The lightweight and slim nature of diffractive opticalelements (DOEs) has led to a large range of applica-tions. These include Gaussian to super-Gaussianbeam conversion,1 intracavity diffractive mode selec-tion,2 optical interconnection,3 wavelength separa-tion,4 and displays.5
As the use of DOEs has become ever more popular,there has been a concurrent increase in the develop-ment of the design algorithms used to optimize theirphase profiles. The earliest design methods6 claimedefficiencies of approximately 75% and an image spotintensity variation of �15%. Methods used today,such as the symmetrical iterative Fourier transformalgorithm7 or simulated annealing,8 can give effi-ciency percentages in the high 90s and nonuniformi-ties below 1%.
As developments in the design algorithms con-tinue, it increasingly becomes the case that the majorfactor contributing to losses in efficiency and in-creases in nonuniformity is not the ability of the al-gorithm to optimize the phase profile, but the errors
introduced by the fabrication process. As algorithmdevelopments offer diminishing improvements in themodeled output, it becomes increasingly likely thatany gain is lost once the element has been fabricated.These issues can become critical for high-power laserapplications. Figure 1 shows a profile designed foruse in fiber coupling of a frequency-doubled Nd:YAG.This design gives very low (1.8%) nonuniformity. Anyincrease in this value can lead to hot spots and there-fore fiber damage. It is hoped that by understandinghow DOE performance is affected by these errors andincorporating this knowledge into the design loop,algorithms that generate gratings with a strongertolerance of fabrication issues can be developed. Toachieve this we will study simple fan-out elements inan attempt to understand the basic processes in-volved.
In this paper we will concentrate on the multimaskreactive ion etching technique for producing DOEs.9The principal sources of fabrication error for thismethod are mask misalignment, etch depth error,and feature rounding. The ultimate aim of analysisinto these errors is to design elements with greatertolerances. The previous work in this field has largelyfocused on improving tolerance to etch depth er-ror,10,11 which is often considered the dominant error,however the relative impact of misalignment androunding errors increases as feature size is reducedand will therefore have an increased influence.
2. Fabrication Process
We begin by outlining the multimask reactive ionetching fabrication technique. Figure 2 illustrates the
A. J. Caley ([email protected]), A. J. Waddie, and M. R. Taghizadehare with the School of Engineering and Physical Sciences, Heriot-Watt University, David Brewster Building, Riccarton, Edinburgh,EH14 4AS, UK. M. Braun is with the Fakultät für Physik, 76131,Karlsruhe, Germany.
Received 10 August 2006; revised 17 November 2006; accepted 1December 2006; posted 18 December 2006 (Doc. ID 73949); pub-lished 3 April 2007.
steps in the process as used at Heriot-Watt Univer-sity.9 First, the substrate is coated with a layer ofphotoresist, an e-beam fabricated binary mask isthen positioned over the substrate and exposed to UVillumination. The resist is developed and baked toleave the desired pattern in hard photoresist. Thesubstrate and photoresist are exposed to reactive ionbombardment, which etches the silica approximatelysix times faster than the photoresist. The remainingresist is removed to leave the etched substrate. In themultimask process this process is repeated with fur-ther masks to give the desired profile. The subse-quent masks are aligned under a microscope tomarkers laid down by the first mask.
A. Misetch
Etch depth error occurs when the substrate is eitheroveretched or underetched for a given mask level,resulting in either too great or too small a phase delaycompared with the desired profile. This problem typ-
ically occurs because of changeable etch rates in thereactive-ion chamber. The accuracy achievable usingthe reactive ion etch chamber is typically 5 nm peretch. It has been noted that to achieve nonuniformitybelow 10% for a DOE designed to operate at 633 nmrequires a total etch depth accuracy of 9–14 nm.12
This corresponds to a phase error of approximately��50. Misetches manifest themselves in greaterzeroth-order energy so the problem is often overcomeusing off-axis designs. Where this is not possible adouble-sided DOE has been proposed to separate sig-nal and noise.13
B. Misalignment
Mask misalignment occurs when the second and latermasks are not aligned precisely with the featuresfrom the first mask. The result of this error is thatfeatures that should be etched are not and featuresthat should not be etched are. This produces incon-sistencies between the fabricated and desired phaseprofiles. Figure 3 illustrates how misalignment gen-erates small feature size anomalies. Comparing the
Fig. 4. Scanning electron microscope image demonstration fea-turing rounding and misalignment.
