Materials for soft matter photonics

Seminar IIb

Author: Urban MurAdvisor: doc. dr. Miha Ravnik

Ljubljana, May 2016

Abstract

In this seminar I present three central concepts to the new developing field of soft matter photonics.First chiral nematics for use in liquid crystal lasers are presented and a brief description of such lasers isgiven. Then I move to liquid crystals in confined volumes and discuss their properties in order to describepolymer dispersed liquid crystals. Finally, use of plasmonic nanoparticle dispersions in soft matter isdiscussed and an example of tuning is presented. Together, the presented concepts are aimed to establisha novel photonic platform based on all-soft matter components as an alternative or complement to solidstate (silicon) photonics.

Materials for soft matter photonics 1 INTRODUCTION

Contents1 Introduction 1

2 Liquid crystal lasers 22.1 Photonic bandgap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Creating a laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Tunability and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

3 Polymer dispersed liquid crystals 43.1 Liquid crystals in confined volumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Light scattering in PDLCs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

4 Use of plasmonic nanoparticles 74.1 Localised surface plasmon resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 Tuning of surface plasmon resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

5 Summary 10

1 IntroductionSoft matter generically encompasses materials that are easily deformed by thermal fluctuations andexternal forces [1]. Predominant physical behaviour in such materials occurs at energy scale, comparableto kBT at room temperature. The behaviour of such materials itself also importantly depends on thephysical structure of molecules and the intermolecular forces are considered responsible for the ordering[2]. Soft matter materials include foams, gels, liquids [2], colloids [3], liquid crystals [4], polymers [1] andmany biological materials [2], like lipid membranes.

Recently, materials for creating biocompatible and even biodegradable electronic and photonic cir-cuits, that could be used in medical applications [5], have become object of interest. Many soft mattermaterials have organic origin and therefore circuits fabricated entirely from soft matter could be used.The materials and applications, described in this seminar, possess key common feature, that they canbe made entirely of soft matter materials and could be potentially used as parts of such circuits.

A distinct soft matter material for use in photonics are liquid crystals (LCs) [4], because of theirbirefringence and high susceptibility to external stimuli that can affect their internal structure. Theyinclude several mesophases between solid and liquid state and may flow like liquids, but at the sametime posses some degree of orientational and positional order, like observed in crystals. They haveelongated, rod-like molecular structure which leads to some interesting optical properties, one of thembeing birefringence.

Best known application of liquid crystals are liquid crystal displays (LCDs) [6]. They are used in awide range of electrotechnical gadgets, like mobile phones, TV-s, computer monitors, clocks and others.LCDs use the birefringent nature of nematic liquid crystals, which are placed between 2 polarizers torotate the polarization of light. Incoming illumination light can be then transmitted or absorbed at thesecond polarizer and by use of active matrix the desired voltage driven intensity pattern is formed onthe screen [7]. However, LCDs only use the most basic optical properties of liquid crystals and are notthe only case of their use in photonics. I present and explain features of liquid crystals that lead to otherinteresting optical applications such as lasers, flexible displays and devices for plasmonic manipulationof electromagnetic waves.

A laser is a device that emits light through a process of optical amplification based on the stimulatedemission of electromagnetic radiation. To achieve amplification of light, gain in gain medium must belarger than losses. Therefore, light must pass through medium several times that sufficient number ofphotons is emitted. Resonance cavities, typically constructed from mirrors, are used. Liquid crystallasers [9–11] are lasers that use liquid crystals as resonator cavity. In a certain state - chiral nematicstate - liquid crystals form a bandgap for visible light, which means that the light with certain wavelengthcannot propagate through the material and resonator is formed. LC lasers attract interest as microscopic

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Materials for soft matter photonics 2 LIQUID CRYSTAL LASERS

light sources with the ability to emit coherent light. They are capable of high power outputs at smallresonator sizes and also allow wideband tunability.

Polymer dispersed liquid crystals (PDLC) are composed of 2 types of well known soft matter materials:liquid crystals and polymers. They consist of small (0.1− 100µm) droplets of liquid crystals embeddedin a polymeric film [7]. The result is a polymeric material with liquid crystal droplets filling its holes. Ithas optical properties of liquid crystals and mechanical advantages of polymers and thus can be used insome unique optical applications like smart glass and windows, flexible displays and light modulators [8].

