HISTORY OF PHOTONICS RELEVANT TO CMOS INTEGRATED CIRCUITS In 1909, Arnold Sommerfeld published his proposed analytical proof of surface polarization waves [3] marking in our history of Photonics the cornerstone of the all nanophotonics is motivated. Sixty years following Sommerfeld’s publication, Chinese physicist Charles Kao published a solution for guiding Sommerfeld’s surface excitations using optical fiber [4] which in 2009 he would also receive a Nobel Prize. Today nanophotonic research is being conducted by many countries for many applications, yet their approach is surprising similar. The majority of resources and funding for nanophotonics is the development of better materials. This point will be further evident in following sections, but for now it should be mentioned that of those resources only a marginal portion is allocated in the direction of CMOS integration. Initially, this discovery was quiet shocking for two big reasons. First of all, in recent years Moore’s law’s famous exponential curve of computing performance and affordability over time has become less exponentially improving and we know one major cause of the bottleneck occurring in integrated circuits is interconnects. Illustrated in figure 1 is a comparison of the performance capability of optical fibers vs coaxial cables. Also in figure 1 is a relation of current nanophotonic waveguide capability compared to optical fiber which has strong implications for what is possible on chips and the potential need for an enhancing technology. Secondly, the CMOS business has been so profitable and so heavily investing in machinery that it seems logical to continue investing as a lot of the infrastructure exists. The answer to the initial shock is illustrated in figures 2. CMOS compatible nanophotonics occupies an extremely narrow space on a wide spectrum of possible use cases and therefore to expect so much of the resources to be allocated so narrowly this early in such a young immature science could greatly delay the achievable possibilities. The following sections, however, will discuss the results of the resources that were allocated for CMOS integrated nanophotonics and the modules that are in development to address Moore’s law.

INTRODUCTION The purpose of the lab was to find the normal modes of rectangular and cylindrical geometries and using that data of these frequencies obtain the speed of sound. Before the lab was started the first 33 modes for the rectangular box were found theoretically up to the (4, 0, 0) mode to compare to our results and the first 17 modes of the cylindrical up to the (4, 2, 0). The physics used to to find the resonant frequencies of the box come from seperation of variables of the pressure wave equation and the frequencies for the cylindrical geometry were found seperating variables again but in cylindrical coordinates to the pressure wave equation and examining the zeroes of the deriviatives of the bessel function. For practial purposes, c was taken to be $345 {s}$ to approximate resonant frequencies.

Writing with a purpose means that you need to make sure your writing communicates amessage. However, doing this effectively might mean taking a few extra steps as you write. Inthis article, we will look at seven ways you can improve your writing and communication skills.

SETUP INFORMATION The Silicon Photomultiplier (SiPM) is operated using a few Ortec Nuclear Instrument Modules in room B309. Specifically, The model 113 preamplifier, the 570 amplifier, and the 927 Aspec MCA that has a USB port in the back of it for data collection. There are 3 SMA ports that are available to use in the front of the box (Figure 1). Currently the third SMA port is not being used but can be in the future and is okay to be left alone. SMA port 1 is used to apply bias to the SiPM. This is to be negatively biased only. The power supply used is a BK Precision 9110. This was specifically used because it allows control of how much current the SiPM will get. In the event the cover of the box is removed, the exposed SiPM won’t be fried. The second and third SMA ports are used for signal out of the SiPM. Specifically, it is wired to read signal out of SMA port 2.

Resonant frequencies measured at (0.69,1.93,2.92,3.69,4.31,4.82) Hz. Q factors varied from the sweeps and from measurements of alpha largely but this can be due to the effect of high amplitudes on the higher frequencies and the natural error in trying to measure the time decaying exponential of a wave that has been turned off from the amplifier.

