The Pennsylvania State University. The Graduate School. College of Engineering THEORY AND APPLICATIONS OF MOLECULAR AND

Transcription

1 The Pennsylvania State University The Graduate School College of Engineering THEORY AD APPLICATIOS OF MOLECULAR AD COLLECTIVE OLIEAR OPTICAL BEHAVIOR OF EMATIC LIQUID CRYSTALS A Thesis in Electrical Engineering by Andres Diaz 4 Andres Diaz Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy May 4

2 The thesis of Andres Diaz was reviewed and approved* by the following: Iam-Choon Khoo Distinguished Professor of Electrical Engineering Thesis Advisor Chair of Committee Francis T. S. Yu Evan Pugh Professor of Electrical Engineering Theresa S. Mayer Associate Professor of Electrical Engineering Vincent H. Crespi Professor of Physics Ruyan Guo Associate Professor of Electrical Engineering Special Signatory Ken Jenkins Professor of Electrical Engineering Head of the Department of Electrical Engineering *Signatures are on file in the Graduate School

3 iii ABSTRACT Two different types of nonlinear effects in nematic liquid crystals have been studied. The first refers to molecular absorption processes in liquid crystals and other similar organic liquids that make them useful for picosecond-nanosecond optical limiting applications. A detailed theoretical study of nonlinear molecular photonic processes accompanying the propagation of short intense laser pulses through a thin organic liquid cell and an organic liquid cored fiber array is presented. A model is proposed to account for the measurements of a recently developed organic liquid, and a comparison with pure two-photon and excited state absorption mechanisms is performed. The second nonlinear effect is the stimulated orientational scattering (SOS) effect of nematic liquid crystals, arising from the interaction of a laser beam with an elastic medium represented in the collective behavior of the liquid crystal molecules. We have demonstrated efficient all-optical polarization conversion of cw. µm laser in nematic liquid crystal films. This SOS effect is mediated by two-beam coupling between the incident polarized laser and its orthogonally polarized noise component scattered by the director axis fluctuations. We report a quantitative theory that accounts for severe pump depletion and high-efficiency polarization conversion and demonstrate good agreement with experimental observation. The existence of complex, time-dependent dynamics of the director motion is also revealed and studied.

4 iv TABLE OF COTETS LIST OF FIGURES...vi LIST OF TABLES...x ACKOWLEDGEMETS...xi Chapter Introduction.... General Background.... Organization of the Material... Chapter Theoretical Background...7. Molecular Photonics and onlinear Absorption Mechanisms onlinear Liquid Cored Fiber Array Organic Liquid L C 6 -doped Isotropic Liquid Crystal (ILC) Pulse propagation through the fiber array..... C 6 -doped L34 liquid system.... Collective onlinear Phenomena in ematic Liquid Crystals: Stimulated Orientational Scattering (SOS)... Chapter 3 Optical Limiting of anosecond and Picosecond Pulses Characterization of Organic Liquid L Optical Limiting in an L34-cored Fiber Array mm long fiber array, pulse width p = 7 ns mm long fiber array, pulse width p = ns mm long fiber array, pulse width p = 66 ps Limiting behavior for C 6 -doped ILC Cored Liquid Consideration of Induced Emission Terms Inclusion of Induced Emission Terms in the L34 model Consideration of Induced Emission in C 6 model Limiting Results for C 6 doped L Chapter 4 Stimulated Orientational Scattering (SOS)...6 Chapter Conclusion...7 Bibliography...73 Appendix A umerical Solution Method for Optical Limiting Simulations...7 A. L34 liquid system...7

5 A.. Pure Two-Photon Absorption Approximation...76 A.. Five-level model for L A..3 Five-level system with induced emission terms for modeling of L34 liquid...8 A. C 6 -doped ILC...8 A.3 Three-level RSA model...84 A.3. RSA model...8 A.3. RSA model with induced emission terms...87 A.4 C 6 -doped L v

6 vi LIST OF FIGURES Fig. -: (a) ematic, (b) smectic, and (c) cholesteric structures... Fig. -: Molecular structure of the constituents making up the liquid crystal E7...3 Fig. -3: Phase diagram of the mixture of two liquid crystals...3 Fig. -4: Alignment of a dichroic dye-doped nematic liquid crystal (a) before and (b) after the switching of an applied electric field...4 Fig. -: Fiber array-based device for optical limiting...8 Fig. -: onlinear limiting effects that take place in the fiber array...9 Fig. -3: Possible coupling situations between the input signal and the liquid core... Fig. -4: Transmission vs. input picosecond laser pulse energy through fibers with ILC core material... Fig. -: Transmission vs. input nanosecond laser pulse energy for different core materials.... Fig. -6: (a) Molecular structure of L34. (b) Linear-absorption spectrum of L34... Fig. -7: Pure two-photon model used to calculate an effective β eff coefficient....3 Fig -8: (a) Effective nonlinear absorption coefficient of L34 as a function of input laser energy obtained from open-aperture z-scan measurement and (b) transmittance curve for a. mm thick cell of bulk-film of L34 (wavelength λ= 3 nm, pulse duration p = 7 ns and beam waist ω o = 6µm) obtained from CREOL...4 Fig. -9: Molecular level model of L34. Main levels:,...4 Fig. -: Molecular structure of the components of an ILC mixture...6 Fig. -: Energy level diagram of C 6. Main levels: S, S and T....7 Fig. -: Experimental setup for generation of self-pumped phase conjugation.... Fig. -3: Wave vectors for the incident beams and scattering noises for (a) general frequency components and (b) for single grating for phase-matched interaction.... Fig. -4: Schematic description of SSOPC by stimulated orientational scattering...3

