Our ongoing research integrating 1.55-µm III-V ridge waveguide gain elements (i.e., diode lasers and semiconductor optical amplifiers) co-axially aligned with and coupled to silicon oxy-nitride waveguides on silicon substrates has made significant strides in the past year.  We are working towards the goal of co-axially coupling III-V laser diodes and semiconductor optical amplifiers with waveguides on Si wafers; to do so, we use techniques consistent with fabricating waveguides on Si-CMOS wafers and integrating the III-V gain elements after all standard front- and back-end Si processing has been completed.

A novel micro-cleaving technique has been used to produce active ridge waveguide platelets on the order of 6 µm thick and 100 µm wide, with precisely controlled lengths (in the current work 300 ± 1.25 µm) and very high-quality end facets.  Typical ridge guide platelet lasers have thresholds under 30 mA.

Passive micro-cleaved platelets have been integrated within dielectric recesses etched through the oxy-nitride (SiO_xN_y) waveguides on a wafer so that the ridge and SiO_xN_y waveguides are co-axially aligned.  Transmission measurements indicate coupling losses are as low as 5 db with air filling the gaps between the waveguide ends, and measurements made through filled gaps indicate that the coupling losses can be reduced to below 1.5 dB with a high index (n = 2.2) dielectric fill.  Simulations indicate that with further optimization of the mode profile in the III-V waveguide, the loss can be reduced to below 1 dB.

We have also performed extensive device design and optimization for co-axial recess integration and have recently completed a comparison of co-axial coupling with the evanescently coupled III-V/Si hybrid integration approach recently introduced by researchers at UCSB and Intel.  The latter comparison revealed that the approach we have taken, co-axial end-fire coupling, and the UCSB/Intel approach, vertical evanescent coupling, are complementary, with each optimal for certain applications.  At the same time it pointed out a number of distinct advantages for co-axial coupling of recess-integrated platelet lasers including higher operating efficiency, smaller device footprint, greater flexibility in choice of materials, lower cost, higher modularity, and easier integration of different wavelength emitters ^[1] .

1. C. G. Fonstad, J. J. Rumpler, E. R. Barkley, J. M. Perkins, and S. Famenini, “Recess integration of micro-cleaved laser diode platelets with dielectric wave-guides on silicon,” in Proc. Novel In-plane Semiconductor Lasers Conference VII, Photonics West 2008, SPIE Conference Proc. vol. 6909O, pp. 1-8.

Optogenetics is commonly used for precision modulation of the activity of specific neurons within neural circuits ^[1] , but assessing the impact of optogenetic neural modulation on millisecond-timescale local and global circuit neural activity remains difficult.  We have developed a novel strategy for designing and fabricating silicon-based microelectrode arrays with customizable electrode locations, targetable to defined neural substrates distributed in a 3-D pattern throughout a neural network in the mammalian brain, and compatible with simultaneous use of a diversity of existing light delivery devices.  Our design of these 3-D electrode arrays provides for both easy electrical and mechanical assembly, and provides for scaling of arrays to up to 1000 neural recording channels and beyond.

Our approach relies upon a number of innovations at the material, structural, electrical, and data acquisition levels.  First, typical silicon-based electrodes that are arranged in a 1-dimensional linear array, or 2-dimensional comb-like fashion, often use linear or tetrode-style electrode locations along the comb’s fingers, with stereotyped spacing and pad sizes.  Our software-driven approach enables variable spacing and pad sizes, so that electrode geometries can be customized to the cellular properties of the brain circuits under investigation.  Second, to support the assembly of such electrode arrays into a 3-dimensional array, we have developed novel electrical and mechanical connector strategies to make the assembly as automated and reliable as possible.  Third, we have developed strategies for amplifying and acquiring data that simplify the use of these probes in an intact, in vivo, mammalian context.  Fourth, we have implemented hybrid electrodes that contain both a low-impedance metallic pad for recording of spike activity, as well as an indium tin oxide (ITO) pad that can report local field potentials (LFPs) without the photo-electrochemical artifacts common in optogenetics.  Finally, these 3-D probes are designed to be easy to use, from design to surgery.  We have developed a user-friendly interface that enables neuroscientists to specify probe geometries based upon neural target geometries and coordinates, and are developing supporting surgical and behavioral strategies for use of such arrays in vivo.