In 1909, Arnold Sommerfeld published his proposed analytical proof of surface polarization waves [11] marking in our history of Photonics the cornerstone of the all nanophotonics is motivated. Sixty years following Sommerfeld’s publication, Chinese physicist Charles Kao published a solution for guiding Sommerfeld’s surface excitations using optical fiber [12] which in 2009 he would also receive a Nobel Prize. Today nanophotonic research is being conducted by many countries for many applications, yet their approach is surprising similar. The majority of resources and funding for nanophotonics is the development of better materials. This point will be further evident in following sections, but for now it should be mentioned that of those resources only a marginal portion is allocated in the direction of CMOS integration. Initially, this discovery was quiet shocking for two big reasons. First of all, in recent years Moore’s law’s famous exponential curve of computing performance and affordability over time has become less exponentially improving and we know one major cause of the bottleneck occurring in integrated circuits is interconnects. Illustrated in figure 1 is a comparison of the performance capability of optical fibers vs coaxial cables. In figure 2 is a relation of current nanophotonic waveguide capability compared to optical fiber which has strong implications for what is possible on chips and the potential need for an enhancing technology. Secondly, the CMOS business has been so profitable and so heavily investing in machinery that it seems logical to continue investing as a lot of the infrastructure exists. The answer to my initial shock is illustrated in figures 3. CMOS compatible nanophotonics occupies an extremely narrow space on a wide spectrum of possible use cases and therefore to expect so much of the resources to be allocated so narrowly this early in such a young immature science could greatly delay the achievable possibilities. The following sections, however, will discuss the results of the resources that were allocated for CMOS integrated nanophotonics and the modules that are in development to address Moore’s law.

ABSTRACT Resonance Cones were observed in a plasma affected by a steady state magnetic field and their angular distribution was measured over a set of electric radio frequencies from the time varying oscillation of a short antenna. The angles were compared to the theoretical values under the cold electron approximation and found to match within a few degrees. Resonance cones at smaller angles were also observed, as predicted by theory for warm electron temperatures where $}{T_{e}}\ll1$[3].

Market SizeHealthy TissueDiseased TissueCurrent TreatmentsApproaches to Tissue EngineeringMy Approach

IXYS COLORADO LASER DIODE DRIVER MODULES IXYS sells a variety of laser diode driver modules. I have disregarded those that had potentiometer controls, large form factor, and closed design. All driver modules have mounting pads provided to mount laser diode directly to driver unless otherwise stated. PCO-7110 Series Laser Diode Driver Module The PCO-7110 series driver module is a compact pulsed laser diode driver designed to provide fast and high current pulses for applications needing nanosecond pulses and up to 50 A pulse output current. The PCO-7110 model comes in 6 different models, two models that may closely match with this project. Their specifications are listed in the excel sheet. The PCO-7110-40-4 laser diode driver module providing pulse currents to 40 A, fixed pulse widths of 4 ns, and at frequencies up to 50Khz. The PCO-7110-50-15 provides pulse currents to 50 A, fixed pulse widths of 15 ns, and frequencies up to 11 Khz. Both models allow for current monitoring by oscilloscope. Output currents for PCO-7110 models are adjusted by varying applied voltage (Vin). Output current depends on available charge of capacitor CPFN, shown in figure 2 in an equivalent circuit diagram.

The redshifts and emission-line strengths for XXXX Galaxies in 29 fields of the WFC3 Infrared Spectroscopic Parallels (WISP) Survey are presented.

INTRODUCTION Radio emission from the ionosphere can produce a whistling sound in the audio frequency that can be heard[1]. The whistling sounds are described as groups of descending tones which are called the whistler mode. When lightning hits the southern hemisphere it produces a range of radio waves, some of which can travel along the earths magnetic field lines from the southern hemisphere to the northern hemisphere[1].These waves are called extraordinary waves .The extraordinary waves emit two solutions to the wave equation named L and R waves.The L and R refer to left and right hand circularly polarized. The waves that describe the whistling sound are R waves and they will be detected in the north and the different frequencies of these waves will travel at different speeds. For $\omega<}{2}$ the phase and group velocities increase with frequency, where $={m}$ is the electron cyclotron frequency[1]. Due to this, the lower frequencies will arrive at the northern hemisphere later than the higher frequencies will, causing the descending tone in the whistler mode. These R waves waves that travel along the magnetic field lines are called whistler waves and these waves can only propagate for $\omega<{2}$. This lab seeks out to measure the dispersion relation of the whistler waves and to find the wave patterns theoretically and experimentally in the inductively coupled plasma device using Appletons equation.

