Time and Frequency Evolution of. the. Precursors in Dispersive Media and. their Applications

Transcription

1 Time and Frequency Evolution of the Precursors in Dispersive Media and their Applications by Reza Safian Department of Electrical and Computer Engineering University of Toronto Copyright 2008 c by Reza Safian

2 Abstract Time and Frequency Evolution of the Precursors in Dispersive Media and their Applications Reza Safian PhD Electrical and Computer Engineering University of Toronto Until now, few rigorous studies of the precursors in structures exhibiting superluminal group velocities have been performed. One dimensional photonic crystals (1DPC) and active Lorentzian media are among the ones which are able to exhibit superluminal propagation. In the first part of the thesis we have studied the evolution of the precursors in active Lorentzian media and 1DPC. The problem of the propagation of the precursors in active Lorentzian media is addressed, by employing the steepest descent method to provide a detailed description of the propagation of the pulse inside the dispersive medium in the time domain. The problem of the time and frequency evolution of the precursors in 1DPC is studied, using the finite-difference time-domain (FDTD) techniques in conjunction with joint time-frequency analysis (JTFA). Our study clearly shows that the precursor fields associated with superluminal pulse propagation travel at subluminal speeds. It is also shown that FDTD analysis and JTFA can be combined to study the dynamic evolution of the transient and steady state pulse propagation in dispersive media. The second part of the thesis concentrates on the applications of the precursors. An interesting property of the precursors is their lower than exponential attenuation rate inside a ii

3 lossy dielectric, such as water. This property of the precursors has made them an interesting candidate for applications such as ground penetrating radar and underwater communication. It was recently pointed out that a pulse which is generated inside of water and assumes the shape of the Brillouin precursor would be optimally suited for long range propagation in water (described by the single-pole Debye model). Here, we have considered the optimal pulse propagation problem, accounting for the interaction of the pulse with the air/water interface at oblique incidence. In addition, we argue that pulse excitations which are rough approximation of the Brillouin precursor will eventually evolve into the Brillouin precursor itself shortly after they enter water. Therefore, the excitation of a long-propagating pulse is not sensitive to its shape. Finally, we studied the performance of the optimized pulse in terms of the energy of the scattered field from an object inside water. Based on the simulation results the optimized pulse scattered field has higher energy compared to pulses with the same energy and different temporal distribution. The FDTD technique is employed in all the simulations. iii

4 Acknowledgements I would like to express my deep and sincere gratitude to my supervisors, professor Mojahedi and professor Sarris. Their wide knowledge and logical way of thinking have been of great value for me. Their understanding, encouraging and personal guidance have provided a good basis for the present thesis. During this work I have collaborated with many colleagues for whom I have great regard, and I wish to extend my warmest thanks to all those who have helped me with my work in the electromagnetic group. I owe my loving thanks to my wife. She has lost a lot due to my research abroad. Without her encouragement and understanding it would have been impossible for me to finish this work. My special gratitude is due to my parents and their loving support. iv

5 Contents 1 Introduction History of the Precursors in Dispersive Media Physical Interpretation of the Precursors Information Velocity and Superluminal Velocity The Low Attenuation Rate of the Precursors Thesis Objectives Precursors and Superluminal Velocities Applications of the Precursors Thesis Structure Analytical and Numerical Methods used in Calculation of the Precursors in Dispersive Media Steepest Descent Method [30] Finite Difference Time Domain Method [66] Dispersion Implementation in FDTD Perfectly Matched Layer in FDTD Joint Time-Frequency Analysis (JTFA) [74] v

6 3 Precursor Fields in One Dimensional Photonic Crystal Introduction Ultra Short Modulated Gaussian Pulse Propagating in a Single Resonance Lorentzian Medium Time and Frequency Evolution of the Precursors in a One Dimensional Photonic Crystal Summary and Conclusions Asymptotic Description of Wave Propagation in Active Lorentzian Medium Introduction Lorentzian Medium with Inverted Atomic Population Superluminal Propagation in Active Lorentzian Medium Plane Wave Propagation in Dispersive Media: An Integral Description Asymptotic Analysis of the Pulse Propagation in Active Lorentzian Medium Evolution of X(ω, θ) in the Complex ω Plane Numerical Determination of the Location of the Saddle Points Asymptotic Calculation of the Field A(z, θ) Comparison of the Passive and Active Lorentzian Summary and Discussion On the Optimization of Electromagnetic Pulse Generation for Low Attenuation Rate Inside Water Introduction vi

7 5.2 The Effects of the Shape of the Excitation on Attenuation Properties of the Double Brillouin Pulse Propagation of the Double Brillouin pulse at Oblique Incidence Summary and Conclusion Scattering of the Double Brillouin Pulse from an Object Inside Water Introduction Problem Statement and Computational Modeling Comparing the Scattering of Double Brillouin Pulse from Dielectric Targets with Modulated Rectangular and Gaussian Pulses Comparing the Scattering of the Double Brillouin Pulse with Sinusoidal Pulses Summary and Discussion Conclusions Summary of the Thesis Future Work Publications vii

8 List of Figures 1.1 Lorentz oscillator model of a dielectric The complex refractive index n(ω): (a) Real part of n(ω), and (b) imaginary part of n(ω). The boxes denote the anomalous dispersion region. The medium parameters are: ω 0 = , ω p = , δ = , which are the same as the parameters used in Sommerfeld and Brillouin analysis [9] Generation of the precursors as the transient part of the transmitted field in the passive Lorentzian medium Schematic presentation of the interaction of light with matter Fast-light pulse propagation through a vacuum (solid-line), and through a medium (dashed line) [43] Reshaping of the pulse in dispersive media Detection in a superluminal channel. The dashed line shows the signal which has traveled length L through dispersive medium and the solid line shows the signal which has traveled length L through free space Attenuation of the Brillouin precursor compared to the exponential attenuation [60] A typical example of a first order saddle point viii

9 2.2 A typical example of (a) first order saddle point (b) second order saddle point. The arrows show the direction of the steepest descent path at each saddle point (a) Time distribution (b) joint time-frequency distribution (c) frequency distribution of the modulated Gaussian pulse after propagating 0.5µm through the dispersive Lorentzian medium. (θ = θ ct 0 /z) (a) Time distribution (b) joint time-frequency distribution (c) frequency distribution of the modulated Gaussian pulse after propagating 1µm through the dispersive Lorentzian medium. (θ = θ ct 0 /z) The physical structure of the 5-slab 1DPC (d s = 1.27 cm, d a = 4.1 cm, n s is the refractive index of the slabs) Pulse advancement in the 5-slab 1DPC Wigner-Ville distribution of the output pulse with a smooth front propagated in the 5-slab 1DPC (a) Envelope of the modulated Gaussian excitation with (E 2 (t)) and without (E 1 (t)) enforced front. (b) Input reflection coefficient (S 11 ) of the 5-slab 1DPC and frequency distribution of the excitations with (E 2 (f)) an without(e 1 (f)) enforced front Wigner-Ville distribution of the output pulse with the enforced front excitation in the 5-slab 1DPC. Precursors appear in early part of the pulse. Both low and high frequencies are concurrently present in the precursors Normalized Gaussian pulse with enforced front at the output of the 1DPC compared to the same pulse that has traveled the same distance in free space ix

