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Today's @momentumdash mantra is "Be grateful." I usually mock things like this but considering the times it's a helpful reminder that among all the uncertainty I'm lucky to have good friends & family. Whatever comes we'll get through this, together. #ThursdayThoughts #covid19UK
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Thank you @momentumdash for bringing a little beauty to my everyday with your Chrome add-on. It's a small thing, but every time I open a tab and see a stunning picture of nature it reminds of the wonders of this planet. We need this. #gratitude
One of the easiest, most powerful ways I've changed my daily life is by using the @momentumdash plugin for Chrome. It shows you your to-do list right on your homepage, so whenever you get sidetracked, you get a reminder when you pull up a new tab about what you're supposed to focus on. Then it asks you, what's your focus for today? and you get to set your intention and keep being reminded of it. The beautiful imagery & the quotes are all a plus, also.
If you are not already using @momentumdash , you should. Install the extension to see a new quote every day. Favorite the quote and/or tweet it out. If anything, enjoy the various pictures of various (beautiful) places each day #travel #InspirationalQuotes #mottos #LifeLessons
I have been using #Momentum for years now. Getting a new wallpaper and a quote every morning is just so refreshing. Recently I started using the focus field to put my screen cast focus. Just a small tweak, but adds a lot of value. Thank you @momentumdash for this awesome extension
My @momentumdash this morning is gorgeous. For those that don't know it's an extension that displays a picture and other info when you open a new tab in chrome. My 'todo' for the day is to conquer it.
Yes, once you insert your momentum card into a slot machine, that will activate your Monthly Reward Credit and you will see the balance through your My Momentum account online. As you use the credits, your balance will be reflected on your account.
You will upgrade or maintain your card level based on the new Status Points Goals required at each card level. Just as before, you can check how many Status Points you have earned online, at the Momentum Desk or now at the Momentum Promotional Kiosks located on the gaming floors.
One recently explored approach to significantly increase the capacity of free-space links is to spatially multiplex multiple orthogonal modes using a single transmitter/receiver aperture pair, in which each mode carries an independent data stream. Orthogonality ensures that the modes can be efficiently multiplexed at the transmitter, spatially co-propagated and are demultiplexed at the receiver with minimal modal crosstalk.
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Structured light has gained much interest in increasing communications capacity through the simultaneous transmission of multiple orthogonal beams. This paper gives a perspective on the current state of the art and future challenges, especially with regards to the use of multiple orbital angular momentum modes for system performance enhancement.
The wavefronts, intensity profiles, and phase profiles of orbital angular momentum (OAM) modes l = 0, 1, 2, and 3. The OAM mode with a nonzero order has a donut shape intensity profile and helical phasefront. The size of the ring in the intensity profile grows with l. We note that p+1 represents the number of concentric amplitude rings and p=0 is shown.
This orthogonality is of crucial benefit for a communications engineer. It implies that multiple independent data-carrying optical beams can be multiplexed and simultaneously transmitted in either free-space or fiber, thereby multiplying the system data capacity by the total number of beams (see Figure 2). Moreover, since all the beams are in the same frequency band, the system spectral efficiency (i.e., bits/s/Hz) is also increased. These multiplexed orthogonal OAM beams are a form of mode-division multiplexing (MDM), a subset of space-division multiplexing , , , .
A key issue in almost any MDM communication system is dealing with intermodal power coupling and deleterious inter-data-channel crosstalk. There are many causes of modal coupling and crosstalk, including the following for free-space OAM-multiplexed optical communication links:Turbulence: Atmospheric turbulence can cause a phase differential at different cross-sectional locations of a propagating beam. Given this phase change distribution in a changing environment, power can couple from the intended mode into other modes dynamically (e.g., perhaps on the order of milliseconds) , .
