HOME 1. Electric field vector 2. Mathematical expression 3. The nature of light 4. Polarizer 5. Late child 6. Intensity and polarization 7. literature
What is polarized light (light intensity and polarization)?
I will explain the polarization of light. This is difficult for high school students to understand, but it is deeply related to the fact that light is a transverse wave.
1. Electric field vector of light (electromagnetic wave)
(1) Direction of electric field vector
The electric field vector determines the properties of light. Light traveling in one direction (electromagnetic waves) Separate article "Propagation of electromagnetic waves" 2. (2) and “Emission of electromagnetic waves by linear oscillators (electric dipoles)” 2. As explained in (3), changes in the electric field vector E, which is perpendicular to the direction of travel, proceed as waves.
At that time, a magnetic field vector exists in relation to the electric field vector, and its direction is 90 degrees to the right of the electric field vector in a plane perpendicular to the direction of travel, with the direction of rotation as the axis of rotation. Furthermore, as explained in a separate paper, the ratio of the magnitudes of the electric field vector and the magnetic field vector always takes a fixed value. This is one expression of the properties of electromagnetic fields suggested by Maxwell's equations in electromagnetism.
Since the magnetic field vector always has a certain fixed relationship with the electric field vector, the electric field vector can be used as a representative when discussing the properties of light (electromagnetic waves). In the future, we will discuss its properties in terms of electric field vectors.
The electric field vector is a type of vector, so even if the electric field vectors in various directions are mixed at a certain point at a certain moment of the traveling light (electromagnetic wave), they are combined vectorially and end up perpendicular to the direction of travel. The electric field vector is concentrated in one direction within the plane. In other words, the electric field vector in any of the diagrams below is a vector of a certain size that points in only one direction perpendicular to the direction of travel, once the time and location are determined. This is why light is said to be a transverse wave.
However, we are now assuming that the light is heading in the positive direction of the z-axis, and the above figure shows the electric field vector at a certain moment at a certain position when looking at light (electromagnetic waves) coming from the positive direction of the z-axis. ing. At that time, the electric field vector must be in a plane perpendicular to the z-axis (direction of travel), but it can point in any direction within that plane.
At that time, for example, when you stop time and observe the spatial changes in the electric field vector as shown in the figure below, you may change its direction so that it rotates around in places.
Furthermore, although the length of the electric field vector E is drawn in the same way in the diagram above, the magnitude (amplitude) of the electric field vector may vary randomly over time and location. Also, in the figure, the electric field vector at a certain moment changes in a spiral with the same pitch depending on the location, but the pitch of the spiral may change randomly depending on the location and time. Of course, the direction of rotation of the spiral may be reversed, or the direction may change depending on location or time. The hit can be changed arbitrarily. In any case, all of these electromagnetic waves are transverse waves that satisfy Maxwell's electromagnetic field equations.
In fact, there is a separate article entitled "Emission of electromagnetic waves by linear oscillators (electric dipoles)" 2. As you will understand if you read (3), if you appropriately adjust the motion of the charges that emit electromagnetic waves (the charges that make up the Hertzian vector, which is an electric dipole), the magnitude of the electric field vector can be changed to any waveform. It can generate electromagnetic waves that change, and by appropriately rotating and changing the direction of vibration of the Hertz vector, it is possible to emit light (electromagnetic waves) that rotates and changes the direction of the electric field in any direction in a plane perpendicular to the direction of travel.
The figure only depicts the electric field vector at a certain moment on the z-axis, but in reality, vectors in the same direction and the same magnitude are spatially distributed. Please read it that way.
(2) Linearly polarized light, circularly polarized light, elliptically polarized light
As emphasized in the previous section, the electric field vector points in only one direction, but if you observe the electric field vector passing through it at a certain location, its magnitude also changes in that direction (in the plane perpendicular to the direction of travel). direction) can also be changed arbitrarily.
