Group Index dependence on refractive index

In summary, the group index, Ng, is a function of wavelength, λ, which means that the index of refraction, n, used in the formula may also vary with λ. If n does not change significantly over the range of wavelengths, then Ng is approximately equal to n. However, if the derivative dn/dλ is not small, then the choice of n becomes crucial and the concept of group index may be redundant with the index of refraction. This is because the group index refers to a broad wavepacket, and if n varies greatly with λ, it may be difficult to define a group velocity.
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In Ng (the group index for a range of wavelengths), there is an index of refraction n used, but if the medium is dispersive, meaning n is a function of wavelength λ, which n is used? Is it some kind of an average? Or does n not change much over this range of wavelengths? if it doesn't change much, then

Ng = n - λ(dn/dλ)

would be approximately

Ng ~ n

So, it makes sense to me that dn/dλ must not be really small, because otherwise Ng ~ n and the concept of group index is redundant with the concept of the index of refraction. but if dn/dλ is not really small, then the choice of n (the first term in Ng) becomes crucial, does it not?
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  • #2
##N_g## is a function of ##\lambda## too: ##N_g(\lambda)##, so you should use the respective value ##n(\lambda)## in the formulas you gave.
Take in mind that the group index refers to a wavepacket which is very broad, so that the spread of ##\lambda## in Fourier space is arbitrary small. If n depends strongly on ##\lambda##, the wavepacket will deform while traveling through the medium and it will be difficult to define a group velocity, or, you have to regard it as a superposition of broader wavepackets which travel with slightly different group velocities.
 
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Thanks! Makes total sense!
 

1. What is the relationship between group index and refractive index?

The group index is a measure of how fast light travels through a medium compared to its speed in a vacuum. It is dependent on the refractive index of the medium, which is a measure of how much the medium bends light. Generally, the higher the refractive index, the higher the group index.

2. How does the group index affect the propagation of light in a medium?

The group index affects the speed at which light travels through a medium, which in turn affects the amount of time it takes for light to propagate through the medium. This can impact the overall transmission of light through the medium and can also cause dispersion, where different wavelengths of light travel at different speeds.

3. What factors can influence the group index of a medium?

The group index is primarily influenced by the material properties of the medium, specifically the refractive index. However, other factors such as temperature, pressure, and the wavelength of light can also impact the group index. In some cases, the group index can also be altered through the use of external fields, such as electric or magnetic fields.

4. How is the group index of a medium measured?

The group index can be measured using a variety of techniques, such as interferometry or time-of-flight measurements. These methods involve sending a light pulse through the medium and measuring the time it takes for the pulse to propagate through a known distance. By comparing this time to the time it would take for light to travel through a vacuum, the group index can be calculated.

5. What are the practical applications of understanding the group index dependence on refractive index?

Understanding the relationship between group index and refractive index is important in various fields, including telecommunications, optics, and materials science. It can help in the design and development of optical devices, such as lenses and fibers, and also aid in the characterization of materials with specific optical properties. Additionally, a deeper understanding of this relationship can lead to advancements in technologies such as fiber optics communication and imaging systems.

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