Alberto Parini Foton
email: alberto.parini@univ-rennes1.fr

Yann G Boucher

Ring-assisted modal conversion for on-chip mode-division multiplexing signal processing

Keywords: Optical networks on-chip, mode division multiplexing, microring resonators, photonic integrated circuits, silicon photonics, scattering parameters

We present a system composed of a two-mode waveguide coupled to a single-mode racetrack resonator. In this configuration, each resonance can be exploited to achieve a complete even-to-odd (odd-to-even) mode power conversion, thus allowing transparent channel switching in mode-multiplexed (MDM) networks. The spectral response of the system is evaluated by extending the single-mode scattering parameters formalism to the multi-mode propagation domain.

I. INTRODUCTION

As energy issues are increasingly affecting the scalability of high-performance multi-core microprocessor architectures, the implementation of optical links (Optical Network-on-Chip ONoC) between computational cores is foreseen as a possible way to increase data throughput while, at the same time, reduce the overall microprocessor power consumption [START_REF] Bergman | Photonic Network-on-Chip Design[END_REF].

The majority of the ONoCs solutions proposed so far in literature rely for their operation on wavelength-division multiplexing (WDM) techniques, hence exploiting the wavelength as a degree of freedom for channel allocation. Nevertheless, the effective use of WDM for on-chip signal networking is clearly subordinate to the availability of silicon-compatible laser sources, eventually multi-wavelength. Although some recent results confirmed the technological feasibility of silicon photonic transceivers including III-V bonded laser sources [START_REF] Zhang | 8x8x40 gbps fully integrated silicon photonic network on chip[END_REF], the integration of active cavities on silicon remains a challenging task, with the existing technological solutions providing contrasting performance in terms of available optical power and electric-to-optical power conversion efficiency.

To reduce the number of laser sources required on the chip, or to optimize the use of the available ones, a possible solution relies on the exploitation of a further dimension inherent to optical communications, namely the modal order [START_REF] Ding | On-chip two-mode division multiplexing using tapered directional coupler based mode multiplexer and demultiplexer[END_REF]. In mode-division multiplexing (MDM) systems, communication channels are allocated, on the same wavelength, to the various propagation modes of a multi-mode waveguide. Therefore, MDM techniques make available an additional degree of freedom (the transverse modal spatial distribution) for the design of on-chip integrated optical systems [START_REF] Luo | WDM-compatible mode-division multiplexing on a silicon chip[END_REF][START_REF] Wang | On-chip silicon 8-channel hybrid (de)multiplexer enabling simultaneous mode-and polarization-divisionmultiplexing[END_REF].

The deployment of a complete MDM-based on-chip interconnection network clearly requires the design of a full set of new functional building-blocks (routers, add-drops, mode converters) able to selectively process optical signals with respect to the modal order. In this work, we address in particular the issue of transparent mode-conversion, and we explore the operation of a configuration consisting of a two-mode waveguide laterally coupled to a single-mode resonator. The presence of the resonator breaks the translational invariance of the field propagation in the two-mode waveguide, thus enabling a cross-power exchange between the even and odd modes of the waveguide, which would be otherwise remain orthogonal.

II. SYSTEM LAYOUT AND MODAL TRANSFER FUNCTION

The layout of the investigated configuration is presented in Fig. 1. A two-mode waveguide (WG) is coupled by means of directional coupler of length to the single-mode racetrack resonator (MRR) of total perimeter . The waveguide WG is designed to guide the fundamental TE mode with a transversal even symmetry (TE0) and a first order TE mode with a transversal odd symmetry (TE1), the two modes being characterized by their respective propagation constants and , with . The (single) mode circulating in the microring MRR has propagation constant  and its losses are modeled by the parameter By taking advantage of a formal analogy holding between a two-mode waveguide and a system of two single-mode coupled waveguides [START_REF] Boucher | Analytical model for the coupling constant of a directional coupler in terms of slab waveguides[END_REF], the coupling zone highlighted by the dashed rectangle in Fig 2 (Top), is replaced by a ternary coupler between three single-mode waveguides, as sketched in Fig. 2(Bottom). This substitutions brings back the initial hybrid single/multi-mode propagation problem to an homogeneous single-mode one. Within this new layout, WG1 is the single-mode waveguide belonging to the microring, while WG2 and WG3 are the two single-mode waveguides which result from the decomposition of the original two-mode section. The governing parameters of the transformed system are the propagation constant , which is common to all the waveguides, and the two coupling constants , accounting for the power exchange between WG1 and WG2 and WG2 and WG3, respectively.

After a straightforward application of the (single-mode) scattering parameter approach, the transfer matrix relating the mode amplitudes on the input ( ) and output ) ports of the system finally reads as:

( ) [ ] ( ) (1)

[ ] ( )

For the explicit form of the elements of [ ] we refer the reader to Ref. [START_REF] Boucher | Resonant modal conversion in a two-mode waveguide[END_REF].

All the elements of the matrix [ ] present an explicit dependence on the spectral parameter , hence the degree of mode mixing introduced by the system heavily depends on the working wavelength. In the case of a transparent medium , and for an operating wavelength corresponding to one of the microring with m integer), the diagonal terms of [ ] vanish. The matrix assumes an anti-diagonal symmetry which enables a complete TE0 to TE1 (respectively, TE1 to TE0) mode switching. More explicitly, a TE0 mode on the input port is converted to a TE1 one as it transit through the system and vice versa. On the antiresonance wavelengths

) the vanishing terms in [ ] are the diagonal ones and any incident modal state is transferred to the output port unaffected.