Fig. 1. Beam shaping applied to fiber coupling. Example showncontains 512 � 512 pixels and 16 phase levels and is designed foroperation with a frequency-doubled Nd:YAG source at 532 nmcapable of delivering 20 mJ in 10 ns pulses. For the design shownefficiency is 88% and nonuniformity 1.8%.
Fig. 2. Steps involved in two mask reactive ion etching of aDOE.
Fig. 3. Effect of misalignment on the DOE phase profile. (a) In-correctly aligned mask overlying the photoresist coated substrate.(b) After UV illumination misalignment is transferred to the resistetching, which in turn copies the misalignment to (c) the substrate.(d) The desired profile.
final profiles in Figs. 3(c) and 3(d) demonstrates theprofile errors that can occur. Misalignment featurescan be seen in the scanning electron microscope im-
age shown in Fig. 4. A mask misalignment of 3 �m9
can be considered typical, this figure is independentof feature size and thus becomes a greater issue forelements designed with smaller pixels.
C. Feature Rounding
It is generally the case that design algorithms pro-duce profiles based on square pixels with sharp cor-ners and of consistent size. In reality both thephotolithographic process and the e-beam writingtechniques cause rounding of features and lead tovariations in pixel size. The scanning electron micro-scope image shown in Fig. 4 illustrates how thisfeature rounding influences the structure of DOEfeatures.
Fig. 5. Effect of misetch on efficiency for (a) 4 � 4 grating and (b) 5 � 5 grating and on nonuniformity for (c) 4 � 4 grating and (d)5 � 5 grating.
Table 1. Specifications for Modeled, Error-Free Elements
There are a number of other sources of fabricationerror, which we will not be examining in detail in thispaper. Overexposure during the UV illuminationstage can result in an enlargement of the features.This is primarily an issue for deep etches where thephotoresist thickness varies, necessitating longer ex-posure times for the thicker resist regions and there-fore overexposing the thinner regions. A rotation ofthe binary mask relative to the substrate during thealignment stage is essentially another form of mis-alignment and produces similar errors in the profile,it is however harder to model and is currently beyondthe scope of our modeling capability. Nonverticalsidewalls and rounding in the vertical plane are alsoissues that lead to discrepancies between the actualand desired profiles.
Although not caused by the fabrication processother sources of error are interference effects. Theseoccur when the order separation approaches the spotsize. Waddie and Taghizadeh14 have shown that thiscan be accounted for during the design stage.
3. Modeling of Errors
The first stage of the modeling process is to designphase profiles for use in the analysis. This was car-ried out using the closed form simulated annealingalgorithm.8 The algorithm was used to design a num-ber of different gratings to allow analysis of how theyare affected by systematic errors. The gratings de-signed are all quantized to 16 phase levels, requiringfour mask levels corresponding to phase delays of��8, ��4, ��2, and � for our design wavelength; thisis fairly standard in the multimask etching processand also allows us to investigate the relative impor-tance of different masks. In this paper we refer to themasks corresponding to a ��8, ��4, ��2, and � etchas level 0, level 1, level 2, and level 3, respectively.The profiles that we have investigated are 2 � 2,3 � 3, 4 � 4, 5 � 5, 6 � 6, 7 � 7, and 8 � 8 fan outs.The modeled efficiency and nonuniformity for theerror-free elements together with the element dimen-sions are shown in Table 1. All the designs are on-axis. Odd elements include the zeroth order and aremade up from adjacent “on” orders. For even ele-ments alternate orders are “on,” and the zeroth orderis not included.
