Control of the propagation direction of Bound States in the continuum (BICs) in liquid crystal waveguides

Master in Photonics “Photonics BCN” MASTER THESIS (Treball de final de Màster) Control of the propagation direction of Bound States in the continuum (BICs) in liquid crystal waveguides Irena Stolić Supervised by Dr.Marlin Baral and Dr.Prof.David Artigas García, ICFO Presented on 21th July 2024 Registered at Control of the propagation direction of Bound States in the continuum (BICs) in liquid crystal waveguides Irena Stolić Nonlinear optical phenomena, The Barcelona Institute of Photonics Science, Av. Carl Friedrich Gauss 3, Castelldefels (Barcelona) 08860, Spain E-mail: [email protected] Abstract. A planar waveguide comprises multiple different media layers, where light is confined and transmitted in a film or core sandwiched by cladding media. Light line frequency is an important feature of a waveguide, while waves with frequencies below the light line form a finite number of guided modes that can propagate in the film, the ones above the light line form a continuum of radiation modes. This thesis will study the bound states in the continuum (BICs) in a planar anisotropic waveguide, which are modes that do not radiate energy away from the film, despite having frequencies above the light line of the waveguide. Two experiments were conducted utilizing reflection spectroscopy to analyze a nematic liquid crystal-based anisotropic planar waveguide. Nematic liquid crystal (LC) is a state of matter that exhibits orientational order but not positional order. The aim was to demonstrate the existence of polarization separable (PS) BICs and the tunability to interferometric (INT) BICs by applying an AC voltage. Keywords: Bound state in continuum, nematic liquid crystal, anisotropic planar waveguide 1. Introduction Bound states in the continuum (BICs) are waves that remain localized despite coexisting with radiating waves in the continuous part of the spectrum in which waves typically carry energy away.1 BICs were first predicted in the field of quantum mechanics2, and later shown to be a general wave phenomenon that exists in many systems that support wave propagation of electromagnetic waves, acoustic waves in air, water waves, and elastic waves in solids.1 Since the first demonstration of photonic BICs, there has been a great interest in their theoretical predictions in various photonics systems and experimental detection.3 In this thesis, we will be focusing on studying photonic BICs in the anisotropic planar waveguides. 1.1. Anisotropic media Anisotropic media is characterized by having a directional dependency on macroscopic optical properties.3 It can be found in crystalline or liquid-crystalline form. Crystalline structures are anisotropic media when they have a well-defined crystal structure (both positional and orientational order) which are non-cubic and their properties vary with different crystallographic orientations. Liquid-crystalline structures do not necessarily exhibit positional order, as positions of the anisotropic molecules can be completely random, however, they do have orientational order, as anisotropic molecules are orientated along one direction.4 Liquid crystals (LC) exist in mesophase, a fluid state of matter that is intermediate between solid and liquid.4 There are several types of liquid crystals such as nematic, smectic, and cholesteric LC (see Fig. 1). Nematic liquid crystals have orientational order but not positional order, their molecules are orientated along the same direction and located at random positions.4 The average orientation direction of the molecules in a liquid crystalline phase is known as the director. Smectic liquid crystals have an orientational order, while their positional order is in one dimension. The orientation of all molecules is along the director and molecules are arranged in layers within which their positions are random.4 In the cholesteric liquid crystal, molecules have a long-range orientational order but no positional order, however, the molecules persist to form a helix about the perpendicular axis.4 A useful property of nematic liquid crystal is that the orientation of its molecules can be controlled by external perturbations, such as applying an electric field, magnetic field, temperature, etc. In the anisotropic media, the refractive index is a second-rank tensor.4 A coordinate system can always be chosen so that off-diagonal elements of the tensor are zero, in which case diagonal elements are called principal refractive indices 𝑛1, 𝑛2, and 𝑛3.4 Uniaxial crystal is a special case of anisotropic media which has two equal refractive indices that are different from the third one 𝑛1= 𝑛2= 𝑛𝑜≠ 𝑛3= 𝑛𝑒.4 These refractive indices are called ordinary and extraordinary and