7 Fig. -: Dependence of the transmitted e-wave (black dots) and the stimulated o-wave power (white dots) on the input e-wave power....4 Fig. 3-: Molecular structure level diagrams of (a) low-energy model and (b) modified model for organic crystal L34... Fig. 3-: Comparison of limiting curves for different model parameters (pulse width p =7ns; linear absorption coefficient α g =.36 cm - ; ground population density =.4787x ) for (a) pure two-photon model and proposed model with fixed α exc = 4 and (b) proposed model with varying parameters α exc, and Fig. 3-3: Comparison of experimental data and theoretical model after proper adjustment of the optical parameters of the molecular level model of Figure Fig. 3-4: Medium and high-energy behavior of the proposed model for varying values of and 3. Pure two-photon absorption (TPA) results are also shown...3 Fig. 3-: Intensity profile and population densities for an input energy of µj for a beam incident on a.mm thick bulk-film cell with the parameters shown in Table Fig. 3-6: Temporal propagation of a µj pulse through a.mm cell for (a) a pure two-photon model; (b) TPA + excited state model (no transition 3 to allowed); and (c) the complete model of Figure 3-(b) with =.ns and 3 = ps...33 Fig. 3-7: Limiting curve for a 7ns pulse through a 3 mm L34-cored fiber array. (a) Low energy and experimental results; (b) higher energies...3 Fig. 3-8: Pulse propagation intensity profiles for a 7 ns pulse through a 3 mm L34-cored fiber at different input energies for TPA + ESA model with (a) =.ns and 3 =ps, (b) =.ns and 3 =ps, and (c) =.ns and no transition 3 ; and for (d) pure TPA model...37 Fig. 3-9: Energy propagation of a 7 ns pulse (input energy E in =.7mJ) through a 3mm L34-cored fiber as a function of distance for different models Fig. 3-: Limiting curve for a ns pulse through a mm L34-cored fiber array. (a) Low energy and experimental results; (b) higher energies....4 Fig. 3-: Limiting curve for a 66 ps pulse through a mm L34-cored fiber array. (a) Low energy and experimental results; (b) higher energies....4 vii

8 Fig. 3-: Comparison between limiting curves for a C 6 -doped ILC core fiber array for pulses of different duration, for a mm length Fig. 3-3: Dynamics of the normalized population densities for levels S o, S, S, T and T, for (a) low (3 µj) and (b) high (3 µj) input intensities for a 6 ps pulse going through a mm length fiber array...4 Fig. 3-4: Three energy-level model for modeling of induced emission from intermediate band S -S x to ground level S...46 Fig. 3-: Comparison of limiting curves between TPA+ESA and TPA+ESA+IE models (7 ns pulse through a 3 mm L34-cored fiber array). (a) Low energy; (b) higher energies.... Fig. 3-6: Pulse propagation intensity profiles for a 7 ns pulse through a 3mm L34-cored fiber at different input energies for TPA + ESA + IE model with (a) =.ns and 3 =ps, (b) =.ns and 3 =ps, and (c) =.ns and no transition 3... Fig. 3-7: Comparison between TPA+ESA and TPA+ESA+IE models for energy propagation of a 7 ns pulse (input energy E in =.7mJ) through a 3mm L34- cored fiber as a function of distance.... Fig. 3-8: Limiting curve showing the effect of IE terms on the C 6 molecule for low energies. Different pulse widths through a 3mm fiber are shown...4 Fig. 3-9: Limiting curves for high input energies when IE terms are included in a C 6 -doped ILC cored fiber. The length of the fiber is 3mm... Fig. 3-: Energy evolution and dynamics of a µj pulse as it propagates through a 3mm length C 6 -doped ILC cored fiber....6 Fig. 3-: Limiting curves for a 7ns pulse and varying concentrations of C 6 in a C 6 -doped L34 thin film (.mm thickness)....7 Fig. 3-: Limiting Curves for L34+C 6 liquid-core fiber array configuration for different input energy intervals (3mm length and 7ns pulse width)...9 Fig. 3-3: Optical limiting results for nanosecond laser pulses using 3 mm L34- cored fiber array with opaque cladding at λ=3 nm. F detection optics. Two-photon absorption coefficient β=4 cm/gw used in the models...6 Fig. 3-4: Propagation of a 3 µj pulse through a 3mm fiber for a C 6 -doped L34 core. Different concentrations of C 6 are shown....6 viii

9 Fig. 4-: (a) Depiction of a linearly polarized laser incident on a planar-aligned nematic liquid crystal. (b) Optical wave vectors k x and k y and grating wave vector condition for maximum conversion Fig. 4-: Experimental setup for SOS experiment Fig. 4-3: Calculated output e- and o-waves as a function of the input field. Input laser is an e-wave. Also shown is the experimental measured transmitted waves as a function of input laser power for a focused diameter of 4 µm Fig. 4-4: (a)-(g) Output field dynamics, Fourier Transform, and phase diagram of angle coefficients for different values of input e-wave field. Coefficients are given in mrads....7 ix

10 x LIST OF TABLES Table -: Parameters used in numerical simulation of laser beam propagating through L34 filled fiber...6 Table -: Photophysical parameters of C 6 model used in numerical simulation...8 Table 3-: Simulation parameters and absorption coefficients used in the proposed model....3 Table 3-: Linear transmission for.mm thin cell and varying concentrations of C Table 3-3: Linear transmission for 3mm L34+C 6 liquid-core fiber array....8

11 xi ACKOWLEDGEMETS I wish to express my gratitude to my thesis advisor, Professor I. C. Khoo, for his patient guidance and encouragement throughout the course of this research. I would also like to thank my thesis committee members for their advice and suggestions. I would like to extend my appreciation to my present and previous colleagues Mike Wood, Min-Yi Shih, Pao Chen, Jianwu Ding, Yana Williams and Kan Chen. I would also like to express my thanks to my family and friends. So I walk on uplands unbounded And know that there is hope For that which Thou didst mold out of dust To have consort with things eternal The Dead Sea Scrolls

