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Research ArticlePHYSICAL SCIENCES
Metal nanoparticle film–based room temperature Coulomb transistor
Svenja Willing1,2,*,
Hauke Lehmann1,*,
Mirjam Volkmann1 and
Christian Klinke1,†
1Institute of Physical Chemistry, University of Hamburg, 20146 Hamburg, Germany.
2Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany.
↵†Corresponding author. Email: klinke{at}chemie.uni-hamburg.de
Science Advances 14 Jul 2017:
Svenja Willing
Institute of Physical Chemistry, University of Hamburg, 20146 Hamburg, Germany.Deutsches Elektronen-Synchrotron (DESY), 22607 Hamburg, Germany.
Hauke Lehmann
Institute of Physical Chemistry, University of Hamburg, 20146 Hamburg, Germany.
Mirjam Volkmann
Christian Klinke
For correspondence: klinke@chemie.uni-hamburg.de
Single-electron transistors would represent an approach to developing less power–consuming microelectronic devices if room temperature operation and industry-compatible fabrication were possible. We present a concept based on stripes of small, self-assembled, colloidal, metal nanoparticles on a back-gate device architecture, which leads to well-defined and well-controllable transistor characteristics. This Coulomb transistor has three main advantages. By using the scalable Langmuir-Blodgett method, we combine high-quality chemically synthesized metal nanoparticles with standard lithography techniques. The resulting transistors show on/off ratios above 90%, reliable and sinusoidal Coulomb oscillations, and room temperature operation. Furthermore, this concept allows for versatile tuning of the device properties such as Coulomb energy gap and threshold voltage, as well as period, position, and strength of the oscillations.
In 1958, an article (1) reported on the unexpected observation of a negative differential resistance in germanium p-n junctions. What started out as an anomaly in the known current-voltage characteristics of a diode led to the development of a new technology and resulted in a Nobel Prize for the author, Leo Esaki. Even beyond the technological point of view, studying tunnel processes enables the attainment of fundamental physical knowledge as Esaki mentioned in his Nobel Prize lecture (2). Today, especially single-electron applications (3, 4) are a proof of how quantum mechanical tunneling can enable new functionalities in conventional devices (5). Despite being similar to a common field-effect transistor, the single-electron transistor (SET) does not rely on the semiconductor band gap but instead on the Coulomb energy gap. The resulting transfer characteristics exhibit periodic on and off states known as Coulomb oscillations, which might render new applications possible in the future. The working principle of the SET is well known both experimentally (6–11) and theoretically (12) and has been implemented in silicon devices (13–16). Nevertheless, using this effect in an actual device that not only yields reproducible and predictable results but can also be fabricated in an inexpensive, easy, and industrially applicable way has remained a challenge. Wet-chemical synthesis is the best-suited approach to preparing monodisperse nanoparticles with tunable properties such as size and interparticle distance. They are solution-processable, stable, and unstrained (17, 18). However, reliably contacting single or multiple particles is not easily achieved. For example, bifunctional linker molecules (19, 20), electrostatic trapping (21), and induced dipoles (22) are known approaches to catching a single nanoparticle or even assemblies thereof between readily fabricated electrodes and to demonstrating SET function (23, 24). In semiconductor-nanoparticle arrays, the coherence of the superlattice determines whether Coulomb charging or band-like transport governs the electrical transport (25). In extended metallic nanoparticle films, the presence of the effect of Coulomb blockade can be extracted by analysis of the transport characteristics (26, 27). To include high-quality nanoparticles as a two-dimensional (2D) array into an electrical device, we use a unique combination of both top-down and bottom-up techniques. By means of the Langmuir-Blodgett method (28–33), the particles' tendency to self-assemble is exploited to create an ordered, homogeneous monolayer of evenly spaced nanoparticles. The position, shape, and size of the monolayer can be easily and precisely controlled by a resist mask. With the Langmuir-Blodgett method as a link between bottom-up and top-down, we can take advantage of the reliability and accessibility of physical techniques, such as lithography and sputter deposition, for the general device fabrication while still profiting from the chemically synthesized and uniformly assembled high-quality nanoparticles. Using confined metal nanoparticle films as a conductive channel in field-effect transistor geometry allows for inducing Coulomb oscillations via a gate electrode. This work demonstrates an alternative device concept that renders thorough and systematic investigations possible. Understanding parameter dependencies and influences allows for tailoring the devices to the needs of future applications and improvement in performance. We present Coulomb transistors with on/off ratios of more than 90%, highly tunable properties, and an unprecedented clarity of Coulomb oscillations up to room temperature.
Device preparation
Colloidal cobalt-platinum (CoPt) nanoparticles (fig. S1) with diameters of (2.3 ± 0.2) nm, (3.5 ± 0.3) nm, and (4.0 ± 0.3) nm were synthesized via the hot injection method. CoPt was chosen because of the highly tunable and well-established synthesis (34) and because the high amount of platinum in the particles avoids oxidation and electromigration. The particle size can easily be adapted by varying the ratio of the platinum source and the amine that is used as one of two ligands. These ligands keep the particles solution-processable, and their chain length defines the particle spacing and thus the tunnel barrier width. In our case, oleylamine and oleic acid were mainly used as ligands resulting in a particle spacing of about 1.9 nm (medium and small particles). Good results have also been obtained using the shorter-chained versions decylamine and decanoic acid that lead to a separation of 1.4 nm (large particles). The electrodes were prepared on Si/SiO2 substrates by a combination of electron-beam lithography including a marker-alignment procedure, metal evaporation of titanium and gold, and sputter deposition of additional SiO2 as gate dielectric (refer also to Materials and Methods). The nanoparticle array was prepared and included onto the sample geometry by the Langmuir-Blodgett method (setup in fig. S2) (28–33). For this, the particles are casted onto a liquid subphase and carefully forced together by barriers until they constitute a densely packed, hexagonally ordered monolayer (30, 35), which is monitored via in situ surface-pressure measurements. Lifting the former submerged substrate and thereby collecting the particles on the surface transfers the monolayer onto the devices. The electrical characterization was performed under vacuum atmosphere in a closed cycle cryogenic probe station at temperatures between 4.5 K and room temperature. An exemplary nanoparticle monolayer, micrographs of the final devices, and a scheme can be found in Fig. 1.
Fig. 1 Device design.
(A) Schematic drawing. (B) Transmission electron micrograph of the CoPt nanoparticle monolayer as assembled by the Langmuir-Blodgett method. (C and D) Light micrographs of the final devices with three pairs of source and drain contacts as well as a local back-gate electrode. The gap between the leads and thus the length of the nanoparticle channel is 1, 2, or 3 μm (top to bottom).
This fabrication route is not only simple and scalable but also especially highly tunable. On the one hand, the particles can be adjusted in size and shape as well as their interparticle spacing (17, 18, 30, 32). On the other hand, the monolayer can be patterned by lithography to define stripe-shaped nanoparticle arrays. Adjusting the length and width controls the total current as expected according to the varying number and length of possible conduction paths. Using this stripe-shaped monolayer channel allows full electrostatic control over all particles at once from the perpendicular direction without screening or stray currents (33). However, beyond that, the new device concept exposes further dependencies that can be used to adjust also working bias or periodicity: A local back-gate geometry is used, which allows fabrication of all electrodes before particle deposition. Consequently, the pristine ordering in the monolayer is preserved, and interferences due to further preparation steps or trap states in a capping dielectric can be avoided (figs. S3 and S4). Because of the increased quality of transfer characteristics, it was possible to observe clear Coulomb oscillations at about 210 K with the very same particles as in the study of Lehmann et al. (33), where 150 K was the highest temperature. Furthermore, this enables a systematic analysis of the parameters' influence on the device performance. Among others, the effect of temperature, lateral bias voltage, dielectric thickness, and gate position on the transport and on the device performance has been investigated.
Coulomb blockade and electrical transport
Applying a bias voltage to the nanoparticle channel shows that conduction in this system is not purely metallic but is governed by tunnel barriers between individual particles. The transport is hindered due to the potential barrier given by the charging energy (12)
(1)required to overcome the electrostatic repulsion when an additional electron transfers onto a particle. The high charging energy results from the particles' inherent small self-capacitance C, which is proportional to the particle radius r. A detailed discussion of the capacitance and an estimation of its different contributions can be found in text S1. From this, a charging energy of about EC = 100 meV is calculated for the medium-sized particles. During transport, this Coulomb blockade leads to the emergence of discrete energy levels separated by a Coulomb energy gap of 2·EC. Charge carriers can tunnel through the barriers to percolate through the monolayered channel. However, elastic tunneling is highly sensitive to the energy difference between the initial and final states (36) and thus to the electrons' energy given by the bias voltage and temperature. Synthesizing the nanoparticles by colloidal chemistry allows for a precise tailoring of the system's energy levels. Very small and uniform capacitances can be achieved with monodisperse nanoparticles. The tunnel barrier width can be adjusted through the interparticle spacing defined by the organic ligands.
Thorough investigations have been conducted with the medium-sized particles of 3.5 nm diameter and will be presented in the following if not stated otherwise. Figure 2 (A and B) displays the output characteristics of an exemplary device. For low temperatures (compare the lower inset in Fig. 2A), no transport takes place up to a certain voltage that delivers the energy needed to overcome the Coulomb blockade. This so-called threshold voltage decreases with increasing temperature, which is very well visible in the logarithmic plot of the respective differential conductances in Fig. 2B. At around 90 K (bold green line in Fig. 2, A and B), there is no observable Coulomb blockade regime anymore as can be seen from the appearance of a finite zero-bias conductance. Even at room temperature, the characteristic is still slightly nonlinear. The monolayer system is well suited to investigate the temperature transition of the transport mechanism (31–33). In accordance with previous observations, electrical transport at low temperatures takes place mainly through tunneling (direct or Nordheim-Fowler tunneling), where the current is strongly dependent on the source-drain voltage (37). At elevated temperatures, transport is increasingly due to thermally activated processes like electron hopping. The current-voltage dependencies of both mechanisms are exemplarily fitted to the data in the two insets in Fig. 2A (more detailed in fig. S5).
Fig. 2 Output characteristics.
(A) Current-voltage curves for different temperatures show the transition from a system governed by Coulomb blockade to almost ohmic conduction. Lowest and highest temperatures are included as insets to provide a good comparability and are fitted with the respective transport mechanism. (B) Same data plotted logarithmically as differential conductance over bias voltage. The decrease of the blockade-regime width, changes in threshold voltage, and increase in zero-bias conductance (at 90 K and above) can be seen. (C) The Arrhenius plot allows for a linear fit of the logarithmic zero-bias conductance outside of the Coulomb blockade regime. The slope indicating the activation energy is similar for all channel lengths. (D) The activation energy decreases exponentially with the applied electric field and is not dependent on channel length or gating geometry.
In addition, the analysis of the temperature dependency of the differential conductance G can give further information. The zero-bias conductance at temperatures T above the Coulomb blockade regime follows the equation
The parameter ν can take different values depending on the underlying transport type. Although ν = 1/3 and ν = 1/2 would suggest variable-range hopping (38, 39), our data are best fitted with ν = 1. This Arrhenius-type behavior can be seen in Fig. 2C and indicates nearest-neighbor hopping. In the case of Arrhenius behavior, the differential zero-bias conductance
(3)can be used to determine the system's activation energy EA, where kB is the Boltzmann constant. For the medium-sized particles (r = 1.75 nm), this yields EA = 61 meV. The dependency on the particle size will be discussed later. Applying a lateral bias voltage will lower the effective barrier for the charge carriers and thus the activation energy. Using the differential conductance at nonzero source-drain voltage values and fitting the temperature dependency as before allows for obtaining these activation energies. As shown in Fig. 2D, the activation energy decays roughly exponentially with the applied electric field independent of channel length or gating geometry. Overall, the transport investigations support the monodispersity and supercrystallinity of the particle monolayer.
Coulomb oscillations and tunability
An additional gate electrode separated from the channel by a dielectric layer allows for influencing the transport capacitively. Tuning the gate voltage results in a field effect that continuously shifts the energy levels of the particles. Whenever neighboring energy states align, tunneling is most probable, and a maximum current flow can be observed. Exemplary transfer characteristics are displayed in Fig. 3 (temperature dependency) and Fig. 4 (source-drain voltage dependency; a 2D pseudocolor plot can additionally be found in fig. S6). They show these periodic Coulomb oscillations with, for extended particle systems, unprecedented regularity and clarity (32, 33). To quantify the results, we calculated the relative on/off ratios between main minimum and neighboring maximum (for consistency always on the right side) and these can be found in the graphs as percentages.
Fig. 3 Temperature dependency.
(A) The local back gate induces clear oscillations with reproducing features during a bidirectional gate-voltage sweep for different temperatures. (B) The highest found on/off ratios for three different channel lengths measured with a locally gated sample decay exponentially with increasing temperature. (C) A sample gated with the global substrate shows very periodic oscillations with similar percentages. (D) Smaller nanoparticles (2.3 nm diameter) lead to higher on/off ratios and more pronounced oscillations at higher temperatures. Minor changes in bias voltage adjust the current and render the curve perfectly smooth. Even above room temperature, clear Coulomb oscillations are visible and efficiencies of 5% could be reached. A larger distance to the gate electrode and smaller particles elongate the oscillation period from an average of 24.3 V (A) to 54.7 V (C) and 96.9 V (D). Note that the origin of the higher frequency oscillations seen in (C) and (D) is not entirely clear; they are assumed to result from some external influence and only appeared from time to time.
Fig. 4 Voltage dependency.
(A) Oscillations induced by a local gate for different bias voltages at a temperature of 10 K. For the lowest shown voltage, an averaged curve has been added as a guide to the eye. The shape and distinctive features of each curve are reproducing at different bias voltages. A shift in minimum position can very well be seen. (B) Corresponding output characteristics and differential conductance plot. The colored lines mark the bias voltages displayed in (A). (C) The bias voltage dependency of the on/off ratio illustrates the different stages of electrical transport in the system. The maximum performance is reached around the threshold voltage.
Sweeping the potential of a local 0.5-μm-wide back gate below a 1-μm-long conduction channel resulted in the transfer characteristics shown in Fig. 3A. It can be seen that the current clearly oscillates and that there is only little noise and hysteresis between the two sweep directions. All main features reproduce for different temperatures, and there is one pronounced minimum, which will be discussed in detail later. At low temperatures, on/off ratios of about 70% could be observed with the medium-sized particles. With increasing temperature, the electrons' energy and thus the overall current are higher as more and more conduction paths are accessible for transport. However, this multitude of conduction paths cannot be simultaneously switched as efficiently because of statistical variations of threshold voltage and Coulomb energy. This results in a quasi-constant background current, and the relative on/off ratio decreases exponentially with temperature as displayed in Fig. 3B. Three devices are compared, differing only in channel length. Although the differences are not large, there is a tendency that shorter channels yield higher on/off ratios with the same-sized back gate.
Using the global silicon substrate as back gate, which stretches over the whole channel length in contrast to the local gate, yields Coulomb oscillations with even higher regularity and periodicity (Fig. 3C). The transfer characteristics appear sinusoidal, and there is no pronounced main minimum anymore. The device performance is about the same compared to the local gate case (Fig. 3A). Note that clear oscillations can be observed at temperatures of up to at least 240 K. Beyond that, both gating geometries show evidence of room temperature Coulomb oscillations. However, the obtained percentages were negligible.
