ExactCFbend.H

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/* Copyright 2022-2026 The Regents of the University of California, through Lawrence
 *           Berkeley National Laboratory (subject to receipt of any required
 *           approvals from the U.S. Dept. of Energy). All rights reserved.
 *
 * This file is part of ImpactX.
 *
 * Authors: Chad Mitchell, Axel Huebl
 * License: BSD-3-Clause-LBNL
 */
#ifndef IMPACTX_EXACTCFBEND_H
#define IMPACTX_EXACTCFBEND_H
 
#include "particles/ImpactXParticleContainer.H"
#include "particles/integrators/Integrators.H"
#include "mixin/alignment.H"
#include "mixin/beamoptic.H"
#include "mixin/dynamicdata.H"
#include "mixin/lineartransport.H"
#include "mixin/spintransport.H"
#include "mixin/named.H"
#include "mixin/nofinalize.H"
#include "mixin/pipeaperture.H"
#include "mixin/thick.H"
#include "mixin/TrackedVector.H"
 
#include <AMReX_Extension.H>
#include <AMReX_REAL.H>
#include <AMReX_Print.H>
#include <AMReX_GpuComplex.H>
#include <AMReX_Math.H>
 
#include <cmath>
#include <memory>
#include <stdexcept>
#include <vector>
 
namespace impactx::elements
{

    struct CFbendCoefficients
    {
        mixin::TrackedVector<amrex::ParticleReal> k_normal;
        mixin::TrackedVector<amrex::ParticleReal> k_skew;
    };
 
    struct ExactCFbend
    : public mixin::Named,
      public mixin::BeamOptic<ExactCFbend>,
      public mixin::LinearTransport<ExactCFbend>,
      public mixin::Thick,
      public mixin::Alignment,
      public mixin::NoFinalize,
      public mixin::PipeAperture,
      public mixin::SpinTransport
    {
        static constexpr auto type = "ExactCFbend";
        using PType = ImpactXParticleContainer::ParticleType;
 
        using DynamicData = mixin::GPUDataRegistry<CFbendCoefficients>;

        ExactCFbend (
            amrex::ParticleReal ds,
            std::vector<amrex::ParticleReal> k_normal,
            std::vector<amrex::ParticleReal> k_skew,
            int unit,
            amrex::ParticleReal dx = 0,
            amrex::ParticleReal dy = 0,
            amrex::ParticleReal rotation_degree = 0,
            amrex::ParticleReal aperture_x = 0,
            amrex::ParticleReal aperture_y = 0,
            int int_order = 2,
            int mapsteps = 1,
            int nslice = 1,
            std::optional<std::string> name = std::nullopt
        )
        : Named(std::move(name)),
          Thick(ds, nslice),
          Alignment(dx, dy, rotation_degree),
          PipeAperture(aperture_x, aperture_y),
          m_unit(unit), m_int_order(int_order), m_mapsteps(mapsteps),
          m_id(DynamicData::allocate_id())
        {
            if (m_int_order != 2 && m_int_order != 4 && m_int_order != 6)
                throw std::runtime_error("ExactCFbend: The order used for symplectic integration must be 2, 4 or 6.");
 
            m_ncoef = int(k_normal.size());
            if (m_ncoef != int(k_skew.size()))
                throw std::runtime_error("ExactCFbend: normal and skew coefficient arrays must have same length!");
 
            auto& coef = DynamicData::emplace(
                m_id,
                std::move(k_normal),
                std::move(k_skew)
            );
            m_k_normal_h_data = coef.k_normal.host_const().data();
            m_k_skew_h_data = coef.k_skew.host_const().data();
        }

        void reverse () { Thick::reverse(); }

        using BeamOptic::operator();

        void compute_constants (RefPart const & refpart)
        {
            using namespace amrex::literals; // for _rt and _prt
 
            Alignment::compute_constants(refpart);
 
            // Ensure dynamic coefficient data is on GPU and we have fresh pointers to it
            auto const & coef = *DynamicData::get(m_id);
            m_k_normal_d_data = coef.k_normal.device_const().data();
            m_k_skew_d_data = coef.k_skew.device_const().data();
 