are the refractive index experienced by waves with polarization perpendicular and parallel to the optic axis, respectively. Birefringence is one of the features of the anisotropic medium wherein the propagating electromagnetic waves of different polarizations have different dielectric responses.5 It can be calculated as the difference between 𝑛𝑒 and 𝑛𝑜. Positive uniaxial crystals have an extraordinary refractive index greater than the ordinary refractive index 𝑛𝑒> 𝑛𝑜, while the opposite is true for the negative uniaxial crystals 𝑛𝑒< 𝑛𝑜.4 Figure 1. Different types of liquid crystals are presented: a) nematic; b) smectic; and c) cholesteric LC. 1.2. Modes in a waveguide An optical waveguide is a structure for confining and transmitting light. Planar dielectric optical waveguide consists of three layers, longitudinally extended along the 𝑦 axis in the shape of a slab, as shown in Figure 2. The film is the central layer, positioned between two cladding layers; the top layer (cover) and the bottom layer (substrate).5 A step-index waveguide has values of refractive index that change abruptly between layers.5 Optical waveguides are normally guided systems, but they can also be anti-guided systems. In guided systems, film is made of high refractive index media, while substrate and cover are made of low refractive index media.4 Light is confined in the film only in one direction and is guided utilizing total internal reflection, the radiation of energy into the cover and substrate is minimized.5 In the anti-guided system, the film is made of low refractive index media, and the substrate or the cover is made of higher refractive index media. The light wave is not confined in the film, it radiates from the film into the media with a higher refractive index.3 For a single homogenous medium, solutions of Maxwell´s equations exist for any value of propagation constant smaller than the product of the refractive index of the medium, 𝑛, by the vacuum wavenumber, i.e., 𝑘𝑦< 𝑛𝑘0.3 For a planar waveguide structure that consists of multiple different materials, the light wave is not allowed to have every value of propagation constant 𝑘𝑦, but just a specific set of 𝑘𝑦 values, due to the need to satisfy the boundary conditions.3 The supported modes fall into two groups, a finite number of discrete guided modes, and leaky modes, which are improper solutions of Maxwell’s equation that fall within the continuum of radiation modes.3 Leaky modes provide an excellent description of the field in the vicinity of the waveguide, while the imaginary part of the propagation constant describes the radiation losses. A transverse mode in a waveguide is a specific transverse field distribution pattern that is fixed along the longitudinal propagation axis 𝑦 and satisfies the wave equation, meaning that amplitude and polarization are maintained constant.5 Light can be guided in one or multiple modes.5 The optical modes that are supported in the waveguide are features that are influenced by the properties of the optical media of layers and the wave equation boundary conditions. The modes can normally be either transverse electric-TE (S polarized in special case) or transverse magnetic-TM (P polarized) depending on the direction of the electric and magnetic field w.r.t the direction of propagation.5 However, waveguides involving anisotropic media can also support hybrid modes, which have all six components of the electric and magnetic fields.6 In Figure 2, an asymmetric planar dielectric step-index waveguide is introduced, which is constant along the 𝑦 and 𝑧 axes, while along the 𝑥 axis layers of different materials are placed. The refractive indices of film, substrate, and cover are denoted as 𝑛𝑓, 𝑛𝑠 and 𝑛𝑐 respectively. The propagation direction is along the 𝑦 axis. To define the optic axis orientation, Polar angle 𝜃 is defined with respect the 𝑥 axis, and azimuthal angle 𝜙 gives rotation about 𝑥 axis and is defined with respect the 𝑦 axis.3 When no voltage is applied, both the optic axes of the film and substrate are in the y-z plane, then the azimuthal and polar angles of both the film and substrate remain the same, a situation known as anisotropy symmetry. The anisotropy-symmetry breaking takes place when, either the film’s azimuthal or polar angle is changed with respect to the substrate. This can be achieved for a planar waveguide with a liquid crystal film by applying an AC voltage. The polar-symmetry breaking happens when voltage is applied in the orthogonal direction (out of plane) and the polar angle 𝜃 is changed, whereas, the azimuthal symmetry-breaking situation is achieved by applying an in-plane voltage. Figure 2. Waveguide cell and the used coordinate system featuring 𝑦 axis as the propagation direction, layers of different materials along the 𝑥 axis, and