Low-Cost EEG-based Seizure Monitoring Hat with Improved Patient ComplianceTiffany Kyu, Hilary Yen, Tiana Baghdikian, Julius Yee, Phillip Cox, Chang-Min HurAbstract—Traditional epileptic seizure monitoring is achieved in a hospital setting; however, it is limited by its cost, lack of mobility, and constant maintenance. Advances in biotechnology have enabled seizure monitoring in nonclinical environments, however, these devices are expensive and obtrusive. We present a low-cost EEG-based seizure monitoring hat that promotes long-term patient compliance through its comfortable and socially-acceptable design without compromising signal quality. This device utilizes Biopac’s dry electrode technology, and we validate the efficacy of its signal acquisition compared to the gold standard wet Ag/AgCl electrodes used in clinical settings. Sewn fabric integrated into a plastic frame achieves high comfort and subtlety. This device is effective in capturing and identifying seizure signals with a 98% accuracy utilizing a custom signal processing algorithm with FFT and the Neural Pattern Recognition toolbox in MATLAB. Our prototype overcomes many of the current clinical and market limitations and achieves long-term and continuous acquisition of seizure signals throughout a patient’s daily life.Index Terms—EEG, dry electrodeI. IntroductionEpilepsy is the fourth most common neurological disorder in the United States, and it is characterized by unpredictable, recurrent, and unprovoked seizures \cite{Alarc_n}. 65 million people around the world suffer from epilepsy where one-third of these people live with seizures that cannot be controlled by anti-epileptic drugs \cite{Baker_1997}. Uncontrollable seizures can even be fatal in cases of Sudden Unexpected Death in Epilepsy (SUDEP) [1, 2, 4, 5-7]. There is no known cause of SUDEP, but preliminary studies have shown that seizures may be a precipitating factor. Early SUDEP research has identified that improving active monitoring of seizures, especially at night, can enable a rapid human response (i.e. shoulder tapping) that could potentially prevent SUDEP [1, 5, 6]. This work aims to develop an at-home wearable EEG (electroencephalogram) monitoring device.Standard hospital EEG monitoring requires constant replenishment of conductive gels, which can cause skin discomfort [2, 3, 4]. In addition, hospital monitoring is costly and requires long-term inpatient care. Current available EEG epilepsy monitoring devices fail to promote active seizure monitoring throughout patients’ daily lives due to their obtrusive design. This research overcomes this limitation and provides epileptic patients with a dry electrode-based EEG monitoring device that is comfortable, low-cost, and socially acceptable. Our device provides near real-time and continuous seizure monitoring and hopes to improve the scientific community’s understanding of the relationship between seizures and SUDEP.In this study, an EEG hat is proposed for long-term measurement and improved patient compliance. First, we evaluated various commercially available dry electrodes focusing on the electrode-skin impedance, attenuation, noise, movement artifacts, and accuracy. Dry electrodes are chosen because they reduce impedance presented by hair through protrusions that penetrate to the scalp [10-12]. In addition, the use of dry electrodes allows us to measure biopotentials without preparation of the skin and replenishment of conductive gel. We use a circuit tested by Sullivan et al to increase the signal by a gain of 1000, and filter out signals with a bandpass filter with a cutoff between 1Hz and 100Hz [9]. Acquired data is Fourier transformed, filtered to extract features, and is then identified as seizures by the Neural Pattern Recognition toolbox in MATLAB®. Hardware, data acquisition algorithms, and electrodes are combined into an adjustable plastic frame to promote patient compliance through a socially acceptable and comfortable design.II. Materials and MethodsA. Evaluation of Dry ElectrodesTwo types of electrodes were studied in this project: the Biopac® 10mm Ag/AgCl post dry electrode and the Cognionics® Flexible Dry EEG sensor as shown in Fig. 1. Dry electrodes are designed to operate without electrolyte or conductive gel [12]. Electrode characteristic data was compared to hospital-grade wet electrodes. With each electrode and test, data was collected using a human skin and tissue substitute, represented by a sausage. In these experiments, we fed signals of varying frequencies (30, 10, 5 Hz) and amplitudes (150, 80, 20mV) to simulate a seizure. The dry electrodes and Ag/AgCl wet electrode disc were consistently placed 1 cm from the edge of the system, and they were fixed to the model using Smith and Nephew Hypafix®. A current was applied, using the 33500B series waveform generatorFig.1: The Biopac® 10mm Ag/AgCl (upper right) and Cognionics® Flexible Dry EEG sensor (upper left) were compared to a standard wet electrode (bottom).(Keysight Technologies, Westlake Village, CA), to the system from a constant height and distance from each electrode to ensure reliable and reproducible results. The electrode output was directly read by the InfiniiVision 3000A X-series oscilloscope (Keysight Technologies, Westlake Village, CA). Using this method, we can measure attenuation, noise, and movement artifact [9, 11]. Experiments were repeated with this phantom utilizing an artificial wig, mimicking hair, in order to study the effects of its interference.B. Development and Integration of HardwareThe EEG circuit was developed according to the schematic used by Sullivan et al as shown in Fig. 2 [9]. The circuit consists of two operational amplifiers that result in a net gain of 1000. Signals are passed through a bandpass filter with cutoff frequencies of 1Hz and 100Hz.As the circuit was designed for use with dry electrodes the circuit uses the INA116, an