In Fig. 3 we report the evolutions of the and output mode amplitudes for the case of a purely TE0 excitation on the input port ( ), calculated via the Eq. ( 1) as a function of the (normalized) spectral parameter . As expected, on the microring resonance the TE0 mode power on the input port is completely transferred to the TE1 mode on the output port ( ), while on the anti-resonances no conversion takes place. Moreover, between the resonance peaks anti-resonance conditions, an intermediate degree of mode mixing is achievable.

III. CONCLUSIONS

We presented a configuration consisting of a two-mode waveguide coupled to a single-mode microring resonator. The presence of the microring breaks the symmetry of the system thus enabling cross-mode conversion when operating on one of the resonances. Thanks to its mode-processing capabilities, this configuration is suitable to find application as a functional building-block in MDM photonic integrated circuits. 3: Evolution against the spectral parameter β of the TE0 (bTE0 -blue solid line) and TE1 (bTE1 red solid line) mode amplitudes on the output port of the system when on the input port is injected a purely TE0 mode (aTE0=1, aTE1=0). On each microring resonance the TE0 mode is completely converted to the TE1 one.

Figure 1 :

 1 Figure 1: The system under investigation consists of a two-mode (TE0,TE1) waveguide (WG) coupled along a distance d to a single-mode racetrack resonator (MRR) of total perimeter L.

Figure 2 :

 2 Figure 2: (Top) After substitution of the two-mode waveguide WG with an equivalent coupler between two single-mode ones (Bottom), the interaction zone highlighted by the green-dotted rectangle can be modeled as a ternary directional coupler between three single-mode waveguides (WG1,WG2 and WG3).

Figure

  Figure3: Evolution against the spectral parameter β of the TE0 (bTE0 -blue solid line) and TE1 (bTE1 red solid line) mode amplitudes on the output port of the system when on the input port is injected a purely TE0 mode (aTE0=1, aTE1=0). On each microring resonance the TE0 mode is completely converted to the TE1 one.

ACKNOWLEDGMENT

This work is supported by the Labex CominLabs (French National Research Agency program "Investing for the Future" ANR-10-LABX-07-01) within the framework of the "3D-Optical-ManyCores" project.

http://www.3d-opt-many-cores.cominlabs.ueb.eu/


M Cordeiro

P Viana

J Ruela

M Body

B Cousin

D Bommart

G Ferrari

M Strambini

W Bernet

E Müller

M Laurentin

B Algayres

S Daulard

I Hoentsch

T Marx

THE ASSET ARCHITECTURE -INTEGRATING MEDIA APPLICATIONS AND PRODUCTS THROUGH A UNIFIED API

Applications and products currently available for the broadcasting market are vertically integrated or proprietary. They are based on components requiring specific and costly development to interoperate and do rely typically on a single manufacturer or system integrator. Hence they are not fully compliant with broadcasters' requirements.

ASSET is an European funded project whose main goal is to overcome the limitations of custom specific implementations of a digital system for TV content creation. These limitations are generally due to the misfit of interfaces between software layers, proprietary APIs of equipment from different vendors and lack of generalized middleware for multimedia content management with openly defined interfaces.

Besides presenting the ASSET proposed architecture and concepts, this paper describes the prototype under development to test and demonstrate the project proposals.

INTRODUCTION

The main goal of the ASSET project (IST-2001-37379 -Architectural Solutions for Services Enhancing digital Television) [1] [START_REF] Viana | A Unified Solution for the Integration of Media Applications and Products in Broadcaster Environments -The ASSET Architecture[END_REF] is to overcome the limitations for custom specific implementation of a complete digital system for TV content creation. These limitations are generally due to lack of connectivity and interoperability between equipment and between applications, the lack of generalised middleware for multimedia content management with openly defined interfaces and the misfit of interfaces between software layers or between the proprietary APIs of the equipment.

Technical solutions available on the market are vertically integrated or proprietary. They are based on components requiring specific and costly development to interoperate and do rely typically on a single manufacturer or system integrator. Therefore they are not compliant with requirements of end-users (in particular, the broadcasters).

These limitations are usually due to:  Databases of different type with respect to access and data mode  Lack of a standard for multimedia Content exchange  Proprietary APIs and protocols to control and interconnect components and applications

The ASSET project is defining and developing a software architecture and the corresponding technologies necessary for a unified management of digital TV content which covers the complete operational workflow, including acquisition, creation, editing, control, storage, broadcasting, publishing and archiving of digital TV content.

ASSET is therefore defining and/or implementing:

 A harmonised data model and interfaces for accessing and manipulating the multimedia objects covering the essence (audio and video) and the associated metadata, based on existing standardisation work.  A Command and Control System between the different ASSET components.

ASSET is defining a middleware layer, based on distributed technology, in order to expose an interface for controlling the different ASSET devices and servers.  A reliable way to interchange audio-visual program material, system and descriptive metadata between the different ASSET components as integral part of a production and content management system. [START_REF] Smpte | Material Exchange Format (MXF); File Format Specification[END_REF], standard data models for describing essence, XML [START_REF] W3c | Extensible Markup Language (XML) 1.0 (Second Edition), W3C Recommendation[END_REF] and distributed systems technologies, for defining the concept of an Asset Middleware. Demonstration of the approach is based on the development of a simple workflow of a Newsroom Platform prototype based on the ASSET architecture and API.

ASSET ARCHITECTURE

The ASSET architecture is based upon a software frameworkthe ASSET frameworkcomposed by a set of three standard interfaces and protocols for applications and products working together in an integrated environment. Applications communicate with the framework using the ASSET Public API, which allows them:

 to communicate with any other application,  to access the public services provided by the framework,  to use additional functionalities implemented by third party integrators as aggregated services.

Products from different manufacturers are integrated in the ASSET framework either natively (using an ASSET agent) or via an ASSET proxy. They are controlled and managed through the Media ASSET Bus API, based on XML service schema definitions. This API ensures an easy and seamless integration of devices and products in the framework.

