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Tallinn University of Technology Institute of Cybernetics Propagation characteristics of coherent optical waves in a stratified medium with Kerr nonlinearity Koherentsete optiliste lainete levi karakteristikud Kerri mittelineaarsusega kihilises keskkonnas by Liis Rebane Thesis Submitted to Tallinn University of Technology for the Degree of Bachelor of Science Supervisors: Dr. Pearu Peterson, Institute of Cybernetics at TUT Prof. Dr. Yasuo Tomita, Dept. of Electronics Engineering, University of Electro-Communications, Tokyo, Japan Tallinn 2004 Deklareerin, et käesolev väitekiri, mis on minu iseseisva töö tulemus, on esitatud Tallinna Tehnikaülikooli bakalaureusekraadi taotlemiseks ja selle alusel ei ole varem taotletud akadeemilist kraadi. Bakalaureusekraadi taotleja L. Rebane: Juhendaja Dr. P. Peterson: Kokkuvõte Käesolevas töös uuritakse koherentsete optiliste lainete levi Kerri mittelineaarsusega perioodilistes materjalides. Peatähelepanu all on mittelineaarsed efektid: bi- ja multistabiilsus ning stabiilne piirjuht, mille korral sisend-väljund intensiivsuste graafik on monotoonne ja ülevalt tõkestatud. Analüüsi läbiviimiseks kasutatakse Maxwelli võrranditest tuletatud paaris-võrrandite süsteemi, mille muutujateks on struktuuris edasi- ning tagasiiikuvate lainete kompleks-amplituudid. Paarisvõrrandite süsteemi abil leitakse statsionaarsed lahendid perioodilises materjalis, mis koosneb kahest erinevate lineaarsete ja mittelineaarsete murdumisnäitajatega kihide massiividest. Töö kuuendas osas rakendatakse käsitletud teooriat CdSe-nano-osakeste lisandiga fotopolümeer struktuurile ning uuritakse optiliste lainete ülekande omadusi. Töös käsitletakse nelja erinevat murdumisnäitajate jaotust. Kõigi juhtude korral on sisend-väljund intensiivsuse graafik globaalselt multistabiilne. Selliseid struktuure kasutatakse kõige sagedamini optilisteks ümberlülitusteks madalamalt ülekandetasandilt kõrgemale. Näidatakse, et käsitletud CdSe-lisandiga fotopolümeeris saavutatakse multistabiilne käitumine juba madalatel sisendintensiivsustel. Töös jõutakse järeldusele, et stabiilse ülevalt tõkestatud piirjuhu saavutamiseks tuleb muuta struktuuri murdumisnäitajate jaotust. See oleks antud materjali korral teostatav kasutades hologrammi kirjutamisel erineva intensiivsusega valgust või muutes hologrammi kirjutamis-kiirte vahelist nurka. Abstract The propagation of coherent light through a nonlinear periodic optical structure with periodic linear and nonlinear refractive indices is studied. Nonlinear coupled mode equations are derived from Maxwell equations to investigate input-output transmission regimes of optical structures. Both stable and multi-stable behavior are analyzed including alloptical limiting — highly nonlinear stable transmission with monotonic and one-to-one input-output power relationship that has upper threshold value. Periodic optical structures can be applied in optical communication systems as uniform switches, logic elements, or optical limiters. In Chapter 6 this theory is applied to CdSe-nanoparticles dispersed photopolymer holograms. We consider four different index distributions and find that all of them exhibit globally multi-stable behavior. Such structures are usually applied as switching devices between lower-transmissive and higher-transmissive states. We have shown that multistable behaviour at very low intensities can be achieved in given structure, especially for longer gratings. Limiting behavior could be achieved by using different intensities for hologram recording, or changing the angle between the writing beams. 1 Acknowledgments I am happy to take this opportunity to express my gratitude to my supervisor Prof. Yasuo Tomita for introducing me to this subject and helping me during this research with his ideas, explanations and comments. Special thanks to my supervisor Dr. Pearu Peterson for his constant help and patience. I look forward to working together in the future. I also thank Mervi Sepp for valuable collaboration, Koji Furushima for his distribution calculations, and Prof. Andrus Salupere and other staff of the Centre of Nonlinear Studies for valuable discussions and help during the preparation of this thesis. This research was supported by ETF grant 5767. 2 Contents 1 Introduction 1.1 Optical communication systems . . . . . . . . . 1.1.1 Optical fibres . . . . . . . . . . . . . . . 1.1.2 Photonic crystals and their applications systems . . . . . . . . . . . . . . . . . . 1.2 Statement of the problem . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . 2 General properties of wave dynamics 2.1 Introduction . . . . . . . . . . . . . . . . . . 2.2 Phase velocity, group velocity and dispersion 2.3 Nonlinearity . . . . . . . . . . . . . . . . . . 2.4 Solitons . . . . . . . . . . . . . . . . . . . . 3 Electromagnetic theory of light 3.1 Introduction . . . . . . . . . . . . . 3.2 Maxwell equations . . . . . . . . . 3.3 Electromagnetic waves in vacuum . 3.4 Electromagnetic waves in dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in optical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . communication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 5 6 6 7 . . . . 8 8 9 10 10 . . . . 11 11 11 12 14 4 Optical communication systems 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Optical fibres . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Linear theory of light propagation in fibres . . . . . . . . 4.2.2 Nonlinear theory of light propagation and optical solitons 4.3 All-optical signal processing . . . . . . . . . . . . . . . . . . . . 4.3.1 Photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . in fibres . . . . . . . . . . . . . . . . 16 16 16 17 18 19 19 5 Light propagation in periodic structures 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 5.2 Weak light assumption and linear periodic medium 5.2.1 Bragg reflection . . . . . . . . . . . . . . . . 5.2.2 Linear coupled mode equations . . . . . . . . . . . . . . . 21 21 22 22 22 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 5.4 5.5 5.6 5.2.3 Dispersion relation and photonic band-gap . . Intense light in a nonlinear periodic medium . . . . . 5.3.1 Kerr effect . . . . . . . . . . . . . . . . . . . . 5.3.2 Nonlinear coupled mode equations . . . . . . 5.3.3 Optical bistability . . . . . . . . . . . . . . . . 5.3.4 Bragg and gap solitons . . . . . . . . . . . . . 5.3.5 Nonlinear Schrödinger limit . . . . . . . . . . Gratings with periodic nonlinear index . . . . . . . . Stationary solutions . . . . . . . . . . . . . . . . . . . 5.5.1 Balanced nonlinearity management: nnl = 0 . 5.5.2 Unbalanced nonlinearity management: nnl 6= 0 5.5.3 Stationary transmission regimes . . . . . . . . Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 25 25 26 26 27 28 29 30 31 34 34 36 6 Transmission properties of nanoparticles (CdSe) dispersed photopolymer hologram 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Holography with photopolymers . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Grating formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Mathematical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Data for numerical experiments . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Results for nCdSe = 2.55 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Diffusion dominant case (R=10) . . . . . . . . . . . . . . . . . . . . 6.5.2 Photopolymerization dominant case (R=0.01) . . . . . . . . . . . . 6.6 Analyzes for nCdSe = 2.728 . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Diffusion dominant case (R=10) . . . . . . . . . . . . . . . . . . . . 6.6.2 Photopolymerization dominant case (R=0.01) . . . . . . . . . . . . 6.7 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 38 38 38 39 40 41 42 42 47 50 50 52 53 7 Summary 55 A Derivation of the expression for index modulation 56 B Derivation of coupled mode equations 58 C Derivation of stationary coupled mode system 60 D Program listing D.1 Analytical algorithm for balanced nonlinearity management (nnl = 0) . . . D.2 General numerical algorithm based on backward finite difference scheme . . 62 62 63 4 Chapter 1 Introduction 1.1 Optical communication systems The objective of any communication system is the transfer of information from one point to another. Most often, the information transfer is accomplished by modulating the information onto an electromagnetic carrier wave, which is then transmitted to the destination, where the electromagnetic wave is received and the information is recovered. Such systems are categorized by the location of carrier frequency in electromagnetic spectrum: radio systems, microwave or millimeter systems, and optical communication systems. In optical systems the carrier frequency is selected from the optical region, which usually includes infrared and visible frequencies. In any communication system, the amount of information transmitted is directly related to the bandwidth of a modulated carrier (frequency extent, which shows how many pulses per second could be sent into a fibre and be expected to emerge intact at the other end), which is generally limited to a fixed portion to a carrier frequency itself. Thus, increasing the carrier frequency also increases the available transmission bandwidth and therefore the information capacity of the overall system. Using higher carrier frequencies generally increases the capability of the system to achieve higher power densities, which leads to improved performance, making the usage of optical frequencies in communication systems to be a very appealing option [8]. 1.1.1 Optical fibres The intensive research for the use of light as a carrier of information began in 1960’s when the laser technology as a source of coherent light became available. Initially the transmission distances were very short, but soon the enhanced techniques for manufacturing high quality glass lead to the development of low-loss dielectric waveguides; and optical communication systems became a serious alternative to electrical communications. Optical fibres carry signals with much less energy loss than copper cables and with much higher bandwidth, meaning that fibres can carry more channels of information over longer distances and with fewer repeaters required. In addition to the high data transmission rate 5 and low transmission losses, the main advantages of the usage of optical fibers instead of usual coaxial fibres are small size, low weight, flexibility, and high security. In order to design fully integrated optical circuits, that would replace electro-optical devices, many other optical components in addition to optical fibres are needed. These optical components are used for swiching, pulse shaping, and amplifying of optical pulses. The current information technology is based on our ability to control the flow of electrons in a semiconductor. For all-optical signal processing we need to achieve a similar control over photons [3, 8]. 1.1.2 Photonic crystals and their applications in optical communication systems Periodic optical structures or photonic crystals are dielectric arrays that selectively transmit or reflect light at various wavelengths. In the form of diffraction gratings they have been used over hundred years to separate color components of a light beam. In optical communication systems photonic crystals serve as optical equivalents to semiconductors in electronics, both having energy band structure. Such photonic crystals could solve many of the problems that currently limit the speed and capacity of optical-communication networks. Nonlinear photonic crystals may bring to reality the vision of light controlling light in micro-scale photonic circuits - the analog of present day electronic integrated circuits where electrons control electrons. Given the impact that semiconductor materials have had on every sector of society, photonic crystals could play even a greater role in future optical-communications industry [14, 15]. 