The role of beam waist in Laguerre-Gauss expansion of vortex beam

Laguerre-Gauss (LG) modes represent an orthonormal basis set of solutions of the paraxial wave equation. LG are characterized by two integer parameters $n$ and $\ell$ that are related to the radial and azimuthal profile of the beam. The physical dimension of the mode is instead determined by the beam waist parameter $w_0$: only LG modes with the same $w_0$ satisfy the orthogonality relation. Here, we derive the scalar product between two LG modes with different beam waists and show how this result can be exploited to derive different expansions of a generic beam in terms of LG modes. In particular, we apply our results to the recently introduced Circular Beams, by deriving a previously unknown expansion. We finally show how the waist parameter must be chosen in order to optimize such expansion.Comment: 5 page

Katolickie wychowanie, wiara i autorytet

The text aims to show that authority and obedience are essential within Catholic education. In fact, Faith is “obedience” to the Truth as says the Second Vatican Council and the Truth is endowed with “authority” because God Himself is embodied in Jesus Christ. It is necessary to stress that – according to the Christian faith – this dependence upon God is not a kind of slavery but rather a loving relationship to the Father which sets humans free. This happens because man has an original dignity coming from his creation “in God’s image”. Catholic education must make people able to practice self-dominion (through the experience of authority) in order to choose only what is good so as to obey to the truth of human identity. Tekst ma na celu wykazanie, że autorytet i posłuszeństwo są niezbędne w katolickim wychowaniu. W rzeczywistości wiara jest „posłuszeństwem” wobec Prawdy, jak mówi Sobór Watykański II, a prawda jest obdarzona „autorytetem”, ponieważ sam Bóg jest ucieleśniony w Jezusie Chrystusie. Konieczne jest wyjaśnienie, że – zgodnie z wiarą chrześcijańską – odwieczna zależność od Boga nie jest rodzajem niewolnictwa, ale miłosnym związkiem z Ojcem, czyniącym człowieka wolnym. Jest tak dlatego, ponieważ oryginalna godność człowieka pochodzi ze stworzenia go „na obraz Boży”. Wychowanie katolickie powinno nauczyć panowania nad sobą (w relacji z autorytetem), aby móc wybrać tylko to, co jest dobre i być wiernym prawdziwej istocie człowieka. 

Christian Faith as an Important Element of Culture and Integral Education

The article, referring to the words of the Letter to Diognetus on Christians as the soul of the world, points to the importance of the Christian faith and its believers – Christians in Europe and in the world, especially in the field of culture and education, which are characterized by an integral approach. This problem is analysed in successive three stages: 1) first, integrity is indicated as an essential feature of the Christian faith and education that has been clearly marked from the very beginnings of Christianity; 2) shows faith as the basic expression of human identity, also consistent with the freedom that gives its identity; 3) critically analyses the secularity, indicating in this context the need to promote the integrity of the upbringing of the person and the cultural and social benefits that the Christian faith can bring. Especially during the present loss of the ability to distinguish between important and less important things, when apart from the need to rethink the idea of secularity, the possibility of a positive culmination is pointed out. Christian faith can be an important element of integral culture and integral education in action for humanization. In this action, reason and faith, as “two wings” can lead to true wisdom.Wiara chrześcijańska jako istotny element kultury i integralnego wychowaniaArtykuł nawiązując do słów Listu do Diogneta o chrześcijanach jako duszy świata, wskazuje na znaczenie wiary chrześcijańskiej i jej wyznawców – chrześcijan dla Europy i dla świata zwłaszcza w obszarze kultury i wychowania, które charakteryzują się podejściem integralnym. Tę problematykę analizuje się w następujących po sobie trzech etapach: 1) najpierw wskazuje się na integralność jako istotną cechę wiary i wychowania chrześcijańskiego zaznaczającą się wyraźnie już od samych początków chrześcijaństwa; 2) ukazuje wiarę jako podstawowy wyraz ludzkiej tożsamości, zgodny także z wolnością, która nadaje jej tożsamości; 3) poddaje krytycznej analizie laickość, wskazując w tym kontekście na potrzebę promowania integralności wychowania osoby oraz na kulturowe i społeczne korzyści, jakie zdolna jest wnosić wiara chrześcijańska. Zwłaszcza w obecnej utracie zdolności rozróżniania rzeczy ważnych i mniej ważnych, gdy obok potrzeby przemyślenia idei laickości wskazuje się na możliwości pozytywnego jej zwieńczenia, wiara chrześcijańska może stanowić istotny element integralnej kultury i integralnego wychowania w działaniu na rzecz humanizacji. W tym działaniu rozum i wiara, jako „dwa skrzydła”, mogą prowadzić do prawdziwej mądrości