Plasmonic based soft matter, like plasmonic nanoparticles dispersed in liquid crystals, uses surfaceplasmon resonance [12] to manipulate electromagnetic radiation. Chemical synthesis and production ofparticles, that interact with visible light, has only become possible recently. Due to numerous differenttypes of particles and their formations, they promise a wide variety of modulations. Furthermore,nanoparticle dispersions in soft matter can be easily tuned by external factors [13] and could be used forcreating wave guides and even metamaterials [14].

The seminar consists of 3 main sections, each describing one of the forms of materials and its ap-plications: chiral nematics for creating lasers, PDLCs and plasmonic nanoparticle dispersions. In eachsection physical properties that lead to unique advantages of described material are briefly presented andpossible applications are discussed.

2 Liquid crystal lasersLiquid crystal lasers are lasers that use liquid crystals as resonance cavities for lasing. The key featurefor creating lasers, possessed by liquid crystals, is photonic bandgap (PBG) for visible light because ofits ability to control light at different frequencies. It is usually observed in cholesteric liquid crystals.

2.1 Photonic bandgap

Figure 1: a) Periodical change in direction of nematic director.The periodicity is only half a pitch. b) Photonic bandgap, asobserved in transmission spectrum. Circularly polarized lightwith the same handedness as pitch is totally reflected betweenλ⊥ and λ‖. λc marks the central bandgap wavelength [9].

Cholesteric liquid crystal (CLC) is avariant of nematic liquid crystal whichcan be formed by elongated chiralmolecules or by adding chiral molecules- chiral dopants - into material. Chi-ral molecules have no internal planesof symmetry and can be imaginedlike screw-shaped molecules. LocallyCLC looks very similar to nematic LC,however the preferred direction of ne-matic director n varies periodically in atwisted configuration as shown in Fig.1a[9]. If the director rotates about singleaxis, a helix is formed and the nematicdirector can be written as

n = (cos θ, sin θ, 0), (2.1)θ = q0z + const, (2.2)

where q0 is the wave number of nematicdirector and can be expressed with a pitch of a helix P as q0 = ±2π/P . Pitch is usually determinedby surface anchoring. The sign of q0 distinguishes between left- and right-handedness. Because of theinvariance of nematic director (n = −n) the periodicity of director is only half a pitch and is in the rangefrom 100 nm to several µm [15].

In uniaxial birefringent medium, like liquid crystals, the direction of n also determines the directionsof principle optical axes, 1 major and 2 minor, with corresponding dielectric constants ε‖ and ε⊥ andrefractive indexes no and ne. Therefore the helical modulation of both observables is present in CLCs.An analytical solution of wave propagation in helical structures can be obtained [16]. It turns out thatin case when nP ≈ λ, where n is average refractive index, the circularly polarised light with the same

2

Materials for soft matter photonics 2 LIQUID CRYSTAL LASERS

handedness as the helix is totally reflected. The reflection occurs in the frequency range of

1λ‖∝ |q0|

no<ω

c0<|q0|ne∝ 1λ⊥

, (2.3)

no = √ε‖, ne =√ε⊥, ne < no, (2.4)

for the propagation along the helix, due to fulfilled Bragg reflection condition [17]. The dispersion relationcan be obtained [10] and because of the resemblance to bandgaps in semiconductors the phenomena iscalled photonic bandgap (Fig.2). Photonic bandgap can also be observed in case of linearly polarizedincident light, because it can always be written as a superposition of two circularly polarized waves withopposite handedness. One component is than transmitted and the other one reflected. It can also be

Figure 2: Dispersion relations a) in vacuum b) in chiral nematic for normal incidence c) in chiral nematicfor oblique incidence [10].

shown, that for oblique incidence higher order reflections occur (Fig.2c) [18].The frequency of the reflected light depends entirely on the pitch of the helix and the difference

between ordinary and extraordinary refractive indexes, which plays a significant role in tuning of liquidcrystal lasers. Potential lasing mode exist at each local maximum of transmission spectrum, as shown inFig.1b, where band edge resonances occur.

2.2 Creating a laser

Figure 3: Structure of liquid crystal laser. Pumpbeam (green) is directed at an angle, so that it doesnot overlap with emitted beam (red) [11].

