public abstract class CalculationEngine
extends java.lang.Object
CalculationEngine performs all the common numerical calculation when
computing machine parameters from simulation data. There are common data produced
by ring simulations and linac simulation, however, the interpretation is different.
It is up to the child classes to interpret the results of these calculations.| Constructor and Description |
|---|
CalculationEngine() |
| Modifier and Type | Method and Description |
|---|---|
protected R6 |
calculateAberration(PhaseMatrix matPhi,
double dblGamma)
Convenience function for returning the chromatic aberration coefficients.
|
protected R4 |
calculateDispersion(PhaseMatrix matPhi,
double dblGamma)
Taken from
TransferMapState. |
protected PhaseVector |
calculateFixedPoint(PhaseMatrix matPhi)
Taken from
TransferMapState. |
protected Twiss[] |
calculateMatchedTwiss(PhaseMatrix matPhiCell)
Taken from
TransferMapState. |
protected R3 |
calculatePhaseAdvance_old(PhaseMatrix matPhi,
Twiss[] twsInit,
Twiss[] twsFinal)
Deprecated.
Does not determine the phase quadrants correctly and therefore cannot produce
an accurate phase advance
|
protected R3 |
calculatePhaseAdvance(PhaseMatrix matPhi,
Twiss[] twsInit,
Twiss[] twsFinal)
Taken from
TransferMapState. |
protected R3 |
calculatePhaseAdvPerCell(PhaseMatrix matPhiCell)
Taken from
TransferMapTrajectory. |
protected R3 |
calculateTunePerCell(PhaseMatrix matPhiCell)
Taken from
TransferMapTrajectory. |
protected Twiss3D |
computeTwiss(TwissProbe probe,
PhaseMatrix matPhi,
double dW)
Deprecated.
This is the algorithm component for a
TwissProbe and should
not be used as the dynamics elsewhere. It has been moved to the
TwissTracker class. |
protected PhaseVector calculateFixedPoint(PhaseMatrix matPhi)
Taken from TransferMapState.
Calculate the fixed point solution vector representing the closed orbit at the
location of this element.
We first attempt to find the fixed point for the full six phase space coordinates.
Let Φ denote the one-turn map for a ring. The fixed point
z ∈ R6×{1} in homogeneous phase space coordinates
is that which is invariant under Φ, that is,
Φz = z .
This method returns that vector z.
Recall that the homogeneous transfer matrix Φ for the ring
has final row that represents the translation Δ of the particle
for the circuit around the ring. The 6×6 sub-matrix of Φ represents
the (linear) action of the bending magnetics and quadrupoles and corresponds to the
matrix T ∈ R6× (here T is linear).
Thus, we can write the linear operator Φ
as the augmented system
Φ = |T Δ |, z ≡ |p| ,
|0 1 | |1|
where p is the projection of z onto the ambient phase space
R6 (without homogeneous the homogeneous coordinate).
coordinates).
Putting this together we get
Φz = Tp + Δ = p ,
to which the solution is
p = -(T - I)-1Δ
assuming it exists. The question of solution existence falls upon the
resolvent R ≡ (T - I)-1 of T.
By inspection we can see that p is defined so long as the eigenvalues
of T are located away from 1.
In this case the returned value is the augmented vector
(p 1)T ∈ R6 × {1}.
When the eigenvectors do contain 1, we attempt to find the solution for the transverse phase space. That is, we take vector p ∈ R4 and T ∈ R4×4 where T = proj4×4 Φ. The returned value is then z = (p 0 0 1)T.
matPhi - the one-turn transfer matrix (full-turn matrix)protected R3 calculatePhaseAdvPerCell(PhaseMatrix matPhiCell)
Taken from TransferMapTrajectory.
Calculates the phase advance for the given transfer map assuming periodicity, that is, the given matrix represents the transfer matrix Φ of at least one cell in a periodic structure. The Courant-Snyder parameters of the machine (beam) must be invariant under the action of the transfer matrix (this indicates a periodic focusing structure where the beam envelope is modulated by that structure). One phase advance is provided for each phase plane, i.e., (σx, σz, σz). In the case the given map is the transfer map for one cell in a periodic lattice, in that case the returned value is the phase advance per cell, σcell of the periodic system.
When the given map is the full turn map of a ring then the phase advances are the betatron phases for the ring. Specifically, the sinusoidal phase that the particles advance after each completion of a ring traversal, modulo 2π (that is, we only take the fractional part).