instrumental amplifier with guard pins to protect high impedance inputs [9]. There is also a reset circuit to protect against offset currents. Both measures are put in place to alleviate problems which arise from high impedances inherent to dry electrodes.The INA116 instrumental amplifier (Texas Instruments, Dallas, TX) and LT6010 operational amplifier (Linear Technology, Milpitas, CA) are used. The INA116 was chosen for its low input bias current of 3fA, low input current noise of 0.1fA/(Hz)½, and integrated guard pins. The LT6010 was chosen for its low noise (14nV/Hz1/2). Both components draw minimal power, as the circuit only uses 10.5mW. Signals are transmitted remotely to a computer via a Wi-Fi component attached to the circuit.The circuit was tested by inputting known signals, using a 33500B series waveform generator (Keysight Technologies, Westlake Village, CA), with frequencies between 10-1 Hz to 104 Hz and observed using a InfiniiVision 3000A X-series oscilloscope (Keysight Technologies, Westlake Village, CA) to ensure signals are properly filtered and to observe mid-band gain.C. Signal Processing AlgorithmAfter the signal is amplified and filtered by the circuit, we must differentiate between seizure and non-seizure signals. The steps we took are illustrated in Fig. 3. A low pass filter is used on input EEG signal files. Fast Fourier Transform (FFT) is then applied to decompose the signal into the amplitude and frequency domain [13, 14]. The decomposed signal is separated into five feature categories: δ, 𝜃, α, 𝛽, γ, which have bandwidths of 0.5 - 3 Hz, 3 - 8 Hz, 8 - 12 Hz, 12 – 38 Hz, and 38 – 42 Hz respectively, as shown in Fig. 4. These five features are used to obtain a power spectrum at a certain period of time. The neural pattern recognition toolbox in MATLAB® is then used to distinguish between the seizure and non-seizure signals [18]. Using Bonn University’s EEG Database, the neural network is trained to detect potential seizure waves. The EEG database consists of 100 files per patient, each of which contains 4096 data points. EEG data from two patients: one with seizures and one without seizures is used. The mean of the absolute value of all the points in aFig.2: EEG amplifying circuit. The two operational amplifiers create a gain of 1000. The reset circuit protect against offset current. The shield is driven by the guard pins, and minimizes the noise the electrode picks up. [9]Fig.3: Signal Processing Flowchart. The original EEG signal is subjected to various steps in order to identify seizures.Fig. 4: The raw EEG signal of a patient is filtered via FFT and separated into the five feature categories δ, 𝜃, α, 𝛽, and γ.frequency band wave was calculated. The neural pattern recognition toolbox uses the first 75 files for both patient’s data to train and define seizure and non-seizure pattern classes. Scalar target values are set to either 1 or -1, which indicates whether an input is classified as seizure or non-seizure. Once the neural network is trained, the remaining 25 files for both patients are tested for seizure detection accuracy. With the algorithm trained, it can now detect whether a patient is having a seizure or not.D. EEG Cap AssemblyThe EEG-based seizure-monitoring hat is comfortable and inconspicuous in its design. The hemisphere-shaped hat frame is made of plastic with an adjustable strap to accommodate individuals with heads of various sizes (Fig. 6). Twelve dry electrodes are securely attached to the angled supports, with two electrodes on each support and an arc length of three inches from each other. These 12 electrodes are placed in specific reference positions F3, AFZ, F4, FC1, FZ, FC2, CP1, CP2, P3, PZ, P4, and POZ as depicted in Fig. X, which is labeled according to the 10-20 International System of Electrode Placement. These electrode positions are sufficient for accurate EEG readings as shown from literature [11]. The reference and ground electrodes are placed at the back of the ear labeled A1 and A2 in Fig. 5.Electrode leads and wires are securely attached to the angled supports to reduce wire movement. 5 mm diameter holes in the plastic frame permit electrode leads and wires to be threaded through and aligned on the outside of the head frame. These wires terminate with the connection to the flexible circuit attached at the front of the frame as labeled (Fig. 6).50% cotton and 50% polyester fabric is sewn to surround the inside and outside of the plastic frame, improving user comfort and maintaining an inconspicuous design. On the inside of the frame, the fabric is sewn around the electrodes to prevent diminished signal acquisition. Epileptic patients can place the hat on their head (Fig. 6) and easily adjust the strap to ensure a tight fit while still permitting movement. Patients can wear this EEG-based seizure monitoring hat in their daily environment.III. Results and DiscussionA. Highest Performing Dry ElectrodeEach electrode was characterized for their signal quality through attenuation, noise, and movement artifact to determine which is most comparable to the current standard of wet electrodes. Attenuation was calculated as already reported by Chen et al [13]. The observed negative attenuations reflect an increase in signal as it travels through the electrode. Additionally, lower amplitudes have a smaller attenuation, which can be explained by an increase in noise artifact at these low values. Fig. 7 shows the averaged values and standard deviations of attenuation measurement results for each of the three aforementioned electrodes. Results showed that the attenuation of the Biopac® electrode is similar to that of the conventional wet electrode for different amplitudes and frequencies that mimic a seizure, and this electrode’s result isFig. 5 Montage of dry electrode placement on patient scalp labeled according to the International System of Electrode Placement.