12 Chapter Introduction. General Background Liquid crystals are organic compounds in a state (or mesomorphic phase) intermediate between that of an isotropic liquid and an anisotropic crystalline solid, and were discovered in 888 by F. Reinitzer and O. Lehman. During the last thirty years, new interest in these materials has brought an endless amount of applications in commercial products, in industry, in research, and even in the arts []. In particular, the unique optical properties of liquid crystals make them very suitable for use in optical signal processing systems and in highly nonlinear optical devices. The molecules (or group of molecules) in liquid crystals are elongated and rodlike, and are usually arranged in layers that can slide against each other, giving these substances liquid-like properties such as viscosity and surface tension. According to the degree of molecular order which they present, three main classes of liquid crystals have been distinguished: nematic, smectic, and cholesteric. In the nematic class, the molecules are arranged with the direction of their long axis (or director) parallel to each other, and although there is no apparent ordering of the centers of gravity of each molecule, there is orientational ordering that accounts for the crystal-like properties. In the smectic class, molecules are further organized into layers, forming a stack of two-dimensional, liquidlike, one-molecule-thick layers. In the third class, cholesterics, molecules are also

13 organized in layers. Within each layer, there is no positional ordering, but the directors are all parallel to each other and to the plane of the layers. The molecules are essentially flat, with a side chain of methyl groups (CH 3 ) projecting upward from the plane of each molecule. This configuration causes the director of each molecule to be displaced slightly from the corresponding direction in adjacent layers forming an overall helical path (the displacement between adjacent layers averages minutes of arc). The structure of the three classes of liquid crystals is given in Figure -. (a) (b) (c) Fig. -: (a) ematic, (b) smectic, and (c) cholesteric structures Most applications of liquid crystals, especially in industry, involve the use of mixtures of two or more pure liquid crystals. By choosing the components of the mixture, the different optical properties, dielectric anisotropies, and viscosities may be tailored to

14 those required. An example of a mixture made up of four liquid crystals is E7 from EM Chemicals, whose composition is shown in Figure -. 3 Fig. -: Molecular structure of the constituents making up the liquid crystal E7 A mixture of liquid crystals also helps to extend the temperature ranges for the various mesophases, as compared to the pure liquid crystals. This behavior is shown in Figure -3 for a mixture of two types of liquid crystals with different melting (crystal nematic transition) and clearing (nematic isotropic transition) points. Fig. -3: Phase diagram of the mixture of two liquid crystals.

15 4 Another procedure is to dope liquid crystals with dissolved concentration of dyes, altering the linear and nonlinear optical properties of the liquid crystal. The dye will increase the absorption of the liquid crystal for some specified wavelength, and may also affect the orientation of the host liquid crystal if the dye molecules undergo some physical or orientational change due to photon absorption. Another way the dye molecules interact with the liquid crystal is through the guest-host effect. Absorption of a dichroic dye molecule is greater when the optical field is polarized parallel to its long axis than when polarization is perpendicular to it. When used in a cell with a nematic liquid crystal, the dye molecules, due to their elongated shape, may be reoriented by the host liquid crystal. Therefore transmission of the cell may be switched with the application of an external field (Figure -4). (a) Fig. -4: Alignment of a dichroic dye-doped nematic liquid crystal (a) before and (b) after the switching of an applied electric field. (b)

16 . Organization of the Material Two different types of nonlinear effects in nematic liquid crystals will be considered. The first refers to nonlinear absorption at a molecular level, arising from the specific molecular photonics of the liquid crystal or of dyes with which the liquid crystal cell has been doped. This type of nonlinear phenomena is used in optical limiting for protection against laser pulses. The second nonlinear mechanism that has been studied and modeled is collective nonlinear behavior arising from the short range interaction between liquid crystal molecules responsible for the elasticity of the medium. When interacting with a polarized laser beam, a dipole moment will be induced on the LC molecules by the oscillating field that in turn, after interacting with the field, will result in a torque acting on the molecule. A complete model of molecular reorientation has to take into account the elasticity of the medium (responsible for the collective behavior) and its interaction with the electromagnetic field. Optical effects due to molecular reorientation induced by a laser beam, and the effects of this reorientation on the incident and scattered beams, can be observed in highly nonlinear liquid crystal materials for low input intensities (tens of mw) [][3]. Liquid crystals are also particularly well-suited for the study of this effect from the visible up to the near-infrared regimes due to their large dielectric anisotropy ( ε = ε ε ~ ) and almost zero absorption at these wavelengths. a e o The necessary theoretical background for the qualitative study of these two phenomena is given in Chapter Chapter 3 provides a detailed account of nonlinear molecular photonic processes accompanying the propagation of short laser pulses through organic nonlinear liquids. In

17 6 particular, the absorption mechanisms of organic liquid L34, synthesized at Penn State, are studied. A molecular photonics model based on a modified two-photon absorption mechanism is derived, and the time evolution of the population densities and the beam intensity as a pulse travels through the liquid are obtained. Studies are also made of materials where excited state absorption processes dominate (C 6 -doped isotropic liquid crystals). The dynamical evolution of molecular energy levels and laser intensity is studied by the numerical solution of the coupled nonlinear differential equations for a variety of situations using observed molecular and optical parameters. The results are applied to the case of a beam propagating in a nonlinear liquid-cored fiber array used in a proposed optical limiting device. Chapter 4 studies all-optical polarization conversion in nematic liquid crystal films. This stimulated orientational scattering effect is mediated by two-beam coupling between the incident polarized laser and its orthogonally polarized noise component scattered by the director axis fluctuations. The existence of complex, time-dependent dynamics of the director motion is revealed, and a complete model for the director dynamics derived from the basic equations is presented. Finally, areas for future work and research and conclusions are presented in Chapter.