To improve the device performance even further, the adjustability of the system can be used in yet another way. By decreasing the particle size, the self-capacitance is lowered and the charging energy is increased. Hence, Coulomb oscillations become more pronounced and remain visible up to higher temperatures. Figure 3D shows results from a device with smaller nanoparticles of 2.3 nm diameter gated again by the global substrate. At the lowest measured temperature of 6 K, a relative on/off ratio of more than 90% could be obtained. Moreover, sinusoidal oscillations were clearly detectable at room temperature. The on/off ratio was six times higher than observed before, and values of up to 5% could be measured.
Further investigations demonstrate the influence of the bias voltage on the oscillations at 10 K. This can exemplarily be seen in Fig. 4A; the chosen voltage values are also marked by the vertical lines in the respective output characteristics in Fig. 4B (more data are given in fig. S7). The lowest displayed bias voltage of VSD = −9 V is clearly still inside the blockade regime. Nevertheless, weak oscillations can be seen in the red curve. Although the signal is very noisy, this shows how the gate's field effect can lower the energy barrier so that current flow is periodically possible. Increasing the bias voltage enables more conduction paths. However, similar to the effect of a high temperature, the paths all vary slightly in their energy-level configuration and cannot be blocked simultaneously, leading to lower relative on/off ratios. Thus, the best ratios were obtained for bias voltages near the threshold voltage, as depicted in Fig. 4C. This subfigure summarizes exemplary device performances at a temperature of 6 K for different bias voltages. For each curve, there is a blockade regime, then an onset of oscillatory behavior, a maximum of the on/off ratio around the threshold voltage, followed by a slow decrease. The longer the conducting channel, the higher the respective oscillation onset and threshold voltage because the total bias is distributed over more particles. Again, it can be seen that the best performance was observed in the device with the shortest channel.
The four oscillations shown in Fig. 4A are shifting toward more negative gate voltages as the applied bias changes in the same direction. In the pronounced main minimum of the gate-voltage sweep, not only the energy-level configuration is most unfavorable for transport, but also the overall electric-field gradient between the three contacts is disadvantageous for elastic tunneling. Depending on the exact location of the gate electrode under the channel and on the gate width, a simple model (32) suggests that this main minimum is at Vgate ≈ VSD/2. At this gate voltage, the linear potential gradient is least disturbed and remains very unfavorable for resonant tunneling so that the current is minimal. The potential gradient between source and drain is simulated and evaluated for two different gate positions based on this model in fig. S8. To investigate this effect in more detail, we fabricated structures with three parallel narrow local back gates, as can be seen in the optical micrograph in Fig. 5A. They have the same width, are separated from the channel by the same amount of dielectric and therefore only differ in their lateral placement, and influence the transport along the channel from different positions.
Fig. 5 Oscillation shift.
(A) Optical micrograph of a device with a 3-μm-long channel and three parallel 500-nm-wide gate electrodes. (B) Each gate can be used to induce oscillations. At the same bias voltage, the main features are shifted. (C) Plotting the position of the minimum with respect to the applied bias voltage yields a linear correlation that is independent of temperature. Each gate position corresponds to a certain slope. The measured data roughly match the simulated slopes but reveal a shift in the lithographic alignment.
Applying the same bias voltage to source and drain and using each one of the gates alone while leaving the other two uncontacted (floating) results in a shift in the oscillations of about 16 V depending on the used gate's position (Fig. 5B). It is now possible to use the gate voltage corresponding to the oscillation minimum as a marker for the shift and plot it for different bias voltages (Fig. 5C). Independent of the temperature, there is a linear correlation between the gate voltage at the minimum and the applied bias voltage. It was found that the slope of this line can be directly connected to the gate's position. The theoretical slopes from simple simulations (fig. S8) are also given in Fig. 5C. Although they are all somewhat smaller than those obtained from the data, the difference is about the same for all three lines. This leads to the conclusion that there might have been a shift in the lithography process, and the gates are positioned slightly more to the right than intended. Note that these gates are still quite wide compared to the channel. The correlation between gate placement and oscillation is valuable for applications: The position of a narrow local gate can be directly used to shift the resulting transfer characteristics or, vice versa, the shift can be used to determine the gate position. For an application that requires the working voltage to be at a certain point of the oscillation, for example in the minimum for current stabilization, this offers an easy, accessible tuning mechanism to adapt the functionality even if the bias voltage is given. Extending the study with the three gates to different channel lengths (fig. S9) confirms the tendency that a better effect is obtained if the gate stretches over a larger fraction of the channel and adds that a gate electrode has no influence anymore if it is placed right next to the channel underneath the gold leads. In summary, the above observations show that depending on the focus of the desired application, one could either use a narrow local gate to tune the position of the oscillations or a wide gate that covers the whole channel area to maximize the on/off ratio.
The device geometry determines many of the system's characteristic properties; however, the nanoparticle diameter has an even stronger influence. The usage of wet-chemically synthesized colloidal nanoparticles makes small sizes and precise tailoring possible. In Fig. 6A, the size distributions of three batches of particles are shown. They range between 2.3 nm and 4.0 nm: The smaller the particles, the higher the self-capacitance and thus the charging energy. This can be seen directly in the transfer characteristics: Sweeping the gate voltage from one maximum to the next means adding the required energy to overcome the corresponding Coulomb energy gap. The period of oscillation becomes longer for smaller particles. This correlation can be seen very well in Fig. 6B. For each particle size, a point cloud of oscillation period measurements is given. It was found that the period did not depend on channel length, temperature, or bias voltage, which is in line with the theory. The horizontal lines mark the average value also given as a number on the right. There is also a second influence on the periodicity: The thickness of the dielectric material, meaning the separation of channel and gate electrode, determines the strength of the electric field in the nanoparticles. A thinner dielectric strongly increases the field effect and also shortens the oscillation period (compare red lines in Fig. 6B). Although a thin dielectric also improves device performance, it still needs a certain thickness to act as barrier against leakage currents.
Fig. 6 Particle size comparison.
(A) Size distributions including Gaussian fits. a.u., arbitrary units. (B) Period of oscillation for different channel lengths, temperatures, and bias voltages. The horizontal lines mark the average value for each set of data points. The period seems to be only dependent on particle size and separation between channel and gate electrode. With these two parameters, it can be tuned over a wide voltage range. (C) Bias voltage dependent on/off ratios at low temperature for the three sizes. The best performance can be obtained by using small particles. (D) The activation energy is inversely proportional to the particle diameter. The results of this work supplement the data from another publication (31).
Increasing the Coulomb energy by decreasing particle size also leads to more pronounced oscillations. They remain visible up to a higher temperature and exhibit higher on/off ratios. Figure 6C summarizes device performance at the lowest measured temperature for the three particle sizes. Simply changing from the medium particles to the small ones increased the on/off ratio by about 15% and enabled sinusoidal room temperature oscillations. As discussed in the beginning of this work, the system's activation energy can be determined from the Arrhenius plots of the differential zero-bias conductance. Being strongly related to the Coulomb energy itself, the activation energy is also proportional to the inverse particle diameter. In Fig. 6D, the three data points from particles investigated in this work extend the data from previous work (31). The linear fit matches the relation very well and confirms the expected behavior.
Fabricating the gate electrode before particle deposition leaves the particle array that constitutes the conducting channel undisturbed and the Coulomb oscillations can be observed with little interference. Therefore, the influence of different gating techniques and parameters could be reliably investigated. It has been observed that even the sputter-deposited silicon dioxide underneath the particles can influence the transfer characteristics. Charging of trap states in the gate dielectric leads to some reversible hysteresis effects if the applied gate voltage is very high (see fig. S3). For later applications, some form of sealing will be required. It was found that polymer systems such as resist or wax do not interfere with the device's function (see fig. S4).
We demonstrate a new device concept that allows for easy and industry-compatible fabrication and a high tunability to match the needs of transistors or future applications. The device parameters can especially influence working temperature and bias, as well as oscillation period and strength. The use of colloidal nanoparticles ensures the tunability of size and tunnel barrier width, monodispersity, very small capacitances and thus high Coulomb energies, and inexpensiveness. In the metal nanoparticle-based devices, Coulomb oscillations are observed with an unprecedented clarity and reliability, even at room temperature.
Nanoparticle synthesis
CoPt nanoparticles were synthesized according to the work done by Shevchenko et al. (40) with some modifications. The mainly investigated particles were monodisperse with a diameter of (3.5 ± 0.3) nm and a composition of 93.2% Pt and 6.8% Co [analysis of the (111) x-ray diffraction (XRD) peak by means of Vegard's law]. In the first step, 33 mg(0.084 mmol) of the platinum source platinum(II) acetylacetonate [Pt(acac)2] was dissolved in 4 ml of the solvent 1-octadecene. To assist in the reduction of the platinum, 131.9 mg (0.51 mmol) of 1,2-hexadecanediol was added. For shape and size control during particle nucleation and growth, 2.22 ml (6.72 mmol) of oleylamine and 0.44 ml (1.4 mmol) of oleic acid were added. The solution was heated to 80°C and stirred at this temperature for at least an hour under vacuum conditions. In the second step, 43.7 mg (0.126 mmol) of the cobalt source dicobalt octacarbonyl [Co2(CO)8] dissolved in 0.6 ml of 1,2-dichlorobenzene in an ultrasound bath was injected rapidly into the solution that has been heated to 160°C under nitrogen atmosphere. This so-called hot injection was followed by thermal decomposition of the cobalt precursor. The mixture immediately turned black and was continuously stirred at the injection temperature for 2 hours. During cool down, 5 ml of toluene was added. Then, the particles were washed three times through precipitation with 2-propanol and methanol, phase-separated in a centrifuge at 7000 revolutions per minute (4492g) for 4 min, and subsequently resuspended in toluene. In the final step, they were filtered through a polytetrafluoroethylene syringe filter with a porousness of 0.2 μm. Other particle sizes can be realized by altering the ratio of amine and platinum source. Various interparticle distances can be achieved by substituting longer- or shorter-chained ligands.
Doped silicon wafers of roughly 1 cm2 with 500 nm of thermal oxide were used as substrates. For electrode preparation, they were covered with the positive resist poly(methyl methacrylate) (PMMA) (dissolved in chlorobenzol) via spin-coating. Standard electron-beam lithography [Quanta Scanning Electron Microscope (FEI) and CAD software ELPHY Plus (Raith)] and metal evaporation of 2 nm of titanium as an adhesion layer and 23 nm of gold created the electrodes and contact pads as well as markers to align the different layers during lithography. As a gate dielectric, an isolating layer of about 130 nm of silicon dioxide was deposited via ion-beam sputtering [Precision Etching and Coating System model 682 (Gatan)] on top of the back-gate electrode, while the contact pad was protected with PMMA that could later be removed. In the last step, another resist mask was created, which defines the conduction channel as a stripe between source and drain electrode. Some samples did not have a local back gate but were measured using the global silicon substrate as a gate. In these cases, the pristine thermal oxide was only 300 nm thick.
Langmuir-Blodgett method
Monolayer preparation was done using the Langmuir-Blodgett trough from KSV NIMA, as seen in fig. S2. The silicon substrates with readily fabricated electrodes were attached to a holder together with a transmission electron microscope (TEM) grid to investigate the film quality afterward. They were submerged into the subphase diethylene glycol. The particles were washed again, dissolved in toluene, and carefully brought upon the subphase surface with a microliter syringe (starting with 300 to 400 μl of nanoparticles in toluene, as described under synthesis for a trough area of 120 cm2). Because of the hydrophobic character of the ligand shell, they will stay on the surface and spread as far as possible while the solvent evaporates. With two computer controlled barriers, they were slowly forced together with a speed of 1 mm/min; the compression was monitored by in situ surface-pressure measurement. The slope changes of the resulting isothermal curve and empirical known values of the required difference in surface pressure enabled the estimation of the point of a densely packed monolayer without holes or overlap. The barriers kept the film at this pressure for 2 hours of relaxation time. Then, the substrates were carefully lifted and slowly broke the surface at an angle of 105°, which allowed collection of the particle film (30).
Particle and sample characterization
The CoPt nanoparticles were investigated with a TEM after synthesis and film preparation. During each Langmuir-Blodgett process, a TEM grid was loaded with particles in parallel to the actual silicon substrates. From these micrographs, the particle diameter, size distribution, spacing, and monolayer quality were assessed. Exemplary batches of particles were measured with XRD to verify the crystal structure and composition. All throughout the fabrication process, the substrates, electrodes, resist masks, and the final sample were frequently checked with a light microscope. Furthermore, in exemplary samples, the heights of the electrodes and of the sputtered dielectric were measured with an atomic force microscope.
Electrical characterization
The main measurements were carried out in a closed cycle cryogenic probe station (model CRX-4K, Lake Shore Cryotronics) with a Keithley 4200-SCS parameter analyzer. Characterization was done at residual gas pressures of 10−6 mbar at room temperature to 10−7 mbar at low temperature. The sample stage can be cooled down to 4.5 K. Contacting was done with up to six micromanipulatable needle probes with a 20-μm-radius beryllium-copper tip. Another pin was connected to the sample stage and was used to apply a voltage to the substrate and thus to the global back gate. Signal transmission was achieved through triaxial cables to the Keithley's source measure units. Currents down to about 100 fA can be detected, whereas everything below was considered as noise. For output characteristics, the gate was left floating or grounded, whereas the bias voltage was varied stepwise between ±20 V. Similarly, for transfer characteristics, the gate voltage was varied stepwise in different intervals depending on the sample with a fixed bias voltage. Parameter sweeps were always done bidirectionally to verify the results.
text S1. Capacitance.
fig. S1. Particle characterization.
fig. S2. Monolayer preparation.
fig. S3. Hysteresis.
fig. S4. Capping.
fig. S5. Transport mechanism.
fig. S6. 2D plot of the device characteristics.
fig. S7. Voltage-dependent oscillations.
fig. S8. Position of minimum.
fig. S9. Influence of gate position.
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Acknowledgments: We thank A. Barck for XRD measurements. Funding: We thank the European Research Council (ERC) for funding an ERC Starting Grant [project 2D-SYNETRA (304980), Seventh Framework Programme FP7]. The German Research Foundation DFG provided financial support and granted the project KL 1453/9-1 and the Cluster of Excellence "Center of ultrafast imaging CUI." Author contributions: S.W., H.L., and C.K. conceived the main concepts. S.W. and H.L. performed electron-beam lithography and the electrical measurements. M.V. performed the nanoparticle synthesis and characterization. S.W. and M.V. prepared the Langmuir-Blodgett films. C.K. performed the simulations. S.W., H.L., M.V., and C.K. analyzed the measurement data, discussed the results, and wrote the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
You are going to email the following Metal nanoparticle film–based room temperature Coulomb transistor
By Svenja Willing, Hauke Lehmann, Mirjam Volkmann, Christian Klinke
Science Advances 14 Jul 2017 : e1603191
A new transistor concept exploits the colloidal synthesis of metal nanoparticles and their Coulomb charging energy.