1.2 Statement of the problem In this the work the transmission properties of nonlinear periodic optical structures and their applicability as all-optical signal-processing elements are studied. In particular the main aims of this work are: • To analyze the light transmission properties of a medium with periodic grating and intensity dependent refractive index on the basis of Pelinovsky’s theory [14]. • To apply this theory to nano-particles dispersed photopolymer holograms that have been proposed and intensively investigated by Prof. Yasuo Tomita’s Photonics Group at University of Electro-Communications, Tokyo, Japan, and to examine their applicability in optical communication systems. 6 1.3 Outline This thesis consists of seven chapters and four appendices. In Chapter 2 some basic concepts of wave theory are explained. In Chapter 3, electromagnetic theory of light is introduced, including derivation of Maxwell’s wave equations for dielectrics. In Chapter 4 optical communication systems are described in more detail, including soliton propagation in optical fibres and the concept of photonic crystals. In Chapter 5 light propagation in periodic structures is analyzed including both linear and nonlinear theories for weak and intense light, respectively. Maxwell’s wave equations are simplified further to a system for complex amplitudes of forward and backward propagating electro-magnetic waves — coupled mode system. Chapter 6 contains the main results of this thesis. The theory, which is described in Chapter 5, is applied to nanoparticles dispersed photopolymer holograms and the transmission characteristics of these structures are calculated. Finally, Chapter 7 gives a brief summary and the further perspectives of the work. 7 Chapter 2 General properties of wave dynamics 2.1 Introduction Light can be modelled as the propagation of electromagnetic waves in space. In this Chapter some terminology of wave dynamics, which is used in further analyzes, is introduced. It is very complicated to define a wave in general. The most common and widelyused definition is: a wave is a disturbance propagating in a medium. Such definition is not completely satisfactory since standing waves do not propagate. Another feature that makes definition difficult is due to the interaction of wave with the medium through which it is passing. A wave on the surface of water passes and leaves the medium unchanged. In contrast, a chemical wave usually leaves the reacting species involved in a different chemical state after it has passed by. For these reasons we can’t have a unique definition for a wave. Wave phenomena is simply viewed as generic set of phenomena with many similarities [2] or as a state moving into another state [11]. Media for wave propagation can be categorized as follows: • Linear medium - the superposition principle holds - different waves at any particular point in the medium can be added to form other solutions. • Nonlinear medium - the solutions to the wave-equations do not form a vector space and cannot be superposed to produce a new solutions. • Bounded medium - finite in extent (otherwise unbounded). • Uniform medium - its physical properties are unchanged at different points. • Isotropic medium - its physical properties are the same in different directions. While most of the waves need a medium for propagation, electromagnetic waves can also exist in vacuum. 8 2.2 Phase velocity, group velocity and dispersion Phase velocity is the rate at which the phase of the wave propagates in space or the velocity at which the phase of single frequency component of the wave will propagate. It is given in terms of the wave’s frequency ω and wave vector k by: v= ω . k (2.1) The phase velocity generally describes only the propagation of monochromatic waves. In general case waves occur as a mixture of a number of component frequencies, and the motion of such a pulse is described by the group velocity, which is the rate that changes in amplitude (known as the envelope of a wave) will propagate. Since the group velocity is the velocity of maximum amplitude of the envelope, it can also be interpreted as the velocity at which energy or information is conveyed along a wave. The group velocity is defined in terms of the wave’s angular frequency ω and wave number k by: vg ≡ dω . dk (2.2) If the phase-velocities of different frequency components are equal, then the component frequencies and their superposition (or group) would travel with the same velocity and the profile of the envelope wave remains constant. A medium in which phase velocity is independent of frequency is called non-dispersive medium. If the phase-velocity is frequency dependent, the envelope of the wave will become distorted as it propagates - this effect is known as group velocity dispersion, which causes a short pulse of light to spread in time as a result of different frequency components of the pulse travelling at different velocities. A medium in which the phase velocity is frequency-dependent, is known as dispersive medium and is described by a dispersion relation, which expresses the variation of ω as a function of k. Since ω = kv where v is the phase velocity, the group velocity can be given by: vg = d dv dv dω = (kv) = v + k =v−λ dk dk dk dλ (2.3) where k = 2π/λ is the wave number. If dv/dλ is positive, so that vg < v, we have a normal dispersion; for negative dv/dλ, and vg > v, we have anomalous dispersion. The result of group velocity dispersion, whether negative or positive, is ultimately temporal spreading of the pulse. This makes dispersion management extremely important in optical communications systems based on optical fiber, since if dispersion is too high, a group of pulses representing a bit-stream will spread in time and merge together, rendering the bit-stream unintelligible [2]. 9 2.3 Nonlinearity For linear systems we can use the superposition principle i.e. if y1 and y2 are solutions of a linear equation, then so is a1 y1 + a2 y2 for any constants a1 and a2 . It means that separation of variables and integral transform methods allow us to determine the solution. For nonlinear systems superposition principle is not valid and the standard mathematical techniques mentioned above, are not applicable [2]. The most interesting aspect of nonlinear phenomena is how the effect of a nonlinear process is modified by that of a linear process (dispersion, diffusion). In general, linear and nonlinear processes have opposing effects. In contrast to dispersion or diffusion, nonlinearity leads to the concentration of a disturbance. [9] 2.4 Solitons Soliton is a stable wave structure that can exist due to a specific equilibrium of dispersive (linear) and focusing (nonlinear) effects. Solitons emerge unchanged from collisions with each other, regaining their asymptotic shapes, magnitudes, and speeds. Solitons are found in many non-linear physical phenomena, as they are solutions to many different nonlinear differential equations (the KdV equation, the Sine-Gordon equation, the nonlinear Schrödinger equation etc.). Few examples of physical systems where solitons occur include nonlinear optics, hydrodynamics, plasma physics, protein models, high energy physics, and solid state physics [5]. Solitons in optics have been widely studied for their applications in distortionless signal transmission in fibers and many other nonlinear applications in optical communication systems. Optical solitons differ somewhat from the classical KdV soliton, which describe the solitary wave of a wave; the optical soliton is the solitary wave of an envelope of a light wave [9]. 10 Chapter 3 Electromagnetic theory of light 3.1 Introduction This Chapter introduces the electromagnetic theory of light, which is based on Maxwell equations. The aim is to derive Maxwell wave equation that describes the propagation of light waves in dielectrics, and will be later used to derive the equations for coupled mode system — the main set of equations used in the analyzes of this thesis . A consistent theoretical explanation of optical phenomena can be described jointly by Maxwell electromagnetic theory and the quantum theory of light. While the quantum theory describes the interactions between light and matter i.e. the absorption and emission of light, Maxwell theory is most useful for treating the properties of propagation of light, which is also the subject in this thesis. According to electromagnetic theory, light is an electromagnetic radiation in the range of wavelengths from infrared to ultraviolet. The three basic characteristics of light waves are amplitude (intensity), frequency (color), and polarization (angle of vibration). 3.2 Maxwell equations The propagation of electromagnetic waves is described by four Maxwell equations: ∇ · D = ρ, ∇ · B = 0, (3.1) (3.2) ∇×H = J+ (3.3) ∂D ∂t ∂B , ∇×E = − ∂t (3.4) where ρ is the free electric charge density (in units of C/m3 ) without dipole charges bound in a material, H is magnetic field strength (in units of A/m), J is current density, E is electric field (in units of V /m), D is electric displacement field (in units of C/m2 ), and B is magnetic flux density (in units of tesla, T ), also called as magnetic induction. 11 These four equations express how electric charges produce electric fields (Gauss law, Eq.(3.1)), the experimental absence of magnetic charges (Eq.(3.2)), how currents produce magnetic fields (Ampère law, Eq.(3.3)), and how changing magnetic field produces electric field (Faraday law of induction, Eq.(3.4)), respectively. 3.3 Electromagnetic waves in vacuum We consider electromagnetic waves in vacuum - a linear, isotropic, non-dispersive medium. In vacuum, the D and H fields are related to E and B by: D = ε0 E, B = µ0 H, (3.5) (3.6) where ε0 and µ0 are electrical permittivity and magnetic permeability, respectively, for vacuum. Also there are no currents or electric charges present in vacuum (i.e. ρ = 0 and J = 0). With these conditions we obtain the source-free Maxwell’s equations: ∇ · E = 0, ∇ · B = 0, (3.7) (3.8) ∇ × B = ε0 µ0 (3.9) ∂E , ∂t ∂B , ∇×E = − ∂t (3.10) This system has a trivial solution E = B = 0. Maxwell equations for vacuum can also be written in a form of linear wave equations for E and B: ∇2 E = 1 ∂2E , c2 ∂t2 ∇2 B = 1 ∂2B c2 ∂t2 (3.11) √ where c = 1/ µ0 ε0 . This is a set of linear differential equations which means that the amplitude of two interacting waves is a simple superposition. Therefore the behavior of a wave can be analyzed by breaking it up into components. The equation of this form has a well known general solution (usually referred as d’Alembert solution): E(r, t) = E+ (k · r − c|k|t) + E− (k · r + c|k|t) (3.12) where r is position vector, k is wave vector and E+ and E− are arbitrary vector functions which can be determined, for example, from specifying initial conditions. This solution represents two counter-propagating waves, which do not interact with themselves nor with each-other. Both of these solution components can be represented as a superposition of their Fourier’ components [2]. The argument (k · r + c|k|t) = p is called the phase of a wave. Evidently p is constant in space-time if dp/dt = 0, i.e. kdr/dt = ±c|k|. An observer moving with velocity c in the 12 direction of a wave vector k will always notice the same phase, meaning that c represents phase velocity of a light wave. For electromagnetic waves in a vacuum the velocity c is √ given by: c = 1/ µ0 ε0 ≈ 3 × 108 m/s. This led Maxwell to a postulate (that was proved later) that light is an electromagnetic wave [6]. Let us consider electromagnetic plane wave propagating in the direction of a wave vector k. Then electric and magnetic fields are: E(r, t) = E0 ei(kr−ωt) + c.c. B(r, t) = B0 ei(kr−ωt) + c.c. (3.13) (3.14) where we have denoted ω = c|k| and E0 and B0 are constant vectors. Note that from ∇ · E = i(E0 · k)ei(kr−ωt) + c.c. = iE · k = 0 (3.15) follows that E⊥k. Similarly from ∇ · B = 0 follows B⊥k. From Eq. (3.10) we have ∇ × E = i(k × E = −iωB. (3.16) So 1 (k × E) (3.17) ω In conclusion, we have shown that the three vectors E, B and k form a pairwise perpendicular set of vectors (see Fig. 3.1) B= E k H Figure 3.1: Electromagnetic field in vacuum The k − E plane is called as plane of polarization. In special case when the plane of polarization does not change, it is said that wave is linearly polarized. In general the plane of polarization rotates as time t varies and then it is said that the wave is either circularly or elliptically polarized [2]. 