## Abstract

Building on the optimal-rotation-angle method, an algorithm for the design of inter-element coherent arrays of diffractive optical elements (DOEs) was developed. The algorithm is intended for fan-out DOE arrays where the individual elements fan-out to a sub-set of points chosen from a common set of points. By iteratively optimizing the array of elements as a whole the proposed algorithm ensures that the light from neighbouring elements is in-phase in all fan-out points that are common to neighbouring DOEs. This is important in applications where a laser beam scans the DOE array and the fan-out intensities constitute a read-out of information since the in-phase condition ensures a smooth transition in the read-out as the beam moves from one DOE to the next. Simulations show that the inter-element in-phase condition can be imposed at virtually no expense in terms of optical performance, as compared to independently designed DOEs.

©2004 Optical Society of America

## 1. Introduction

A diffractive optical element (DOE), kinoform, or computer generated hologram is a reflective or transmissive optical component that introduces a spatially varying phase modulation onto an incoming laser beam. By carefully designing the phase modulating pattern, the wavefront of the incident beam can be altered to accomplish for instance deflection, beam splitting (fan out), beam shaping or focusing.

Over the years, several algorithms for the design of DOEs have been developed. Iterative Fourier transform methods [1, 2] and direct search methods [3–5] are two of the more common classes. The optimization method used in this paper is called the optimal rotation angle (ORA) method [6] and will be treated more thoroughly in section 2. It is iterative but does not use spatial Fourier transforms. Like most other algorithms it treats the phases of the target spots, i.e. the phase of the optical field in the fan-out positions, as free parameters that can be manipulated to achieve a high diffraction efficiency.

In some applications, several different DOEs are used together in an array. For instance, the optical analog-to-digital converter scheme demonstrated in [7–9] (see fig. 1) sweeps a laser beam over an array of fan-out DOEs to generate the corresponding digital bit-pattern on an array of detectors. Each element illuminates a different combination of the detectors, and as the beam moves from one element to another the two neighbouring elements are simultaneously illuminating a common set of detectors. To get diffraction limited spots on the detectors, the light from both elements must then be in-phase. Instead of optimizing each element independently of the others, this implies that the whole array must be optimized at once, keeping the light from all neighbouring elements in-phase. This problem was observed in [7, 8] and later a simple solution to the problem was attempted in [9].

In this paper the problem is more thoroughly addressed and an algorithm that uses the few remaining degrees of freedom to optimize the performance of the array is presented. The algorithm was designed with the above mentioned application in mind but is readily generalized to handle other constraints on the output phase.

## 2. The optimal rotation angle method

The optimal rotation angle method (ORA) as formulated in [6] is based on the Helmholtz-Kirchhoff rigorous scalar diffraction integral for the evaluation of the electric field behind the DOE. It can be used for any one-, two- or three dimensional spatial configuration of the target spots and modifications of the algorithm have shown e.g., that it is possible to design DOE structures that are less sensitive to linear depth errors of the DOE surface relief [10]. With some minor changes in the notation, the following section summarizes the essential parts of sections 2 and 3 in ref. [6].

It is assumed that the DOE is divided into rectangular cells, pixels, within which the relief height, or equivalently the imposed phase modulation, is approximately constant. The first part of the algorithm consists of the evaluation of the contribution to the optical field from each pixel to each desired fan-out position. In the following we will frequently refer to a fan-out position as a “target spot” position. The contribution from pixel *k* to the field in the position of the *m*:th fan-out can be written

where ${A}_{k}^{\mathit{\text{inc}}}$
and ${\phi}_{k}^{\mathit{\text{inc}}}$
are the known amplitude and phase values of the incident field in the center of pixel *k*, and *φ*_{k}
the amount of phase modulation imposed by the DOE, which is the quantity to be optimized. *A*_{km}
exp(*jφ*_{km}
) is the complex transfer function from pixel *k* to target spot *m*. If the field incident on pixel *k* is approximated to have a constant amplitude and a linearly varying phase the scalar diffraction integral can be solved and the transfer function is found to be