Quantized vs. Analog Feedback for the MIMO Downlink: A Comparison between Zero-Forcing Based Achievable Rates

We consider a MIMO fading broadcast channel and compare the achievable ergodic rates when the channel state information at the transmitter is provided by analog noisy feedback or by quantized (digital) feedback. The superiority of digital feedback is shown, with perfect or imperfect CSIR, whenever the number of feedback channel uses per channel coefficient is larger than 1. Also, we show that by proper design of the digital feedback link, errors in the feedback have a minor effect even by using very simple uncoded modulation. Finally, we show that analog feedback achieves a fraction 1 - 2F of the optimal multiplexing gain even in the presence of a feedback delay, when the fading belongs to the class of Doppler processes with normalized maximum Doppler frequency shift 0 <= F <= 1/2.Comment: Submitted to ISIT, January 2007. 5 page

Aristotle and pedagogy

Aristotle’s metaphysics, ethics and psychology can help to interpret pedagogy from a “scientific” point of view. Naturally, it is not a question of considering the science of education as a natural science born during modernity; the main difference is that the object of pedagogy is actually a subject, i.e. the human being, notably the free human. That is why an ancient thinker like Aristotle can promote pedagogy through theoretical reflection. In fact Aristotle clearly indicates human goals which even nowadays can guide human education and action

On the properties of Circular-Beams

Circular-Beams were introduced as a very general solution of the paraxial wave equation carrying Orbital Angular Momentum. Here we study their properties, by looking at their normalization and their expansion in terms of Laguerre-Gauss modes. We also study their far-field divergence and, for particular cases of the beam parameters, their possible experimental generation.Comment: 5 page

Rheological Behavior of Microemulsions

We study the stationary and transient behaviors of the microemulsion phase subjected to a shear flow. The system is described by a diffusion-convective equation which generalizes the usual Cahn-Hilliard equation. Non-linear terms are treated in a self-consistent approximation. Shear, first and second normal stresses are calculated as momenta of the structure factor. Shear thinning is observed in stationary conditions. After a newtonian regime at small values of the shear rate, the excess viscosity decreases when the shear rate becomes of the order of the inverse of the relaxation time of the system without flow. In transient regimes, when the flow is applied starting from a quiescent state, we find that the shear stress reaches a maximum before decreasing to a constant value.Comment: 10 figures, 2 of which have format jp

Optimized Training and Feedback for MIMO Downlink Channels

We consider a MIMO fading broadcast channel where channel state information is acquired at user terminals via downlink training and channel feedback is used to provide transmitter channel state information (CSIT) to the base station. The feedback channel (the corresponding uplink) is modeled as an AWGN channel, orthogonal across users. The total bandwidth consumed is the sum of the bandwidth/resources used for downlink training, channel feedback, and data transmission. Assuming that the channel follows a block fading model and that zeroforcing beamforming is used, we optimize the net achievable rate for unquantized (analog) and quantized (digital) channel feedback. The optimal number of downlink training pilots is seen to be essentially the same for both feedback techniques, but digital feedback is shown to provide a larger net rate than analog feedback.Comment: To appear at IEEE Information Theory Workshop, Volos, Greece, June 200