. INTRODUCTIONiffractive optical elements (DOEs) are optical devices
hat produce desired diffraction patterns from an incom-ng laser beam by phase distortion. The lightweight andlim nature of these elements makes them ideal for manyaser applications, including Gaussian to super-Gaussianeam conversion,1 intracavity diffractive mode selection,2
An area of considerable research activity has been theevelopment of DOEs that produce different patterns de-ending on the wavelength of the incident light. Develop-ent of such DOEs is a challenge, as these elements are
nherently wavelength specific. Two approaches haveeen used in overcoming this problem. The first is to de-ign DOEs for operation in the Fresnel region.6–8 In thisegion both phase modulations caused by the DOE struc-ure and those caused by the differing path lengths fromhe element to the image plane are wavelength depen-ent. These two mechanisms can be varied independentlynd thus give the required degree of freedom to generateifferent output patterns for different wavelengths.9
The disadvantage of exploiting Fresnel region mecha-isms is that the image becomes very sensitive to the po-ition of the image plane. To overcome this, Fraunhofferegion methods have been developed. Dammann10 first in-roduced the use of phase delays beyond 2� to give the re-uired extra degree of freedom; this is illustrated in Fig.. For any given pixel in the element, the phase delay isiven by
�� = 2��n��� − 1�h/�, �1�
here n��� is the wavelength-specific refractive index, hs the etch depth for the pixel, and � is the wavelength of
ight we are considering. It is assumed that the refractivendex of the surrounding medium is 1. Choosing h1� to in-rease the phase delay by 2� essentially has no effect onhe diffraction pattern produced by �1. The difference be-ween the phase produced by using h and h1� for a secondavelength, �2, is
2��n��2� − 1��1
�n��1� − 1��2. �2�
For most cases of �2, this is not an integer multiple of�, so, for a given phase delay in �1, there is a choice ofwo phase delays for �2. Increasing the available mul-iples of h1� increases the available choices for �2. Thisechnique has been exploited for the design of harmoniciffractive lenses11 and dual-wavelength patternormation.12
We have recently described13 an alternative algorithmor designing Fraunhoffer region DOEs with greater than� phase delays for multiple-wavelength pattern forma-ion. This method included the extension of bias phase14
o two wavelengths. In this paper we intend to reviewow the technique is applied. In particular, the benefits ofpplying the bias phase enhancement to improve the re-ulting output patterns are investigated. The choice ofavelength in a single-wavelength DOE is not of particu-
ar importance to the design process; the same phase pro-le can be reproduced for different wavelengths by chang-
ng the etch depths. This is not true in the case of dual-avelength DOEs, where the selection of the wavelengthsffects what phase is available. In the final section of thisaper we present an analysis of how the selection of theair of wavelengths used in the design, which we termur wavelength choice, affects the quality of the resultingmages produced by the DOE.
06 Optical Society of America
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194 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 A. J. Caley and M. R. Taghizadeh
. METHODe are aiming to design a single DOE that will produce
ne output image at a given wavelength and a completelyifferent output image for a second wavelength; these twoavelengths are specified prior to beginning the designrocess. Our method is a two-step process. The first steps to determine the optimum unquantized profiles for eachavelength and its associated pattern. This can bechieved by using any of the many algorithms availableor designing DOEs for a single wavelength. We have cho-en the well-known Gerchberg–Saxton algorithm15 fol-owed by the modified iterative Fourier transformlgorithm.16 This combination has been shown to give ex-ellent efficiency, and nonuniformity results after only ahort run time for pattern formation with a singleavelength.1 Running this first step gives �1 and �2, theesired phases for �1 and �2, respectively.The second step is to find the quantized profile that
roduces the desired pattern for each wavelength. To dohis, values of h that cause phase delays greater than 2�re considered. At the beginning of this step the maxi-um value of h and the number of quantization levels are
pecified. These choices are limited by fabrication issues.aghizadeh et al.17 give a general discussion of fabrication
ig. 1. DOE cross section with single-wavelength etch depths h.he option of adding h1� to the etch depth, increasing the phaseelay for �1 by 2�, introduces a choice in the phase delay for �2.
ig. 2. Two-color quantization. Numbers 1–8 show the availableuantization levels. Open circles indicate etch depths that give ahase delay equivalent to �1. Solid circles indicate etch depthsiving a phase delay equivalent to �2. �1 and �2 indicate theuantization error in terms of depth. The method tries to mini-ize the quantization error in terms of phase for bothavelengths.
ssues. Analysis of the bounds for two color elements isimited; our earlier paper13 has shown that etch depthsp to 8� for a 16-level element with a primary wave-
ength of 633 nm and feature sizes of 10 �m can be fabri-ated successfully by using the reactive-ion etching tech-ique.Figure 2 illustrates how the quantization is carried out.
ividing the maximum depth into the number of levelsives the available values of h, represented by the num-ers 1–8 in the figure. The quantization is carried out onach pixel independently. The depths h1 and h2, requiredo give �1 and �2, respectively, can be calculated by rear-anging Eq. (1). As discussed above, adding h1� to h1 and2� to h2 give alternative etch depths that produce theame effective phase delay. As shown in Fig. 2, multiplesf h1� and h2� can be added until the total etch deptheaches the maximum depth limit specified.