            // access reference particle values to find relativistic factors
            m_beta = refpart.beta();
            m_ibeta = 1_prt / m_beta;
            m_ibetgam2 = 1_prt / amrex::Math::powi<2>(refpart.beta_gamma());
            m_brho = refpart.rigidity_Tm();
 
            amrex::ParticleReal const * k_normal = m_k_normal_h_data;
 
            // find magnetic field and curvature; normalize units to MAD-X convention if needed
            amrex::ParticleReal curvature = m_unit == 1 ? k_normal[0] / m_brho : k_normal[0];
            m_rc = curvature == 0.0 ? 0.0_prt : 1.0_prt/curvature;
 
            // arc length and angle of the current slice
            m_slice_ds = m_ds / nslice();
        }

        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void operator() (
            amrex::ParticleReal & AMREX_RESTRICT x,
            amrex::ParticleReal & AMREX_RESTRICT y,
            amrex::ParticleReal & AMREX_RESTRICT t,
            amrex::ParticleReal & AMREX_RESTRICT px,
            amrex::ParticleReal & AMREX_RESTRICT py,
            amrex::ParticleReal & AMREX_RESTRICT pt,
            [[maybe_unused]] uint64_t const & AMREX_RESTRICT idcpu,
            RefPart const & AMREX_RESTRICT refpart
        ) const
        {
            using namespace amrex::literals; // for _rt and _prt
 
            // numerical integration parameters
            amrex::ParticleReal const zin = 0_prt;
            amrex::ParticleReal const zout = m_slice_ds;
            int const nsteps = m_mapsteps;
 
            // initialize phase space 6-vector
            amrex::SmallVector<amrex::ParticleReal, 6, 1> particle{
                x, px, y, py, t, pt
            };
 
            // call integrator to advance through slice (int_order = 2 or 4, otherwise default 2)
            if (m_int_order == 2) {
                integrators::symp2_integrate_particle(particle,zin,zout,nsteps,refpart,*this);
            } else if (m_int_order == 4) {
                integrators::symp4_integrate_particle(particle,zin,zout,nsteps,refpart,*this);
            } else if (m_int_order == 6) {
                integrators::symp6_integrate_particle(particle,zin,zout,nsteps,refpart,*this);
            }
 
            // assign updated values
            x = particle(1);
            px = particle(2);
            y = particle(3);
            py = particle(4);
            t = particle(5);
            pt = particle(6);
        }

        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void map1 (amrex::ParticleReal const tau,
                   amrex::SmallVector<amrex::ParticleReal, 6, 1> & particle,
                   amrex::ParticleReal & zeval,
                   [[maybe_unused]] RefPart const & AMREX_RESTRICT refpart) const
        {
            using namespace amrex::literals; // for _rt and _prt
            using namespace amrex::Math;  // for powi
 
            amrex::ParticleReal const x = particle(1);
            amrex::ParticleReal const px = particle(2);
            amrex::ParticleReal const y = particle(3);
            amrex::ParticleReal const py = particle(4);
            amrex::ParticleReal const t = particle(5);
            amrex::ParticleReal const pt = particle(6);
 
            amrex::ParticleReal xout = x;
            amrex::ParticleReal pxout = px;
            amrex::ParticleReal yout = y;
            amrex::ParticleReal pyout = py;
            amrex::ParticleReal tout = t;
            amrex::ParticleReal ptout = pt;
 
            // treat the special case of zero field
            if (m_rc==0_prt) {
               // compute the radical in the denominator (= pz):
               amrex::ParticleReal const inv_pzden = 1_prt / std::sqrt(
                   powi<2>(pt - m_ibeta) -
                   m_ibetgam2 -
                   powi<2>(px) -
                   powi<2>(py)
               );
 
               // advance position and momentum (exact drift)
               xout = x + tau * px * inv_pzden;
               yout = y + tau * py * inv_pzden;
               tout = t - tau * (m_ibeta + (pt - m_ibeta) * inv_pzden);
 
               // assign updated momenta
               pxout = px;
               pyout = py;
               ptout = pt;
 
            } else {
 
               // assign intermediate quantities
               amrex::ParticleReal const pperp2 = powi<2>(pt)-2.0_prt * m_ibeta * pt - powi<2>(py)+1.0_prt;
               amrex::ParticleReal const px2 = powi<2>(px);
 