10 3.9 Propagation of the Gaussian pulse inside the 1DPC with different number of slabs The values of U for the precursor and the main part of the pulse for different number of slabs The real part of the refractive index for the two-level atomic medium in inverted state (solid line) and ground state (dashed line) Branch points and branch cuts for the active single resonance Lorentzian medium Pulse advancement in active medium at z = 1 µm (the time difference between the peaks of the pulse is t = ps) Contours of X(ω, θ) on the upper half of the complex ω-plane for θ = (Color online) Contours of X(ω, θ) on the upper half of the complex ω-plane for θ = Contours of X(ω, θ) on the upper half of the complex ω-plane for θ = Locations of the upper half plane saddle points as a function of θ Locations of the upper half plane saddle points as a function of θ (a) 1 < θ < θ 1, (b) θ > θ The part of the total sampled field 1µm inside the active medium due to the saddle points Total sampled field at 1µm inside the active medium for (a) 1 < θ < (θ = is an arbitrary value greater than θ 1 ); (b) < θ < 25 calculated using the steepest descent (SD) and inverse fast Fourier transform (IFFT) methods (δ = , ω 0 = Hz, ω p = Hz, ω c = Hz).. 62 x

11 5.1 Temporal structure of (a) single Brillouin pulse (SB) (b) Double Brillouin pulse (DB) The time distribution of the (a) Single Brillouin pulse(sb), (A is the one percent time bandwidth of the single Brillouin pulse) and (b) single trapezoid pulse (E 0 is the maximum amplitude of the pulse and A and B are the bottom and top lengths of the single trapezoid pulse, respectively) The time distribution of the (a) Single Brillouin pulse (SB) and single trapezoidal pulses with different amplitudes (b) Double Brillouin pulse (DB) and double trapezoidal pulses with different amplitudes (E 0 defines the maximum amplitude of the pulse) Frequency distribution of the Double Brillouin pulse (DB) and double trapezoidal pulses with different amplitudes The normalized peak amplitude of the double Brillouin and double trapezoidal pulses at consecutive observation points inside the dispersive medium (E 0 defines the maximum amplitude of the pulse and DB pulse is the double Brillouin pulse) Slope of the data presented in Fig. 5.5 (E 0 defines the maximum amplitude of the pulse and DB pulse is the double Brillouin pulse) Temporal evolution of the single Brillouin pulse (E 0 = 0.7) The sampled field inside water at z = 5z d for the double Brillouin pulse and double trapezoidal pulses with different amplitudes (E 0 defines the maximum amplitude of the pulse and DB pulse is the double Brillouin pulse) Two dimensional FDTD computational domain which models the infinite interface between the air and the Debye medium xi

12 5.10 The normalized peak amplitude of the double Brillouin pulse at consecutive observation points inside the dispersive medium for different incident angles as compared to the exponential decay (a) The absolute value of the transmission function for the interface between free space and water (normalized) (b) The phase of the transmission function for the interface between free space and water The normalized Fourier transform of the double Brillouin propagated one absorption depth inside water as the incident angle changes at the air-water interface and the Fourier transform of the double Brillouin pulse propagated in the infinite dispersive medium The normalized peak amplitude of the modified double Brillouin pulse at consecutive observation points inside the dispersive medium for different incident angles Receiver and transmitter configuration FDTD computational domain (a) The time distribution of the double Brillouin pulse (b) The frequency distribution of the double Brillouin pulse The time distribution of the (a) modulated Gaussian pulse (b) modulated rectangular pulse. (c) The frequency distribution of the Modulated Gaussian and rectangular pulses (normalized) (a), (b), and (c), energy norm E of the scattered field when the excitation is double Brillouin pulse for d = λ c, 5λ c, and 10λ c, respectively, for different sizes of the object. The object is a dielectric with ɛ r = xii

13 6.6 (a), (b), and (c), energy norm E of the scattered field when the excitation is modulated Gaussian pulse for d = λ c, 5λ c, and 10λ c, respectively, for different sizes of the object. The object is a dielectric with ɛ r = (a), (b), and (c), energy norm E of the scattered field when the excitation is modulated rectangular pulse for d = λ c, 5λ c, and 10λ c, respectively, for different sizes of the object. The object is a dielectric with ɛ r = Maximum of the energy norm E of the scattered field for different locations of the dielectric square target inside water, when the excitation is a) double Brillouin pulse b) modulated Gaussian pulse c) modulated rectangular pulse (a) Time distribution of the sinusoidal pulse (b) frequency distribution of the double Brillouin pulse and one of the sinusoidal pulses with carrier frequency at 420 MHz Maximum energy of the scattered pulse for double Brillouin and sinusoidal pulses for different sizes of the object. The object is a dielectric with ɛ r = 10. Each of the curves shows the maximum scattered energy of the sinusoidal pulse at different frequencies for a specific object location. Each of the cross signs shows the maximum scattered energy of the double Brillouin pulse for a specific location of the object (D/λ c ) which is shown as a subscript (DB 15, DB 20, DB 25, DB 30 ). 96 xiii

14 Chapter 1 Introduction 1.1 History of the Precursors in Dispersive Media Wave propagation through a linear, temporally dispersive medium has been a complex and sometimes controversial topic since the 19th century. The earliest research on the problem of wave propagation in dispersive media was carried out by Hamilton [1] in 1839 when the concept of the group velocity seems to have been first introduced. Subsequently, Rayleigh expanded the work by distinguishing and clarifying the concept of phase and group velocities [2, 3]. The turning point of the theory of the dispersive properties of dielectric media was the introduction of the classical models by Debye, Lorentz, and Drude [4 6]. Sommerfeld and Brillouin were also among the early researchers who continued this line of work by studying wave propagation in dispersive media [7 10]. They used the asymptotic method of steepest descent to describe the propagation of a unit step-modulated signal with a constant carrier frequency in a semi-infinite, single resonance, passive Lorentz medium. The purpose of their analysis was to find the velocity at which the signal propagated. Their analysis 1

