/** * * * <h2>Remove Emittance Growth Through an RF Gap</h2> * * <p>Method to modify the covariance matrix when simulating emittance growth through RF * accelerating gaps. (The method only considers the case of propagation through an <code> * IdealRfGap</code> element). If the <code>IElement</code> argument is any other type of element, * nothing is done. The argument <code>matTau</code> is the covariance matrix after the usual * propagation through the <code>elem</code> element. * * <p>Note that this method is essentially the complement of the method {@link * EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix)}. Whereas <code> * addEmittanceGrowth()</code> augments the momentum elements of <b>σ</b>, this method * reduces them by the same amount. Specifically, let <i>x</i> be either transverse phase space * variable. The emittance growth effect is achieved by first multiplying the element <x'|x> * of the RF gap transfer matrix <b>Φ</b> by the factor * <i>F<sub>t</sub></i>(Δ<i>φ</i>) returned by method {@link * EnvelopeTrackerBase#compTransFourierTransform(double)} (see {@link * EnvelopeTrackerBase#modTransferMatrixForEmitGrowth(double, PhaseMatrix)}). Currently this * action is done in {@link #compTransferMatrix(double, EnvelopeProbe, IElement)}. Once the * covariance matrix <b>τ</b> is back-propagated by the modified transfer matrix <b>Φ</b>, * the moment <<i>x'</i><sup>2</sup>> is reduced by the result of this function. * * <p>The discussion below is taken directly from {@link * EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix)}. It is applicable * here if the emittance is reduced by Δ<<i>x'<sub>f</sub></i><sup>2</sup>> rather * than increased by it. * * <p>The before gap and after gap transverse RMS divergence angles, <i>x'<sub>i</sub></i> and * <i>x'<sub>f</sub></i>, respectively, are related by the following formula: <br> * <br> * <<i>x'<sub>f</sub></i><sup>2</sup>> = * Δ<<i>x'<sub>f</sub></i><sup>2</sup>> + <<i>x'<sub>i</sub></i><sup>2</sup>> * <br> * <br> * where Δ<<i>x'<sub>f</sub></i><sup>2</sup>> is the emittance growth factor given by * <br> * <br> * Δ<<i>x'<sub>f</sub></i><sup>2</sup>> ≡ * <i>k<sub>t</sub></i><sup>2</sup> * <i>G<sub>t</sub></i>(<i>φ<sub>s</sub></i>,Δ<i>φ</i>) * <<i>x<sub>i</sub></i></i><sup>2</sup>>. <br> * <br> * where <i>G<sub>t</sub></i>(<i>φ<sub>s</sub></i>,Δ<i>φ</i>) is the transverse * 3-dimensional emittance growth function, and <i>x<sub>i</sub></i> represents the before-gap * position for <em>either</em> transverse phase plane. The action of this method is described by * the original equation. * * <p>The resulting action on the before gap and after gap transverse RMS emittances, * <i>ε<sub>t,i</sub></i> and <i>ε<sub>t,f</sub></i>, respectively, is now * described by the following formula: <br> * <br> * <i>ε<sub>t,f</sub></i><sup>2</sup> = * <i>η</i><sup>2</sup><i>ε<sub>t,i</sub></i><sup>2</sup> + * Δ<i>ε<sub>t,f</sub></i><sup>2</sup> <br> * <br> * where <i>η</i> is the momentum compaction due to acceleration <br> * <br> * <i>η</i> ≡ * <i>β<sub>i</sub>γ<sub>i</sub></i>/<i>β<sub>f</sub>γ<sub>f</sub></i> <br> * <br> * and Δ<i>ε<sub>t,f</sub></i> is the emittance increase term <br> * <br> * Δ<i>ε<sub>t,f</sub></i><sup>2</sup> ≡ * Δ<<i>x'<sub>f</sub></i><sup>2</sup>> * <<i>x<sub>f</sub></i></i><sup>2</sup>><sup>2</sup>. <br> * <br> * There are analogous formulas for the before and after gap longitudinal plane emittances * <i>ε<sub>z,i</sub></i> and <i>ε<sub>z,f</sub></i>, respectively, with * <i>G<sub>t</sub></i>(<i>φ<sub>s</sub></i>,Δ<i>φ</i>) replaced by * <i>G<sub>z</sub></i>(<i>φ<sub>s</sub></i>,Δ<i>φ</i>) and * <i>x</i><sub>(<i>f,i</i>)</sub> replaced by <i>z</i><sub>(<i>f,i</i>)</sub>. * * <p><strong>NOTES</strong>: CKA <br> * · Since we are modeling the RF gap as a thin lens, only the momentum (divergance angle) * is modified, <<i>x</i><sup>2</sup>>, <<i>y</i><sup>2</sup>>, and * <<i>z</i><sup>2</sup>> remain unaffected. Thus, <<i>x<sub>f</sub></i><sup>2</sup>> * = <<i>x<sub>i</sub></i><sup>2</sup>> and <<i>z<sub>f</sub></i><sup>2</sup>> = * <<i>z<sub>i</sub></i><sup>2</sup>> and may be computed as such in the above. <br> * · The <<i>x'</i><sup>2</sup>> element is modified by the formula <br> * <br> * <<i>x'</i><sup>2</sup>> = <<i>x'</i><sup>2</sup>> + * <i>c<sub>eg</sub></i><<i>x</i><sup>2</sup>> <br> * <br> * where <i>c<sub>eg</sub></i> is the emittance growth coefficent. There are similar equations for * the other phase planes. The emittance growth coefficents are computed in the base class <code> * EnvelopeTrackerBase</code> by the methods <code>emitGrowthCoefTrans(EnvelopeProbe, IdealRfGap) * </code> and <code>emitGrowthCoefLong(EnvelopeProbe, IdealRfGap)</code>. * * <p><strong>NOTES</strong>: (H. SAKO) <br> * · Increase emittance using same (nonlinear) procedure on the second moments as in * Trace3D. * * @param iElem <code>IElement</code> element for exceptional processing * @param probe <code>IProbe</code> object associated with correlation matrix * @param matTau correlation matrix after (normal) propagation thru <code>elem</code> * @return covariance matrix of <code>probe</code> after adjusting for emittance growth * @throws ModelException unknown/unsupported emittance growth model, or unknown/unsupported phase * plane * @see #compTransferMatrix(double, EnvelopeProbe, IElement) * @see EnvelopeTrackerBase#compTransFourierTransform(double) * @see EnvelopeTrackerBase#compLongFourierTransform(double) * @see EnvelopeTracker#addEmittanceGrowth(EnvelopeProbe, IElement, PhaseMatrix) * @author Hiroyuki Sako * @author Christopher K. Allen */ private PhaseMatrix removeEmittanceGrowth(EnvelopeProbe probe, IElement iElem, PhaseMatrix matTau) throws ModelException { // Check for RF Gap if (!(iElem instanceof IdealRfGap)) return matTau; if (!this.getEmittanceGrowth()) return matTau; // Get the synchronous phase and compute the phase spread IdealRfGap elemRfGap = (IdealRfGap) iElem; double W = probe.getKineticEnergy(); double dW = elemRfGap.energyGain(probe); probe.setKineticEnergy(W - dW); double phi_s = elemRfGap.getPhase(); double dphi = this.effPhaseSpread(probe, elemRfGap); // Compute the divergence angle increment coefficients // (emittance growth coefficients) double dxp_2; // transverse divergence angle augmentation factor double dzp_2; // longitudinal divergence angle augmentation factor // if (this.getEmitGrowthModel() == EmitGrowthModel.TRACE3D) { // // dxp_2 = this.emitGrowthCoefTrans(probe, elemRfGap); // dzp_2 = this.emitGrowthCoefLong(probe, elemRfGap); // // } else { double Gt = this.compEmitGrowthFunction(PhasePlane.TRANSVERSE, phi_s, dphi); double kt = elemRfGap.compTransFocusing(probe); dxp_2 = kt * kt * Gt; double Gz = this.compEmitGrowthFunction(PhasePlane.LONGITUDINAL, phi_s, dphi); double kz = elemRfGap.compLongFocusing(probe); // double gf = elemRfGap.gammaFinal(probe); // double gf_2 = gf*gf; // dzp_2 = kz*kz*Gz/(gf_2*gf_2); // dzp_2 = gf_2*gf_2*kz*kz*Gz; dzp_2 = kz * kz * Gz; // } probe.setKineticEnergy(W); // Compute new correlation matrix // Transverse planes double x_2 = matTau.getElem(0, 0); double xp_2 = matTau.getElem(1, 1); double xp_2eg = xp_2 - dxp_2 * x_2; matTau.setElem(1, 1, xp_2eg); double y_2 = matTau.getElem(2, 2); double yp_2 = matTau.getElem(3, 3); double yp_2eg = yp_2 - dxp_2 * y_2; matTau.setElem(3, 3, yp_2eg); // Longitudinal plane double z_2 = matTau.getElem(4, 4); double zp_2 = matTau.getElem(5, 5); double zp_2eg = zp_2 - dzp_2 * z_2; matTau.setElem(5, 5, zp_2eg); return matTau; }
