Diffractive optical elements have an optical function-ality that far surpasses that of conventional optics,and they are applied extensively for optical comput-ing and material processing tasks.1,2 The most com-mon form of diffractive optical element used is thediffractive phase element ~DPE!. DPE’s employphase-only modulation. They are characterized byhigh efficiency, while maintaining good optical func-tionality. DPE’s are increasingly used to sculpt theintensity of an input wave front after a specified prop-agation transformation, i.e., beam shaping.
Of the general class of beam-shaping DPE’s, weconsider only a specific type: pattern-formation el-ements ~PFE’s!. PFE’s are space invariant, possess-ing rapidly varying, pseudo-random phase profilesthat reconstruct discontinuous wave fronts in the farfield. On operation in the focal plane, PFE’s areidentical to large-scale fan-out devices, producingnoise-free output patterns constructed from manyclosely packed focal spots.3
Applications for PFE’s include display, target illu-mination, and laser welding. The large number ofdiffraction orders that can be manipulated effectivelyallows for the formation of complicated intensity pat-terns. For display, these patterns can be imagedsuch that individual focal spots are not resolved bythe eye, giving the illusion of a uniform image.
For high-power applications, PFE’s are especially
The authors are with the Department of Physics, Heriot-WattUniversity, Edinburgh, EH14 4AS, UK.
Received 19 May 1997; revised manuscript received 3 September1997.
9132 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997
valuable. Many applicable substrates are available,e.g., fused silica for UV to near-infrared wave-lengths,4 with very high laser damage thresholds.In material processing, such elements can be useddirectly with spatially incoherent sources, such as theexcimer laser, to produce smooth output profiles.There are also instances in which high-frequencymodulation of the output intensity ~because of theclosely packed spots! does not reduce the usefulnessof the element, e.g., owing to thermal smoothing ef-fects.5
In Section 2 we examine the design of PFE’s. Spe-cial attention is paid to the phase-encoding schemeemployed within the design process, which governshow a sampled phase profile is formed into a realiz-able surface-relief structure. We consider the choiceof encoding scheme with regard to the computationaleffort that it requires, and its overall effect on ele-ment performance. A new phase-encoding schemeis demonstrated that is well suited for efficient off-axis pattern formation. In Section 3 we considerDPE design for an on-axis pattern formation with apixelated encoding scheme, and an off-axis patternformation with both pixelated encoding and the newencoding scheme. In Section 4 corroborating exper-imental results are presented.
2. Theory
The output profile of a DPE in the far field is found byuse of the Fourier integral.6 This can be stated as
U~m, n! 5 * *2`
`
exp@iu~x, y! 2 i2p~mx 1 ny!#dxdy,
(1)
where U~m, n! is the amplitude of the individualdiffraction orders and u~x, y! is the phase profile of the
DPE. The intensity profile of the diffraction ordersis commonly known as the power spectrum.
The computational effort needed to solve the Fou-rier integral is proportional to N4, where N is thecharacteristic size in each direction of the signal win-dow. For the PFE’s considered in this paper thesignal window consists of many diffraction orders~N . 512!. In this case the calculation of the Fourierintegral is extremely laborious, inhibiting its use forPFE design.
The Fourier integral is equivalent to the Fouriertransform ~FT!. Many fast Fourier transform ~FFT!algorithms exist that perform a discrete version of theFT in a computationally efficient manner. For N 3N sample points the computational effort needed tocalculate the FFT of a two-dimensional ~2-D! DPE is7
2N2 log~N!. The FFT becomes ~many! orders ofmagnitude faster than the equivalent integral calcu-lation for large N. Because of this speed, many al-gorithms have been developed that apply the FFT inthe design of DPE’s for wave-front transformation.One notable example is the iterative Fourier trans-form algorithm ~IFTA!, which is applied in this paper.
The major drawback of the FFT comes from itsrepresentation of the DPE as a sampled phase profile.Current fabrication technology requires a DPE with asurface relief that is formed from flat structures. Tolet us represent this DPE with a sampled phase pro-file, it would have to be heavily oversampled. This isimpractical owing to the large number of sampledphase sections typically used in PFE’s.
A. Phase-Encoding Schemes
The phase-encoding scheme determines how a sam-pled phase profile is translated into a realizable DPE.The phase-encoding scheme assigns the sampledphase to distinct physical structures, e.g., regularsquares ~pixels!. There is diffraction from the edgesof these structures, which can limit the performanceof the final element. In general, we can consider thechosen surface relief as adding a further modulationfunction to the output pattern, which is dependent onthe phase profile of the DPE. This can be expressedas
Pmn 5 dmn@u~x, y!#uFT$exp@iu~x, y!#%u2, (2)
where u~x, y! is the sampled DPE profile and Pmn isthe power spectrum of the ~m, n!th diffraction order.This modulation function represents the correction tothe power spectrum that is due to diffraction effectsfrom the surface-relief profile that are not present inthe sampled version.