               // trigonometric evaluations
               amrex::ParticleReal const slice_phi = tau / m_rc;
               auto const [sin_phi, cos_phi] = amrex::Math::sincos(slice_phi);
 
               // determine if particle lies within the domain of map definition
               if (pperp2 > px2)
               {
                   amrex::ParticleReal const pperp = std::sqrt(pperp2);
                   amrex::ParticleReal const pzi = std::sqrt(powi<2>(pperp) - powi<2>(px));
                   amrex::ParticleReal const rho = m_rc + x;
 
                   // update momenta
                   pxout = px * cos_phi + (pzi - rho / m_rc) * sin_phi;
                   pyout = py;
                   ptout = pt;
 
                   // angle of momentum rotation
                   amrex::ParticleReal const px2f = powi<2>(pxout);
                   // determine if particle lies within domain of map definition
                   if (pperp2 > px2f)
                   {
                       amrex::ParticleReal const pzf = std::sqrt(powi<2>(pperp)-powi<2>(pxout));
                       amrex::ParticleReal const theta = slice_phi + std::asin(px/pperp) - std::asin(pxout/pperp);
 
                       // update position coordinates
                       xout = -m_rc + rho*cos_phi + m_rc*(pzf + px*sin_phi - pzi*cos_phi);
                       yout = y + theta * m_rc * py;
                       tout = t - theta * m_rc * (pt - 1.0_prt * m_ibeta) - slice_phi * m_rc * m_ibeta;
 
                   } // else { amrex::Print() << "Warning: outside map domain." << "\n"; }
               } // else { amrex::Print() << "Warning: outside map domain." << "\n"; }
            }
 
            // push particle coordinates
            particle(1) = xout;
            particle(2) = pxout;
            particle(3) = yout;
            particle(4) = pyout;
            particle(5) = tout;
            particle(6) = ptout;
 
            zeval += 0_prt;
        }

        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void map2 (amrex::ParticleReal const tau,
                   amrex::SmallVector<amrex::ParticleReal, 6, 1> & particle,
                   amrex::ParticleReal & zeval,
                   [[maybe_unused]] RefPart const & AMREX_RESTRICT refpart) const
        {
            using namespace amrex::literals; // for _rt and _prt
            using namespace amrex::Math;  // for powi
 
            amrex::ParticleReal const x = particle(1);
            amrex::ParticleReal const px = particle(2);
            amrex::ParticleReal const y = particle(3);
            amrex::ParticleReal const py = particle(4);
            amrex::ParticleReal const t = particle(5);
            amrex::ParticleReal const pt = particle(6);
 
            amrex::ParticleReal xout = x;
            amrex::ParticleReal pxout = px;
            amrex::ParticleReal yout = y;
            amrex::ParticleReal pyout = py;
            amrex::ParticleReal tout = t;
            amrex::ParticleReal ptout = pt;
 
            amrex::ParticleReal const * k_normal = m_k_normal_d_data;
            amrex::ParticleReal const * k_skew = m_k_skew_d_data;
 
            // defined dimensionless variables scaled by radius of curvature
            amrex::ParticleReal rho = 1_prt + x / m_rc;
            amrex::ParticleReal y_scl = y / m_rc;
            amrex::ParticleReal ln_rho = std::log(rho);
 
            // initialize the momentum kick
            amrex::ParticleReal dpx = 0_prt;
            amrex::ParticleReal dpy = 0_prt;
 
            // loop over array of multipole coefficients
            for (int m = 1; m < m_ncoef; m++) {
 
