We present a systematic study of the force-free field equation for simple axisymmetric configurations in spherical geometry and apply it to the solar active regions. The condition of separability of solutions in the radial and angular variables leads to two classes of solutions: linear and nonlinear force-free fields (NLFF). We have studied these linear solutions and extended the nonlinear solutions for the radial power law index to the irreducible rational form n = p/q, which is allowed for all cases of odd p and cases of q p for even p, where the poloidal flux ψ∝1/rn and the field B∝1/r n + 2. We apply these solutions to simulate photospheric vector magnetograms obtained using the spectropolarimeter on board Hinode. The effectiveness of our search strategy is first demonstrated on test inputs of dipolar, axisymmetric, and nonaxisymmetric linear force-free fields. Using the best fit, we build three-dimensional axisymmetric field configurations and calculate the energy and relative helicity with two independent methods, which are in agreement. We have analyzed five magnetograms for AR?10930 spanning a period of three days during which two X-class flares occurred and found the free energy and relative helicity of the active region before and after the flare; our analysis indicates a peak in these quantities before the flare events, which is consistent with the other results. We also analyzed single-polarity regions AR 10923 and 10933, which showed very good fits to potential fields. This method provides useful reconstruction of NLFF and input fields for other numerical techniques.
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