Solve Free Electron Laser Homework Equations

[Raz91]

Homework Statement

Homework Equations

2. (b)

I didn't understand how can I find the app. wavelength

The Attempt at a Solution

I tried : the electron move over λ_u with velocity β so the wavelength is λ≈λ_u /β
but its not the right answer.

my lecturer told me I should consider also the dopler

I only know that the answer is :
λ≈λ_u /(2*γ²)

γ= gamma
THANK YOU!

 

[mfb]

I tried : the electron move over λ_u with velocity β so the wavelength is λ≈λ_u /β

This is the distance the light can travel while the electrons move by 1 structure length - but then the electrons moved forwards as well. What is the difference between those two distances?

 

[Raz91]

the electron moves λu while the light moves λ≈λu /β
so the difference is λu [1/β-1]

so how can I use this fact to find the wave length?

I tried use the approximation β≈1-1q/(2*γ^2) for ultra-relativistic electrons
but still didnt get the right answer...thank you

 

[mfb]

1/β is also 1/(1+β-1) where β-1 is small, so you can express that fraction as ~1-β+1
That should lead to the right answer.

 

[paranormal]

I would suggest first reviewing the basic principles of Free Electron Lasers (FELs) and their operating mechanisms. This will help in understanding the equations and their application in solving for the wavelength. Additionally, it is important to carefully read and understand the homework statement to ensure that all necessary information is being considered in the solution.

In this case, the equation for the wavelength of an FEL is given by λ ≈ λu/(2*γ2), where λu is the undulator period and γ is the Lorentz factor. This equation can be derived from the basic principles of FELs, which involve the interaction between electrons and an undulator.

To solve for the wavelength, you will need to know the value of the undulator period and the Lorentz factor. The Lorentz factor can be calculated using the velocity of the electrons (β) and the speed of light (c) as γ = 1/√(1-β^2/c^2). The value of β can be determined from the given information about the electron's movement over a distance of λu.

It is also important to consider the Doppler effect in the solution, as your lecturer mentioned. This effect can cause a shift in the wavelength of the emitted radiation due to the movement of the electrons. To account for this, you can use the relativistic Doppler formula, which takes into account the velocity of the electrons and the angle between the direction of motion and the direction of emission.

Overall, solving for the wavelength in this case involves understanding the basic principles of FELs, using the given equations, and considering the effects of relativistic motion and the Doppler effect. I hope this helps in your understanding and solving of the homework equations.