The shaping of conventional laser beam profiles intoarbitrary far-field intensity patterns by means ofphase-only scalar-domain diffractive optical elements~DOE’s! has found application in many diverse areasof optical processing, ranging from the generation ofarrays of low-power optical beams for parallel opticalcomputing1,2 to the creation of specific beam profilesfor high-power laser processing of materials.3
In this paper we consider high-power laser beam-shaping applications in which the separation be-tween adjacent diffraction orders is small. Weexamine the consequences of interference betweenadjacent mutually coherent diffraction orders and de-termine the conditions that must be satisfied to min-imize such interference. It should be noted that formost high-power beam-shaping applications ofDOE’s, the optical system in which they will be de-ployed is fixed; and it would be difficult, costly, orboth to alter, for example, the focal lengths and ma-terials or the laser wavelength and beam widths ofthe lenses. Therefore any minimization of interfer-ence effects must be achieved by alteration of theDOE.
The complex amplitude profile generated by a DOE
The authors are with the Department of Physics, Heriot-WattUniversity, Riccarton, Edinburgh EH14 4AS United Kingdom.A. J. Waddie’s e-mail address is [email protected].
Received 8 February 1999; revised manuscript received 18 May1999.
imaged through a Fourier lens of focal length f asshown in Fig. 1 is given by
p~x, y! 5
A expFjp~x2 1 y2!
lf Gjlf *
2`
`
* P~u, v!
3 expF2j2p
lf~ux 1 yv!dudvG , (1)
where ~u, v! is a point in the DOE plane and ~x, y! isa point in the focal plane of the lens. The DOEtransmission function P~u, v! for the phase-only ele-ments considered in this paper is
P~u, v! 5 exp@ j2pnd~u, v!yl#, (2)
where d~u, v! is the surface relief profile of the DOE.The intensity and phase distributions in the focalplane of the lens are given by the modulus and argu-ment of the amplitude distribution, respectively.
There are a number of general design specificationsthat any DOE must satisfy, irrespective of the par-ticular beam-shaping task to be performed and theoptical system in which the element will be deployed.These are the overall efficiency ~h0!, the reconstruc-tion error ~DR!, and the minimum feature size ~dmin!.
he reconstruction error, which is a measure of thedelity of the generated far-field intensity pattern tohe target intensity pattern, is defined as
DR 5 maxm,n[}
U1 2Imn
a
ImntU , (3)
where Imna is the actual and Imn
t is the target inten-sity in the ~m, n!th diffraction order and } is the setof diffraction orders needed to generate the desired
1 October 1999 y Vol. 38, No. 28 y APPLIED OPTICS 5915
fti
tradtbtms
t
w
5
far-field intensity profile. The target intensities Imnt
are determined by one dividing the desired overallefficiency of the DOE ~h0! by the number of orders in}. The lowest acceptable limit on both the overallefficiency and the reconstruction error are typicallyspecified by the DOE designer, but the minimumfeature size is usually limited by the photolitho-graphic process used in the manufacture of the DOE.For example, in the case of the in-house facilities atHeriot-Watt University, the minimum feature size islimited to ;1 mm.
There are several different optimization tech-niques4–6 that can be used to calculate the DOE sur-ace relief profile which is needed to generate thearget intensity pattern in the Fourier plane of themaging lens. These include simulated annealing,7,8
direct binary search,9 iterative Fourier-transform al-gorithms,10 and finite-element methods.11 Withthese optimization techniques, used either singly orjointly, high-efficiency ~.65%!, low reconstructionerror ~,2%! far-field intensity patterns have beengenerated12 for a wide range of applications. Theone-dimensional ~1-D! separation of the diffractionorders is given by
Sab 5 Uf lH a@T2 2 ~al!2#1y2 2
b@T2 2 ~bl!2#1y2JU , (4)
where f is the focal length of the optical system usedo image the far-field intensity pattern, T is the pe-iod of the DOE, l is the illuminating wavelength,nd a and b are the diffraction order numbers. Theiffraction order numbers are limited by the need forhe deviation of a diffracted order from the normal toe less than 610° for the scalar-domain approxima-ions to remain valid. Any orders that are deviatedore than this require a rigorous diffraction analy-
is13 and are out of the scope of this paper. This
Fig. 1. ~a! Schematic of the DOE imaging system showing theparameters used throughout this paper. ~b! Far-field intensitydistribution of a single diffracted spot. s is the diffraction-limitedspot size.
