def infer_angular_variance_spherical(var_r, phi_c, var_q_min):
    """Calculate angular variance given properties of a beam with quadratic phase corrected by a lens.

    The lens which removes the spherical component has focal length f = -k*var_r/(2*phi_c). This means that -phi_c is the
    phase of the quadratic component at r  =  var_r**0.5.

    The formula is (19) of Siegman IEEE J. Quantum Electronics,  vol. 27 1991,  with k = 2*pi/lambda. For consistency with
    the rest of the program,  I use slightly different definitions. The derivation is on p120 of Dane's Fathom notebook 2.
    It is valid for any refractive index.

    Args:
        var_r (scalar): real space variance.
        phi_c (scalar): real-space curvature - see above.
        var_q_min (scalar): angular variance of the beam after its curvature has been removed.

    Returns:
        var_q (scalar): angular variance of the beam after the lens.
    """
    var_q = var_q_min + 4*phi_c**2/var_r
    return var_q