Control number of FFT modes in longitudinal space charge solver

[austin-hoover]

This PR gives the option to restrict the number of modes used in longitudinal space charge kicks. From Chao (3.7), the formula for the energy gain from longitudinal impedance $Z_{0}^{\parallel}(\omega)$ for particle at position $z$ is

$$ V(z) = -\int_{z}^{\infty} \rho(z') W_0'(z - z') dz'. $$

In the frequency domain (3.10):

$$ V(z) = -\frac{1}{2 \pi} \int_{-\infty}^{\infty} e^{i \omega z / c} Z_0^{\parallel}(\omega) \tilde{\rho}(\omega) d\omega. $$

From the FFT, the solver uses $N / 2$ frequencies/modes, where $N$ is the number of longitudinal bins.

PyORBIT3/src/spacecharge/LSpaceChargeCalc.cc

Lines 211 to 227 in e4ac4ce

	double LSpaceChargeCalc::_kick(double angle)
	{
	// n=0 term has no impact (constant in phi)
	// f(phi) = _FFTMagnitude(1) + sum (n = 2 -> N / 2) of
	// [2 * _FFTMagnitude(i) * cos(phi * (n - 1) +
	// _FFTPhase(n) + _chi(n))]
	double kick = 0.;
	double cosArg;
	for(int n = 1; n < nBins / 2; n++)
	{
	cosArg = n * (angle + OrbitConst::PI) + _fftphase[n] + _chi[n];
	kick += 2 * _fftmagnitude[n] * _z[n] * cos(cosArg);
	}
	return kick;
	}

This PR lets the user restrict the number of modes used in the energy kick, acting as a low-pass filter.

[austin-hoover]

Example with decreasing number of modes:

rotation_fft_modes.mov

[austin-hoover]

Wait, that formula is the net energy loss of the beam. The space charge impedance is purely imaginary and doesn't contribute. Closing this as I need to review the formula used for the energy kick.

[austin-hoover]

The correct expression is Chao Eq. (7), which gives the voltage for a particle at position $z$ in CGS units:

$$ V(z) = -\int_{z}^{\infty} \rho(z') W_0'(z - z') dz'. $$

In the frequency domain (3.10):

$$ V(z) = -\frac{1}{2 \pi} \int_{\infty}^{\infty} e^{i \omega z / c} Z_0^{\parallel}(\omega) \tilde{\rho}(\omega) d\omega. $$

[austin-hoover]

Updated formula in original comment.