In order to achieve band edge laser emission inliquid crystals a gain medium must be present.Usually it is a form of rare earth or a dye, whichcan be dispersed into liquid crystal or it can bea subunit of LC molecules [9]. CLC forms theresonance cavity with the reflectivity high enough,to exceed the threshold gain.

Pump laser excites dye molecules. At thebandgap edge, where the photon group velocityapproaches zero (slope in the dispersion relation)the density of states is maximal. Stimulated emis-sion occurs on one of the two edges of the bandgap,where the density of states is the largest, follow-ing the Fermi’s golden rule. This can also be ex-plained in terms of the photon lifetime. In themiddle of the bandgap the lasing is not possible,since the propagation of photons is prohibited andeven the spontaneous emission is suppressed. Ifthe emission spectrum of the emitter overlaps withthe bandgap edge, the emitted light suffers reflec-

tion and is amplified. Lasing occurs without use of reflective mirrors, always in a direction parallel tothe helix, equally in forward and backward direction, due to symmetry of helical structure (Fig.3).

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Materials for soft matter photonics 3 POLYMER DISPERSED LIQUID CRYSTALS

2.3 Tunability and applicationsLC lasers are particularly useful laser sources because they can be easily tuned by external stimuli, likethermal heating, mechanical stress, spatial variation and electric field. The tuning range is limited bythe region where the dye gain spectrum and the bandgap of the CLC overlap.

A mechanical deformation can directly affect the pitch and cause a shift of the reflection band.Consequently a shift of the wavelength of the laser emission occurs. It has been shown, that biaxialstretching can result in a shift of wavelength up to 150 nm [19]. The same principle is used in thermallyinduced tuning, where the mechanical deformation is caused by thermal expansion. However, changes intemperature can also significantly affect the structure of liquid crystal and the value of its birefringence.

Spatial tuning of the pitch can be achieved by formation of a pitch gradient in combination withdifferent dyes. It can be created by using different processes throughout the cell filling. If a liquid crystalwith cholesteric dopant is used, the pitch can be modified by changing the concentration of dopant [15].Shifts of emitted light frequencies can be then achieved by simply moving the pump laser along the liquidcrystal cell. A shift of wavelength from 370 nm to 680 nm is reported to be observed in such system [20].

Wide variety of tuning capabilities and the fact, that no external resonance cavity is needed, makesLC lasers suitable for many applications. Maybe the most obvious is creating a display from LC lasers.With the appropriate tuning, red green and blue laser emission can be produced in a same devicesimultaneously. Because of high spectral purity, microscopic cavities and no need to use polarizing plates,such displays could produce better resolution and lower energy costs as usual LCDs. Polymerization ofLCs could lead to flexible lasers [11]. It has been shown, that certain lasing modes can also occurin spherical cells, which can lead to creation of 3D liquid crystal lasers [15] that could be used inbioengineering and medicine [21].

3 Polymer dispersed liquid crystalsIn many practical applications, for example in LCDs, liquid crystals are confined by external forces. Incombination with anchoring effects, they align LC molecules in certain directions. Materials, where thedroplets are dispersed in and confined with a polymeric matrix are known as polymer dispersed liquidcrystals (PDLCs).

3.1 Liquid crystals in confined volumesFor the purpose to understand their optical applications, PDLCs can be presented as a material composedof isotropic solid phase (polymer) and droplets of anisotropic liquid (liquid crystals). The configurationof liquid crystal inside the droplets can depend on the size and shape of the droplet, surface anchoringand applied external fields.

Optical properties of liquid crystals are highly influenced by the director field n(r), which can bedetermined by application of elastic theory and minimizing the total free energy density. Total freeenergy in confined volume can be written as

F =∫V

(fe + fmf + fef )dV +∫S

fsidS, (3.1)

where fe represents bulk elastic free energy density, fmf and fef are terms due to magnetic and electricfield and fsi describes the anchoring at boundary surfaces [7]. For sufficiently smooth variations of theorder parameter, fe can be expanded in powers of the spatial derivatives of the order parameter and ifuniform scalar order parameter is assumed, terms related to its gradient disappear. Frank, Nehring andSaupe [22,23] introduced the classical notation for fe:

fe =12K11(∇ · n)2 + 1

2K22[n · (∇× n)]2 + 12K33[n× (∇× n)]2

−∇ · [(K13n(∇ · n)]−∇ · {(K22 +K24)[n(∇ · n) + n× (∇× n)]}(3.2)

where the splay (K11), twist (K22), bend (K33), splay–bend (K13) and saddle–splay (K24) temperaturedependent Frank elastic constants are used. Last two terms represent surface terms as they can betransformed into surface integral, contribute to the stability of the director configuration and can often