18 Chapter Theoretical Background. Molecular Photonics and onlinear Absorption Mechanisms Several optical materials and devices have been studied to protect the eye or optical sensors from over-exposure to lasers [4]. The requirements imposed to these devices are rather challenging, due to the wide temporal and spectral characteristics of lasers. For example, in order to protect the eye, the laser radiation that reaches the retina should be. J/cm, which corresponds to a laser energy of µj (considering the eye as an optical system with focused-intensity gain of ). The device should also response and act within the duration of the pulse, which means employing all-optical techniques for pico-second and nano-second limiting applications. Self-limiting effects using nematic liquid crystals films have been studied [], but have the disadvantages of being polarization selective, having a large scattering loss, and a short interaction length. On the other hand, isotropic liquid crystals (ILC) are not polarization selective, have very low optical losses due to orientational fluctuations, and hence a longer interaction length [6]... onlinear Liquid Cored Fiber Array One of the devices that has been proposed for the optical limiting application makes use of a fiber array filled with nonlinear ILC in the core [7]. The use of a fiber

19 8 array overcomes the limitation of focused optics solutions in which the nonlinear material is placed in the focal plane of the optical system to utilize the increase intensity. In this configuration, the interaction length is very limited, putting a boundary on the total limiting effect. With the fiber array, the incident wave is guided and the interaction length may be increased to several millimeters. A schematic of the device is shown in Figure -. Input scenery and laser Input lens onlinear optical fiber array scenery laser Collection lens sensor nonlinear absorption Absorbed by opaque glass cladding Fig. -: Fiber array-based device for optical limiting. The external scenery (and incident laser beam) is focused by an input lens on the entrance glass window of the fiber array. The image then propagates through the fiber array, where the nonlinear absorption takes place. A collection lens is placed at the output of the fiber. For a normal low intensity and safe input, a certain amount of absorption proportional to the input intensity takes place. evertheless, enough light goes through in order for the eye to see (or for the sensor to pick up an input). If instead a high intensity laser is aimed at the device, the input is clamped to a safe level by the very high absorption processes that take place in the fiber array.

20 The nonlinear effects responsible for the limiting behavior of the device are summarized in Figure -. 9 entrance glass window onlinear liquid film Laser from input optics and systems bubble formation laser focal plane opaque glass cladding absorbs lossy waveguide modes onlinear liquid core nonlinear guided-mode extinction (lossy mode) nonlinear absorption nonlinear scattering (density, thermal, orientational) opaque glass cladding absorbs lossy waveguide modes nonlinear defocusing Fig. -: onlinear limiting effects that take place in the fiber array. The nonlinear absorption due to a two-photon transition from the ground level of the core molecule to an excited electronic state is primarily responsible for the limiting effect. Other mechanisms that also contribute to the limiting are nonlinear guided mode extinction (fiber in lossy mode ), nonlinear scattering due to density, thermal and orientational effects, nonlinear defocusing at the entrance of the fiber array, and absorption of lossy modes due to an opaque glass cladding that surrounds the cores. Furthermore, a poor input coupling (see Figure -3) may decrease the output intensity even further.

21 Front of fiber array opaque glass cladding liquid guiding core good input coupling poor input coupling Fig. -3: Possible coupling situations between the input signal and the liquid core. The molecular photonic transitions responsible for most of the limiting may be understood with the aid of a molecular level model, and may include effects such as excited state absorption (ESA), reverse saturable absorption (RSA), or multiphoton absorption. Different proposed energy diagrams are given below for each of the nonlinear liquids studied (L34 and C 6 -doped ILC). Experimental results obtained using this device are given in Figure -4 for a picosecond laser for ILC filled fiber arrays of several lengths. These are compared with analytical and numerical results.

22 Fig. -4: Transmission vs. input picosecond laser pulse energy through fibers with ILC core material. The nonlinear effects inherent of ILC are enough to guarantee a low enough output. For nanosecond pulses, ILC does not provide the necessary limiting, and more highly nonlinear materials are needed. Some of the materials used are L34 (an organic core liquid) and fullerene-doped ILC (ILC+C 6 ). The output vs. input energy curves for mm length fiber arrays using several core liquids are shown below ( Figure -): Output energy (µj) 8 ILC 6 4 ILC + LC-x L34 ILC + C Input energy ( µj ) Fig. -: Transmission vs. input nanosecond laser pulse energy for different core materials.

23 .. Organic Liquid L34 L34 is a single constituent liquid synthesized at Penn State with molecular structure shown in Figure -6 (a). It has a refractive index n=.6, and it is slightly yellow. The absorption constant at 3 nm is cm -. The linear absorption spectrum of L34 is given in Figure -6 (b). Fig. -6: (a) Molecular structure of L34. (b) Linear-absorption spectrum of L34. Characterization of the molecular and optical parameters of L34 has been done through picosecond and nanosecond z-scans and time delayed picosecond pump-probe studies conducted at UCF CREOL (Center for Research and Education in Optics and Lasers at the University of Central Florida) using the second harmonic (λ = 3 nm) of a Q-switched and mode-locked d:yag laser [,]. The pulse width of the laser is ps with a beam waist ω o of µm at the focus. A -µm-thick bulk film of L34 is used. Curve fitting gives the two-photon absorption coefficient β=4. cm/gw for laser energies

24 of. and. µj, and appears to be quite constant and around this value up to a few µj, indicating this value is very close to the intrinsic two-photon absorption coefficient. In the nanosecond regime, z-scan measurements of the same sample show a much larger nonlinear absorption coefficient. A Q-switched nanosecond d:yag laser at 3 nm, with pulse duration of 7 ns and beam waist ω o =6 µm is used. For laser energies up to µj, an effective two-photon absorption coefficient β eff is obtained by fitting the z- scan measurements using a purely two-photon absorption model, shown in Figure > Excited Electronic states onlinear effective two photon transition β α g Single photon transition 4> > High lying ro-vibrational manifold of ground electronic states Ro-vibrational manifold of ground electronic states Fig. -7: Pure two-photon model used to calculate an effective β eff coefficient. The effective two-photon absorption coefficient has values ranging from cm/gw at low laser energy (. µj) to nearly cm/gw at higher energies ( µj), indicating the presence of higher order nonlinear processes [,3]. Figure -8 shows the effective two-photon absorption coefficient reported in [] for nanosecond pulses, and the corresponding transmittance curve.

