Typesofdispersion inopticalfiber

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In the spherical coordinate system, a point \(P\) in space (Figure \(\PageIndex{9}\)) is represented by the ordered triple \((ρ,θ,φ)\) where

Dispersive prisms are used in a variety of scientific and technical applications, such as spectroscopy, where they are used to analyze the composition of materials based on their spectral signature. They are also used in optics and telecommunications, where they can be used to control the dispersion of light and correct chromatic aberration in lenses.

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In the \(xy\)-plane, the right triangle shown in Figure \(\PageIndex{1}\) provides the key to transformation between cylindrical and Cartesian, or rectangular, coordinates.

After that, a series of polishing and smoothing steps of the optical surfaces are needed.  This can take several iterations depending on the optical tolerances requested by the client and their application.  At this stage, antireflection coatings, filters, and metallic layers can be added to achieve the required performance.

b. Substitute \(r^2=x^2+y^2\) into equation \(r^2+z^2=9\) to express the rectangular form of the equation: \(x^2+y^2+z^2=9\). This equation describes a sphere centered at the origin with radius \(3\) (Figure \(\PageIndex{7}\)).

Figure 1 shows one of the most common reflective prisms geometries.  In general, if the number of reflective faces is even, we will be creating an upright image, while an even number of reflective surfaces  will create an inverted image.

In conclusion, designing a dispersive prism involves considering several critical factors, including the choice of material, the shape of the prism, the angle of incidence, the size of the prism, and the coating. By carefully considering these factors, it is possible to design an effective and efficient dispersive prism that provides high-quality light separation.

where:D = dispersion of the prism (in degrees)n = refractive index of the prism materialA = apex angle of the prism (in degrees)

In each of the following situations, we determine which coordinate system is most appropriate and describe how we would orient the coordinate axes. There could be more than one right answer for how the axes should be oriented, but we select an orientation that makes sense in the context of the problem. Note: There is not enough information to set up or solve these problems; we simply select the coordinate system (Figure \(\PageIndex{17}\)).

Use the equations in Converting among Spherical, Cylindrical, and Rectangular Coordinates to translate between spherical and cylindrical coordinates (Figure \(\PageIndex{12}\)):

Rhomboid Prisms create an output beam that is displaced from the input beam, but it doesn’t change the direction of the beam, nor does it invert the image.

As when we discussed conversion from rectangular coordinates to polar coordinates in two dimensions, it should be noted that the equation \(\tan θ=\dfrac{y}{x}\) has an infinite number of solutions. However, if we restrict \(θ\) to values between \(0\) and \(2π\), then we can find a unique solution based on the quadrant of the \(xy\)-plane in which original point \((x,y,z)\) is located. Note that if \(x=0\), then the value of \(θ\) is either \(\dfrac{π}{2},\dfrac{3π}{2},\) or \(0\), depending on the value of \(y\).

d. To identify this surface, convert the equation from spherical to rectangular coordinates, using equations \(y=ρ\sin φ\sin θ\) and \(ρ^2=x^2+y^2+z^2:\)

If this process seems familiar, it is with good reason. This is exactly the same process that we followed in Introduction to Parametric Equations and Polar Coordinates to convert from polar coordinates to two-dimensional rectangular coordinates.

Cylindrical and spherical coordinates give us the flexibility to select a coordinate system appropriate to the problem at hand. A thoughtful choice of coordinate system can make a problem much easier to solve, whereas a poor choice can lead to unnecessarily complex calculations. In the following example, we examine several different problems and discuss how to select the best coordinate system for each one.

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Porro prisms (either stand-alone or in higher-degrees configurations) are usually used to change the orientation of an image.  They are usually used as erectors in optical instruments like binoculars, telescopes, and microscopes where there are space restrictions.  The degree of a porro system will depend on how many axes the image needs to be altered in

Let \(P\) be a point on this surface. The position vector of this point forms an angle of \(φ=\dfrac{π}{4}\) with the positive \(z\)-axis, which means that points closer to the origin are closer to the axis. These points form a half-cone.