               // normalize units to MAD-X convention if needed
               amrex::ParticleReal kn = m_unit == 1 ? k_normal[m] / m_brho : k_normal[m];
               amrex::ParticleReal ks = m_unit == 1 ? k_skew[m] / m_brho : k_skew[m];
 
               int n = m+1;  // multipole order
 
               // define dimensionless multipole coefficients (scaled by radius of curvature)
               kn = std::pow(m_rc, static_cast<amrex::ParticleReal>(m)) * kn;
               ks = std::pow(m_rc, static_cast<amrex::ParticleReal>(m)) * ks;
 
               switch (n) {
 
                  case 1:   // dipole-like contribution (not used - contributing to map 1)
                     dpx += -rho * kn;
                     dpy += ks;
                     break;
                  case 2:   // quadrupole-like contribution
                     dpx += -rho*ln_rho*kn + y_scl*rho*ks;
                     dpy += y_scl*kn + 0.5_prt*(rho*rho - 1_prt)*ks;
                     break;
                  case 3:  // sextupole-like contribution
                     dpx += 0.25_prt*rho*(1_prt + 2_prt*y_scl*y_scl - rho*rho + 2_prt*ln_rho)*kn + y_scl*rho*ln_rho*ks;
                     dpy += 0.5_prt*y_scl*(rho*rho - 1_prt)*kn + 0.25_prt*(1_prt - 2_prt*y_scl*y_scl - rho*rho + 2_prt*rho*rho*ln_rho)*ks;
                     break;
                  case 4:  // octupole-like contribution
                     dpx += -0.25_prt*(rho - powi<3>(rho) + (rho - 2_prt*y_scl*y_scl*rho + powi<3>(rho))*ln_rho)*kn
                            -1_prt/12_prt*y_scl*rho*(3_prt + 2_prt*y_scl*y_scl - 3_prt*rho*rho + 6_prt*ln_rho)*ks;
                     dpy += -1_prt/12_prt*y_scl*(-3_prt + 2_prt*y_scl*y_scl + 3_prt*rho*rho - 6_prt*rho*rho*ln_rho)*kn
                            -1_prt/16_prt*((-1_prt + 4_prt*y_scl*y_scl - rho*rho)*(-1_prt + rho*rho) + 4_prt*rho*rho*ln_rho)*ks;
                     break;
                  case 5:  // decapole-like contribution
                     dpx += -1_prt/192_prt*rho*(8_prt*powi<4>(y_scl) - 24_prt*powi<2>(y_scl)*(-1_prt + rho*rho)
                            + 3_prt*(-5_prt + 4_prt*powi<2>(rho) + powi<4>(rho))
                            +12_prt*(-1_prt + 4_prt*y_scl*y_scl - 2_prt*rho*rho)*ln_rho)*kn
                            +1_prt/12_prt*y_scl*rho*(3_prt*(-1_prt + rho*rho) + (2_prt*y_scl*y_scl - 3_prt*(1_prt + rho*rho))*ln_rho)*ks;
                     dpy += -1_prt/48_prt*y_scl*((-1_prt + rho*rho)*(4_prt*y_scl*y_scl - 3_prt*(1_prt + rho*rho)) + 12_prt*rho*rho*ln_rho)*kn
                            + 1_prt/192_prt*(3_prt + 8_prt*powi<4>(y_scl) + 12_prt*rho*rho
                            - 15_prt*powi<4>(rho) + 24_prt*y_scl*y_scl*(-1_prt + rho*rho)
                            +12_prt*rho*rho*(2_prt - 4_prt*y_scl*y_scl + rho*rho)*ln_rho)*ks;
                     break;
                // default:
                //   TODO: print a warning here that order > 5 is not currently supported
               }
 
            }
 
            // advance momentum by a step of length tau
            pxout = pxout + tau * dpx;
            pyout = pyout + tau * dpy;
 
            //update the particle coordinates
            particle(1) = xout;
            particle(2) = pxout;
            particle(3) = yout;
            particle(4) = pyout;
            particle(5) = tout;
            particle(6) = ptout;
 
            zeval += tau;
        }
 

        template<typename T_Real>
        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void map3 (amrex::ParticleReal const tau,
                   amrex::SmallVector<T_Real, 6, 1> & particle,
                   amrex::SmallVector<T_Real, 3, 1> & particle_spin,
                   amrex::ParticleReal & zeval,
                   RefPart const & AMREX_RESTRICT refpart) const
        {
            using namespace amrex::literals; // for _rt and _prt
            using namespace std; // for cmath(float)
            using amrex::Math::powi;
 
            // reference particle relativistic factors
            amrex::ParticleReal const gamma_ref = refpart.gamma();
            amrex::ParticleReal const gyro_anomaly = refpart.gyromagnetic_anomaly;
 