15 led to the discovery of two wave phenomena that precede the arrival of the main signal in a dispersive medium, which they named forerunners or precursors. The first precursor (Sommerfeld precursor) determines the earliest-time behavior of the signal, and contains its highest frequency components. The second precursor (Brillouin precursor) contains the lowest frequency components of the signal. Sommerfeld also showed that the front of the pulse that is, the moment when the field first becomes nonzero propagates precisely at the speed of light in vacuum, c. He stressed that a very sensitive detector should be able to register the front of the pulse and hence measure a propagation speed of c for the signal, independent of the medium in which the signal is propagating. Subsequent to the work by Sommerfeld and Brillouin several other methods have been developed in order to describe the wave propagation in different dispersive media [11 16]. An excellent review of the methods of approximate solutions for one-dimensional wave propagation can be found in [17]. Oughstun and his colleagues have carried out extensive research generalizing and enhancing Sommerfeld and Brillouin s work [18 29]. Using modern asymptotic techniques [30, 31], they were able to improve significantly both the qualitative and quantitative description of this classical problem in both single and double resonance Lorentz media. Although Sommerfeld and Brillouin precursors were predicted as early as 1914, experimental observations of forerunners are few and only qualitative; mainly dealing with electromagnetic waves in a dielectric at microwave frequencies [32] or in water or GaAs at optical frequencies [33 35]. Recently, in a series of experiments, propagation of the precursors was claimed to be linked to the preservation of Einstein s causality during superluminal light pulse propagation in the region of anomalous dispersion of dielectrics [36]. Sommerfeld 2

16 Electron e - Spring p x Nucleus e + Microscopic Dipoles Figure 1.1: Lorentz oscillator model of a dielectric. forerunners have also been predicted in various dispersive media such as biological [37] or viscoelastic [38]. 1.2 Physical Interpretation of the Precursors What happens when an electromagnetic pulse propagates through a linear dispersive dielectric? Different frequency components interact with the medium differently, thereby experiencing different phase velocities or absorption; i.e. the pulse suffers from dispersion. The origin of the dispersion can be understood as follows (here we are only concerned with the temporal dispersion) ; i.e. the frequency dependency of the phase velocity and attenuation. Materials consist of electric dipoles, as shown in Figure 1.1, which can be treated approximately as classical harmonic oscillators. This is known as the Lorentz model of a dielectric (Lorentz oscillator) [39]. In Figure 1.1, an electron is bound to a fixed nucleus at a distance x by a force obeying Coloumb s Law (e is the absolute electron charge and p is the microscopic dipole moment). The distance x oscillates in time due to an external electric 3

17 field E( r, t). There is a restoring force which obeys Hook s law. The oscillation experiences damping characterized by a damping rate δ. These properties are quantified by the equation of motion, which is given by [40] d 2 x dx + 2δ dt2 dt + ω2 0 = e E( r, t). (1.1) m Assuming time harmonic fields (E( r, t) = E( re iωt )), the solution to Eq. (1.1) is given by x(ω) = (e/m)e( r, ω) ω 2 0 ω 2 i2δω. (1.2) The macroscopic polarization of the medium P ( r, ω) is obtained using Eq. (1.2) P ( r, ω) = Np( r, ω) = Nex(ω) = α p E( r, ω), (1.3) where N is the number of dipoles (atoms or molecules) per unit volume, α p is the complex electric polarizability and p( r, ω) = ex(ω) is the microscopic dipole moment. Comparing Eqs. (1.2) and (1.3) we can easily identify α p of the Lorentzian atom to be α p (ω) = Ne 2 /m ω 2 0 ω 2 i2δω. (1.4) The macroscopic polarization, P ( r, ω) is related to the electric displacement as D( r, ω) = ɛ(ω)e( r, ω) = ɛ 0 E( r, ω) + P ( r, ω). (1.5) From Eqs. 1.3 and 1.5, the complex refractive index n(ω) is given by n(ω) = ɛ(ω)/ɛ 0 = 1 + ω 2 p ω 2 0 ω 2 i2δω, (1.6) where, ω p = Ne 2 /m is the plasma frequency and ɛ(ω) is the dielectric constant. The real and imaginary part of the complex refractive index (n r (ω) and n i (ω)) are plotted in Figs. 4

18 3 2.5 (a) (b) Real(n(ω)) Imag(n(ω)) ω(hz 1e16) ω(hz 1e16) Figure 1.2: The complex refractive index n(ω): (a) Real part of n(ω), and (b) imaginary part of n(ω). The boxes denote the anomalous dispersion region. The medium parameters are: ω 0 = , ω p = , δ = , which are the same as the parameters used in Sommerfeld and Brillouin analysis [9]. 5

19 Incident field Transmitted field Dielectric Material Arrival of the first forerunner Arrival of the second forerunner Arrival of the main signal Figure 1.3: Generation of the precursors as the transient part of the transmitted field in the passive Lorentzian medium. 1.2(a) and (b), respectively, for the set of parameters, ω 0 = , ω p = , and, δ = They are related to each other through the Kramers-Kronig relations as [41] ɛ r(ω) = π ɛ r (ω) = 2ω π where, n(ω) = ɛ r (ω) = ɛ (ω) iɛ (ω). 0 0 ω ɛ r (ω ) (ω ) 2 ω 2 dω (1.7) 1 ɛ r(ω ) (ω ) 2 ω 2 dω (1.8) The imaginary part n i (ω) is related to the absorption or gain of the wave as it propagates through the medium. On the other hand, the real part of the refractive index n r (ω) defines the dispersion of the medium. Figure 1.2(a) shows that n r (ω) is an increasing function of the frequency except for the region indicated by the dashed-line rectangle, where the slope is negative. The frequency range of negative slope is the so-called region of anomalous dispersion, and the region of positive slope is known as normal dispersion. Each spectral component of the initial pulse propagates through the dispersive medium with its own phase velocity, therefore the phase relationship between different spectral com- 6

20 ponents of the pulse changes as the pulse propagates inside the medium. Furthermore, each monochromatic component of the pulse is attenuated or amplified at its own rate. Therefore, the relative amplitude of different spectral components changes as well. One manifestation of these effects is the formation of the precursor fields preceding the main part of the pulse. Figure 1.3 shows an example of the generation of optical precursors in the passive Lorentzian medium [9,39,42]. Generally, when a sharp edge of a pulse propagates through a dispersive medium, the transmitted field has a transient signal followed by the main signal. The transient behavior consists of the precursors. There is another way to understand the origin of this transient behavior. Let us consider the transient interaction between light and matter as shown in Fig. (1.4). An incident pulse has a front where the field is turned on. When the pulse enters a dispersive medium, the incident field polarizes a collection of microscopic dipoles, thereby creating a macroscopic polarization. The macroscopic polarization generates a radiative field, which interferes with the incident light. The modified field interacts with the collection of dipoles again, and the above events are repeated. As a result, it takes a finite time for the steady-state polarization to build up. Hence, the medium can not respond to the instant turn-on of the front. Based on this argument, Sommerfeld predicted that the first part of a steep edge pulse passes through the medium as if the medium was an optical vacuum [9]. 1.3 Information Velocity and Superluminal Velocity Sommerfeld and Brillouin s classification of different wave velocities in terms of phase, group, energy, and precursor velocities continues to be the standard today. Group velocity (v g ) is 7