An ASSET Private API is defined to access core services of the framework (common services) that enable the workability of this integrated environment. This API is only used internally in order to guarantee the overall system integrity.

Figure 1 illustrates the different components of

ASSET Components

The ASSET Architecture defines a number of concepts, components and functions that enable the implementation of an ASSET Compliant Framework. The core of the ASSET Framework consists of three main components:

 The ASSET Public Services expose the mandatory services of the ASSET framework to the applications and to the aggregated services through the ASSET Public API. These services provide a minimum set of multimedia functionality, sufficiently rich and extensible to not limit the system efficiency. They ensure the consistency and integrity of the system by handling the internal logic (e.g. access right, resource allocation, etc.) required for each operation.  The ASSET Common Services provide implementation of key infrastructure requirements such as security, logging, notification, resource management, etc. This allows a uniform and single implementation of these services throughout the framework. They expose a private API that can be only used by ASSET Public Services.  The ASSET Function Services provide an abstraction of functionalities (encoder, recorder, player, etc.) to the ASSET public services. They hide the specificities of the different interconnected products (e.g., for a public service, a VTR output and a Video Server output are considered as two system-wide logical output ports).

At the Application and Business Logic layer, three components are introduced:

 The ASSET Compliant Applications are the top level ASSET software components. They use the ASSET Public API to access services provided by the ASSET Framework and, optionally, by ASSET aggregated services.  The ASSET Services Library is a software component included in (or linked with) an application, which makes it compliant with the ASSET framework and gives it access to the ASSET public services.  The ASSET Aggregated Services implement additional business logic on top of public services (or even other aggregated services). They register in the framework as new services available for ASSET compliant applications (e.g., complex workflows may be specified as aggregated services and then be made available to other connected applications or aggregated services).

At the product layer, we call product a manageable hardware or software component that implements one or several common functions (e.g., most of the Video Server products implement a Recorder function, a Player function and a Storage function) and logical components (logical ports, repositories, etc.). Two ways exist to make a product compatible with the ASSET framework:

 An ASSET Compliant Product is a product that is managed by the framework through a built-in ASSET Agent.  A Legacy Product is a product that has not (or cannot have) a built-in ASSET agent.

The ASSET framework can nonetheless manage such a product through an external software module called an ASSET Proxy (e.g., a VTR is not capable of including a built-in ASSET agent and has to be connected through an ASSET Proxy).

Media ASSET Bus

Most of the ASSET framework is built on top of a software bus called the Media ASSET Bus or MAB.

The goal of the MAB is to provide support for the integration of the widest range of products within the ASSET framework, independently of the underlying operating system environment and protocols.

The MAB defines a set of standard interfaces and synchronization processes, which ensure a seamless interoperability between ASSET components and products. These interfaces allow any third party media application or product to be integrated with the Media ASSET Bus by developing a simple software adapter in an agent or a proxy. A MAB Software Development Kit (SDK) is provided in order to facilitate this task. It gives a uniform way of connecting devices, registering services and exchanging messages within the ASSET framework.

The integrated environment offered by the MAB is based on a Transport Abstraction Layer (TAL) that takes care of message exchanges. Messages have XML format, thus ensuring flexibility and adaptability of the ASSET solution.

The concept of the Media ASSET Bus is illustrated in Figure 2. Different ASSET components and products are interconnected via software adapters on top of the MAB SDK.

DEMONSTRATION PLATFORM

The demonstration platform, which is being implemented as part of the ASSET project, has the purpose of showing the feasibility and simplicity of integrating different broadcast systems via the ASSET middleware. It should also serve as a reference system, where end users, administrators and integrators can test and verify the implemented functionality against the requirements, thus validating the result of the project.

The demonstrator uses one client and two providers of different manufacturers to emulate a simple workflow of a News Platform. The client is a newsroom application that controls the ingest process and lists the content stored on the online video server. The providers are an ingest server and an online storage system provided by a video server. Both the providers and the client application are connected to the ASSET middleware. All the Command & Control operations and the data exchange are controlled by the components of the ASSET middleware.

Interaction between the different components used in the demonstrator is presented in Figure 3. b. Metadata PS: is the central access point for accessing, creating and modifying any kind of metadata in the system (content management metadata, descriptive metadata, structural metadata).

MAB Transport Abstraction Layer

Recorder

c. Repository PS: provides the interface for the applications to access and to modify content stored in any repository (online, near line).

1. Common Services a. Service Repository CS: provides a mechanism for services (Common and Function services) to register to the framework so that they can be addressed and accessed by other services or applications.

b. Notification CS: provides a subscription mechanism for other services and applications to be notified upon events like creation and modifications to objects.

c. UMID Generator CS: provides a unique material identifier according to the SMPTE standard [START_REF] Smpte | Unique Material Identifier (UMID)[END_REF].

d. Content Management Metadata CS: maintains the link between a UMID and the repositories where the material is located.

1. Function Services a. Record FS: provides an abstraction layer (entities, data exchanged) for controlling Audio/Video input ports (e.g. sending differed commands to control the recording of a media asset). In the demonstrator scenario the Record public service uses the Record function to control the input port of the acquisition server.

b. Repository FS: provides an abstraction layer (entities, data exchanged) for controlling all the storage repositories of a system. The repository for the demonstrator is an online storage video server storing the material ingested by the Acquisition service.

Demonstration Hardware Architecture

The demonstrator components are connected using different network technologies as presented in Figure 4. Figure 4 presents the network architecture of the demonstrator platform.

Register Record and Repository Functions -Use Case

Some use cases were developed to model the behaviour and the interaction between different modules in the demonstrator. Figure 5 presents the Record and Repository FS registration into the ASSET framework. The following steps are executed: A0 -All the PS subscribe for notifications to the Notification CS in order to receive notifications they are interested in -the Metadata PS to receive all modifications regarding the creation, deletion, move, and modification of content, and the Record and Repository PS to receive notifications about the corresponding newly registered functions.