text:
Scores catalog
Saturday 28 December 2013, by Laurent (Vnouk) // Works
Partition des 24 préludes
The first Vsevolod Petrovitch's score published was the 24 preludes for piano, in 1970, by Mouzitchna oukraïna publishing house, Kiev. In his essay "Ich bin schon längst tot", Komponisten im Gulag : Vsevolod Zaderackij und Aleksandr Veprik published in Osteuropa, june 2007 (Germany), Jascha Nemtsov wrote a main catalog of Zaderatsky's published scores and manuscripts. In his book "Per Aspera …" Vsevolod Vsevolodovitch Zaderatsky proposes the first exhaustive catalog of the published scores which we propose you there. This catalog will be up to date with the new publications.
24 preludes for piano (1934). Mouzitchna oukraïna publishing house, Kiev, 1970
To the Russian Soldier's Poem : 7 pieces from the 10 (1946/47). Composer publishing house, Moscow, 1973
Collection of 3 cycles for piano : "Porcelains cups" (1930), "Micro-Lyrics" (1929), "Miniatures' album" (1928). Mouzitchna oukraïna publishing house, Kiev, 1979
Sonate for piano N°3 in F sharp minor (1939). Mouzitchna oukraïna publishing house, Kiev, 1982
Selected works from the 24 preludes et fugues for piano : C major, A minor, G major, G minor, F major, C minor, B minor (1937/38). Mouzitchna oukraïna publishing house, Kiev, 1983
Miniature's album (1928), 4 pieces cycle for piano (missing "The March"). C. F. Peters Music Publishing House, Leipzig, 1983
Sonatine (from the 6 pieces cycle for piano, for child) (1944). In the collection "Easy sonatines for piano, from ukrainiens composers ". Mouzitchna oukraïna publishing house, Kiev, 1985
5 Pieces for piano, for child, (from the 6 pièces cycle ) : Sonatine, Rondo, Jester, Waltz, Gunner's march (1944). Spolom publishing house, L'viv, 2003
Motherland (1946), 7 pieces suite for piano. Spolom publishing house, L'viv, 2003
The silver shower (1948). In the collection "Works for piano from L'viv composers". Spolom publishing house, L'viv, 2003
Concerto for Domra and orchestra (piano) (1947). Spolom publishing house, L'viv, 2003
Sonates N°1 and N°2 for piano (1928). Spolom publishing house, L'viv, 2003
Sonate N°3 for piano. Spolom publishing house, L'viv, 2003
Concerto for violin and orchestra (piano) (1952). Spolom publishing house, L'viv, 2003
Rhapsodies for piano : "Caucasian rhapsody " (1944), "Russian rhapsody" (1947). Spolom publishing house, L'viv, 2003
Child's Concerto for piano and orchestra N°1 (piano) (1946). Spolom publishing house, L'viv, 2003
Child's Concerto for piano and orchestra N°2 (piano) (1948/49). Spolom publishing house, L'viv, 2003
The front (1944), 10 pieces suite. Spolom publishing house, L'viv, 2003
Legends (1944), 5 pieces cycle. Spolom publishing house, L'viv, 2003
Pieces for piano, selected works : "The little wreath", "Oriental album" (1940), "Spanish dances", "Sea front", "Persan danse" (1940), "Album's page" (1940), "Epitaph" (1940). Spolom publishing house, L'viv, 2003
Melodies on poems from russian poets : "The path" (1949), "To the chesnut son" (1949), "The flask" (1949), "Victoria Regia" (1944), "Sentimental evening" (1946), "Fortune-telling" (1931), "Dark river" (1949), "Light is dimming in your window" (1946), "Coachman" (1932), "The Cat and the Lazybones" (1934), "Bad weather" (1948), "The Evening star" (1948), "Mandolina" (1948), "Sleep, white snow Field" (1939), "Lullaby" (1944), "Baku" (1944), "One Tenth" (1944), "Queen of Clubs" (1944), "Persian song" (1949), "The Orange Light is my world" (1931), "The factory", "Western front" (1931), "Don river old song" (1932), "Mother's heart" (1934). Composer publishing house, Moscow, 2003
Pieces for piano, selected works : 3 préludes and fugues (C major, A minor, G major) (1937/38), 4 pieces from the cycle of "Miniatures' album" (missing "The march") (1928), "The silver shower" (1948), "Album's page" (1940). "Deka-BC" publishing house, Moscow, 2004
24 preludes for piano (1934). "Deka-BC" publishing house, Moscow, 2006
Chamber symphony, for a 9 instruments orchestra (1940). Composer publishing house, Moscow, 2007
"De Profundis" (1944), vocal cycle for tenor and piano on Ilya Sadofief's poems. Composer publishing house, Moscow, 2011
Sonata for french horn and piano (1948), Compozitor Publishing House, Saint-Petersburg, 2013
Motherland (1946), suite for piano. "Deka-BC" Publishing House, Moscow, 2015