The simulation of mask misalignment is carriedout in the following way: The profile is split into itsdifferent mask components, and each of the pixels inthe phase profile is then expanded to A � B subpix-els. Generally we use 10 � 10 as this gives a goodbalance between computation time and resolution oferror analysis. To model the changes due to maskmisalignment each mask is shifted by an integernumber of subpixels. This process allows differentmisalignments to be applied to each of the masks.The alterations to the phase profile resulting fromrounding of corners are simulated by “turning off” thesubpixels located at the corners of the features asso-ciated with each mask to leave a circular outline. The
degree of rounding modeled is determined by thenumber of pixels turned off at each corner. The accu-racy of the approximation to a circular feature islimited by the number of pixels available, in a similarway to drawing pixelated circles using computer soft-ware. Finally, the phase profile is reconstructed withthe errors incorporated by summing the differentmask components. Misetch is incorporated at thispoint by applying a weighting factor to the phase foreach mask during the reconstruction. The resultingoutput is modeled by computing the Fourier trans-form of the profile. For the work described in thispaper, misalignments in both the x and y directionsare considered. Both negative and positive misalign-ments in each direction are investigated. The degreeof rounding will be varied from 1 to 10 where 1 rep-resents a single subpixel at the corner being turnedoff, and 10 represents a fully circular feature.
To compare the quality of the output from DOEswith varying degrees of rounding and misalignmentwe use the parameters nonuniformity, �R, and effi-ciency, �. We define the nonuniformity as
�R � max�F�X, Y� � G�X, Y�Gmax
�, (1)
where F�X, Y � is the resulting intensity of the diffrac-tion order in position �X, Y �, G�X, Y � is the desiredintensity for the diffraction order at �X, Y � and Gmax
Fig. 6. Influence of misalignment on efficiency for 6 � 6 grating.
Table 2. Second Derivative of Parabolic Fit to Misetch Efficiency Plots
is the peak intensity in the desired output pattern,and efficiency as
� ��p F�X, Y�� g�x, y�
, (2)
where p is the set of desired “on” diffraction orders,F�X, Y � is the resulting intensity of the diffractionorder in position �X, Y �, is the set of all pixels in theelement, and g�x, y � is the input intensity at the pixellocated at �x, y �.
A. Misetch
Misetch analysis was carried out on each mask. Themisetch was varied from ���16 to ��16 in steps of��160 for the mask being considered and set to zerofor the other three masks. This corresponds to arange of �42 to �42 nm for 633 nm light and a fusedsilica substrate. We use the notation that a positivemisetch occurs when the profile is underetched. Fig-ures 5(a) and 5(b) show the effect of misetch of eachmask level on efficiency for the 4 � 4 and 5 � 5gratings. The curves generated are of a parabolic
form, although it should be noted that the turningpoint of the curves is generally not located at thepoint of zero misetch. This is not as unexpected as itmay seem. The algorithm used to design the DOEsforces a quantized profile with an even distribution ofthe available levels. Ballüder and Taghizadeh15
showed that improved results can be achieved bynonuniform quantization, and the results shown heresupport this. By analyzing the second derivative ofthe parabolic function we can determine how quicklythe efficiency drops with increased misetch. The re-sults of this analysis are shown in Table 2 and give aconsistent trend. Level 0 is more affected by misetchthan level 1 which is more affected than level 2. Level3 is the mask most affected by misetch for even de-signs and least influenced for odd designs. This can beexplained as it is well known10 that misetch producesexcess zeroth-order energy. For the odd gratings thisis included in the desired intensity of the efficiencycalculation, for even gratings it is not. This impliesthat misetch of the � mask is the main driver behindexcess zeroth-order intensity.
Fig. 7. Influence of misalignment on nonuniformity for (a) 5 � 5 grating, (b) 7 � 7 grating, and (c) 6 � 6 grating.
Fig. 8. Influence of feature rounding on efficiency for 6 � 6grating.
Fig. 9. Influence of feature rounding on nonuniformity for 6 � 6grating.
Analyzing the nonuniformity variation with misetchsupports the observations from the efficiency results,producing two distinct trends, one for odd gratings andone for even gratings. Figures 5(c) and 5(d) show thenonuniformity data for misetch of the 4 � 4 and5 � 5 gratings. On this occasion we can see a lineardependence on misetch rather than the parabolic de-pendency for efficiency. For all the cases studied wesee that increasing the misetch of level 2 increases thenonuniformity more rapidly than level 1, which in turnincreases more rapidly than level 0. Once again theinfluence of level 3 is dependent on whether the zerothorder is included in the desired orders; where it isincluded, as is the case in Fig. 5(d) we see a dramaticincrease in the nonuniformity influence of level 3 andconsequently a large increase in the nonuniformity ofthe desired output. Where the zeroth order is not oneof the desired orders, as in Fig. 5(c), level 3 has theleast influence on the nonuniformity.
