Dr. Robertson's Research
The following links take you to the web pages associated with my research interests.
Photonic Band Structure
Photonic band structure refers to the modification of the propagation properties of electromagnetic waves travelling through a periodically modulated dielectric. As an example consider light traveling through a regularly spaced array of spherical glass beads. The effects of scattering and interference of the light by the glass beads clearly would result in a change in the propagation of the waves. The alteration in the propagation properties is particularly significant when the wavelength of the light is approximately equal to the spacing between the beads. In this regime photonic band gaps--frequency intervals in which no photon modes are allowed--can be created for appropriately designed dielectric arrays. The ability to create volumes of space in which no photons of a given band of energies can exist has a number of fundamental and applied consequences. To find out more about the subject you can check out the special issues of the Journal of the Optical Society of America B, Volume 10, 1993 and the Journal of Modern Optics, 41, 1994, or the following internet sites at UCLA, Iowa State University, and Redstone Arsenal.
My current projects in the area of photonic band structure center around two projects. The first is the development and exploration of new analysis tools for calculating the response of Photonic Band Gap Arrays. My particular interest is in the temporal response and I am at present using a time domain simulator based on the transmission line matrix method (TLM) to model electromagnetic wave propagation in dielectric arrays.
Surface Elelctromagnetic Waves on One-Dimensional Photonic Band Gap Materials
The second aspect of my photonic band structure research concerns the use of an attenuated total-internal-reflection (ATR) configuration capable of measuring optical frequency surface electromagnetic wave generation at the surface of photonic crystals. The ATR technique uses a prism in a reflection configuration as shown in the following figure.
I performed the first experiments that detected surface waves on two-dimensional photonic crystals some years ago [Optics Letters, 18#7, 528-530, 1993.]. Those experiments used the ATR method but they were performed at microwave frequencies for which two- and three-dimensional photonic crystals could be easily fabricated. Making photonic band gap arrays with fundamental gaps at optical frequencies is a current challenge to the field of photonic band gap research, but one that is being met successfully by a number of groups. I hope to have a novel tool for probing these samples as they become available. For now, I use a commercially made Bragg reflector--a one-dimensional photonic band gap material--to verify the performance of the system.
The figure below shows the experimental ATR reflectivity of a Bragg stack at three different wavelengths. For clarity, the curves have been offset vertically.
The narrow dip at the highest angle corresponds to the excitation of surface waves at the air/Bragg stack interface, whereas the other dips correspond to lossy modes guided within the multilayer stack. The angular position of the reflectivity minima at each wavelength can be directly related to the wave vector of the corresponding mode. By varying the wavelength of the incident light and finding the corresponding angles of coupling, it is possible to reconstruct the surface wave and guided wave dispersion relations. In principle, the guided mode dispersion should permit one to determine the effective index of the photonic band gap material at the particular wavelength/angle of coupling. This information would permit the determination of the photonic dispersion relation governing propagation of radiation in the dielectric array for frequencies outside the forbidden band gap. The surface modes exist only within the region of the band gap.
Finally, the work described here pertains to electromagnetic waves. A similar process can occur for acoustic waves, a topic that my student Jake Rudy and I explored experimentally in the summer months of 1997. With very simple equipment and home made samples we measured acoustic stop bands and determined the acoustic band structure in two-dimensional arrays. This work is described in more details on my acoustic research page.
Acknowledgement: This work is funded by a grant from the Research Corporation.
Acoustic Band Gap and Acoustic Metamaterials
As described in the section on photonic band structure, the effects of scattering and interference can result in photonic band gaps--frequency intervals over which propagation of light is forbidden. Many researchers realized that a similar phenomenon should be possible for acoustic waves propagating in a periodic scattering array. In the summer of 1997 my student, Jake Rudy, and I measured the acoustic stop bands in two-dimensional periodic arrays using an impulse response technique. The frequencies of the stop-band centers are consistent with simple theoretical considerations based on the spacing between layers along the high symmetry directions of the regular arrays. The phase information extracted from the Fourier analysis is used to construct the acoustic dispersion relation for frequencies above and below the stop band. Measurements were made on both square and triangular scattering arrays.