However, such general things are difficult to handle, so here we will discuss the case where a plane wave traveling in a straight line vibrates harmonically in time (in other words, sin and cos). Various situations can be considered even for this type of harmonic vibration.
1. linear polarized light
This is the case as shown in Figure 1-1 in the previous section. This is because the electric field vector always repeats harmonic oscillation in the same direction when viewed from the direction in which the light (electromagnetic waves) is coming. In other words, it is one of the ones shown in Figure 1-2. In such a case, the light is said to be linearly polarized. However, the electric field vector in this case can also be decomposed into an x component Ex and a y component Ey. Then, we think that each of them moves forward while vibrating at the same frequency with no phase difference. In this way, the difference in the direction of the electric field vector can be expressed by the difference in the combination of the amplitudes of its x component Ex and y component Ey.
As an example, if we illustrate the state of the electric field at a certain time of linearly polarized light oscillating in the second and fourth quadrants of e or f in the figure above,
become that way. Similar decomposition and synthesis can be done in other cases as well.
At that time, if you rotate the x- and y-axis directions, the values of maximum amplitude Ex0 and Ey0 will change. However, conversely, this means that by adjusting the directions of the x and y axes, any linearly polarized light can be expressed as a combination of vibrational components with the same amplitude in two orthogonal directions.
At this time, it is meaningless to ask whether real light exists in a state where it is decomposed into x and y components, or as a composite vector of them.
This is because the x and y components can only be definitively inferred in the process of explaining the various experiments using polarizers and retardators, which will be explained in Chapters 4 and 5. No one can know whether the ingredients are separated or not.
In other words, there is no way to know in which direction the xy coordinates, which are decomposed into x and y components, are oriented with respect to the z-axis until you experiment, so it can actually be taken in any direction.
This can also be said when decomposing circularly polarized light and elliptically polarized light into their components below.
This is what I wonder most about when learning about polarized light, but the only way to understand it is that light (electromagnetic field) is like that. We will revisit this point in Chapter 3.
2. Circularly polarized light and elliptically polarized light
If the two components of the electric field (x component: Ex and y component: Ey) described in the previous section are not in phase, the combined electric field vector moves around on one ellipse. Various cases occur depending on the angle of phase difference between both vibrations.
Figure 1-6 shows the temporal changes in the electric field when light (electromagnetic waves) traveling in the positive direction of the z-axis is viewed from the positive direction of the z-axis while the location is fixed, with the phase difference changed.
Here, the amplitudes Ex0 and Ey0 of the x and y components are the same, but of course they may be different. However, as we will prove later, any elliptically polarized light can be created by combining the x component: Ex and the y component: Ey such that the amplitude always satisfies Ex0 = Ey0 by appropriately adjusting the direction of the (x, y) coordinates. Since it can be expressed, such generalization is not very meaningful.
In case g in Figure 1-6, the spatial distribution of the electric field when time is stopped is
becomes. We can say that the light of g moves parallel to the positive direction of the z-axis over time while the electric field maintains this spatial distribution. At this time, the direction in which the tip of the electric field vector rotates may be reversed, which is the case c in Figure 1-6.
At this time, the method of decomposition changes depending on how the x and y coordinates are taken. Regarding the relationship with changes in the slope of coordinates, see 2. This will be explained in section (1).
(3) Polarization plane and rotation direction
1. Polarization plane of linearly polarized light
In this article, the direction in which the electric field vector oscillates will be referred to as the polarization direction, or the plane that includes the electric field vector and the direction of travel will be referred to as the plane of polarization.
If we pay attention to the physical effects of vectors that represent fields, we can say that it is reasonable to adopt E as a vector that represents light, since the electric field is the direction in which charges are swayed.