B. Misalignment
Each mask layer has been analyzed with misalign-ments ranging from �10 to 10 subpixels, equating to�1 to 1 full pixel misalignment. This analysis wascarried out on misalignment in the x direction, y di-rection, and diagonally in the xy direction. As eachmask is misaligned all other masks are given zeromisalignment. Intuition would suggest that increas-ing the misalignment error will decrease the qualityof the output. Misalignment in the positive and neg-ative directions will give similar results, and errors infabrication will have a greater impact the deeper theetch depth. However, our results indicate that not allof these assumptions are consistently valid.
In every case we have observed, misalignment ofthe ��8 mask has the least influence on both effi-ciency and nonuniformity. Similarly the ��4 maskconsistently has more influence than the ��8 maskbut less than both deeper masks for both qualitymeasures. However, the � mask does not consistentlyhave a greater effect than the ��2 mask. For effi-ciency results the majority of the cases we have stud-ied show the � mask to have the most influence, theonly exceptions being the 2 � 2 and 4 � 4 fan outs,however the difference in the influence between the �and ��2 masks is relatively small. Again with theexception of the 2 � 2 and 4 � 4 gratings the effi-ciency plots are generally fairly symmetric as shownin Fig. 6 for the 6 � 6 grating.
An analysis of the effect of misalignment on non-uniformity, as defined by Eq. (1), shows much greater
variation than the efficiency. The results can be cat-egorized into 3 groups. First, those closest to the ex-pected result where the � level has the greatest effecton the non-uniformity and misalignment in the pos-itive and negative directions give similar results. The5 � 5 element misaligned in the y direction shown inFig. 7(a) is a good example of this. The second cate-gory exhibits a strong degree of asymmetry, wherethe direction of misalignment is important and oftenhas the ��2 mask having the biggest influence on theresult, at least in one direction. A typical example ofthis is shown in Fig. 7(b). In the final category weobserve a crossing of the dominant level. This is il-lustrated in Fig. 7(c) where the � level dominates forsmall misalignments, but the ��2 level becomes dom-inant as the misalignment increases.
C. Feature Rounding
Simulations of rounding errors were carried out onall seven elements. This was done by applying therounding to each mask level separately (with therounding for the other levels set to zero) and alsoapplied to all four mask levels simultaneously, whichis a more realistic approach but does not allow us toexamine which levels are most affected. It was ob-served that the influence of rounding errors on thefan-out quality had a generally consistent pattern forthe different designs. Efficiency is least influenced byrounding on the ��8 mask followed by the ��4 maskfor all the elements studied. In all cases other thanthe 2 � 2 and 4 � 4 elements the � level has thegreatest influence on efficiency followed by the ��2mask. In the 2 � 2 and 4 � 4 elements this hierarchyis reversed. Figure 8 illustrates the effect of roundingon the efficiency for the 5 � 5 element. In addition tothe above observations on the relative importance of
Fig. 10. Experimental setup.
Fig. 11. CCD image of the 5 � 5 array used for nonuniformitycalculations.
the different levels a parabolic trend can be seen. Thisis expected because of the r2 dependence of area. Thisobservation implies that doubling the rounding error(for example, by halving the feature size) will in-crease the effect of rounding on efficiency by a factorof 4. As we expect, the case where all levels sufferfrom rounding has a greater reduction in efficiency,being approximately equal to the sum of the individ-ual layers. The loss of efficiency caused by the mostinfluential layer with the maximum degree of round-ing is consistently in the region of 5% or 6%. The only
exception to this is the 4 � 4 element where a 12%reduction is observed.