Representative data from these experiments is shown in the figures below. The first figure shows the Fourier amplitudes of an acoustic pulse that has travelled through air (solid circles) and the amplitudes of a pulse passed through a two-dimensional regular array of scattering cylinders (made up of 1 m lengths of electrical conduit!). The figure demonstrates that a segment of the Fourier components has been removed over a band of frequencies centered about 3500 Hz.
Using Fourier analysis it is possible (with some limitations) to determine the acoustic dispersion relation--essentially the speed at which a specific frequency travels through the array. This information extracted from the experiments is shown in the figure below which plots angular frequency (y-axis) versus wave vector (x-axis). The frequency range in which there are no data points lie within the forbidden acoustic band gap.
A complete report of this work has been published in the Journal of the Acoustic Society of America in August 1998.
Since that early work on free space measurements of acoustic band gap materials, my students and I have conducted a series of measurements on one-dimensional acoustic band structures in waveguides made from conventional PVC plumbing pipe. This line of research has been an invaluable source of undergraduate research thesis projects over the years. The breadth of the work we have done is too extensive to give a comprehensive overview here; full details are available from my publication list. However, here is a brief introduction to the sort of work we have done.
Almost all of our measurements are impulse response measurements in which the effects of reflections from the ends of the waveguide can be excluded by time-windowing. We also employ the method of coherent averaging where the test impulse, created by the computer is played and recorded many times. Through the use of triggered data acqisition the timing of the start of the recording is the same from one pulse to the next. By adding these successive test pulses the random background noise averages out whereas the signal adds coherently with each additional pulse. This technique leads to very high signal to noise signals in the time domain. We export the time domain data to MATLAB and use Fourier analysis to determine the characteristics of filters under test.
The figure below shows a schematicof the test configuration. The sound impulses are created in MATLAB and then played through the computer's sound output. One channel contains the audio impulse whcih is directed to an amplifier and then to a speaker coupled to the end of the PVC pipe waveguide. The other channel of the stereo output is used as a trigger signal to make sure that the acquisition of the voltage signal from the microphone always starts at the same time with respect to the output of the audio pulse.
A typical acoustic filter created from different diameter PVC plumbing pipe and couplers is shown in the figure below (the bottom part is a close-up of the upper showing typical dimensions). The regular pattern of equal length large and small diameter pipe corresponds to an alternating high and low acoustic impedance pattern that results in a series of acoustic band gaps.
The figure below shows the analysed results of measurements on two different pipe filters eavch with band gaps centered at about 500 Hz and 1500 Hz. These filters had one section in the center that was of a differnt length than the others. The effect of this "defect" in the acoustic filter results in a narrow trnasmission state within the band gap. The frequency of the defect state within the band gap depends on the size of the defect.
Finally, more recently we have been investigating the the effect of attached resonant structures such as helmholtz resonators or loop filters on the waveguide transmission. The aim of these studies is to create compact filter structures that do not depend of being some multiple of the wavelength as is the case in acoustic band gap materials. A second aim is that sub-wavelength resonant structures are potential building blocks for acoustic metamaterials.
The figure below shows the schematic arrangement for measuring the effect of a side loaded Helmholtz resonator on transmission down a waveguide. A Helmholtz resonator is an acosutic simple harmonic oscillator--the equivalent of a mass on a spring. The spring function is provided by the air trapped in the bottle and the mass by the air in the neck of the resonator. At resonance there is interference such that the transmission past the Helmholtz resonator is strongly attenuated. We made measurements both on single resonators and a sequesnce of 4 sequentially detuned resonators. The detuned resonators had transmission dips that overlapped to give a fairly broad stop band.
The figure below shows the experimental transmission through 4 detuned resonators.
The theoretical transmission through 4 detuned Helmholtz resonators is plotted in the figure below
More complete descriptions of this work and other related acoustic research can be forund from my publication list. If you are interested in the impulse response method with coherent averaging you should particularly look at the article in the Journal of the Acoustical Society of America by me and my student Josh Parker.
Sensors based on Surface Electromagnetic Waves in Band Gap Multilayers
What are surface plasmons?
A surface plasmon (SP) is a collective excitation of the electrons at the interface between a conductor and an insulator. Surface plasmon phenomena crop up in a number of different scenarios: the energy loss of electrons propelled through thin metal foils; the colorful appearance of suspensions of small metallic particles; and the dips in the intensity of light reflected from metal coated diffraction gratings. My interest is in the interaction of light with SPs in thin films, and the use of this interaction to make optical modulators and sensors and as a means to characterize the properties of dielectric overlayers on the metal.
