Response Matrix

The beam response is defined as r_j = du/dq_j, where u is the vector of orbit distortion at the BPMs caused by a kick angle of q by the steerer magnet j. The r_j can be combined to a matrix R_ij = du_i/dq_j. The theoretical Orbit Response Matrix (ORM) is calculated as,

Orbit response matrix R_ij measures the change in the transverse orbit position at some location s_i, caused by a transverse kick (typically, with a corrector magnet) at some other location s_j. The response matrix can be measured in a real accelerator or computed with an accelerator code

For the dispersion response matrix, we consider two classes of perturbation, the ordinary chromatic terms and those due to field errors. Considering the perturbations at quadrupoles and the sextupole locations, the Dispersion Response Matrix (DRM) can be written as,

Skew Quadrupole Response Matrices

A horizontal orbit offset through a skew quadrupole will generate a deflection in the vertical plane through an angle Dq_y = k_1slx. This will generate a vertical orbit distortion of Dy such that,

With x=D_x. DP/P and y=D_y. DP/P,  one can write the Skew Quadrupole response matrix as

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