A1 -The Record and Repository Proxies are registering to the ASSET framework. The Service Repository CS creates the corresponding function services acting as client stubs for the proxies. After that, the corresponding PS are notified about the new registrations through the notification common service. In the general case, the public services would read the pre-defined configuration from the Configuration Management CS. In the demonstrator scenario the configuration for the proxies are stored in configuration files read by the corresponding PS. The PS initialises the corresponding proxy by calling the function service' Initialise command passing the configuration in the XML data parameter.

A2 -After its registration, the Repository proxy sends a notification describing its content. This notification is received by the Metadata PS, which updates the Content Management CS information regarding the content location (repository) and media assets structures. The software architecture and the main ASSET concepts are described and a simplified prototype that is expected to test and validate the project approach is presented.

Figure 3 -

 3 Figure 3 -ASSET demonstration platform components

Figure 4 -

 4 Figure 4 -ASSET network architecture for the demonstrator platform

AFigure 5 -

 5 Figure 5 -ASSET demonstration Registration Scenario

  The partners of the ASSET project (HP, THOMSON, Dalet a.n.n, INESC Porto, INRIA, IRT, FPDI, SHS Multimedia) are exploiting open standards and emerging concepts and technologies like MXF

A SSET Public AP I Media ASSET Bus API ASSET Services L ibrary ASSET Aggregated Service ASSET Aggregat ed Service Application & Business Logic Layer ASSET Framework

	the ASSET architecture.
	ASSET Compli ant
	Applicati on
	Common Services • Service Repositor y • Security • Notification • Debug/logging • Management • Session • Transaction • License	ASS ET Privat e A PI	ASSET Public Service ASSET Functi on Service	ASSET Public Service ASSET Function Service	ASSET Public Service ASSET Function Service
	ASSET Proxy		ASSET Agent		ASSET Agent
	Legacy Protocol	ASSET Compliant Product	ASSET Compli ant Product
	Legacy Product
					Product Layer
	Figure 1-ASSET Architecture

MAB SDK MAB SDK MAB SDK MAB SDK MAB SDK MAB SDK

	Recorder	Player		Player	Metadata
	Functi on	Function	Public	Public	Public
	Service	Service	Service	Service	Service
			MAB SDK	MAB SDK
	VTR Proxy	Video Server Agent	Archive Server Agent
			ASSET Compliant	ASSET Compliant
	VTR		Video Server	Archive Server
	XML based ASSET messages Legacy protocol
	Figure 2 -Media ASSET Bus

Acquisition&Ingest

ASSET Framework

Service Consumer Newsroom Application Video Server Repository Function Service Proxy Proxy Cmd&Ctrl via MAB SDK Content/File Exchange Content Management Common Service Record Public Service Cmd&Ctrl via HTTP and XML

		Metadata	Repository
		Public	Public
		Service	Service
	UMID Generator
	Common Service
	Notification
	Common Service
	Service Repository Common Service	Record Function
		Service

This work is expected to improve the interconnection between equipment and media applications in digital TV environments, solving some of the problems users and system integrators face currently.


P Fraundorf

Piecewise-Continuous Nano-Worlds Online

The familiar 1D barrier-tunneling problem, when converted to a trough-scattering problem instead, leads us to the strong-phase-object deBroglie phase-lag. This in turn reminds us of the exact Coulomb-scattering solution to the Schrödinger equation, and opens the door to real-time online electron-optic simulations with which students may discover: (a) ways to acquire and interpret data from modern multi-million dollar tools of electron-optics, (b) nano-characterization of matter on the atomic scale, (c) visualizations intermediate between direct and reciprocal space for harmonic-analysis of spatial-periodicities, and (d) important modern uses for crystal defects, including crystal surfaces themselves. CONTENTS I. Introduction 1 II. The deBroglie phase-lag 1 III. Coulomb/Rutherford scattering 2 IV. Electron-optics simulation 2 V. Nano-worlds online 3 VI. Discussion 5 A. underlying math 5 1. phase-lag and projected potential 6 2. point-charge scattering 6 3. electron-optics phase/amplitude contrast 7

I. INTRODUCTION

The one-dimensional barrier-tunneling problem is very popular in modern physics texts because it is: (a) analytically solvable, (b) connected to practical problems like alpha-decay, and (c) a beautiful illustration of the role of uncertainty in behaviors that would be classically disallowed. What is seldom mentioned, however, is that the same piecewise-constant potential setup, using a potential trough instead of a barrier, also makes contact with emerging science and technology. We show further that the exact Schrödinger-equation solution to the Coulombscattering problem (approximated classically by Rutherford) can similarly be inverted to reinforce this result.

In this note, in particular, we describe how that piecewise-constant potential-trough calculation gives rise to the strong-phase-object deBroglie phase-lag. We then show how calculations using this phase-lag are being used a) Electronic mail: pfraundorf@umsl.edu (with help from modern electronic technology) to provide students with a real-time window on the world of nanomaterials, as well as on ways currently under development for exploring that world with help from high-speed electrons.

II. THE DEBROGLIE PHASE-LAG

This is the classic one-dimensional problem: Define the vacuum wavevector k = 2π/λ v = √ 2mU / and the specimen wavevector K = 2π/λ s = 2m(U -V )/ , where U is the incident particle energy and V is potentialenergy associated with our particle in the "specimenregion" e.g. of width L.

In the barrier-tunneling case 0 < U < V , so as to make the wave classically disallowed in the specimen region. In the trough-scattering case, V < 0 < U so that wave-vector (hence momentum p = k = h/λ) of our particle actually increases in the specimen-region, and therefore our particle's wavelength (e.g. in [meters per cycle]) decreases. It's not surprising then that the size of this wavelength decrease is directly related to the number of deBroglie phase cycles which take place in traversing a specimen-region of thickness L.