$$\times \frac{4{A}_{\mathit{k}}^{\mathit{inc}}}{{r}_{\mathit{km}}^{c}}\mathrm{exp}\left(j{k}_{0}{r}_{\mathit{km}}^{c}\right)\frac{\mathrm{sin}\left({\tilde{k}}_{x}\frac{a}{2}\right)\mathrm{sin}\left({\tilde{k}}_{y}\frac{b}{2}\right)}{{\tilde{k}}_{x}{\tilde{k}}_{y}}$$

where *L*_{km}
=*z*_{m}
-*z*_{k}
, *k*
_{0}=|*k*|=2*π*/*λ*
_{0}, ${r}_{\mathit{km}}^{c}=\sqrt{{\left({x}_{m}-{x}_{k}\right)}^{2}+{\left({y}_{m}-{y}_{k}\right)}^{2}+{\left({z}_{m}-{z}_{k}\right)}^{2}}$ is the distance from the center of pixel *k* to target spot location *m*, *k̃*_{x}
=*k*_{x}
+*k*
_{0}(*x*_{k}
-*x*_{m}
)/${r}_{\mathit{\text{km}}}^{c}$
, *k̃*_{y}
=*k*_{y}
+*k*
_{0}(*y*_{k}
-*y*_{m}
)/${r}_{\mathit{\text{km}}}^{c}$
, *a* and *b* are the side lengths of a pixel and *k*_{x/y/z}
is the *x/y/z*-component of the *k*-vector of the field incident on pixel *k*. See also fig. 2a. Note that the transfer function does not depend on the DOE phase modulation *φ*_{k}
. Thus, *A*_{km}
and *φ*_{km}
can be pre-calculated before the optimization loop is entered.

Having established the influence from each pixel on the field in each spot, the optimization can be carried out. In the optimal-rotation-angle method the phase modulation in each pixel is iteratively altered to maximize the power into the desired spot locations. Fig. 2b shows the complex amplitude plane where the field *U*_{m}
in one target spot location is shown together with the contribution from one pixel, *U*_{km}
. With Φ
_{km}
being the difference in phase between the total field in spot *m* and the contribution to the same from pixel *k*, the figure shows that the absolute value of the field in point *m* is altered by Δ|*U*|
_{km}
=*A*_{km}${A}_{k}^{\mathit{\text{inc}}}$
cos(Φ
_{km}
-Δ*φ*_{k}
)-*A*_{km}${A}_{k}^{\mathit{\text{inc}}}$
cos(Φ
_{km}
) when the phase modulation in pixel *k* is altered by Δ*φ*_{k}
.

In each iteration of the algorithm the phase in each pixel is changed by an optimal value Δ${\phi}_{k}^{\mathit{\text{opt}}}$
in order to maximize the sum of Δ|*U*|
_{km}
for all spot locations, i.e., finding the Δ*φ*_{k}
that maximizes ∑
_{m}
Δ|*U*|
_{km}
|pixel *k*. It can be shown that the maximum is found by calculating

$${{S}_{2}=\Sigma}_{m}{A}_{\mathit{km}}\mathrm{sin}{\Phi}_{\mathit{km}}$$

$${\alpha}_{k}=\mathrm{arctan}({S}_{2}/{S}_{1})$$

$$\begin{array}{cc}\Delta {\phi}_{k}^{\mathit{opt}}={\alpha}_{k}& \mathrm{if}{\phantom{\rule{.2em}{0ex}}S}_{1}>0\\ \Delta {\phi}_{k}^{\mathit{opt}}={\alpha}_{k}+\pi & \mathrm{if}\phantom{\rule{.2em}{0ex}}{S}_{1}<0\\ \Delta {\phi}_{k}^{\mathit{opt}}=\pi /2& \mathrm{if}{\phantom{\rule{.2em}{0ex}}S}_{1}=0\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{S}_{2}>0\\ \Delta {\phi}_{k}^{\mathit{opt}}=-\pi /2& \mathrm{if}\phantom{\rule{.2em}{0ex}}{S}_{1}=0\phantom{\rule{.2em}{0ex}}\mathrm{and}\phantom{\rule{.2em}{0ex}}{S}_{2}<0\end{array}$$