21 EM Field Microscopic Dipole p Macroscopic Polarization P Figure 1.4: Schematic presentation of the interaction of light with matter. the most controversial among these velocities. It is related to the real part of the complex refractive index at the carrier frequency according to [39] v g = c n r (ω c ) + ω c (dn r (ω c )/dω). (1.9) The assumption in the definition of the group velocity is for the signal to be narrowband in frequency domain. In the case that a signal has a broadband spectral distribution, depending on the dispersion of the medium, the definition for group velocity may or may not be valid [42]. 8

22 Figure 1.5: Fast-light pulse propagation through a vacuum (solid-line), and through a medium (dashed line) [43]. 9

23 As we see in the denominator of Eq. (1.9), v g < c for dn/dω > 0 where the medium is normally dispersive. However, it is possible for v g to be greater than c (superluminal) or negative for dn/dω < 0 in the anomalous dispersion region. The interpretation of these abnormal group velocities has triggered lengthy debates among scientists. An interesting point is that these abnormal velocities are realistic and they have been calculated and demonstrated in experiments at microwave frequencies [44 49], at optical frequencies [50 53], in the single-photon limit [54, 55], and even in media with a negative index of refraction [56, 57]. As an example, Fig. 1.5 shows the comparison of Gaussian pulses propagating through vacuum and a medium which supports superluminal group velocity [43] when the carrier frequency is set to the resonance frequency ω c = ω 0. The pulse advancement is about 27.4 ns compared to the pulse propagating through vacuum. At this point, one may ask the following question: Is relativistic causality, demanded by Einstein s theory of special relativity preserved in the case of superluminal group velocity? More specifically, has the information traveled faster than the speed of light in vacuum under these circumstances? Group velocity can be considered as the velocity of the peak of the pulse. An important point is that there is no causal connection between the peak of the input and output pulses. Because of dispersion, the shape of the pulse changes as it propagates through the medium and the peak shifts to earlier times. Figure 1.6 shows this pulse reshaping phenomenon inside a dispersive medium without any attenuation or amplification. The pulse which has traveled through the dispersive medium is compared with a pulse which has traveled the same length through free space. The peak of the reshaped pulse appears in the output sooner than the free space companion pulse. 10

24 Figure 1.6: Reshaping of the pulse in dispersive media. Detection Level (1) Through dispersive medium Through free space (2) (3) time Figure 1.7: Detection in a superluminal channel. The dashed line shows the signal which has traveled length L through dispersive medium and the solid line shows the signal which has traveled length L through free space. 11

25 Another important point is that the characteristics of the detector play and important role in the definition of information velocity. For example, consider a detection scheme based on pulse amplitude sensing, by a threshold detector. Figure 1.7 shows three different detection levels in the case that the group velocity is superluminal. If we consider level (1) or (2) as the detection level, the pulse is detected faster in a superluminal channel compared to free space. But, if we bring the detection level low enough (level (3)) to detect the front of the pulse, then both signals will be detected at the same time and the detection is luminal. Therefore, the superluminal detection of the pulse does not mean superluminal transfer of information. In conclusion, complete and precise knowledge of the evolution of all parts of the pulse (front, precursors, main part) helps us to define the information velocity more accurately. This motivates the study of the front and precursors in temporally and structurally dispersive media. Defining the information velocity is beyond the scope of this thesis and it is mentioned only as a motivation for studying precursors. 1.4 The Low Attenuation Rate of the Precursors The interesting characteristic of the precursors is that their time and frequency evolution is different from that of the steady state part of the pulse. The less than exponential decay rate of the precursors in Lorentzian or Debye media is an example of these properties [25,58,59]. For a Debye-type dielectric, the dynamical field evolution is eventually dominated by the Brillouin precursor as the pulse propagates inside the medium. The peak amplitude in the Brillouin precursor decays only as the square root of the inverse of the propagation distance. 12

26 Peak Amplitude z/z d Figure 1.8: Attenuation of the Brillouin precursor compared to the exponential attenuation [60]. This is shown in Fig. 1.8, which compares the attenuation of the Brillouin precursors inside water, modeled as a Debye medium, with an exponential attenuation. This property of the precursors makes them an interesting candidate for applications such as remote sensing and detection in dispersive media. Therefore, studying precursors is also motivated because of their application potentials. 1.5 Thesis Objectives The thesis has two main objectives. The first objective is to study the precursors in media which support superluminal group velocity. As explained in section 1.3, this part is motivated by the ambiguities about the information velocity. The second objective is about the application of the precursors. Specifically, studying the exploitation of the low attenuation rate of the precursors for long range propagation and remote sensing in Debye type media. 13

27 In the following the undergone studies are briefly explained Precursors and Superluminal Velocities As mentioned earlier, extensive research on the propagation of the precursors in passive Lorentzian media has been carried out [18 29]. Another example of a dispersive medium which can support superluminal propagation is active Lorentzian medium. The possibility of superluminal propagation in such a medium was first considered by Chiao [61, 62]. Furthermore, there has been experimental observation of superluminal velocities in active media [53, 63]. The part which is lacking is the calculation of the precursors in such a case. Here, we have calculated the evolution of the precursors in an active Lorentzian medium using the steepest descent method. A more complex dispersive structure which exhibits superluminal behavior is the one dimensional photonic crystal (1DPC). The 1DPC structure exhibits abnormal group velocities due to structural dispersion that is, dispersion due to geometrical features. In the microwave domain, Mojahedi et al. have measured superluminal group velocities for the wave propagation through the bandgap of a 1DPC [64]. To our knowledge, there has been no research studying the propagation of the precursors in 1DPC. This is partially due to difficulties in applying the asymptotic analysis to these structures. In this thesis we have calculated the time and frequency evolution of the precursors in a 1DPC using a combination of the Finite Difference Time Domain (FDTD) and Joint Time Frequency Analysis (JTFA). 14