4

Materials for soft matter photonics 3 POLYMER DISPERSED LIQUID CRYSTALS

be neglected even in confined volumes. The free energy densities due to external electric and magneticfield for uniaxial nematic liquid crystals can be written as

fef = −ε02 ∆ε(E · n)2 fmf = − 12µ0

∆χ(B · n)2 (3.3)

where ε0 and µ0 are electric and magnetic permittivity in a vacuum, ∆ε = ε‖− ε⊥ and ∆χ = χ‖−χ⊥ aredielectric and diamagnetic anisotropies and B and E are local, not external, fields. The surface interaction(anchoring) free energy can be described by the angle between orientation of nematic director n and thepreferred direction ns, characterized by θs, and by the orientation of preferred direction relative to thesurface. In case of homeotropic anchoring, where ns is perpendicular to the surface, surface interactionenergy can be written as

fsi = 12Wθ sin2 θs. (3.4)

Equations listed above show, that the determination of director field has no general solution. There-fore numerical calculations or certain approximation must be made. Firstly, spherical droplets, char-acterized by their radius Rd and a preferred director direction at zero external field n∗d are assumed.Average director field inside the droplet is defined as nd = 〈n〉. Also one-elastic constant approximation(K11 = K22 = K33 = K) can be assumed.

In a cylindrical system the director field can be described by the angle between the nematic directorn(ρ, φ, z) and the droplet director nd, noted as function θn(ρ, φ, z). It has been shown [24] that theminimization of free energy leads to following nonlinear partial differential equation

∇2θn −(

1ξ2 −

1ρ2

)cos θn sin θn = 0, (3.5)

where ξ is the field correlation length given by

ξ = ξe =(

K

ε0∆ε

)1/2 1E

or ξ = ξm =(Kµ0

∆χ

)1/2 1B

(3.6)

for electric (insulator) and magnetic regime respectively and presents a measure of the range of direc-tor field, induced by anchoring. Numerical solving of this differential equation and applying differentanchoring and field strength, measured by ratio F = Rd/ξ leads to different nematic director configura-tions. Zero field (F = 0) and strong anchoring give bipolar (planar anchoring) or radial (homeotropicanchoring) configuration and further removing of one-constant approximation lead to axial and thoroidalconfiguration (see Fig.4).

Figure 4: Different configurations of director field in spherical liquid crystal droplets. a) and b) occur incase of zero field and strong anchoring and c) and d) occur in case of emergent elastic anisotropy [7].

3.2 Light scattering in PDLCsThe most important optical property of PDLCs is the difference between light intensity transmittedthrough the film in ON and OFF state, referring on the voltage applied (see Fig.5). It is usuallydescribed by contrast ratio T = Imax/Imin between maximum (ON-state) and minimum (OFF state)transmitted light intensities at normal incidence. T also varies with the light wavelength λ and must beused carefully as the incidence of the light is not always perpendicular to the film.

5

Materials for soft matter photonics 3 POLYMER DISPERSED LIQUID CRYSTALS

The distribution of droplet directors ndi over all droplets and the director field n(ρ, φ, z) inside singledroplets have siqnificant role in transmittance and have to be considered in calculations. To maximize T ,the OFF-state transmittance must be reduced which can be achieved by using large optical anisotropy∆n = ne − no. Also ON-state transmittance can be increased by matching ordinary LC and polymerrefractive indexes.

Figure 5: Structure of PDLC film in OFF and ON state. In OFF-state, there is no applied voltage on thefilm and droplets are randomly oriented. LC molecules inside droplets occupy configurations, specifiedby theory of LCs in confined volumes without external field. Light is heavily scattered. In ON-statevoltage is applied to the film. Droplet directors ndi align in the direction of external field and also LCmolecules inside droplets become more ordered. Because ordinary LC and polymer refractive indexesmatch, droplets do not scatter light and high transmittance is achieved [7].

