Design of diamond turned holograms by the nonlinear conjugate gradients method

Ultraprecision diamond turning is an established technique, mainly used for the fabrication of free form optics and mechanical components. By the use of Piezo ceramics as servo actuator, this technique offers cutting depth modulations with accuracies suitable for the fabrication of relief holograms [?]. In a previous work, a first approach for the design of diamond turned holograms (DTH) was presented, which provide the reconstruction of arbitrary intensity distributions in the far field [?]. In [?], an improved design algorithm, based on the generalized projections method (GPM) [?] was established, taking into account properties of the fabrication process, such as the shape of the used diamond tool. Here, we investigate a different optimization approach, the nonlinear conjugate gradients (NCG) method. The NGC method allows the minimization of an error metric representing the errors in the optical reconstruction. For the implementation of the algorithm, the gradient of with respect to the design parameters must be derived. For this derivation, a theorem was developed, allowing the omission of certain terms in the gradient, thereby reducing the effort of its derivation and calculation. This theorem is not only applicable to the design of DTHs, but is generally valid for a class of error metrics called summed-distance errors (for a definition see [?]), which also belongs to. For a verification of the approach, two DTHs have been designed using 100 iterations of the GPM and the NCG method, respectively. Fig. ?? shows measurements of the reconstructed far field intensities obtained from these elements. Also, the spatial distribution of the reconstruction error is shown. By the use of the NCG method, is reduced by 16% compared to the GPM.