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Polar Equations of Conics

Polar Equations of Conics

A conic section can be defined in polar coordinates as the set of all points whose distance from a fixed point, known as the focus, bears a constant ratio to their distance from a fixed line, known as the directrix. This constant ratio is called the eccentricity. This definition unifies all types of conic sections—ellipses, parabolas, and hyperbolas—under a single framework. When the focus is positioned at the origin of the polar coordinate system, a single polar equation can describe any conic...

Measuring Reaction Rates

Measuring Reaction Rates

Polarimetry finds application in chemical kinetics to measure the concentration and reaction kinetics of optically active substances during a chemical reaction. Optically active substances have the capability of rotating the plane of polarization of linearly polarized light passing through them—a feature called optical rotation. Optical activity is attributed to the molecular structure of substances. Normal monochromatic light is unpolarized and possesses oscillations of the electrical field in...

Graphs of Polar Equations

Graphs of Polar Equations

The polar coordinate system represents points using a distance from a central point (the pole) and an angle from a reference direction (the polar axis). Unlike rectangular coordinates, polar coordinates are ideal for graphing curves with radial symmetry or periodic behavior.Some general forms of graphs in polar coordinates include the following:Equation of a Circle (Centered at the Pole):A graph where the radius remains constant for all angles traces a circle centered at the pole:Equation of a...

Polar Coordinates: Problem Solving

Polar Coordinates: Problem Solving

Directional radiation patterns are central to antenna analysis, as they illustrate how signal strength varies with direction. These patterns are often modeled using polar plots, where the radial distance from the origin represents signal intensity at a given angle. A commonly used idealized form is the four-lobed rose curve, which captures the concept of directional beams in a simplified mathematical form.The four-lobed rose curve, described by r = cos⁡(2θ), features four symmetric lobes, each...

Polar Coordinate System

Polar Coordinate System

The polar coordinate system provides a natural way to describe points in the plane when distances and directions are more meaningful than horizontal and vertical displacements. It is especially useful for modeling non-rectangular regions such as circles and spirals, where symmetry about a center point is easier to express than it is in a rectangular grid. A familiar example is a ship’s plan position indicator, which marks detected targets as dots positioned relative to the ship at the display’s...

Dielectric Polarization in a Capacitor

Dielectric Polarization in a Capacitor

The presence of a dielectric medium in a capacitor not only changes the voltage and capacitance but also affects the electric field. In general, dielectrics can be of two types: polar and nonpolar. In a polar dielectric, the positive and negative charges in the molecules are separated by a distance and hence have a permanent dipole moment. In contrast, no such charge separation exists in a nonpolar dielectric, however the nonpolar molecules get polarized in the presence of an external electric...

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Related Experiment Video

Updated: Jul 13, 2026

Determination of the Excitation and Coupling Rates Between Light Emitters and Surface Plasmon Polaritons

Determination of the Excitation and Coupling Rates Between Light Emitters and Surface Plasmon Polaritons

Published on: July 21, 2018

Compact ellipsometer employing a static polarimeter module with arrayed polarizer and wave-plate elements.

Takashi Sato¹, Takeshi Araki, Yoshihiro Sasaki

¹Photonic Lattice, Inc., Aoba Incubation Square, Aoba, Aramaki, Aoba-ku, Sendai 980-0845, Japan. sato@photonic-lattice.com

|August 7, 2007

Summary

A new portable ellipsometer utilizes a compact static polarimeter for rapid, repeatable measurements of thin silicon dioxide films. This technology offers potential for integration into manufacturing equipment.

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Area of Science:

Optical Engineering
Materials Science
Metrology

Background:

Ellipsometry is a crucial technique for characterizing thin films.
Existing ellipsometers can be bulky and expensive, limiting their application in manufacturing.
There is a need for compact, fast, and repeatable ellipsometry solutions.

Purpose of the Study:

To develop a portable and compact ellipsometer module.
To demonstrate high repeatability and measurement speed.
To assess the potential for integration into manufacturing equipment.

Main Methods:

Development of a portable static polarimeter.
Utilizing an arrayed polarizer, arrayed wave plate, and CCD image sensor.
Measurement of silicon dioxide (SiO2) films on silicon (Si) wafers.

Main Results:

Achieved a high level of repeatability in measurements.
Demonstrated a fast measurement speed of 0.3 seconds.
Successfully measured SiO2 films with thicknesses ranging from 2 to 300 nm.

Conclusions:

The developed portable ellipsometer with a compact static polarimeter is effective for thin film characterization.
The system exhibits excellent repeatability and speed, suitable for industrial applications.
Integration of optical source and receiver can lead to an ultracompact module for diverse manufacturing and measurement instruments.