In order to solve the scattering problem certain approximations have to be done. First we assumethat the wave length of the incoming and scattered light is the same and far field approximation isused. In such case a scattering differential and total cross section can be derived and they depend onthe dielectric tensor, which is influenced by nematic director field in the observed volume. It is usuallywritten as a function of scattering matrix. Derivation of the cross sections is explained in Ref. [24].

The ON state transmittance is maximized when ordinary LC and polymer refractive indexes matchand this is usually the case in PDLCs. The situation is often referred as scattering on optically softscatterers. Anomalous diffraction approximation, known also as Rayleigh-Gans approximation can beused, if we assume that the scatterers are also optically small (kR � 1, where k is the wave vector ofthe light and R is the droplet radius) [25]. Optical transmittance for a slab of scatterers, as a model forPDLC, is usually obtained by numerical simulation and is influenced by various parameters.

In Fig.6 the dependence of transmittance on the degree of order of droplets and molecules insidedroplets is presented. Droplet order parameter Qd describes the distortions of LC molecules alignmentinside a single droplet, in respect to the uniform configuration. It is influenced by external fields andelastic theory. Qd = 1 represents uniform configuration and Qd = 0 represents random isotropic config-uration of molecules inside droplet. Values of Qd for different droplet configurations can be calculated.Similarly, order parameter Q describes ordering of droplets. Q = 1 represents the state where droplets’optical axis are aligned in the same direction (like ON-state in Fig.5) and Q = 0 represents randomorientation of droplets’ axes. Graphs show that the transmittance highly depends on these parameters.

6

Materials for soft matter photonics 4 USE OF PLASMONIC NANOPARTICLES

Figure 6: Influence of parameters on transmittance of PDLC film. a) and b) show the dependence oftransmittance T on different droplet orientations. c) and d) show the dependence of T on the ordering ofmolecules inside droplets. One can observe that the influence of Q is more important, but also influenceof Qd can not be neglected, especially in state, where droplets are perfectly aligned. A maximumin transmittance is observed in d), where the refractive indexes of droplet and medium are matched.Curves were calculated numerically for bipolar droplet configuration [25].

3.3 Applications

Figure 7: Example of smart glass. Thedifference in transmittance between OFFand ON state is shown [28].

PDLCs possess some interesting characteristics, that makethem advantageous for use in optics over pure liquid crys-tals. For example, they do not require rigid boundaries tohold them together, like glass plates, so they can be producedin large films. The amount of liquid crystals in PDLCs islower, so the costs of production is reduced, as liquid crys-tals are usually expensive materials. The variation of lighttransmission is obtained by changing scattering cross sec-tions and not by polarizers (like in LCDs). Therefore thewhole light intensity from the source can be used by deviceand also heating is minimal.

Probably the most popular use of PDLCs is smart glass.By placing PDLC film between 2 glass plates the device canbe easily switched between transparent and opalescent mode(Fig.7). Also reverse mode PDLCs have been developed byapplying liquid crystals with negative sign of dielectric anisotropy, so that the glass is transparent inoff state, and even self-adjusting smart windows have already been introduced [27]. Smart windows areoften used in architecture, automotive and aircraft industry and in army. Other common applicationsare spatial light modulators, direct view displays, projection displays and others.

4 Use of plasmonic nanoparticlesThe functioning of plasmonic nanoparticles bases on the surface plasmons and plasmon resonances.In order to modulate electromagnetic radiation, they have to be smaller than the wavelength of theelctromagnetic wave spectrum in which they operate. Nanoscales have to be obtained for modulation ofvisible light, which is often the interest.

Combining anisotropic nanoparticles with liquid crystals can create interesting configurations of par-ticles, that are influenced and can also be controlled by nematic director and thus the plasmonic responsecan be tuned [14].

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Materials for soft matter photonics 4 USE OF PLASMONIC NANOPARTICLES

4.1 Localised surface plasmon resonanceMaterials that possess a negative real and small positive imaginary dielectric constant are capable ofsupporting a surface plasmon resonance (SPR). This resonance is a coherent oscillation of the surfaceconduction electrons excited by electromagnetic radiation. If the radiation interacts with a particle,much smaller than the wave length, plasmon oscillates locally around the particle with a localised surfaceplasmon resonance frequency (LSPR) [12]. This happens because mean electron free path in materials,used for producing nanoparticles (often noble materials are used) are larger in comparison to size ofparticles. Therefore no scattering is expected in bulk and all interaction happen on surface. As thewave passes the particle, the electron density in the particle is polarized to one surface and oscillates inresonance with the frequency of incoming wave, as shown in Fig.8.