As we did with cylindrical coordinates, let’s consider the surfaces that are generated when each of the coordinates is held constant. Let \(c\) be a constant, and consider surfaces of the form \(ρ=c\). Points on these surfaces are at a fixed distance from the origin and form a sphere. The coordinate \(θ\) in the spherical coordinate system is the same as in the cylindrical coordinate system, so surfaces of the form \(θ=c\) are half-planes, as before. Last, consider surfaces of the form \(φ=c\). The points on these surfaces are at a fixed angle from the \(z\)-axis and form a half-cone (Figure \(\PageIndex{11}\)).

The dispersion of a prism is typically measured as the angular separation between two spectral lines of a particular wavelength. The following formula, which assumes a thin-prism, can be used to calculate the dispersion of a prism:

Reflective prisms present lower optical power losses than equivalent systems made with mirrors and are usually easier to align due to the fact that a single element is used instead of several.

Plot the point with spherical coordinates \((2,−\frac{5π}{6},\frac{π}{6})\) and describe its location in both rectangular and cylindrical coordinates.

b. Equation \(φ=\dfrac{5π}{6}\) describes all points in the spherical coordinate system that lie on a line from the origin forming an angle measuring \(\dfrac{5π}{6}\) rad with the positive \(z\)-axis. These points form a half-cone (Figure \(\PageIndex{14}\)). Because there is only one value for \(φ\) that is measured from the positive \(z\)-axis, we do not get the full cone (with two pieces).

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Conversion from cylindrical to rectangular coordinates requires a simple application of the equations listed in Conversion between Cylindrical and Cartesian Coordinates:

c. Equation \(ρ=6\) describes the set of all points \(6\) units away from the origin—a sphere with radius \(6\) (Figure \(\PageIndex{15}\)).

The point with spherical coordinates \((8,\dfrac{π}{3},\dfrac{π}{6})\) has rectangular coordinates \((2,2\sqrt{3},4\sqrt{3}).\)

Because Sydney lies south of the equator, we need to add \(90°\) to find the angle measured from the positive \(z\)-axis.

Modal dispersion

The formulas to convert from spherical coordinates to rectangular coordinates may seem complex, but they are straightforward applications of trigonometry. Looking at Figure \(\PageIndex{10}\), it is easy to see that \(r=ρ \sin φ\). Then, looking at the triangle in the \(xy\)-plane with r as its hypotenuse, we have \(x=r\cos θ=ρ\sin φ \cos θ\). The derivation of the formula for \(y\) is similar. Figure \(\PageIndex{10}\) also shows that \(ρ^2=r^2+z^2=x^2+y^2+z^2\) and \(z=ρ\cos φ\). Solving this last equation for \(φ\) and then substituting \(ρ=\sqrt{r^2+z^2}\) (from the first equation) yields \(φ=\arccos\left(\dfrac{z}{\sqrt{r^2+z^2}}\right)\). Also, note that, as before, we must be careful when using the formula \(\tan θ=\dfrac{y}{x}\) to choose the correct value of \(θ\).

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In geography, latitude and longitude are used to describe locations on Earth’s surface, as shown in Figure \(\PageIndex{16}\). Although the shape of Earth is not a perfect sphere, we use spherical coordinates to communicate the locations of points on Earth. Let’s assume Earth has the shape of a sphere with radius \(4000\) mi. We express angle measures in degrees rather than radians because latitude and longitude are measured in degrees.

Dove prisms are right angle prisms with their top part removed.  They can be used to invert images. It is possible to coat the side where light is reflected for optical sensing applications.

The latitude of Columbus, Ohio, is \(40°\) N and the longitude is \(83°\) W, which means that Columbus is \(40°\) north of the equator. Imagine a ray from the center of Earth through Columbus and a ray from the center of Earth through the equator directly south of Columbus. The measure of the angle formed by the rays is \(40°\). In the same way, measuring from the prime meridian, Columbus lies \(83°\) to the west. Express the location of Columbus in spherical coordinates.

There are actually two ways to identify \(φ\). We can use the equation \(φ=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right)\). A more simple approach, however, is to use equation \(z=ρ\cos φ.\) We know that \(z=\sqrt{6}\) and \(ρ=2\sqrt{2}\), so

The point with cylindrical coordinates \((4,\dfrac{2π}{3},−2)\) has rectangular coordinates \((−2,2\sqrt{3},−2)\) (Figure \(\PageIndex{5}\)).

