Analytical formula

We develop analytical expressions for the magnetic vector  potential  and  the  magnetic  field  of  modulated  double helical  coils.  It  is  assumed  that  the  coils  are  tightly  wound, and  can  be  approximated  by  two  infinite  thin  current  sheets. From each coil parametric equation, the surface current density is derived by considering the turn-to-turn spacing. The field inside and  outside  each  coil  is  then  calculated  by  decomposing  the surface current density into three contributions. By summation, the field of any modulated double helical coil can be obtained. As a validation, we evaluate the field of a superconducting double helical quadrupole. A good agreement is found between analytical and numerical results. From the field, the main coil parameters can  be  quickly  estimated  with  a  fair  accuracy.  This  can  be useful for the design an optimization of advanced magnets, both conventional and superconducting.