Figure 8: Schematic diagram illustrating a localised surface plasmon in a nanosphere. [12]

The oscillation frequency of electrons is determined by four factors: the density of electrons, theeffective electron mass, and the shape and size of the charge distribution [26], which directly correspondsto shape and size of nanoparticle. Analytical solution can be derived for the case of small spheres byapplication of quasi-static approximation. If a� λ, where λ is the wavelength of incident light and a isa radius of the sphere, the electric field appears static around the nanoparticle and a solution is obtainedform static Maxwell equations (time dependence is neglected) [26]. Following equation for a field outsideof the particle can be derived in case of incoming wave in direction of unit vector z:

Eout = E0z−(εin − εoutεin + 2εout

)a3E0

(zr3 −

3zr5 (xx + yy + zz)

), (4.1)

where x, y and z are standard basis unit vectors, x, y and z are coordinates in space, εin is the dielectricconstant of nanosphere and εout the dielectric constant of surroundings. The second term represents thepolarization of the conduction electron density in form of induced dipole field .

εin strongly depends on the frequency of the incoming light and if a condition εin = −2εout is met,a resonance occurs. The EM field is enhanced relative to the incoming field. As seen from previouscondition, negative values of ε have to be achieved. In metals ε exhibits resonance response, which canbe explained by Helmholz-Drude classical bead-spring model [29]. At frequencies slightly above resonancethe response of electron cloud around atoms, the polarization, is directed in the opposite direction as theexcitation, which leads to ε < 0. In case of silver and gold, a resonance occurs at visible light frequenciesand therefore these two elements have good potential for optical applications. Numerical solutions canbe obtained for other shapes of nanoparticles.

4.2 Tuning of surface plasmon resonanceLiquid crystals appear as candidates for self-assembly of nanoscale materials [30]. In recent years manyapplications of liquid crystals combined with nanoparticles have appeared, for example tuning of plas-monic nanostructures in metals and reduction in the magnetic fields required to switch the LC moleculeswhen embedded with ferromagnetic nanoparticles [31].

8

Materials for soft matter photonics 4 USE OF PLASMONIC NANOPARTICLES

Tuning of plasmonic nanostructuires can be achieved via changing εout in Eq. (4.1). Because ofthe resonance condition, given in section 4.1, the resonance frequency shifts and a broad spectrum ofincoming light can be modulated with the same material.

Due to anisotropic structure of liquid crystals, εout also depends on the angle of incident light andthe nematic director. However, even the most optimized geometries of nanoparticles only lead to shiftsof an order of 10 nm, caused by changes of external refractive index because of birefringence [13].

If a 1D array of nanoparticles is formed, very narrow geometric resonance (GR) occurs. Because ofdipole coupling interactions between particles, the tuning effects can be enhanced. Nanoparticles can bepresented as dipoles, folowing (4.1) and polarization of each particle with its center at position ri can bewritten as

Pi = αEloc(ri), (4.2)where Eloc is a sum of incident field and retarded fields of all other dipoles and α is dipole polarizabilityof single particle [32]. Eloc is given by:

Eloc(ri) = E0 expik·ri +∑j

expik·rij

r3ij

(k2rij × (rij ×Pj) + 1− ikrij

r2ij

× [r2ijPj − 3r2

ij(rij · Pj)]), (4.3)

where rij is vector between 2 dipoles and the first term represents the incident field. By putting (4.3) in(4.2), we get a set of complex linear equations. Numerical solution for Pi can be obtained [13] and anextinction spectrum for whole array can be calculated. As we see from (4.3), polarizations of particlesaffect each other, which can lead to enhancement and occurrence of GR. GR is then more sensitive tochanges in refractive index of surrounding media than LSPR for single particle.

Figure 9: a) Schematic representation of 1D array of gold nanospheres, embedded in nematic liquidcrystal. b) Measured extinction spectrum for nanospheres with radius R = 50 nm and interparticledistance d = 520 nm. The shift of resonance can be observed [13].

