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What is "phase velocity"
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Problem description:

5. Phase velocity

The phase velocity of light in medium can exceed the speed of light in vacuum in some frequency bands. Phase velocity refers to the "propagation velocity" corresponding to the phase lag of a continuous sine wave (assuming that the signal propagates for a long time and reaches a stable state) after propagating in a medium for a certain distance. Obviously, a simple sine wave cannot convey information. In order to transmit information, it is necessary to modulate the slowly varying wave packet on the sine wave. The propagation speed of this wave packet is called group velocity, which is less than the speed of light. (Translator's Note: Sommerfeld and Brillouin's research on pulse propagation in the medium proves that the propagation speed of a signal with an initial time of [zero before a certain moment] in the medium cannot exceed the speed of light. )

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Analysis:

Group velocity refers to the envelope propagation velocity of electromagnetic waves. Is actually the actual speed of electromagnetic waves.

Phase velocity is the speed at which electromagnetic waves propagate. Generally speaking, it is the speed at which the electromagnetic wave shape changes forward. In waveguides, the phase velocity is usually greater than the group velocity.

Figuratively speaking, if you use an electric drill to drill a hole in a solid wall, you will feel that the thread of the electric drill bit seems to advance at a high speed when rotating, but this is only your illusion, because what you see is the "phase speed" of the thread. Although it is very fast, your electric drill slowly pushes into the wall, which means that the overall speed of the electric drill is "group speed". If the wall is hard, your electric drill can't get in at all. The speed of the electric drill is "0", but you always feel that the electric drill keeps getting in through the thread of the electric drill.

Group velocity and phase velocity

When a radio wave propagates in a medium, if the dielectric constant ε of the medium has nothing to do with the frequency, the propagation speed of the wave has nothing to do with the frequency. This medium is called a non-dispersive medium. On the contrary, if the ε or propagation velocity v of a medium is related to frequency, it is called a dispersive medium.

The formula of monochromatic wave propagation velocity is derived from equal plane propagation, so it is called phase velocity.

Phase velocity: the velocity v=c/n of the isophase plane (such as the wave crest surface or the wave trough surface) where c is the speed of light in free space and n is the refractive index of the medium to the electromagnetic wave of this frequency.

The signal of an actual system always consists of many frequency components. In dispersive media, each monochromatic component will propagate at a different phase velocity, so it is difficult to determine the propagation velocity of the signal in dispersive media. Therefore, the concept of group velocity is introduced, which describes the energy propagation speed of signals. For the ionosphere (the earth's atmosphere is divided into troposphere, stratosphere, ionosphere and magnetosphere from bottom to top), because the refractive index n < 1, the phase velocity of radio waves is greater than the speed of light c, which is not contradictory to the theory of relativity, because the phase velocity only represents the speed of phase transition and does not represent the real propagation speed of electromagnetic wave energy. The group velocity in free space is always less than the speed of light c.

Group velocity: the speed at which the synthetic signal of many sinusoidal electromagnetic waves with different frequencies propagates in the medium. The amplitude and phase of sine waves with different frequencies are different, and the phase velocity in dispersive media is also different, so the shape of the synthesized signal will change in different spatial positions. Group velocity represents the propagation velocity of energy.

The following is excerpted from Advanced Optics, edited by Zhao Jianlin and published by National Defense Industry Press.

(Page 16)

Equal phase plane and phase velocity of monochromatic plane wave: the point multiplication kr of wave vector k and position coordinate vector r reflects the phase delay of electromagnetic wave in the process of spatial propagation, so the spatial point * * * where kr= constant is usually called equal phase plane. The velocity vφ at which an isoplane moves along its normal direction is called the phase velocity.

Obviously, the isosurface of plane wave is a cluster of parallel planes in space, which is orthogonal to the direction of wave vector K everywhere, so the direction of its phase velocity vφ is the same as K. 。

It can be seen that the phase velocity of plane wave is the speed of light V in wave equation, but it should be noted that only in isotropic homogeneous media can the speed of light be equal to the phase velocity.

(27 pages)

Group velocity and phase velocity:

The wave velocity v=v/n determined by the wave equation reflects the wave propagation velocity. Due to dispersion, light waves with different frequencies propagating in the same medium have different phase velocities, that is, different spectral components contained in the same optical signal cannot propagate synchronously in the dispersive medium. In this way, the problem arises. When we observe the optical signal emitted from a certain point in a space far away from the light source, the optical signals with different frequencies received at the same time are actually emitted by the light source at different times. It is now assumed that the optical signal propagating along the Z axis consists of monochromatic plane waves with two frequency components. The amplitudes and vibration directions of the two light waves are the same, and the light vibration at a certain point in space (time t) can be respectively vibrated as follows:

If △ω=(ω2-ω 1)/2 and △k=(k2-k 1)/2,

ω0=(ω2+ω 1)/2 and k0=(k2+k 1)/2, which respectively represent the circular frequency, wave number difference, average circular frequency and average wave number of two monochromatic light waves.

Visible vibration is a polychromatic plane wave modulated by △ω low frequency, with an average frequency of ω0. When the plane wave propagates at the phase speed ω0/ k0, the modulated wave also propagates at the speed△ω△ω/△ kK ... This speed reflects the propagation speed of light wave energy, so it is called the group velocity of light wave in dispersive media. And expressed as vg. To show this difference, the phase velocity is usually expressed by vP. Obviously, when the frequency difference △ω is very small, the group velocity is actually the derivative of the time circle frequency to the space circle frequency (wave number).

It can be seen from equation (1) and equation (2) that in dispersive media, the group velocity is not equal to the phase velocity (dvp/dλ≠0, vg≠vp), and it is in the normal dispersion region.

(dvp/dλ > 0, dn/dλ < 0), and the group velocity is less than the phase velocity (VG < VP); In the anomalous dispersion region (dvp/dλ < 0, dn/dλ > 0), the group velocity is greater than the phase velocity (VG > VP). Only in a dispersion-free medium or vacuum (dvp/dλ=0, dn/d λ=0), the group velocity is equal to the phase velocity (vg=vp).

The following content is taken from Electromagnetic Fields and Waves, edited by Feng Enxin 1999 1 Edition, page 142.

According to the trajectory of the invariant point of electromagnetic wave propagation in space, the signal speed, that is, the phase speed, can be calculated. In an ideal medium, the phase velocity of electromagnetic wave is only related to the medium parameters.

The following is excerpted from Radio Communication edited by Gao Jianping Zhang Zhixian published by Northwestern Polytechnical University Press, page 65.

The propagation characteristics of single frequency sinusoidal uniform plane electromagnetic wave in space (medium or conductor) are studied. The results show that the phase velocity of wave has nothing to do with frequency and is equal to the propagation velocity of energy in the medium. In a conductor, the phase velocity of a wave is related to the frequency.

In communication system, in order to transmit information, single frequency SUPW (called carrier) must be modulated in a certain way. The modulated wave (containing multiple frequency components) carries information to be transmitted to the receiving end through the information channel.

To illustrate the problem, suppose that there are two groups of single-frequency SUPW propagating along the direction, with equal amplitude and different frequencies, and their angular frequencies are respectively