constant material properties in the 𝑧 direction. Additionally, in case no voltage is applied, the optic axis is positioned along the 𝑦 axis. The applied voltage can result in optic axis orientation expressed in terms of polar and azimuthal angles.3 For an incident wave of wavelength 𝜆0, after it enters the medium of relative dielectric permittivity 𝜖𝑟 , the value of the wavenumber constant is3: 𝑘 = 2 · 𝜋 𝜆0 · √𝜖𝑟 = 𝜔0 𝑐· 𝑛 = 𝑘0𝑛 (1) The momentum of the wave 𝑘 󰇍 can be decomposed into two components along the 𝑥 and 𝑦 axes, 𝑘 󰇍 = 𝑘𝑥𝑥 + 𝑘𝑦𝑦 , where 𝑥 and 𝑦 are unit vectors. The 𝑦 component is 𝑘𝑦= 𝑘 · sin𝜃 , and the 𝑥 component is 𝑘𝑥= 𝑘 · cos𝜃, where 𝜃 is the angle. Further, normalizing components for each material 𝜅𝑥=𝑘𝑥 𝑘0 and 𝑁 = 𝑘𝑦 𝑘0 , where N is the mode effective index, it is a dimensionless parameter that has the meaning of the refractive index that the mode senses and it can be referreded to as normalized propagation constant, we get (𝜅𝑥𝑓)2+ 𝑁2= 𝑛𝑓2.3 For internal reflection to happen, angle 𝜃 has to be greater than the critical angle of each surface 𝜃𝑓𝑠 =sin−1 𝑛𝑓 𝑛𝑠 and 𝜃𝑓𝑐 =sin−1 𝑛𝑓 𝑛𝑐 .5 The light line of a material 𝜔 = 𝑐𝑘0 𝑛 is a cutoff frequency.4 Waves with frequencies below the light line cannot be transmitted to the cladding and are evanescent, while waves with frequencies above the light line can propagate in the cladding.4 For a waveguide where 𝑛𝑓> 𝑛𝑠 , 𝑛𝑐 , there are guided modes and radiation modes. Guided modes happen if 𝜃 > 𝜃𝑓𝑐 , 𝜃𝑓𝑠 in other words, if the effective index N is larger than the maximum of 𝑛𝑠 and 𝑛𝑐 , 𝑁 > 𝑚𝑎𝑥(𝑛𝑠,𝑛𝑐).3 These are modes that have frequencies below the light line in the cover and the substrate and because of that remain confined in the film. Guided modes are discrete and exist only if the propagating wave forms constructive interference with itself while propagating, which depends on the 𝑘𝑥 component. Radiation modes appear if 𝑁 < 𝑚𝑎𝑥(𝑛𝑠,𝑛𝑐) and form a continuum of frequencies above the light line. Within the continuum of radiation modes, there are resonances with high confinement in the film which are described by the so called leaky modes. These are the improper solutions of Maxwell’s wave equation where the imaginary part of the solution mathematically represents the radiation loss.3,5 They are characterized by energy being coupled to the radiation channel and being radiated away from the film.3 Radiation channels are waves that radiate energy away from the film, they can appear in the cover, substrate, or both.3 1.3. Bound states in the Continuum in anisotropic media In the anisotropic planar waveguides, BICs are modes that don´t radiate energy away and remain localized in the film, even though their frequencies are above the light line of the waveguide, in the part of the spectrum that corresponds to the continuum of radiation modes.3 For BICs, the coupling to the continuum of radiation modes in the substrate or cover via the radiation channel does not happen. One way of explaining this phenomenon is approached by the leaky mode formalism.3 Leaky modes are modes above the light line, that are partially confined inside the film, unlike other radiation modes.3 As a leaky mode propagates inside the film along the 𝑦 axis, it attenuates exponentially, because it couples to the radiation modes in the substrate, cover, or both.3 Consequently, there is exponential energy growth in the transverse direction in the immediate surrounding, because of the flux consideration, as energy conservation law applies.3 This loss of energy is described by the imaginary part of the complex propagation constant, 𝑘𝑦= 𝑁𝑘0, of the leaky mode.3 Semi-leaky modes are characterized by having only one radiation channel in the cladding, provided by either ordinary or extraordinary wave, while the other wave is evanescent.3 For a semileaky mode to exist in the waveguide with an anisotropic substrate and isotropic film, 𝑁 needs to be situated between the refractive indices of the substrate and below the refractive index of the film such that one of the tails exponentially decays to give a guiding profile and the other tail has an exponential rise corresponding to the radiation in the continuum.3 Similarly, in case both film and substrate are anisotropic, semi-leaky modes happen for 𝑁 being between any refractive indices of the film and the substrate, as shown in Figure 3.3 Generally, both guided and leaky modes can be supported by planar uniaxial waveguides when 𝑛𝑜𝑠 < 𝑛𝑒𝑓.3 However, the case of Figure 3 left corresponds to an antiguiding structure as 𝑛𝑜𝑠 > 𝑛𝑒𝑓, and only leaky modes are supported. Additionally, pure transverse Figure 3. Refractive indices for the isotropic film are shown on the left and for the anisotropic film on the right. In the case of an anisotropic substrate, ordinary and extraordinary