How Do You Calculate Phase and Group Velocity from Refractive Index Data?

In summary, the conversation discusses using a graph of n(lambda) vs lambda to estimate Vphase and group velocity in a dispersive medium with an index of refraction of 1.45 at a wavelength of 1000. The method involves finding the phase velocity using n = c/vphase = ck/omega and then using the derivative of omega with respect to k to find the group velocity. The group velocity is determined by vgroup = vphase(1 + lambda/n * (dn/dlambda)). The derivative can be estimated by finding the
  • #1
stunner5000pt
1,461
2
given is a graph of n(lambda) vs lambda where n is the index of refraction
N(1000) =1.45

Estimate Vphase and group velocity using the above info.

i know that [tex] n = \frac{c}{v_{\phi}} = \frac{ck}{\omega} [/tex]

i can't simply susbtitute into that above relation because the lambda given is that lambda in the vacuum and thus the lambda would be different in the medium. Also oncei found v phase how would i go about finding group velcoity, since group velocity is a derivative, do i simply pick off two points on the graph and find the slope?

But first i need to figure out phase velocity which i can't get and i don't know if its right

Please help!
 
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  • #2
i know that [tex] V_{\phi} = \frac{\omega}{k} [/tex]
where omega is the angular velocity of the wave and is 2 pi / T
and k is the wave number 2pi / lambda
however since it is in another medium with index of refraction 1.45 would the lambda be different??

would i have to use the following relation to figure out the lambda in that respective medium isnce the wavelength given is that wavelength in a vacuum (right?)
[tex] \frac{n_{1}}{n_{2}} = \frac{v_{2}}{v_{1}} = \frac{\lambda_{2}}{n_{1}} = \frac{f_{2}}{f_{1}} [/tex]
and since in a vacuum n1 = 1 then i can find the lambda 2 the lambda in the medium?

then i back substitute in the above relation and figure out the frequency as well.
im not quite sure where this leads to... i am going off on a tangent
im given n (wvelegnth) not n(angular frequency) which isn eeded to find the phase velocity
 
  • #3
stunner5000pt said:
given is a graph of n(lambda) vs lambda where n is the index of refraction
N(1000) =1.45

Estimate Vphase and group velocity using the above info.

i know that [tex] n = \frac{c}{v_{\phi}} = \frac{ck}{\omega} [/tex]

i can't simply susbtitute into that above relation because the lambda given is that lambda in the vacuum and thus the lambda would be different in the medium. Also oncei found v phase how would i go about finding group velcoity, since group velocity is a derivative, do i simply pick off two points on the graph and find the slope?

But first i need to figure out phase velocity which i can't get and i don't know if its right

Please help!
SOLUTION HINTS:
Problem presents {n = n(λ)} for a given dispersive medium.
Continue in direction you've begun to determine "vphase" and "vgroup" in the dispersive medium:

[tex] 1: \ \ \ \ \ v_{phase} \ \ = \ \ \frac{c}{n} \ \ = \ \ \frac{\omega}{k} [/tex]

[tex] 2: \ \ \ \ \ \Longrightarrow \ \ \ \omega \ \ = \ \ \frac{ck}{n} [/tex]

[tex] 3: \ \ \ \ \ \ \ \ \Longrightarrow \ \ \ \frac{d\omega}{dk} \ \ = \ \ \frac{c}{n} \ \ - \ \ \frac{ck}{n^{2}}\left(\frac{dn}{dk}\right) \ \ = \ \ \frac{c}{n}\left(1 \ - \ \frac{k}{n}\left(\frac{dn}{dk}\right) \right) [/tex]

[tex] 4: \ \ \ \ \ \ \ \Longrightarrow \ \ \ \mathbf{v_{group}} \ \ = \ \ \mathbf{v_{phase} }\left ( 1 \ - \ \frac{k}{n} \left ( \frac{dn}{dk} \right ) \right ) \ \ = \ \ \mathbf{v_{phase} }\left ( 1 \ + \ \frac{\lambda}{n} \color{red} \left ( \frac{dn}{d\lambda} \right ) \color{black} \right ) [/tex]

Determine relationship between "vphase" and "vgroup" in the dispersive medium from Eq #4, where derivative in red is evaluated from the given function {n = n(λ)}.



~~
 
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  • #4
so for number 4 the derivative is calculated by using two points and finding the slope of the graph (it did ask for an estimate, after all) ?

is that the right way?
 
  • #5
stunner5000pt said:
so for number 4 the derivative is calculated by using two points and finding the slope of the graph (it did ask for an estimate, after all) ?

is that the right way?
Without seeing the graph, it's difficult to judge. However, your method would be reasonable if the graph were approx linear. Incidentally, you should find that {(dn/dλ) < 0}


~~
 
Last edited:

Related to How Do You Calculate Phase and Group Velocity from Refractive Index Data?

What is the index of refraction Vphase?

The index of refraction Vphase is a measure of how much light bends as it passes through a material. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the material.

How is the index of refraction Vphase measured?

The index of refraction Vphase is typically measured using a device called a refractometer. This instrument measures the angle at which light is bent as it passes through a material and uses this information to calculate the index of refraction.

What factors affect the index of refraction Vphase?

The index of refraction Vphase is affected by various factors, including the type of material, its density, and its temperature. Generally, the denser the material and the lower its temperature, the higher the index of refraction will be.

What are some applications of the index of refraction Vphase?

The index of refraction Vphase has a wide range of applications in various fields. It is essential in optics and the design of lenses, as well as in the study of materials and their properties. It is also used in industries such as telecommunications, medicine, and manufacturing.

How does the index of refraction Vphase relate to the speed of light?

The index of refraction Vphase and the speed of light are inversely related. As the speed of light decreases in a material, the index of refraction increases. This is because light travels slower in denser materials, leading to a higher index of refraction.

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