Finite-Difference Time-Domain Simulations

As a result of ever-increasing computational power, the finite-difference time-domain (FDTD) method has become a popular tool to simulate propagation of electromagnetic radiation through arbitrary media. The accuracy of FDTD and the ease with which it is parallelized make it even more attractive for larger, more realistic simulations. However, most applications are in the engineering sciences with few major applications in problems of a purely scientific nature--though there are exceptions. The algorithm is based on Maxwell's modifications to Faraday's, Ampere's and Gauss's law. For a more complete introduction, see my paper "Finite Difference Time Domain (FDTD) Simulations of Electromagnetic Wave Propagation Using a Spreadsheet," which illustrates FDTD with Excel spreadsheets.

I developed an extended FDTD method that simulates phonon-polariton propagation. In this case, energy in the electromagnetic field couples to lattice motion in a ferroelectric or IR active crystal. You can download polariton_FDTD here. We have used this to simulate phonon-polariton generation, propagation, and real-space probing.

In collaboration with Andrea Cavalleri of Oxford University, we have used polariton_FDTD to simulate femtosecond X-ray diffraction from phonon-polaritons excited in lithium tantalate. The results were published in Nature, September 2006. Phonon polariton in lithium tantalate were excited by 800-nm laser pulses impinging onto the sample at the same 22-degree angle of incidence of the femtosecond X-ray pulses, as dictated by the Bragg condition. A time and space-dependent pattern of phonon-polaritons was generated in the sample by impulsive stimulated Raman scattering (ISRS). The time dependent 006 X-ray reflection from lithium tantalate, which gave a direct measurement of the 1.5 THz lattice distortions associated with phonon-polaritons of Ag symmetry, was measured and compared with simulations using polariton_FDTD. Simulated lattices displacements were used to calculate the expected 006-diffraction signal, which agreed well with X-ray measurements. Kinematic diffraction is a good approximation in this case due to the homogeneous optical excitation, occurring over tens of microns beneath the surface of transparent lithium tantalate.

References

A.Cavalleri, S. Wall, C. Simpson, E. Statz, D. W. Ward, K. A. Nelson, M. Rini, R.W. Schoenlein, "Tracking the motion of charges in a terahertz light field by femtosecond X-ray diffraction," Nature 442 664-666, (2006).

A. Cavalleri, S. Wall, M. Rini, C. Simpson, N. Dean, M. Khalil, E. Statz, D. W. Ward, K. A. Nelson, and R. W. Schoenlein, "Lattice Motions from THz Phonon-polaritons measured with Femtosecond X-ray Diffraction.," in 15th International Conference on Ultrafast Phenomena, OSA Technical Digest Series (CD) (Optical Society of America, 2006), paper WB4.

David W. Ward and Keith A. Nelson, "Finite Difference Time Domain (FDTD) Simulations of Electromagnetic Wave Propagation Using a Spreadsheet," Computer Applications in Engineering Education, 13 (3), 213-221, (2005).

David W. Ward, Eric Statz, Nikolay Stoyanov, and Keith A. Nelson, "Simulation of Phonon-Polariton Propagation in Ferroelectric LiNbO3 Crystals ," in Engineered Porosity for Microphotonics and Plasmonics: MRS Symposium Proceedings, Vol. 762, edited by R. Wehrspohn, F. Garcial-Vidal, M. Notomi, and A. Scherer (Materials Research Society, Pittsburgh, PA, 2003), pp.C11.60.1-6.

Negative Refractive Index Materials

In 1968, Veselago suggested that a material with a negative index of refraction would enjoy certain peculiar properties. Foremost among these was negative refraction, where a ray incident on the interface of such a material would refract on the same side of the normal rather than away from it. The subject lay dormant until 1999, when Pendry proposed designs for magnetic metamaterials that were subsequently implemented by Shelby et al., along with a dispersed electric metamaterial, to demonstrate negative refraction in 2001. Despite the fact that no naturally occurring negative index material is available, there has been a surge in interest, particularly with regard to the prospect of creating a perfect lens.