The point with rectangular coordinates \((1,−3,5)\) has cylindrical coordinates approximately equal to \((\sqrt{10},5.03,5).\)

A dispersive prism is an optical element used to break up light into its different wavelength components – a phenomenon discovered by Sir Isaac Newton. By doing this, the prism separates light of varying wavelengths, with longer wavelengths (red) deflecting at a lesser angle than the shorter ones (violet). This phenomenon occurs because the prism’s refractive index varies by the wavelengths of the light.

The radius of Earth is \(4000\)mi, so \(ρ=4000\). The intersection of the prime meridian and the equator lies on the positive \(x\)-axis. Movement to the west is then described with negative angle measures, which shows that \(θ=−83°\), Because Columbus lies \(40°\) north of the equator, it lies \(50°\) south of the North Pole, so \(φ=50°\). In spherical coordinates, Columbus lies at point \((4000,−83°,50°).\)

Right angle prisms are usually used to deviate the direction of light by 90-degrees. It’s possible to use a right angle prism in a Porro configuration when light is incident through the prism’s hypotenuse.  Light will be deflected 180-degrees and flipped.

\[ \begin{align*} ρ^2 &=x^2+y^2+z^2=(−1)^2+1^2+(\sqrt{6})^2=8 \\[4pt] \tan θ &=\dfrac{1}{−1} \\[4pt] ρ&=2\sqrt{2} \text{ and }θ=\arctan(−1)=\dfrac{3π}{4}. \end{align*}\]

Dispersionof light

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Let the center of Earth be the center of the sphere, with the ray from the center through the North Pole representing the positive \(z\)-axis. The prime meridian represents the trace of the surface as it intersects the \(xz\)-plane. The equator is the trace of the sphere intersecting the \(xy\)-plane.

Use the second set of equations from Conversion between Cylindrical and Cartesian Coordinates to translate from rectangular to cylindrical coordinates:

The manufacturing of prisms usually involves several steps.  Starting with the chosen glass, a series of cuts are done to form a basic prism shape. This stage usually ends in a rough draft of the final product.  The prism will have the shape requested but of poor optical performance.

\[ \begin{align*} x &=ρ\sin φ\cos θ \\[4pt] &=8 \sin\left(\dfrac{π}{6}\right) \cos\left(\dfrac{π}{3}\right) \\[4pt] &= 8\left(\dfrac{1}{2}\right)\dfrac{1}{2} \\[4pt] &=2 \\[4pt] y &=ρ\sin φ\sin θ \\[4pt] &= 8\sin\left(\dfrac{π}{6}\right)\sin\left(\dfrac{π}{3}\right) \\[4pt] &= 8\left(\dfrac{1}{2}\right)\dfrac{\sqrt{3}}{2} \\[4pt] &= 2\sqrt{3} \\[4pt] z &=ρ\cos φ \\[4pt] &= 8\cos\left(\dfrac{π}{6}\right) \\[4pt] &= 8\left(\dfrac{\sqrt{3}}{2}\right) \\[4pt] &= 4\sqrt{3} \end{align*}\]

An interesting application of prisms is the change of the incident beam dimensions.  This is caused exclusively by the geometry of the prism (e.g. the angle of the incident vs refracted faces), and not the focusing elements or collimating effects like in a lens.  Anamorphic lenses are usually configured in pairs to keep the beam traveling along the optical axis.

c. To describe the surface defined by equation \(z=r\), is it useful to examine traces parallel to the \(xy\)-plane. For example, the trace in plane \(z=1\) is circle \(r=1\), the trace in plane \(z=3\) is circle \(r=3\), and so on. Each trace is a circle. As the value of \(z\) increases, the radius of the circle also increases. The resulting surface is a cone (Figure \(\PageIndex{8}\)).

Materialdispersion

In the cylindrical coordinate system, a point in space (Figure \(\PageIndex{1}\)) is represented by the ordered triple \((r,θ,z)\), where

Waveguide dispersion

a. When the angle \(θ\) is held constant while \(r\) and \(z\) are allowed to vary, the result is a half-plane (Figure \(\PageIndex{6}\)).