13 3.4 Electromagnetic waves in dielectrics Dielectrics are not passive carriers of electromagnetic waves and Maxwell equations need to be modified in order to describe the wave dynamics in dielectrics. Although in dielectrics, electrons are bound sufficiently tightly to the nuclei of the constituent atoms and can not produce a current (J=0), the distribution of electrons around each nucleus is deformed when the electromagnetic field is applied to the material. This leads to a modification of the permittivity and magnetic permeability of the material and Maxwell equations become: ∇ · D = ρ, ∇ · B = 0, (3.18) (3.19) ∇×H = J+ (3.20) ∂D , ∂t ∂B , ∇×E = − ∂t (3.21) where µ0 H = B − M, D = ε0 E + P, (3.22) P(E) = ε0 (χe E + χ(2) EE + χ(3) EEE + ...) (3.23) where P is electric dipole moment per unit volume or polarization and M is magnetic dipole moment per unit volume or magnetization. In the following we assume that there are no extra charges other than internal charges related to atomic structure, so ρ = 0. For non-magnetic materials M = 0 and the response of the material depends on polarization P. In general, polarization depends on electric field. For small but finite fields we can expand P into Taylor series where χe is electric susceptibility of the material and χ(i) are i-th order nonlinear susceptibility tensors. For perfectly isotropic medium χ2 vanishes for symmetry reasons P(−E) = −P(E). The third order nonlinearity (also known as Kerr nonlinearity) tensor χ(3) is nonzero and gives rise to nonlinear polarization effects. In the case of linear polarization, that is, the vector of electric field stays in a polarization plane (say x − z plane), we can write P(E) ≈ ε0 (χe + χ(3) |E|2 )E = ε(z, |E|2)E − ε0 E, (3.24) where we have defined ε(z, |E|2 ) = ε0 (1 + χe + χ(3) |E|2 ). So, D = ε(z, |E|2 )E, B = µH, (3.25) (3.26) and ∇ · D = ε∇ · E + ∇ε · E = 0. 14 (3.27) We consider the medium that is only slightly inhomogeneous along the z-axis, so that ∂ε/∂z ≪ ε and ∇ε ∇·E =− · E ≈ 0. (3.28) ε Now we can assume that E is perpendicular to the z-axis and take E = (E, 0, 0) where E = E(z, t). To obtain the wave equation for the electric field component E, we take the curl of (3.21) and replace ∇ × B by using (3.19). This yields to relation: 1 2 2 ∂ D ∂ 2 0 , ∇ × (∇ × E) = −µ0 2 = −µ0 2 ε(z, |E| )E · (3.29) ∂t ∂t 0 where ε(z, |E|2) is related to refractive index n of the material by ε(z, |E|2 ) = n2 (z, |E|2 )ε0 . On the other hand, 1 2 ∂ ∇ × (∇ × E) = ∇(∇ · E) − ∇2 E = − 2 E · 0 . (3.30) ∂z 0 We assume that the intensity of the wave, |E|2 , is slowly varying compared to the variations in the carrier of E, so that we arrive to nonlinear Maxwell wave equation in a form: ∂ 2 E n2 (z, |E|2 ) ∂ 2 E − =0 ∂z 2 c2 ∂t2 √ where E(z, t) is a scalar electric field and c = 1/ µ0 ε0 is the speed of light. 15 (3.31) Chapter 4 Optical communication systems 4.1 Introduction In this Chapter the structure of a general optical communication system is described and the importance of the photonic crystal technology is explained. Complete all-optical communication is achievable when electro-optic components that are currently used in optical communication systems, are replaced by all-optical signal-processing elements. For this purpose, the development of new all-optical devices is necessary. Usage of photonic crystal techniques is one of the most promising opportunities for these applications. Advantages and problems of optical systems The principal advantages in communicating at optical frequencies are: (i) wider available transmission bandwidth due to higher carrier frequency, (ii) ability to concentrate the power in extremely narrow beams, and (iii) significant reduction in component sizes because of the extremely small optical wavelengths. The main problems in development of optical networks are the technological difficulties associated with reliable and cheap connections, and the development of an optical circuit technology that would match the potential data-rates of the fibres. The speed of these electronically controlled circuits is usually the limiting factor on the bit-rate of optical systems. The difficulty of connection and high-cost of associated circuitry result in optical fibres being used only in very high bit-rate communication in this moment [8]. 4.2 Optical fibres Most optical fibers consist of the core (cylindrical insulator), which is surrounded by the cladding (a layer of another insulator with slightly lower refractive index). The material is designed to be optically transparent and absorb as little light as possible. The main advantages of optical fibres compared to widely used coaxial cables are [2, 8]: 16 • Potentially huge bandwidth. In practice it has not been realized yet and the bandwidth of individual fibre is almost the same as high quality coaxial cable. However, it is possible to lay many hundreds of optical fibres in the same cable cross-section as a single coaxial cable. • Potentially high data transmission rate. Standard optical fibre transmitters can send up to 109 bits of information per second by switching a laser beam on and off (data transmission rates for coaxial cables are typically in the order of 106 bits per second). • Small size, low weight, and high flexibility. Optical fibres have very small diameters (about 120µm). • Good electrical isolation. Optical fibres are almost completely immune to external fields and do not suffer from cross-talk, radio interference, etc. • High security. It is difficult to tap into an optical line. • Low transmission losses (0.2 − 0.5 dB/km compared to about 7 dB/km for ordinary coaxial cables). 4.2.1 Linear theory of light propagation in fibres A plane wave, which is incident on the interface of two surfaces, will experience total internal reflection when it is incident almost parallel to the interface. Otherwise, most of the wave is transmitted into the second medium. Due to the wave nature of light, only certain modes (standing oscillating electromagnetic waves, which are defined by the wave geometry) would be guided by the fiber. A mono-mode fibre is a fibre that only has one acceptable mode per frequency. A multi-mode fibre has a number of possible modes. Light attenuation in a wave-guide has a number of sources. Absorption of light occurs in the glass and this decreases with frequency. Scattering of light from internal imperfections within the glass - Rayleigh scattering - increases with frequency. Waveguide imperfections account for low-level loss that is approximately constant with wavelength. Bending the waveguide changes the local angle of total internal reflection and loss increases through the walls. A combination of all these effects results in a minimum absorption in the 0.8µm to 1.8µm wavelength region, that are used for transmission. In addition to attenuation, optical waveguides also suffer from dispersion. The dispersion has two sources: firstly there is a dispersion due to the different modes propagating in the fibre (modal dispersion) and material dispersion or chromatic dispersion, due to the dielectric properties of glass. Dispersion effects cause the pulse spreading, which sets a limit to the bandwidth of the fibre. 17 4.2.2 Nonlinear theory of light propagation and optical solitons in fibres The description of pulse transmission in optical fibres explained above was based on linear theory. By making the use of nonlinear response of an optical fibre (this requires sufficiently intense laser light), the effects from Kerr nonlinearity arise that tends to compress the pulse while dispersion tends to broaden it. When these two competing effects balance, the formation of stable optical pulses - optical solitons - is possible [2]. Optical solitons in fibers are different from the solitons of KdV equation, which describe the solitary wave of a wave. In optical communications optical pulses are used and their shape represents an envelope of a light wave. The soliton propagating in an optical fiber is the solitary wave of an envelope of a light wave, not a soliton of a wave itself. These solitons are referred as envelope solitons. Soliton formation results from group velocity dispersion (its dependence on the wavelength of light) and the third order nonlinear effect (wavelength dependence on the intensity of a wave). The model equation which describes the envelope soliton propagation is known as the nonlinear or cubic Schrödinger equation. The origin of nonlinear properties of a light wave comes from nonlinear Kerr effect, which is caused by the deformation of the electron orbits in glass molecules due to the electric field of light. As a result, the refractive index of a fibre becomes intensity dependent: n(I) = nL + n(2) I (4.1) where nL is the linear component of refractive index. n(2) , which represents a change in refractive index due to Kerr effect, has a very small value. The reason why such a small change in refractive index becomes important, is that modulation frequency ∆ω, which is determined by the inverse of a pulse width, is much smaller than the frequency of a wave ω. Consequently, the group velocity dispersion, which is produced by ∆ω is also small. As a result the relative change in the wave number due to the group dispersion becomes comparable to the nonlinear change. In order for those effects to become significant, the wave distortion due to the fiber loss should be less than these small effects. This requires that the fiber loss rate per wavelength of light should be less than 10−10 , meaning that the fiber loss rate of the fiber should be less than 1 dB per km. The concept of optical solitons could be extremely useful in optical communications. The width of an optical pulse used in most of the optical communication systems, which are in practical use in this moment, is approximately 1 nanosecond. In this case the major distortion of the pulse results from the fiber loss. We can overcome this problem by using repeaters at every several tens of kilometers. When the pulse width is decreased in order to increase the transmission rate to the level of 10 picoseconds, the separation between two repeaters is no longer determined by the fiber loss, but by the group velocity dispersion of the pulse. An optical soliton, which is produced by the balance between the nonlinear effect and group velocity dispersion, produces no distortion caused by the dispersion. However, soliton pulses are not immune to fiber losses and consequently the pulse width of a soliton will expand. Due to that, a soliton transmission system also requires pulse to be repeated. 18 In contrast with the linear system, the reshaping can be achieved utilizing only optical amplifiers (i.e. Raman effect) [9]. 4.3 All-optical signal processing Although optical fibers are widely used today, switching, pulse shaping, and amplifying of optical pulses and optical data streams is most often accomplished by expensive optical/electronic devices, which convert optical pulses to electrical pulses, process the electrical pulse trains within electronic circuit and then convert the processed electrical pulses back to light. In fiber-optic networks the use of all-optical processing instead of expensive and slow electro-optic devices yields to huge savings in both power and cost. In contrast with applications in information storage, where one could be limited, for example, to lowintensity regime, optical signal processing operation devices, which are uniformly stable for all pertinent incident intensities, can be developed. All-optical devices include: optical amplifiers, optical limiters, optical switches, devices that can multiplex many independently modulated wavelength channels onto a single physical medium, optical logic elements etc. The development of all-optical signal-processing elements has been a widely studied topic in recent years. There is considerable current debate as to whether optics could completely replace the widely-used electronic technology [3, 14, 17]. 