Need some help regarding phase and group velocity of a wave

In summary, the conversation discusses the relationship between group velocity and phase velocity in the context of working with lasers and fiber optics. The formula for calculating group velocity is v_g = v_p * sin(A), where v_g is the group velocity, v_p is the phase velocity, and A is the incident angle. The concept is illustrated with a picture and it is clarified that group velocity refers to the velocity of energy flow along the fiber direction.
  • #1
ehj
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I'm currently working with lasers and in relation to that, fiber optics. In the book I'm reading there's a section regarding modes and their "group velocity". The text claims that there is the following relation:
[tex] v_g=v_p * sin(A) [/tex]
where [tex] v_g [/tex] is the group velocity and
where [tex] v_p [/tex] is the phase velocity and A is the incident angel as showed on the picture, although denoted as theta there.
From the picture you can see that the relation is reached by using trigonometry, and [tex] v_g [/tex] is in this case simply the velocity in the direction along the fiber. My question is, if this is the general idea of group velocity? I thought group velocity was the velocity of the "amplitude wave" in a wave packet? Am I wrong, or are there just different meanings of group velocity?
 
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  • #2
Here is the picture!
 

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  • #3
v_g found that way would be the velocity of the envelope of a wave packet.
It is the velocity of energy flow along the fiber direction.
 

What is the difference between phase velocity and group velocity?

Phase velocity refers to the speed at which the phase of a single frequency component of a wave travels. Group velocity, on the other hand, refers to the speed at which energy is transmitted by a wave. In other words, phase velocity describes how quickly the wave appears to be moving, while group velocity describes how quickly the wave carries energy.

How are phase velocity and group velocity related?

Phase velocity and group velocity are related through the dispersion relation, which is a mathematical relationship between the frequency, wavelength, and wave speed. In general, the group velocity is equal to the phase velocity only in certain cases, such as in a vacuum or in a medium with no dispersion.

What factors affect the phase and group velocity of a wave?

The phase and group velocity of a wave can be affected by various factors, including the medium through which the wave is traveling, the frequency of the wave, and the presence of any obstacles or boundaries. For example, in a medium with a higher refractive index, the phase velocity of a wave will be lower.

How can we measure the phase and group velocity of a wave?

The phase velocity of a wave can be measured by measuring the wavelength and frequency of the wave and then calculating the ratio between them. The group velocity can be measured by observing the movement of the wave's energy or by using specialized equipment such as interferometers.

Why is understanding phase and group velocity important in the study of waves?

Phase and group velocity are important concepts in the study of waves because they help us understand how energy is transmitted and how waves behave in different environments. They also play a crucial role in many practical applications, such as in telecommunications and signal processing.

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