Although Δ${\phi}_{k}^{\mathit{\text{opt}}}$
is the optimum change in phase for one individual pixel, a full change of the pixel phase modulation by this amount can sometimes cause instability since commonly “one iteration” of the design algorithm constitutes the optimization of all the pixels in the DOE without intermediate reevaluation of the quantity Φ
_{km}
used in the optimization. This can lead to an over-compensation and therefore instability but can simply be remedied; in this work we made the phase change more gentle by using a modified value for the optimized phase modulation simply as Δ${\phi}_{k}^{\mathit{\text{opt}}\mathit{.}\mathit{\text{mod}}}$
=*const*·Δ${\phi}_{k}^{\mathit{\text{opt}}}$
where *const*≤1.

Due to the fact that it is slightly more favourable to send light into certain target spot locations, the so optimized DOE will in general not produce a fan-out with uniform intensity. To get exactly the desired intensities the sums *S*
_{1} and *S*
_{2} are modified as

$${S}_{2}={\Sigma}_{m}{w}_{m}{A}_{\mathit{km}}\mathrm{sin}{\Phi}_{\mathit{km}}$$

where *w*_{m}
is a weighting factor that is updated according to

where *I*_{m}
is the intensity in target spot location *m* and *p* is a constant that determines the weight adjustment speed. The exact value of *p* is not very critical but should be set sufficiently low to avoid instability. Typically, *p* is set to 0.1–0.4. In fig. 3a the flow of the complete algorithm is outlined. The iterative loop is repeated until the desired quality constraints are met.

## 3. Extension of ORA to DOE-arrays

In this section we consider diffractive structures that consist of several DOEs and therefore add a third index, *n*, to the appropriate variables to indicate the *n*:th DOE. The complex transfer function now has to be calculated from each pixel in each DOE to every target spot location. The target spot locations, as before indexed by *m*, are the same for all elements in the array although their desired intensities can be different from one DOE to another.

In Fig. 3 the array-extended algorithm is outlined side-by-side with the original ORA algorithm. Steps 1–3 as well as the last step are identical apart from the index *n* added to account for the set of DOEs. Step 4 is altered to equalize power across the DOE elements and last but not least, a fifth step that connects the phases across elements to satisfy the array-constraints is added. In the ADC case this step makes sure that spots illuminated by neighbouring DOEs are in-phase.

*step 4 - weight update*

The weights of the spots, *w*_{mn}
, for those *m* and *n* for which ${I}_{\mathit{\text{mn}}}^{\mathit{\text{desired}}}$
>0, are updated so as to equalize the target spot power from each individualDOE according to eq. 5. Further, to equalize the target spot power from the different DOEs, some power is redirected into a special “beam dump” spot. The weight for this spot is updated in the following way:

$$2.\phantom{\rule{.2em}{0ex}}{w}_{n}^{\mathit{BD}}:={w}_{n}^{\mathit{BD}}-min\left({w}_{n}^{\mathit{BD}}\right)$$

where *I*_{n}
=∑
_{m}
|*I*_{mn}
|^{2} is the efficiency of DOE *n* and *Ī* the average efficiency. *b*_{c}
determines the convergence speed and should be chosen small enough to maintain stability, typically *b*_{c}
≈1. The first step in eq. 6 adjusts the weights to get equal efficiency and the second step renormalizes the smallest beam-dump weight to zero to get the best possible efficiency.

During the first few iterations the spot intensities fluctuate rapidly as the phase modulation of the DOEs develop from their original random values. Therefore, the weight adjustment algorithm is first turned on after a certain number of iterations when the spot intensities have been stabilized by the main ORA optimization (step 6).

*step 5 - inter-element coherence*

Generally, the phases *φ*_{mn}
=arg(*U*_{mn}
), which can be viewed as the target phases of the ORA algorithm in step 6, do not fulfill the array specific requirements. The proposed method of establishing the desired inter-element coherence is to replace *φ*_{mn}
by some other phases *φ′*_{mn}
to make the ORA algorithm “aim” for other phases than the actual ones in a way suitable to the specific application. The determination of target phases *φ′*_{mn}
from the previously calculated phase values *φ*_{mn}
is referred to as a “merger”. Taking the ADC application as an example, in the following two different mergers are studied.

