The sampling constraints for Fresnel propagation are strict. Particularly, the angular-spectrum method is best suited for propagating only short distances. The key problem is wrap-around, caused by aliasing. Several approaches to mitigating these effects have been proposed. Most of these approaches center around spatially attenuating and filtering the optical field. For example, Johnston and Lane describe a technique in which the free-space transfer function is filtered and the grid size is based on the bandwidth of the filter.41 After this step, they set the sample interval based on avoiding aliasing of the quadratic phase factor just like in Sec. 7.3.2.
Johnston and Lane's choice of spatial-filter bandwidth works, but it is somewhat indirectly related to specific wrap-around effects. This book covers a more direct approach. For fixed D1, δ1, D2, and δ2, we must satisfy constraints 1, 3, and 4 from Ch. 7. Generally, Δz is fixed, too; it is just a part of the geometry that we wish to simulate. Often, the only free parameter is N, and for large Δz the constraints dictate large N. Sometimes the required N is prohibitively large, like N > 4096. Usually the culprit is constraint 4, which is only dependent on the propagation method, not the fixed propagation geometry. If constraint 4 is satisfied, it remains satisfied if we shorten Δz while holding N, δ1, δ2, and λ fixed. Consequently, this chapter develops a method of using multiple partial propagations with the angular-spectrum method to significantly relax constraint 4. To illustrate the propagation algorithm, we first begin with two partial propagations in Sec. 8.2 and then generalize to n - 1 partial propagations (n planes) in Sec. 8.3.
At first this may sound like a good solution, but multiple partial propagations are mathematically equivalent to a single full propagation. The extra partial propagations just take longer to execute. The key difficulty that we want to mitigate is wrap-around caused by aliasing. The variations in the free-space transfer function, given in Eq. (6.32), become increasingly rapid as Δz increases. Therefore, wraparound effects creep into the center of the grid from the edge. With partial propagations, we can attenuate the field at the edges of the grid to suppress the wrap-around all along the path.