To illustrate how negative refraction might arise, we consider the path of least of action for a stream of photons in vacuum (n1 = 1) incident on a homogeneous, isotropic material with an index of refraction, n2 . The most probable path is determined by the path of stationary phase, which corresponds to an extremum in the spatial derivative of the total travel time through all possible paths. In the case of positive refraction, this is just Fermat’s principle of least time. From the diagram in Figure 1(a), the optical path length in vacuum from source A to the interface point O is c1t1 = AO = sqrt(a^2 + y^2) , and in the semi-infinite material from O to B it is c2t2 = OB =sqrt(b^2 + (l − y)^2). To find the extremum in the time of travel from A to B, we form d(t1 + t2 )/dy = 0, with a and b fixed at arbitrary values, which upon substituting for the optical path lengths gives n1y/c0 sqrt(a^2 + y^2) = n2(l −y)/c0 sqrt(b^2 + (l − y) ^2) , with l −y ≥ 0. Recognizing the trigonometric relations sin(theta1 ) = y/sqrt(a^2 + y^2) and sin(theta2 ) = (l − y)/ sqrt(b^2 + (l − y) ^2 ), this can be rewritten in the familiar form of Snell’s law: n1 sin(theta1 ) = n2 sin(theta2 ). If we postulate that the photons refract to the other side of the normal, as depicted in the figure below, then l − y <=0, which implies that the angle of refraction becomes −theta2. The extremum for this optical path gives n1y/c0 sqrt(a^2 + y^2) = −n2 |l − y|/c0 sqrt(b^2 + (l − y) ^2) , which can only be satisfied if n2 < 0. Since n1sin(theta1 ) > 0 and sin( −theta2 ) = − sin(theta2 ), Snell’s law is found to be valid for both positive and negative refraction, provided we allow for the possibility of a negative index of refraction. The curvature becomes d^2 (t1 + t2 )/dy^2 = n1a^2 /[c0(a^2 + y^2 )^(3/2) ] + n2b^2 /[c0(b^2 +(l −y)^2)^(3/2)]. For n2 > 0, the curvature is positive, indicating that Fermat’s result indeed gives the minimum time (and distance). Interestingly, with n2< 0, the curvature is zero when |n2 | = |n1|, indicating a saddle point, and is negative for |n2 | > |n1 |, indicating a maximum. In terms of least time, this result is not meaningful. However, the principle of least action, based on Feynman’s path integral approach to quantum mechanics, supplanted the principle of least time. Negative curvature is then properly interpreted as the path of stationary phase, which in the framework of the least action principle determines the proper path for refraction, thereby removing any ambiguity as to whether negative refraction is physical.

I demonstrated (read here) that the disparity between the frequency-dependent phase velocity of light in a material and the constant velocity of photons in a vacuum is resolved by recognizing that photons impinging on a medium can drive resonances in that medium, which may radiate and contribute to the total scattered field. When there is a difference in phase between the source and radiated field, the wavefronts at the detector will appear to be advanced or retarded with respect to the source, and it is on this basis that the concept of negative velocity is explained. Using a microscopic model for dilute media, I illustrated the physical origins of negative refraction.

One of the most interesting phenomena that follows from negative refraction is the prospect of a superlens, that is a lens that can image with resolution below the diffraction limit. Pendy proposed that evanescent fields grow rather than decay in negative refractive materials. For the case of n=-1 and with no loss, this would suggest the possibility of a perfect lens with infinitesimal resolution. In collaboration with Kevin Webb, we showed that a perfect lens is not possible because negative refractive materials are dispersive, with real and imaginary components of the susceptibility, and hence negative refractive lenses are intrinsically lossy. This limitation does not preclude a better, or even super, lens. We demonstrated the effect of loss expected in a real negative refractive lens and illustrated the limited evanescent field growth associated with a superlens (read here).