One can of course also calculate the various reflected and transmitted wave amplitudes (relative the amplitude incident from the left) by equating wavefunctions and wavefunction spatial 1st-derivatives at specimen-region entry and exit points. This has been done for a wide range of V /U values in Figure 1. The diagonal bands in the lower right quadrant show clearly the deBroglie phase-lag running in the negative z-direction roughly in proportion to V /U for V < 0.

For application to high energy (U = 60 to 300 [keV]) electrons, condensed-matter attractive-potentials (like work functions) are in the negative tens of Volts so that |V /U | ≪ 1. The exact phase shift relative to V = 0 case is plotted (solid line) for unit-arguments in Fig. 2, along with two approximations. The first approximation (dashed) is simply the difference in wave-numbers (e.g. in [radians per meter]) times the "specimen-thickness" L, which assumes that the first order effect on the transmit-FIG. 1. One-dimensional wavefunctions (in logarithmic complex-color online) of an energy U wave incident from the left on a variable-V/U barrier/trough between 0 < z < 1. In each of these panels, the horizontal axis runs (left to right) from -6 ≤ z ≤ +6; if we set particle mass and Planck's constant to 1, while the vertical axis shows step-potentials ranging (bottom up) from -6 ≤ V /U ≤ +6. In the rightmost panel, the dotted horizontal-band is the region of positive barrier still lower than the particle energy (transmission still classically allowed), while the dotted vertical-band marks the barrier/trough region itself.

ted phase is simply the wavelength shift during traversal.

This in turn can be Taylor-expanded around the high particle-energy U limit of V /U → 0 to get the phaseshift of ∆φ ≃ -(V /U ) mU/2L/ . This gives us the straight line (dot-dashed) in Fig. 2 which is proportional to "projected-potential" Φ = V L e.g. in [eV-m].

For a specimen-potential which varies along a beampath in the z-direction, this might be written as Φ = L 0 V [z]dz. This prompts us to ask, however, whether such a phase-lag (with lateral variation in x and y) will also emerge if the trough for a plane-wave flowing through a three-dimensional solid is not flat, but instead shaped like the attractive Coulomb-potential for electrons passing through a set of atomic-nuclei. Amazingly, another of the rare classical exact-solutions to the Schrdinger equation, with a potential appropriately inverted, comes to the rescue once again.

III. COULOMB/RUTHERFORD SCATTERING

It's often forgotten that the classical Rutherford scattering problem is connected to an exact quantummechanical solution to the Schrödinger equation in threedimensions, which perhaps by accident 1 gives the same result as the classical calculation e.g. for atomic-number (Z) dependent nuclear back-scatttering. As in the case of the tunneling example discussed above, however, this problem is generally framed as a barrier-problem i.e. for predicting what happens when a positive-ion encounters a positively-charged atomic nucleus.

Once again, something also interesting emerges when we look at the "attractive" or "potential-well" version of the same calculation. As shown in Fig. 3, just as in the 1D case the interaction e.g. of an incoming electron with a localized positive charge also gives rise to a deBroglie FIG. 2. Phase-shift estimates using (solid) an exact piecewisecontinuous potential example, (dashed) the wavenumberdifference approximation, and (dot-dashed) the first term in the approximation's Taylor series around the V /U ≃ 0 or "high-energy U particle" approximation. phase-lag. Of course the Coulomb potential of atomic nuclei in condensed matter is screened at Ångström distances by the field of the electron cloud, but the principle underlying the observed phase-lag none-the-less remains that seen in our trough-scattering calculation above.

IV. ELECTRON-OPTICS SIMULATION

Thus introductory quantum-mechanics predicts an exit-surface deBroglie phase-lag, which varies depending on the projected-potential of atoms along the electrontrajectory, when an electron passes through a thin slice of condensed matter. Several issues remain, however. First, how can we convert that phase lag into a '"detected" map of projected-potential across the transverse coherencewidth of an '"individual" electron's wave-packet! Secondly, how does this projected-potential model fare both qualitatively and quantitatively as the specimen's thickness increases?

The answers to these questions are well established in the technical literature 3-5 , so we won't elaborate on them here except to illustrate the "Scherzer defocus" condition 6 (Fig. 4) for converting the phase-lag map at the exit surface into an intensity-map downstream, and then to mention two calculations that physics undergraduates might enjoy.

The "Scherzer strategy" yields useful image intensities for spatial-frequencies out to the "point-resolution maximum". It does this primarily by moving the first dark/light intensity Fourier-phase inversion away from the origin in frequency-space (the "DC-peak"), associated with "spherical-aberration transfer-function zeros" of a convex magnetic lens, until a new pair of transferfunction zeros begins to insert itself into the region between DC-peak and that first-zero.

For sufficiently thin (weak phase-object) specimens, where the deBroglie phase lag is much less than 2π, Scherzer-defocus yields intensities that are inversely proportional to projected-potential i.e. where openings between atoms are bright and columns of atoms in projection look dark. The strong phase-object (arbitrary phase-lag) analysis discussed in addition shows a subset of thickness effects, but only incompletely considers multiple scattering.

The first enjoyable calculation may be to verify that electron-phase contrast is obtained from individual electrons. Not only are electrons fermions that obey Pauli exclusion, making them unwilling to share a single coherent state like photons do in a LASER, but typi-cal high-energy transmission-electron-microscope (TEM) specimen-currents are quite low i.e. in the nano-Ampere range.