refractive indexes are nos and nes, and in the case of an anisotropic film, they are nof and nef. The blue line represents the mode propagating in the film and the cover, the red line represents the extraordinary wave that gets confined, while the green line represents the ordinary wave which in this case serves as the radiation channel. electric (TE) and transverse magnetic (TM) modes are supported in case the light propagates along one of the principal axes of the dielectric tensor, otherwise, modes are hybrid.3 There are two types of BICs in anisotropic planar structures, polarization separable (PS) BICs and interferometric (INT) BICs.3 PS BICs are a subset of symmetry-protected BICs whose origin is related to the polarization of the radiation channel in the cladding being orthogonal to the polarization of the propagating mode.7 In the case of positive uniaxial film and negative uniaxial substrate, the polarization of the mode propagating along the 𝑦 axis is TE, and the polarization of the radiation channel in the substrate is TM, so the mode can’t couple to the radiation channel of the orthogonal polarisation.3 The PS BICs exist only for propagation directions along the principle axes, wherein the mode is pure TM or TE, otherwise, modes are hybrid for any other propagation direction.3 PS BICs exist for all wavelengths for which there is a leaky mode.3 INT BICs occur for propagation directions that are offaxis or in other words that are not along any of the principle axes.3 The origin of INT BICs is due to the destructive interference at the interface between the film and the cladding.3 INT BICs are interesting, because the modes are hybrid, and their polarization, propagation direction and existence frequency can be tuned by carefully choosing the waveguide parameters. Solving the dispersion equation gives the solutions for 𝑁 where the modes exist. The dispersion equation gives the values of N where leaky and guided mode (only leaky in anti-guiding structures) exists. However, it is not a sufficient condition to provide information about the BICs existence and location in the parameter space, particularly INT BICs.3 Therefore, an additional auxiliary condition is imposed, which states that the amplitude of the radiation channel needs to be zero.3 Planar waveguides with isotropic film and anisotropic substrate can support PS BICs, but not INT BICs, while planar waveguides with both film and substrate being anisotropic, can support both PS BICs and INT BICs.3 In the case of the planar waveguide with anisotropic film, PS BICs are only supported when the cladding and film optic axes are aligned and parallel to the waveguide interface (anisotropic symmetric structures) and for light propagating along one of the principal axes.7 INT BICs can exist for anisotropy symmetry broken structures and when light is propagating off-axes direction. In the present thesis, we demonstrate the existence of PS BICs in a liquid crystal-based waveguide using reflection spectroscopy. Additionally, we explore the tunability of the BICs by applying an AC voltage to the system in the 𝑥 direction perpendicular to the interface planes. Applying an out-of-plane AC voltage resulted in the breaking of polar symmetry and thus, PS-BICs disappears, becoming leaky modes or INT-BICs. 2. Implementation 2.1. Material properties Table 1. Materials used to form a waveguide ( *values for refractive indices taken from the website 8 ) temperature (˚C) wavelength (nm) Ordinary refractive index Extraordinary refractive index 5PCH LC 25 632 1.4870 1.6070 *PVA Room temperature 632 1.4814 – *Calcite Room temperature 632 1.6557 1.4849 *SF11 Room temperature 632 1.7787 – The optical waveguide used in the experiment has a calcite substrate, 5PCH liquid crystal (LC) for the anisotropic film and air is the cover. To couple the modes at the film-substrate interface using reflection spectroscopy, SF11 hemisphere prism is used. Therefore, the LC film waveguide has SF11 plate as the top cover which is a continuation of the refractive index of SF11 prism for prism coupling at the film-substrate interface. When the cell is used in the experiment, it is heated before use to maintain the mesophase state of the LC. The index-matching liquid of 𝑛 = 1.78 is filled between the SF11 prism and the cell. 2.2. Sample preparation SF11 and calcite are cut into plates with dimensions 10𝑚𝑚 ×10𝑚𝑚. The plates are cleaned with the acetone. Next, transparent Indium Tin Oxide (ITO) electrodes are deposited on the surfaces of SF11 and calcite plates. The thickness of the electrodes is directly proportional to the time of spattering, so the calculated thickness of the electrodes for 24 minutes of exposure is around 0.08μm. One side of each plate is exposed to UV light for 20 minutes to make their surfaces more