Of course, the magnetic field vector also exerts a force on the moving charge, but its effect is small compared to the electric field vector. A magnetic field exerts a force (Lorentz force F=q[v×B]=μ0q[v×H]) only when the charge is in motion. In the MKSA unit system, B is 1/c times E, and v/c is usually a small value compared to 1, so this effect can usually be ignored. [Of course, even if the unit system changes, the actual circumstances do not change. Separate paper “Light pressure [radiation pressure]” 1. Please see "Why the unit system of electromagnetism is difficult". ]
In that sense, it is appropriate to define the polarization direction by the vibration direction of the electric field vector.
However, historically, the direction in which the magnetic field vector oscillates has been called the polarization direction (polarization plane). For example, Planck's theory of thermal radiation follows this definition. Therefore, when reading old literature, pay attention to which meaning the plane of polarization is used. If you confuse the definition of the plane of polarization, you will not be able to understand the meaning.
2. Clockwise and counterclockwise circularly polarized light and elliptically polarized light
In classical optics books, the left and right sides of circularly polarized light are determined as follows. When incoming light (electromagnetic waves) is viewed from a fixed point, if the electric field vector passing through that fixed point rotates counterclockwise over time, it is called left-handed circular (elliptical) polarization, and if it rotates clockwise, it is called left-handed circular (elliptical) polarization. This is called right-handed circular (elliptical) polarization. With this definition, the light in Figure 1-7 is left-handed circularly polarized light. In classical optics, left and right circularly polarized light is determined as shown in Figure 1-8 below.
This corresponds to the direction of rotation of the electric field vector when viewed from the perspective of observing the incoming light.
However, modern elementary particle theory (quantum theory) considers light to be an aggregate of light quanta. If we consider the case where it hits an electron that is stationary in front of it while traveling along with the light quantum, in classical optics, left-handed circularly polarized light can apply a clockwise electric field to the stationary electron when viewed in the direction of travel. It turns out. In other words, the electrons are induced to move clockwise when viewed in the direction in which the light travels. Therefore, when viewed from the perspective of a traveling photon, it is easier to understand that this case is called right-handed circularly polarized light. In other words, in modern physics based on the quantum theory of light, right-handed circularly polarized light and left-handed circularly polarized light are determined as shown in Figure 1-9 below.
This corresponds to the direction of rotation (spin) when viewed from the perspective of a traveling light quantum.
As you can see, the definitions of right-handed circularly polarized light and left-handed circularly polarized light differ depending on the book, so be careful about which meaning they are used in.
HOME 1. Electric field vector 2. Mathematical expression 3. The nature of light 4. Polarizer 5. Late child 6. Intensity and polarization 7. literature
2. Mathematical expression of harmonically oscillating light (electromagnetic waves)
Here, we will discuss the case where a plane wave traveling in a straight line vibrates harmonically (that is, sin, cos) with respect to time and space. In other words, the rectangular coordinate components of E and B are
Consider the wave shown in .
Here, τ is the changing part of the phase term
represents. See here for the meaning of these expressions. See here for an animation showing how waves propagate.
(1) Elliptically polarized light
The direction of the unit vector s indicating the propagation direction is taken as the z-axis. In that case, since light (electromagnetic waves) is a transverse wave, the electric field vector E has no z component and only x and y components. Let's examine the curve drawn by the end points of the electric field vector at a point in space.
The electric field vector E at the point of interest is generally
It can be expressed as In order to eliminate τ from the first two equations, we transform them to
is obtained. If we square these pieces and add them, we will set δy−δx=δ.
is obtained. This formula has the general form of a quadratic curve on orthogonal coordinates (x, y) on a plane, Ax2 + 2Hxy + By2 + 2Gx + 2Fy + C = 0, which was explained in the separate article "Properties of Quadratic Curves". As explained there, its shape is determined by its coefficients. That is, when AB-H2>0, it becomes an ellipse, when AB-H2<0, it becomes a hyperbola, and when AB-H2=0, it becomes a parabola. In this case
Therefore, it becomes an ellipse. Therefore, it is called elliptically polarized light (ellip
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