Students might in that context use electron-energy to determine the speed of e.g. typical 100[keV] microscope electrons, and show that individual electrons in a nano-Ampere beam will be something like an inch apart in the direction of propagation. An electron-energy spread and hence energy-uncertainty at the parts-per-million level will increase the longitudinal coherence-length' of these electrons, but not nearly enough to allow them to overlap in the specimen even if we had a way to get them to share a common deBroglie phase. Hence on average there is only one electron traversing the specimen at a time, so all wave-interference (including diffraction effects) is being done with a single quantum-particle.

In order to map projected-potential laterally across multiple atom-columns in a condensed-matter specimen, of course, the transverse coherence-width of individual electrons must be expanded to many times the distance between atoms as well. This is done in modern electronoptical systems by reducing the transverse-momentum uncertainty of electrons at the specimen, largely by choice of a locally bright electron-source (like a LaB 6 or fieldemission gun) to begin with.

The second calculation involves the difference in calculation-time between strong-phase-object and morequantitative "multi-slice" image simulation. The strongphase-object approach, as we'll see, simulates a wide range of phenomena qualitatively. For quantitative simulation, however, one must propagate the incident beam through multiple slices of the specimen.

For an n × n pixel image field this is generally done in an n × n complex-coefficient "reciprocal space". This means that each slice past the first requires that one do n 2 more strong-phase-object calculations. For a minimally useful 256 × 256 pixel field, this means that simulations will take some integral multiple of 65, 536 times longer, and hence be difficult to do in real time even on systems with relatively fast processors.

V. NANO-WORLDS ONLINE

The simplicity of the deBroglie phase-lag and the power of the resulting strong-phase-object approximation, combined with the speed of fast-Fourier-transform (FFT) algorithms implemented in javascript on modern processors, gives us an ability to simulate online in realtime the exploration of nanomaterials with high energy electrons. This article is not the place to detail the many types of explorations that are now possible online, in part because they make contact (beyond the worlds of electron scattering and electron optics) with condensed matter topics like: (a) crystallography and bulk/surface defect-crystallography, (b) pair and pair-pair correlations in non-crystalline materials, as well as (c) the behavior of individual molecules in the neighborhood of surfaces, FIG. 4. Electron phase contrast transfer function and wavefield image analysis: At left find maps of contrast-transfer versus spatial-frequency without (left-half) and with (right-half) spherical aberration, as a function of imaging-system defocus in the vertical direction. At right find images (left-half) and image power-spectra (right-half) showing contrast transfer zeros at Gaussian and Scherzer defocus. In the thin-specimen limit, cyan bands (color online) in the panel at left show specimen-regions of increased projected-potential as darker, while red bands show contrast as reversed. FIG. 5. Fig. 5: Brightfield high-resolution TEM (phase) and darkfield scanning-TEM (amplitude) screenshots of an "unknown" specimen in an online simulator 2 , with the same defocus, orientation, and darkfield-aperture. FIG. 6. A hemoglobin molecule in an online simulator, with a likely alpha-helix (circle at top) viewed end on in the absence of the vacuum re-arrangement, beam damage, and the need for a "support" that might get "in the way" experimentally.

of vacuum, of high energy electrons. Such explorations will be discussed by ourselves and others under separate cover.

However this does not make these developments irrelevant to undergraduate physics students and teachers. Far from it, since such simulators give students phenomenological access (at least qualitatively) to a wide range of phenomena, including small-wavelength (flat Ewaldsphere) diffraction from three-dimensional single and poly crystalline unknowns, strain imaging tools (including both diffraction contrast and digital darkfield interferometry), an ability to form darkfield images of selected periodicities, to switch between deBroglie phase-contrast and Z-contrast imaging modes, to get 2D "Fourier-phase information" about specimens that is impossible to obtain directly from diffraction, etc. Figure 5, for example, shows two-screen shots of a near-Scherzer-focused online simulator examining an Au/ZnO 2 bi-crystal tilted so that the phase-interface is side-on. The left-half of the figure provides a strongphase-object approximation deBroglie phase-contrast image of atom columns in projection down that orientation (top-left), and diffraction pattern of the crystal (topright), a power-spectrum of the phase-contrast image (bottom-right), and the complex-color digital-darkfield image (obtained from the image Fourier transform) of periodicities selected by an aperture (white circle) in the power spectrum.

The right-half of the figure in effect reverses the path of electrons through the electron-optic system, which by reciprocity allows us to record (with comparable resolution) an incoherently-illuminated high-angle annular darkfield image of the same specimen. The physics of this type of image is somewhat different, and not the topic of this paper here. Nonetheless students and teachers can get a visceral feel for the qualitative differences between these imaging modes, whether or not they later learn to understand, and/or apply, such tools for nanoworld detective work of their own downstream.

VI. DISCUSSION

As discussed above, the piecewise-constant potential calculation used in undergraduate physics courses to explain quantum tunneling can also yield for us insight into the deBroglie phase-lag which is central to modern electron-based tools for experimental study of condensed matter. We try to make this connection here so that teachers, and textbook authors in the future, can put the connection to use.

There is also an exciting future in physics education for the kinds of electron-image simulators to which this deBroglie phase-lag connects. Students already get some visceral experience with electron-diffraction and electronimaging as it is used today to provide insight into a wide range of condensed matter challenges in the physical and life sciences. In the latter context Fig. 6 shows what the alpha-helix in a hemoglobin molecule might look like edge-on, if we can find a way to either average over or lessen the effects of vacuum and ionization damage in real electron microscopes.

A wide range of other interesting "nano-specimens" remain yet to be assembled. In addition to specimentilting, future incarnations of these online simulators will allow for other imaging and analysis modes, specimen translation, and even zooming from nanometer to millimeter field-widths on procedurally-generated specimens.