redpajama_set_name: RedPajamaCommonCrawl


Excuse me for being stupid but I can't seem to work out how to rotate an object perpendicular to its surface. Currently I can only rotate an object perpendicular to a reference plane. Which becomes useless once an object has been rotated to an angle differing with the reference planes.
I'm sure there's a simple answer to this but I've tried searching the web with no avail!
Try the Transform tool which lets to move /rotate an object about it's own axis, OR use the "define reference plane" from the a face of the object from which you can rotate.


text:
\section{Introduction}

Photonic lattices (PLs) are currently at the focus of extensive research for two main reasons. Firstly, for their flexibility in simulating various phenomena, especially those related to tight-binding Hamiltonians \cite{PLRev1,PLRev2,PLRev3,PLRev5}, and secondly for their potentials
as building blocks of quantum circuits for all-optical information processing and networking \cite{3DPLs,PLRev4,SoS},
and their role in related studies on quantum random walks \cite{PLrandomWalk} and boson sampling \cite{PLbosonFilter}.

PLs can be fabricated in a doped silica multilayer structure on a silicon substrate \cite{PLRev4,SoS}, as well as by means of femtosecond laser-writing techniques in the bulk  of glasses \cite{3DPLs,PLRev3}. Both of these techniques allow one to exploit Kerr nonlinearity in order to achieve certain  tasks. In addition, fabrication of waveguides in  LiNbO$_3$ and KTP by means of  etching techniques \cite{PDC} allows for integrated sources of non-classical light, paving thus the way toward integrated quantum chips, where the generation of entangled photons, their transmission, and their processing take place on the same chip.

The faithful transfer of signals  is a necessary precondition for further developments in these directions, and the problem has attracted considerable interest in recent years \cite{Segment,Lon10,PSTrealization,JogPRA,FSTrealization,protocol,Gor04,prl08,ReviewsPST,EdVolume,HamEng,GenericBends}.
The typical evolution of an initially well-localized wavepacket  in an ideal finite PL (i.e.,  in the absence of disorder and dissipation), is characterized by spreading,  reflections from the boundaries, and interference phenomena that give rise to diffraction effects \cite{Lon10}.
At relatively high intensities \cite{remark1}, such distortion effects can be avoided by using discrete solitons as information carriers \cite{PLRev1,PLRev2,PLRev3,3DPLs}. Alternatively, in the linear regime (i.e., for input powers below the threshold for the existence of discrete solitons \cite{remark1}), one may resort to  the segmentation of appropriate lattice sites \cite{Segment}, or to the engineering of judicious couplings between adjacent sites \cite{Lon10,PSTrealization,JogPRA,FSTrealization,protocol,Gor04,prl08}. The latter scenario has been also studied thoroughly in the context of quantum networks, and various solutions have been proposed \cite{ReviewsPST,EdVolume, HamEng, GenericBends}.

In any case, however,  the reliable transfer of signals between two distant nodes of a network
is not sufficient for large-scale  information processing and networking. To this end, one has to be able to perform reliably and efficiently more complex signal manipulations such as routing, splitting, switching, etc. Such communication tasks can be performed only in higher-dimensional geometric arrangements, where the presence of  bends at different angles are inevitable \cite{3DPLs,Chr01,ExpBend,prl08,GenericBends}.

The transfer of  signals around bent nonlinear (discrete-soliton) PLs has been investigated to some extent in the literature \cite{3DPLs,ExpBend,Chr01}. On the contrary, the only analogous study for the case of linear networks with engineered couplings has been limited to a generic theoretical model and small bend angles \cite{GenericBends}, so that the effects of bending  can be treated as a small perturbation  to the unbent chain. The regime of bend angles for which this approximation is justified could not be assessed within the generic model of Ref.  \cite{GenericBends}, since it depends strongly on the details of the physical system under consideration. One of the purposes of the present work is to address this question, in the framework of a considerably more elaborate theoretical model that pertains to a two dimensional (2D)  linear PL with engineered couplings, taking into account possible anisotropy effects. To mimic the asymmetry of the waveguides, typically present in experimental realizations, we consider waveguides of an asymmetric rectangular shape.  Moreover, in contrast to Ref.  \cite{GenericBends}, our formalism takes into account all the couplings beyond nearest-neighbours, between any two waveguides. The parameters used throughout our simulations are typical for PLs written in the bulk of glasses,  as this technique has the advantage of being maskless, fast, and rather versatile, allowing for fabrication of optical circuits with 3D layouts, and  waveguides of  controllable transverse profile.  Furthermore,  the  specific coupling configuration we consider, has been implemented recently in this experimental set-up \cite{PSTrealization}.