The first merger simply averages the phase for each spot position over all elements, i.e., *A*exp(*iφ′*_{mn}
)=∑
_{n}
exp(*iφ*_{mn}
), in which case *φ′*_{mn}
evidently is independent of *n*. With a 5 bit ADC and inverse word generation this gives 10 phase groups, one for each spot position. The second merger method is a bit more advanced and assigns one phase group for every group of neighbouring elements that illuminates the same spot, see fig. 4. As an example, one can see that the spot location where the most significant bit (MSB) is read out should have a high intensity (a“one”) where any of the DOEs numbered 16–31 is illuminated. The phases of the optical field in this position from these DOEs should then be the same, to avoid negative inter-DOE interference, and these phases thus constitute a phase group labelled “2” in fig. 4. Within each phase group one again finds the target phase value as the average of the calculated phase values, from the preceding iteration, within the group.

To generate the full 5-bit Gray code and its inverse using the second merger method, 36 groups are formed. Obviously, the second merger approach is a relaxation compared to the first since the design problem then uses 36 free phase values, one for each phase group, to solve the problem rather than 10. Of course, the design is still more constrained than in the independent design of each DOE for which there would be 32·5=160 free phases.

## 4. Simulated results

To analyze the impact of the inter-element coherence demand on the performance of the array, the two merger methods were compared to the original “no-merger”ORA method for the previously mentioned ADC application. For the statistical evaluation, 300 different design problems were chosen, differing in their positions of the ten target spots. Each problem was solved by each of the three design methods. For all the methods, 100 iterations were used and for each of the 2^{5}=32 DOEs that comprise the entire diffractive structure 32×32 pixels were used in the calculations.

As can be seen in table 1, neither of the inter-element coherent design methods has a significant negative effect on the diffraction efficiency. The table lists the diffraction efficiency of the least efficient DOE, among the 32, averaged over the 300 different problems. The merger approach lowers this value by less than one percent. Moreover, the standard deviation of the efficiencies is small, as is the uniformity error of the whole array among the “on” spots, defined as

Another 200 iterations did not improve the efficiency noticeably but did reduce the uniformity errors somewhat in all cases. However, all these quality measures only consider each DOE separately. Of utmost importance for the function of the entire DOE array is the quantity shown in the last column, which is the average phase difference when comparing all pairs of target spots lit by neighbouring DOEs. Obviously, this value is 0.5*π* for independently designed DOEs, while the merger approach in the modified ORA algorithm reduces it to a mere 1% of its original value. Finally, we notice that the 36-merger design performs slightly better in every respect than does the 10-merger design. This is expected because the 36-merger design uses more degrees of freedom.

To demonstrate the results of the proposed design method, a simulation of a beam sweeping over the mentioned 32-element array similar to the one in fig. 1 was performed. The array was constructed by repeating each 32×32 pixel element four times in each direction, making the effective DOE size 128×128 pixels. This repetition was performed to decrease the target spot sizes and isolate the spots from each other. The illumination was taken as a Gaussian beam with a 1/*e*
^{2} intensity diameter corresponding to the 128×128 element width.

In Fig. 5, a movie clip showing the target spot plane can be found alongside a movie clip from the same simulation for a DOE array designed with the no-merger method. The results are also summarized in fig. 6 that shows the spot intensities as functions of the laser beam position for the same two simulations. To simulate small, but finite sized, high-speed detectors the intensities were integrated over an area with a diameter of approximately the visible spot size in Fig. 5.

## 5. Conclusion

The optimal rotation angle algorithm [6] was extended to allow the generation of inter-element coherent DOE arrays. It was shown that the performance increase for applications that depend on such coherence can be substantial and that the negative impact on efficiency and uniformity due to the added constraints is negligible compared to the original ORA algorithm.

## Acknowledgments

This research was supported by the Chalmers Center for High-Speed Electronics and Photonics (Swedish Foundation for Strategic Research, SSF) as well as the Swedish Research Council.

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