1. X. Han and E. S. Boyden, “Multiple-color optical activation, silencing, and desynchronization of neural activity, with single-spike temporal resolution,” PLoS ONE, vol. 2, no. 3, p. e299, Mar. 2007.

The use of magnetic forces to improve fluidic self-assembly of micro-components has been investigated using Maxwell 3D to model the forces between Ni thin films on semiconductor device micro-pills and Sm-Co thin films patterned on target substrates ^[1] .  Orienting and restraining forces on pills far in excess of gravity are predicted, and it is found that the fall-off of these forces with pill-to-substrate separation can be engineered through the proper design of the Sm-Co patterns to retain only properly oriented pills ^{[1] [2]} .

Micro-scale hybrid assembly is a potentially important way of doing heterogeneous integration, i.e., of integrating new materials on silicon integrated circuits to obtain functionality not readily available from silicon device structures alone, and fluidic self-assembly is an attractive way to automate micro-scale assembly.  A serious limitation of fluidic self-assembly, however, is the lack of a good method for holding properly assembled components in place and accurately positioned until all of the components have been assembled and permanently bonded in place.  We have shown, based on our modeling, that suitably patterned magnetic films can be used to provide the forces necessary to retain, and to accurately orient and position, assembled micro-components.

Our motivation for pursuing micro-scale hybrid assembly is our general interest in doing optoelectronic integration, specifically of vertical cavity surface emitting lasers (VCSELS), edge-emitting lasers (EELs), and light emitting diodes (LEDs), with state-of-the-art, commercially processed Si-CMOS integrated circuits.  Our ongoing research integrating these devices on silicon described elsewhere in this report provides the context for this work and illustrates the types of applications we envision for magnetically assisted self-assembly using the results of this study.

Assembly experiments to verify and demonstrate the theoretical predictions are currently in progress using two sizes of 6-µm-thick pills (50 µm by 50 µm and 50 µm by 100 µm) and a variety of magnetic thin film patterns.  Recesses with different dimensions are also being studied ^[2] .

1. D. Cheng, “Theoretical and experimental study of magnetically assisted fluidic self assembly,” M.Eng. Thesis, Massachusetts Institute of Technology, Cambridge, June 2008.
2. D. Cheng, J. J. Rumpler, J. M. Perkins, M. Zahn, C. G. Fonstad, E. S. Cramer, R. W. Zuneska, and F. J. Cadieu, “Use of patterned magnetic films to retain and orient micro-components during fluidic assembly,” Journal of Applied Physics, vol. 105, p. 07C123, 2009.

Optoelectronic devices intimately integrated on silicon integrated circuits have long been sought for optical intercon-nect applications, optical communications modules, and–more recently–neural stimulation and sensing.  Toward this end we have recently demonstrated a new heterogeneous integration technique for integrating vertical cavity surface emitting lasers (VCSELs) and edge-emitting laser diodes (EELs) on silicon CMOS integrated circuits ^{[1] [2]} .

Fully processed and tested oxide-aperture VCSELs emitting at 850 nm have been fabricated as individual “pills” 55 µm in diameter and 8 µm tall with a disk contact on the n-type backside and a ring contact on the p-type, emitting top-side.  Similarly, 1.55-µm emitting micro-cleaved cavity EEL platelets 5 µm thick, 150 µm wide, and 300 µm long have also been fabricated.  Using a custom micro-pipette vacuum pick-up tool, these micro-laser pills and platelets have been placed on contact pads at the bottom of recesses etched though the dielectric over coating on a Si chip, and batch solder-bonded in place using a custom pressurized-diaphragm bonding apparatus.  Back-end processing of the chip then continues with surface planarization, contact via formation, and interconnect metal deposition and patterning.  An example of a completely integrated VCSEL pill appears in Figure 1.

No adverse effects are seen from fabricating laser diodes as freestanding pills and platelets.  Devices integrated in this manner show the same high performance as devices left on their native substrates and in fact have superior thermal characteristics, largely due to the better thermal conductivity of Si over that of GaAs and InP.