hydrophilic. The drops of PVA are put on the hydrophilic surfaces where they spread to cover almost the whole surface. Afterwards, the spin-coating takes place. For each plate, spin coating is done for 1 minute at the speed of the spin of 3000 rpm and the result is a coated surface of the expected thickness of 50-60nm. The coated samples are then put in the oven for one hour at the temperature of 90 ˚C and then for 2 hours at the temperature of 120 ˚C, to harden the coated PVA. Afterward, the coated samples are rubbed with a velvet cloth on a rotating roller. Rubbing aims to reorient the long chains of the PVA polymer along the rubbing direction because the rubbed surface will later ensure the planar alignment of the liquid crystal that will be put in contact with it in the following steps. The rubbing takes place in the direction of the optic axis of the birefringent substrate which is already defined before the procedure. After that, the formation of a cell takes place, the plates are put onto one another very precisely, with a spacer between them. The pressure is applied on the top and bottom surfaces, and then the cell is put into the vacuum chamber for 180 minutes for the layer of air between the plates to be removed, after which two out of four sides of the cell are sealed with the NOA61 glue and the UV lamp. For the glue to further harden, the cell is put into the UV chamber for the duration of 12 hours. The 5PCH LC in the form of the liquid is put inside the glass plates utilizing capillary forces. The sample is then left at the temperature of 120 ˚C for half an hour, to spread along the space between the cover and the substrate, after which the temperature is slowly decreased to 50˚C and the cell remains at this temperature for 3 hours. Finally, the last two edges are sealed following the same procedure that was conducted for sealing the first two edges. In this thesis, we are exploring the properties of an anisotropic waveguide, with positive uniaxial film, negative uniaxial substrate, and anisotropic cover (air acts as a cover). This is an anti-guiding structure that doesn´t support any guided modes because the highest refractive index of the structure is in the substrate and not in the film, it is the ordinary refractive index of the substrate. Type of the liquid crystal used in the experiment is nematic, and the type of waveguide cell that is formed is called 𝜋-cell. 3. Measurement 3.1. Experimental setup The setup used for reflection spectroscopy utilizing prism coupling is a modified OttoKretchmann configuration.9 The input light from the He-Ne laser (𝜆 = 632.8 𝑛𝑚) is guided by an optical fiber to the fiber collimator. Collimated light passes through the polarizer where 𝑝 or 𝑠 polarisation can be selected. As the next component, a lens focuses light onto the cell. Light falls onto the hemispheric prism made out of SF11 so the prism coupling takes place and index-matching liquid is placed between the prism and the waveguide. Due to the path difference of reflected waves by the different sample layers, interferometric fringes will be formed. Reflected light passes through the precision slit, collimating lens, and falls onto the beam splitter. After the beam splitter, part of the light is transmitted and falls on the CCD camera, and part of the light is reflected and falls on the photomultiplier tube (PMT). The signal from the PMT is collected by the oscilloscope, while the signal from the camera is directly read by the custom PC application. For applying voltage, the electrodes of the cell are connected to the waveform generator. The waveguide is placed on a rotating stage which rotates 360° during the prism coupling, to vary the optic axis orientation. The setup is controlled using the in-house developed LABVIEW software, which was modified for this project to automate the rotation stage and collect output intensity through the CCD camera. Figure 4. The experimental setup used for reflection spectroscopy of the waveguide cell. 3.2. Prism coupling A high refractive index optical prism is used as a surface coupler for coupling incident light to the film in the optical waveguide. As we are interested in the interface between the film and the substrate, the prism is made of SF11 glass, the same material as the top plate of the waveguide. In this way, when index-matching liquid is used, only a thin layer of ITO and PVA separates the prism from the liquid crystal film. Thus, in this way, the prism is closer to the film which has the mode with a given frequency (𝜔), polarization, and momentum. Therefore, efficient prism coupling is ensured by phase matching in the direction of propagation a phenomenon known as frustrated TIR evanescent coupling, following the phase-matching condition 𝑁 = 𝑛𝑝∙sin𝜑 , wherein np is the