4.3.1 Photonic crystals Photonic crystals are highly periodic artificial structures that have attracted much attention in recent years. They serve as conducting media for electromagnetic waves and can be designed to control and manipulate the propagation of light. In general they have a similar role in optical communication systems as semiconductors in electronic signal processing, being even more flexible because we have far more control over the properties of photonic crystals than we do over the electronic properties of semiconductors. Since the periodic variation of photonic crystals can often be controlled and even engineered, limited only by the available micro-fabrication technologies, such systems offer many promising applications both for the purpose of studying fundamental effects in nonlinear dynamics and optical device engineering [14, 16]. It is well known from the quantum theory of solids that the energy spectrum of an electron in a solid consists of bands separated by gaps. The periodic arrangement of ions on a lattice gives rise to the energy band structure in semiconductors. Energy bands control the motion of charge carriers through the crystal. Similarly, in a photonic crystal, the periodic arrangement of refractive index variation controls how photons are able to move through the crystal. Replacing the ions on a lattice creates regions of low refractive index within a high-refractive index material or vice-versa. Photons react to the refractive index contrast in an analogous manner to the way electrons react when confronted with a periodic potential of ions. In both cases a range of allowed energies and a band structure characterized by an energy gap or photonic band gap appears. The basic physical reason for 19 the rise of gaps lies in the coherent multiple scattering and interference of waves inside the crystal – Bragg resonance of the waves with the crystal structure. The basic mathematical theory that describes how the band gaps arise in periodic dielectric and acoustic media was essentially constructed by using the Floquet-Bloch theory [1]. Since photonic band-gaps occur in the spectrum of light waves due to periodicity of the medium and described Bragg resonance between incident light and the grating, lowintensity light with frequencies inside the photonic bandgap can not propagate. If the medium is nonlinear (a nonlinear photonic crystal) and sufficiently intense light is used, the photonic band gaps are shifted and light still can be transmitted through the grating due to the effects of Kerr nonlinearity. Nonlinear photonic crystals can be used to control the propagation of light. The most useful applications include: • Formation of a narrow band filter for selecting a particular wavelength. • Pulse shaping and compression using Bragg soliton effects. • Photonic crystal defects which can slow or even stop light pulses propagating in the crystal. • The use of temporal solitons — stable pulses which result from combination of Kerr nonlinearity and chromatic dispersion. By using these effects, the ultimate goal of nonlinear photonic crystal technique - the all-optical processing of light pulses - will hopefully be achievable [17]. 20 Chapter 5 Light propagation in periodic structures 5.1 Introduction In this chapter light propagation in periodic structures is compared with light propagation in uniform media. The aim is to derive the coupled-mode system from Maxwell equations that allows us to analyze the transmission properties of periodic optical structures (photonic crystals). Modelling of time-dependent responses of photonic crystals in three spatial dimensions can be analytically difficult in the framework of Maxwell equations and is considerably simplified when using the coupled-mode approach. Coupled-mode equations are typically derived for two counter-propagating waves in one spatial dimension and at the lowest band gap of Bragg resonances. One-dimensional periodic medium In the following calculations it is assumed that a laser pulse is incident on a sample of infinite extent whose surface coincides with the xy plane. If the wave fronts are infinite planes parallel to the xy-plane, then the simple one-dimensional picture can be used for describing the evolution of the pulse [12]. It is assumed that medium has uniform average Figure 5.1: Linear periodic structure 21 index of refraction n0 and periodic index modulation ∆n with grating period Λ (see Fig. 5.1). In many cases (including calculations in this thesis) it is assumed that the modulation amplitude of the index of refraction is small (∆n ≪ n0 ) that will allow several further simplifications. 5.2 5.2.1 Weak light assumption and linear periodic medium Bragg reflection In the case of weak incident light the properties of prescribed periodic medium and the medium with uniform index are very similar: the light, which is reflected off from various interfaces is mutually out of phase and, as a consequence, the light propagates through the structure essentially unimpeded as is the case for uniform medium. The crucial difference occurs when the lights wavelength satisfies the Bragg condition - the condition of cumulative reflections - which is given by: λB = 2n0 Λ, (5.1) where λB is Bragg wavelength, n0 is average linear refractive index of the material and Λ is the grating wavelength. The Bragg law specifies that when the light wavelength equals to Bragg wavelength (λ = λB ), exactly half of a wavelength of light in medium fits into each period of the grating. Consequently, light-beams that are Fresnel reflected off interfaces of the grating, are all in phase and this leads to a strong reflected wave. Light, which exactly satisfies the Bragg condition, can not propagate inside the structure because it is entirely reflected off. Light with a wavelength nearly satisfying the Bragg condition slows down since it is reflected back and forth between the periodic layers instead of being transmitted directly through the material. This slowing down strongly depends on the wavelength of light - different colors propagate with different velocities, i.e. a very strong dispersion occurs. The range of wavelengths, where the incident light is strongly reflected is called photonic bandgap and its width is determined by the index contrast, as is shown later (Eqs. (5.13) and (5.9)). The properties of periodic structures are often given in terms of the (angular) frequency ω rather than wavelength. The expression for Bragg frequency can be found from Eq.(5.1) in a form: ω0 = πc no Λ (5.2) where c is the speed of light in vacuum [16]. 5.2.2 Linear coupled mode equations Let us consider an one-dimensional optical structure and assume the incident light is coherent and linearly polarized. The direction of light propagation is chosen to be along the 22 z-axis. The scalar electric field is described by the linear Maxwell wave equation, which is derived in Section 3.4: ∂ 2 E n2 (z) ∂ 2 E − 2 = 0, (5.3) ∂z 2 c ∂t2 where n2 (z) is a periodic function. We assume nearly monochromatic electric field that will be represented as a slowly varying envelope of a highly oscillatory carrier wave. The slow modulation in space and time of the plane wave envelope in an uniform medium is induced by a weak periodicity component of the refractive index, so that: n(z) = nln + ∆n(z). (5.4) Where nln is average index of refraction and ∆n is periodical variation. Since ∆n is a periodic function, it can be expanded into the Fourier series. In the following analyzes we consider only the first order of the expansion (derivation of Fourier’ expansion for refractive index in nonlinear periodic medium is given in Appendix B) and ∆n can be expressed as: ∆n(z) ≈ 2n0k cos(2k0 z), (5.5) where k0 = 2π/λB denotes Bragg wave-number [17]. If we would neglect the periodicity of the refractive index, the solution to (5.3) would be: E(z, t) = E+ ei(k0 z−ω0 t) + E− ei(k0 z+ω0 t) + c.c, (5.6) where E+ and E− are the amplitudes of the field components propagating to the right and left, respectively. With the coupled mode approximation we can use the same form of a solution, but include small variation in E+ and E− , because the small perturbation of ε could scatter the wave going to the right causing it to go to the left and vice versa. So the solution to (5.3) is searched in the following form [16]: E(z, t) = E+ (z, t)ei(k0 z−ω0 t) + E− (z, t)ei(k0 z+ω0 t) + c.c. (5.7) Using these assumption, a linear coupled mode system can be derived (see Appendix B and linearize)[17]: ∂E+ nln ∂E+ +i + κE− = 0, ∂z c ∂t nln ∂E− ∂E− +i + κE+ = 0, −i ∂z c ∂t i where κ≡ ω0 n20k . 2nln c 23 (5.8) (5.9) 5.2.3 Dispersion relation and photonic band-gap To obtain the dispersion relation for the linear system (5.8), we search for the envelope functions of the form: E± (z, t) = A± ei(Qz−Ωt) (5.10) where Q ≡ k − ko is the wave number and Ω ≡ ω − ω0 is the frequency of the envelope. Coupled mode equations then reduce to a set of algebraic linear equations: A+ nln Ω/c − Q κ = 0. (5.11) A− κ nln Ω/c + Q Solving this eigenvalue problem leads to dispersion relation: p nln Ω/c ≡ ∆ = ± κ2 + Q2 . (5.12) This relation is plotted in Figure 5.2 and clearly shows that there is a frequency region (a band gap)in which plane wave solutions of the form Eq. (5.10) cannot exist. Ω=ω−ω0 δω Q=k−k0 Figure 5.2: Dispersion relation From the dispersion relation we can also determine the width of the gap: δω = 2κc/nln . (5.13) Combining it with Eq.(5.9), δω turns out to be proportional to the lowest Fourier’ component of the grating [16]. We can also define the normalized group velocity of the wave by: v± = p dω d∆ = = ± 1 − κ2 /∆2 dt dQ 24 (5.14) At the edges of the photonic band gap (∆ → κ), the group velocity is v = 0; and far from the gap (∆ → ∞) it equals to the group velocity of the uniform medium (normalized group velocity is 1). For intermediate values of the detuning, the group velocity has a value between these extreme values. For each eigenvalue in Eq. (5.11), there is an eigenvector, which describes the mixing of the forward and backward propagating mode by the grating. Eigenvectors describe the eigen-states of the field inside the grating. Since the grating is periodic, the eigen-states are Bloch functions. We define the eigenvectors in terms of group velocity v in upper and lower branch. The Bloch functions are then given by: p p p p 1 + v/2, 1 − v/2 , v− = 1 − v/2, − 1 + v/2 . v+ = (5.15) Note that the Bloch functions are normalized and that Bloch functions on the two different branches and at the same Q are orthogonal. Far from the Bragg resonance (v = 1) and the effect of the grating is small, and and the eigen-states of the system are decoupled forward- and backward propagating waves. At the band edges (v = 0), the Bloch functions are standing waves [17]. Since the group velocity is now frequency dependent, the grating is dispersive. For typical gratings, the group velocity varies between 0 and v within a wavelength range of less than a nanometer, leading to a huge dispersion near the edges of the photonic bandgap. 5.3 5.3.1 Intense light in a nonlinear periodic medium Kerr effect Intense laser sources allow us to generate optical fields that may change dielectric constants in a photonic crystal. It makes possible to tune the photonic crystal reflection and transmission bands by simply varying the intensity of the incident light. In this work we have considered third order nonlinear material. The influence of third order nonlinearities to light propagation is called nonlinear Kerr effect, and is often described by nonlinear refractive index n2 . The expression for refractive index of one-dimensional Kerr nonlinear photonic crystal is given by: n(z) = n0 + n2 I + ∆n(z), (5.16) where n0 is the linear refractive index, n2 is the Kerr refractive index, I is the light intensity, and ∆n(z) is the linear periodic refractive index modulation along the z-axis. This is quite good approximation for a light propagating through the optical structure with a grating written in the core (optical fibers and optical elements), or for a waveguide with a periodic variation in its thickness [17]. Kerr nonlinearity causes the intensity dependent influences of the wave on itself, so called self-interaction effect, wherein a pulse changes the medium within which it propagates [12]. 