Antiderivative

In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F' = f. The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative. Antiderivatives are often denoted by capital Roman letters such as F and G. Antiderivatives are related to definite integrals through the second fundamental theorem of calculus: the definite integral of a function over a closed interval where the function is Riemann integrable is equal to the difference between the values of an antiderivative evaluated at the endpoints of the interval. In physics, antiderivatives arise in the context of rectilinear motion (e.g., in explaining the relationship between pos

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Photoinitiator-free Laser Fabrication of Ultra-compact, Low-loss Waveguides in Polydimethylsiloxane

Christophe Moser, Giulia Panusa, Demetri Psaltis, Ye Pu, Jieping Wang

Owing to its excellent elasticity, wide spectral range of transparency, and outstanding chemical and thermal stability, polydimethylsiloxane (PDMS) is an elastomer of great technological importance, being widely used in the fabrication of microfluidic and optofluidic devices in particular. Compact, low loss optical waveguides are crucial in such devices for a dense integration of optical functionalities. Compared with photolithographic methods, multiphoton laser direct writing through photopolymerization has shown great promise in the fabrication of three-dimensional (3D) optical waveguides in PDMS without the limit of planar structures. Here we demonstrate a multiphoton laser direct writing process that produces low-loss, ultra-compact waveguides in PDMS. The fabrication employs a self-initiated multiphoton polymerization of phenylacetylene infused in PDMS without additional photoinitiator. The elimination of the photoinitiator results in an excellent refractive index homogeneity and thus a low propagation loss due to scattering, as well as a greater biocompatibility by involving the fewest possible chemicals (monomer only) in the polymerization reaction. Our characterizations show that the waveguides fabricated as such are on average 1.3 mu m wide with a refractive index contrast of 0.06 and a propagation loss of 0.03 dB/ cm in the spectral band of 650-700 nm. Our technique will enable a broad range of applications spanning from wearable photonics to chip-scale optical interconnects.

SPIE-INT SOC OPTICAL ENGINEERING2018

Photoinitiator-free multi-photon fabrication of compact optical waveguides in polydimethylsiloxane

Christophe Moser, Giulia Panusa, Demetri Psaltis, Ye Pu, Jieping Wang

Compact, low loss flexible optical waveguides are crucial in optofluidic and microfluidic devices for a dense integration of optical functionalities. We demonstrate the fabrication of compact optical waveguides in polydimethylsiloxane through multiphoton laser direct writing using phenylacetylene as the photosensitive monomer. Our fabrication technique employs photo-induced radical chain polymerization initiated by the monomer molecule itself without a photoinitiator. Because of the dense pi-electrons in phenylacetylene, we achieved a high refractive index contrast (Delta n >= 0.06) between the waveguide core and the PDMS cladding. This allowed for efficient waveguiding with a core size of 1.3-mu m with a measured loss of 0.03 dB/cm in the spectral band of 650-700 nm. (C) 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

OPTICAL SOC AMER2019

MATH-101(en): Analysis I (English)

We study the fundamental concepts of analysis, calculus and the integral of real-valued functions of a real variable.

MATH-101(a): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

MATH-101(c): Analysis I

Étudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles d'une variable.

Integral

In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the

Function (mathematics)

In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the dom

Trigonometric functions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate