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terative algorithm for the design of free-spaceiffractive optical elements for fiber coupling

artin J. Thomson, Jinsong Liu, and Mohammad R. Taghizadeh

We present a design method based on the Gerchberg–Saxton algorithm for the design of high-performance diffractive optical elements. Results from this algorithm are compared with results fromsimulated annealing and the iterative Fourier-transform algorithm. The element performance is com-parable with those designed by simulated annealing, whereas the design time is similar to the iterativeFourier-transform method. Finally, we present results for a demanding beam-shaping task that wasbeyond the capabilities of either of the traditional algorithms. The element performances demonstrategreater than 85% efficiency and less than 2% uniformity error. © 2004 Optical Society of America

OCIS codes: 050.1970, 070.2580.

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. Introduction

ver the years, advances in computing power havenabled the application of diffractive optical elementso increasingly complex beam-shaping problems. Toate, the main design algorithms include simulatednnealing1,2 �SA� and the iterative Fourier-transformlgorithm3–5 �IFTA�, which are well documented andested. Recently, genetic algorithms6 have becomeore widely used as a design technique, either in con-

unction with another method7 or on their own. How-ver, each of these techniques encounters difficultieshen one is tackling demanding beam-shaping prob-

ems such as the uniform illumination problem fornertial confinement fusion8 or the design of multifunc-ional diffractive optical elements producing homoge-ized inputs to individually mounted fibers.9SA is a computationally intensive algorithm that is

mplemented with a closed-form solution of the Fou-ier integral. Changes that degrade the perfor-ance of the element are accepted with a probability

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The authors are with the School of Engineering and Physicalciences, Heriot-Watt University, David Brewster Building, Ric-arton, Edinburgh, EH14 4AS UK. The e-mail address for M. J.homson is m.thomson@hw.ac.uk.Received 7 August 2003; revised manuscript received 19 Decem-

er 2003; accepted 19 December 2003.0003-6935�04�101996-04$15.00�0© 2004 Optical Society of America

996 APPLIED OPTICS � Vol. 43, No. 10 � 1 April 2004

here �E is the change in a defined merit functionnd T is a temperature parameter. By carefully se-ecting the annealing parameters and slowly reduc-ng the temperature throughout the optimization, wean produce an excellent solution for the design prob-em. The major drawback is that the computationalime required scales as �MN�2, where M and N arehe number of pixels in the x and y directions, respec-ively. This dependence prevents use of SA for de-igns containing a large number of pixels.Genetic algorithms are capable of producing high-

erformance elements. In addition, it is possible tose a fast Fourier transform �FFT� to calculate theerformance, so it is a reasonably fast algorithm.he usefulness for elements with a large number ofixels is limited by the fact that several designs areequired at each step, making the memory require-ents too high.In 1972, Gerchberg and Saxton10 proposed an iter-

tive scheme for phase retrieval that has been used iniffractive optic design. A schematic of the algo-ithm is shown in Fig. 1. The algorithm uses aFT11 to propagate between the element plane and

he output plane. This produces a much faster al-orithm than SA because it reduces the dependencen the number of pixels to 2�MN�log2�MN�. Con-traints are applied in each plane, causing the algo-ithm to converge toward the required performance.or the purposes of this paper, the Gerchberg–Saxton

GS� algorithm refers to phase constraints that di-ectly quantize the phase to the required number ofevels and an output constraint that forces the targeto match the desired output intensity. The phase iseft unchanged by the output constraints. The hard