25 5.3.2 Nonlinear coupled mode equations Described structure now incorporates both photonic band dispersion and optical Kerr nonlinearity. The envelope of slowly varying fields with appropriate carrier frequency and amplitude are governed by nonlinear coupled mode equations, which can be obtained from the nonlinear Maxwell wave equation Eq. (3.31): ∂ 2 E n2 (z, |E|2 ) ∂ 2 E − = 0, ∂z 2 c2 ∂t2 (5.17) where the refractive index n(z, |E|2 ) is given by: n(z, |E|2 ) = n0 + nnl |E|2 + 2n0k cos 2k0 z. (5.18) The derivation is given in Appendix A (here we have temporarily neglected the periodic variation in nonlinear refractive index, but it canpbe easily included). Note that nnl = (n0 /(2η0 )n2 where n2 is given in SI units and η0 ≡ µ0 /ε0 ∼ 377Ω is the wave impendence in vacuum. Light, that is incident to the medium is assumed to lie within the linear forbidden band and then the two counter-propagating waves are strongly coupled. The intensity-dependent refraction of the optical material supports the resulting transmission of light through the periodic structure. All the effects due to deep gratings are neglected and the nonlinear coupled mode equations are simply the generalization of Eq. (5.8) (derivation is given in Appendix B) and have a general form : ∂E+ n0 ∂E+ +i + κE− + Γ(|E+ |2 + 2|E− |2 )E+ = 0 ∂z c ∂t n0 ∂E− ∂E− +i + κE+ + Γ(|E− |2 + 2|E+ |2 )E− = 0, −i ∂z c ∂t i (5.19) where Γ = 3ω02 χ(3) /(2k0 c2 ) is the nonlinear coefficient. Nonlinear terms in (5.19) describe the influence of nonlinear changes in the grating refractive index to the wave propagation of interest. In Kerr-nonlinear material waves affect their own propagation (self-phase modulation) and the wave propagating in the opposite direction (cross-phase modulation) [16]. 5.3.3 Optical bistability The calculations of Winful et al. [23] show that the field inside the grating (the solution to the nonlinear coupled mode equations) can be written in terms of Jacobi elliptic functions. At the center of the band gap, the solution has a form: I/S = nd[2 cosh (χ)κz|1/ cosh2 (χ)], (5.20) where I = |E+ |2 + |E− |2 is the total intensity, and S = |E+ |2 − |E− |2 is the energy flow through the grating. nd(z|m) is one of the Jacobi elliptic functions and χ satisfies the 26 equation: sinh χ = 3ΓS/(4κ). For real arguments, nd is a real positive periodic function that varies between 1 and 1/ tanh χ. If the field’s envelope function is periodic, then so must be the electric field, up to a phase detuning. Therefore, if the period of the field envelope equals the length of the grating, then the fields at the front and the back of the grating are identical, resulting in the zero reflectivity vanishes at certain powers or lengths of the medium. Described behaviour shows that the structure becomes transparent at certain intensities (S = I), while for other values of the intensity S < I. Detailed analyzes shows that the transmissivity of the structure is no longer a single-valued function, but consists of a lowertransmission branch corresponding to linear regime and of a higher transmission branch. Such effect is termed as bistability in the input-output power relationship and means that the state of the system is not uniquely determined by the input. Stability of these solutions must be determined by the analyses of full time-dependent equations. This shows that the lower branch is always stable, but the upper branch is often unstable. For cases in which the upper branch is stable, the system exhibits bistable switching [17]. 5.3.4 Bragg and gap solitons Another class of solutions to coupled mode equations is a class of pulse-like solutions that are referred as Bragg solitons. They result from the combination of linear dispersion of the grating and nonlinear change in dielectric constant at a high light intensity. These pulses do not change in shape in spite of the strong dispersion (that otherwise would quickly broaden an arbitrary pulse at low intensities). As shown previously, the uniform Kerr nonlinearity leads to optical bistability and to switching pulsations between lower-transmissive and higher-transmissive stationary states. The grating may become transparent in the gap, switching from total reflection to high transmissivity. The switchings occur as regular and, sometimes, irregular oscillations, arising at the left end of the grating, where the light is illuminated, and travelling to the right end of the gratings. These pulsations of light in the optical grating display a series of localized pulses travelling across the periodic structure with constant speed, which are called Bragg solitons. The input pulse is transmitted through the periodic structure without any change in its shape and amplitude. In simple, one-dimensional geometries Bragg solitons are very similar to the regular solitons in uniform optical fibers. In both of the structures the wave gains its stability through a counter-balancing effect of the group velocity dispersion and the effect of the nonlinearity. The difference is that for the solitons of uniform medium the group-velocity dispersion is primarily due to the underlying dispersion of the uniform material, while for a gap soliton, it is due to the photonic band structure. Optical dispersion for wavelengths near to the photonic bandgap are nearly 6 orders of magnitude larger than for propagation in a uniform material. Large dispersion with nonlinear changes in the refractive index results in soliton formation in length scales of only in centimeters. 27 Gap soliton is an optical pulse that propagates at the wavelength within the photonic bandgap for long distances without distortion at any velocity 0 − c. Bragg soliton is formed when the pulse wavelengths are either inside or partially outside the gap (even at wavelengths nearly outside the gap). The first experimental observation of Bragg solitons in a fiber with Bragg grating was performed in 1996 [4] under laser pulse irradiation at a frequency near (but not inside) the photonic band gap. The nonlinear optical pulses can be used for all-optical switching, soliton lasers, pulse compressors, optical buffers, and storing devices [16, 17]. 5.3.5 Nonlinear Schrödinger limit Nonlinear coupled mode equations can be reduced to nonlinear Schrödinger equation (NLSE) by using the method of multiple scales [10] if the regime of weak nonlinearities is assumed. The electric field of nonlinear coupled mode equations is assumed to be the perturbation of linear coupled mode equations. The solutions to nonlinear coupled mode equation are constructed as slow modulation of Bloch waves on appropriately long spatial and temporal scales. With small dimensionless parameter µ, which is the measure of the amplitude of electric field E± , spatial and temporal variables are scaled as: tj = µj t, zj = µj z, j = 1, 2, ... (5.21) Finally we search for the solution of the nonlinear coupled mode equations (5.19) of the form: E = (µa(z1 , z2 ; ..., t1 , t2 , ...)v+ + µ2 b(z1 , z2 , ...; t1 , t2 , ...)v− )ei(Qz−V ∆+ t) , (5.22) where E is a vector with components E± , a and b are the envelope functions and the v+ mode dominates the v− mode. By finding the derivatives up to the order µ3 , we can achieve the NLSE in a form: ic ∂a ω2 ∂ 2 a Γ + + (3 − v 2 )|a|2 a = 0 2 n0 ∂τ 2 ∂ξ 2 (5.23) where ω2 is the quadratic dispersion, and τ = t and ξ = z − v ncln t are the moving coordinates. Equation (5.23) is regarded as the NLSE, because it has a structure of the Schrödinger equation with Γ2 (3 − v 2 )|a|2 as a self-induced potential. Since a is a complex function, travelling wave solution is expected to have an oscillatory modulation. The fundamental soliton solution of the NLSE is a sech-shaped wave, which acts as an envelope to the highly oscillatory carrier wave. The NLSE plays an extremely important role in the theory of evolution of slowly varying wave trains in stable weakly nonlinear systems and it occurs in many other physical areas, not only in nonlinear optics. Similarly to the KdV and Sine-Gordon equations its travelling wave envelope-solutions are solitons [5, 17]. 28 5.4 Gratings with periodic nonlinear index In previous section we used an assumption that the analyzed structure has a periodic linear and uniform nonlinear index of refraction. A number of entirely new applications are possible if both linear and nonlinear components of the refractive index are modulated. Periodically nonlinear optical materials generally exhibit the phenomena of bistability and multi-stability, and do not lead to memoryless operation. They do not exhibit saturation of the transmitted intensity and may undergo chaotic behavior. Many operations in signal-processing are required to be independent of the past state of the channel [14, 15]. In structures with periodical variation in nonlinear refractive index, stable limiting behavior is possible in their input-output transmission characteristic: the transmitted intensity is bounded by the asymptotic limiting value and no switching to a state of higher transmittance takes place. This stable highly-nonlinear effect is termed as all-optical limiting. Structures with a periodic and nonlinear refractive index variation, which exhibit stable all-optical limiting, represent an important example of optical signal-processing elements - passive optical limiters, which are most commonly used as protective devices, but also in optical logic circuits, optical signal processing, optical sensing, and optical fiber communications [3]. Generalization of coupled mode theory Mathematical theory for gratings with a periodic and nonlinear index variation is very similar to the coupled mode theory discussed previously in this chapter. The expression for the refractive index now has an additional term for nonlinear periodic index and is expressed as: n(Z, |E|2 ) = nln + nnl |E|2 + 2n0k cos kZ + 2n2k |E|2 cos kZ (5.24) (derivation is given in Appendix A). The structure can still be analyzed with the coupled mode system, which now has a form: ∂E+ ∂E+ i + n0k E− + nnl (|E+ |2 + 2|E− |2 )E+ (5.25) + ∂Z ∂T +n2k [(2|E+ |2 + |E− |2 )E− + E+2 E−∗ ] = 0, ∂E− ∂E− −i + n0k E+ + nnl (2|E+ |2 + |E− |2 )E− + − ∂Z ∂T +n2k [(|E+ |2 + 2|E− |2 )E+ + E−2 E+∗ ] = 0. p where E± are the normalized complex amplitudes for the electric field defined as E±′ / 2η0 I0 /nln , where E±′ are the complex amplitudes of the electric field and I0 is reference intensity, and Z = ω0 z/c and T = ω0 t/nln are the normalized spatial coordinate and time, respectively. Also, note that nnl and n2k are normalized such that they are non-dimensional ′ ′ ′ ′ as nnl = nnl I0 2η0 /nln and n2k = n2k I0 2η0 /nln , where nnl and n2k are nonlinear index coefficients used in previous chapters (Derived in Appendix B). 29 5.5 Stationary solutions In this Section we study the stationary solutions of light transmission in finite optical grating. The normalized length of the grating is L such that Z ∈ [0, L]. The boundary conditions are given at both ends of the structure (Z = 0 and Z = L) by: |E+ (0)|2 = Iin , |E− (0)|2 = Iref , |E+ (L)|2 = Iout , |E− (L)|2 = 0, (5.26) where Iin , Iref , and Iout are the normalized intensities of incident, reflected, and transmitted waves, respectively. The backward propagating wave vanishes at the right end of the grating. The intensity flow through the structure is conserved and satisfies the equation: |E+ (Z)|2 − |E− (Z)|2 = Iin − Iref = Iout (5.27) Stationary coupled mode system is obtained by assuming that ∂E± /∂T = 0 in (5.25). It can be written in terms of the intensity of reflected wave Q = |E(Z)|2 and the phase mismatch Ψ = Arg(E− (Z)) − Arg(E+ (Z)) in a form: p ∂Q = −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)], (5.28) ∂Z ∂Ψ cos Ψ 2 [n0k (Iout + 2Q) + n2k (Iout + 8Iout Q + 8Q2 )] = −3nnl (Iout + 2Q) − p ∂Z Q(Iout + Q) (see derivation in Appendix C). The stationary coupled mode system can also be written in Hamiltonian form: dE± ∂H = ±i ∗ , dZ ∂E± (5.29) where the real-valued Hamiltonian is given by: 1 H = [n0k +n2k (|E+ |2 +|E− |2 )](E+∗ E− +E+ E−∗ )+ nnl (|E+ |4 +4|E+ |2 |E− |2 +|E− |4 ). (5.30) 2 If we replace E± in Eq. (5.30) by Eq. (C.1), then the Hamiltonian can be written in terms of Q and Ψ: p 1 2 H = 2 Q(Iout + Q) cos Ψ[n0k + n2k (Iout + 2Q)] + 3nnl Q(Iout + Q) + nnl Iout . (5.31) 2 2 The boundary conditions are satisfied for H = 1/2nnl Iout when Q(z) and Ψ(z) are related by the equation: p −3nnl Q(Iout + Q) . (5.32) cos Ψ = 2[n0k + n2k (Iout + 2Q)] Coupled mode system given by Eq. (5.28) can be written in a single equation either for Q(z) or for Ψ(z) by using Eq. (5.32). Such a reduced equation depends on several parameters: nnl , n0k , n2k and Iout and we consider different possibilities separately. 