There exist many generalized encoding schemesthat offer a large degree of freedom in the form of thesurface-relief structure. Such encoding schemes aredesigned to permit maximum variation in thesurface-relief profile while maintaining a form thatcan be expressed in a simple manner. Throughparametric optimization of these relief structures,better overall performances can be obtained thanwith a phase-encoding scheme with less freedom.
One common example is the geometry of trapezoidsin evenly spaced stripes8; this has been applied suc-cessfully to 2-D fan-outs.9 The power spectrum cor-rection for the relief structure must be found bymeans of integration. For this reason, further opti-mization is again laborious, limiting the use of arbi-trary encoding schemes for these problems.
B. Pixelated Encoding Scheme
The most direct encoding scheme is the pixelated one,where each sample point represents one pixel on theDPE. For a pixelated element made up of M 3 Npixels, the modulation function for the ~m, n!th dif-fraction order is the well-known
dmnp 5 sinc2~myM!sinc2~nyN!, (3)
where sinc~x! 5 sin~px!y~px!, as shown in Fig. 1.The modulation term in this case is not dependent onthe phase structure of the DPE, so the correction tothe FFT does not require an integral calculation.
For on-axis pattern formation, the effect of the pix-elated structure is to reduce the diffraction efficiencyslightly and to distort the profile of the output pat-tern. By including the modulation term within thepropagation transformation, by means of Eq. ~2!, onecan remove the distortion in the output pattern.Any efficiency loss cannot be regained in this manner,since the efficiency reduction is a consequence of theeffect of diffraction from the physical structure of theelement. As the size of the output pattern in rela-tion to the size of the signal window increases, themaximum efficiency decreases, and the distortion inthe output pattern also increases. The small scale ofthis efficiency loss, as is shown in Fig. 2, for smalloutput patterns within the signal window ~;0.25!,does not seriously inhibit the use of a pixelated struc-ture.
Fig. 1. Profile of the modulation function dmn arising from thepixelated encoding geometry.
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C. Angled-Pixels Encoding Scheme
The resultant distortion in the far field from pixelatedencoding for elements that generate off-axis patternsis more severe than that of the on-axis case. Thedistortion of the output pattern increases rapidlywith the distance that the signal pattern is placed offaxis. The associated efficiency loss becomes a sig-nificant problem for even very small displacementsfrom the optical axis. Figure 3 details this loss for asquare intensity pattern, 1⁄4 the size of the signalwindow, displaced fully, diagonally, and off axis.
On realization of a DPE, any errors in the elementprofile incurred during fabrication result in noise dis-tortion in the output profile centered on axis.10 Forimproving DPE performance it is well known thatmoving the pattern off axis will separate it from themajority of noise distortion in the output profile.11
Despite the lower efficiency obtained, off-axis diffrac-tive elements remain a desirable option.
The efficiency loss found in off-axis elements isagain derived from the physical structure of the ele-ment. We examine an alternative encoding geome-try that will be more favorable for this class ofelement, on the basis of the geometry of trapezoidalfeatures in evenly spaced stripes. This is an en-hancement of the pixelated phase-encoding scheme,where the M 3 N individual pixel structures aredeformed in one direction to form arbitrary shapetrapezoids. The geometry of one such structure isdetailed in Fig. 4~a!, where the ~ j, r!th trapezoid of
Fig. 2. Maximum efficiency of an on-axis, pixelated DPE as afunction of the pattern window size.
Fig. 3. Maximum efficiency of the off-axis, pixelated DPE as afunction of the displacement of the pattern window from the opticalaxis.
9134 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997
phase height ujr is bound between a stripe defined at~r 1 1!yN and ryN, where N is the total number ofstripes, and has corner coordinates @$ajr, ryN %, $bjr,~r 1 1!yN %, $a~ j11!r, ryN %, and $b~ j11!r, ~r 1 1!yN %# inthe x–y plane. There are M such trapezoids in eachstripe.