28 1.5.2 Applications of the Precursors The peak amplitude of the Brillouin precursor in a Debye medium decays as the square root of the inverse of the propagation distance, as opposed to the exponential decay of the steady state part of the pulse. A pulse which has the Brillouin precursor time distribution was proposed as a near optimal excitation inside unbounded triply distilled water at normal incidence [60]. By fitting experimental data from water, soil and many other dispersive media to the Debye model, the possibility of using precursors in applications such as underwater communications and underground detection can be studied. We have used the FDTD method to study the behavior of the near optimal pulse of [60] in the case that the dispersive medium is bounded and there is an interface between free space and the dispersive region. On the other hand, although the propagation of precursors has been studied extensively, the scattering of precursors and ultimately their application for detection studies has not. Here, using the FDTD method we have studied the scattering of the Brillouin precursor from metallic and dielectric objects inside water. The energy of the sampled scattered field from the near optimal pulse is compared to the scattered energy of modulated Gaussian and rectangular pulses. 1.6 Thesis Structure This thesis is comprised of three parts. In the first part, which consists of chapter 2, a brief introduction of the numerical and analytical methods used is given. The numerical techniques are FDTD and JTFA, whereas the steepest descent method is our analytical technique of choice. The second part of the thesis, which consists of chapters 3 and 4, 15

29 describes precursors in active Lorentzian media and in 1DPCs. In chapter 3, a combination of the FDTD and JTFA is used to calculate the evolution of precursors in 1DPC. In chapter 4, the steepest descent method is used to calculate the propagation of precursors in active Lorentzian media. The third part of the thesis, which describes applications of the precursors, includes chapters 5 and 6. In chapter 5, the FDTD method is used to study the propagation of the near optimal pulse inside water and its interaction with an air/water interface at oblique incidence. In chapter 6, the scattering of the near optimal pulse from different objects inside water is studied. 16

30 Chapter 2 Analytical and Numerical Methods used in Calculation of the Precursors in Dispersive Media In this thesis, the analytical technique of the steepest descent method and the numerical techniques of FDTD method and JTFA have been employed to analyze the theoretical and applied aspects of precursor fields evolution in dispersive media. This chapter is dedicated to a brief presentation of these methods. 2.1 Steepest Descent Method [30] The steepest descent method is used in the study of the asymptotic behavior of integrals of the form, I(z) = q(ω)e zp(ω) dω as z, (2.1) C 17

31 Figure 2.1: A typical example of a first order saddle point. where, q and p = u + iv are analytic functions of the complex variable ω. The path of integration C has endpoints at infinity. Along C, the parameter z is assumed to be real and positive. The basic idea is that we can deform the contour C to a new contour C, using Cauchy s theorem [65]. In this approach we require C to be the steepest descent path which goes through the saddle points of p. For an M th order saddle point at ω sp, p (n) (ω sp ) = 0, n = 1... M and p (M+1) (ω sp ) 0, where, p (n) (.) is the n th order derivative of p with respect to ω. Figure 2.1 shows a typical first order saddle point. On the steepest descent path the imaginary part of the function p is constant. Therefore, we can write Eq.(2.1) as, I(z) = e C izv q(ω)e zu(ω) dω. (2.2) When the contour of integration C is deformed into a steepest descent path C, we can determine the leading order asymptotic behavior of I(z) from the behavior of the integrand in the vicinity of the local maxima of u(ω). The simplest case is when p(ω) has only one first order saddle point at ω sp (p (ω sp ) = 18

32 0, p (ω sp ) 0) on C and q(ω) has no singularities near this first order saddle point. Then, it can be shown that the asymptotic approximation of the integral Eq.(2.1) is given by [30], [25], 2π I(z) zp (ω sp ) q(ω sp)e zp(ωsp), z. (2.3) For isolated saddle points of higher order, the asymptotic approximation of the integrand is given by [30] I H (z) I H (z) = + ω sp q(ω)e zp(ω) dω, z, (2.4) [ (M + 1)! ] 1/(M+1)q(ωsp )e zp(ω Γ(1/(M + 1)) sp), z, (2.5) p (M+1) (ω sp ) (M + 1)z1/(M+1) where M is the order of the saddle point. The assumption is that the function q(ω) has no singularities near the M th order saddle point. The (M + 1) th root in the first factor is chosen such that {[ (M + 1)! ] 1/(M+1) } arg = α arg(dω) p (M+1) ωsp, (2.6) (ω sp ) where dω denotes an element along the steepest descent path, which begins at ω sp and ends in an appropriate valley at (in the valley p(ω) < p(ω sp ) ). Figures 2.2 (a) and (b) show typical examples of a first and a second order saddle point, respectively. In these cases, the assumption is that the contour of integration does not include any singularities. When the function q(ω) has one or more poles and these poles are crossed when the original contour of integration is deformed to C, the poles are encircled. Thus, according to Cauchy s theorem [65], the integral representation of I(z) is given as I(z) = I sp (z) + I c (z), (2.7) where I c (z) = 2πiΓe zp(ω sp). (2.8) 19

33 " Saddle Point " Saddle Point Valley Valley Valley Valley Valley ' Valley ' (a) (b) Figure 2.2: A typical example of (a) first order saddle point (b) second order saddle point. The arrows show the direction of the steepest descent path at each saddle point. and, I sp (z) is the contribution of the saddle points which can be calculated using Eq.(2.3) or (2.5). The function Γ is the summation of the residues of the poles that were crossed. When the deformed path C encircles N poles, Γ is defined as Γ = N γ i, (2.9) i=1 where γ i = lim [(ω ωpole)q(ω)]. i (2.10) ω ωpole i 2.2 Finite Difference Time Domain Method [66] The FDTD method solves Maxwell s equations by approximating the curl equations using centered finite differences for both the temporal and spatial derivatives, and then marches the fields through time to obtain the time domain representation. All the FDTD update equations are based on Maxwell s curl equations D t = H, (2.11) 20

34 B t = E, (2.12) where, E and H are electric and magnetic vector fields and, D and B are electric flux density and magnetic flux density, respectively. The simulations in chapter 3 are one dimensional (E y, H z 0) and the ones in chapter 5 and 6 are two dimensional (E y, H x, H z 0). The one dimensional equations can be easily derived from the two dimensional equations by removing the appropriate field component. For a two dimensional transverse electric (T E z ) (E y, H x, H z 0) field, equations (2.11) and (2.12) are simplified to D y t = H x z H z x, (2.13) µ 0 H x t µ 0 H z t = E y z, (2.14) = E y x. (2.15) All the two dimensional simulations in the chapters 5 and 6 are in T E z mode. The equation for the transverse megnetic (T M z ) mode can be found in [66]. Equations (2.13) and (2.15) are the ones used in one dimensional simulataions of chapter Dispersion Implementation in FDTD Several techniques have emerged for the treatment of dispersive media with FDTD [67]. In this thesis we have used the auxiliary differential equation (ADE) approach. The ADE method was introduced by Kashiva et al. [68,69], Joseph et al. [70], and Gandhi et al. [71]. In this method, the frequency dispersion is implemented by concurrently integrating (in time) an ordinary differential equation that relates D y (t) and E y (t). The time domain equation is 21