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uantization employed in the GS algorithm causeshe algorithm to stagnate before a useful solution isound. The result of this is that the algorithm pro-uces high-efficiency elements but with poor unifor-ity.The IFTA is based on the GS algorithm with aodification made to the phase constraints in the

lement plane to overcome the stagnation problem.he modification is described in depth byyrowski,3–5 but amounts to the introduction of an

terative quantization where the direct quantizations performed only during the final iteration. Thisllows the phase values to cross the threshold levelsuring the cycle and break the stagnation. Thisodification improved the uniformity in comparisonith the GS algorithm but still does not allow designs

o be performed with uniformities of much less than0%. A direct binary search, with SA at a temper-ture of zero, can be used to improve the uniformityharacteristics, but these improvements come at thexpense of computational speed.In 1992, Prongue et al.12 suggested an algorithm

ased on the GS algorithm, with a modification to theutput constraint, as opposed to the element con-traints modified in the IFTA. This was extendedecently by Liu and Taghizadeh8 to produce Gaussiano super-Gaussian beam shapers with high efficiencynd low mean square errors. In this paper we usehis principle to design Fourier plane beam-shapinglements13 that produce discrete diffraction orders inhe far field. The results from this new design pro-ess are compared with the traditional techniques ofA and the IFTA. Elements are then demonstrated

or a design problem that proved beyond the scope ofhe traditional algorithms.

. Modified Algorithm

he algorithm described by Liu and Taghizadeh8 isffectively a three-step design process. The firsttep generates an estimation for the required phaserofile with geometrical optics. This initial phasestimation is then optimized with the GS algorithmsee Fig. 1� until stagnation is reached. At thisoint, the output constraint is modified, and the uni-

Fig. 1. Schematic of the GS algorithm.

ormity is optimized until a satisfactory solution ischieved.For the research described here, the algorithm was

educed to a two-step design process in which in therst step we optimize the efficiency of the grating and

n the second step we optimize the uniformity.For the first step, a random phase distribution is

sed as the input to the GS algorithm, which runsntil a stable solution is reached. The phase con-traints used in the algorithm state that the elementust be phase only, i.e., no amplitude dependence,

nd quantized to Q levels. Q � 2N where N is theumber of masks used in a binary optics fabricationrocess.The output constraint in the GS algorithm is F�u,

� � T�u, v�, where F�u, v� is the output constraintpplied in the design process and T�u, v� is the de-ired irradiance in each diffraction order. The phaseemains unchanged.

At the second stage, the output constraint is mod-fied to yield

Fmod�u, v� � T�u, v� � c1�T�u, v� � G�u, v�� (1)

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G�u, v�� c2

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here G�u, v� is the irradiance produced by the phaserofile from the previous iteration and c1 and c2 areonstants to be selected in the design process. A fewore iterations are carried out until the element per-

ormance is acceptable. Which constraint to use andsuitable constant are chosen at each new iteration.n examination of the modified constraints shows

hat, when the output is too high, the target is low-red and vice versa. A graphic illustration is pro-ided in Fig. 2.

. Application to Fourier Beam-Shaping Elements

. Comparison with Simulated Annealing and theterative Fourier-Transform Algorithm

he performance of the algorithm was tested againstA and the IFTA for several different elements. Theesign characteristics that we used for comparisonere diffraction efficiency and uniformity, or recon-

truction, error. The diffraction efficiency � is de-ned as

� � G�u, v�. (3)

he uniformity error �R is given by

�R � max�1 �G�u, v�

T�u, v�� . (4)

The three designs used for comparison are aat-top beam shaper, an element producing four setsf diffraction orders in a 2 2 configuration with aat top at each set of orders, and the Heriot-Watt

1 April 2004 � Vol. 43, No. 10 � APPLIED OPTICS 1997

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niversity crest �shown in Fig. 3�. All designs con-ain 16 phase levels.