There are several approaches to potentially mitigate coupling and crosstalk in free-space OAM-multiplexed systems, including (see Figure 4):Electrical digital signal processing (DSP): Crosstalk due to modal coupling has many similarities to crosstalk that occurs in multiple-transmitter-multiple-receiver (i.e., multiple-input multiple-output, MIMO) radio systems . Multiple optical modes are similar to parallel radio frequency (RF) beams that experience crosstalk. Similar to electronic DSP that can undo much of the crosstalk in MIMO RF systems, these DSP approaches could also be used for mitigating OAM modal crosstalk .
As compared to RF links, optics in general can provide: (a) more bandwidth and higher data capacity due to the higher carrier wave frequency, and (b) better beam directionality and lower divergence, thus making eavesdropping more difficult . When incorporating MDM using OAM multiplexing, such optical links can potentially achieve capacity enhancement and increased difficulty to eavesdropping. This lower probability of intercept stems from the issue that any misalignment causes intermodal coupling, such that it is extremely difficult for an off-axis eavesdropper to recover the signals, and even an on-axis eavesdropper would need to know the modal properties in order to recover the data, again fairly difficult. In addition, these free-space applications share some common desirable characteristics, including: (1) low size, weight and power (SWaP), which can be alleviated by advances in integrated OAM devices ; and (2) accurate pointing, acquisition and tracking (PAT) systems, which helps limit modal coupling and crosstalk .
These advantages have generated interest in free-space MDM communications in the following scenarios:Atmosphere: OAM multiplexing can potentially benefit communication to: (a) unmanned aerial vehicles, for which distances may be relatively short range and a key challenge is to miniaturize the optical hardware, and (b) airplanes and other flying platforms, for which distances may require turbulence compensation and highly accurate pointing/tracking , ,  (see Figure 5).
Should both modal indices be used?: Structured beams that are from a modal basis set can generally be described by two modal indices, such that the beam can be fully described by these coordinates. For example, LG modes have l (azimuthal) and p (radial) components, whereas HG beams have n (horizontal) and m (vertical) components. However, the vast majority of publications concerning MDM-based free-space optical communications utilized only a change in a single modal index for the different OAM beams. Specifically, each beam commonly had a different l value but the same p=0 value , , . This one-dimensional system can accommodate many orthogonal beams, but a system designer could also use the other beam modal index in order to possibly achieve a larger two-dimensional set of data channels. This two-dimensional approach was shown experimentally for LG and HG beams , . It is important to note that a significant challenge is the sufficient capture of the beam at the receiver aperture to ensure accurate phase recovery and orthogonality along both indices .
Separate from using optical beams, free-space communication links can take advantage of mode multiplexing in many other carrier-wave-frequency ranges to increase system capacity. For example, OAM can be manifest in many types of electromagnetic and mechanical waves (see Figure 8), and interesting reports have explored the use of OAM in millimeter, acoustic, and THz waves .
MDM can be achieved in both free-space and fiber, with much of the transmitter and receiver technology being similar. However, the channel medium is different, which gives rise to the following distinctions:There is no beam divergence in light-guiding fiber.
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Time-independent form of the Schrodinger equation states$$\hat H\psi=E\psi$$For a Hamiltonian in form of$$\hat H=\frac\hat p^22m$$Which indicates a free particle, In the position space is routine and starts with plugging in the momentum operator in position space as$$\hat p=-i\hbar\frac\partial\partial x$$And we can obtain eigenvalues and eigenfunctions as$$E=\frac\hbar^2k^22m$$$$\psi^+(x)=e^ikx\space\space,\space\space\psi^-(x)=e^-ikx$$$$\psi(x)=A\psi^++B\psi^-$$I also know we can derive the wavefunction in the momentum space with a Fourier transform. But I want to solve the SE in the momentum space. So$$\hat H\tilde\psi(p)=E\tilde\psi(p)$$$$\fracp^22m\tilde\psi(p)=E\tilde\psi(p)$$$$\tilde\psi(p)(\fracp^22m-E)=0$$One answer is the same as the previous method$$E=\frac\hbar^2k^22m$$But here $\tilde\psi(p)$ can be any function of $p$. But we know it should be the same as the result of the Fourier transform on $\psi$. 350c69d7ab