Introduction and Background

a) Overlap integral given by \[ \eta=^{\infty} } ^{output} ^{input} dx\] The paper by tong et al, explains that they adjusted the overlap until the output is maximized. I think that for that occur then say the input was a gaussian E = E(0)eax² where E(0) would be a central maximum, then overlap adjusted until the output was E(0), making the overlap integral just integrating a gaussian. b) Using the mode solver for this part. I wanted to see how modes would look after LP₀₁, not single moded. So I went back to the paper and looked at the equation for the diameters that allow for single mode operation, $D< {\pi }$, where n₁ is 1 for air, and n₀ is the index of refraction for the medium in use. From the tong paper, the second page of the paper, or pg.817 of the nature journal it was published in, index of refraction for Silica is said to be n₀ = 1.46. This gives NA = 1.063, using λ = 633nm, the max diameter is then given to be 454.6 nm for single mode operation. So I will put diameters larger then this for non single modes. Using 600 nm for the diameter. The image generated looks just like two very sharp gaussians c) I believe this is due to when the diameter of the wire is decreased below the wavelength its supposed to be guiding, more of the light is guiding outside the wire as a surface wave. So for 1550 nm , looking at the graph for loss, starting at wire diameters of 1200 nm we are already operating below the wavelength we want to guide so as the diameter decreases , more light is outside the wire leading to more loss. Compared to the 633 nm wavelength, you can see the increase in loss occurs when operating below 633 nm diameter wire but we are lower loss from 1200 nm until we get to 633 nm d) To go along with this, the loss mechanism is from surface contamination. The silica wires are’t perfectly uniform. By virtue of that, when light is guided by surface waves it is more susceptible to the surface contaminations.