The technique demonstrated in this work offers numerous other advantages over alternative heterogeneous integration techniques.  Both the devices to be integrated, and the target circuit wafers, are fabricated under optimal conditions and are pre-tested and screened prior to integration to insure high yield.   Significantly, many different types of devices can be integrated on the same IC wafer, a feature unique to this approach.  Furthermore, the integration process effectively avoids thermal expansion mismatch limitations and wafer diameter mismatch issues, and it is compatible with parallel assembly techniques, such as fluidic self-assembly.

1. J. M. Perkins, and C. G. Fonstad, “ Full recess integration of small diameter low threshold VCSELs within Si-CMOS ICs,” Optics Express, vol. 16, no. 18 pp. 13955-13960, 2008.
2. J. J. Rumpler and C. G. Fonstad, Jr., “Continuous-wave electrically pumped 1.55 µm edge-emitting platelet ridge laser diodes on silicon,” IEEE Photonics Technology Letters, vol. 21, pp. 827-829, 2009.

Professor Ed Boyden uses light to precisely control neural activity.  His lab has invented safe, effective ways to deliver light-gated membrane proteins to neurons and other excitable cells (e.g., muscle, immune cells, pancreatic cells, etc.) in an enduring fashion, thus making the cells permanently sensitive to being activated or silenced by millisecond-timescale pulses of blue and yellow light, respectively ^[1] .  This ability to modulate neural activity with a temporal precision that approaches that of the neural code itself holds great promise for human health, and his lab has developed animal models of epilepsy and Parkinson’s disease to explore the use of optical control to develop new therapies.

Professors Boyden and Fonstad have established a collaborative effort to use heterogeneous integration techniques developed in Fonstad’s laboratory to construct miniature linear probes to deliver light to activate and silence neural target regions along their length as desired ^[2] .  The goal is to develop mass-fabricatable multiple light guide microstructures produced using standard microfabrication techniques.  Each probe is a 200- to 250-micron-wide insertable micro-structure comprising many miniature lightguides running in parallel and delivering light to many points along the axis of insertion.  Such a design maximizes the flexibility and power of optical neural control while minimizing tissue damage. If 2-D arrays of such probes are built, multiple colors of light can be delivered to 3-dimensional patterns in the brain, at the resolution of tens to hundreds of microns, thus furthering the causal analysis of complex neural circuits and dynamics.  Such devices will allow the substrates that causally contribute to neurological and psychiatric disorders to be systematically analyzed via causal neural control tools.  Given recent efforts to test such reagents in nonhuman primates, these devices may also enable a new generation of optical neural control prosthetics, contributing directly to the alleviation of intractable brain disorders.

The initial light-guide structures have been fabricated from silicon oxynitride clad with silicon dioxide (Figure 1), and tests show excellent transmission of light with no visible loss in the taper and bend regions of the patterns ^[2] .  Significantly, the novel 90˚ bend invented to direct light laterally out the side of the narrow probe functions as designed ^[2] .  The optical sources for initial tests with the probe are independent laser modules coupled to one end of a fiber-optic ribbon cable (Figure 2).  The other end of the ribbon cable is butt-coupled to the inputs of the probe via a standard fiber-optic connector ferrule.  This allows for increased modularity and control in initial probe-testing.  Work on the fabrication of visible-emitting platelet laser diodes to be integrated on a similar ferrule mating to the guides is in process.

1. X. Han and E. S. Boyden, “Multiple-color optical activation, silencing, and desynchronization of neural activity, with single-spike temporal resolution,” PLoS ONE, vol. 2, no. 3, p. e299, Mar. 2007.
2. A. N. Zorzos, E. S. Boyden, and C. G. Fonstad, “A multi-waveguide Implantable probe for light delivery to distributed brain targets,” Applied Optics Letters vol. 35, no. 12, pp. 4133-4135, Dec. 2010.

Figure 1: Triplet exciton transport in tetracene crystals with varying time delays. a, The prompt fluorescence profile. The image was taken with camera integration with a time delay of 0–0.5 μs upon laser excitation. b, c, d, e, Delayed fluorescence profiles with time delays of 0.5–1 μs, 1–2 μs, 2–3 μs, and 3 μs– upon laser excitations, respectively. f, Cross-section of images in a, b, c, d, e. The area of cross-section is denoted in image a.

