SUPPLEMENTARY NOTES Companion report is F49620-95-1-0432 20000503 094 12a DISTRIBUTION /AVAILABILITY STATEMENT Full version of both reports available online at http://optics.colorado.edu/~kelvin/Iinks/soliton_report.ps.gz UL 12b. DISTRIBUTION CODE UL 13. ABSTRACT (Maximum 200 Words) High-bandwidth optical Communications will greatly benefit from optical switches since they could eliminate the optical/electronic conversion. Optical logic gates allowing data regeneration, gain, cascadability,

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... d allow even more complex all-optical routing functions. In this work we report on an in-depth study of an optical logic gates based on spatial and spatio-temporal solitons. Optical solitons that propagate long distances without change, act as the natural carrier of binary data due to their stability to perturbations and intrinsic threshold. The non-diffracting nature of spatial optical solitons lends to their use in a class of angular deflection logic gates in which a weak signal can alter the propagation of a strong pump in order to change the device state from high to low, thereby implementing a controlled inverterwhich is cascadable to produce logically-complete, multi-input NOR. Reduced forms of the multi-dimensional, nonlinear spatio-temporal wave equation are solved numerically to study the spatial collision and dragging interactions between orthogonally-polarized spatial solitons and spatio-temporal solitary waves. These three-terminal angular deflection gates, provide complete logic-level restoration, fanout greater than two with large noise margin, and cascadability. In addition, the spatiotemporal logic gates are expected to have pj switching energies using enhanced nonlinear media, and ps switching times through temporal pipelined operation. This thesis deals with the topic of ultrafast, all-optical switching and logic using spatial and spatio-temporal solitons. The main focus is on the accurate analytical and numerical modeling of the interaction among solitons in geometries that allow for logically-complete, cascadable logic gates with fanout and level restoration. Detailed modeling will assist in subsequent experimental investigation by identifying regions of stability, robust operation and reliability, and candidate material systems. The work in this thesis makes significant contributions in the areas of fundamental nonlinear spatio-temporal propagation and optical switching and logic. The first contribution is the result of a multiple-scales derivation directly from Maxwell's equations, which results in a first-order, multi-dimensional, vectorial differential equation that is accurate beyond the standard paraxial and slowly-varying envelope and amplitude approximations. This result is fundamental to the studies of this thesis and will have application in other areas as well, such as optical communications, short-pulse passively mode-locked lasers, and atmospheric pulse propagation. The second contribution arises from the systems-level approach taken to optical switching and logic. Most optical switching technologies do not satisfy the basic requirements for a logic gate, thereby rendering them of little use in applications beyond simple, smele-staee switching operations. The novel logic gates studied here satisfy the necessary requirements and thus have the potential to make an impact in areas of ultra-high speed switching and logic systems. The desirable properties of these logic gates are facilitated by the non-diffracting and/or non-dispersing nature of optical solitons. Outline The remainder of this chapter briefly presents background in device requirements and optical switching and logic. Section 1.2 provides motivation and a brief overview of promising applications for optical logic. Section 1.3 covers generic requirements for switching and logic devices and additional specific requirements for optical devices. A brief review of other contemporary optical switching devices and their shortcomings is presented in section 1.4. Logic devices which specifically take advantage of the properties of optical solitons are discussed in section 1.5. which also provides an introduction to the optical soliton logic cates.studied in this thesis. Detailed background on optical solitons is given in Chapter 2. First, section 2.1 provides historical background in solitary wave and soliton phenomena. Section 2.2 covers some preliminary details that lead directly to the discussion of optical solitons and solitary waves. The following sections then discuss spatial (sec. 2.3), temporal (sec. 2.4), and spatio-temporal (sec. 2.5) optical solitons. Chapter 3 derives the multi-dimensional vector wave equations necessary for the numerical simulations of the soliton-based logic gates presented in later chapters. Section 3.1 covers a standard treatment of a fully vectorial, nonlinear Helmholtztype wave equation along with background on the linear and nonlinear susceptibilities. This second-order wave equation is unsuitable for the purposes of this thesis due to the difficulty in numerically propagating an initial field distribution because of the fast time and space scales in the equation. Instead, section 3.2 derives an asymptotic vector nonlinear