An algorithm for optimizing the designs of diffractive optics is based on constrained dynamics and the reciprocity of electromagnetic propagation. This algorithm can contribute to realization of the potential of diffractive optics, which offer capabilities greater than those of refractive optics. Because the diffractive-optics design-optimization problem is equivalent to determining the diffractive limit of phase-only propagation compensation by adaptive optics, the algorithm could be used to create control algorithms for adaptive optics. In addition, the mathematical techniques and physical principles used in developing the algorithm could be used in solving the problem of phase retrieval for characterizing wavefronts.

The design and optimization of diffractive optics are generally more complex and computationally intensive than are the design and optimization of refractive optics. Prior to the development of the present algorithm, most of algorithms used for the same purpose were based on propagation only in the forward direction; that is, from the diffractive optical element(s) to the diffraction plane. In each such prior algorithm, an initial guess (often a random configuration) of the design of the diffractive optical element(s) is made, and stochastic integration techniques are used to improve the initial guess by generating more random configurations and then deciding which ones are better and headed in the direction of the desired optimized design. Such algorithms are computationally inefficient (and hence time-consuming) because they must generate many tentative design configurations, most of which must be discarded because they are far removed from design specifications.

The computational burden associated with generating many random configurations makes it impractical to produce designs with more than a limited number of degrees of freedom per phase-template pixel; that is, phase templates generated by such algorithms can contain only a few quantized levels or steps in phase. However, it is possible to manufacture diffractive optical elements with continuous phase profiles.

Hence, what is needed to realize the potential of diffractive optics is a computationally efficient algorithm that generates phase templates with continuous phase profiles. The present algorithm satisfies this need. Because random configurations are not used and physical quantities are treated as continuous in the first place, there are no restrictions on the number of degrees of freedom per phase-template pixel.

Mathematically, reciprocity of electromagnetic propagation is a consequence of the symmetry of Maxwell's equations. The present algorithm takes advantage of the reciprocity of electromagnetic propagation to propagate design information both from the diffractive optical elements forward to the diffraction plane and from the diffraction plane backward to the diffractive optical element(s). In so doing, the algorithm utilizes maximally the information available on both the diffractive-element and diffraction planes. Moreover, because the algorithm is ultimately based on Maxwell's equations, it can, in principle, be applied to a wide variety of diffractive optical elements, including those for which vector diffraction analysis is necessary.

The constrained-dynamics aspect of the algorithm makes it possible to solve the design-optimization problem through successive approximation. Each iteration brings the design solution closer to the optimized design solution, and no randomness is involved.

*This work was done by S. Enguehard and B. Hatfield of Applied Mathematical Physics Research, Inc., for *Marshall Space Flight Center. For further information, contact the company at (781) 862-6357.

*In accordance with Public Law 96-517, the contractor has elected to retain title to this invention. Inquiries concerning rights for its commercial use should be addressed to*

###### Applied Mathematical Physics Research, Inc.

420 Bedford Street

Suite 230

Lexington, MA 02420

*Refer to MFS-31427*

##### Photonics Tech Briefs Magazine

This article first appeared in the November, 2001 issue of *Photonics Tech Briefs* Magazine.

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