The paper is organized as follows.  Our theoretical model, together with various aspects
of the system under consideration, is presented in Sec. \ref{sec2}. Section \ref{sec3} is devoted to our simulations, with an extensive discussion of our results.
It is shown that by engineering the couplings between nearest-neighbours one can achieve   faithful transfer of signals between the first and the last  waveguide of a bent PL that operates in the linear regime, for bend angles at least up to $90^\circ$. For sharper bends, the detrimental effects of the bending become very pronounced rapidly, especially in the case of asymmetric waveguides, and can be suppressed by introducing a defect at the corner site, while keeping all the other parameters in the lattice constant. Our simulations suggest that in this way, for the particular coupling configuration under consideration, faithful transfer of signals for bend angles up to $60^\circ$ ($120^\circ$ with respect to the unbent chain), is possible. Qualitative as well as quantitative aspects of the defect required for suppression of  bending effects at different angles are also discussed, and our main results are compared to related results for nonlinear networks, that rely on solitonic information carriers. In the last Sec. \ref{sec4}, we summarize our findings and discuss certain issues that remain open.

\begin{figure}
\centering
\includegraphics[scale=0.9]{Configuration-eps-converted-to.pdf}
\includegraphics[scale=0.8]{RIscheme-eps-converted-to.pdf}
\caption{(Color online)
The system under consideration: a bent quantum chain of $N$ rectangular waveguides. Light enters the structure at $z=0$ in the first waveguide, and as it propagates along $z$ it couples to other waveguides.
Our task is the faithful (ideally perfect) transfer of the input signal to the $N$th waveguide at the exit $(z=L)$ of the structure. The lower scheme shows a cross section of a rectangular waveguide of area $\Delta x\times \Delta y$,
and refractive index  $n_g$. }
\label{fig1}
\end{figure}

\section{Physical system and modelling}
\label{sec2}
Various 2D configurations of laser-written buried photonic lattices in glasses, have been demonstrated and studied experimentally \cite{PLRev1,PLRev2,PLRev3,3DPLs,ExpBend,PSTrealization}. Typically, the wavenumber along the propagation direction for each waveguide can be  controlled by adjusting the corresponding size and/or the refractive-index change. Moreover, the coupling between two neighboring waveguides drops exponentially with their separation, and the precise form of the exponential law is determined by the details of the experimental set-up. Knowing the precise form of this exponential  law one can engineer various configurations of waveguides that perform certain tasks, such as the non-dispersive transfer of signals between  two waveguides of a PL, for input light with specific properties (i.e., wavelength, polarization) \cite{PSTrealization,FSTrealization}. In most cases, experimental observations have been shown to be in excellent agreement with the predictions of coupled mode theory, and other theoretical models that rely on the Helmhotlz equation for scalar fields.

\begin{figure}
\centerline{\includegraphics[scale=0.55]{EigenmodePairWG.jpg}}
\caption{(Color online)
Eigenmodes of two rectangular waveguides and the geometry of their overlap.
The two identical waveguides (see rectangles) are shown together with the
electric field ${\mathcal E}_x^{(j)}(x,y)$. Parameters:   $\Delta x = 6\mu$m, $\Delta y =2\mu$m, $n_s=1.444$, $\delta n = 10^{-3}$, $\lambda = 800 $nm, $r = 30\mu$m, $\theta = \pi/4$.}
\label{fig2}
\end{figure}

In the present work we are interested in the transfer of signals between the two outermost waveguides of a bent array of linear waveguides.
The 2D arrangement under consideration is depicted in Fig. \ref{fig1}, and pertains to $N$ identical rectangular waveguides, of cross section $\sigma = \Delta x \times \Delta y$ and length $L$.
Light of specific wavelength $\lambda$, is injected in the first waveguide  at $z=0$, and couples to the neighbouring waveguides as it propagates along $z$.
Our task is to achieve faithful (ideally perfect) transfer of the signal from the first to the $N$th waveguide at the exit of the
structure i.e., at $z=L$, for a given value of the permanent bend angle $\theta\geq 0$.

To this end, for a given value of $\theta$ we engineer the distances between neighbouring  waveguides aiming ideally at a configuration of the coupling constants of the form
\begin{eqnarray}
G_{m,l} = \left \{
\begin{array}{ll}
\frac{\pi}{2L} \sqrt{(N-M)M}, &   |m-l|=1\\
0, &  \textrm{otherwise},
\end{array} \right.
\label{coupling_j}
\end{eqnarray}
for $m,l \in[1,N]$ and $M\equiv\min\{m,l\}$. This is a centrosymmetric configuration with respect to the central waveguide(s) of the PL \cite{ReviewsPST,EdVolume,protocol,Gor04,prl08}. In practise, the
couplings beyond nearest neighbors are never zero, but they must be negligible relative to the
nearest-neighbor couplings.  For an unbent PL $(\theta = 0)$, such engineering has been demonstrated recently by two different groups in the framework of PLs in the bulk of fused silica \cite{PSTrealization}. It requires knowledge mainly on the spatial dependence of the coupling, as well as on the refractive index profile for each waveguide  and the form of the excited eigenmodes at a given operation wavelength $\lambda$.  In the following we will investigate whether the coupling configuration (\ref{coupling_j}) can be  implemented in the same manner,  in bent PLs (i.e., for $\theta >0$), with symmetric or asymmetric waveguides. Before we address this question, let us present our model for the waveguides.

\subsection{Eigenmodes of independent waveguides}
In various physical realizations the waveguides have an asymmetric elliptic profile, which is reflected in the observed modal field distribution at a given wavelength and field polarization, as well as in the dependence of the coupling on the waveguide separation \cite{PLRev3,3DPLs,FSTrealization,AsymWG}. Although, as discussed in Sec. \ref{sec4}, the design of symmetric waveguides is possible with current technology, throughout the present work we present results for waveguides with  asymmetric shape, since in this case the presence of bends turns out to be more pronounced than for symmetric waveguides.  Analogous simulations for symmetric waveguides have also been performed, and we will refer to related findings wherever necessary, pointing out the main differences from the case of asymmetric waveguides.

We consider  rectangular waveguides with $\Delta x \geq \Delta y$, corresponding to the major and minor axes of the  elliptic profiles typically observed experimentally.  For the analysis of the waveguides we follow the Marcatili's approach, which is widely used in photonics and optoelectronics \cite{Marc69,book1,book2}. A cross-sectional view of the waveguides under consideration is shown in the lower Fig. \ref{fig1}, where $n_g$ and $n_s$ denote the refractive indices for the core and the cladding (substrate) respectively. Typically, their difference is very small $(\sim 10^{-3})$ and thus  the refractive index distribution for the $j$th rectangular  waveguide is well approximated by
\begin{eqnarray}
n_j^2(x,y) \approx n_{j;x}^2(x)+n_{j;y}^2(y) + {\cal O}(n_g^2 - n_s^2)
\end{eqnarray}
with
\begin{eqnarray}
 n_{j;x}^2(x) = \left \{
\begin{array}{ll}
n_g^2/2, &  |x| \leq \Delta x/2\\
n_s^2-n_g^2/2, &  |x| > \Delta x/2
\end{array} \right.
\end{eqnarray}
and
\begin{eqnarray}
 n_{j;y}^2(y) = \left \{
\begin{array}{ll}
n_g^2/2, &  |y| \leq \Delta y/2\\
n_s^2-n_g^2/2, &  |y| > \Delta y/2
\end{array} \right. ,
\end{eqnarray}
where we have assumed that the waveguide is centred at $(x,y)=(0,0)$.
In the following, for the sake of simplicity and without loss of generality we set
\begin{eqnarray}
n_g = \frac{n_s}{\sqrt{1-2\delta n}}
\end{eqnarray}
where $\delta n$ is the modification of the refractive index.
The refractive index of the shaded (corner) areas in the lower scheme of Fig. \ref{fig1}, is approximated by $\sqrt{2n_s^2-n_g^2}\approx n_s(1-\delta n-{\cal O}(\delta n^2))$. Clearly, for $\delta n\sim 10^3$, one has $\sqrt{2n_s^2-n_g^2}\approx n_s$.

The dimensions of the waveguides and the associated  refractive-index modulations adopted throughout this work are within the range of values one typically
finds in experiments pertaining to laser-written waveguides in  glasses.  There are some quantitative differences, however, since the adopted rectangular profile is not expected to capture precisely all the features of the modes observed in experiments (e.g., precise form of modes,
 penetration depth, etc). One of the key features in most of the experiments is that only one (the lowest) eigenmode is excited at the operation wavelength. Hence, the waveguide parameters we consider here are such that the lowest mode of the rectangular waveguide is  excited  (i.e., $\mathbb{E}_{1,1}^{(x)}$	or $\mathbb{E}_{1,1}^{(y)}$), which means that the electric field exhibits only one  peak along both x- and y-axis directions. The main field components for modes $\mathbb{E}_{1,1}^{[x(y)]}$ in the $j$th waveguide are  $E_{x(y)}^{(j)}$  and $H_{y(x)}^{(j)}$,  with  the electric field polarized along the $x(y)$ direction, respectively.

The focus of the present work is on the effects of bends and to this end
we ignore various types of possible imperfections so that all the estimated losses in our simulations can be attributed only to bending effects. Furthermore it is sufficient to consider the  mode $\mathbb{E}_{1,1}^{(x)} $ in  the following analysis, since the calculations for the $\mathbb{E}_{1,1}^{(y)}$  mode are the same.
We will return to this point in the concluding remarks of the present work.

The components of the electric $({\bm E}^{(j)})$ and magnetic $({\bm H}^{(j)})$ fields that prevail in the $j$th waveguide are \cite{Marc69,book1,book2}
\begin{eqnarray}
&& E_x^{(j)} = {\mathscr E}_{x}^{(j)}(x,y) \exp[i(\omega t+\beta_j z)] \\
&& H_y^{(j)} = {\mathscr H}_{y}^{(j)}(x,y) \exp[i(\omega t+\beta_j z)],
\end{eqnarray}
 whereas $H_x^{(j)} = 0$. From Maxwell's equations one has:
\begin{eqnarray}
\label{Hy_eq}
&&\frac{\partial^2 {\mathscr H}_y^{(j)}}{\partial x^2}+ \frac{\partial^2 {\mathscr H}_y^{(j)}}{\partial y^2}
+ [k^2 n_j(x,y)^2 - \beta_j^2] {\mathscr H}_y^{(j)} = 0,\\
&& {\mathscr E}_x^{(j)} \approx \frac{\omega\mu_0}{\beta_j} {\mathscr H}_y^{(j)},
\label{Ex_eq}
\end{eqnarray}
where $\omega$ and $k$ are the frequency and the wavenumber of the input light, whereas $\beta_j$ is the
wavenumber along the propagation direction of the waveguide (z-axis).
Equation (\ref{Hy_eq}) can be solved numerically, or analytically for the model under consideration (see appendix).  In Fig. \ref{fig2} we show
the electric field  ${\mathscr E}_x(x,y)$ as obtained in our simulations, for a particular set of parameters. Clearly, the asymmetry of the waveguides is also reflected in the eigenmodes, and as will be seen later on, it also affects the coupling between adjacent waveguides. It has to be emphasized here, however, that the depicted modal profile is for the sake of illustration only, and pertains to the particular parameters given in the caption. The quantitative aspects of the
modal profile may change e.g., by changing the dimensions of the waveguide, the wavelength of the light, etc.