In the experiment with gold nanospheres, embedded in liquid crystals, shifts of wavelength up to100 nm have been observed [13]. Nanospheres with radius R = 50 nm have been dispersed in liquidcrystal at interparticle distance d = 520 nm. Layout of the experiment is shown in Fig.9a. The incidentplane wave, polarized along x-axis, was propagated along z-axis. The optical axis lied in x-y plane,rotated by angle φ with respect to the y-axis. Its direction was modulated by external electric field. Fordifferent angles between the polarization of the incident wave and the optical axis, the extraordinarywave sees angle dependent refractive index of the medium

next(φ) = none√n2e cos2 φ+ n2

o sin2 φ(4.4)

where ne and no are ordinary and extraordinary indexes. Difference between refractive indexes of liquidcrystal was ∆n = ne − no = 0.19. Resulting extinction spectrum, shown in Fig.9b, indicated the shift ofGR frequency by ≈ 100 nm, if the optical axis was rotated by 90 ◦.

This relatively simple experiment shows that even small changes in refractive index can lead totunability of localised surface plasmon resonance, when appropriate dispersion of plasmonic nanoparticlesis used. Such effects can be potentially used for sensing and switching [13].

9

Materials for soft matter photonics REFERENCES

5 SummaryIn the seminar I have shown, that soft matter posses many materials that show interesting potential foruse in photonics. By using some more complex principles, soft matter materials can be used in numerous,more advanced applications.

Liquid crystal lasers show promising future in medicine and other biological applications becauseof their close relation with biological soft matter tissues. By using polymer dispersed liquid crystals,liquid crystal dispersion is stabilized and becomes easily tunable, which leads to optical applications thatare not possible by only using twisted nematic, like smart windows or flexible displays. They also donot require any rigid boundaries to hold them together and can be fabricated entirely from soft mattermaterials. Physical processes in LC lasers and PDLCs are well known, but theirs advantages are not yetfully exploited and used in applications.

Plasmonic nanoparticles have only been developed recently. By optimizing the structure and tuningthe response, soft matter dispersion of plasmonic nanoparticles could lead to development of light guides,modulators and even metamaterials.

Finally, this seminar was aimed to provide selected research directions in modern soft matter sciencein relation to modern applications in complex optics and photonics.

References[1] Jones, R. (2002). Soft condensed matter. Oxford: Oxford University Press.

[2] Hamley, I. (2000). Introduction to soft matter. Chichester: Wiley.

[3] Dhont, J. (1996). An Introduction to Dynamics of Colloids. Burlington: Elsevier.

[4] Gennes, P. and Prost, J. (1993). The physics of liquid crystals. Oxford: Clarendon Press.

[5] Irimia-Vladu, M., Troshin, P., Reisinger, M., Shmygleva, L., Kanbur, Y., Schwabegger, G., Bodea, M.,Schwodiauer, R., Mumyatov, A., Fergus, J., Razumov, V., Sitter, H., Sariciftci, N. and Bauer, S. (2010).Biocompatible and Biodegradable Materials for Organic Field-Effect Transistors. Adv. Funct. Mater., 20(23),pp.4069-4076.

[6] Schadt, M. (1997). Liquid crystal materials and liquid crystal displays. Annual Review of Materials Science,27(1), pp.305-379.

[7] Vicari, L. (2003). Optical applications of liquid crystals. Bristol: Institute of Physics Pub.

[8] Pirs, J., Olenik, M., Marin, B., Zumer, S. and Doane, J. (1990). Color modulator based on polymer dispersedliquid-crystal shutters. J. Appl. Phys., 68(8), p.3826.

[9] Coles, H. and Morris, S. (2010). Liquid-crystal lasers. Nature Photonics, 4(10), pp.676-685.

[10] Quan Li., (2012). Liquid Crystals Beyond Displays: Chemistry, Physics, and Applications. John Wiley &Sons.

[11] Ford, A., Morris, S. and Coles, H. (2006). Photonics and lasing in liquid crystals. Materials Today, 9(7-8),pp.36-42.

[12] Willets, K. and Van Duyne, R. (2007). Localized Surface Plasmon Resonance Spectroscopy and Sensing.Annual Review of Physical Chemistry, 58(1), pp.267-297.

[13] Li, J., Ma, Y., Gu, Y., Khoo, I. and Gong, Q. (2011). Large spectral tunability of narrow geometricresonances of periodic arrays of metallic nanoparticles in a nematic liquid crystal. Appl. Phys. Lett., 98(21),p.213101.