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

One of the most common images for any that has study optics is that of Sir Isaac Newton with a beam of white light going through a glass prism and a rainbow coming out on the other side.  It is one of the most famous optical experiments, not only for its simplicity, but also because it helped Newton set the foundations for his corpuscular theory of light.

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The design of a dispersive prism is critical to its functionality and can significantly impact the quality of the light separation. This article will discuss essential parameters for designing a dispersive prism and provide guidelines for creating an effective and efficient device.

Spherical coordinates are useful in analyzing systems that have some degree of symmetry about a point, such as the volume of the space inside a domed stadium or wind speeds in a planet’s atmosphere. A sphere that has Cartesian equation \(x^2+y^2+z^2=c^2\) has the simple equation \(ρ=c\) in spherical coordinates.

Notice that these equations are derived from properties of right triangles. To make this easy to see, consider point \(P\) in the \(xy\)-plane with rectangular coordinates \((x,y,0)\) and with cylindrical coordinates \((r,θ,0)\), as shown in Figure \(\PageIndex{2}\).

Reflective prisms can be used in imaging systems.  Due to the total internal reflection, light entering the prism can undergo multiple reflections until they reach an output face.  It is possible to add a reflective surface so the prism behaves as a beam splitter.  Reflective prisms are used to reduce the physical size of an optical system, to redirect the direction of light, and to reform the orientation of an image.

When we expanded the traditional Cartesian coordinate system from two dimensions to three, we simply added a new axis to model the third dimension. Starting with polar coordinates, we can follow this same process to create a new three-dimensional coordinate system, called the cylindrical coordinate system. In this way, cylindrical coordinates provide a natural extension of polar coordinates to three dimensions.

The use of cylindrical coordinates is common in fields such as physics. Physicists studying electrical charges and the capacitors used to store these charges have discovered that these systems sometimes have a cylindrical symmetry. These systems have complicated modeling equations in the Cartesian coordinate system, which make them difficult to describe and analyze. The equations can often be expressed in more simple terms using cylindrical coordinates. For example, the cylinder described by equation \(x^2+y^2=25\) in the Cartesian system can be represented by cylindrical equation \(r=5\).

Right-angle roof prisms are usually used in binoculars or when a right angle deflection of an image is required. The image is deflected left-to-right not top-to-bottom.

Plot the point with spherical coordinates \((8,\dfrac{π}{3},\dfrac{π}{6})\) and express its location in both rectangular and cylindrical coordinates.

a. The variable \(θ\) represents the measure of the same angle in both the cylindrical and spherical coordinate systems. Points with coordinates \((ρ,\dfrac{π}{3},φ)\) lie on the plane that forms angle \(θ=\dfrac{π}{3}\) with the positive \(x\)-axis. Because \(ρ>0\), the surface described by equation \(θ=\dfrac{π}{3}\) is the half-plane shown in Figure \(\PageIndex{13}\).

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That small prism that Newton was using is just one of many applications that prisms have in many optical systems.  In the article we will try to describe the four  types of prisms: Dispersive (like the one Newton was using), Reflective, rotation and displacements. As well as some consideration on the manufacturing of prisms.

\[ \begin{align*} r&=ρ \sin φ \\[4pt] &= 8\sin \dfrac{π}{6} \\[4pt] &=4 \\[4pt] θ&=θ \\[4pt] z&=ρ\cos φ\\[4pt] &= 8\cos\dfrac{π}{6} \\[4pt] &= 4\sqrt{3} .\end{align*}\]

Dispersionof lightthrough prism

We choose the positive square root, so \(r=\sqrt{10}\).Now, we apply the formula to find \(θ\). In this case, \(y\) is negative and \(x\) is positive, which means we must select the value of \(θ\) between \(\dfrac{3π}{2}\) and \(2π\):

\[\begin{align*} x &=r\cos θ=4\cos\dfrac{2π}{3}=−2 \\[4pt] y &=r\sin θ=4\sin \dfrac{2π}{3}=2\sqrt{3} \\[4pt] z &=−2 \end{align*}. \nonumber \]

A technician is in charge of supervising and evaluating each stage.   Some prism geometries can be bought off-the-shelf but for specific applications or custom made optics, it usually requires a considerable amount of time for testing and manufacturing.