The power spectrum for a trapezoidal encodedDPE, for m Þ 0, is
Pmn 51
~2pmN!2 U(r51
N
(j51
M [(expFiSujr 2 2pnrN DG
3 $exp@2ipm~ajr 1 bjr!#
3 sinc@m~bjr 2 ajr! 1 nyN#
2 exp@2ipm~aj11r 1 bj11r!#
3 sinc@m~bj11r 2 aj11r! 1 nyN#%)] . (4)
If the edges of each trapezoid are arranged in thesame direction, and at the same angle, then for largeM Eq. ~2! approximates
Pmn >sinc2~md 1 nyN!
~2pmN!2 U(r51
N
(j51
M [(expFiSujr 2 2pnrN DG
3 $exp@2ipm~ajr 1 bjr!#
2 exp@2ipm~aj11r 1 bj11r!#%)]U2
, (5)
where d is the translation of each pixel edge, suchthat
d 5 ~bjr 2 ajr!. (6)
Fig. 4. ~a! Detail of one structure for the phase-encoding geome-try of trapezoids in evenly spaced stripes. ~b! Detail of one stripefor the angled-pixels phase-encoding geometry.
If each trapezoid is considered to be equally spaced@Fig. 4~b!#, then the power spectrum further reduces to
Pmn >sinc2~md 1 nyN!sinc2~myM!
~MN!2
3 U(r51
N
(j51
M HexpFiSujr 2 2pnrN
22pmj
M DGJU2
. (7)
This is more recognizable as the discrete FT of thephase profile sampled at every feature, multiplied bya new modulation function da~m, n!, which is inde-pendent of the element phase profile. That is, thisnew geometry maintains exactly the phase profilecontribution term as found from the IFTA but altersthe modulation function such that
dmna 5 sinc2~md 1 nyN!sinc2~myM!. (8)
The altered modulation function is the product of twosinc-squared functions, as before, but the n-directionterm does not necessarily peak on axis. Figure 5shows the altered modulation function when d 521yN.
With d chosen carefully, the altered modulationfunction can overlap the desired pattern to a greaterextent than can the previous modulation function.This translates to a gain in efficiency of the fabricatedDPE. Since dmn
a is independent of the elementphase profile, the propagation transformation can bealtered without requiring integration.
3. Design
We design a number of on- and off-axis elementsusing a modified IFTA. A phase-encoding geometryis chosen that takes into account the associated mod-ulation function correction to the FFT.
A. Iterative Fourier Transform Algorithm
The IFTA is widely used12–14 to design DPE’s. Thedesired output pattern for the DPE is defined only in
Fig. 5. Profile of the altered modulation function dmna for the
angled-pixels encoding scheme ~d 5 21yN!.
terms of an intensity profile, and so the phase isunimportant. The IFTA takes advantage of thisphase freedom to find an input wave front that formsthe desired intensity pattern on propagation to theoutput plane. We performed the optimization bymoving between the input and the output planes ofthe system, using the FFT as the propagation trans-formation. At each plane operators are applied thatrepresent the required wave-front attributes: uni-form intensity at the DPE, and the desired intensitypattern at the output. A full mathematical descrip-tion of the IFTA is given by Wyrowski.15
We design a number of on- and off-axis elements,using the IFTA with a modified propagation trans-formation to represent the chosen phase-encoding ge-ometry. We look at the unencoded and the pixelatedencoding cases for the on-axis case and also at theangled-pixels encoding case for off-axis pattern for-mation. The expected output profile for each ele-ment is evaluated with a Fraunhofer integralcalculation. The diffraction efficiency and the non-uniformity are used to characterize the performanceof the DPE. The diffraction efficiency h is defined asthe total normalized intensity in the output pattern,W:
h 5 (m,n[W
Pmn, (9)
where Pmn is intensity of the ~m, n!th diffractionorder. The nonuniformity DR measures signal fidel-ity. It is defined as
DR 5Pmax 2 Pmin
Pmax 1 Pmin, (10)
where Pmax and Pmin are the maximum and the min-imum intensity values, respectively, for the “on” dif-fraction orders within the pattern. Note that a lowDR is desirable.
B. On-Axis Element Design
The effect of the alteration to the design algorithmwas examined for a DPE with an output profile of anon-axis, uniform square. The pattern size was~127 3 127! within a signal window size of ~M, N! 5512. The two pixel-encoded, 8-level solutions, withand without signal compensation, are compared byuse of the Fraunhofer integral in Table 1. The fidel-ity of the standard IFTA solution is poor when en-coded as a pixelated element. When the modulationfunction correction is applied within the IFTA, thereis a dramatic improvement in signal fidelity. We
Table 1. Comparison of Design Algorithms for On-Axis PatternFormation
PropagationTransformation
EncodingGeometry
Efficiency~%!