Finally, the normalization we have adopted  throughout this work implies that the power carried by the eigenmode  of the $j$th waveguide along the propagation direction is \cite{EdVolume,book1,book2, book3}
\begin{eqnarray}
P_j = \frac{1}{2}\int \int \Re \left [({\bm E}^{(j)}\times {\bm H}^{(j)\star} )\cdot \hat{\bm z}\right ]\,  dx dy = 1\,\textrm{(Watt)}.
\end{eqnarray}

\subsection{Coupled-mode theory}

As shown in Fig. \ref{fig2}, when two waveguides are brought close together, their  optical modes overlap. For sufficiently small overlaps, the electromagnetic field distribution for either of the neighbouring waveguide does not differ substantially from the one for an isolated waveguide, and the propagation characteristics of the coupled waveguides can be analyzed by means of  the coupled-mode theory, the details of which can be found almost in every textbook on photonics and optoelectronics (e.g., see \cite{EdVolume,book1,book2, book3}).
For the sake of completeness, here we sketch the main steps of the approach.

The total electric field in a configuration of $N$ evanescently coupled identical waveguides  is well approximated by the superposition
\begin{eqnarray}
{\bm E}(x,y,z) = \sum_{j=1}^N A_j(z) {\mathscr E}_x^{(j)}(x,y)\exp[i(\omega t - \beta_j z)] \hat{\bm x},
\label{totalEF}
\end{eqnarray}
where ${\mathcal E}_x^{(j)}$ are determined by Eq. (\ref{Ex_eq}).
Substituting  this expression into the wave-equation
\begin{eqnarray}
\left [ {\nabla}^2
+ \frac{\omega^2}{c^2}n^2(x,y)
\right ] {\bm E}(x,y,z)  = 0,
\end{eqnarray}
where $n(x,y)$ the refractive-index distribution of the entire  structure  of the coupled
waveguides, and following standard well-known steps one obtains a closed set of differential equations for the amplitudes $A_j(z)$
\begin{eqnarray}
\frac{d{\mathfrak A}}{dz} = {\mathfrak  J}\cdot  {\mathfrak A}
\label{eom}
\end{eqnarray}
where ${\mathfrak A}\equiv (A_1, A_2, \ldots, A_N)^T$ and ${\mathfrak  J}$ is an $N\times N$  matrix with all the diagonal elements equal to zero, and the off-diagonal elements given by
\begin{eqnarray}
{\mathfrak  J}_{m,l} = {\mathscr  J}_{m,l}\exp[i(\beta_m-\beta_l)z],
\end{eqnarray}
with  the coupling between the $m$th and the $l$th waveguide  given by
\begin{eqnarray}
 {\mathscr  J}_{m,l} = \frac{\omega \varepsilon_0}{4} \int \int dx dy\,
{\mathscr E}_x^{(m)*}(x,y)  \Delta n_l^2 (x,y) {\mathscr E}_x^{(l)}(x,y),\nonumber\\
\label{eq:coupling}
\end{eqnarray}
where $\Delta n_l^2\equiv n^2(x,y) - n_l^2(x,y)$, with $n_l(x,y)$ the refractive-index  profile for the $l$th waveguide alone.

Before we focus on the behaviour of the couplings for the particular setup under consideration,  it is worth mentioning that in addition to the coupling between adjacent waveguides, in the framework of coupled-mode theory one also obtains corrections to the propagation wavevector $\beta_l$ of the $l$th waveguide, due to the presence of the adjacent waveguides, as well as the so-called
``butt-coupling" coefficients. Such terms are typically much smaller than ${\mathscr  J}_{m,l}$ and thus their effects are neglected here \cite{book1}.

\subsection{Coupling constants}
Equation (\ref{eq:coupling}), shows that the coupling between two waveguides originates from the overlap between the corresponding
eigenmodes. Hence, one expects the asymmetry of the eigenmodes  to be also reflected in the couplings. Consider
the directional coupler of Fig. \ref{fig2}. The corresponding coupling between the two waveguides is plotted  in Fig. \ref{fig3}(a) as a function  of the separation $r$
(measured for the centers of the waveguides), at a fixed angle $\theta$. For any value of $\theta$ the  dependence of the coupling on $r$ is well approximated by an exponential of the form
\begin{eqnarray}
 {\mathscr  J}_{m,l} (r,\theta) = \mu(\theta)\exp[-\xi(\theta) r],
 \label{exp_law}
\end{eqnarray}
and it is anisotropic since the details of the exponential drop with increasing $r$ depend on $\theta$. Indeed, as shown in Fig. \ref{fig3}(b), for fixed $r$ the coupling varies by a factor of 3, as we change the angle $\theta$ from $0$ to $\pi/2$. The relative position of the curves for different $\theta$ in Fig. \ref{fig3}(a), depends strongly on the specific parameters under consideration (e.g., wavelength of light, refractive-index modulation, height and width of the rectangular shape).
In the case of symmetric waveguides, for example, the spread of the curves for various $\theta$ is considerably smaller \cite{remark3}. Varying the refractive-index profile of the waveguides, one can change the precise form of the modal distribution [see Fig.  \ref{fig2}], and thus the values of the parameters $\mu,\xi$ in Eq. (\ref{exp_law}) for a given $\theta$.

In any case, the crucial point is that the present model captures the dependence of the coupling  on both $r$ and $\theta$ (see Eq. \ref{exp_law}), which is also what one has in practise. The precise mathematical form of this dependence on $\theta$ is not crucial for what follows and actually, it is never used explicitly. The key point is that such a type of anisotropy allows for the engineering of couplings at different coupling angles $\theta$, by adjusting the distance $r$ (e.g., see \cite{AsymWG}).  For the implementation of a centrosymmetric  coupling configuration [such as the one in Eq. (\ref{coupling_j})] in a bent chain with $\theta > 0$, the separations $r_{j,j+1}$ between waveguides with indices below the index of the corner site $C$, have to be different from the separations for  waveguides with indices above $C$. In other words, a centrosymmetric coupling configuration does not imply centrosymmetric distribution of the separations $r_{j, j+1}$  in the case of an anisotropic spatial dependence of the coupling.

\begin{figure}
\centerline{\includegraphics[scale=0.37]{Coupling_vs_distance_angle-eps-converted-to.pdf}}
\caption{(Color online)
Geometry of the coupling between two rectangular waveguides (see Fig. \ref{fig2}).
(a) The coupling as a function of the separation $r$ at different angles $\theta$.
(b) The coupling as a function of the angle $\theta$ at different separations $r$.
Other parameters as in Fig. \ref{fig2}.
}
\label{fig3}
\end{figure}

\begin{figure*}
\centering
\includegraphics[scale=0.8]{AllHamiltonianPlots_ColorBar-eps-converted-to.pdf}
\caption{(Color online) The relative strength of the couplings between different identical waveguides ${\mathscr J}_{i,j} [\max{\{{\mathscr J}_{i,j}\}}]^{-1}$,  for a bent chain of $N=9$ waveguides with $r_{j,j\pm 1}$ such that ${\mathscr J}_{j,j\pm 1}$is given by Eq. (\ref{coupling_j}),
and bend angles: (a) $\theta = 0$, (b) $\theta = 16 \pi/32$,
(c) $\theta = 19 \pi/32$, (d) $\theta = 20 \pi/32$. Other parameters as in Fig. \ref{fig2}. }
\label{fig4}
\end{figure*}

\begin{figure*}
\centering
\includegraphics[scale=1,clip]{AllOutputNoOpt.jpg}
\caption{(Color online) Intensity distribution at the output ($L=10$ cm) of a bent chain with $N=9$ identical
rectangular waveguides (also shown on each plot).  Bend angles: (a) $\theta = 16 \pi/32$,  (b) $\theta = 18 \pi/32$,
(c) $\theta = 19 \pi/32$, (d) $\theta = 20 \pi/32$. The intensity is measured in units Wcm$^{-2}$ and the total power
in the sample at any $z$ is normalized to the input power.  Other parameters as in Fig. \ref{fig2}. Note the
different scale in the $y$ axis.}
\label{fig5}
\end{figure*}

\section{Light transport through bent photonic lattices}
\label{sec3}

Throughout our simulations we worked on a three-dimensional grid in a sufficiently large box so that reflections from the boundaries are absent. The total electric field at a given point  $(x,y,z)$
was estimated according to Eq. (\ref{totalEF}), through the  solution of Eqs. (\ref{Hy_eq}), (\ref{Ex_eq})  and (\ref{eom}).  Working with different parameters,  we have reached similar conclusions and for the sake of concreteness, in this section we present results  pertaining to a bent PL consisting of $N=9$ nearly identical waveguides of length $L=10$ cm.
The waveguides are written in the bulk of a glass with $n_s=1.444$ and the associated  refractive index change is $\delta n = 10^{-3}$, whereas their cross-section is $\sigma=6\times 2\mu{\rm m}^{2}$.  For a given $\theta\geq 0$, the distances between successive waveguides $r_{j,j\pm 1}$ are engineered  so that  the nearest-neighbour couplings ${\mathscr J}_{j,j\pm 1}$ are well approximated by Eq.  (\ref{coupling_j}) for $L=10$ cm and $N=9$.  Light of wavelength $\lambda = 800$ nm, and sufficiently low power so that nonlinear effects are negligible,  is injected in the first waveguide at $z=0$, and couples to the other waveguides as it propagates along $z$.

\subsection{Coupling constants}

From a theoretical point of view, the propagation of light in the PL is determined by Eqs. (\ref{eom}), and in particular by the coupling matrix ${\mathfrak J}$.  When the distance between non-neighbouring waveguides is sufficiently large, couplings beyond nearest neighbours  are negligible and thus the coupling matrix ${\mathfrak J}$  has basically tridiagonal form. Recent experiments on the realization of the coupling configuration (\ref{coupling_j}) in unbent PLs  $(\theta = 0)$ have shown that the assumptions underlying the coupled-mode theory, as well as the assumption of nearest-neighbour couplings of the form (\ref{coupling_j}) can be fulfilled experimentally for a moderate number of waveguides, and thus faithful transfer between the two ends of the unbent chain has been observed \cite{PSTrealization}. For an unbent chain of a given length, these assumptions are expected to break down only for a large number of waveguides  (e.g., see related discussion in the work of Bellec {\em et al}. \cite{PSTrealization}).