The basic computation is
σ = cos-1[½ Tr Φαα] ,
where Φαα is the 2×2 block diagonal
of the the provided transfer matrix for the α phase plane,
and Tr Φαα indicates the trace of matrix
Φαα.
matPhiCell - transfer matrix for a cell in a periodic structureprotected R3 calculateTunePerCell(PhaseMatrix matPhiCell)
Taken from TransferMapTrajectory.
Calculates the fractional tunes for the given transfer map assuming periodicity. That is, the Courant-Snyder parameters of the machine (beam) must be invariant under the action of the transfer matrix (this indicates a periodic focusing structure where the beam envelope is modulated by that structure). One tune is provided for each phase plane, i.e., (νx, νz, νz). The given map must in the least be the transfer map for one cell in a periodic lattice. In that case the returned value is the fractional phase advance per cell, σcell/2π of the periodic system.
When the given map is the full turn map of a ring then the tunes are the betatron tunes for the ring. Specifically, the fraction of a cycle that the particles advance after each completion of a ring traversal, modulo 1 (that is, we only take the fractional part).
The basic computation is
ν = (1/2π) cos-1[½ Tr Φ] ,
where Φ is the 2×2 diagonal block of the transfer matrix
corresponding to the phase plane of interest.
The calculations are actually done by the method
calculatePhaseAdvPerCell(PhaseMatrix)
matPhiCell - transfer matrix for a cell in a periodic structureprotected R3 calculatePhaseAdvance(PhaseMatrix matPhi, Twiss[] twsInit, Twiss[] twsFinal)
Taken from TransferMapState.
Compute and return the particle phase advance for under the action of the given phase matrix when used as a transfer matrix.
Calculates the phase advances given the initial and final Courant-Snyder α and β values provided. This is the general phase advance of the particle through the transfer matrix Φ, and no special requirements are placed upon Φ. One phase advance is provided for each phase plane, i.e., (σx, σz, σz).
The definition of phase advance σ is given by
σ(s) ≡ ∫s [1/β(t)]dt ,
where β(s) is the Courant-Snyder, envelope function, and the integral
is taken along the interval between the initial and final Courant-Snyder
parameters.
The basic relation used to compute σ is the following:
σ = sin-1 φ12/(β1β2)½ ,
where φ12 is the element of Φ in the upper right corner of each
2×2 diagonal block, β1 is the initial beta function value (provided)
and β2 is the final beta function value (provided).
matPhi - the transfer matrix that propagated the Twiss parameterstwsInit - initial Twiss parameter before application of matrixtwsFinal - final Twiss parameter after application of matrix@Deprecated protected R3 calculatePhaseAdvance_old(PhaseMatrix matPhi, Twiss[] twsInit, Twiss[] twsFinal)
Taken from TransferMapState.
Compute and return the particle phase advance for under the action of the given phase matrix when used as a transfer matrix.
Calculates the phase advances given the initial and final Courant-Snyder α and β values provided. This is the general phase advance of the particle through the transfer matrix Φ, and no special requirements are placed upon Φ. One phase advance is provided for each phase plane, i.e., (σx, σz, σz).
The definition of phase advance σ is given by
σ(s) ≡ ∫s [1/β(t)]dt ,
where β(s) is the Courant-Snyder, envelope function, and the integral
is taken along the interval between the initial and final Courant-Snyder
parameters.
The basic relation used to compute σ is the following:
σ = sin-1 φ12/(β1β2)½ ,
where φ12 is the element of Φ in the upper right corner of each
2×2 diagonal block, β1 is the initial beta function value (provided)
and β2 is the final beta function value (provided).
matPhi - the transfer matrix that propagated the Twiss parameterstwsInit - initial Twiss parameter before application of matrixtwsFinal - final Twiss parameter after application of matrixprotected Twiss[] calculateMatchedTwiss(PhaseMatrix matPhiCell)
Taken from TransferMapState.
Calculates the matched Courant-Snyder parameters for the given period cell transfer matrix and phase advances. When the given transfer matrix is the full-turn matrix for a ring the computed Twiss parameters are the matched envelopes for the ring at that point.
The given matrix Φ is assumed to be the transfer matrix through
at least one cell in a periodic lattice. Internally, the array of phase advances
{σx, σy, σx}
are assumed to be the particle phase advances through the cell for the matched
solution. These are computed with the method
calculatePhaseAdvPerCell(PhaseMatrix)
The returned Courant-Snyder parameters (α, β, ε) are invariant
under the action of the given phase matrix, that is, they are matched. All that
is required are α and β since ε specifies the size of the beam
envelope. Consequently the returned ε is NaN.
matPhiCell - transfer matrix Φ for a periodic cell (or full turn)NaN)
for each phase planeprotected R6 calculateAberration(PhaseMatrix matPhi, double dblGamma)
Convenience function for returning the chromatic aberration coefficients.