Nonuniformity~%!
FFT Pixelated 89 29dmn
p FFT Pixelated 88 7
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note that the diffraction efficiency is not significantlyaffected by this modification to the design process.
C. Off-Axis Element Design
For an examination of the performance of the newencoding scheme, three off-axis DPE’s were designedwith the same desired output pattern. The signalwindow is again chosen to be of size ~M, N! 5 512.The chosen binary intensity pattern ~Fig. 6! is of size128 3 128 orders and is placed fully off axis by 4 ordersin each direction. The theoretical performance ismeasured by use of the Fourier integral, Eq. ~1!. Theperformance of the pixelated encoding scheme, withand without the modulation function correction, iscompared with that of the angled-pixels encoding ge-ometry in Table 2.
The pixelated solution produces the best fidelity forthe output profile. For the angled-pixels geometrythe efficiency is improved over the pixelated geome-try by approximately 5%, and the fidelity was slightlydiminished owing to the approximations made in themodulation function calculation. Although these el-ements have a fidelity that is suitable for most appli-cations, a parametric optimization could still beapplied to the surface-relief profile to increase thesignal fidelity if required. Note, however, that onecalculation of the diffraction pattern of the elementwith the Fraunhofer integral required many timesthe computational effort of the whole design proce-dure with the IFTA, demonstrating the exceedinglylaborious nature of any further optimization process.
Fig. 6. Binary pattern of 128 3 128 pixels used in the designprocess.
Table 2. Comparison of the Design Algorithms for Off-Axis PatternFormation
PropagationTransformation
EncodingGeometry
Efficiency~%!
Nonuniformity~%!
FFT Pixelated 80 39dmn
p FFT Pixelated 78 5dmn
a FFT Angled pixels 83 9
9136 APPLIED OPTICS y Vol. 36, No. 35 y 10 December 1997
4. Experimental Results
A number of pattern-formation elements were de-signed with the modified IFTA for a pixelated encodinggeometry. All the elements were 8-phase-level struc-tures, made up of 512 3 512 pixels, and reconstructeda binary intensity image of size 128 3 128 orders.Two elements produced on-axis patterns, the first auniform grid and the second a uniform cross. All el-ements were fabricated in fused silica16 with a periodof 2 mm. The output efficiency was measured for theelements for a He–Ne laser ~l 5 633 nm!.
Figures 7 and 8 are photographs of the recon-structed output profiles for each on-axis element.The films in both cases were overexposed to highlightthe noise surrounding the signal pattern. Both ele-ments have an experimental diffraction efficiency of84%, excluding Fresnel losses, to compare with anexpected value of 89%.
Two other elements were designed, each recon-structing the same binary intensity pattern ~Fig. 6!fully off axis by 4 orders in each direction. The first
Fig. 8. Photograph showing an on-axis grid pattern output. Thegrid structure is formed from lines composed of 127 3 3–5 diffrac-tion orders.
Fig. 7. Photograph showing an on-axis cross-pattern output.The cross structure is formed from lines composed of 127 3 11diffraction orders.
element employed the pixelated encoding geometry,and the second used angled pixels. A photograph ofthe reconstructed image for the angled-pixels ele-ment is shown in Fig. 9. The photograph was sim-ilarly overexposed so it would clearly show the noisesurrounding the signal and the faint mirrored imagereflected about the optical axis. The pixelated ele-ment had an efficiency of 70% compared with the 75%obtained from the angled-pixels case, excludingFresnel losses in both cases. As predicted, the newencoding geometry produced an '5% increase in ef-ficiency of the element. The measured efficiency ofall the elements was slightly less than the theoreticalvalues. This reduction in performance can be attrib-uted to inaccuracies in the DPE relief structure in-curred during the fabrication process.9
5. Conclusion
The importance of the phase-encoding scheme chosento represent the sampled phase profiles obtained withthe IFTA has been shown. A new encoding geome-try consisting of angled pixels was introduced; it wasshown to have higher diffraction efficiency than themore common pixelated encoding for off-axis patternprojection. A fabricated element was demonstrated,with the angled-pixels structure, which recon-structed a complex binary intensity pattern, off axis,with a diffraction efficiency of 75%. The diffractionefficiency was shown to increase by approximately5% over an equivalent pixelated element.
Fig. 9. Photograph showing the reconstructed pattern from Fig.6, positioned diagonally off axis by 4 diffraction orders.
The authors thank N. Ross and A. Waddie for theircontributions to the fabrication of the diffractive ele-ments featured in this paper.
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