refractive index of the prism.10 3.3. The experiment Two separate experiments were conducted, both utilizing the modified Otto-Kretchmann configuration shown in Figure 4, and the mentioned waveguide sample. Figure 5. (a) Top view of the sample on the rotating stage. (b) The experimental setup. The first experiment that was conducted was measuring the reflected waves using the CCD camera. The light was directed onto the waveguide sample and reflected light that fell onto the camera was collected. The sample was positioned onto the rotating stage, and measurements were conducted for different values of azimuthal angle 𝜙, achieved by rotating the sample in a sweep from 0° to 360°, with an increment of 1°. Additionally, the whole process was repeated for different values of the incident angle 𝜑, as a sweep from 34° to 36° degrees. Following this, in the image processing stage, images were stitched together both horizontally (to concatenate all the azimuthal angles taken) and vertically (to concatenate all the measured angles of incidence), to create an image where the horizontal axis corresponds to the azimuthal angle, and vertical axis corresponds to the angle of incidence or 𝑁. The final image would demonstrate the various modes supported by the waveguide along with the existence of PS BICs. As the start position of the rotating stage doesn’t correspond to the zero value of the azimuthal angle, from the final image we need to generate the useful part which corresponds to the values of the azimuthal angle from 0° to 180°. In the second experiment, the waveform generator was used to apply the AC voltage onto the electrodes and generate the voltage on and off states. The applied voltage induces a change in the polar angle position of the optic axis which causes the polar-symmetry breaking. The applied voltages were a sweep in the range of amplitude values from 1𝑉 to 10𝑉 with the 1𝑉 increment. The signal was sinusoidal with the 1𝑘𝐻𝑧 frequency. After passing the beam splitter, part of the light falls onto the PMT, which is connected to the oscilloscope. Two measurements were carried out, first highest order mode and then the second highest order mode were examined one by one, by changing the incidence angle values. On the oscilloscope, we can detect the change in the output intensity when the voltage is applied. This would help us understand the switching dynamics of the BICs in the corresponding modes. 4. Results and Discussion 4.1. Results of the first experiment Figure 6 demonstrates the results of the reflection spectroscopy when a TE (S polarized) wave is incident on the waveguide. We observe the change in reflected intensity due to the different azimuthal angles for multiple angles of incidence. The red dashed lines represent values of ordinary and extraordinary refractive indices of the substrate. As we can see, by changing the azimuthal angle, the ordinary refractive index remains constant, while the extraordinary refractive index is a convex function. The black dashed lines represent values of ordinary and extraordinary refractive indices of the film. Figure 6. In the final image, reflection images are stitched together, so that the horizontal axis represents different values of the azimuthal angle, and the vertical axis represents different angles of incidence. Since it is a multimode system, we observe various leaky modes for different refractive indices and each of the modes corresponds to the specific effective index, N which are the solutions to Maxwell’s equations. In the experimental Figure 6, when a TE wave is incident on the waveguide and we collect the reflected image. The leaky modes are shown as reflection dips as there is transmission through the sample due to the mode radiation to the substrate. However, we observed the solutions of the leaky mode exist in the region where the extraordinary refractive index of the film is greater than the extraordinary refractive index of the substrate only. The rest of the reflection dips are some Brag interferences where either transmission or reflection is favor depending on the incident angle and le local value of the extraordinary refractive indices. When the optic axis becomes perpendicular to the direction of propagation (at Φ = 90°), we observe the blue bands (reflection dips) of leaky mode start to disappear resulting in maximum reflection at Φ = 90° for the higher order mode. In this case, a PS-BIC with TE polarization exists, which is orthogonal to the ordinary (or TM) radiation channel, resulting in total reflection. This is in agreement with the literature, where BICs have been demonstrated as zeros of radiation amplitudes and experimentally they cannot be coupled from the continuum, therefore, showing a maximum reflection points where the leaky mode reflections vanish.