35 obtained from the inverse Fourier transform of the complex permittivity expression ɛ(ω) = ɛ r (ω)ɛ 0 = D y(ω) E y (ω), (2.16) where, ɛ 0 is the free space permittivity and ɛ r (ω) is the relative permittivity of the medium. For an order-m dispersion (M is the number of the poles of ɛ(ω)), the computational model becomes a three step recursive process, which is given as [66, 67] H n+ 1 2 x (i, k ) = 1 Hn 2 x (i, k ) + t [ ] Ey n (i, k + 1) Ey n (i, k), (2.17) µ 0 z H n+ 1 2 z (i + 1 2, k) = 1 Hn 2 z (i + 1 t [ ], k) Ey n (i + 1, k) Ey n (i, k), (2.18) 2 µ 0 x D n+1 y (i, k) = Dy n (i, k) + t z t x [ H n+ 1 2 x (i, k ) Hn+ 1 2 x (i, k 1 2 ) ] [ H n+ 1 2 z (i + 1 2, k) Hn+ 1 2 z (i 1 2, k) ], (2.19) E n+1 y (i, k) = f(dy n+1,..., Dy n M+1 ; Ey n,..., Ey n M+1 ). (2.20) In these equations, n is the time step and i,k specify the grid point in the computational domain along the x and z directions, respectively. The simulations in chapter 3 are in single resonance Lorentzian medium, whose complex permittivity is given by ɛ(ω) = ɛ + (ɛ s ɛ )ω 2 0 ω 2 0 ω 2 + 2iδω. (2.21) Here, ω 0 is the resonance frequency, ɛ and ɛ s are the permittivities at very high and very low frequencies, and δ is the damping constant of the dispersive lossy dielectric. If we take the inverse Fourier transform of Eq. (2.21) the result can be written in the discretized form as, y (i) = A 1 Dy n+1 (i) + A 2 Dy n (i) + A 3 Dy n 1 (i) + A 4 Ey n (i) + A 5 Ey n 1 (i), (2.22) E n+1 22

36 where, the coefficients A 1 to A 5 are defined as [66] A 1 = ω2 0 t 2 + 2δ t + 2 ɛ s ω δɛ t + 2ɛ, (2.23) A 2 = 4 ɛ s ω δɛ t + 2ɛ, (2.24) A 3 = ω2 0 t 2 2δ t + 2 ɛ s ω δɛ t + 2ɛ, (2.25) A 4 = 4ɛ ɛ s ω δɛ t + 2ɛ, (2.26) A 5 = ɛ sω 2 0 t 2 + 2δɛ t 2ɛ ɛ s ω δɛ t + 2ɛ. (2.27) The simulations in chapter 5 and 6 are in the Rocard-Powles-Debye medium, whose complex permittivity is given by [60] ɛ(ω) = ɛ + (ɛ s ɛ ) (1 iωτ)(1 iωτ f ), (2.28) where, τ is the relaxation time with an associated friction time τ f. If we take the inverse Fourier transform of Eq.(2.28) the discretized eqaution would be E n+1 y (i, k) = B 1 Dy n+1 (i, k) + B 2 Dy n (i, k) + B 3 Dy n 1 (i, k) + B 4 Ey n (i, k) + B 5 Ey n 1 (i, k),(2.29) If we define ρ 2 = 1/ττ f and ν = (τ + τ f )/ττ f, the coefficients B 1 to B 5 will be [66] B 1 = ρ2 t 2 + ν t + 2 ɛ s ρ 2 + νɛ t + 2ɛ, (2.30) B 2 = 4 ɛ s ρ 2 + νɛ t + 2ɛ, (2.31) B 3 = ρ2 t 2 ν t + 2 ɛ s ρ 2 + νɛ t + 2ɛ, (2.32) B 4 = 4ɛ ɛ s ρ 2 + νɛ t + 2ɛ, (2.33) B 5 = ɛ sρ 2 t 2 + νɛ t 2ɛ ɛ s ρ 2 + νɛ t + 2ɛ. (2.34) 23

37 2.2.2 Perfectly Matched Layer in FDTD The Perfectly Matched Layer (PML) enables us to achieve efficient and accurate solution of electromagnetic wave problems in unbounded regions. The PML absorbers can be developed for media having Debye and Lorentzian dispersive behavior [66]. Here, the eqautions for the two dimensional PML which is used in the simulations in chapter 5 and 6 are briefly presented. where, For a TE wave the time harmonic Ampere s law within the PML can be expressed as B x B z = µ 0µ r µ H x H z, (2.35) H x z H z x = jωɛ 0ɛ r ɛe y, (2.36) and, µ = ɛ = S x S z, (2.37) S z /S x 0 0 S x /S z, (2.38) S x = 1 + δ x jωɛ 0, S z = 1 + δ z jωɛ 0. (2.39) The values δ x and δ z are calculated based on the parameters of the PML [66]. For a wave impinging upon the PML at an angle θ (relative to the normal to the interface), the reflection coefficient can be computed using transmission line analysis, yielding R(θ) = e 2σηd cos θ. (2.40) Here, η, σ and d are the PML s characteristic wave impedance, conductivity, and thickness, respectively. The value R(θ) is referred to as the reflection error and it increases as θ 24

38 increases. At grazing angle (θ = 90 ), the value of R is one and the PML is completely inactive. Several profiles have been suggested for grading σ. In this thesis, the polynomial grading is used as the PML profile. Assuming the z direction is normal to the interface the polynomial grading of σ is given by σ(z) = (z/d) m σ max then, R(θ) = e 2ησmaxd cos θ/(m+1). (2.41) Typically, 3 m 4 has been found to be optimal for many FDTD simulations [72], [73]. The PML thickness (d) is usually 5 or 10 cells. Based on a 10 cell PML the optimum value for σ is σ opt max 0.8(m + 1), (2.42) η where, is the spatial discretization in the direction of the propagation [66]. Introducing the parameter Q y = ɛ 0 ɛ r S x E y, Eq.(2.36) can be written as H x z H z x = jωɛs zq y. (2.43) For a dispersive medium with a relative permittivity ɛ r (ω) = ɛ(ω)/ɛ 0, the relation between Q y (ω) and D y (ω) is given as, Q y (ω) = ɛ(ω)/ɛ 0 D y (ω), (2.44) where, the equation for ɛ(ω) is Eq.(2.28). Hence, the discritized FDTD update equations for the electric field are [66] Q n+1 y (i, k) = 2ɛ 0 σ z t 2ɛ 0 + σ z t Qn y(i, k) + [ H n+ 1 2 x (i, k ) Hn+ 1 2 x (i, k 1 2 ) z 2ɛ 0 t 2ɛ 0 + σ z t 25 1 Hn+ 2 z (i + 1, k) 1 2 Hn+ 2 x z (i 1 2, k) ], (2.45)