916 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999
limit on the usable diffraction orders enables Eq. ~4!o be simplified to
Sab 5 Uf l
T~a 2 b!U . (5)
The extension of Eq. ~5! to cover the two-dimensional~2-D! separation between orders ~m, n! and ~a, b!produces the following:
Smanb 5f l
T@~m 2 a!2 1 ~n 2 b!2#1y2. (6)
Generally, the period of a scalar-domain DOE is ofthe order of 1–10 mm for visible and near-infraredwavelengths, producing nearest-neighbor separa-tions of ;50–500 mm for a typical optical processingsystem.
2. Coherent Interference
It is desirable for beam-shaping elements to producea uniform far-field output with the individual ordersmerging together to form a continuous profile. Thismerging of the separate orders is usually achieved byone increasing the period of the DOE ~T!. As theperiod of the DOE increases, the separation betweenadjacent diffraction orders @given by Eq. 5!# decreasesuntil the individual orders cannot be discerned, pro-ducing a smooth far-field beam profile, provided thatat least one full period of the DOE is illuminated.However, as the diffraction order separation relativeto the diffraction-limited spot size decreases, inter-ference effects between the diffracted orders can de-grade the uniformity of the generated intensityprofile. The critical parameter governing this deg-radation in uniformity is the relative spot separationand is defined as the ratio of the 1-D separation ofadjacent diffraction orders to the diffraction-limitedspot size:
sR 5f lyT
1.22f lyD5
D1.22T
. (7)
The effect of diffraction-order interference on a DOEfar-field intensity profile is demonstrated in Fig. 2,which shows a section of the far-field output of a 1 3100 fan-out element ~h 5 78.2%, DR 5 0.18%! de-signed to produce a continuous line 1 cm long at 500mm. In this experiment we varied the relative orderseparation by altering the diameter of the illuminat-ing beam ~l 5 633 nm! and thus changing thediffraction-limited spot size. Figure 2 demonstratesthe effect that interference has between adjacent dif-fraction orders on the uniformity of the output. ThesR 5 1.47 curve has a variance of 0.0056 compared
ith variances of 0.0062 and 0.008 for the sR 5 0.97and sR 5 1.22 curves. The lower variance corre-sponds to a lower nonuniformity demonstrating thatchanging the relative spot separation produces achange in the observed nonuniformity of the grating.
The interference of two mutually coherentdiffraction-limited spots is governed by the spatial
tot
h
flvastrtetrvTdn
8y
separation of the spots and their relative phases.Assuming that both spots are equally bright, we cangive the 1-D intensity cross section across the spotcenters by14
I~x! 5 U2 J1@p~x 2 x0!ys#
p~x 2 x0!ys
1 exp~ jf!2J1@p~x 2 x1!ys#
p~x 2 x1!ysU2
, (8)
where J1 is a Bessel function of the first kind; s 51.22f lyD is the radius of the diffraction-limited spotof the optical system under consideration; x0 and x1are the center of the first and second spots, respec-tively; and f is the relative phase between both spots.Figure 3 shows the interference between two equally
Fig. 2. Section of output from a 1 3 100 fan-out element of period.4 mm. The beam diameters used were 10, 12.5, and 15 mm,ielding spot separations of 0.97s, 1.22s, and 1.47s, respectively.
Fig. 3. Interference between two equally bright spots for differentrelative phases.
bright spots with spot center separations of s and 2sfor different relative phases between the beams.
To calculate the interference effects between dif-fraction orders, Eq. ~8! is expanded in two dimensionso include more than two spots. It is assumed thatnly the eight nearest-neighbor orders lying closest tohe ~m, n!th order produce significant interference
effects. This assumption is justified by reference toFig. 3 which shows that, for a spot separation of s, theinterference effect at x 5 62s is minimal. The in-tensity at the peak of the ~m, n!th diffraction-orderspot is given by
Imn 5 U (j5m21
m11
(k5n21
n11
2hjk
J1$p@~m 2 j!2 1 ~n 2 k!2#1y2sR%
p@~m 2 j!2 1 ~n 2 k!2#1y2sR
3 exp@ j~ujk 2 umn!#U2
, (9)
where Smjnk is the order separation given by Eq. ~6!,jk is the uninterfered intensity in the ~ j, k!th
diffraction order, ujk is the phase of the ~ j, k!th dif-raction order, and s is the radius of the diffraction-imited spot. Figure 4 is a plot of the maximumariation in intensity at different relative beam sep-rations for the full 2-D nearest-neighbor case de-cribed by Eq. ~9! and for the 1-D three-order casehat is defined by setting k 5 n in Eq. ~9!. Theelative phase of each of the nearest-neighbor beamso the central beam was allowed to take one of 20qually spaced values between 0 and 2p. The cen-ral intensity was then calculated and the processepeated for all the permutations of these 20 phasealues to determine the largest intensity variation.he intensity variation in Fig. 4 is strongly depen-ent on the relative order separation and is still sig-ificant at moderately wide separations ~sR . 5s!.
