(np)|normalised intensityshort wavelength edgelong wavelength edge0.7 0.75 0.8 0.85 0.9 0.95 1 1.050123456 Reflectivity of the CLC structurereduced wavelength  = /(np)Reflectivityshort wavelength edge0.7 0.8 0.9 1 1.1 1.2 1.3 1.400.10.20.30.40.50.60.70.80.91 Reflectivity of the CLC structurereduced wavelength  = /(np)Reflectivity0.7 0.8 0.9 1 1.1 1.2 1.3 1.400.10.20.30.40.50.60.70.80.91 Reflectivity of the CLC structurereduced wavelength  = /(np)Reflectivity0.82 0.84 0.86 0.88 0.9 0.92 0.94-25-20-15-10-50510 Density of States (DOS) of the eigenmode E1reduced wavelength || = |/(np)|normalised density of statesshort wavelength edge0.82 0.84 0.86 0.88 0.9 0.92 0.94024681012141618 Density of States (DOS) of the eigenmode E1reduced wavelength || = |/(np)|normalised density of statesshort wavelength edge0.82 0.84 0.86 0.88 0.9 0.92 0.9401234567 Density of States (DOS) of the eigenmode E1reduced wavelength || = |/(np)|normalised density of statesshort wavelength edge(a) (c) (b) (a) (c) (b) (a) (b) (c) (d) In the lossless case, the DOS increases with thickness in a parabolic fashion, which is also a well known result for wave propagation in periodic layered media[4]. The DOS decreases rapidly with losses, as also discussed in [4,5], where a relation of the form                (3.8) is introduced to explain the observed behaviour. The exponentially decaying term represents the incorporation of losses from the case of the Fabry-Perot resonator as an exponential multiplicative factor in the mirror reflectivities. In the figures that follow, we depict the dependence of losses on the cell thickness by introducing a small imaginary factor    in the framework of the original analysis in [1], that modifies accordingly the real and imaginary part of the transmission coefficient.   Figure 3.4: Maximum DOS for a CLC cell (as a function of its thickness) for the eigenwave     (a) with birefringence        and a loss factor of         (b) with birefringence        and a loss factor of         (c) with birefringence         and a loss factor of         .  The curves fit qualitatively the experimental data reported in [4] and [5], as to a first order approximation, the lasing threshold energy is inversely proportional to the density of states. The threshold energy, then, appears to follow the suggested form of the relation                 (3.9) in [5]. The slope efficiency is also found to have the same behaviour with the DOS. As it can be deduced, the DOS decreases dramatically with losses. When losses tend to zero, the parabolic profile is approached [4, 5]. This is similar to conventional DFB structures [4]. The DOS increases also with increased refractive index difference between alternating layers in a quarter-wave stack [6].   4. Periodic layered media: calculation of the DOS and comparison with the CLCs Although CLCs differ substantially from a quarter dielectric stack, the DOS seems to exhibit a similar behaviour as noted in [4].We use a transfer matrix method in order to calculate the DOS, assuming that each layer of the dielectric stack is represented by a unimodular translation matrix [7].When a plane wave is incident on a periodic medium, a Bloch wave is generated [7].The analysis results in the formulation of an eigenvalue problem where the resulting eigenvalues are the expressions for the Bloch wavenumber and the eigenvectors have the complex amplitudes of the travelling plane waves as their elements. If the frequency of the impinging wave falls in the region of the stop band, then an evanescent wave is generated that cannot propagate in the medium (analogous to the case of the imaginary ‘reduced index’ of the first propagating eigenwave in the CLC cells). The periodic layered medium is depicted in the figure that follows             Figure 4.1: Schematic representations of a periodic layered medium consisting of alternating layers of two different transparent materials with refractive indices    and    and thicknesses   and   respectively. For TE waves the density of states is calculated again with the aid of (3.4) for a transmission coefficient [7]                                    (4.1)       with                                                                                       (4.2)                                (4.3)                                                                   (4.4) and                                                                                                                              (4.5) 0 10 20 30 40 50 60024681012141618Peak Density of States SWE vs. cell thicknessSample thickness  