            // initialize the three components of the axis-angle vector
            T_Real lambdax = 0_prt;
            T_Real lambday = 0_prt;
            T_Real lambdaz = 0_prt;
 
            T_Real const x = particle(1);
            T_Real const px = particle(2);
            T_Real const y = particle(3);
            T_Real const py = particle(4);
            //T_Real const t = particle(5); // not used
            T_Real const pt = particle(6);
            T_Real sx = particle_spin(1);
            T_Real sy = particle_spin(2);
            T_Real sz = particle_spin(3);
 
#if AMREX_DEVICE_COMPILE
            const amrex::ParticleReal *const k_normal = m_k_normal_d_data;
            const amrex::ParticleReal *const k_skew = m_k_skew_d_data;
#else
            const amrex::ParticleReal *const k_normal = m_k_normal_h_data;
            const amrex::ParticleReal *const k_skew = m_k_skew_h_data;
#endif
 
            // defined dimensionless variables scaled by radius of curvature
            amrex::ParticleReal rho = 1_prt + x / m_rc;
            amrex::ParticleReal y_scl = y / m_rc;
            amrex::ParticleReal ln_rho = std::log(rho);
 
            // Initialization of magnetic field variables:
            T_Real Bx = 0_prt;
            T_Real By = 0_prt;
            T_Real Bz = 0_prt;
 
            // loop over array of multipole coefficients
            // NOTE:  Here we start at m = 0 in order to include the dipole contribution to the spin
            for (int m = 0; m < m_ncoef; m++) {
 
               // normalize units to MAD-X convention if needed
               amrex::ParticleReal kn = m_unit == 1 ? k_normal[m] / m_brho : k_normal[m];
               amrex::ParticleReal ks = m_unit == 1 ? k_skew[m] / m_brho : k_skew[m];
 
               int n = m+1;  // multipole order
 
               // define dimensionless multipole coefficients (scaled by radius of curvature)
               kn = std::pow(m_rc, static_cast<amrex::ParticleReal>(m)) * kn;
               ks = std::pow(m_rc, static_cast<amrex::ParticleReal>(m)) * ks;
 
               // Below, we use the consistency relationships Bx = dpy/rho and By = -dpx/rho.  See T. Zolkin, PRAB 20, 043501 (2017).
 
               switch (n) {
 
                  case 1:   // dipole-like contribution (not used - contributing to map 1)
                     Bx += ks;
                     By += rho * kn;
                     break;
                  case 2:   // quadrupole-like contribution
                     Bx += y_scl*kn + 0.5_prt*(rho*rho - 1_prt)*ks;
                     By += -(-rho*ln_rho*kn + y_scl*rho*ks);
                     break;
                  case 3:  // sextupole-like contribution
                     Bx += 0.5_prt*y_scl*(rho*rho - 1_prt)*kn + 0.25_prt*(1_prt - 2_prt*y_scl*y_scl - rho*rho + 2_prt*rho*rho*ln_rho)*ks;
                     By += -(0.25_prt*rho*(1_prt + 2_prt*y_scl*y_scl - rho*rho + 2_prt*ln_rho)*kn + y_scl*rho*ln_rho*ks);
                     break;
                  case 4:  // octupole-like contribution
                     Bx += -1_prt/12_prt*y_scl*(-3_prt + 2_prt*y_scl*y_scl + 3_prt*rho*rho - 6_prt*rho*rho*ln_rho)*kn
                            -1_prt/16_prt*((-1_prt + 4_prt*y_scl*y_scl - rho*rho)*(-1_prt + rho*rho) + 4_prt*rho*rho*ln_rho)*ks;
                     By += -(-0.25_prt*(rho - powi<3>(rho) + (rho - 2_prt*y_scl*y_scl*rho + powi<3>(rho))*ln_rho)*kn
                            -1_prt/12_prt*y_scl*rho*(3_prt + 2_prt*y_scl*y_scl - 3_prt*rho*rho + 6_prt*ln_rho)*ks);
                     break;
                  case 5:  // decapole-like contribution
                     Bx += -1_prt/48_prt*y_scl*((-1_prt + rho*rho)*(4_prt*y_scl*y_scl - 3_prt*(1_prt + rho*rho)) + 12_prt*rho*rho*ln_rho)*kn
                            + 1_prt/192_prt*(3_prt + 8_prt*powi<4>(y_scl) + 12_prt*rho*rho
                            - 15_prt*powi<4>(rho) + 24_prt*y_scl*y_scl*(-1_prt + rho*rho)
                            +12_prt*rho*rho*(2_prt - 4_prt*y_scl*y_scl + rho*rho)*ln_rho)*ks;
                     By += -(-1_prt/192_prt*rho*(8_prt*powi<4>(y_scl) - 24_prt*powi<2>(y_scl)*(-1_prt + rho*rho)
                            + 3_prt*(-5_prt + 4_prt*powi<2>(rho) + powi<4>(rho))
                            +12_prt*(-1_prt + 4_prt*y_scl*y_scl - 2_prt*rho*rho)*ln_rho)*kn
                            +1_prt/12_prt*y_scl*rho*(3_prt*(-1_prt + rho*rho) + (2_prt*y_scl*y_scl - 3_prt*(1_prt + rho*rho))*ln_rho)*ks);
                     break;
               }
 
