Undulator magnetic field expression seems wrong

[delaossa]

Hello,

I have been revising the UndulatorAtom class and have noticed that there seems to be a couple of inconsistencies:

1. The general expression for the magnetic field seems to be wrong.
   In the code this is written:

    Bx = -B0 * (kx / ky) * np.sin(kx_x) * np.sinh(ky_y) * np.cos(kz_z)
    By = B0 * np.cos(kx_x) * np.cosh(ky_y) * np.cos(kz_z)
    Bz = -B0 * (kz / ky) * np.cos(kx_x) * np.sinh(ky_y) * np.sin(kz_z)

But, from S. Reiche's thesis, Eq. (2.1), I get:

    Bx = B0 * (kx / ky) * np.sinh(kx_x) * np.sinh(ky_y) * np.cos(kz_z)
    By = B0 * np.cosh(kx_x) * np.cosh(ky_y) * np.cos(kz_z)
    Bz = -B0 * (kz / ky) * np.cosh(kx_x) * np.sinh(ky_y) * np.sin(kz_z)

See here:

ocelot/ocelot/cpbd/elements/undulator_atom.py

Line 124 in 39bda26

Bx = k1 * np.sin(kx_x) * sinhy * cosz # // here kx is only real

In this piece of code kx is always set to zero and, in this case, the two expressions yield the same result.
However, for consistency, I think that this should be fixed.

2. Depending on the end_poles parameter, a phase shift is introduced: np.cos(kz_z + ph_shift).
   However, this phase shift is only applied to the np.cos(kz_z) part in By, but not to the np.sin(kz_z) part in Bz.
   This would certainly lead to a wrong focusing effect when using the RK method.

Perhaps is easier to explain the issue through this PR #262.

[sergey-tomin]

Hi,
1. Both sets of formulas are correct. However, one should not forget that, in reality, the field in the horizontal direction should decay; for that, $k_x$ must be an imaginary number. In that case, we have $\cosh(i x) = \cos(x)$. For the $B_x$ component, it is a bit different. We have originally:

$$ B_x = B_0 \frac{k_x}{k_y} \sin(k_x x) ...., $$

which will transform to:

$$ B_x = B_0 \frac{i k_x}{k_y} \sinh(i k_x x)... = -B_0 \frac{k_x}{k_y} \sin(k_x x)... $$

As you can see, now we have the same formulas as in Ocelot. By the way, apparently I tried to provide a hint (to myself) with the comment "# // here kx is only real.". I will include an explanation in the doc string.
2. Regarding the issue with $\sin(k_z z)$, it seems you are right. This shift also has to be in the $B_z$ field. Thanks for the tip.

I see you submitted a PR, so you have a choice: either you modify your PR and remove the changes regarding point 1 and leave the changes regarding point 2, or I can do it. Please let me know.

Cheers,
Sergey.

[delaossa]

Hi Sergey,

Oh I see! I missed that...
Thanks a lot for the explanation!

About the PR, I only did it so one could see better the issue by comparing the two versions.
I think that it is better if you take care of point 2.
I am not very satisfied how the phase shift is implemented anyway...