30 5.5.1 Balanced nonlinearity management: nnl = 0 From Eq. (5.32) follows that cos Ψ = 0 whenever nnl = 0, meaning that Ψ = ±π/2. At the right end of the structure the boundary conditions Q(L) = 0 and Q(Z) ≥ 0 for Eq. (5.28) must be satisfied. The sign of dQ/dZ at Z = L depends only on the signs of n0k + n2k Iout and cos Ψ at Z = L. As the slope must be negative at the right boundary, the phase factor needs to satisfy the following boundary conditions: π , if n0k + n2k Iout ≥ 0, 2 π Ψ(z) = − , if n0k + n2k Iout < 0. 2 Ψ(z) = (5.33) We may assume without loss of generality that the first layer is focusing and the second layer is defocusing - i.e. nnl1 > 0 and nnl2 < 0. In this assumption, according to Eq.(A.7) n2k always has a positive value. Now we can interpret negative values of n0k as out of phase matching, and positive values of n0k as in-phase matching between linear and Kerr nonlinear refractive indices. Slightly differently from analyzes of Pelinovsky [14] we consider two cases defined in Eq. (5.33) separately. The periodic structure with zero net average Kerr nonlinearity is now analytically tractable. Solution for n0k ≤ n2k Iout and Ψ(Z) = π 2 The coupled mode system Eq. (5.28) is reduced to the ordinary differential equation for Q: p dQ (5.34) = −2 Q(Iout + Q)[n0k + n2k (Iout + 2Q)]. dZ Now we can separate the variables: dQ p = −4n2k dZ, Q(Iout + Q)(Q + a) (5.35) where we have denoted a = ( nn0k + Iout )/2. 2k Direct integration of Eq. (5.35) gives: 1 p a(Iout − a) sin −1 L (Iout − 2a)(Q + a) + 2a(a − Iout ) = −4n2k (L − Z) Iout |Q + a| Q(Z) (5.36) Now we can acquire the analytical solution for reflected intensity Q(z): Q(z) = where θ = p Iout (n0k + n2k Iout ) sin2 θ , n2k Iout cos 2θ − n0k 2 n22k Iout − n20k (L − Z). 31 (5.37) Q(z) is defined if the denominator of Eq. (5.37) is nonzero. Output intensity Iout needs to satisfy the condition Iout ≤ Ilim , where Ilim is the upper threshold for output-intensity and satisfies the equation: q n0k 2 ≤ 1. (5.38) −1 ≤ cos (2 n22k Ilim − n20k L) = n2k Ilim It is seen from the Figure 5.3 below that the reflected intensity Q is monotonically decreasing between Z = 0 and Z = L as expected. −4 3 x 10 Reflected intensity 2.5 2 1.5 1 0.5 0 0 5 10 Z 15 20 Figure 5.3: Z-dependence of reflected intensity through the structure (L=20), where nnl = 0, n0k = −0.02, n2k = 1 and Iout = 0.025 Solution for n0k > n2k Iout and Ψ(Z) = −π/2 The coupled mode system given by Eq. (5.28) is now reduced to: p dQ = 2 Q(Iout + Q)[n0k + n2k (Iout + 2Q)], dZ (5.39) and the solution has a generalized form: Q(z) = Iout (n0k + n2k Iout ) sinh2 φ , n0k − n2k Iout cosh 2φ (5.40) p 2 (L − Z). With the change of variables Φ = −iθ the where φ = −iθ = n20k − n22k Iout solution (5.40) reduces to the solution (5.37) for n0k ≤ n2k Iout . 32 Transmission characteristics Transmission curves for balanced nonlinearity management can be calculated either analytically (using the Matlab program in Appendix D.1) or by general numerical algorithm (described in Section 5.6 and D.2). 0.03 0.025 n =−0.02 0k n =0 0k n0k=0.02 Iout 0.02 0.015 0.01 0.005 0 0 0.01 0.02 0.03 0.04 0.05 Iin Figure 5.4: Transmission curves for different linear gratings where nnl = 0, n2k = 1 and L = 50 1 Iout/Ilim 0.8 L=20 L=50 L=200 0.6 0.4 0.2 0 0 0.5 1 Iin/Ilim 1.5 2 2.5 Figure 5.5: Transmission curves for different grating lengths. nnl = 0 n0k = −0.02, and n2k = 1 In-phase gratings (lower curves in Figure 5.4) exhibit simple all-optical limiting, while for out of phase gratings (curve for n0k = −0.02) the dependence Iout (Iin ) exhibits the S-shaped profile, which is still one-to-one function and is bounded by its limiting value Ilim . 33 Transmission properties also depend on the grating length L. As it can be seen from the Figure 5.5, case L = 20 is suitable for optical limiting. When the structure is sufficiently long (e.g. L=200), the S-shaped transmission curve becomes a simple two-step map between the lower-transmissive and higher-transmissive limiting state and is suitable for logic operations. Generally bistability never occurs in the limit nnl = 0. 5.5.2 Unbalanced nonlinearity management: nnl 6= 0 Combining (5.28) and (5.32), we obtain a single differential equation for reflected intensity Q(Z): 2 dQ = Q(Iout + Q)(4[n0k + n2k (Iout + 2Q)]2 − 9n2nl Q(Iout + Q)). (5.41) dZ Explicit solutions to Eq. (5.41) are given in terms of elliptic functions [16] and can be analyzed numerically. The condition for the limiting behavior, i.e. Q(0) → ∞ for Iout → Ilim < ∞, can be found from the connecting relation (5.32). True all-optical limiting regime exists when the condition | cos Ψ| ≤ 1 is satisfied, i.e when n2k ≥ 3nnl 4 (5.42) Otherwise, no limiting regime is possible and the system is either bistable or multi-stable. Transmission characteristics Curves, which fit to the domain given by Eq. (5.42), show limiting behaviour, while curves which violate it, exhibit multi-stable behaviour (illustrated in Figure 5.6). For sufficiently strong out-of-phase gratings which satisfy the condition (5.42), multistability takes place in low and high intensities, but transmission curve still has a limiting value (n0k = −0.04). This behavior is illustrated on Figure 5.7 and is called locally multistable limiting. It occurs for n0k ≤ −0.03 Generally for nnl 6= 0 true all-optical limiting is supported within domain given by Eq. (5.42) by the in-phase and weakly out of phase gratings [15]. 5.5.3 Stationary transmission regimes As a conclusion, the described system has three types of stationary transmission regimes: • Stable limiting regime, which is uniform for all pertinent incident intensities. Transmission curve Iout (Iin ) is a one-to-one function. Occurs for gratings with balanced nonlinearity management (nnl = 0) and unbalanced nonlinear gratings (nnl 6= 0) in the domain (5.42) for in-phase or weak out-of-phase gratings [15, 13]. 34 0.4 III 0.35 Transmitted intensity 0.3 0.25 II 0.2 Ib 0.15 0.1 Ia 0.05 0 0 0.2 0.4 0.6 Incident intensity 0.8 Figure 5.6: Transmission curves with parameters: L = 20, n2k = 1, and (Ia ) nnl = 1, n0k = 0, (Ib ) nnl = 0, n0k = −0.15, (II) nnl = 1, n0k = −0.15, and (III) nnl = 1.4, n0k = 0. 0.08 0.07 n0k=−0.04 Transmitted intensity 0.06 0.05 0.04 n0k=−0.02 0.03 0.02 n0k=0 0.01 0 0 n0k=0.02 0.02 0.04 0.06 Incident intensity 0.08 Figure 5.7: Unbalanced nonlinearity management with different linear gratings, where nnl = 1, n2k = 1 and L = 50. n0k = −0.04 illustrates locally multi-stable limiting. 35 • Locally multi-stable limiting, which exhibits branching for small and intermediate values of Iin , but is bounded by the limiting value of Iout for large values of Iin . Occurs in strongly out of phase unbalanced nonlinear gratings where the constraint (5.42) is still held. • Multi-stable regime has several stationary transmission regimes for the same value of incident intensity Iin . Occurs in strongly unbalanced nonlinear gratings when constraint (5.42) is violated. 5.6 Numerical method Backward finite difference scheme was used to find the time independent solutions of E± (Z) of the coupled mode system, starting from the boundary conditions at the right end Z = L and iterating back to the left end Z = 0 [15]. The complex valued stationary coupled mode system for E± (Z) can be decomposed into the real and imaginary parts by substituting: E+ = u + iω, E− = v + iy (5.43) The coupled mode system can now be written in terms of real functions u, ω, v and y in a form: ∂u + n0k y + fu ∂z ∂y + n0k u + fy − ∂z ∂ω − + n0k v + fω ∂z ∂v + n0k ω + fv ∂z = 0 (5.44) = 0 (5.45) = 0 (5.46) = 0, (5.47) where the nonlinear functions are given by: fu = nnl (u2 + ω 2 + 2v 2 + 2y 2)ω + n2k [(u2 + 3ω 2 + v 2 + y 2)y + 2uωv] fy = nnl (2u2 + 2ω 2 + v 2 + y 2)v + n2k [(u2 + ω 2 + 3v 2 + y 2)u + 2ωvy] fω = nnl (u2 + ω 2 + 2v 2 + 2y 2 )u + n2k [(3u2 + ω 2 + v 2 + y 2)v + 2uωy] fv = nnl (2u2 + 2ω 2 + v 2 + y 2 )y + n2k [(u2 + ω 2 + v 2 + 3y 2)ω + 2uvy] (5.48) (5.49) (5.50) (5.51) The functions u, ω, v, and y are calculated on the grid points of spatial coordinate z: z = zn = n δz, n = 0, 1, ..., N, (N + 1) (5.52) where δz = L/(N + 1) is the space step size and the grid has N interior points and two end points at z = 0 and z = L = (N + 1)δz. 36 The boundary conditions at the right end of the structure (z = L) are given by: p u(L) = Iout , ω(L) = 0 (5.53) v(L) = 0, y(L) = 0 (5.54) Now the coupled mode system can be solved by iteration method (program code is given in Appendix D.2). [15] 37 Chapter 6 Transmission properties of nanoparticles (CdSe) dispersed photopolymer hologram 6.1 Introduction In this Chapter we analyze the transmission properties of a photopolymer material that is dispersed with semiconductor nanoparticles (CdSe - cadmium selenide), reported by Tomita et. al. [20, 22, 18, 19], which is holographically illuminated by Nd:YAG laser to form a stable structure with periodical variation in its index of refraction. The wave dynamics in this structure is analogous to a photonic crystal and could be analyzed by the computational model of Pelinovsky, described in Ref. [14] (see Chapter 5). 6.2 6.2.1 Holography with photopolymers Grating formation A conventional photopolymer (without nanoparticles) consists of mobile light-sensitive monomers, immobile polymers, and light-insensitive immobile binders. During the holographic exposure monomers are inhomogeneously polymerized depending on the intensity pattern of the incident light. In bright regions monomers polymerize more intensively, leading to the density gradient of monomers between the bright and dark regions. As a result monomers diffuse from the dark to the bright regions. Index grating (a hologram) is formed by periodical changes in monomer concentration (transient grating) and polymer concentration (permanent grating). Photopolymers with radical photopolymerization have high refractive index contrast (∆n ≈ 10−2 ), which is determined by the difference of refractive indices between un-reacted and reacted components, but at the same time it possesses undesired large photopolymerization shrinkage (∼ 10%). 38 If we added secondary mobile but photo-insensitive component (i.e., inorganic nanoparticles) to monomers, ∆n is considerably increased [18, 19, 20, 22]. Now, in addition to the above-mentioned photopolymerization process, photo-insensitive mobile nanoparticles would diffuse from the dark to the bright regions due to the chemical potential difference of monomers and nanoparticles in space[19]. Due to a large difference in refractive indices of polymers and nanoparticles (where we assume high contrast materials), a high-contrast grating (a hologram) can be formed. In Figure 6.2 blue squares represent mobile nanoparticles. Spatial mass-transfer of both monomers and nanoparticles during holographic exposure is responsible for the observed grating formation. Figure 6.1: Formation of a hologram in conventional photopolymer structure Figure 6.2: Formation of a hologram in nanoparticles-dispersed photopolymer structure Since nanoparticles such as II-VI semiconductor (e.g., CdSe) quantum dots with large third-order nonlinearities (optical Kerr effect) are employed, then the hologram would be nonlinear - i.e., its background and spatially modulated indices of refraction have optical Kerr nonlinearities[21]. The functional form of such linear and nonlinear refractive indices will depend on the density distributions of polymers and nanoparticles after holographic exposure [7]. In the following calculations it is assumed that the index modulation has sinusoidal form (instead of its distorted form) for simplicity. Such a structure may be described by the computational model of Pelinovsky [14]. 