Exciton transport is universal in every kind of organic optoelectronic devices – including organic light-emitting diodes (OLEDs) and organic solar cells ^[1] . In organic bilayer photovoltaic devices, exciton diffusion limits donor/acceptor thicknesses, restricting sufficient absorptions of photovoltaic materials. Exciton diffusion of singlet excitons (total spin of 0) is usually limited to tens of nanometers, significantly smaller than the absorption length at the visible spectrum (a ~ 1mm) ^[2] . However, triplet excitons (total spin of 1), having disallowed-transition to ground states, are capable of moving much longer distances, up to several mm ^[3] . The long-range triplet exciton transport can allow us to build more efficient solar cells.

Despite their long intrinsic lifetime, triplet diffusion is disorder-limited in amorphous or polycrystalline organic semiconductors ^[2] . Organic single crystals, however, provide defect-free environment where triplet excitons can diffuse over long distances without being quenched by defects ^[3] .

Long-range triplet exciton transport has been reported before in organic acene crystals. However, in previous studies, triplet exciton diffusion was measured using indirect methods, such as probing polarization- and wavelength-dependent photoconductivity ^[4] . In this work, we perform direct imaging of triplet excitons by monitoring delayed fluorescence in two archetypical organic single crystals: tetracene and rubrene crystals. The comparison between tetracene and rubrene crystals is interesting since they exhibit similar molecular structures but differ in crystal structures. Our study will contribute to a better understanding of long-range exciton transport and benefit the power conversion capability of organic solar cells by overcoming exciton diffusion bottlenecks.

1. P. Peumans, A. Yakimov, and S. R. Forrest, “Small molecular weight organic thin-film photodetectors and solar cells,” Journal of Applied Physics, vol. 93, pp. 3693, 2003.
2. R. R. Lunt, N. C. Giebink, A. A. Belak, J. B. Benziger, and S. R. Forrest, “Exciton diffusion lengths of organic semiconductor thin films measured by spectrally resolved photoluminescence quenching,” Journal of Applied Physics, vol. 105, pp. 053711, 2009.
3. M. Pope and C. E. Swenberg, Electronic Processes in Organic Crystals and Polymers. Oxford University Press, New York, 1999.
4. H. Najafov, B. Lee, Q. Zhou, L. C. Feldman, and V. Podzorov, “Observation of long-range exciton diffusion in highly ordered organic semiconductors,” Nature Materials, vol. 9, pp. 938, 2010.

Figure 1: Illustration of ray tracing under (a) normal (spontaneous) and (b) lasing (stimulated) conditions within an LSC. (c) Typical laser transfer characteristic, showing the dramatic change in efficiency above the threshold.

Many on-chip optical applications, including spectroscopy ^[1] ; sensing ^{[2] [3]} ; nonlinear optics ^{[4] [5] [6]} ; and optical communications require high-finesse ^{[7] [8]} , high-quality factor (high-Q) micro-lasers. Such lasers must be exceptionally transparent at their emission wavelength. But if high-Q micro-lasers exhibit correspondingly weak absorption at the pump wavelengths, they are challenging to excite. Here we demonstrate micro-ring lasers that exhibit Q > 5.2 × 10⁶ and a finesse of > 1.8 × 10⁴ with a direct-illumination, non-resonant pump.  The micro-rings are coated with a combination of three organic dyes. This cascaded combination of near and ultimately far field energy transfer reduces material-losses by a factor of more than 10⁴, transforming incoherent light to coherent light with high quantum-efficiency. The operating principle established here is general and can enable fully integrated on-chip, high-finesse micro-lasers without the complications of coupled pump and emitter resonators ^[9] .

We are now working on lasing luminance solar concentrators (LSC) or solar powered lasers based on the cascaded energy concept. Above threshold, all the fundamental properties of an LSC improve. Specifically, (i) the brightness of the lasing LSC can be orders of magnitude larger than conventional solar concentrators; (ii) stimulated emission enhances the photoluminescent efficiency; (iii) it increases the trapping efficiency of the LSC; (iv) and stimulated emission decreases self-absorption. A solar powered laser also, in essence, converts a portion of the incoherent solar spectrum into a coherent source. This conversion enables the solar powered laser light to be frequency converted in nonlinear crystals, allowing harvesting of more of the solar spectrum via efficient upconversion and downconversion for high efficiency photovoltaics. Figure 1 shows ray tracing of a below- and an above-threshold LSC.