[14] Pratibha, R., Park, K., Smalyukh, I. and Park, W. (2009). Tunable optical metamaterial based on liquidcrystal-gold nanosphere composite. Opt. Express, 17(22), p.19459.

[15] Humar, M. and Musevic, I. (2012). Liquid-crystal microdroplets as optical microresonators and lasers.Ljubljana: [M. Humar].

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Materials for soft matter photonics REFERENCES

[16] Dreher, R. and Meier, G. (1973). Optical Properties of Cholesteric Liquid Crystals. Phys. Rev. A, 8(3),pp.1616-1623.

[17] Yang, Y., Kee, C., Kim, J., Park, H., Lee, J. and Jeon, Y. (1999). Photonic defect modes of cholestericliquid crystals. Physical Review E, 60(6), pp.6852-6854.

[18] Takezoe, H., Ouchi, Y., Hara, M., Fukuda, A. and Kuze, E. (1983). Experimental Studies on ReflectionSpectra in Monodomain Cholesteric Liquid Crystal Cells: Total Reflection, Subsidiary Oscillation and ItsBeat or Swell Structure. Jpn. J. Appl. Phys., 22(Part 1, No. 7), pp.1080-1091.

[19] Finkelmann, H., Kim, S., Munoz, A., Palffy-Muhoray, P. and Taheri, B. (2001). Tunable Mirrorless Lasingin Cholesteric Liquid Crystalline Elastomers. Adv. Mater., 13(14), pp.1069-1072.

[20] Chanishvili, A., Chilaya, G., Petriashvili, G., Barberi, R., Bartolino, R., Cipparrone, G., Mazzulla, A.,Gimenez, R., Oriol, L. and Pinol, M. (2005). Widely tunable ultraviolet-visible liquid crystal laser. Appl.Phys. Lett., 86(5), p.051107.

[21] Humar, M. and Hyun Yun, S. (2015). Intracellular microlasers. Nature Photonics, 9(9), pp.572-576.

[22] Nehring, J. (1971). On the Elastic Theory of Uniaxial Liquid Crystals. The Journal of Chemical Physics,54(1), p.337.

[23] Frank, F. (1958). I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc., 25, p.19.

[24] Zumer, S. and Doane, J. (1986). Light scattering from a small nematic droplet. Phys. Rev. A, 34(4), pp.3373-3386.

[25] Kiselev, A., Yaroshchuk, O. and Dolgov, L. (2004). Ordering of droplets and light scattering in polymerdispersed liquid crystal films. Journal of Physics: Condensed Matter, 16(41), pp.7183-7197.

[26] Kelly, K., Coronado, E., Zhao, L. and Schatz, G. (2003). The Optical Properties of Metal Nanoparticles:The Influence of Size, Shape, and Dielectric Environment. ChemInform, 34(16).

[27] Cupelli, D., Pasquale Nicoletta, F., Manfredi, S., Vivacqua, M., Formoso, P., De Filpo, G. and Chidichimo,G. (2009). Self-adjusting smart windows based on polymer-dispersed liquid crystals. Solar Energy Materialsand Solar Cells, 93(11), pp.2008-2012.

[28] America.pink. (2016). Smart glass. [online] Available at: http://america.pink/smart-glass_4073205.html[Accessed 13 Apr. 2016].

[29] Cai, W., Chettiar, U., Kildishev, A. and Shalaev, V. (2007). Optical cloaking with metamaterials. NaturePhotonics, 1(4), pp.224-227.

[30] Hegmann, T., Qi, H. and Marx, V. (2007). Nanoparticles in Liquid Crystals: Synthesis, Self-Assembly, DefectFormation and Potential Applications. Journal of Inorganic and Organometallic Polymers and Materials,17(3), pp.483-508.

[31] Abdulhalim, I. (2012). Liquid crystal active nanophotonics and plasmonics: from science to devices. Journalof Nanophotonics, 6(1), p.061001.

[32] Zhao, L., Kelly, K. and Schatz, G. (2003). The Extinction Spectra of Silver Nanoparticle Arrays: Influenceof Array Structure on Plasmon Resonance Wavelength and Width †. The Journal of Physical Chemistry B,107(30), pp.7343-7350.

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