39 D n+1 y (i, k) = B 1 Q n+1 y (i, k) + B 2 Q n y(i, k) + B 3 Q n 1 y (i, k) + B 4 Dy n (i, k) + B 5 Dy n 1 (i, k),(2.46) Ey n+1 (i, k) = 2ɛ 0 σ x t 2 [ ] 2ɛ 0 + σ x t En y (i, k) + Dy n+1 (i, k) Dy n (i, k), (2.47) 2ɛ 0 + σ x t and the discretized equations for the magnetic fields are x (i, k ) = 1 Bn 2 x (i, k ) t [ ] Ey n (i, k) Ey n (i, k + 1), (2.48) z B n+ 1 2 H n+ 1 2 x (i, k ) = 2ɛ 0 + σ x t 1 µ 0 (2ɛ 0 + σ z t) Bn+ 2 x (i, k ) 2ɛ 0 σ x t 1 µ 0 (2ɛ 0 + σ z t) Bn 2 x (i, k ) + 2ɛ 0 σ z t 1 µ 0 (2ɛ 0 + σ z t) Hn 2 x (i, k + 1 ). (2.49) 2 The concept of a PML reduces to a regular matched absorber in one dimensional simulations. First-order Mur s absorbing boundary conditions have been used in chapter 3 simulations, where one dimensional waves propagate in free space. The complete description of the Mur s boundary condition can be found in [66]. The final descritized equation for first order Mur s boundary condition are E n+1 y (i) = E n y (i 1) + S 1 S + 1 [En+1 y (i 1) E n y (i)], (2.50) E n+1 y (1) = E n y (2) + S 1 S + 1 [En+1 y (2) E n y (1)], (2.51) where, S is the Courent stability number. For the wave which propagtes in the +z-direction, eqautions (2.50) and (2.51) terminte the end and the begining of the one dimensional computational domain, respectivly. 2.3 Joint Time-Frequency Analysis (JTFA) [74] In the field of computational electromagnetics, JTFA has been mainly employed for remote sensing and scattering problems [75,76]. In this thesis, the Wigner Ville Distribution (WVD) 26

40 is chosen among different distributions because it provides relatively high resolution in the time-frequency plane [74]. The WVD of a signal x(t), with a Fourier transform X(f) can be defined either as: W x (t, f) = + x(t + τ/2)x (t τ/2) exp( i2πfτ)dτ, (2.52) or as: W x (t, f) = + X(f + ξ/2)x (f ξ/2) exp( i2πξt)dξ, (2.53) where denotes complex conjugate. While WVD provides high resolutions in time and frequency, the drawback is the relatively large interference terms present in the distribution. These can be troublesome, since they may overlap with the signal and thus make it difficult to visually interpret the WVD image. However, these terms must be present for the good properties of the WVD (marginal properties, instantaneous frequency, group delay, and so on) to be attained. The interference terms are rather easily (visually) recognizable due to their oscillatory structure. Moreover, these interference terms do not appear in the steady-state representations of the signal; therefore, Fourier transform of the signal can also be used to identify the cross terms. 27

41 Chapter 3 Precursor Fields in One Dimensional Photonic Crystal 3.1 Introduction In recent years, advances in the area of photonic crystals have created a need to examine electromagnetic pulse propagation in materials with geometrical dispersion [77]. Even though the electromagnetic propagation in two and three dimensional photonic crystals is complicated and demands extensive calculations, some of the physical characteristics of these structures can be understood by considering one dimensional photonic crystals (1DPC). One dimensional photonic crystals consist of alternating layers of materials with different refractive indices. Because of this periodic modulation of the index, the spectral transmission function of 1DPC exhibits pass bands and stop bands. These dispersion properties of 1DPC which are due to its inhomogeneous structure dominate material dispersion. As a consequence, the evolution of the precursors, which depends on the dispersive behavior of the 28

42 1DPC, is different from the cases of the Lorentzian or Debye media where the medium is homogenous. As the consequence of the dispersion, there is no unique definition of velocity in dispersive media such as 1DPC. Each of the energy, group, or precursor velocities has a different definition and assumptions. As one calculates the energy velocity as the ratio of the Poynting vector and stored electromagnetic energy, or group velocity as the speed of the peak of the pulse, the assumption is that the steady state has been reached and the medium has stored the energy provided by the field. Many researchers have tried to theoretically analyze these velocities in 1DPC based on these assumptions [50, 51, 54, 55, 78 86]. On the other hand, the precursors temporal and spectral distribution and their propagation in space are quite different than the considerations associated with definitions for energy or group velocities. These differences are mainly due to the fact that precursors are transient responses while any definition of group or energy velocities are non-transients. To our knowledge, there has been no work on the precursors in 1DPCs. This problem is addressed in this chapter by means of FDTD and JTFA. This chapter is organized as follows. In Sec. 3.2, the combined FDTD and JTFA is used to study the evolution of the precursors in a single resonance Lorentzian medium subject to a modulated Gaussian excitation. The FDTD results agree very well with those obtained from the asymptotic analysis of [24]. This agreement reinforces the conclusion that combining FDTD and JTFA provides a robust and versatile method for studying the precursor fields in complex structures such as 1DPC. In Sec. 3.3, the superluminal propagation of a modulated Gaussian pulse inside a 1DPC with various conditions on the front is presented. These simulations confirm that the precursors travel with a speed lower than the speed of light. 29

43 It is also shown that the attenuation rate for the precursors is lower than the attenuation rate for the rest of the pulse, hence providing the possibility of further penetration into the dispersive medium. 3.2 Ultra Short Modulated Gaussian Pulse Propagating in a Single Resonance Lorentzian Medium To confirm the validity of our approach, the FDTD method is used in conjunction with Wigner-Ville JTFA to study the precursor fields in a Lorentzian medium. We consider the propagation of a modulated Gaussian pulse inside a passive Lorentzian medium that occupies the half-space z 0. The input modulated Gaussian pulse propagating in the positive z direction is given by [ S(t) = exp ( t t ] 0 ) 2 sin(ω c t), (3.1) T s where ω c is the carrier frequency. The pulse is temporally centered around time t 0 > 0 at the z = 0 plane with a full width at the 1/e point given by 2T s. The medium is characterized by a passive single resonance Lorentzian with complex refractive index n(ω) = ( ω 2 ) p 1/2. 1 (3.2) ω 2 ω iδω Here, ω 0 is the resonance frequency, ω p is the plasma frequency, and δ is the damping constant of the dispersive, lossy dielectric. The material absorption band is defined over the approximate angular frequency domain ( ω0 2 δ 2, ω1 2 δ 2 ), where ω 1 = ω0 2 + ωp. 2 The material dispersion is normal (n r (ω) increases with increasing frequency) over the approximate angular frequency domains [0, ω 2 0 δ 2 ] and [ ω 2 1 δ 2, ], below and above the 30