/** * * * <h2>Compute Transfer Matrix Including Space Charge</h2> * * <p>Computes the back-propagating transfer matrix over the incremental distance <code>dblLen * </code> for the beamline modeling element <code>ifcElem</code>, and for the given <code>probe * </code>. We include space charge and emittance growth effects if specified. * * <p><strong>NOTE</strong>: (CKA) <br> * · If space charge is included, the space charge matrix is computed for length <code> * dblLen</code>, but at a half-step location behind the current probe position. This method is * the same technique used by Trace3D. The space charge matrix is then pre- and post- multiplied * by the element transfer matrix for a half-step before and after the mid-step position, * respectively. <br> * · I do not know if this (leap-frog) technique buys us much more accuracy then full * stepping. * * @param dblLen incremental path length * @param probe beam probe under simulation * @param ifcElem beamline element propagating probe * @return transfer matrix for given element * @throws ModelException bubbles up from IElement#transferMap() * @see EnvelopeTracker#compScheffMatrix(double, EnvelopeProbe, PhaseMatrix) * @see EnvelopeTracker#transferEmitGrowth(EnvelopeProbe, IElement, PhaseMatrix) * @see EnvelopeTracker#modTransferMatrixForDisplError(double, double, double, PhaseMatrix) */ private PhaseMatrix compTransferMatrix(double dblLen, EnvelopeProbe probe, IElement ifcElem) throws ModelException { // Returned value PhaseMatrix matPhi; // transfer matrix including all effects // Check for exceptional circumstance and modify transfer matrix accordingly if (ifcElem instanceof IdealRfGap) { IdealRfGap elemRfGap = (IdealRfGap) ifcElem; double dW = elemRfGap.energyGain(probe, dblLen); double W = probe.getKineticEnergy(); probe.setKineticEnergy(W - dW); PhaseMatrix matPhiI = elemRfGap.transferMap(probe, dblLen).getFirstOrder(); if (this.getEmittanceGrowth()) { double dphi = this.effPhaseSpread(probe, elemRfGap); matPhiI = super.modTransferMatrixForEmitGrowth(dphi, matPhiI); } matPhi = matPhiI.inverse(); probe.setKineticEnergy(W); return matPhi; } if (dblLen == 0.0) { matPhi = ifcElem.transferMap(probe, dblLen).getFirstOrder(); return matPhi; } // Check for easy case of no space charge if (!this.getUseSpacecharge()) { matPhi = ifcElem.transferMap(probe, dblLen).getFirstOrder(); // we must treat space charge } else { // Store the current probe state (for rollback) EnvelopeProbeState state0 = probe.cloneCurrentProbeState(); // ProbeState state0 = probe.createProbeState(); // Get half-step back-propagation matrix at current probe location // NOTE: invert by computing for negative propagation length PhaseMap mapElem0 = ifcElem.transferMap(probe, -dblLen / 2.0); PhaseMatrix matPhi0 = mapElem0.getFirstOrder(); // Get the RMS envelopes at probe location CovarianceMatrix covTau0 = probe.getCovariance(); // covariance matrix at entrance // Move probe back a half step for position-dependent transfer maps double pos = probe.getPosition() - dblLen / 2.0; PhaseMatrix matTau1 = covTau0.conjugateTrans(matPhi0); CovarianceMatrix covTau1 = new CovarianceMatrix(matTau1); probe.setPosition(pos); probe.setCovariance(covTau1); // space charge transfer matrix // NOTE: invert by computing for negative propagation length PhaseMatrix matPhiSc = this.compScheffMatrix(-dblLen, probe, ifcElem); // Compute half-step transfer matrix at new probe location PhaseMap mapElem1 = ifcElem.transferMap(probe, -dblLen / 2.0); PhaseMatrix matPhi1 = mapElem1.getFirstOrder(); // Restore original probe state probe.applyState(state0); // Compute the full transfer matrix for the distance dblLen matPhi = matPhi1.times(matPhiSc.times(matPhi0)); } if (ifcElem instanceof IdealMagQuad) { // sako put alignment error in sigma matrix // NOTE the use of negative displacements for back-propagation IdealMagQuad elemQuad = (IdealMagQuad) ifcElem; double delx = -elemQuad.getAlignX(); double dely = -elemQuad.getAlignY(); double delz = -elemQuad.getAlignZ(); matPhi = this.modTransferMatrixForDisplError(delx, dely, delz, matPhi); } return matPhi; }