The minimum interference occurs when the peak ofone order lies near the zeroes of the Bessel functionsdescribing the intensity profiles of the surrounding
Fig. 4. Maximum variation in beam intensity with relative orderseparation for the 2-D nearest-neighbor and 1-D three-order cases.
1 October 1999 y Vol. 38, No. 28 y APPLIED OPTICS 5917
e5m
5
orders. By inspection of Fig. 4, the jth minimum ofthe 1-D three-order curve lies at
sRj 5 ~ j 1 0.229!, (10)
where j is an integer $1. The minima defined by Eq.~10! coincide with mimina of the 2-D nearest-neighbor curve that can be considered to be global inthe sense that the minima are significantly below thesurrounding maxima of the curve. Equation ~10!was derived by inspection because an analytic solu-tion to the zeroes of Eq. ~9! depends heavily on theprecise far-field phase profile of the element beinginvestigated. This is different for every DOE as theoptimization techniques that are used exploit free-dom in the far-field phase to generate the desiredfar-field intensity pattern. When the DOE is de-signed to have a relative order separation lying nearone of these minima, the interference effects on thefar-field output can be reduced. For example, de-creasing the number of orders used to generate thedesired far-field profile results in a smaller DOE pe-riod and therefore a larger relative order separation.
By applying Eq. ~9! to the theoretical far-field out-puts of a number of different beam-shaping elements,one can gain some measure of the influence that in-terference has between adjacent orders on the non-uniformity of the DOE. Figure 5 shows thevariation in the nonuniformity with relative orderseparation for the beam-shaping elements specifiedin Table 1. The variation in nonuniformity in Fig. 5
Fig. 5. Variation in DOE nonuniformity with relative order sep-aration for three different beam-shaping elements.
918 APPLIED OPTICS y Vol. 38, No. 28 y 1 October 1999
follows the same general trend observed in Fig. 4with the minima given by Eq. ~10! being valid to areasonable approximation. It appears that there issignificant ~.10%! nonuniformity in the DOE outputfor relative order separations below 2. Above thisvalue the nonuniformity obeys an exponentially de-caying sinusoidal curve with minima in nonunifor-mity below 2% at the relative order separations givenby Eq. ~10!.
To demonstrate the benefits of one using a smallernumber of orders to generate a particular far-fieldoutput, two DOE’s were designed for deployment in alaser materials processing application and to producean annular flattop 144 mm across at f 5 108 mmunder illumination from a UV ~l 5 355-nm! source.The first used the 66th orders as the extrema of theannulus and the second used the 69th orders. Theperiods of the DOE’s were calculated to be T6 5 3.195mm and T9 5 4.793 mm, corresponding to relativeorder separations of 1.45 and 0.96, respectively.Figure 6 shows the theoretical far-field output for the66th- and 69th-order DOE’s with the interferenceffects excluded and included. As predicted by Fig., the DOE with relative order spacing closest to theinima given by Eq. ~10! is least affected by the
interference between adjacent orders. The interfer-ence effect on the 66th-order DOE is still significant~;5%! but it is qualitatively lower than that observedfor the 69th-order DOE.
3. Conclusions
We have demonstrated that interference between ad-jacent diffraction orders can have a significant effecton the uniformity of DOE’s. The precise variation inthe uniformity with relative order separation de-pends on the relative phases of the individual ordersand therefore varies tremendously from DOE to
Fig. 6. Comparison between annular flattop beam-shaping ele-ments with output from the grating when interference effects areignored and output when they are included.
gratings for array generation,” J. Opt. Soc. Am. A 7, 1514–1528
DOE. However, there are relative order separationsat which the interference between adjacent diffrac-tion orders is minimized and is effectively indepen-dent of the relative phases between the orders. Byensuring, where possible, that DOE’s are designedsuch that their relative order separations lie nearthese minima, one can largely eliminate the degra-dation of uniformity caused by interference.
The authors acknowledge the assistance of P.Blair, B. Layet, and N. Ross in the research outlinedin this paper.
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