44 absorption band respectively, while it is anomalous (n r (ω) decreases with increasing frequency) over the approximate angular frequency domain [ ω0 2 δ 2, ω1 2 δ 2 ] containing the absorption band. The Brillouin choice of medium parameters which describes a highly absorptive material is used in the numerical calculations (ω 0 = Hz, δ = Hz, and ω p = Hz). The modulated Gaussian pulse with parameters ω c = Hz, T s = 1 fs, and t 0 = 3 T s is a wideband pulse that its frequency distribution covers the anomalous dispersion region of the medium. We have used the space and time discretization parameters, δ x = λ 0 /400 and δ t = sδ x /c, where s = 0.9 is the Courant stability number and λ 0 is the vacuum wavelength at the carrier frequency of the modulated pulse [87]. First order Mur s boundary condition is used to terminate the computational domain at both ends [66]. Since the input carrier frequency ω c lies within the medium absorption band, the frequency components of the propagating field that are within this band will be significantly attenuated. As the consequence of propagation, the input Gaussian evolves into two Gaussianshaped pulses, the first containing the high frequency oscillations, while the second contains the low frequency oscillations. These first and second pulses are recognized as the Sommerfeld and Brillouin precursors, respectively. Figures 3.1 and 3.2 show the time and frequency evolution of the total field due to the input Gaussian pulse at propagation distances z = 0.5 µm and z = 1 µm, respectively. The high frequency oscillations in the beginning of the signal in Fig. 3.1(a) is the Sommerfeld precursor that has completely evolved. The oscillations following the Sommerfeld precursor are part of the Brillouin precursor that has not completely evolved at this observation 31

45 Figure 3.1: (a) Time distribution (b) joint time-frequency distribution (c) frequency distribution of the modulated Gaussian pulse after propagating 0.5µm through the dispersive Lorentzian medium. (θ = θ ct 0 /z). point. The JTFA in Fig. 3.1(b) shows that the frequency evolution of the Sommerfeld precursor begins with high frequencies that are above the absorption band and decreases as θ (θ = θ ct 0 /z) increases. At one point around θ = 1.5 the frequency begins increasing towards the carrier frequency from below the absorption band due to the Brillouin precursor. In Fig. 3.2 the pulse has traveled deeper inside the medium and both Sommerfeld and Brillouin precursors are fully present. The Sommerfeld precursor has similar frequency evolution as in Fig. 3.1, and around θ = 1.5 the Brillouin precursor appears, whose frequency evolution starts from low frequencies and increases towards the carrier frequency. In both JTFA plots, there are high intensity components in the absorption band. These components are the spurious cross terms in the WVD. As stated earlier, to have a good frequency 32

46 Figure 3.2: (a) Time distribution (b) joint time-frequency distribution (c) frequency distribution of the modulated Gaussian pulse after propagating 1µm through the dispersive Lorentzian medium. (θ = θ ct 0 /z). resolution these cross terms are unavoidable; however they can be easily identified by their oscillatory appearance. Integrating these cross-terms over frequency results to zero, as shown from the absence of frequency components of the signal in the band. The bandwidth and resolution of the JTFA is necessarily limited by those of the FDTD technique. However, the FDTD bandwidth is chosen to be wide enough to include the spectral content of the excitation pulse up to frequencies with a power spectral density that is 30 db lower than its maximum. The results presented here are in good agreement with those calculated using asymptotic techniques in [24] (approximately 2 percent error at the peak of the pulse). Our results show that a combination of FDTD as a time domain technique, and JTFA as a post processing 33

47 n s d a d s Air Dielectric Figure 3.3: The physical structure of the 5-slab 1DPC (d s = 1.27 cm, d a = 4.1 cm, n s is the refractive index of the slabs). technique can be used to provide a comprehensive view of the pulse evolution in both time and frequency domains. 3.3 Time and Frequency Evolution of the Precursors in a One Dimensional Photonic Crystal In this section, a combination of FDTD and JTFA has been used to study the propagation of the precursors in a 1DPC. To confirm the accuracy of the FDTD method, we have analyzed the superluminal propagation of the modulated Gaussian pulse through the 1DPC based on the physical experiment by Mojahedi et. al. [64]. Figure 3.3 shows the physical structure of the 1DPC. It consists of five dielectric slabs with the width of 1.27 cm. The slabs are separated by 4.1 cm air-gaps. The dielectric model in the 1DPC is a single Lorentzian with parameters ω 0 = Hz, δ = Hz, and ω p = Hz. We have used the space and time discretization parameters, δ x = λ 0 /50, δ t = sδ x /c where λ 0 34

48 E(z,t) (V/m) Through 1DPC t=479ps E(z,t) (V/m) Through free space Time(ns) Figure 3.4: Pulse advancement in the 5-slab 1DPC. is the vacuum wavelength at the resonance frequency of the medium and s = 0.9 is the Courant stability number. Due to the fine discretization of the medium in 1DPC the abrupt change of the dielectric constant at the air-dielectric slab interfaces does not contribute significant numerical errors. Mur s first order boundary condition is used to terminate the computational domain at both ends [66]. The structure is excited with a modulated Gaussian pulse with parameters f c = 9.6 GHz, T s = 3.11 ns, and t 0 = 3 T s. The center frequency of the modulated Gaussian is inside the bandgap of the 1DPC. The frequency components of the pulse outside the bandgap are negligible. Figure 3.4 shows the pulse that has traveled the same distance through the 1DPC and free space. The peak of the pulse that has traveled through the 1DPC appears in the output 479 ps sooner than the companion pulse traveling through the same distance in vacuum. It is also important to note that the photonic crystal pulse has been greatly attenuated (approximately 14 db.) 35

49 Figure 3.5: Wigner-Ville distribution of the output pulse with a smooth front propagated in the 5-slab 1DPC. The group velocity of the pulse propagating through the 1DPC of length L pc is given by v g = L pc /τ g, (3.3) where τ g is the time associated with traversing the 1DPC, also known as group delay. The time difference ( t) between the peaks of the pulses that have traveled through 1DPC and free space can be used to calculate the group delay according to τ g = L pc c t. (3.4) The length of the 1DPC is L pc = cm and based on the simulation results t 479 ps, therefore, group velocity in the 1DPC is v pc = 2.7c. The computational results are in good agreement with the experimental results in [64] (8 percent error for t). Figure 3.5 shows the Wigner-Ville distribution of the aforementioned smooth pulse propagated through the 1DPC. It is similar to the time-frequency distribution of the pulse that 36