wave equation directly from Maxwell's equations which is first-order in the propagation coordinate and depends only on the scales of the slowly-varying envelope, and is thus more suitable for numerical propagation. The importance of the results obtained in this chapter arises from the higher-order nature of the derived equation, which extends beyond the approximations made in the well-known multi-dimensional nonlinear Schrödinger (NLS) equation, resulting in the capability to describe propagation of vector optical field envelopes with very broad spectral content in both the spatial and temporal frequency domains, including the effects of higher-order nonlinearities. Chapter 4 describes the split-step numerical method used to solve the vector nonlinear wave equations. The basis of Final Report AFOSR Spatial Soliton Interactions, S. Blair & K. Wagner, U. Colorado, Boulder 6 this method is to treat linear and nonlinear (inhomogeneous) propagation in separate steps, performing linear propagation in its natural Fourier domain and nonlinear propagation in the real-space domain. For a small step size, this splitting is a good approximation. Section 4.1 covers multi-dimensional linear spatio-temporal diffraction, which can be solved without approximation as an initial value problem. Noting that linear and nonlinear propagation are not separable, section 4.2 derives a split-step formulation which is approximately second-order accurate in step size. This formulation is then applied separately to (1+1)-D spatial and (2+l)-D spatio-temporal nonlinear propagation, to be used in later chapters. Finally, section 4.3 discusses the accuracy of the method as it applies to problems of later chapters. The use of one-dimensional spatial soliton interactions for logic gates is presented in Chapter 5. The first section covers the basic soliton interaction geometries useful for three-terminal, restoring logic, noting that the collision and dragging geometries using orthogonally polarized solitons, which are of the general class of angular deflection gates, provide the best performance in terms of large gain with high contrast. Section 5.3 then examines the effects of linear and two-photon absorption on the propagation of a single spatial soliton and develops appropriate figures-of-merit for use in comparing the suitability of nonlinear materials for soliton logic gates. The collision and dragging logic gates are then compared in the presence of absorption using the material parameters of fused silica, where the spatial dragging gates generally perform better because of their shorter gate lengths. The final section (5.4) computes the transfer functions of the collision and dragging gates and determines the maximum fanout that these gates can provide in a cascaded system. The results of this chapter provide guidance for the spatio-temporal logic gates of the next chapter, which is of ultimate interest. " Logic gates based on the interactions between two-dimensional spatio-temporal solitary waves is the subject of Chapter 6, which have the potential for greater than THz switching rates with nJ to pJ switching energies. Section 6.1 discusses the propagation of a single spatio-temporal solitary wave with higher-order linear and nonlinear effects as derived in Chapter 4. This section shows numerically that stabilized solitary waves suitable for logic gates should exist. With these results, section 6.2 studies the vectorial interaction between these spatio-temporal solitary waves with emphasis on the dragging interaction. For completeness, section 6.3 presents results on cascaded logic. It is shown that the dragging logic gates can provide reasonable fanout in a cascaded, logically-complete, system, but also that, ultimately, the performance of the logic gate will be limited by the effects of Raman scattering. Finally. Chapter 7 concludes, noting that this thesis provides the theoretical and numerical development that is necessary to study a novel class of all-optical logic gates. Numerical simulations have shown that these gates satisfy the system requirements that suggest their use beyond simple, single-stage operation, and paves the way for future theoretical and experimental work on ultratast. all-optical logic systems. Optical Communications Networks With the advent of high-bandwidth optical communications [4], high-speed switching technology becomes a necessity. Because information will be transmitted optically, it makes sense to explore optical technologies and devices to perform (or at least assist Final Report AFOSR Spatial Soliton Interactions, S. Blair & K. Wagner, U. Colorado, Boulder 7 in) the switching necessary to multiplex, demultiplex, and route information to the correct destination. Future bandwidth needs will be too great for an electronic-only solution to the switching problem. An advantage of optical devices in communications systems is the ability to switch and/or perform logic at the bit rate. This means that switching and logic operations are performed at and scale with the rate of incoming data, which is especially important in Tb/s communications switching and routing applications. Conversely, interconnected electronic gates cannot arbitrarily scale with the data rates and may