25 4 Effective two-photon absorption. Limiting curve (l=. mm).8 β eff (x -8 cm/w) Transmittance.6.4 ο=7 ns (FWHM)... Input Energy (µj) Input Energy (µj) (a) (b) Fig -8: (a) Effective nonlinear absorption coefficient of L34 as a function of input laser energy obtained from open-aperture z-scan measurement and (b) transmittance curve for a. mm thick cell of bulk-film of L34 (wavelength λ= 3 nm, pulse duration p = 7 ns and beam waist ω o = 6µm) obtained from CREOL. The molecular photonic level model for L34 proposed in [8] accounts for linear, two-photon, intermediate and excited state absorptions in this material, and is shown in Figure -9. Excited state absorption α exc 3 High lying electronic excited state Excited Electronic states onlinear two photon transition β 4 High lying ro-vibrational manifold of ground electronic states α g Single photon transition Ro-vibrational manifold of ground electronic states Fig. -9: Molecular level model of L34. Main levels:,

26 The rate equations derived from this model describing the temporal evolution of the molecular level population densities are given by [7]: t t β I = hν βi = hν A A α g I + hν A α exci + hν A (.) where A is the total molecular number density, and the s and α s denote the respective level population densities and linear absorption coefficients. The nonlinear (two-photon) absorption coefficient is given by β. To adequately describe the physics of the absorption process for nanosecond or shorter laser pulses, it is sufficient to consider only the dominant two-photon absorption (TPA) process originating from the ground state, and the transition from the excited state to high lying states connected by large single photon transition moments (the excited-state absorption ESA effect). Longer time scales will have to include other effects such as thermal, density, trans-cis isomerization, and order parameter changes. The parameters used in the numerical modeling of the dynamics of the molecular levels of L34 are given in Table - [8]

27 Table -: Parameters used in numerical simulation of laser beam propagating through L34 filled fiber. Wavelength λ 3 nm Absorption coefficients: α g.36 cm - α exc 4 - cm - β 4 and.9 cm/gw Relaxation time ps through ns. x cm -3 Core Diameter - µm Fiber Length thin cell and 3 - mm 6..3 C 6 -doped Isotropic Liquid Crystal (ILC) ILC is a mixture of two nematic and two isotropic liquid crystals, with molecular structure shown in Figure -. The mixture is colorless, with a refractive index n=.4 at 3 nm, a linear loss constant of ~. cm - and a scattering loss of ~ cm -. RO COO M4 R = C 6 H 3 R = C 8 H 7 R COO M4 R = C 3 H 7 R = C H M4 = CH 3 CH CHCH CH 3 Fig. -: Molecular structure of the components of an ILC mixture The optical limiting behavior exhibited by fullerenes (C 6 and C 7 ) has been attributed to reverse saturable absorption (RSA), which occurs when the absorption cross section of the excited state is greater than that of the ground state. Most studies on C 6

28 7 have been performed at 3 nm, although some extensions to the UV region have been done [4]. The theoretical model used for the study of RSA in fullerene solutions is a five level model involving the ground state S, the first excited state S, higher excited singlet states represented by S, and lowest T and higher T triplet states, as shown in Figure -. S σ n T S X x σ T Tn S k ST n T T σ o T S Fig. -: Energy level diagram of C 6. Main levels: S, S and T. Molecules are excited from the ground state S by laser radiation to a vibrational substate S x of the first electronic excited singlet state S (or of higher-lying singlet states) with an absorption cross section σ o. From this state, the molecules decay rapidly ( x ~picoseconds) to the singlet state S. This level relaxes either to the ground state (with rate / o ) or to the triplet state T with intersystem crossing rate k ST. Absorption of laser radiation may further excite the molecules in levels S or T to higher energy states S and T, with absorption cross-sections σ and σ T respectively. These excited states decay

29 rapidly to states S and T. A summary of the values of the different model parameters and the experimental conditions where these were obtained is given in Table -. 8 Table -: Photophysical parameters of C 6 model used in numerical simulation Parameter 38 nm (Excimer) 337 nm (Excimer) 34 nm (d:kgw) cell thickness mm n r : refractive index.46 for Toluene at o C t p : fwhm pulse duration ns 7 ns 6 ns Beam Intensity (from ~ mj ~.7 mj ~. mj laser) Beam Area 7.x - cm.3x -3 cm 9x -4 cm Incident fluence kept < J/cm T: small signal C transmittance through mm cell C C: concentration C 6.8 x -3 M.36 x -3 M.67 x -3 M C 7. x -3 M.4 x -3 M.9 x -3 M k ST ~ 9 s - C 6 33 ns T (saturated air) C 7 73 ns C 6 ~4 us us T (lower than -4 M) C 7 ~9 us 3 us C 6 > ns T (for exp conditions in C 7 > 9 ns ref ) : lifetime of S C 6.3±. ns - Indep of wavelength & concentration C 7.7±. ns C 6. ns C 7.7 ns C 6.4 ns C 7.66 ns C 6. ns C 7.6 ns ~ ps ~ ps ~ ps X ~ ps ~ ps ~ ps n ~ ps ~ ps ~ ps Tn derived k ST = φ C 6 C 7.3 ns.78 ns This table is based in the results given by J. Barroso, et. al. in [4] (and references therein) where a good summary of own and previous results for C6 and C7 at 38nm, 337nm, and 34nm is given. They also observe that no RSA effect is present in the 337nm wavelength.

