What are the main applications of ellipses in optics?

1. Remember that one of the polarized lights is elliptical polarized light: the magnitude and direction of the light vector change regularly during propagation, and the end point of the light vector moves along the elliptical trajectory. The linearly polarized light produced by the polarizer becomes a special elliptical polarized light after being oriented to a certain 1 4 wave plate.

2. There is a refractive index ellipsoid in the geometric representation of the optical properties of the crystal, and its equation is x2 nx2+y2 ny2+z2 nz2=1

A refractive index ellipsoid has the following properties.

1) Any sagittal diameter r=nd emitted from the refractive index ellipsoid (d is the unit vector of the value distribution of the crystal refractive index in the d direction of each light wave in crystal space), that is, the direction of the sagittal diameter represents a vibration direction of the light wave d vector, and its sagittal diameter length represents the refractive index of the light wave vibrating in the direction of the d vector.

2) The center of the perfractive index ellipsoid is a plane perpendicular to a given wave normal direction k, and it is an ellipse with the cross-section of the ellipsoid, then the major and minor axis directions of the ellipse are the directions d of the two allowable light waves d corresponding to k'、d'', and the length of the long and minor semi-axes is equal to the refractive index n of these two light waves'、n''

3) The relationship between d, e, k, and s is obtained by using a refractive index ellipsoid. i.e., the e vector is the normal of the ellipsoid tangent plane at the end of the d vector; Since d, e, k, and s are coplanar, and d is perpendicular to k, and e is perpendicular to s, the corresponding direction of k and s can be obtained, and k is perpendicular to the tangent plane at the intersection of s s and ellipsoid.

I said landlord, the problem of your text is not only a bit professional, but also very bad, because it involves geometric figures and directionality, and it is difficult to explain it with words alone. I'm just trying to give you an answer based on my own understanding and memory, but I'm getting old. There's no merit and hard work, right? Hehe.

It is recommended that you take a look at the "Applied Optics" and "Optics Handbook", which are more detailed.

An ellipsometry measures film thickness and refractive index.

The basic idea of elliptic measurement is that the linearly polarized light generated by the polarizer becomes a special elliptical polarized light after being oriented to a certain 1 4 wave plate, and when it is projected on the surface of the sample to be measured, as long as the polarizer takes the appropriate light transmission direction, the linearly polarized light will be reflected on the surface of the sample to be measured According to the polarization state changes of the polarized light before and after reflection, including the change of amplitude and phase, many optical properties of the sample surface can be determined

The optical properties of the ellipse, the mirror of the ellipse is based on the long axis of the ellipse, the ellipse rotates 180 degrees to form a three-dimensional figure, and its inner surface is all made into a reflective surface, hollow.

Elliptical mirrors reflect all the light emitted from one focal point to another, and elliptical lenses have the effect of concentrating light, such as reading glasses, magnifying glasses, and farsighted glasses.

The basic properties of ellipses

In mathematics, an ellipse is a curve in a plane around two focal points such that for each point on the curve, the sum of the distances to the two focal points is constant. Therefore, it is a generalization of a circle, which is a special type of ellipse with two focal points at the same position. The shape of an ellipse, represented by its eccentricity, for an ellipse can be any number from 0, the limit case of a circle to anything close but less than 1.

An ellipse is a closed conical section: a planar curve that is intersected by a cone and a plane. The ellipse has many similarities with the other two forms of conical sections:

Parabola and hyperbola, both are open and unbounded. The cross-section of a cylinder is elliptical unless the section is parallel to the axis of the cylinder.

An ellipse can also be defined as a set of points such that the ratio of the distance of each point on the curve to the distance of a given point to the distance of the same point on the curve is a constant. This ratio is called the eccentricity of the ellipse. An ellipse can also be defined in this way, an ellipse is a collection of points, and the sum of the distances from the points to the two focal points is a fixed number.

Ellipses are common in physics, astronomy, and engineering.

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Problem description: How to prove that when a ray of light from one focal point in an ellipse is reflected on the ellipse, the reflected light must pass through the other focal point?

And what are the applications of the properties of ellipses?

Analysis: There are many proofs.

Physical Method: Fermat's Principle: Light travels from one point in space to another along a path with an extreme optical path (maximum, minimum, or constant).