ultimately be limited to speeds of about 50 Gbit/s [5] . Optical switching and logic devices are well suited for time-division multiplexed (TDM) data transmission [6] , m which many low bandwidth channels are interleaved into individual time slots within a single high bandwidth channel. Here, the aggregate data rate may exceed 100 Gb/s [7] , which is well beyond the performance expectation of practical, low power, electronic or optoelectronic switching networks. Therefore, at the very least, optical switching technology will find use in the intermediate processing layer in which the individual channels are optically separated (demultiplexed) from the single, highspeed transmission channel, for subsequent processing in the electronic/optoelectronic domain. An additional area of use is in data regeneration within an optical transmission line, which requires an optical logic gate. Given sufficient noise margins, an opticaltogic gate regenerates a noisy or attenuated data stream with fresh pulses that are derived from a power supply or clock, such that liming, amplitude, shape, etc., are completely restored. This type of all-optical repeater may eliminate the need for high-speed, high-power electronic repeaters in long-haul transmission. Currently, the most popular multiplexing technique for long-haul communications and local-area optical networks is wavelength division multiplexing (WDM), in which each low bandwidth source channel is assigned a slot in the optical frequency spectrum. WDM has the advantage that multiplexing and demultiplexing can be performed by spectrally selective, passive, optical elements. However, due to the difficulties in developing a large array of stable wavelength sources, the individual channels in a WDM system may exceed data rates of 10 Gb/s [8, 9] , which may be of sufficient bandwidths that other advantages of optical switching and logic devices, such as the elimination of the optoelectronic conversion process [10], potentially lower power, and the use of deep optical circulating delay lines [11] for contention resolution, may play a role in the choice of implementation technology. More complex operations are required in packet switched optical networks [10, 12] . In these networks, at each node, a packet header is decoded, which determines the direction in which to route the packet. Header recognition is a simple digital correlation operation, but must be performed at the bit rate. Typical implementations based on optical switches [13] perform this operation in parallel, which results in a 1//V loss where TV is the number of bits in the header. In order to reduce the latency at each node and avoid the fanout loss, a digital comparator, or shift register, could be used instead, based on optical logic gates. In addition, optical logic could be used for contention resolution, real-time encryption/decryption, and other complex tasks as well. Therefore, intelligent digital optical processing may'play a significant role in the development in the next generation of high-speed optical communications networks. Digital Optical Computing and Processing Determining what role optics should play in general-purpose computing is an open question. The technological lead, resources, and infrastructures that electronics enjoy may be insurmountable for general-purpose optical computing. Even a compelling optical technology may not be enough to impact the future of digital computing, but there is potential in niche areas which are discussed in this section. Simple digital optical logic circuits have been demonstrated [14, 15] and proposed [16] , which pave the way for more complex systems. However, many of these studies have been based on optical switches or gates that do not completely satisfy the requirements for digital logic. For example, the nonlinear Fabry-Perot etalon [17] is a two-terminal device which must be biased about an operating point with a holding beam and is very sensitive to variations in device parameter (i.e. the transfer characteristics vary from device to device). All of these factors preclude their use in complex systems. The logic gates studied in this thesis, on the other hand, do not suffer from these problems and could serve as fundamental building blocks in more complex systems. In the absence of high space-bandwidth optical interconnections, bit-serial computation and signal processing [18] is a promising application area, especially when the problems of interest can be processed in a highly pipelined manner [19]. Genera! purpose optical computers [20] have even been demonstrated using the bit-serial approach. One potential limitation for ultra-fast digital optical computing is the memory store. It has been shown, however, that an optical delay line can be used to implement a general machine [16] , at the cost of increased latency. These tradeoffs are based on the transformations of computational origami [21]. Even more computational capacity can be realized with the combination of optical logic gates with optical interconnection, for which the power of the interconnection network allows for the efficient mapping of problems that would be difficult to process electronically [22]. Final Report AFOSR Spatial Soliton Interactions, S. Blair &K. Wagner, U. Colorado, Boulder 8 Requirements for Digital Switching and Logic Devices Routing switches and logic elements must possess certain properties in order to function properly within a system. The major distinction made here between switching elements and logic gates is that switching elements physically direct electrons or photons from one or more input ports to one or more output ports, while logic gates replace electrons or photons on the input ports with "new" electrons or photons derived from a power supply which then appear at the output ports. The requirements for a switching device are much less stringent than those for a logic device and will be discussed first. Additional requirements for an optical logic device are discussed in the following section. Switching Devices Optical routing switches are typically used just in the first switching stage to reduce data bandwidth to a level that electronics can handle in subsequent stages. These switching elements use a control to physically direct light from one or more input ports to one or more output ports and thus can be used to implement multiplexing and demultiplexing functionality. The routing decision is based either on the intensity of the signal inputs or by the presence of an externally derived control which is independent of the switching fabric. Therefore, the control is typically not of the same format as the data, meaning that the output of the switch cannot directly serve as the control for another switch [3] . As a result, this type of routing switch has fewer requirements than a logic gate. The most important requirement of a switching device is high contrast operation. In binary transmission, the contrast determines the threshold level which separates the high and low states. Higher contrast gives larger noise margin, resulting in lower bit-error-rate (BER). For the output of a switching element, the overall contrast depends both on the contrast of the data stream (the difference in signal level between the two binary states which depends on the modulation ratio at the input of the transmission line and on the transmission line itself) and the contrast of the switching operation. Another important requirement is switch transparency. If the switch is lossy or only a small fraction of the input is diverted to the output by the switching action, then high-speed, single-shot detection may become impossible, resulting in information loss. This does not mean that the switching operation must provide gain, though. For high-contrast output, saturation of an external amplification stage can be used to restore the data signal levels, but the gain recovery time may limit the data rate and introduce inter-symbol interference, the contrast at the output of the amplifier may be reduced, and complete logic level restoration (as discussed later for optical devices) is not obtained. The amount of gain is also limited due to amplified-spontaneous emission (ASE) noise, which increases the noise floor. Routing switches do not have signal level restoration (in the absence of external amplification or level shifting) and the outputs may degrade due to loss or crosstalk and therefore BER may suffer from a long cascade of switches. If data pass through many levels of switching, such as in a multi-level implementation of an N:N crossbar or the binary tree structure required for 1:N or N:l multiplexors or demultiplexers constructed from 1:2 or 2:1 switches, then gain and level restoration as mentioned previously may be a necessity, subject to the limitations discussed in addition to ASE noise. Three-terminal operation also becomes necessary when many levels of switching elements are used such that the operation of the switch in unidirectional, which prevents any light entering the output ports from affecting the operation of the switch. Therefore, in order to implement more complex, multi-level switching fabrics, or to handle data-dependent (i.e. self-routing) switching operations, switching elements must have the additional properties and fulfill the more demanding requirements of a logic gate. It is in these applications that most all-optical switching devices are inadequate, as discussed in section 1.4. Logic Devices A digital logic gate performs a Boolean operation on one or more binary inputs. All inputs and outputs are of the same physical format thus allowing control to be distributed throughout the switching fabric [3] such that one data stream can control another. The logic gate completely restores signal integrity and timing by replacing the electrons or photons at the input with new electrons or photons from the power supply that go to the output. Logic levels (i.e. 0 or 1 in digital logic) are physically represented by signal levels, which may be voltage or current for electronics; or for the representationally richer case of optical logic, amplitude, phase, polarization, or color, and are differentiated by a threshold nonlinearity. Information is carried by the logic level and is determined by the interpretation of the signal level (based on the threshold). Since the signal level can be altered during a logic operation, small noise or loss can accumulate throughout the computation and cause information loss. A digital