6.2.2 Applications Holographic recording using photopolymers is well known for its high refractive-index contrast (∆n ≈ 10−2 ) and high recording sensitivity. Holography with the usage of pho39 topolymers has found many important applications in photonics, including holographic optical elements, holographic memories, optical interconnects, narrow-band optical filters, waveguides, electrically switchable Bragg gratings, and photonic crystals [21]. Inclusion of inorganic nanoparticles as functional components for holographic applications would increase the index contrast further (∆n ≥ 10−1 ). This allows us to use smaller writing-intensities, the suppression of polymerization shrinkage, the increased dimensional stability, and the addition of new functionalities to photopolymers (e.g., nonlinearities)[21]. 6.3 Mathematical model The system of the photopolymer hologram is mathematically identical to an optical device that has finite length and a number of alternating layers with different linear refractive indices and different Kerr nonlinearities (the properties of such structures are described in Section 5.3). Total length of the structure is given by l = NΛ, where Λ is a grating period, and N is the number of layers. Using averaged approximation for the weak fields, the refractive index is given by: n(z, |E|2 ) = n0 (z) + nnl (z)|E|2 + O(|E|4) (6.1) where n0 (z) is a linear refractive index and nnl (z) is an intensity dependent index-modulation caused by Kerr nonlinearity. n0 (z) and nnl (z) are considered to be constant within each layer. Figure 6.3: Mathematical model of the nano-particles dispersed photopolymer hologram For two-layer model, the Fourier coefficients of (n(z, |E|2 )) can be evaluated from constant linear and nonlinear indices in each layer. Coupled mode system (5.25) (derived in Appendix B) is used to model the transmission properties of CdSe-dispersed photopolymer holograms. Since this structure does not have balanced nonlinearity management (nnl 6= 0), the explicit solutions to coupled mode equations are given in terms of elliptic functions [16] and can be solved only numerically 40 (numerical model is described in Section 5.6 and the Matlab code is given in Appendix D.2). 6.4 Data for numerical experiments Second harmonic component of Nd:YAG laser (green light, λ =532nm) was used during hologram formation. Resultant grating spacing after holographic exposure is Λ = λm / cos q, where q is the half angle between two recording beams and λm = λ/nln is the recording wavelength in the medium. nln is average index of refraction and equals about 1.9 in the given material. By choosing the appropriate value for q, hologram can be written for both fundamental wavelength of Nd:YAG laser (λ = 1064nm) and second harmonic component (λ = 532nm). Corresponding grating spacing equals to either 0.28µm (for λ = 1064nm) or 0.14µm (for λ = 532 nm). Realistic grating lengths would be in the range of 30µm to few millimeters. For the study of transmission properties, intensities of 1, 10 and 100 MW/cm2 are considered. These values are experimentally obtainable by using high intensity light pulses from Q-switched Nd:YAG laser with pulse-width of about few nanoseconds. Since such a pulse width covers the whole structure, the pulse can be interpreted as quasi-continuous incident wave in the model. In CdSe-dispersed photopolymer holograms there are two types of index distributions: the diffusion-dominant case (R = 10) and the photopolymerization dominant case (R = 0.01). Parameter R is defined as the ratio of the diffusion rate to photopolymerization rate, i.e. large values R implies the dominance of diffusion of monomers and nanoparticles compared to the photopolymerization. R depends on material parameters and recording intensity (R decreases with the increase of recording intensity). Also we need to consider two different cases for the values of refractive indices, because the index of refraction of CdSe nanoparticles has been reported to be either 2.55 or 2.728 in the green. In following numerical calculations we consider all four combinations of R and nCdSe . Distributions of monomers, polymers, and nanoparticles were calculated by mutual diffusion model [7]. Normalized parameters The nonlinear coupled mode equations, given by Eq. (5.25) are given in normalized space and time variables (Z and T ). Linear uniform refractive index nln only appears in normalized time variable T = ω0 t/nln , so that nln influences only time dependent dynamics of the system, but stationary solutions are nln -independent. Spatial variable Z is also normalized as Z = zω0 /c (see Appendix B) and the normalized grating length L is related to grating length l by L = lω0 /c = 2πl/λ = kl, where ω0 is the light frequency, λ is the optical wavelength and k is the wave-number of the incident light. To match with the realistic grating lengths defined above, we need to consider normalized grating lengths within the interval 200 - 6000 (in this analyzes we have considered the cases L = 200 and L = 500). 41 As shown in Section 5.4, nonlinear refractive indices and intensities (I) are normalized with respect to the reference intensity I0 = 100MW/cm2 . Note that the following relations are used: nnl = II0 (n2 )nl and n2k = II0 (n2 )2k , where (n2 )nl and (n2 )2k are average and periodic components of nonlinear index n2 . In numerical experiments normalized intensities of I = 0.01, I = 0.1 and I = 1, are used (corresponding to actual intensities of 1, 10 and 100 MW/cm2 , respectively). 6.5 6.5.1 Results for nCdSe = 2.55 Diffusion dominant case (R=10) n0k = 0.10402 2 nonlinear indices in cm /W I0 n2 (n2 )nl 8.0608 × 10−10 0.080608 (n2 )2k 2.5813 × 10−10 0.025813 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.5 1 Incident intensity 1.5 Figure 6.4: Multi-stable transmission for L = 200. See Figs. 6.5 and 6.6 for the behavior near the origin of the transmitted intensity. 42 −17 7 x 10 Transmitted intensity 6 5 4 3 2 1 0 0 0.5 1 Incident intensity 1.5 Figure 6.5: The first multi-stable branch for low output intensities (part of the Fig. 6.4). −14 5 x 10 Transmitted intensity 4 3 2 1 0 0 0.02 0.04 0.06 Incident intensity 0.08 0.1 Figure 6.6: The second multi-stable branch for low output intensities (part of the Fig. 6.4). We can see, that multistability occurs for input intensities as low as Iin = 0.01 43 2.5 I f I b Intensity 2 1.5 1 0.5 0 0 50 100 Distance (Z) 150 200 Figure 6.7: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the grating. L = 200, Iin = 1, Iout = 0.986. The values correspond to the positive slope of multi-stable curve. The light transmission at high transmissivity exhibits high amplitude oscillations inside the structure. 1.8 I f I 1.6 b 1.4 Intensity 1.2 1 0.8 0.6 0.4 0.2 0 0 50 100 Distance (Z) 150 200 Figure 6.8: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the grating. L = 200, Iin = 1, Iout = 0.448. The values correspond to the negative slope of multi-stable curve and give unstable solutions. 44 1 Intensity 0.8 0.6 0.4 0.2 0 0 50 100 Distance (Z) 150 200 Figure 6.9: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the grating at lower intensity transmissive state. The graphs of If and Ib are overlapping in the figure (Ib ≈ If ). L = 200,Iin = 0.1 and Iout = 2.8 · 10−18 . 1.6 1.4 If Ib 1.2 Intensity 1 0.8 0.6 0.4 0.2 0 0 50 100 Distance (Z) 150 200 Figure 6.10: Intensity of the forward- and backward propagating waves inside the grating for L = 200, Iin = 0.1, and Iout = 0.0155. Oscillations occur at higher intensity transmissive state 45 Longer gratings 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.5 1 Incident intensity 1.5 Figure 6.11: Multi-stable transmission in longer gratings — L = 500. Generally longer gratings have higher density of multi-stable branches. −22 Transmitted intensity x 10 4 3 2 1 0 0 0.5 1 Incident intensity 1.5 Figure 6.12: Multi-stable transmission for L = 500 at low output intensities (part of Fig. 6.11). Multistability occurs at lower output intensities in longer gratings. 46 6.5.2 Photopolymerization dominant case (R=0.01) n0k = 0.024955 nonlinear indices in cm2 /W I0 n2 (n2 )nl 8.0958 × 10−10 0.080958 (n2 )2k 6.1929 × 10−11 0.0061929 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 Incident intensity 0.8 1 Figure 6.13: Multi-stable transmission for L = 200. See Fig. 6.14 for the behavior near the origin of the transmitted intensity. 47 −3 1 x 10 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.05 0.1 0.15 Incident intensity 0.2 0.25 Figure 6.14: Multi-stable transmission for L = 200 at low output intensities (part of the Fig. 6.13). Multistability doesn’t occur at Iin = 0.01 (compare with Fig. 6.6 for diffusion dominant gratings where such behavior is observable). 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 Incident intensity 0.8 1 Figure 6.15: Multi-stable transmission in longer grating (L = 500). Transmission curve has higher density of multistability branches compared to Fig. 6.13 for L = 200. Such effect is also observable in diffusion dominant gratings (see Figs. 6.4 and 6.11) 48 0.12 Intensity of the forward wave 0.1 0.08 0.06 0.04 0.02 0 0 50 100 Distance (Z) 150 200 Figure 6.16: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the grating for L = 200, Iin = 0.1, and Iout = 1.9 · 10−5 at low intensity transmissive state. The graphs of If and Ib are overlapping in the figure (Ib ≈ If ) and the structure transmits only a very small fraction of incident light. Intensity of the forward wave 0.25 0.2 0.15 0.1 0.05 0 0 50 100 Distance (Z) 150 200 Figure 6.17: Intensities of forward- (If ) and backward (Ib ) propagating waves inside the grating for L = 200, Iin = 0.1, and Iout = 3.1 · 10−4 at first multi-stable branch state. The graphs of If and Ib are overlapping in the figure (Ib ≈ If ). 49 6.6 Analyzes for nCdSe = 2.728 The behaviour is qualitatively identical to the case nCdSe = 2.55. In this Section only transmission figures are plotted, discussions are analogical to previous Section. 6.6.1 Diffusion dominant case (R=10) n0k = 0.1233 nonlinear indices in cm2 /W I0 n2 (n2 )nl 7.0358 × 10−10 0.070358 (n2 )2k 2.253 × 10−10 0.02253 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.5 1 Incident intensity 1.5 Figure 6.18: Multi-stable transmission for L = 200. See Figs. 6.19 and 6.20 for the behavior near the origin of the transmitted intensity 50 −8 1 x 10 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 Incident intensity 2 Figure 6.19: Multi-stable transmission for L = 200 at first multistability branch (part of the Fig 6.18). −6 3 x 10 Transmitted intensity 2.5 2 1.5 1 0.5 0 0 0.02 0.04 0.06 Incident intensity 0.08 0.1 Figure 6.20: Multi-stable transmission for L = 200 at low output intensities (part of the Fig 6.18). Multistability is not achievable for Iin = 0.01 for such index distribution. 51 6.6.2 Photopolymerization dominant case (R=0.01) n0k = 0.029582 nonlinear indices in cm2 /W I0 n2 (n2 )nl 7.0664 × 10−10 0.070664 (n2 )2k 5.4054 × 10−11 0.0054054 1 Transmitted intensity 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 Incident intensity 1 1.2 Figure 6.21: Multi-stable transmission for L = 200. See Figs. 6.22 and 6.23 for the behavior near the origin of the transmitted intensity. 52 −4 Transmitted intensity x 10 2 1 0 0 0.1 0.2 0.3 Incident intensity 0.4 Figure 6.22: Multi-stable transmission for photopolymerization dominant grating with nCdSe = 2.55, L = 200 at low output intensities (part of the Fig. 6.21). 0.01 Transmitted intensity 0.008 0.006 0.004 0.002 0 0 0.02 0.04 0.06 Incident intensity 0.08 0.1 Figure 6.23: Multi-stable transmission for photopolymerization dominant grating with nCdSe = 2.55, L = 200 at low output intensities (part of the Fig. 6.21). 6.7 Discussions All the analyzed structures clearly exhibit globally multi-stable behaviour. This is in good agreement with analytical model described in Ref. [14]. For nnl 6= 0 the limiting behavior is possible if the condition 3|nnl | n2k ≥ (6.2) 4 is satisfied. 53 Evaluating n2k and nnl from Sections 6.5 and 6.6 clearly shows that condition given by Eq. (6.2) is not satisfied for considered CdSe-dispersed photopolymer hologram with strongly unbalanced nonlinear grating. Stable limiting regime does not occur for these material parameters and this multi-stable behaviour is also predictable by the analytical model. We have shown that given structure generates efficient multi-stabilities at very low intensities and long grating lengths that are practically achievable. During wave transmission through the grating, intensities of forward- and backward propagating waves are monotonic functions at lowest transmissive state of multi-stable transmission curve. but show oscillatory behaviour at higher transmissive states. Such behaviour could be usable for bistable or multi-stable switching, all-optical logic, or memory operations. However, we have not examined their stabilities yet and some of the transmission regimes might be unstable, leading to chaotic behaviors. Only full stability analyzes would show whether observed light transmission does indeed survive under real-life disturbances. We found that longer grating lengths lead to increased number of multistability branches in input-output power relationship. Also multistability occurs at smaller input intensities. In this work we have analyzed only shorter gratings (normalized lengths of 200 and 500), but in practice grating lengths up to few millimeters (L = 6000) is achievable. Such long gratings should be analyzed in future. My experience has shown that these computation require very high accuracy and longer computation time. Stable all-optical limiting can’t be achieved in given material without changing the material parameters and index distributions, namely, we should increase n2k compared to nnl (condition given by Eq. (6.2) must be satisfied). Index distributions are generated during holographic exposure and resultant grating is characterized by the parameter R, as described in Section 6.4. Since R depends on hologram recording intensity and grating spacing, it can be varied during hologram writing and it is thus possible to engineer the material in order to achieve the behaviour of interest. Finally it needs to be mentioned that actually the nonlinear uniform index of refraction influences the Bragg-matching condition - i.e. the Bragg-matching frequency depends on the average index of refraction of a periodic structure. In previously described mathematical model, which we used, the Bragg-mismatch term ”∆k” has been neglected in the coupled-wave equations, as the nonlinear refractive index is assumed to be much smaller than nonlinear refractive index. Since in considered structure the values for average linear refractive indices are between 1.9 and 2, while the maximum value for (n2 )nl I0 is about 0.08, nonlinear index is about 4% of the linear index. It might slightly affect the result because the value of nonlinear index is more than 1% of linear index. 54 Chapter 7 Summary We have studied the coherent light transmission in Kerr nonlinear periodic structures and analyzed light transmission properties in stationary regime — true all optical limiting and multi-stable transmission. True all-optical limiting is best achieved when the Kerr nonlinearity is compensated exactly across the alternating layers, i.e., when the net average nonlinearity is zero. It can also be achieved for gratings with unbalanced nonlinearity management. Generally there are three factors that affect the performance of these optical structures: mismatch between linear and nonlinear refractive indices, the length of the structure, and net average nonlinearity. We have analyzed transmission properties of CdSe-nanoparticles dispersed photopolymer holograms and found that they exhibit global multistability in input-output power relationship, including lower intensities. Such structures can be used as optical switches or logic elements. Also we can model the material parameters during hologram recording in order to meet the conditions that are necessary for all-optical limiting behaviour. Further perspectives of this work include: (i) performing the stability analyzes in order to estimate the applicability of the structure in real life disturbances; and (ii) improving the numerical algorithm in order to study longer gratings (currently such calculations are very time-consuming). I have completed the task of numerically analyzing light transmission properties in CdSe-dispersed photopolymer holograms and performed the theoretical analyzes of nonlinear periodic structures and their applications in optical communication systems, which gives me sufficient background for further studies at this field. 55 Appendix A Derivation of the expression for index modulation If we consider the structure described on Figure 6.3, where the index modulation has a rectangular form, and make the assumption that the variations in the refractive index of the grating are much smaller than the average index and the spacial variations of indices n0 (z) and n2 (z) have the same spatial symmetry centered at Z = 0 (index modulation is an even function of Z), the index modulation can be described by the Fourier’ series of a square wave. General expression for index modulation is given by: n(Z, |E|2 ) = n0 (Z) + n2 (Z)|E|2 (A.1) Fourier expansion for even square wave with the grating period Λ is given by: ∞ ∞ 4 X cos (2m − 1)2πZ/Λ 4 X cos (2m − 1)kZ J(Z) = = π m=1 2m − 1 π j=1 2m − 1 (A.2) is the grating wave number. where k = 2π Λ Now the linear and nonlinear refractive indices can be written in a form: ( 4k+1 ; Λ, p ∈ Z n01 , if Z ∈ 4k−1 n01 + n02 n01 − n02 2 2 n0 (Z) = (A.3) + J(Z) = 2 2 n02 , elsewhere ( 4k+1 nnl1 , if Z ∈ 4k−1 ; λ, p ∈ Z nnl1 + nnl2 nnl1 − nnl2 2 2 n2 (Z) = + J(Z) = (A.4) 2 2 nnl2 , elsewhere Inserting these values to general index expression (A.1) and considering only m = 1 values of J(Z) gives: n(Z, |E|2 ) = nln + nnl |E|2 + 2n0k cos kZ + 2n2k |E|2 cos kZ 56 (A.5) where: n01 + n02 , 2 n01 − n02 = π nln = n0k nnl1 + nnl2 , 2 nnl1 − nnl2 = . π nnl = n2k 57 (A.6) (A.7) Appendix B Derivation of coupled mode equations Let us consider the nonlinear Maxwell’s wave equation (derived in Section 3.4) ∂ 2 E n2 (z, |E|2 ) ∂ 2 E − = 0, ∂z 2 c2 ∂t2 (B.1) with refractive index (derived in Appendix A): n(z, |E|2 ) = nln + 2n0k cos kz + nnl |E|2 + 2n2k |E|2 cos kz, (B.2) where k = 2π/Λ and Λ is the grating period. We assume that the electric field to consist of two counter-propagating components with slowly varying amplitudes: E(z, t) = E+ (z, t) exp[i(k0 z − ω0 t)] + E− (z, t) exp[i(k0 z + ω0 t)], (B.3) where ω0 = ck0 /nln is the frequency of incident light and k0 = 2πnln /λ is the corresponding wave number. Maxwell wave equation simplifies when the spatial variation of linear grating and the variations due to nonlinearity are small compared to the average index of refraction nln , i.e. |n0k |, |nnl ||E|2, |n2k ||E|2 ≪ nln . If we also assume that E± is slowly varying in space and time: |∂E± /∂z|, |∂E± /∂z| ≪ |E± |, then the amplitudes E± (z, t) satisfy the coupled-mode system near the resonance frequency (when the optical wavelength λ matches the period of the structure: λ = 2nln Λ). First let us find the derivatives: ∂2E ∂E+ ∂E− = (2ik0 − ko2 E+ )ei(k0 z−ω0 t) + (−2ik0 − k02 E− )ei(k0 z+ω0 t) + c.c. (B.4) 2 ∂z ∂z ∂z ∂E+ ∂E− ∂2E = (−2iω0 − ω02 E+ )ei(k0 z−ω0 t) + (−2iω0 − ω02 E− )ei(k0 z+ω0 t) + c.c. 2 ∂t ∂t ∂t Here we have neglected the second derivatives because of the slow variation approximation. We also approximate the expression for n2 : n2 ≈ n2ln + 2nln (n0k cos kz + nnl |E|2 + n2k |E|2 cos kz) 58 (B.5) The Bragg condition is satisfied if k = 2k0 . We now insert Eq.(B.4) and Eq.(B.5) to Eq.(B.1) and collect the terms with ei(k0 z−ω0 t) and ei(k0 z+ω0 t) . The coupled mode equations can be given in a form: ∂E+ ∂E+ i + n0k E− + nnl (|E+ |2 + 2|E− |2 )E+ (B.6) + ∂Z ∂T +n2k [(2|E+ |2 + |E− |2 )E− + E+2 E−∗ ] = 0 ∂E− ∂E− −i + n0k E+ + nnl (2|E+ |2 + |E− |2 )E− + − ∂Z ∂T +n2k [(|E+ |2 + 2|E− |2 )E+ + E−2 E+∗ ] = 0 where Z = ω0 z/c and T = ωt/nln are the normalized spatial coordinate and time parameter, respectively. For n2k = 0 and nnl = 0, the system represents coupled mode equations for linear Bragg gratings. 59 Appendix C Derivation of stationary coupled mode system The coherent light transmission through the photonic grating is stationary when the amplitudes E± do not depend on time, i.e. ∂E± /∂T = 0 in coupled mode equations (B.6). The coupled mode system can be written in terms of the intensity of the reflected wave Q(Z) and the phase mismatch Ψ(Z) when using the change of variables: p Iout + Qei(Φ−Ψ) , (C.1) E+ (z) = p iΦ E− (z) = Qe . Substituting (C.1) in the coupled mode system (B.6) gives: p ∂Q ∂Φ ∂Ψ p 1 Iout + Q + n0k QeiΨ + − − i √ ∂Z ∂Z 2 I + Q ∂Z p out p nnl (Iout + 3Q) Iout + Q + n2k Q[(2Iout + 3Q)eiΨ + (Iout + Q)e−iΨ )] = 0 p p 1 ∂Q p ∂Φ −i √ + Q + n0k Iout + Qe−iΨ + nnl Q(2Iout + 3Q) ∂Z 2 Q ∂Z p +n2k Iout + Q[(Iout + 3Q)e−iΨ + QeiΨ ] = 0 (C.2) (C.3) Imaginary part of (C.2) is: 1 p p ∂Q + n0k Q sin Ψ + n2k Q[(2Iout + 3Q) sin Ψ − (Iout + Q) sin Ψ] = 0 (C.4) 2 Iout + Q ∂Z √ that yields to differential equation for reflected intensity Q: p ∂Q = −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)] ∂Z 60 (C.5) The real part of Eqs. (C.2) and (C.3) give a system: s ∂Φ ∂Ψ Q − + = cos Ψ[n0k + n2k (3Iout + 4Q)] − nnl (Iout + 3Q), (C.6) ∂Z ∂Z Iout + Q s Iout + Q ∂Φ = − cos Ψ(n0k + n2k (Iout + 4Q)) − nnl (2Iout + 3Q). (C.7) ∂Z Q Which can be reduced to a single differential equation for Ψ in a form: ∂Ψ cos Ψ 2 [n0k (Iout + 2Q) + n2k (Iout + 8Iout Q + 8Q2 )]. (C.8) = −3nnl (Iout + 2Q) − p ∂Z Q(Iout + Q) Equations (C.5) and (C.8) form a system of stationary coupled mode system for Q and Ψ: p ∂Q (C.9) = −2 Q(Iout + Q) sin Ψ[n0k + n2k (Iout + 2Q)], ∂Z ∂Ψ cos Ψ 2 = −3nnl (Iout + 2Q) − p [n0k (Iout + 2Q) + n2k (Iout + 8Iout Q + 8Q2 )]. ∂Z Q(Iout + Q) 61 Appendix D Program listing D.1 Analytical algorithm for balanced nonlinearity management (nnl = 0) clear format short n_2k=1;%modulation of nonlinear refractive index n_0k=-0.02;%modulation of linear refractive index Z=0; L=50;%length of the structure I_out=linspace(0.0001,0.0283);%vector for output intensities %calculating transmission (input-output) relationship of n_0k for k=1:100 if(abs(n_0k)<=n_2k*I_out) I_in(k)=(I_out(k)*(n_0k+n_2k*I_out(k))*(sin(sqrt(n_2k^2*I_out(k)^2-n_0k^2)... *(L-Z)))^2)/(n_2k*I_out(k)*cos(2*sqrt(n_2k^2*I_out(k)^2-n_0k^2)*... (L-Z))-n_0k)+I_out(k); else I_in(k)=(I_out(k)*(n_0k+n_2k*I_out(k))*(sinh(sqrt(-n_2k^2*I_out(k)^2+n_0k^2)... *(L-Z)))^2)/(-n_2k*I_out(k)*cosh(2*sqrt(-n_2k^2*I_out(k)^2+n_0k^2)*... (L-Z))+n_0k)+I_out(k); end end plot(I_in,I_out) xlabel(’I_{in}’) ylabel(’I_{out}’) 62 D.2 General numerical algorithm based on backward finite difference scheme clear L =100; n_0k= 0.024955; %modulation of linear refractive index n_2k = 0.0061929; %modulation of nonlinear refractive index n_nl = 0.080958; %average nonlinear index N = 400; %number of mesh points for the grating K =400; %number of points for output intensity I_start =0; %the starting value for transmitted intensity (|A_{+}(0)|^2) I_fin =1; %output intensity (|A_{+}(L)|^2) dz = L/(N + 1); % the space step size z = 0 : dz : L; % the vector for the grid points between z = 0 and z = L dI = (I_fin-I_start)/K; %transmitted intensity step size I_out = (I_start + dI) : dI : I_fin; %the vector for transmitted intensities % boundary conditions at the right boundary (output intensity) u(N+2,:) = sqrt(I_out); w(N+2,:) = zeros(1,K); v(N+2,:) = zeros(1,K); y(N+2,:) = zeros(1,K); for n = (N+1):-1:1 uu = u(n+1,:); ww = w(n+1,:); vv = v(n+1,:); yy = y(n+1,:); % steady-state coupled mode equations: u(n,:) = uu + dz*( n_0k*yy + n_nl*(uu.^2+ww.^2+2*vv.^2+2*yy.^2).*ww n_2k*((uu.^2+3*ww.^2+vv.^2+yy.^2).*yy+2*uu.*ww.*vv)); w(n,:) = ww - dz*( n_0k*vv + n_nl*(uu.^2+ww.^2+2*vv.^2+2*yy.^2).*uu n_2k*((3*uu.^2+ww.^2+vv.^2+yy.^2).*vv+2*uu.*ww.*yy)); v(n,:) = vv - dz*( n_0k*ww + n_nl*(2*uu.^2+2*ww.^2+vv.^2+yy.^2).*yy n_2k*((uu.^2+ww.^2+vv.^2+3*yy.^2).*ww+2*uu.*vv.*yy)); y(n,:) = yy + dz*( n_0k*uu + n_nl*(2*uu.^2+2*ww.^2+vv.^2+yy.^2).*vv n_2k*((uu.^2+ww.^2+3*vv.^2+yy.^2).*uu+2*ww.*vv.*yy)); end I_in = u(1,:).^2 + w(1,:).^2; x=linspace(0,1); plot(I_in,I_out,’k’,x,x,’k-.’); xlabel(’Incident intensity’) ylabel(’Transmitted intensity’) 63 + + + + Bibliography [1] D. 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