Backward Cherenkov Cones

In 1968, Veselago suggested that a material with spectrally overlapping negative permittivity and permeability would result in a negative index of refraction, in which a ray refracts to the opposite side of the normal as it would in a positive refractive index material. He also suggested that superluminal charged particles would induce a backwards Cherenkov cone in negative refractive index materials. Here we show that the backward Cherenkov cone predicted by Veselago occurs, and that this radiation mechanism can be achieved using impulsive stimulated Raman scattering (ISRS) through rigorous finite-difference time-domain (FDTD) simulations that have proved accurate in earlier experimental work. In our work, the short optical pulses play the role of the charges in traditional Cherenkov radiation.

This work was never published because I was busy working on other stuff, but it should have been. You can get the final copy of the manuscript here.

References

K.J. Webb, M. Yang, David W. Ward, and Keith A. Nelson, "Metrics for Negative Refractive Materials," Physical Review E, 70, 035602(R), (2004).

David W. Ward, Keith A. Nelson, and Kevin J. Webb "On the Physical Origins of the Negative Index of Refraction," New Journal of Physics, 7, 213 (2005).

K.J. Webb, M. Yang, David W. Ward, and Keith A. Nelson, "Resolution Limits of a Negative Refractive Index Lens," CLEO 2005 (2005).

David W. Ward, Kevin J. Webb, and Keith A. Nelson "Backward Cherenkov Cones in Negative Refractive Index Materials Excited Through Impulsive Stimulated Raman Scattering," unsubmitted (2006).

Polaritonics

Polaritonics is an intermediate regime between photonics and electronics (see figure below). In this regime, signals are carried by an admixture of electromagnetic and lattice vibrational waves known as phonon-polaritons, rather than currents or photons. Since phonon-polaritons propagate with frequencies in the range of hundreds of gigahertz to several terahertz, polaritonics bridges the gap between electronics and photonics. A compelling motivation for polaritonics is the demand for high speed signal processing and linear and nonlinear terahertz spectroscopy. Polaritonics has distinct advantages over electronics, photonics, and traditional terahertz spectroscopy in that it offers a fully integrated platform that supports terahertz wave generation, guidance, manipulation, and readout in a single patterned material.

Polaritonics, like electronics and photonics, requires three elements: robust waveform generation, detection, and guidance and control. Without all three, polaritonics would be reduced to just phonon-polaritons, just as electronics and photonics would be reduced to just electromagnetic radiation. These three elements can be combined to enable device functionality similar to that in electronics and photonics.

To illustrate the functionality of polaritonic devices, consider the hypothetical circuit in the figure below. The optical excitation pulses that generate phonon-polaritons, in the top left and bottom right of the crystal, enter normal to the crystal face (into the page). The resulting phonon-polaritons will travel laterally away from the excitation regions. Entrance into the waveguides is facilitated by reflective and focusing structures. Phonon-polaritons are guided through the circuit by terahertz waveguides carved into the crystal. Circuit functionality resides in the interferometer structure at the top and the coupled waveguide structure at the bottom of the circuit. The latter employs a photonic bandgap structure with a defect (yellow) that could provide bistability for the coupled waveguide.

Waveform Generation

Phonon-polaritons generated in ferroelectric crystals propagate nearly laterally to the excitation pulse due to the high dielectric constants of ferroelectric crystals, facilitating easy separation of phonon-polaritons from the excitation pulses that generated them. Phonon-polaritons are therefore available for direct observation, as well as coherent manipulation, as they move from the excitation region into other parts of the crystal. Lateral propagation is paramount to a polaritonic platform in which generation and propagation take place in a single crystal. A full treatment of the Cherenkov-like terahertz wave response reveals that in general, there is also a forward propagation component that must be considered in many cases.