representation of information avoids this degradation if the signal level is restandardized at each step [23]. Restandardization means that the signal levels are restored to values (that are the same throughout the system) which represent valid logic states. As a result, a small deviation about a valid logic level propagates at most one stage and the logic level remains intact (within the allowable noise margin). The is termed logic level restoration. Final Report AFOSR F49620-95-1-0431, Spatial Soliton Interactions, S, Blair & K. Wagner, U. Colorado, Boulder nMOS inverter supply (Vdd) A transfer function Vdd input output -VtFT input Fieure 1.1: nMOS inverter circuit and associated transfer function. Near the threshold voltage is a region of small-signal gain. Large-signal gain is also provided (V d d > Kh) and the high and low output states are saturated. The properties required by a digital electronic [24] or optical [25] logic device are well known. Perhaps the most important requirements, upon which many other requirements depend, are gain, saturation, and threshold. These characteristics are illustrated in Figure 1 .1, which shows the sigmoidal input-output relationship for a simple nMOS inverter. At the most fundamental level. a region (typically near the input threshold) of differential or small signal gain in the input-output transfer function, where a small change in the input produces a large change in the output, is required (but not sufficient) for a restoring logic gate [26]. Small-signal gain provides sharp switching characteristics and allows for low modulation depth on the input signal biased about the threshold level to produce high contrast switching. Outside the linear region of small-signal gain, saturated levels are key to providing large noise margin by attenuating small variations in the input about valid logic levels (i.e. inputs lying within the "on" and "off" regions shown in the figure), thus producing little change in the output. Restoration of the logic level prevents accumulation of errors due to attenuation or crosstalk by providing fresh gate output levels direct] v from the power supply. In the case of an optical gate, logic level restoration must include, in addition to power levels, beam shape and position, pulse duration, color, polarization, and timing [25]. These levels are standardized throughout the entire system typically by the common power supply and ground. Standardization is possible because small-signal gain and saturated levels in the nonlinear transfer function allow for a wide tolerance to the variation in operational characteristics of the devices in the system [24]. Large-signal gain means that the output of the gate is larger than the input required (at least threshold) to switch the gate. Without large-signal gain, fanout, in which the output of a gate provides inputs to many gates in order to implement arbitrary logic functions, is impossible. Most optical logic gates cannot intrinsically provide large-signal gain, and in fact, the output is typically much smaller than the input. Although a separate, external amplification stage can be used, this may limit the bit rate and result in reduced contrast and increased BER when the amplifier noise exceeds the device noise margins. A logic gate with intrinsic gain, in which a small input controls a large power supply, does not suffer this source of noise. Additional requirements for synthesis of arbitrary logic and switching are logical completeness and cascadability. A complete set of logic functions must include inversion [24]. Inversion is a basic function of MOSFET electronics and the inverter structure forms the basis towards implementing more complex Boolean operations such as multi-input and logically complete NOR. Cascadability means that the output of one gate can directly drive the input to the next and allows the direct implementation of multi-level logic. Logic level restoration and inputs and outputs of the same format are therefore crucial to allowing cascadability. The most successful logic devices have three orthogonal ports (which can be separate in time, space, wavelength, or polarization) and input-output isolation. A three-terminal device, such as an electronic transistor, ensures that the output of the gate depends only on the inputs and the output does not perturb the operation of the gate [24]. This input-output isolation results in one-way operation and prevents downstream noise which enters the output port from affecting the operation of the gate. There are many examples of two-terminal devices in optics. The problem with two-terminal devices is that they work equally well in either direction [25]. The classic examples are the laser [27] (a two-terminal device with gain) and nonlinear Fabry-Perot etalon [17] which, when critically biased [28], can switch either through the input or through an unwanted reflection back into 'cm -/, 1 v ' Final Report AFOSR F49620-95-1 -0431, Spatial Soliton Interactions, S. Blair & K. Wagner, U. Colorado, Boulder 12 /\ input ^ \ output A bar cross Figure 1 .3: Operation of half-beat length nonlinear directional coupler. For low power, or linear, operation, the input pulse exits the cross port (guide 2) while in nonlinear operation, the