30 9 C 6 3. ns derived = φ C 7 7 ns φ : quantum yield C 6.96 Indep of wavelength C 7.9 σ C 6.6x -7 cm.x -6 cm 3.x -8 cm C 7 8.x -7 cm 9.4x -7 cm.x -7 cm σ T C 6 9.x -7 cm 9.x -7 cm.4x -7 cm C 7 9.8x -7 cm 9.4x -7 cm 4.x -7 cm C 6.44x - cm.x -7 cm.6x -7 cm C 7.9x - cm 9.4x -8 cm 6.3x -7 cm Ground state quenching C 6 x 8 M - s - C 7 6x 8 M - s - Triplet-Triplet C 6.8x 9 M - s - annihilation C 7 x 9 M - s - σ Quenching rate constant C 6 x 9 M - s - by oxygen C 7 9x 8 M - s - The -level system is described by the following set of rate equations: S t S t S t T t T t σ I = S hv σ I = S hv S = = S k n ST σ I + S hv T S + T σ T I T = T hv Tn T + T S Sk ST S + n σ T I T T + hv Tn σ I hv S (.) A simplification of the system of equations is possible for not so high incident beam intensities, for which higher singlet and triplet state levels S and T may be assumed to have a negligible population.

31 ..4 Pulse propagation through the fiber array When the laser pulse propagates through the core, the fibers are going to operate in the multimode regime due to the large core-cladding index difference and the core radius. The intensity distribution and molecular level densities within the core are therefore going to be radially uniform and a function of the propagation distance z only. The rate of absorption can then be described by Eq..3: for L34-filled core, and ( z, t) ( z, t) ( ) ( z, t) di dz for the fullerene-doped ILC core. The shape of the incident laser pulse is assumed to be Gaussian, with fwhm pulse duration t p. ( z, t) I( z t) = α g I z, t β I ( z, t) α exc, (.3) A di dz A ( σ S + σ S + σ T St )I (.4) = A.. C 6 -doped L34 liquid system Previous experimental and theoretical studies of C 6 -doped ILC and L34 liquids systems used in fiber arrays have showed the limiting advantages of RSA and TPA processes. A natural extension of this work is to consider a C 6 -doped L34 liquid system, where we expect the nonlinear response to be enhanced due to the interplay of both limiting mechanisms. The two systems of rate equations previously presented may be used, but are now coupled by the propagation equation:

32 di dz α g I + β I + αexc I ( σ S + σ S + σ T St )I A A A = (.) It is clear that only a complete numerical solution may be used to describe the dynamics of such a system, for semi-analytic approaches previously used with C 6 [7] and L34 [8][7] may no longer be applied.. Collective onlinear Phenomena in ematic Liquid Crystals: Stimulated Orientational Scattering (SOS) A second application of the nonlinear properties of liquid crystals is self-starting optical phase conjugation (SSOPC). This process occurs when a single incident laser beam generates its phase-conjugate replica through an optical wave mixing effect in a nonlinear optical material. The conjugated signal originates as coherent scattered noise from the pump laser beam (as a result of scatterers in the crystal, spontaneous Brillouin scattering, etc.) which then interacts with the pump beam and grows into a strong coherent signal. SSOPC has been reported in nematic liquid crystals by several groups [9][]. As shown in Figure -, an incident laser traverses a nematic liquid crystal film twice, as an incident field E (with direction k ) and then as a reflected signal E (direction k ). The incident field generates a noise source field (direction k 3 ) coherent with E. This means E and E 3 may interfere with each other and generate an index grating. Another grating may be produced by E and its coherent noise E 4.

33 Fig. -: Experimental setup for generation of self-pumped phase conjugation. As shown in Figure -3, the scattered noise E 3 and E 4 generally contain several frequency components, and the generated gratings do not coincide. Only for the phase matching condition q = k k3 = k k4 will they share a common grating and thus contribute to each other s growth (Figure -3 (b)). (a) (b) Fig. -3: Wave vectors for the incident beams and scattering noises for (a) general frequency components and (b) for single grating for phase-matched interaction. Two mechanisms to generate SSOPC are stimulated thermal scattering and stimulated orientational scattering [6]. In stimulated thermal scattering, the index change is brought about by a temperature-dependent extraordinary refractive index n e ( T ), while in stimulated orientational scattering, the scattered noise from the incident beams originates from orientational fluctuations of the director axis. Because of the birefringence of nematics, the fluctuations give rise to an extraordinary wave component,

34 as shown in Figure -4 below. The incident optical field propagates as an o-wave in a planar nematic liquid crystal cell. δ n is the orientational fluctuation of the director axis, and n is the perturbed director axis. 3 Fig. -4: Schematic description of SSOPC by stimulated orientational scattering. The generation of an extraordinary wave component means that for stimulated orientational scattering, the reflected beam E has to be orthogonally polarized to the input beam E in order to fulfill the wave-vector matching condition and take advantage of the feedback of the external mirrors. In contrast, in stimulated thermal scattering the incident field E and the generated noise E 3 are co-polarized. Both generation mechanisms for SSOPC have been studied in the case of negligible pump depletion, and the effect has been demonstrated in thin nematic liquid crystals with mw-power cw lasers. Figure - shows the experimental results of the stimulated orientational scattering effect for a µm E7 nematic liquid crystal planar cell at. nm [].

35 4 Fig. -: Dependence of the transmitted e-wave (black dots) and the stimulated o-wave power (white dots) on the input e-wave power. Dynamical behavior of light-induced director reorientation has recently been studied experimentally [,6,7] and theoretically [8,9] for the case of a linearly polarized plane wave incident on a thin cell of homeotropically aligned nematic LCs. As the intensity of the light is increased, the director starts oscillating. A rich variety of complex, time dependent director motions have been revealed showing periodic, quasiperiodic, and chaotic behavior; and several routes to chaos have been reported []. In Chapter 4 I report the results obtained so far for similar studies of the temporal dynamics in the stimulated orientational scattering effect, as well as advantages over the homeotropic cell configuration previously mentioned.