By contrast, the realization of the coupling configuration (\ref{coupling_j}) for bent chains  with $\theta > 0$ has not been investigated in the literature so far. In particular, given that
non-neighbouring waveguides around the corner $(C)$ come closer as one increases $\theta$ (see Fig. \ref{fig1}),
couplings beyond nearest neighbours are expected to increase. There should
exist, therefore, a critical angle $\theta_c$ above which the couplings beyond nearest neighbours become comparable to the nearest-neighbour ones, and thus their effects cannot be neglected.  Our first task here is to estimate the critical angle for the
particular set-up under consideration.  Subsequently, for  angles $\theta>\theta_c$ our task is
to investigate whether it is possible to improve the transfer of the signal between the two outermost waveguides of the PL, without additional extensive engineering \cite{remark4}.

For the reasons explained above, in our formalism the matrix ${\mathfrak J}$ includes the couplings for all possible pairs of waveguides. The relative strengths of the couplings in the matrix ${\mathfrak J}$  (with $\beta_m=\beta_l\,\forall m,l$) are plotted  in Fig. \ref{fig4} for a chain of $N=9$ waveguides, and  for increasing bend angles, with the corner site $C=5$. For the unbent chain $(\theta = 0)$ as well as for $\theta < \pi/2$, the couplings beyond nearest neighbours are at
least one order of magnitude smaller than the nearest-neighbour couplings, and thus they can be safely ignored. For  $\theta = \pi/2$ the strength of the couplings between the waveguides $C- 1$ and $C+1$  has been doubled, whereas couplings
between higher-order neighbours emerge. For $\theta = 19\pi/32$ and $\theta = 20\pi/32$  the couplings between the waveguides $C- 1$ and $C+1$ are comparable to the nearest-neighbour couplings, and the couplings between higher-order neighbours also increase.  These observations suggest that strong deviations from the coupling configuration of the unbent chain are expected for bend angles above $\theta_c=90^\circ$, whereas the deviations for  angles up to $90^\circ$ are not expected to be so pronounced \cite{remark5}.

In the case of symmetric waveguides (i.e., for $\Delta x = \Delta y = 6\mu$m) our simulations show that the couplings beyond nearest neighbours are less sensitive to bends. For example, for $\theta = 20 \pi/32$ we find that the coupling ${\mathscr J}_{C- 1, C+ 1}$  is at least five times smaller than ${\mathscr J}_{C\pm 1,C}$, whereas couplings between higher-order neighbours can be safely ignored.
This is because the confinement of the lowest eigenmodes in all directions turns out to be  stronger than in the case of the asymmetric waveguides with $\Delta x = 6 \mu$m and $\Delta y = 2\mu$m.

\subsection{Output intensity distributions and losses}

As mentioned before,  one expects ideally complete transfer of the light between the two outermost waveguides, when the matrix elements ${\mathfrak J}_{m,l}$ are well approximated by
Eq. (\ref{coupling_j}). Our simulations show that this happens for $\theta\leq \pi/2$ and in Fig. \ref{fig5}(a) we show only the intensity distribution at the output  for the case of $\theta=\pi/2$. As we increase $\theta$ further, couplings beyond nearest neighbours distort the transfer between the two outermost waveguides. For the sake of illustration, in Figs. \ref{fig5}(b-d) we present the intensity distributions at the output of a bent chain for  $\theta = 18\pi/32,\,19\pi/32,$ and $20\pi/32$,  respectively.   One can see that the output intensity is not restricted only to the 9th waveguide, but there are also  non-negligible  fractions in other waveguides, including  the 1st and the 7th one. A clearer quantitative picture can be
obtained by looking at  how the input power is distributed among the waveguides at the output. As shown in Fig. \ref{fig6}(a),
for $\theta = 19\pi/32,$ and $20\pi/32$ we find that 85\% and 60\% of the input power respectively, has been  transferred to the target
waveguide at the output, whereas a significant fraction of the  input light  can be found at the exit of all the other waveguides, but mainly of the first one.  Hence, as shown in Fig. \ref{fig6}(c) (see open circles), the relative losses are about
15\% and 40\%, respectively.

The present scheme that relies on engineered couplings in linear PLs seems to be a bit more robust against bending losses, than schemes that rely on solitonic signals and nonlinear PLs. For instance, the authors of Ref. \cite{Chr01} have estimated that for $\varphi = \pi-\theta = 90^\circ$  solitons suffer about
$5\%$ bending losses, whereas for $\varphi = 70^\circ$ the losses exceed  $38\%$.
As shown in Fig. \ref{fig6}(c), in the present scheme bending losses do not exceed $5\%$ for angles $\theta\lesssim 100^\circ$, whereas for  $\theta \approx 113^\circ$ (corresponding to $\varphi \approx 67^\circ$) bending losses are about $40\%$. For the reasons discussed above, the losses for the same set-up can be reduced considerably by using symmetric waveguides. In this case we find that losses do not exceed 10\% for $\theta \approx 113^\circ$, which suggests that linear PLs with symmetric waveguides considerably outperform nonlinear PLs with solitonic informations carriers.

We turn now to discuss a method for minimizing the bending losses, by introducing a
defect at the corner site.

\begin{figure}
\centering
\includegraphics[width=8.5cm,clip]{AllLosses-eps-converted-to.pdf}
\caption{(Color online) Losses in the transfer of signals  along a bent chain of $N=9$ identical rectangular waveguides.
(a) Fraction of the input power that has been transferred to the $j$th waveguide at the output of the sample
(i.e., at $L=10$ cm ),  for different bend angles. (b) As in (a) with optimized corner site.  (c) The relative losses at the exit of the sample,  and for different bend angles with and without optimized corner. Other parameters as in
Fig. \ref{fig5}.}
\label{fig6}
\end{figure}

\begin{figure}
\centering
\includegraphics[width=8.5cm,clip]{AllOutputOpt.jpg}
\caption{(Color online) As in Fig. \ref{fig5} after optimization of the corner site $C=5$, for (a) $\theta = 19 \pi/32$ and (b) $\theta = 20 \pi/32$.  }
\label{fig7}
\end{figure}

\subsection{Minimization of bending losses}

In the framework of solitonic signals it has been shown that bending losses can be reduced by  introducing a defect at the corner site of the bend \cite{Chr01,ExpBend}. More recently, it was shown that the same method also works in the framework of linear Hamiltonians with engineered couplings \cite{GenericBends}. The generic model of Ref. \cite{GenericBends}, however,  did not allow for an in depth investigation of various quantitative aspects of the defect. The present model allows us to gain further insight into the method, and shed light on crucial questions pertaining to the size and the  refractive-index change of the defect.

We assume that all of the waveguides, but the corner one, are identical, and let us denote by $\beta$ the corresponding propagation wavenumber.  The wavenumber for the corner site will be denoted by $\beta_C$ and let  $\Delta\equiv \beta_C-\beta$ be the detuning of the corner site relative to the other waveguides of the chain.
In our simulations, for a given angle this detuning has been optimised, while keeping all other parameters of the PL fixed, so that the transfer from the first to the last waveguide is maximized (i.e. losses are minimized).

Figures \ref{fig7}(a,b) show the intensity distribution at the output of a bent PL for two different angles after optimization
of the wavenumber for the corner waveguide. Comparing these two figures to the corresponding figures
without optimization [see Fig. \ref{fig5}(c,d)], one sees a clear improvement of the transfer from the first
to the last waveguide. Still, there are fractions of the input signal that are not found at the exit of  the target waveguide, but certainly they are considerably smaller than in  Figs. \ref{fig5}(c,d). Indeed, as shown in Fig. \ref{fig6}(b), for bend angles up to $20\pi/32$, more than about 90\%  of the input power has been transferred to the target waveguide at the output, whereas the relative losses are at least twice smaller than without optimization and they hardly exceed 10\% [see filled squares in Fig. \ref{fig6}(c)]. We see therefore that by introducing a corner defect one can minimize bending losses for fixed $\theta$ in the present linear array with engineered couplings, but the same approach seems to work more efficiently for nonlinear arrays and solitons. According to Ref. \cite{Chr01} bending losses
after the inclusion of defect are restricted to less than approximately $1\%$ for
angles $\theta = 90^\circ$ and $110^\circ$, whereas in our case we find losses approximately 1.4\% for $\theta = 90^\circ$  and $ 5\%$ for $\theta = 110^\circ$.  However, when symmetric waveguides are considered in our scheme, the corresponding bending losses  after optimization do not exceed 1\% for angles up to $\theta = 113^\circ$; a performance that is comparable to (if not better than) the performance of the bent nonlinear PLs considered in Ref. \cite{Chr01}.

The quantitative aspects of the defect are intimately connected to the details of the set-up
under consideration. The optimal values of the detunings that minimize bending losses at various $\theta$ in our model  are given in table \ref{tab1}. Clearly, in all cases $\Delta$ is negative and increases (in absolute value) as we increase the bend angle.  Typically, the wavenumber of a waveguide can be controlled by changing the size of the waveguide,
or by adjusting the associated refractive-index modification. An estimation of the changes required to
achieve some of the detunings discussed here are shown in the last columns of table \ref{tab1}. A close inspection of these values shows that in order to achieve the estimated
optimal detunings the accuracy required in the writing of the waveguides is at least $10^{-5}$
in refractive-index changes and at least $10^{-2}$ cm$^{2}$ in the cross-section of the waveguides. Interestingly enough, the present estimations for the required refractive-index changes are comparable to related estimations for solitonic schemes \cite{Chr01,ExpBend}.

\begin{table}
\caption{\label{tab1} Optimal detunings of the corner site that minimize bending losses at different angles in a linear chain with $N=9$ asymmetric waveguides of length $L=10$ cm. The
detuning is defined as $\Delta = \beta_C -\beta$ where for the parameters under consideration, $\beta\simeq 11.3392\times10^{4}$ cm$^{-1}$.  The strength of the detuning
relative to the coupling  $G_{C,C\pm1}$ (see Eq. (\ref{coupling_j})) is given in the third column. The fourth and the fifth column give the control required on the refractive-index modulation and the size of the corner site relative to $\delta n =10^{-3}$ and $\sigma = 12\mu{\rm m}^2$ respectively, in order to achieve the optimal detunings of the second column.}
\footnotesize\rm
\begin{tabular*}{\textwidth}{@{}l*{15} {@{\extracolsep{0pt plus12pt} } c} }
\br
Angle $(\times \pi/32)$ & $\Delta$ (cm$^{-1}$)  &  $|\Delta|/G_{C,C\pm1}$  & Relative change of refractive index$^1$ & Relative change of size$^2$\\
\mr
18 &  -0.1955 &  0.275  & 0.46 & 0.25\\
19 &  -0.4317  &  0.608  & 1.60 & 0.85\\
20 & -1.0733  &  1.512  & 5.15 & 2.50\\
\br
\end{tabular*}
$^1$Defined as $100\times (\delta n-\delta n_C)/\delta n$.\\
$^2$Defined as  $100\times (\sigma-\sigma_C)/\sigma $.
\end{table}