For example, in the
horizontal phase plane (x,x') these coefficients specify the
change in position Δx
and the change in divergence angle Δx'
due to chromatic dispersion δ within the beam, or
Δx = (dx/dδ) ⋅ δ ,
Δx' = (dx'/dδ) ⋅ δ .
That is, (dx/dδ) and (dx'/dδ) are the dispersion
coefficients for the horizontal plane position and horizontal plane divergence
angle, respectively. The vector returned by this method contains all the analogous
coefficients for all the phase planes.
The chromatic dispersion coefficient vector Δ can be built from
the 6th
column of the state response matrix Φ. However we must pay close
attention to the transverse plane quantities. Specifically, consider again
the horizontal phase plane (x,x'). Denoting the elements of Φ
in the 6th column (i.e., the z' column) corresponding
to this positions as φ1,6 and φ2,6, we can write
them as
φ1,6 = ∂x/∂z' ,
φ2,6 = ∂x'/∂z' .
Now consider the relationship
δ ≡ (p - p0)/p0
= γ2z'= γ2dz/ds
or
∂z'/∂δ = 1/γ2 .
Thus, it is necessary to multiply the transfer plane elements of
Row6Φ by 1/γ2 to covert to
the conventional dispersion coefficients. Specifically, for the horizontal plane
Δx = (dx/dδ) = φ6,1/γ2 ,
Δx' = (dx'/dδ) = φ6,2/γ2 ,
For the longitudinal plane no such normalization is necessary.
matPhi - transfer matrix for which we are computing aberrationsdblGamma - relativistic factor for the beam particles at the location of aberrationprotected R4 calculateDispersion(PhaseMatrix matPhi, double dblGamma)
Taken from TransferMapState.
Calculates the differential coefficients describing change in the fixed point of the closed orbit versus chromatic dispersion. The given transfer matrix Φ is assumed to be the one-turn map of the ring with which the fixed point is caculated. The fixed point is with regard to the transverse phase plane coordinates.
The transverse plane dispersion vector Δ is defined
Δt ≡ -(1/γ2)[dx/dz', dx'/dz', dy/dz', dy'/dz']T .
It can be identified as the first 4 entries of the 6th
column in the transfer matrix Φ. The above vector
quantifies the change in the transverse particle phase
coordinate position versus the change in particle momentum.
The factor -(1/γ2) is needed to convert from longitudinal divergence
angle z' used by XAL to momentum δp ≡ Δp/p used in
the dispersion definition. Specifically,
δp ≡ Δp/p = γ2z'
As such, the above vector can be better described
Δt ≡ [Δx/δp, Δx'/δp, Δy/δp, Δy'/δp]T
explicitly describing the change in transverse phase coordinate for fractional
change in momentum δp.
Since we are only concerned with transverse phase space coordinates, we restrict ourselves to the 4×4 upper diagonal block of Φ, which we denote take T. That is, T = π ⋅ Φ where π : R6×6 → R4×4 is the projection operator.
This method finds that point zt ≡
(xt, x't, yt, y't)
in transvse phase space that is invariant under the action of the ring for a given momentum spread
δp. That is, the particle ends up
in the same location each revolution. With a finite momentum spread of δp > 0
we require this require that
Tzt + δpΔt = zt ,
which can be written
zt = δp(I - T)-1Δt ,
where I is the identity matrix. Dividing both sides by δp yields the final
result
z0 ≡ zt/δp = (I - T)-1Δt ,
which is the returned value of this method. It is normalized by
δp so that we can compute the closed orbit for any given momentum spread.
The eigenvalues of T determine conditioning of the the resolvent (I - T)-1. Whenever 1 ∈ λ(T) we have problems. Currently a value of 0 is returned whenever the resolvent does not exist.
matPhi - we are calculating the dispersion of a ring with this one-turn mapdblGamma - relativistic factor@Deprecated protected Twiss3D computeTwiss(TwissProbe probe, PhaseMatrix matPhi, double dW)
TwissProbe and should
not be used as the dynamics elsewhere. It has been moved to the
TwissTracker class.Advance the twiss parameters using the given transfer matrix based upon formula 2.54 from S.Y. Lee's book.
probe - probe with target twiss parametersmatPhi - the transfer matrix of the elementdW - the energy gain of this element (eV)