36 Chapter 3 Optical Limiting of anosecond and Picosecond Pulses 3. Characterization of Organic Liquid L34 Theoretical studies of the limiting performance of an L34-core fiber array device were initially done employing the molecular level structure shown in Figure 3- (a). The model is found to work and agree with experimental results in the picosecond regime, and for low-input laser energies for nanosecond pulses, as reported in []. For energies above a couple of µj, the simulation shows a continually rising curve, whereas the experimental results clamps to a level near µj. Adjustment of the model to the experimental data requires the introduction of other nonlinear processes, or the use of an effective two-photon nonlinear absorption coefficient. (a) Low-energy model (b) General model onlinear two photon transition β Excited state absorption α exc α g Single photon transition 4 3 High lying electronic excited state Excited Electronic states High lying ro-vibrational manifold of ground electronic states Ro-vibrational manifold of ground electronic states Excited electronic states onlinear two photon transition β α g Single photon transition High lying electronic excited state Excited state absorption α exc 4 3 High lying ro-vibrational manifold of ground electronic states Ro-vibrational manifold of ground electronic states Fig. 3-: Molecular structure level diagrams of (a) low-energy model and (b) modified model for organic crystal L34. 3

37 6 In order to model the higher energy behavior of the experimental limiting curve while keeping the intrinsic two-photon absorption coefficient value of β =4 cm/gw, the model shown in Figure 3- (b) with a modified excited state was introduced. Levels 3 and may correspond to triplet states, also found in other organic molecules. Once the intensity is high enough (a few µj), energy level becomes populated. The molecules in this state can then decay to level (decay constant ns). Absorption from to 3 further enhances the limiting process, and a very small half-life for level 3 ( 3 ps) guarantees the availability of molecules in to keep the limiting process going on. Eventually, for extremely high energies (above mj), level may deplete, though before this happens other effects such as absorption-induced thermal and density effects (often followed by the formation of a bubble at the entrance region of the fiber array) will have kicked in, as discussed in []. The rate equations and intensity propagation equation describing the dynamics of the proposed model are h I t h I t h I t h I t h I h I t exc g exc g ν α ν α ν α ν β ν α ν β + = = = = + = (3.)

38 where the main levels are,, 3, and. 4 is an auxiliary level that is not actually needed in the calculation, but is included in order to have a close set of equations that can be used to verify convergence of the solution by checking invariance of the molecular population. di = α g I β I α exc I (3.) dz A An analysis of how the different model parameters affect the behavior of the limiting curves for a thin (.mm) cell is shown in Figure 3-. All the models exhibit the A A 7 same linear transmission at low energies (less than µj) for which absorption to level 4 is the predominant process. At higher energies, the pure two-photon model of Figure -7 shows a very week limiting action, even when the effective two-photon absorption coefficient of Figure -8(a) is used. For a fixed excited absorption coefficient α exc, an increment in decay time 3 increases the threshold of the limiting curve. A greater value of 3 implies it takes more time for the molecules in energy level 3 to decay to and be available again to aid in the absorption of the incoming radiation. Increasing has a similar effect, for once the molecule has been excited to level, it would be less probable to decay to to start the excited state absorption responsible of the limiting threshold. Figure 3-(b) also shows the effect of a change in the excited state absorption coefficient α exc and how it can bring down the limiting threshold when it is increased. A loss of 4% at the entrance and exit of the cell has been considered in all the limiting curves to account for reflection at interfaces.

39 (a) Output Energy (µj) Pure Two-Photon Model β=4. cm/gw Pure Two-Photon Model β=.9 cm/gw Complete Model: =.ns; no transition 3 Complete Model: =.ns 3 =ps Complete Model: =.ns 3 =ps 8 (b) Output Energy (µj) 3 4 Input Energy (µj) α exc = 4 cm - =.ns 3 =ps α exc = 4 cm - =.ns 3 =ps α exc = 4 cm - =.ns 3 =ps α exc = cm - =.ns 3 =ps 3 4 Input Energy (µj) Fig. 3-: Comparison of limiting curves for different model parameters (pulse width p =7ns; linear absorption coefficient α g =.36 cm - ; ground population density =.4787x ) for (a) pure two-photon model and proposed model with fixed α exc = 4 and (b) proposed model with varying parameters α exc, and 3. Taking the previous effects into consideration, it is possible to fit the model to the experimental data from CREOL. is considered to be no less than.ns in order to have agreement between the proposed model and the results of z-scan measurements in the picosecond regime. Agreement with the experimental data also suggests a value in the order of ps or less for 3. The results are given in Figure 3-3 with an excited absorption

40 coefficient α exc =.7x 4 cm - and decay times =.ns and 3 =.ps. One order of magnitude changes in the decay times are also plotted for comparison. Similarly, the values used in the model are listed in Table 3- below. 9 Output Energy (µj) =.ns 3 =ps =.ns 3 =ps =.ns 3 =ps Experimental Results Input Energy (µj) Fig. 3-3: Comparison of experimental data and theoretical model after proper adjustment of the optical parameters of the molecular level model of Figure 3-.

41 Table 3-: Simulation parameters and absorption coefficients used in the proposed model. Sample Length (cm).e Pulse FWHM p (sec) 7 ns Min Input Intensity (GW/cm ).E Peak Input Intensity (GW/cm ). Wavelength λ (m) 3.E-9 Beam Area (cm ) corresponding to beam waist ω o =6µm..397E-6 Absorption coefficients L34 α g (cm - ) pure.36e α exc (cm - ) pure.7e4 β (cm/gw) pure 4. Relaxation Times: (ns). 3 (ps) (cm -3 ).4787E 3 The modeled medium and high-energy behavior for a.mm L34 cell are depicted in Figure 3-4. It is found that decay time 3 effectively imposes a limit on the dynamic limiting range of the cell (around µj for 3 =ps, and 6 mj for 3 =ps). Above these values, the cell bleaches because of the saturation of energy level 3. (a) Output Energy (µj) Pure TPA model β=4. cm/gw Pure TPA model β=.9 cm/gw =.ns 3 =ps =.ns 3 =ps =.ns 3 =ps =.ns 3 =ps 3 4 Input Energy (µj)

42 (b) 3 Output Energy (µj) Pure TPA model β=4. cm/gw Pure TPA model β=.9 cm/gw =.ns 3 =ps =.ns 3 =ps =.ns 3 =ps =.ns 3 =ps Input Energy (µj) Fig. 3-4: Medium and high-energy behavior of the proposed model for varying values of and 3. Pure two-photon absorption (TPA) results are also shown. The intensity profile of the input beam, and the population densities for an input energy of µj are given in Figure 3-. otice how for this input energy the incoming pulse attenuates uniformly, maintaining a Gaussian profile. The ground population is not depleted (retaining in fact most of the total population), and most of the excited molecules are in level. Intensity Intensity (GW/cm )... / Time(ns). Position(mm)..9 3 Time(ns). Position(mm).