Before closing this section it is worth discussing briefly the reason for the success of the corner defect  in minimizing bending losses.
In the case of solitonic signals and nonlinear PLs it has been conjectured that the detuning
of the corner site relative to the rest of the lattice virtually  removes the corner
site from the lattice \cite{ExpBend}.
Thus, the remaining (identical) waveguides effectively constitute a
smoother link, which is reflected in the improvement of the transport. This explanation does not apply to our setup as we work in the linear regime and the coupling configuration under consideration [see Eq. (\ref{coupling_j})] is rather sensitive to the details of the lattice (i.e., number of waveguides, length, etc). First of all, as shown in table \ref{tab1}, the optimal detunings
are smaller or at most comparable to the couplings of the corner site to its neighbours.
Furthermore, as shown in Fig. \ref{fig8} the corner site may acquire at least 10\% of the input power as the light propagates from $z=0$ to $L$, which is not a  negligible amount. These two observations together suggest that there
is no solid ground for omission of the corner site relative to the others, and thus the derivation of an effective chain cannot be justified.

The coupling configuration under consideration is a member of a large class of state-transfer
Hamiltonians, whose operation relies on the commensurate eigenenergies \cite{ReviewsPST, EdVolume}. As explained in Ref. \cite{GenericBends}, for such a type of Hamiltonians a corner defect  minimizes the bending losses by  rearranging the spectrum that has been distorted by the bend. To confirm this once more, in Fig. \ref{fig9} we plot  the separation between successive eigenvalues of the matrix ${\mathfrak J}$ in Eq. (\ref{eom}), for different angles before and after optimization of the corner site. For the unbent chain ($\theta = 0$)  the eigenvalues are commensurate (i.e., equidistant), and that is why the coupling configuration of Eq. (\ref{coupling_j}), ensures ideally perfect transfer of light between the two outermost waveguides. As we bend the lattice, however, the commensurate nature of the eigenvalues is distorted, and the distortion is getting larger for sharper bends [see Fig. \ref{fig9}(a)].  As depicted in Fig. \ref{fig9}(b), the inclusion of a defect at the corner site of the bend tends to restore the relative position of the eigenvalues (i.e., the deviations from the case of the unbent chain are getting smaller). The remaining deviations at the borders are not of great importance since the contribution of eigenvectors with small/large indices to the evolution of the system is negligible \cite{GenericBends}.

\begin{figure}
\centering
\includegraphics[scale=0.32]{CornerPopulation-eps-converted-to.pdf}
\caption{(Color online) Fraction of the input power that is found at the corner waveguide
at different lengths. The detuning of the corner waveguide has been optimized to minimize bending losses. Other parameters as in Fig. \ref{fig5}.
}
\label{fig8}
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.9]{EigenvaluePlots_v3-eps-converted-to.pdf}
\caption{(Color online) Spectrum of matrix ${\mathfrak J}$ for various bend angles.
(a) Difference between successive eigenvalues before optimization of the corner site.  (b) As in (a),  after optimization of the corner site. Other parameters as in Fig. \ref{fig2} and table \ref{tab1}.
}
\label{fig9}
\end{figure}

\section{Concluding remarks}
\label{sec4}

We have analyzed the effects of bends on the transport of photonic signals between
the two outermost waveguides of a  2D photonic lattice with engineered nearest-neighbour
couplings that operates in the linear regime. In contrast to previous studies, in the present scheme the suppression of dispersion effects and the faithful transport of light  does not rely on Kerr nonlinearities, but rather on  the  engineering  of
judicious couplings between nearest neighbours. It has been shown that our scheme works
reliably for bends at least  up to $\theta_c = 90^\circ$ (with respect to the unbent chain). Sharper bends (with $\theta>\theta_c$)  have been shown to distort the transport, with the distortion being more pronounced for asymmetric waveguides. In this case, one has to find ways for minimizing bending effects, and in this direction it has been shown that the inclusion of a defect at the corner site can be a rather useful approach. Although our findings suggest that the present scheme outperforms its nonlinear (soliton-based) counterparts, further analysis is required for definite conclusions in this respect.

Laser-written buried waveguides in glasses typically have elliptic shape, due to the beam focus, and they exhibit ``form
birefringence", as a result of which fields with different polarizations experience different effective refractive indices \cite{FSTrealization}.
Moreover, due to the formation of self-aligned nanogratings in the material during the irradiation process, one may also have ``material birefringence" \cite{San-etalPRl10,nano-grating}.

Birefringence is a detrimental effect for quantum circuits that are intended for efficient guide and manipulation of qubits that are encoded in the polarization of photons.
In general, the shape of the waveguides can be controlled efficiently by shaping the writing beam using standard
techniques \cite{SymWGwriting}, and thus ``form birefringence" can be, in principle, eliminated. The elimination of ``material birefringence"
is  also  feasible if one chooses the right material/substrate, and the right combination of writing  parameters  (i.e., wavelength, duration and energy of the laser pulses, repetition rate,
objective numerical aperture, translation speed, etc). In this way, the overall birefringence can be reduced
by at least an order of magnitude facilitating thus the design of photonic primitives (e.g.,  directional couplers),
that operate efficiently for polarization-encoded qubits \cite{San-etalPRl10}.
One has to keep in mind, however, that  ``material birefringence" can be useful in the design of crucial polarization-sensitive components of quantum circuits, such as integrated wave plates \cite{Waveplates}, polarization routers \cite{nano-grating}, etc.
In this context, for instance, one can have waveguides that allow for transmission of light with specific polarization, whereas light with the  orthogonal polarization is totally reflected.

For the sake of simplicity, the present analysis of bending losses  has been restricted to one of the lowest polarization modes (the calculations and conclusions for the other polarization are the same given that birefringence is not included in our formalism). Strictly speaking, the present results are valid for optical networks and communication schemes in  which qubits are not encoded in the polarization of the signal. In the case of polarization-based qubits,  the present
results and conclusions are expected to be valid only for photonic lattices that are polarization-independent i.e., non-birefringent.  As mentioned above, currently available technologies allow for fabrication of such a type of lattices \cite{San-etalPRl10}.

In closing it has  to be emphasized that the precise value of the critical angle $\theta_c$  beyond which bending effects cannot be ignored, depends on the details of particular set-up under consideration. The present analysis can be performed for any coupling configuration, and  it is pertinent to on-going  experiments on photonic lattices, in the framework of which
a point-to-point link that relies on the coupling configuration discussed here has been
demonstrated \cite{PSTrealization}. When combined with the ideas of \cite{3DPLs}, the present findings suggest that linear waveguide arrays with  engineered nearest-neighbour couplings can be used as building blocks of fundamental optical primitives that perform more demanding communication tasks such as routing, splitting, blocking, as well as logical functions.

\section{Acknowledgements}
The author acknowledges with pleasure discussions at various times with M. Bellec on experimental  issues pertaining to laser-written photonic lattices. He is also
grateful to P. Lambropoulos, T. Brougham and M. Bellec, for helpful comments on this paper.

redpajama_set_name: RedPajamaArXiv


The other weekend, I had the privilege of attending the National Writers' Conference, organized by Writing West Midlands. It was held at Birmingham University. I'd never actually been to that university campus before and we were hosted in the beautiful and well equipped Bramall Music Building.
I felt a sense of excitement as I walked along the driveway in the pleasant morning sun, with a couple of fellow attendees. And that excitement was not disappointed because throughout the day, we were treated to an inspiring lineup of speakers and presentations that fuelled my happiness to be alive.
While we waited for the first session to convene, I stopped at the bookstand in the upstairs foyer and one of the first books that leaped off the table at me was, 'The Blank Screen – Blogging', by William Gallagher. I've known for a while that I need to revamp, focus, do something that really works with my blog. So I quite readily turned over the £7, swapping it for the said book. And just as I was about to tuck it under my arm, I caught a glimpse of the name and realized that it was by William Gallagher, who I had recently discovered that I'd be working with in the next series of Spark Young Writers' workshops.
'Oh!' I exclaimed to the stall manager and explained the connection. I soon discovered that William was standing across the room not far from me, so I went over and introduced myself.
The only thing is the book was not actually tucked under my arm as I'd thought, or in my bag where perhaps I should have put it but it was clutched to my chest, the front cover exposed. So as I spoke to William—neither of us knew each other previously—there was his book, like a shameless act of name dropping: I'll be working with you as your assistant writer and see I bought your book!
For his part, William didn't hide the fact that he was pleased that I had bought his book and so, of course, since we were on the subject of his book, I made him sign it for me. Well, why not?
I'm not going to go into the ins and outs of everything that the day's sessions covered. Suffice to say, it was a fruitful day. Most inspirational was talking with new people and hearing what things they were writing. What I do want to say though, is that during one presentation I was surprised to realize just how little authors often actually earn from their books when they have traditional publishing contracts. Ever hear about those advances that they pay you? Well, they're not actually all they're cracked up to be. But that's another story.
The day was topped off with an awards ceremony for the Spark Young Writers group. Some of them shared selections from their work with the audience. I was blown away with the excellence and maturity of their work. I ended up feeling somewhat unqualified, yet challenged and excited, to be included to assist on the upcoming schedule of workshops with William. Under his expert guidance, though, I'm sure it's going to be great.
But back to the blog: I need to do something about my blog—something to make it better. I've started reading 'The Blank Screen' already. In it, William says he knows we bought it because of the bit where he says you can make money from your blog. Gosh, is he psychic? But now he's saying you shouldn't focus on writing your blog to make money. He says he's going to tell us how but that that's not what it's about. So I'm reading it, every single word, every single page, and I hope I end up with a better blog out of it; I hope I become an even better writer; and by jove, I hope I do earn some money from it…am I a bad person?