Today's physics teachers and students can help apply their own skills to these developments, which mainly require access to the internet through a web-browser. In fact, these developments in turn will help with development of improved-interfaces to the modern multi-million dollar aberration-corrected instruments now available at selected facilities for researchers with interesting nanoworld puzzles to solve. FIG. 7. Plots of electron potential V [x, y, z] for electron trajectories along two different "columns" through the specimen.

phase-lag and projected potential

This is a standard homework exercise e.g. for modern physics, but it might not hurt to outline some details here. Consider a one-dimensional potential of the form:

V [z] =      0 if z < 0 (region 1: upstream) -eΦ if 0 ≤ z ≤ T (region 2: specimen) 0
if z > T (region 3: downstream) (A1) where Φ is an average projected-potential in the specimen column of interest. This average potential (cf. Fig. 7) depends strongly on the proximity of atomic nuclei in the specimen to the selected column, and will e.g. be much less if the column follows a tunnel between atoms through the structure. Our objective is to examine ways that the transmitted wave reflects changes, from columnto-column, in this projected-potential.

For incident plane waves of the form Ψ[z, t] = Ae ı(kz-ωt) in the positive z-direction with electron kinetic energy E r = eV r , write out the Schrödinger equation for regions 1, 2 and 3 with parameters specified in terms of the quantities given above. The one-dimensional non-relativistic free-particle Schrödinger wave equation, for instance, may be written:

- 2 2m ∂ 2 ψ ∂z 2 + V [z]ψ = ı ∂ψ ∂t , ( A2
)
where ψ is the wave function whose amplitude squared yields a probability density as a function of position and time, the expression ı ∂/∂t on the right is the "energy operator", and the first term on the left represents the classical kinetic-energy operator i.e. the "momentum operator" ı ∂/∂z squared over 2m. The Schrödinger equation basically differs from region to region here only by the addition of the negative but piecewise-constant workfunction potential energy -eΦ which multiplies the wavefunction in region 2.

Next write out the form for wavefunction solutions of that equation in each of these regions, in terms of some yet undetermined constants. The high-energy electron wave in the first two regions might be described for region 1 (upstream) as ψ 1 [z] = Ae ıkz + Be -ıkz , and for region 2 (specimen) as ψ 1 [z] = Ce ıkz + De -ıkz . Here the "'vacuum wavevector" k = 2π/λ v = √ 2meV r / while the "specimen wavevector" K = 2π/λ s = 2me(Φ + V r )/ is larger (implying a smaller electron wavelength λ s in the specimen) because the solid attracts electrons. We'll leave it as an exercise for the reader to do the same for region 3 (downstream), using say F and G for the forward and reverse wave coefficients in each case.

Following this, write out boundary conditions that these constants obey at the boundaries between region 1, 2, and 3. For instance at the z = 0 boundary between regions 1 and 2, equating ψ values gives A + B = C + D while equating ∂ψ/∂z values gives ık(A -B) = ıK(C -D). Do the same thing then for the z=T boundary between regions 2 and 3, and then solve for the "amplitude" of the transmitted wave at z = T as a function of the incident amplitude A, and wave vectors k and K in regions 1 and 2. Also show how this simplifies when V r ≫ Φ. The results are discussed in the text.

At relativistic speeds, electron momentum goes from

p ≃ √ 2m e E k to p = 2E k (E o + E k )/c.
Here rest energy E o ≡ m e c 2 , where c and m e are lightspeed and electron rest-mass, respectively. Hence wavenumber k = 2π λ is changed via the dependence of wavelength on momentum λ = h/p. Also, the ratio between wavelength outside the specimen to wavelength inside the specimen (i.e. the specimen's refractive index) changes subtly as well. Therefore the factor of 1 2Vr , which multiplies projected potential to get the optical path difference, becomes 1

Vr

Eo+E k 2Eo+E k . Otherwise, the above analysis of phase shift remains intact 3,4 .

point-charge scattering

Following e.g. Messiah's Chapter XI on the Coulomb Interactiont 1 , a naked-charge wavefield model may be obtained if the 1/r Coulomb potential term is inserted into Schrödinger's partial differential equation:

- 2 2m ∇ 2 + Z 1 Z 2 e 2 r ψ[ r] = Eψ[ r]. (A3)
This is of course the hydrogen atom potential, but here we'd like the solution in a form that can be broken down into incident and scattered plane wave components. Such a solution may be expressed in terms of the threeargument confluent hypergeometric function 1 F 1 [u|v|w] as:

ψ c = Ae ıkz 1 F 1 - ı ka |1| ık(r -z) , (A4)
where A is an adjustable constant and a ≡ 2 /(Z 1 Z 2 me 2 ) is inversely proportional to the interaction strength. This wave function is the one that is plotted in Fig. 3. The above differential equation might be modified for addition of a neutralizing charge sphere at radius R, by merely stepping the potential for r > R down to zero. Solutions for no potential of course generally look something like Ae ıkz . Since the solutions for both potentials in standalone form are known, one might take the next step with a composite wave model for the neutralized charge by simply joining those two solutions up at the spherical r = R boundary.

Spherically-symmetric atomic-scattering-factors are often used instead to propagate a planar (or convergent/divergent) wavefront though the specimen. For quantitative work, multislice strategies 5 consider the sequential evolution of a scattered wave through atomically thin slices of specimen in sequence.

Regardless, to first order the electron wavefunction (whose transverse coherence is many atoms in width) will exhibit a deBroglie phase-shift (relative to nearby columns) which is from section A 1 proportional to the projected-potential at the exit surface of the specimen. The problem is that an image of the phase-contrast in the exit wave-function cannot be detected directly, and so in the next section we turn to electron optics for a solution to that problem.