Single-molecule Förster Resonance Energy Transfer (smFRET) allows probing intermolecular interactions and conformational changes in biomacromolecules, and represents an invaluable tool for studying cellular processes at the molecular scale. smFRET experiments can detect the distance between two fluorescent labels (donor and acceptor) in the 3-10 nm range. In the commonly employed confocal geometry, molecules are free to diffuse in solution. When a molecule traverses the excitation volume, it emits a burst of photons, which can be detected by single-photon avalanche diode (SPAD) detectors. The intensities of donor and acceptor fluorescence can then be related to the distance between the two fluorophores. While recent years have seen a growing number of contributions proposing improvements or new techniques in smFRET data analysis, rarely have those publications been accompanied by software implementation. In particular, despite the widespread application of smFRET, no complete software package for smFRET burst analysis is freely available to date. In this paper, we introduce FRETBursts, an open source software for analysis of freely-diffusing smFRET data. FRETBursts allows executing all the fundamental steps of smFRET bursts analysis using state-of-the-art as well as novel techniques, while providing an open, robust and well-documented implementation. Therefore, FRETBursts represents an ideal platform for comparison and development of new methods in burst analysis. We employ modern software engineering principles in order to minimize bugs and facilitate long-term maintainability. Furthermore, we place a strong focus on reproducibility by relying on Jupyter notebooks for FRETBursts execution. Notebooks are executable documents capturing all the steps of the analysis (including data files, input parameters, and results) and can be easily shared to replicate complete smFRET analyzes. Notebooks allow beginners to execute complex workflows and advanced users to customize the analysis for their own needs. By bundling analysis description, code and results in a single document, FRETBursts allows to seamless share analysis workflows and results, encourages reproducibility and facilitates collaboration among researchers in the single-molecule community.

This paper describes the methods to calbrate LRO's Diviner Lunar Radiometer Experiment. Like many radiometers, Diviner is sensitive to instrument temperature changes along the orbit of LRO. Regularly executed calibration blocks include instrument pointings to space and towards internal blackbodies at a known temperature. Data from these blocks serve to determine current offsets and current DN to radiance conversion value. A ground calibration campaign served to determine conversion tables over temperature.

Coupled-mode theory is concerned with coupling spatial modes of differing polarizations, distributions, or both. To understand codirectional coupling it is useful to have an understand of background material that builds to codirectional coupling. First, consider coupling normal modes in a single waveguide that is affected by a perturbation. Such case is single-waveguide mode coupling. The perturbation in question is spatially dependent and is represented as ΔP(r), a perturbing polarization. Consider the following Maxwell’s equations \[ \nabla \times E=i \omega \mu_0 H\] \[\nabla \times H=-i \omega \epsilon E-i \omega \Delta P\] Consider two sets of fields (E₁, H₁) and (E₂, H₂), they satisfy the Lorentz reciprocity theorem give by ∇ ⋅ (E₁×H₂*+E₂*×H₁) = −iω(E₁⋅ΔP₂*−E₂*⋅ΔP₁) For ΔP₁ = ΔP and ΔP₂ = 0 and integrating over the result for the cross section of the waveguide in question, we get \[ {\partial z} A_{\nu}(z)e^{i\left(-\right)} z=i \omega e^{-i z} ^{\infty} ^{\infty} E_{\mu}^{*} \cdot \Delta P dxdy \] Evoking orthonormality, we can get the coupled-mode equation \[ \pm }{\partial z} =i \omega e^{-i z} ^{\infty} ^{\infty} E_{\nu}^{*} \cdot \Delta P dxdy \] The plus sign indicates forward propagating modes when Bν > 0 and the minus sign indicates a backward propagating mode with Bν < 0 Many applications are concerned with the coupling between two modes. This coupling between two modes can be within the same waveguide or can be coupled between two parallel waveguides. For a system where we are interested in coupling two modes for either the parallel waveguides case or within the same waveguide, the two modes are described by two amplitudes A and B. The coupled equations are given by \[\pm {\partial z}=i A+ i B e^{i(\beta_b-\beta_a)z} \] \[\pm {\partial z}=i B + i A e^{i(\beta_a-\beta_b)z} \] for the two modes we want to solve. The coupling coefficients are given as part of a matrix C = [Cνμ] and $=[_{\nu \mu}]$ They are given as \[C_{\nu \mu}=^{\infty} ^{\infty} \left( E_{\nu}^{*} \times H_{\mu} + E_{\mu} \times H_\nu \right) \cdot dxdy=c_{\mu \nu}^{*}\] \[ _{\nu \mu}=\omega ^{\infty} ^{\infty} E_{\nu}^{*} \cdot \Delta \cdot E_{\mu} dxdy\] where Δϵ is the perturbation applied to the system. We then simplify the math further by removing the self coupling terms in equations (5) and (6) by expressing our normal mode coefficients by \[A(z)=(z)e^{\pm i ^{z} (z)dz} \] \[B(z)=(z)e^{\pm i ^{z} (z)dz}\] For cases of interest, perturbation will either be independent of z or be a periodic funciton of z. This further reduces our coupling coefficients to \[ \pm }{\partial z}=i e^{i 2 \delta z} \] \[ \pm }{\partial z}=i e^{-i 2 \delta z}\] The parameter of 2δ is the phase mismatch between the two coupled modes. CODIRECTIONAL COUPLING Codirectional coupling is when the coupling of two propagating modes are in the same direction, over some length l, where βa and βb are both greater than zero. Coupling equations to be used are \[}{\partial z}=i e^{i 2 \delta z} \] \[ }{\partial z}=i e^{-i 2 \delta z}\] The general solution of this system is solved as an initial value problem in matrix form is given as $$ (z)\\ (z)\\ = F(z;z_0) (z_0)\\ (z_0)\\ $$ Where F(z; z₀) is the forward coupling matrix and it relates field amplitudes at an intial value of z₀ to those at z. For the most simple case when power is launched into only mode a at z = 0 , giving $(0)=0$. With z = 0 \[(z)=(0)\left( cos \beta_c z -{\beta_c} sin \beta_c z \right) e^{i \delta z} \] \[(z)=(0)\left(}{\beta_c} sin \beta_c z \right) e^{-i \delta z} \] where $\beta_c=\left( +\delta^2\right)^{2}$ Then looking at the power of the two modes when they are completely phase matched, that is when δ = 0 \[{P_a(0)}=|(z)}{(0)}|^2=cos^2 \beta_c z \] \[{P_a(0)}=|(z)}{(0)}|^2=sin^2 \beta_c z\] where κab = κba* Define the coupling efficiency for a length l as $\eta=|^2}{\beta_c ^2} sin^2 \beta_c z$ as well as the coupling length $l_c={2 \beta_c}$. I have plotted the phase matching condition of δ = 0 in figure 1. It is seen that we can only have complete power transfer when we are phase matched between the two coupled modes.