Signal Detection

Direct observation of phonon-polariton propagation was made possible by real-space imaging, in which the spatial and temporal profiles of phonon-polaritons are imaged onto a CCD camera using Talbot phase-to-amplitude conversion. This by itself was an extraordinary breakthrough. It was the first time that electromagnetic waves were imaged directly, appearing much like ripples in a pond when a rock plummets through the water's surface (see figure below). Real-space imaging is the preferred detection technique in polaritonics, though other more conventional techniques like optical Kerr-gating, time resolved diffraction, interferometric probing, and terahertz field induced second harmonic generation are useful in some applications where real-space imaging is not easily employed. For example, patterned materials with feature sizes on the order of a few tens of microns cause parasitic scattering of the imaging light. Phonon-polariton detection is then only possible by focusing a more conventional probe, like those mentioned before, into an unblemished region of the crystal.

Guidance and Control

The last element requisite to polaritonics is guidance and control. Complete lateral propagation parallel to the crystal plane is achieved by generating phonon-polaritons in crystals of thickness on the order of the phonon-polariton wavelength. This forces propagation to take place in one or more of the available slab waveguide modes. However, dispersion in these modes can be radically different than in bulk propagation, and in order to exploit this, the dispersion must be understood.

Control and guidance of phonon-polariton propagation may also be achieved by guided wave, reflective, diffractive, and dispersive elements, as well as photonic and effective index crystals that can be integrated directly into the host crystal. However, lithium niobate, lithium tantalate, and other perovskites are impermeable to the standard techniques of material patterning. In fact, the only etchant known to be even marginally successful is hydrofluoric acid (HF), which etches slowly and predominately in the direction of the crystal optic axis

Femtosecond laser micromachining

Femtosecond laser micromachining is used for device fabrication by milling ``air'' holes and/or troughs into ferroelectric crystals by directing them through the focus region of a femtosecond laser beam (see Fig. 4). This is the first demonstration of convenient, controllable, and rapid large scale damage induced in lithium niobate and lithium tantalate. The advantages of femtosecond laser micromachining for a wide range of materials have been well documented. In brief, free electrons are created within the beam focus through multiphoton excitation. Because the peak intensity of a femtosecond laser pulse is many orders of magnitude higher than that from longer pulse or continuous wave lasers, the electrons are rapidly accelerated and heated to form a plasma. The electrostatic instability, produced by the plasma, of the remaining lattice ions results in ejection of these ions and hence ablation of the material, leaving a material void in the laser focus region. Since multiphoton excited free electrons are always available at the beam focus, highly uniform and repeatable damage confined to the laser focus region results. Also, since the pulse duration and ablation time scales are much faster than thermalization time, femtosecond laser micromachining does not suffer from the adverse effects of a heat-affected-zone, like cracking and melting in regions neighboring the intended damage region. Fabrication of several structures on a variety of size scales is shown below.

We have already fabricated a variety of basic optical elements and demonstrated their functionality by generating and monitoring the propagation of THz radiation within them. The structures include simple waveguides, resonators, high throughput sharp 90-deg turn, Mach-Zehnder interferometric structure, diffraction grating, Y-coupler, and focusing circular and elliptical mirrors. Figure 7 illustrates propagation through a terahertz diffraction grating.

References

"Polaritonic Materials Fabricated and Tested with Ultrashort-Pulse Lasers," David W. Ward, Eric R. Statz, T. Feurer, and Keith A. Nelson, Ultrafast Lasers for Materials Science: MRS Symposium Proceedings, Vol. 850, M.J. Kelley, A. Pique, E.W. Kreutz, and M. Li eds. (Materials Research Society, Pittsburgh, PA, 2004), pp. MM1.2.1-6.

"Terahertz wave generation and propagation in thin-film lithium niobate produced by crystal ion slicing," David W.Ward, Eric R. Statz, Keith A. Nelson, Ryan M. Roth, and Richard M. Osgood, Appl. Phys. Lett. 86, No. 2, 022908 (2005).

"Coherent Control of Phonon-Polaritons in a Terahertz Resonator Fabricated with femtosecond Laser Machining," David W.Ward, Jaime D. Beers, T. Feurer, Eric R. Statz, and K.A. Nelson, Opt. Lett. 29, No. 22, pp. 2671-2673 (2004).