Surface plasmons on a plane surface are non-radiative electromagnetic modes, that is, SPs cannot be generated directly by light nor can they decay spontaneously into photons. The origin of the non-radiative nature of SPs is that the interaction between light and SPs cannot simultaneously satisfy energy and momentum conservation. This restriction can be circumvented by relaxing the momentum conservation requirement by roughening or corrugating the metal surface. A second method is to increase the effective wave vector (and hence momentum) of the light by using a prism coupling technique. The experimental realization of this method is essentially the same as the prism configuration shown in Figure 1 of the Photonic Bandgap Page for coupling to surface waves on photonic band gap arrays.
Why study SPs, you may ask. (Well, you may not have asked, but I'll answer anyway). Early work on SPs explored the fundamental nature of this excitation. My interest at present is in more applied aspects of the SP resonance -- in particular in multiple thin film systems. Most of the interesting SP-mediated effects happen when the metal surface at which the SP is generated is covered with a dielectric thin film. The presence of even very thin films measurably alters the behavior of the SP reflectivity resonance -- typically shifting the incident angle at which resonance occurs and broadening the reflectivity dip. These effects can be used to make devices. For example, if the film is electro-optically active, one can make an optical modulator; chemical changes in the dielectric overlayer can be used to make a chemical sensor.
The characteristics of the SP reflectivity minimum can be analyzed to find the dielectric properties of thin overlayers of materials such as polymers or biomolecules on the surface. This use of surface plasmons for sensing is a commercially important application.
What are surface optical waves on photonic band gap multilayers?
As indicated, sensors based on surface plasmons in metals films are a commercially successful and well-developed technology--Biacore, a division of GE Healthcare is an example of an sensor built entirely on surface plamson technology. Early in my time at MTSU, I realized that the substitution of a suitably designed photonic band gap multilayer in place of the metal film could support surface waves similar to surface plasmons. Through this it is possible to create a sensor with very similar characteristics to surface plasmon sensors but with a number of significant advantages. Two advantages are that a multilayer can be designed to support surface optical waves at any wavelength (surface plasmons are limited by the properties of metal priimarily to the red and infrared) and that the coupling to surface optical waves in multilayers is a much sharper resonance which should translate to a sensor with much higher sensitivity. To appreciate the advantages of PBG materials for surface optical wave applications it is useful to briefly review the properties and optical coupling characteristics of SPs.
Photonic Band Gap Materials. Photonic band gap (PBG) materials are specifically designed periodic composites that can manipulate the properties of light (photons) in much the same manner that semiconductors manipulate the properties of electrons. In PBG materials the coherent effects of scattering and interference in the periodic composite lead to wavelength regions in which the propagation of light is forbidden—so-called photonic band gaps. Most applications make use of this filtering property of the PBG material. However, early researchers (Meade, 1991; Robertson, 1993) recognized that PBG materials support surface electromagnetic waves—optical modes that propagate at the interface of a PBG material. These modes are analogous to surface plasmons (SPs) which are surface electromagnetic waves that can be generated on the surface of metals.
Principles of coupling light to Surface Plasmon Resonance. SP’s are a non-radiative excitation, which means that a SP propagating on a metal surface cannot spontaneously decay into light, nor can light incident directly on a bare metal surface generate SP’s. The limitation results from the fact that although SP’s exist on a metal surface with oscillation frequencies in the optical range, the SP wavelength is always smaller than that of light with the corresponding frequency. The most common method to circumvent this limitation in sensing applications is to use a prism to couple the light with SP’s. In the prism the wavelength of the light is reduced by a factor equal to the refractive index of the prism. Light incident at an appropriate angle (always greater than the angle for total-internal-reflection) creates an evanescent EM field in the vicinity of the reflecting surface that can couple to SP’s. When coupling occurs incident light is converted to SP’s and there is a concomitant reduction in the reflected light. The reflectivity dip as a function of incident angle is the hallmark of coupling to SP’s and it is the sensitivity of the angular position of this reflectivity dip to conditions at the metal surface that is the basis of SP resonance sensors. A schematic of the typical prism configuration for generating surface plasmons is shown in the figure below.