Rectangular coordinates \((x,y,z)\), cylindrical coordinates \((r,θ,z),\) and spherical coordinates \((ρ,θ,φ)\) of a point are related as follows:

\[ \begin{align*} \dfrac{5π}{6} &=\arccos\left(\dfrac{z}{\sqrt{x^2+y^2+z^2}}\right) \\[4pt] \cos\dfrac{5π}{6}&=\dfrac{z}{\sqrt{x^2+y^2+z^2}} \\[4pt] −\dfrac{\sqrt{3}}{2}&=\dfrac{z}{\sqrt{x^2+y^2+z^2}} \\[4pt] \dfrac{3}{4} &=\dfrac{z^2}{x^2+y^2+z^2} \\[4pt] \dfrac{3x^2}{4}+\dfrac{3y^2}{4}+\dfrac{3z^2}{4} &=z^2 \\[4pt] \dfrac{3x^2}{4}+\dfrac{3y^2}{4}−\dfrac{z^2}{4} &=0. \end{align*}\]

In the Cartesian coordinate system, the location of a point in space is described using an ordered triple in which each coordinate represents a distance. In the cylindrical coordinate system, the location of a point in space is described using two distances \((r\) and \(z)\) and an angle measure \((θ)\). In the spherical coordinate system, we again use an ordered triple to describe the location of a point in space. In this case, the triple describes one distance and two angles. Spherical coordinates make it simple to describe a sphere, just as cylindrical coordinates make it easy to describe a cylinder. Grid lines for spherical coordinates are based on angle measures, like those for polar coordinates.

This page titled 12.7: Cylindrical and Spherical Coordinates is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Gilbert Strang & Edwin “Jed” Herman (OpenStax) via source content that was edited to the style and standards of the LibreTexts platform.

The Cartesian coordinate system provides a straightforward way to describe the location of points in space. Some surfaces, however, can be difficult to model with equations based on the Cartesian system. This is a familiar problem; recall that in two dimensions, polar coordinates often provide a useful alternative system for describing the location of a point in the plane, particularly in cases involving circles. In this section, we look at two different ways of describing the location of points in space, both of them based on extensions of polar coordinates. As the name suggests, cylindrical coordinates are useful for dealing with problems involving cylinders, such as calculating the volume of a round water tank or the amount of oil flowing through a pipe. Similarly, spherical coordinates are useful for dealing with problems involving spheres, such as finding the volume of domed structures.

Spherical coordinates with the origin located at the center of the earth, the \(z\)-axis aligned with the North Pole, and the \(x\)-axis aligned with the prime meridian

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What type of optical material will disperse lightin physics

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Let’s consider the differences between rectangular and cylindrical coordinates by looking at the surfaces generated when each of the coordinates is held constant. If \(c\) is a constant, then in rectangular coordinates, surfaces of the form \(x=c, y=c,\) or \(z=c\) are all planes. Planes of these forms are parallel to the \(yz\)-plane, the \(xz\)-plane, and the \(xy\)-plane, respectively. When we convert to cylindrical coordinates, the \(z\)-coordinate does not change. Therefore, in cylindrical coordinates, surfaces of the form \(z=c\) are planes parallel to the \(xy\)-plane. Now, let’s think about surfaces of the form \(r=c\). The points on these surfaces are at a fixed distance from the \(z\)-axis. In other words, these surfaces are vertical circular cylinders. Last, what about \(θ=c\)? The points on a surface of the form \(θ=c\) are at a fixed angle from the \(x\)-axis, which gives us a half-plane that starts at the \(z\)-axis (Figures \(\PageIndex{3}\) and \(\PageIndex{4}\)).

Point \(R\) has cylindrical coordinates \((5,\frac{π}{6},4)\). Plot \(R\) and describe its location in space using rectangular, or Cartesian, coordinates.

This set of points forms a half plane. The angle between the half plane and the positive \(x\)-axis is \(θ=\dfrac{2π}{3}.\)