The optical path length is the product of the distance l passed by light in a homogeneous medium and the refractive index n of the medium, nl. From a constant pathlength, the light is reflected everywhere, always passing through another focal point.

Mathematically: Prove that the perpendicular line of the tangent of a point on an ellipse bisects the angle formed by the two focal radii of that point.

For the ellipse of x2 a 2 + y 2 b 2 = 1, the slope of the tangent of a point p(x0,y0) is (-b 2x0) (a 2y0).

Therefore, the slope of the perpendicular line of the tangent of a certain point and the slope of the two focal radii can be obtained, and then the proposition can be proved by calculating the angle by using the chamfer formula.

There are many acres of applications, the most classic is ultrasonic lithotripsy, the ultrasonic source stool is placed on one focus of the ellipsoidal surface, so that the stones in the body are located on the other focus of the stone can be crushed.

In addition, when showing a movie, the filament is placed on one focal point of Xunji, and the film gate is placed on the other focal point to show the movie.

elliptical lenses, planetary orbits, rotating body orbits; Or use the hand rope to drag something around in the air, and draw an approximate elliptic curve; hyperboloid lenses, reflectors; the curve of the parabola, etc. There are many other optical properties and geometric properties of conic curves.

Optical properties: 1. The elliptical mirror (with the long axis of the ellipse as the axis, the three-dimensional pattern formed by rotating the ellipse 180 degrees, and its inner surface is all made into a reflective surface, hollow) can reflect all the light emitted by a certain focal point to another focal point;

2. Elliptical lenses (some cross-sections are elliptical) have the effect of converging light (also called convex lenses), reading glasses, magnifying glasses and farsighted glasses are all such lenses (these optical properties can be proved by the method of counterproof);

3. The ellipse is a kind of conic curve, that is, the section between the cone and the plane;

4. The circumference of an ellipse is equal to the length of a particular sinusoidal curve in a cycle.

The elliptical mirror (a three-dimensional figure formed by rotating the ellipse 180 degrees on the long axis of the ellipse, and all its inner surfaces are made into reflective surfaces, hollow) can reflect all the light emitted from one focal point to another focal point; Elliptical lenses (some of which are elliptical) have the effect of concentrating light (also called convex lenses), such as reading glasses, magnifying glasses, and farsighted glasses (these optical properties can be proved by the method of rebuttal).

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Analysis: Known ellipse: +=1, its two foci are f(c,0),f'(-c,0), a light or sound wave from one focal point to any point on the ellipse, which is reflected by the ellipse and passes through the other focal point.

then +=1 y=b(1-)=b-

The tangent line of the p-mileage is l:+=1 bxx+ayy=ab

The equation for the straight line pf is y=(x-c) yx-(x-c)y-cy=0

Straight line pf'The equation is y=(x+c) yx-(x+c)y+cy=0 in the hall bucket

The acute angle between the tangent l and the straight line pf is the sharp angle between the normal vector (bx,ay) and (y,-(x-c)).

Tangent L with straight line PF'The acute angle of is the sharp angle between the normal vector (bx,ay) and (y,-(x+c)).

cos===

cos===

cos=cos, both acute angles=

Straight line pf, pf'The angle to the normal of the p-point is equal.

Therefore, the light or sound wave that is emitted from one focal point of the ellipse to any point on the ellipse will pass through the other focal point after being reflected by the pin ellipse.

It is known that the congregation ellipse: +=1, and its two foci are f(c,0),f'(-c,0), then the light or sound wave that is shot from one focal point of the nato to any point on the ellipse, and the ellipse will pass through the other focal point after being reflected by the ellipse. Proof:

Let p(x,y) be the previous point, then +=1 y=b(1-)=b- and the tangent through p is l:+=1 bxx+ayy=ab, and the equation for the straight line pf is y=(x-c) yx-..

Kepler's three laws of planetary motion are also known as Changhu's elliptic law, all the planets orbit around the sun are elliptical, the sun is at a focal point of all ellipses, the light emitted from one focal point of the ellipse, after the ellipse is reflected, the reflected light is converged on another focal point of the ellipse. We call this the optical type of the ellipse.