Prism Coupler. The plot below shows typical reflectivity priofiles for two common surface plasmon metals and for one of our PBG multilayer designs.
The plots are theoretically calculated reflectivity dips for a silver layer (red line), for a gold film (blue line), and for a PBG multilayer (black line). Detecting the change in resonance conditions with binding of entities to the surface is the origin of sensing action in all surface wave sensors. However, the most striking difference in the plots is the angular width of the resonance. This resonance width is associated with the amount of dielectric loss in the material. For sensing applications a sharp resonance translates to higher sensitivity. Thus, for surface plasmon sensors, silver is, in principle, more desirable than gold. However, in practice, because silver is too chemically reactive, gold is the compromise material of choice in almost all commercial applications. In contrast, the resonance width of surface electromagnetic waves in the PBG sample is over an order of magnitude sharper than in silver indicating the usefulness of multilayers as sensors. This difference in resonant width occurs because the PBG material is a composite of two dielectric materials each with loss factors much less than even the lowest loss metals. Although these plots are theoretical I, along with my colleagues Dr. Andrienne Friedli and Dr. Stephen Wright (and a host of undergraduate and graduate students from Physic, Biology, and Chemistry) have demonstrated that these modes do indeed serve as effective sensors of biological and chemical reactions.
This web page is a brief introduction to the sort of sensing work we do. Again for more details refer to my publications page which I try (not always successfully) to keep updated.
Diffractive optics manipulate the properties of a plane wave by selectively retarding
portions of the wave front. This retardation is accomplished with the use of a glass
substrate etched with the appropriate pattern. The diffraction by this etched pattern
sends the light in the desired direction. The use of diffraction rather than refraction
or reflection -- the traditional means by which optical waves are manipulated -- permits
a more versatile and powerful means of steering light.
My interest in diffractive optics, and in the particular class of diffractive optics known as spot array generators, stems from my involvement with the Photonics Group at McGill University and their work on optical backplanes for high performance computing and telecommunication switches. A spot array generator is a diffractive optic component that converts a single point of light into an array of spots, each with a specified intensity and position. In the optical backplane work, spot array generators are used to deliver optical power to arrays of electro-optic modulators. In the work being conducted at McGill, these modulators were GaAs / GaAlAs devices based on the self electro-optic effect. These optical modulators are combined with GaAs or silicon circuitry to provide an interface between conventional electronic circuitry and free-space optical interconnects.
At MTSU, I am developing an ever-growing suite of software tools for designing two-level and multi-level diffractive optic components. In addition to being a research topic of continuing interest, I've also integrated this work into the Advanced Lab for our Physics majors. In the first such project (Spring semester '96), two students worked on writing a Simulated Annealing Design Code for simple diffractive optic spot array generators in Mathematica©. The designs were plotted out on transparencies using a laser printer to make amplitude gratings. These amplitude gratings were then characterized optically in the lab.
Transmission Line Matrix Modeling
The TLM Simulato
The TLM technique is a method pioneered by the microwave engineering community. It implements Huygen's principle on a discrete mesh of transmission lines, the simulation proceeds in time increments equal to the propagation speed between nodes of the mesh. You can learn more about the TLM technique by clicking here. My students and I are currently refining a MATLAB© implementation of the TLM technique which we may make available at the end of the summer (or when we are confident that it is working well).
My most recent research project with the TLM simulator concerned the tunneling of Gaussian pulses through the forbidden gap of a two-dimensional photonic band gap array. My results were published in the Journal of the Optical Society of America (go to the publications page for a reference) and my undergraduate student Jeff Parker wrote a paper on superluminal tunneling in an attenuated total-internal reflection configuration published in the Journal of Undergraduate Research in Physics.
The interesting aspect of the photonic band gap array tunneling is that the pulse tunneling through a finite thickness of photonic crystal is superluminal, that is, the group velocity, as measured by following the position of the peak of the pulse, exceeds the speed of light in vacuum. As others have noted there is no violation of relativity or causality in this process because the attenuation of frequencies within the band gap asssures that the energy velocity is always below the speed of light. This work validated the usefulness of the TLM technique as a modeling tool for time-domain phenomena in photonic band gap systems.
Acknowledgement: This work is funded by a grant from the Research Corporation.
COMSOL Finite Element Simulations (coming soon)
Superluminal Pulse Tunneling (coming soon)