The flat-top beam shaper contains 256 256 pix-ls. The output consists of a circular flat top of 25iffraction orders in diameter, situated 4 orders offxis in x and y so that the fabrication errors do notrovide an intensity spike in the center of the design.The four-way fan-out considers a signal window

ontaining approximately 400 400 orders with theth order exactly in the middle of the square. Thelement was designed with 1500 1500 pixels.The university crest was embedded in a signal win-

ow containing approximately 100 100 orders sit-ated 4 orders off axis in x and y so the 0th order doesot affect the element performance. The designedlement contained 512 512 pixels.The comparison of element characteristics for the

ifferent algorithms is shown in Table 1. Because ofhe large size of the four-way fan-out, a hybrid ap-roach was used instead of SA. This consisted of our

ig. 2. Illustration of the principle of the modification: �a� shows�u, v�, the output from the previous iteration; �b� shows F�u, v�,

he modified constraint; �c� shows T�u, v�, the desired output. Itan clearly be seen that the modified constraint is a mirror imagef the output from the previous iteration about the desired output.

ig. 3. White on black image of the Heriot-Watt University crest.n the design, the white areas are the desired on diffraction orders.

998 APPLIED OPTICS � Vol. 43, No. 10 � 1 April 2004

esigning the initial phase profile using the IFTA andmproving the uniformity characteristics by applying

direct binary search.The information in Table 1 shows that the perfor-ance of the elements designed with the new ele-ents is comparable to those produced with SA.he elements designed by the new algorithm displayslightly higher efficiency than either of the tradi-

ional techniques, possibly because of the two-stepature of the design process. In addition, the timeaken to design the elements is similar to theime taken by the IFTA and is much lower than theime required for SA to reach a satisfactory solution.

. Fiber Coupling Elements

e have previously reported on diffractive elementshat were coupled to individually mounted fibers with00-�m cores, producing a flat-top intensity distribu-ion at each fiber face.9 The elements presentedere for two- and four-way fan-outs. At the time,ight- and ten-way fan-outs were also desirable, buthe time required to complete the designs was pro-ibitive. The development of the modified algo-ithm has removed this time constraint.

The geometry of the connector used is shown in Fig.. The eight-way fan-out couples light to the fibersituated in a ring at the connector edge, whereas theen-way coupler illuminates all fiber faces.

The flat tops are created when the diffraction or-ers are packed as closely as possible before interfer-nce effects occur.14 As a result, the circular flatops at the fiber faces contained 29 diffraction orders

ig. 4. Geometry of the connector used for the eight- and ten-wayan-outs. Situated at the center of each connector site are 400-�more fibers.

Table 1. Comparison of Design Algorithms

Design Algorithm � �%� �R �%� Run Time

Flat top IFTA 91.89 25.93 2 min, 29 sSA 89.85 4.55 24 hModified 92.38 2.42 6 min, 32 s

Four way IFTA 86.76 21.83 2 hHybrid 85.23 4.75 �1 monthModified 88.63 3.58 5 h

Crest IFTA 88.64 81.97 14 minSA 77.62 13.87 10 daysModified 89.81 4.19 28 min

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n diameter. This led to the design requiring phaserofiles with 16 phase levels and more than 4000 000 pixels.The designs took two days to complete with the

rst day used to create the initial profile with the GSlgorithm and the second day devoted to the selectionf constants and the generation of the final designs.

Fig. 5. Simulated performance of the eight-way fan-out.

Fig. 6. Simulated performance of the ten-way fan-out.

he final design for the eight-way fan-out had � �

6.16% and �R � 1.90%. The ten-way fan-outchieved � � 85.83% and �R � 1.82%. Simulatederformances for the eight-way and ten-way fan-outsre shown in Figs. 5 and 6, respectively.

. Conclusion

e have presented an iterative algorithm for theesign of Fourier plane beam-shaping elements,ased on a modification to the output constraint ofhe GS algorithm. The comparisons with SA andhe IFTA have shown that the modified algorithm canroduce diffractive elements that perform as well ashose produced by SA in a time scale comparable tohe IFTA. The power of the modified algorithm haseen demonstrated by elements produced for fiberoupling applications. The modified algorithm com-leted a design of more than 4000 4000 pixels inwo days, achieving element performances of greaterhan 85% efficiency and less than 2% uniformity er-or.

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