1. C. Y. Chao, W. Fung, and L. J. Guo, “Polymer microring resonators for biochemical sensing applications,” IEEE J. of Sel. Top. Quantum Electron, vol. 12, no. 1, pp. 134-142, Jan. 2006.

2. [1]    F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Applied Physics Letters, vol. 80, no. 21,  pp. 4057-4059, April 2002.

3. A. Serpenguzel, S. Arnold, and G. Griffel, “Excitation of resonances of microspheres on an optical fiber,” Opt. Lett., vol. 20, no. 7, pp. 654-656, Apr. 1995
4. R. K. Chang, and A. J. Campillo, Optical Processes in Microcavities. Singapore: World Scientific, 1996.
5. F. Treussart, V. S. Ilchenko, J. F. Roch, J. Hare, V. Lefevre-Seguin, J. M. Raimond, and S. Haroche, “Evidence of intrinsic Kerr bistability of high-Q microsphere resonators in superfluid helium,”  Eur. Phys. J. D., vol. 1, pp. 235-238, Jan. 1998.
6. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultra-low threshold Raman laser using a spherical dielectric microcavity,”  Nature, vol. 415, pp. 621-623, Feb. 2002.
7. B. E. Little, S. T. Chu, P. P. Absil, J. V. Hryniewicz, F. G. Johnson, F. Seiferth, D. Gill, V. Van, O. King, and M. Trakalo, “Very high-order microring resonator filters for WDM applications,”  IEEE Photon. Technol. Lett., vol. 16, no. 10, pp. 2263-2265, Oct. 2004.
8. M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. Little, and D. J. Moss, “Low power continuous-wave nonlinear optics in doped silica glass integrated waveguide structures,” Nature Photonics, vol. 2, pp. 737-740, Nov. 2008.
9. Rotschild, C., Tomes, M., Mendoza, H., Andrew, T. L., Swager, T. M., Carmon, T. and Baldo, M. A., “Cascaded Energy Transfer for Efficient Broad-Band Pumping of High-Quality, Micro-Lasers,” Advanced Materials, vol. 23, 2011

Organic photovoltaics (OPVs) are promising low-cost solar cells: they can be stacked in multi-junctions, and they are compatible with roll-to-roll processing. But as a solar cell’s installation costs are proportional to the area it covers, OPVs’ low efficiencies presently bar their widespread adoption. A significant source of loss in OPVs is the recombination of charges at the donor-acceptor interface: excited electrons combine with holes, returning the system to its ground state, rather than powering an external load. We therefore need to reduce the recombination rates in organic photovoltaics. We consider doing so by taking advantage of spin-disallowed transitions.

Excited states in OPVs come in two flavors of spin: singlets and triplets. Since the ground state is almost always a singlet, quantum mechanical rules prevent triplet excited states from relaxing, so triplets have longer lifetimes—i.e., lower recombination rates. In the absence of spin mixing processes, an OPV that produces electron-hole pairs in the triplet state should be more efficient than an OPV that produces singlets.

To prepare excited states in either the singlet or triplet state, we made a heterojunction solar cell with PTCBI (which produces singlets when excited) and pentacene (which produces triplets when excited ^[1] ). The spectral dependence of optical absorption in the two materials allows us to produce mostly triplets or singlets by exciting the device with 635-nm or 532-nm light, respectively. At room temperature the two show the same behavior; we are now examining the devices at much lower temperatures and under a magnetic field, where the mixing rate between the two states may be low enough to reveal the difference between the two.

1. J. Lee, P. Jadhav, and M. A. Baldo, “High efficiency organic multilayer photodetectors based on singlet exciton fission,” Appl. Phys. Lett., vol. 95, p. 033301, July 2009.