            }
 
            // Scaling magnetic field to units used in the T-BMT representation (e.g. q*B/mc).
 
            Bx = Bx * m_beta * gamma_ref / rho;
            By = By * m_beta * gamma_ref / rho;
 
            // Electric field normalized by q/mc^2:
            T_Real const Ex = 0.0_prt;
            T_Real const Ey = 0.0_prt;
            T_Real const Ez = 0.0_prt;
 
            // Quantities required to evaluate the full Thomas-BMT precession vector.
            T_Real const Pnorm = sqrt(1_prt - 2_prt * pt * m_ibeta + pt*pt);
            T_Real const iPnorm = 1_prt/Pnorm;
            T_Real const Ps = sqrt(Pnorm*Pnorm - px*px - py*py);
            T_Real const gamma = gamma_ref * (1_prt - pt*m_beta);
            T_Real const ux = px * iPnorm;
            T_Real const uy = py * iPnorm;
            T_Real const uz = Ps * iPnorm;
 
            // Design-orbit curvature
            amrex::ParticleReal const h = 1_prt/m_rc;
 
            // Evaluation of the full Thomas-BMT precession vector.
            tbmt_precession_vector(x,ux,uy,uz,gamma,h,gyro_anomaly,Bx,By,Bz,Ex,Ey,Ez,lambdax,lambday,lambdaz);
 
            // Generator of the spin rotation for a single step
            lambdax *= tau;
            lambday *= tau;
            lambdaz *= tau;
 
            // push the spin vector using the generator just determined
            rotate_spin(lambdax,lambday,lambdaz,sx,sy,sz);
 
            // update the spin variables
            particle_spin(1) = sx;
            particle_spin(2) = sy;
            particle_spin(3) = sz;
 
            // The coordinates and momenta are not affected.
 
            zeval += 0_prt;
        }
 

        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void operator() (RefPart & AMREX_RESTRICT refpart) const
        {
            using namespace amrex::literals; // for _rt and _prt
 
            // assign input reference particle values
            amrex::ParticleReal const x = refpart.x;
            amrex::ParticleReal const px = refpart.px;
            amrex::ParticleReal const y = refpart.y;
            amrex::ParticleReal const py = refpart.py;
            amrex::ParticleReal const z = refpart.z;
            amrex::ParticleReal const pz = refpart.pz;
            amrex::ParticleReal const t = refpart.t;
            amrex::ParticleReal const pt = refpart.pt;
            amrex::ParticleReal const s = refpart.s;
            amrex::ParticleReal const brho = refpart.rigidity_Tm();
 
#if AMREX_DEVICE_COMPILE
            amrex::ParticleReal const * k_normal = m_k_normal_d_data;
#else
            amrex::ParticleReal const * k_normal = m_k_normal_h_data;
#endif
            // find magnetic field and curvature; normalize units to MAD-X convention if needed
            amrex::ParticleReal const curvature = m_unit == 1 ? k_normal[0] / brho : k_normal[0];
            amrex::ParticleReal const rc = curvature == 0.0 ? 0.0_prt : 1.0_prt/curvature;
 