For sufficiently thick specimens and/or with incoherent illumination, atomic columns will also exhibit an amplitude decrease due to scattering which is a monotone increasing-function of projected potential. Given the projected-potential this too is relatively easy to approximate roughly.

electron-optics phase/amplitude contrast

Scattering in quantum mechanics results in an electron wave which is both phase shifted, and changed in direction. A classic use of the Born approximation, the WKB method, and partial wave analyses has been calculating the angular dependence of scattering from single atoms, based on the details of charge distribution within -Rutherford's famous discovery of the nucleus was an early consequence thereof. Fortunately for the electrons discussed here, these angular issues are less important because scattering angles are well under 10 milliradians.

If one uses apertures which are large enough to include both scattered and unscattered beams (e.g. in the back focal plane of the objective lens, where exit surface waves are separated according to direction rather than amplitude), then one can imagine imaging the amplitudeadded unscattered and scattered wavefields together (at least over specimen distances less than the transverse coherence width of incident electrons). Hence the first practical requirements for electron phase contrast imaging are thin specimens, moderate coherence width illumination, and large objective apertures.

Under these conditions, the imaged exit surface wave from a thin crystal looks like the incident wave, multiplied by e +iσ , where σ[x, y] ≃ k 2Vr Φ[x, y] varies with lateral coordinate values x and y across the specimen. If the phase angle is small (i.e. much less than π), the multiplying factor can be approximated by e +iσ ≃ 1+iσ.

Imagine a one dimensional specimen with only spatial frequency q = 1/Λ, where Λ is the repeat distance, e.g. so that σ[x] = σ o cos[2πqx]. At the exit surface, the wavefield will have been multiplied by ψ

[x] = 1+iσ o cos[2πqx].
Note that, as long as σ o << 1, the amplitude of the composite wave field is hardly affected by the specimen's periodicity. In light microscopy, this problem is overcome by inserting a Zernike phase plate 4 for shifting the scattered beam only by an additional π/2, resulting (when recombined) in first order amplitude effects.

In the absence of a suitable phase plate, one can instead defocus one's imaging optics so as to sample the wavefield downstream (or virtually, upstream as well) from the exit surface of the specimen, at a point where phase differences at the exit surface interfere with one another to create ripples in intensity which may be photographed. What defocus setting is appropriate depends on the spatial frequencies of interest.

Physically, the scattered wavefield evolves by convolution with the Fresnel propagator, which is oft written (apart from a constant factor 3 ) as e iπ(x 2 +y 2 ) λ∆z

. Here ∆z is the distance from the exit-surface along the beam direction, or the "defocus setting", defined so as to be positive in the downstream (underfocus) direction. In frequency space, this is equivalent to multiplying the wavefield by the propagator's Fourier transform, the transfer function e -iW defined here in terms of the "defocus function" W [∆z] = πλ∆z(q 2 x + q 2 y ). For our single frequency specimen, the imaged amplitude is thus modulated by

ψ = 1 + ie -iW σ o cos[2πqx]. The intensity found in the image becomes I ≡ ψ * ψ = 1 + 2sin[W ]σ o cos[2πqx].
The factor sin[W ], a function of both defocus ∆z and spatial frequency q, thus determines both the strength and sign of intensity ripples corresponding to specimen structure, in the captured image. It predicts dark spots in the neighborhood of large phase shifts (e.g. atom columns), for example when W is near -π/2. We also might want to choose defocus ∆z so that the images are simplest to interpret, e.g. so that sin[W ] is finite and constant for as wide a range of frequencies as possible. To assist with this, we've plotted sin[W ] as a function of transverse spatial frequency q, and defocus ∆z, for unit wavelength on the left hand side of Figure 4.

Spherical aberration in the objective lens also affects the image. The effect is shown graphically on the right hand side of Figure 4. Since zeros in sin[W ] correspond to phase reversals for the corresponding periodicity, a traditional goal has been to select a defocus for which these zero's are moved outward to as high a frequency as possible. As illustrated in the diagram, Scherzer defocus is the setting of choice in this regard.

Recent work has concentrated on focussed-probe illumination in which the aberrations occur upstream of the specimen, which probe in "scanning transmission" (STEM) is rastered across the specimen, resulting generally in more amplitude rather than phase contrast, and on "aberration-correction" which uses controllable aberrations to correct uncorrectable aberrations (like spherical) in one focus plane only. The latter requires more work in focusing, but has allowed sub-Angstrom lateral resolutions, even at beam voltages which are below the knock-on threshold from important light atoms like carbon 7 .

For quantitative microscopy of thicker specimens in real microscopes, of course, other issues must be considered as well: e.g. vibration and electrical instabilities, knowledge of specimen thickness, astigmatism, and the range of illumination angles in the beam. Even then the theory remains quite elegant, and except for the effects of specimen thickness can be visualized by changes in the fingerprint of Fig 4. Of course, this analysis might seem abstract. Au contraire! Recall that amorphous materials in projection are, unlike crystals, likely to show a uniform distribution of periodicities. As a result, the zeros of the imaginary part of the Fresnel transfer function, for the current defocus setting in the right side of Fig 4, show up as rings in the power spectrum from amorphous material in every HREM image.

FIG. 3 .

 3 FIG. 3. Coulomb scattering (in complex-color online) of a negative charge's deBroglie extended wavefield by a fixed positive charge, from one of those rare regular solutions of the Schrödinger equation.

FIG. 8 .

 8 FIG.8. An electron phase contrast image of the internal structure of a 15 wall carbon nanotube, the innermost 6 atomic layers of which form an internal closure.

ACKNOWLEDGMENTS

Thanks to industry and university collaborators across the region for related projects.

Appendix A: underlying math

In order to increase assessibility, in the main body of this paper we've tried to illustrate using quantitative graphical techniques like complex-color, which lets us show the amplitude and phase of a complex number with a single pixel. In this appendix, however, we attempt to connect interested readers to the underlying models and approximations.