CHAPTER 5 5.1 a)2,6,4,2,7,4,0,6,5,0 b)T,T,T,T,H, T,T,T,H,T c)No I only got 2 heads 5.10 7/4, −10% cant be real probabilities for 7/4 is larger than one and we cant have negative probabilities. 5.15 a)1/16 b)4/16 = 1/4 c)6/16 = 3/8 d)4/16 = 1/4 e)1/16 5.21 104/1858 = 5.6% 5.22 26/1858 = 1.4% 5.23 Were looking at the total numbers for Democrat or Republican, and since this is an or questions we add (689 + 469)/1858 = 62.3% 5.24 (593 + 530/1858 = 60.4% 5.25 Following the guided excersiceP(Liberal)=530/1858 = 28.5% P(Democrat)=689/1858 = 37.1% They arent mutually exclusive because you can see that there are people who are both liberal and democrat. P(LibandDem)=(306)/1858 = 16.5% If we dont subtract away the probability of people being democrat and liberal from that equation, you will be over counting. P(LiborDem)=(530 + 689 − 306)/1858 = 49.1% 5.33 a)P(Oddorlessthan3)=3/6 + 2/6 − 1/6 = 4/6 = 66.7% b)P(Oddorlessthan2)=3/6 + 1/6 − 1/6 = 50% 5.40 a)15/50 = 30% b)35/50 = 70% c)30/50=60%$\\ d)20/50=40\%$ e)1 because they are compliments. 5.44 a)1 − 0.23 − 0.41 = 0.36 b)0.41 + 0.36 = 0.77 c)0.41 + 0.23 = 0.64 d)a and c are complementary.16 or more mistakes mean 16 up to 30. 15 or fewer all values from 0 up to 15. If you add these together you get the whole experiment. 5.47 a)306/530 = 57.7% b)104/593 = 17.53% c)The highest percent is with liberal democrats. 5.57 Follow the guided exercise