            // length of the current slice
            amrex::ParticleReal const slice_ds = m_ds / nslice();
 
            // special case of zero field = an exact drift
            if (rc==0_prt) {
               // advance position and momentum (drift)
               amrex::ParticleReal const step = slice_ds / std::sqrt(amrex::Math::powi<2>(pt) - 1.0_prt);
               refpart.x = x + step*px;
               refpart.y = y + step*py;
               refpart.z = z + step*pz;
               refpart.t = t - step*pt;
 
            } else {
 
               // assign intermediate parameters
               amrex::ParticleReal const B = refpart.beta_gamma() /rc;
               amrex::ParticleReal const theta = slice_ds / rc;
 
               // calculate expensive terms once
               auto const [sin_theta, cos_theta] = amrex::Math::sincos(theta);
 
               // advance position and momentum (bend)
               refpart.px = px*cos_theta - pz*sin_theta;
               refpart.py = py;
               refpart.pz = pz*cos_theta + px*sin_theta;
               refpart.pt = pt;
 
               refpart.x = x + (refpart.pz - pz)/B;
               refpart.y = y + (theta/B)*py;
               refpart.z = z - (refpart.px - px)/B;
               refpart.t = t - (theta/B)*pt;
 
            }
 
            // advance integrated path length
            refpart.s = s + slice_ds;
        }
 
 

        AMREX_GPU_HOST AMREX_FORCE_INLINE
        Map6x6
        transport_map (RefPart const & AMREX_RESTRICT refpart) const  // TODO: update as well, but needs more careful placement of calc_constants
        {
            using namespace amrex::literals; // for _rt and _prt
 
            // length of the current slice
            amrex::ParticleReal const slice_ds = m_ds / nslice();
 
            // find beta*gamma^2, beta
            amrex::ParticleReal const betgam2 = amrex::Math::powi<2>(refpart.pt) - 1_prt;
            amrex::ParticleReal const bet = refpart.beta();
            amrex::ParticleReal const ibetgam2 = 1_prt / betgam2;
 
            amrex::ParticleReal const * k_normal = m_k_normal_h_data;
 
            // find magnetic field and curvature; normalize units to MAD-X convention if needed
            amrex::ParticleReal const curvature = m_unit == 1 ? k_normal[0] / m_brho : k_normal[0];
            amrex::ParticleReal const rc = curvature == 0.0 ? 0.0_prt : 1.0_prt/curvature;
            amrex::ParticleReal const kn = m_unit == 1 ? k_normal[1] / m_brho : k_normal[1];
            amrex::ParticleReal const b2rc2 = amrex::Math::powi<2>(bet) * amrex::Math::powi<2>(rc);
 
            // update horizontal and longitudinal phase space variables
            amrex::ParticleReal const gx = kn + amrex::Math::powi<-2>(rc);
            amrex::ParticleReal const omega_x = std::sqrt(std::abs(gx));
 
            // update vertical phase space variables
            amrex::ParticleReal const gy = -kn;
            amrex::ParticleReal const omega_y = std::sqrt(std::abs(gy));
 
            // trigonometry
            auto const [sinx, cosx] = amrex::Math::sincos(omega_x * slice_ds);
            amrex::ParticleReal const sinhx = std::sinh(omega_x * slice_ds);
            amrex::ParticleReal const coshx = std::cosh(omega_x * slice_ds);
            auto const [siny, cosy] = amrex::Math::sincos(omega_y * slice_ds);
            amrex::ParticleReal const sinhy = std::sinh(omega_y * slice_ds);
            amrex::ParticleReal const coshy = std::cosh(omega_y * slice_ds);
 
            amrex::ParticleReal const igbrc = 1_prt / (gx * bet * rc);
            amrex::ParticleReal const iobrc = 1_prt / (omega_x * bet * rc);
            amrex::ParticleReal const igobr = 1_prt / (gx * omega_x * b2rc2);
 
            // initialize linear map matrix elements
            Map6x6 R = Map6x6::Identity();
 