Unlike the efficiency there is no dominant case forthe effect of rounded corners on nonuniformity. Threeof the seven cases, namely the 3 � 3, 6 � 6, and7 � 7 elements, follow the same pattern as the dom-inant case for efficiency, that is the deeper the etchesthe greater the effect; a typical example is shown inFig. 9. For the 4 � 4, 5 � 5, and 8 � 8 elements the��4 mask has become more influential than the ��2mask, although in the 4 � 4 and 8 � 8 case the
Fig. 12. Experimentally acquired data. Efficiency shown at the top of each plot, nonuniformity at the bottom.
standard order of importance applies for higher de-grees of rounding. The 2 � 2 element is the mainexception with the ��2 mask having the least effectfollowed by the ��8 mask; the ��4 mask has mosteffect.
4. Experimental Results
Each of the elements studied has been fabricated,using the multimask process discussed in Section 2,to make a comparison with the modeled results formisalignment. Each element has been fabricated tentimes with target misalignments of �3, 0, 3, and6 �m for the �, ��2, and ��4 masks, and a 10 �mfeature size. The ��8 mask is not misaligned as thisis the level that contains the alignment markers. Tomake our measurements the setup shown in Fig. 10was used. The light meter intensity reading istaken when the beam passes through the DOE andan aperture, which restricts the light incident onthe meter to the signal window. This is divided by theintensity reading when the beam passes through anormal glass slide to give the efficiency. Nonuniformityis determined by taking an image grab of the patternusing a CCD camera; the image from the 5 � 5 arrayis shown in Fig. 11. The relative intensities of theorders are determined from the image grab and usedto calculate the nonuniformity by using Eq. (1).
The efficiency measurements are displayed in Fig.12. The main observations we can make from theefficiency results are as follows: The general trendthat efficiency drops more as misalignment is in-creased is the same as observed in our modeling. It ishard to say this is a linear relationship as observedearlier because we have too few data points for eachelement. The relative impact of the different masksdiffers from that observed in the modeled case; al-most all readings show the ��2 mask having thegreatest impact followed by the � mask. Although, asexpected, the ��4 mask has the least influence, it iscloser than expected to the two deeper levels. A pos-sible explanation for the switch in influence betweenthe � and ��2 masks is that the method of measuringefficiency is not perfectly accurate as it includes alllight within the signal window. For example, thiswould include unwanted zeroth-order energy in theeven gratings. In addition, it is not possible to recreatedifferent errors in the misalignment while keeping theother sources of error perfectly constant. In particularit was observed that the ��2 level of the elementintended to have no error was overetched by 21 nm.
The experimental results for nonuniformity are lessconclusive. It is thought this is again because of inac-curacies in the etch depths, in particular the ��2 levelof the zero misalignment element. Nonuniformity ismuch more sensitive to such an error than efficiency;consequently a number of our plots show improvementwhen they are misaligned. In truth this is more likelyto be attributable to improved etch depth accuracy.Although these experimental results do not verify themodeling they do indicate how variations in othersources of error can have a considerable impact on thequality of the output.
5. Conclusion
We have presented the modeling of misetch, mis-alignment, and feature rounding in the fabrication ofmultilevel DOEs and analyzed the results of theseerrors on a range of fan-out elements. A number ofinteresting observations has arisen from this analy-sis. First, we have seen that the previously observedresult that misetch results in unwanted zeroth-orderenergy can be further refined to state that it ismisetch of the � level that is the dominant source ofthis problem. Furthermore we see that misetch canactually improve efficiency, supporting the use of un-evenly distributed quantization levels. Misalignmenterrors are often strongly asymmetric, which may of-fer a means of increasing tolerance to this error. Itwas also observed that both the � and ��2 levels canbe dominant. The simulation of rounded featuresgenerally followed intuition with an r2 dependenceand deeper etches having greater influence. Havingmodeled rounding, the next step is to include it in thedesign process reducing its impact. A further possi-bility for increasing tolerances is to apply bulk phaseshifts, which do not affect the modelled output butwhich will affect the distribution of pixels beingetched in each mask. Finally, experimental work wascarried out to verify the misalignment results. In thecase of efficiency this was successful at indicating thesame general trends as observed in the modeling;however the nonuniformity results are generally notin agreement. It is believed that this is attributable tothe difficulties in isolating one source of error andkeeping the others constant.
The authors acknowledge an Engineering and Phys-ical Sciences Research Council in the UK studentshipand a Basic Technology Grant. The authors acknowl-edge the support of the Network of Excellence onMicro-Optics for part of this work.
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