In recent years there has been significant interest in broadband, omnidirectional antireflection (AR) nanostructures that minimize Fresnel reflection due to index (impedance) mismatch at an optical interface ^{[1] [2]} .  The reflection can be suppressed by using adiabatic impedance matching implemented as an intermediate material with gradually varying index in the direction of surface normal. Subwavelength patterning is an effective method to implement such a gradient index (GRIN) surface.  However, these recent studies have been mostly restricted to planar surfaces.  Diffractive optical elements such as diffraction gratings, Fresnel zone plates, and holographic optics also suffer from Fresnel reflection losses evidenced as undesirable reflection orders.  Recently, we proposed a new class of GRIN diffractive optics that is capable of suppressing such reflection losses ^[3] .  Using the same GRIN principles, we can demonstrate diffractive elements where the reflected energy can be suppressed.

The proposed concept of the nanostructured GRIN grating is illustrated in Figure 1, where subwavelength tapered nanostructures with period p are integrated on both the ridge and groove of the grating.  Top-view and cross-section micrographs of the fabricated GRIN grating in silicon substrate are depicted in Figure 2. The grating has a period Λ of 5 mm, and the subwavelength cone-shaped pillars have nominal base diameter of 150 nm. The fabricated structure resembles a grating with nano-engineered surfaces. A cross-sectional micrograph is shown in Figure 2(b), where the cone heights on the ridge and groove are 650 and 600 nm, respectively. Some point defects characteristic of nanosphere self-assembly used in the fabrication process can be observed.  Broadband characterization of the fabricated structure indicated suppression by at least two orders of magnitude in the reflected orders of the GRIN grating over a large range of incident angles up to 60º.

1. P. Lalanne and G. M. Morris, “Antireflection behavior of silicon subwavelength periodic structures for visible light,” Nanotechnology, vol. 8, pp. 53-56, Oct. 1997.
2. Y. Kanamori, M. Sasaki, and K. Hane, “Broadband antireflection gratings fabricated upon silicon substrates,” Opt. Lett., vol. 24, no. 20, pp. 1422-1424, Oct. 1999.
3. C.-H. Chang, L. Waller, and G. Barbastathis, “Design and optimization of broadband wide-angle antireflection structures for binary diffractive optics,” Opt. Lett., vol. 35, no. 7, pp. 907-909, April 2010.

Luneburg lens is a gradient index (GRIN) element ^[1] known to produce diffraction-limited focus at the lens edge opposite to an incident plane wave. Despite its usefulness in applications such as radar systems, omnireflectors, or integrated optics, implementing the Luneburg lens in optical frequencies due to the difficulty in producing the desired GRIN profile. In this work, we describe the design and fabrication of a Luneburg lens for operation at near infrared optical frequencies using subwavelength aperiodic nanostructures.

The Luneburg lens is designed using a dielectric periodic square lattice of circular silicon rods with spatially varying parameters and subwavelength features. If the variation in the structure is gradual enough to be considered periodic within the adiabatic length scale, local dispersion relations can be analyzed through established photonic crystal theory ^[2] , and Hamiltonian optics can be used to analyze and design the propagation of light with adequate accuracy ^{[3] [4]} .  Using the developed algorithm less computational power is needed, and it allows for convenient structure optimization.

We designed a Luneburg lens with lattice constant of λ/6 at operating wavelength λ = 1.55 µm and fabricated a planar (2D) implementation using silicon-on-insulator (SOI) substrate.  The structure was patterned using electron-beam lithography and transferred into the device layer using reactive ion etching.  Although this structure suffers slightly from structure anisotropy, through design and optimization we were able to obtain a geometrical waist diameter calculated as λ/3 at the lens focus, as depicted in Figure 1. The fabricated structure, shown in Figure 2, has minimum feature size of around 90 nm, which can be readily achieved by the resolution limits of our lithographic approach. The fabricated lens is being characterized using scanning near-field optical microscope, and initial results demonstrate tight focusing of light.

1. R. K. Luneburg, Mathematical Theory of Optics, Berkeley, CA: University of California Press, 1964.
2. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals Molding the Flow of Light, Princeton, NJ: Princeton University Press, 2008.
3. P. S. J. Russell and T. A. Birks, “Hamiltonian optics of nonuniform photonic crystals,” J. Lightwave Technol., vol. 17, pp. 1982-1988, Nov. 1999.
4. Y. Jiao, S. Fan, and D. A. B. Miller, “Designing for beam propagation in periodic and nonperiodic photonic nanostructures: Extended Hamiltonian method,” Phys Rev E, vol. 70, pp. 036612, Sep. 1999.