pMaximum Density of States0 10 20 30 40 50 6011.522.533.544.55Peak Density of States SWE vs. cell thicknessSample thickness  pMaximum Density of States0 10 20 30 40 50 6011.522.533.544.555.5Peak Density of States SWE vs. cell thicknessSample thickness  pMaximum Density of States(a) (b) (c) The stop-band in these structures is the region where the Bloch wavenumber   is complex. For a quarter-wave stack, each period is an exact half-wave at the stop band centre. The   components of the wavevector read                           (4.6) where the case of oblique incidence and propagation corresponds to     . Below, we depict the DOS and the transmission coefficient for normal and oblique incidence (and propagation) of a TE plane monochromatic wave, as a function of the angular frequency (three stop-bands are shown). In the figures that follow,    denotes that centre of the photonic band-gap. We can observe a shift in the stop-bands as well as an obvious increase in the DOS for oblique propagation.  Figure 4.2: Normalised DOS and transmission coefficient of a 30-period quarter wave Bragg reflector with       and      for normal incidence      ) (left) and for oblique incidence (right). The   component of the wavevector of propagation (remaining constant throughout the medium) is set to                  . The results are in accordance with the graphs given in [6], where the quadratic dependence of the DOS on the cell thickness (for a non absorbing quarter-wave stack) is also reported. Conclusions The aim of this paper is to present important aspects of laser emission in CLC structures, emphasising the significance of the DOS in identifying the key parameters that must be addressed in designing low threshold, high efficiency lasers. One must consider variations in the DOS and transmission characteristics of such structures with a multitude of parameters and situations  (e.g. cell thickness, oblique propagation, different gain profiles of the fluorescent dye molecules amongst others) bearing in mind that CLCs are an absorbing medium for the pumping wave but an amplifying one for the emitted wave. Other DFB structures have proved to be good guides for predicting the behaviour of the DOS in CLCs despite their differences. In particular, the effect of anomalous absorption under the presence of gain is also reported in [1] as well as in the case of conventional DFB structures [7]. Furthermore, determining the DOS will be an essential tool for studying propagation characteristics of oblique modes in CLC structures.  References [1] V. A. Belyakov and S. V. Semenov, ‘Optical Edge Modes in Photonic Liquid Crystals’, JETP, Vol. 109, No. 4, 687-699 (2009) [2] Hl. de Vries, ‘Rotatory Power and Other Optical Properties of Certain Liquid Crystals’, Acta Cryst. 4, 219-226 (1951) [3] J. Schmidtke and W. Stille, ‘Fluorescence of a dye-doped cholesteric liquid crystal film in the region of the stop band: theory and experiment’, Eur. Phys J. B 31, 179-194 (2003) [4] M. F. Moreira, S. Relaix, W. Cao, B. Taheri and B. Palffy-Muhoray, ‘Mirrorless lasing and lasing thresholds in cholesteric liquid crystals’, Liquid Crystal Microlasers, Chapter 12, 223-240, Transworld Research Network,  2010  [5] S. M. Morris, A.D. Ford, G. Gillepsie, M. N. Pivnenko, O. Hadeler and H. J. Coles, ‘The Emission Characteristics of Liquid Crystal Lasers’, J. SID, 14(6),565-573 (2006) [6] J. M. Bendickson, J. P. Dowling and M. Scalora, ‘Analytic expressions for the electromagnetic mode density in finite, one-dimensional, photonic band-gap structures’, Phys. Rev. E53, 4107 (1996)  [7] A. Yariv and P. Yeh, ‘Photonics: Optical Electronics in Modern Communications’, 6th edition, Oxford University Press, 2007, ISBN 0-19-517946-3  0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.505101520253035404550 Density of States (DOS) of the Bragg Reflectorreduced frequency  = /onormalised density of states0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.500.10.20.30.40.50.60.70.80.91 Transmission Coefficient of the Bragg Reflectorreduced frequency  = /omagnitude of the transmission coefficient0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5010203040506070 Density of States (DOS) of the Bragg Reflectorreduced frequency  = /onormalised density of states0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.500.10.20.30.40.50.60.70.80.91 Transmission Coefficient of the Bragg Reflectorreduced frequency  = /omagnitude of the transmission coefficient

Laser emission characteristics of dye-doped cholesteric films

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