            R(1,1) = gx > 0_prt ? cosx : coshx;
            R(1,2) = gx > 0_prt ? sinx / omega_x : sinhx / omega_x;
            R(1,6) = gx > 0_prt ? -(1_prt - cosx) * igbrc : -(1_prt - coshx) * igbrc;
            R(2,1) = gx > 0_prt ? -omega_x * sinx : omega_x * sinhx;
            R(2,2) = gx > 0_prt ? cosx : coshx;
            R(2,6) = gx > 0_prt ? -sinx * iobrc : -sinhx * iobrc;
            R(3,3) = gy > 0_prt ? cosy : coshy;
            R(3,4) = gy > 0_prt ? siny / omega_y : sinhy / omega_y;
            R(4,3) = gy > 0_prt ? -omega_y * siny : omega_y * sinhy;
            R(4,4) = gy > 0_prt ? cosy : coshy;
            R(5,1) = gx > 0_prt ? sinx * iobrc : sinhx * iobrc;
            R(5,2) = gx > 0_prt ? (1_prt - cosx) * igbrc : (1_prt - coshx) * igbrc;
            R(5,6) = gx > 0_prt ?
                slice_ds * ibetgam2 + (sinx  - omega_x * slice_ds) * igobr :
                slice_ds * ibetgam2 + (sinhx - omega_x * slice_ds) * igobr;
 
            // apply the transverse rotation (roll) alignment error
            return rotate_aligned_map(R);
        }
 

        template<typename T_Real=amrex::ParticleReal, typename T_IdCpu=uint64_t>
        AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
        void spin_and_phasespace_push (
            T_Real & AMREX_RESTRICT x,
            T_Real & AMREX_RESTRICT y,
            T_Real & AMREX_RESTRICT t,
            T_Real & AMREX_RESTRICT px,
            T_Real & AMREX_RESTRICT py,
            T_Real & AMREX_RESTRICT pt,
            T_Real & AMREX_RESTRICT sx,
            T_Real & AMREX_RESTRICT sy,
            T_Real & AMREX_RESTRICT sz,
            [[maybe_unused]] T_IdCpu const & AMREX_RESTRICT idcpu,
            RefPart const & AMREX_RESTRICT refpart
        ) const
        {
            using namespace amrex::literals; // for _rt and _prt
 
            // numerical integration parameters
            amrex::ParticleReal const zin = 0_prt;
            amrex::ParticleReal const zout = m_slice_ds;
            int const nsteps = m_mapsteps;
 
            // initialize phase space 6-vector
            amrex::SmallVector<T_Real, 6, 1> particle{
                x, px, y, py, t, pt
            };
 
            // initialize spin 3-vector
            amrex::SmallVector<T_Real, 3, 1> particle_spin{
                sx, sy, sz
            };
 
            // call integrator to advance through slice
            integrators::symp_integrate_particle_spin(particle,particle_spin,zin,zout,nsteps,m_int_order,refpart,*this);
 
            // assign updated values
            x = particle(1);
            px = particle(2);
            y = particle(3);
            py = particle(4);
            t = particle(5);
            pt = particle(6);
            sx = particle_spin(1);
            sy = particle_spin(2);
            sz = particle_spin(3);
        }
 

        using LinearTransport::operator();
 
 
        int m_unit; 
        int m_int_order; 
        int m_mapsteps; 
        int m_id; 
 
        int m_ncoef = 0; 
        amrex::ParticleReal const * m_k_normal_h_data = nullptr; 
        amrex::ParticleReal const * m_k_skew_h_data = nullptr; 
        amrex::ParticleReal const * m_k_normal_d_data = nullptr; 
        amrex::ParticleReal const * m_k_skew_d_data = nullptr; 
 
    private:
        // constants that are independent of the individually tracked particle,
        // see: compute_constants() to refresh
        amrex::ParticleReal m_slice_ds;       
        amrex::ParticleReal m_ibetgam2;       
        amrex::ParticleReal m_beta;           
        amrex::ParticleReal m_ibeta;          
        amrex::ParticleReal m_brho;           
        amrex::ParticleReal m_rc;             
    };
 
} // namespace impactx
 
IMPACTX_GPUDATA_EXTERN(impactx::elements::ExactCFbend)
IMPACTX_PUSH_EXTERN_TEMPLATE(impactx::elements::ExactCFbend)
 
#endif // IMPACTX_EXACTCFBEND_H