43 3 3. x -3. x -3 / 3 / Time(ns) Position(mm) Time(ns) Position(mm) 4 4 / x -.. / Time(ns). Position(mm). 3 Time(ns). Position(mm). Fig. 3-: Intensity profile and population densities for an input energy of µj for a beam incident on a.mm thick bulk-film cell with the parameters shown in Table 3-. If no transition is allowed between the high lying electronic state 3 and state, the limiting action, although slightly better than a pure two-photon transition model, does not compare with the proposed model that clamps the output at a very low level. As with the pure TPA model, at high energies (~ µj) the cell eventually is bleached and no further limiting occurs. Figure 3-6 shows the spatio-temporal behavior of the pulse as it propagates through the cell, as well as the resulting relative populations in the most significant population levels.

44 33 (a) Intensity Intensity (GW/cm ) / / Position(mm). Time(ns) 3 Time(ns). Position(mm). 3 Time(ns). Position(mm). (b) Intensity 3 Intensity (GW/cm ) / / Position(mm). Time(ns) 3 Time(ns). Position(mm). 3 Time(ns). Position(mm). (c) Intensity Intensity (GW/cm ) / / Position(mm). Time(ns) 3 Time(ns). Position(mm). 3 Time(ns). Position(mm). Fig. 3-6: Temporal propagation of a µj pulse through a.mm cell for (a) a pure two-photon model; (b) TPA + excited state model (no transition 3 to allowed); and (c) the complete model of Figure 3-(b) with =.ns and 3 = ps.

45 3. Optical Limiting in an L34-cored Fiber Array 34 A nonlinear liquid-filled fiber core extends the interaction length of a focused beam, making a fiber array an ideal solution to obtain a limiting threshold low enough to be safe to the naked eye. We studied and analyzed the performance of fiber arrays of different length with pulses of varying temporal width, under the configuration given in Figure -. Performance is also compared with a pure two-photon absorber (no additional excited states) mm long fiber array, pulse width p = 7 ns The proposed model was checked against the results reported in [], where a fiber array [7][] with an L34 filled core was used. The length of the array studied was 3mm, with a core radius of µm. The results are presented in Figure 3-7.

46 (a) 3 Output Energy (µj) Linear Transmission 86% TPA TPA+ESA =.ns; no transition 3 TPA+ESA =.ns 3 =ps TPA+ESA =.ns 3 =ps Experimental Results F Optics Input Energy (µj) (b) Output Energy (µj) Input Energy (µj) Linear Transmission 86% TPA TPA+ESA =.ns; no transition 3 TPA+ESA =.ns 3 =ps TPA+ESA =.ns 3 =ps Fig. 3-7: Limiting curve for a 7ns pulse through a 3 mm L34-cored fiber array. (a) Low energy and experimental results; (b) higher energies. The experimental results correspond to measurements taken for a 3mm fiber array with opaque glass cladding for an f/ input and output optical setup. The results are lower to the simulation (and correspond to an effective β of cm/gw) due to additional loss of energy of the pulse through absorption of side scattered radiation by the cladding, and

47 the use of an f/ aperture configuration in both the input and output optics (vs. an open aperture measurement). Depletion of the ground state and absorption of the laser energy are particularly pronounced in the entrance region of the fiber. Decay constant 3 determines the effectiveness of the excited state limiting mechanism, and imposes an ultimate limit for the input intensity that can be successfully attenuated. An excited state not only improves the response by providing a lower clamping energy level, but also makes the system more resistant to bleaching which may also happen in this configuration around mj for a pure TPA model, and 3 mj if an excited state with no allowed transition 3 to is considered. The difference in the three mechanisms may further be understood in Figure 3-8, where the evolution of pulses with three different energies through the fiber array is shown. 36

48 37 4 Position(mm) Time(ns) Position(mm) Time(ns) Position(mm) Position(mm) 3 Time(ns) 3 Time(ns) Position(mm) 3 Time(ns) 3 Time(ns) 4 Time(ns) 4 Intensity (GW/cm ) Position(mm) Intensity (GW/cm ) 3 Intensity (GW/cm ) Position(mm) 4 Time(ns) 4 Time(ns) 3 Intensity (GW/cm ) Position(mm) 4 3 Time(ns) Intensity (GW/cm ) 3 Intensity (GW/cm ) Position(mm) 4 4 Intensity (GW/cm ) Time(ns) (d) 3 Intensity (GW/cm ) Position(mm) (c) (b) 4 Intensity (GW/cm ) (a) 7 µj Intensity (GW/cm ) Intensity (GW/cm ) 7 µj Intensity (GW/cm ) 7 µj Input Energy 4 Position(mm) 3 Time(ns) 4 Position(mm) Fig. 3-8: Pulse propagation intensity profiles for a 7 ns pulse through a 3 mm L34-cored fiber at different input energies for TPA + ESA model with (a) =.ns and 3=ps, (b) =.ns and 3=ps, and (c) =.ns and no transition 3; and for (d) pure TPA model. It can be observed how in Figure 3-8 (a) and (b) most of the attenuation takes place in the entrance of the fiber array, allowing better transmission control (even at very high intensities) for a fixed length, or reduction of the interaction length to obtain a specific output, when compared with liquids modeled in (c) and (d). ote also how the