central portions of the pulse, which exceed the critical intensity, exit the bar port (guide 1) with the wings exiting the cross port. and represents the input intensity for which the light is divided equally between the two output ports [31]. For a half-beat length coupler, the critical intensity results in a n phase shift for the field in guide 1, and 8 = 1 in equation 1.1. Above the critical intensity, the fraction of light exiting the bar port increases towards unity. This two-terminal device performs a switching operation since photons from the data stream are directed into one of two output ports. A half-beat length NLDC is shown schematically in Figure 1 .3, where a low-power data pulse couples to the cross port while a high-power data pulse remains in guide 1 (bar state). The high-power case can be understood intuitively by considering the nonfinear increase in the core index which more strongly guides the light, thus detuning the coupler. Notice that only the central portion of the high-power pulse exceeds the critical intensity and remains in guide 1. This effect is called partial switching, or pulse breakup, and is characteristic of many optical switching devices and results in reduced integrated energy contrast [33]. In fact, partial switching is a problem with all of the non-soliton devices discussed in this section. Solutions to this problem are the use of square pulses [34] such that the intensity is constant across the pulse duration, or the use of temporal solitons [35][36][37], which propagate unchanged and tend to switch as a unit because of uniform phase across the pulse profile [38]. Fmure 1.4 shows a plot of the switching fraction into the bar output port versus normalized peak pulse intensity launched into iiuide 1 for the half-beat length NLDC. The solid curve shows the fraction of the light that remains in the bar state assuming a constant intensity square-top pulse as given directly by equation 1.7a. The curve for the soliton case has similar shape with sharp su itching characteristics [35], but the switching intensity threshold is about twice the critical intensity because the induced nonlinear phase is half that of the plane wave case, as shown in section 2.2. The dashed curve shows the light fraction assuming a-non-soliton secrrO input intensity profile and takes into account partial switching which leads to reduced contrast operation as indicated by the greatly reduced switching fraction. In either case, at low input levels, all of the light switches over to the cross port, while at high inputs, most of the light exits the bar port. Direct cascadability can also be addressed using Figure 1 .4, where the dotted line represents the input threshold for which the output of the bar port exceeds / cril . Figure 1 .5, which shows the transfer function for each port of the device, provides the same information more clearly. As shown in the figure, for a square-top pulse, an input intensity greater than 1.1 / crit is required to exceed the threshold intensity at the bar port. In this case, the output of the bar port can be used as the input to another device resulting in 50^ switching of that second device. Therefore, much higher inputs than 1.1 / crit are necessary to cascade to another device, but. due to the lack of large-signal gain and in the absence of external amplification, any optical losses in the system will eventually result in the degradation of the signal level to the point that switching ceases. The situation is worse in the case of a nonuniform pulse. Here, a peak input intensity of greater than 1.6 / crit is required for at least 50% switching of a subsequent device. Indeed, because of partial switching, attempts at cascading two such devices have met with limited success 139]. A final point to note is that when designed as a half-beat-length coupler in linear operation, the NLDC transfer function has quasi-saturated levels which attenuate fluctuation in the input level and allow for some noise margin. As described so far, the NLDC is a two-terminal device whose switching operation is determined by the intensity of the input signals. A three-terminal version can also be constructed by using a high-power control pulse to switch a low-power data pulse as shown in Figure 1 .6, but this device is not directly cascadable either due to the lack of large-signal gain and inconsistent signal representation (using different wavelengths). The control pulse must be of different central frequency or polarization in order to avoid phase-sensitivity and to extract the data at the output with zero background. Experimental demonstrations have used different frequencies because of the large difference in coherence lengths between orthogonal polarizations [33]. There are still partial switching problems if the control pulse does not have constant amplitude. An additional problem is dispersive walkoff, which will be discussed in more detail for the Kerr gate. A recent demultiplexing demonstration [33] used a longer control pulse length and allowed the data pulse to walk-through the control to achieve complete switching, and therefore, high contrast. Lengthening of the control pulse means that the separation between data pulses needs to be larger, thereby reducing