"Direct Visualization of a Polariton Resonator in the THz Regime," N. S. Stoyanov, T. Feurer, David W.Ward, Eric R. Statz, and Keith A. Nelson, Opt. Express 12, 2387-2396 (2004).

"Polaritonics in complex structures: confinement, bandgap materials, and coherent control," D.W. Ward, E.R. Statz, J.D. Beers, T. Feurer, J.D. Joannopoulos, R.M. Roth, R.M. Osgood, K.J. Webb, and K.A. Nelson, in Ultrafast Phenomena XIV, in press (2004).

"Typesetting of terahertz waveforms," T. Feurer, J.C. Vaughan, T. Hornung, and K.A. Nelson, Opt. Lett. 29, 1802-1804 (2004).

"Phonon-polariton propagation, guidance, and control in bulk and patterned thin film ferroelectric crystals," D.W. Ward, E. Statz, J.D. Beers, N.S. Stoyanov, T. Feurer, R.M. Roth, R.M. Osgood, K.A. Nelson, in Ferroelectric Thin Films XII: MRS Symposium Proceedings, vol. 797, A. Kingon, S. Hoffmann-Eifert, I.P. Koutsaroff, H. Funakubo, and V. Joshi eds. (Materials Research Society, Pittsburgh, PA, 2003), pp. W5.9.1-6.

"Automated two-dimensional femtosecond pulse shaping and phased-array THz generation," J.C. Vaughan, T. Feurer, and K.A. Nelson, in Ultrafast Phenomena XIII, R.D. Miller, M.M. Murnane, N.F. Scherer, and A.M. Weiner, eds. (Springer-Verlag, Berlin 2002), pp. 214-216.

"Field expulsion and reconfiguration in polaritonic photonic crystals," K.C. Huang, P. Bienstman, J.D. Joannopoulos, K.A. Nelson, and S. Fan, Phys. Rev. Lett. 90, 196402 (2003).

"Phonon-polariton excitations in photonic crystals," K.C. Huang, P. Bienstman, J.D. Joannopoulos, K.A. Nelson, and S. Fan, Phys. Rev. B. 68, 075209 (2003).

"Spatiotemporal coherent control of lattice vibrational waves," T. Feurer, J.C. Vaughan, and K.A. Nelson, Science 299, 374-377 (2003).

"Integrated diffractive THz elements," N. S. Stoyanov, T. Feurer, D.W.Ward, and K.A. Nelson, Appl. Phys. Lett. 82, No. 5, (2002).

"Terahertz polariton propagation in patterned materials," N.S. Stoyanov, D.W. Ward, T. Feurer, and K.A. Nelson, Nature Materials 1, 95-98 (2002).

"Direct visualization of phonon-polariton focusing and amplitude enhancement," N.S. Stoyanov, D.W. Ward, T. Feurer, and K.A. Nelson, J. Chem. Phys. 117, 2897-2901 (2002).

"Terahertz polaritonics: Automated spatiotemporal control over propagating lattice waves," R.M. Koehl and K.A. Nelson, Chem. Phys. 267, 151-159 (2001).

"Coherent optical control over collective vibrations traveling at light-like speeds," R.M. Koehl and K.A. Nelson, J. Chem. Phys. 114, 1443-1446 (2001).

"Real-space and real-time imaging of polariton wavepackets," S. Adachi, R.M. Koehl, and K.A. Nelson, J. Lumin. 87-89, 840-843 (2000).

"Real-space imaging of phonon-polaritons," S. Adachi, R.M. Koehl, and K.A. Nelson, Butsuri (Jpn.) 54, 357-363 (1999).

"Direct visualization of collective wavepacket dynamics," R.M. Koehl, S. Adachi, and K.A. Nelson, J. Phys. Chem. A 103, 10260-10267 (1999).

"Real-space polariton wave packet imaging," R.M. Koehl, S. Adachi, and K.A. Nelson, J. Chem. Phys. 110, 1317-1320 (1999).