We are looking for a technique that can be used for aarge range of possible input wavelengths. The result ofhis, together with the limitations on maximum etchepth, is that it is highly unlikely there is an exact solu-ion that fits for both �1 and �2. To get around this weust compromise between the requirements for eachavelength; we have done this on the basis of quantiza-
ion error. The error for a given level and wavelength isefined as
�L,� = ��L,�eff − ���, �3�
here �L,�eff is the effective phase for that level, deter-
ined using Eq. (1) and removing multiples of 2�. Theuantization error for a given level is then considered toe the maximum of �L,�1
and �L,�2. For each pixel the level
hat minimizes this value is selected. This level may note the optimum for either �1 or �2 individually but is theest fit when both are considered.The quality parameters that we will use in this paper
re efficiency, �; mean square error (MSE), and nonuni-ormity, �R. These parameters are defined as
� =
�P
F�x,y�
�E
g�X,Y�, �4�
here P is the set of desired orders, E is the set of all pix-ls, F�x ,y� is the intensity produced at order �x ,y� by theesigned DOE, and g�X ,Y� is the input intensity at pixelX ,Y�.
MSE =
�P
�G�x,y� − F�x,y��2
�P
�F�x,y��2, �5�
here G�x ,y� is the desired intensity at order �x ,y�.
�R = max�F�x,y� − G�x,y�
Gmax� , �6�
here Gmax is the peak intensity in the desired outputattern.
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A. J. Caley and M. R. Taghizadeh Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. A 195
. Enhancementse have considered two enhancements to improve the re-
ults of this technique. The first of these, bias phase, isiscussed in detail in Section 3. The other enhancementhat we have considered is depth bias. Varying the maxi-um phase for a given wavelength varies the allowed
hase levels, as these are calculated by dividing the maxi-um phase by the number of levels. This in turn affects
he spread, between 0 and 2�, of the effective phase, sohe maximum possible phase shift is not always the besthoice. A poor choice may result in bunching of the phaseevels within the 0 to 2� range. An extreme exampleould be a maximum phase of 8� and four levels. In this
ase each level would have the same effective phase of 2�,hus giving no choice of phase. For this example a choicef 7� as the maximum phase would probably be more ef-ective. To find the optimum value for the maximumhase, the quantization, described above, is run with a
ig. 3. Application of bias phase is equivalent to applying ahift of S1 and S2 to the original values of h1 and h2, respectively.he original positions are indicated by the gray circles, the newositions by the open circles for �1 and black circles for �2.
Fig. 4. Desired intensity pattern.
ange of maximum phase values. The choice that opti-izes some preselected quality measure for the whole
utput is selected. The choice of this parameter may de-end on the application; we have used nonuniformity asefined in Section 2. We will refer to this approach asepth bias.
. BIAS PHASEias phase is a technique, introduced by Balluder andaghizadeh,14 to optimize the quantization step for mul-ilevel DOEs. This process exploits the fact that the appli-ation of a constant phase shift to a phase profile pro-uces the same amplitude profile in the image plane.pplying different bulk shifts prior to quantization affects
he quality of the quantization process, effectively shift-ng the position of the available phase levels relative tohe unquantized phase. By careful selection of this shift,he result was optimized.
We have extended this technique by applying it to thewo-color problem. Bulk phase shifts are applied indepen-ently to the two profiles produced by step 1, prior to theuantization stage. Figure 3 shows how the technique ispplied by applying shifts S1 and S2 to the original h1 and2 in turn, thus altering the quantization error for each
evel. It should be noted that the diagram represents therocess for only one pixel and that the values of S1 and S2ust remain constant for all the pixels in the element.herefore, they cannot simply be chosen such that the de-ired etch depths coincide with our available levels. Ashe bias phase is performed prior to quantization, there iso limit to how fine the shifts can be. The only limit on theagnitude of the shift is that after its application the new
hase should not exceed the maximum phase limit.To examine the effects of bias phase, two 1010 order
quare arrays are designed in a 6464 signal window;ee Fig. 4. The algorithm was run twice, once with a maxi-um �1 phase depth of 8� and once with a maximum �1
hase depth of 6.8�. Both designs have 16 levels. The firstquare is for a wavelength of 442 nm and has its lowereft order located at (2,2) relative to the center of the sig-al window; the second square is for 633 nm and has its
ower left order located at �−12,−12� relative to the cen-er. The algorithm was run with 100 different phase shiftsrom 0 to �. The effects of the different shifts on the effi-iency and nonuniformity are shown in Figs. 5 and 6.
The first observation we can make is that bias phaseroduces much more variation in the MSE than the effi-iency. In the examples shown there is typically a 2%ariation in the efficiency but a 30%–50% variation in theSE. Based on this it will generally be desirable to use
ias phase as a tool for optimizing MSE. As expected,imilar features repeat with a period in shift 1 equal to �1nd shift 2 equal to �2; for example the bright features atpproximately (0.1,0.8) and (1.4,1.7) in Fig. 6(a). � is theifference in phase between adjacent levels, so for Fig. 5,1=1.57 and �2=1.08; for Fig. 6, �1=1.34 and �2=0.91.he plots also indicate that the quality measurements forach wavelength tend to be dominated by the choice ofulk shift for that wavelength. However, this is not truen all cases, especially for the MSE. This suggests that theange of shifts could be limited to between 0 and � for
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196 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 A. J. Caley and M. R. Taghizadeh
ach wavelength; however, there is enough interdepen-ence to suggest that each shift should not be optimizedn isolation.
As the character of the MSE graphs is not obviouslyredictable, it is necessary to continue optimizing this pa-ameter by trying a range of evenly spaced shifts withinhe desired range, at least for a first approximation. Arial and improvement approach is less computer inten-ive; however, it is likely to get stuck in local minima. Itould be used as a second pass to optimize the results fur-her.
Modeling of the best and worst results, based on MSE,or a maximum �1 phase depth of 8� are shown in Figs. 7nd 8, respectively. The algorithm works by selecting theias phase that minimizes the worst of the two MSE val-es (i.e., one for each wavelength). It is therefore not sur-rising that while we see a noticeable difference in the
ig. 5. The effect of applying different bulk shifts to the two de-ired phase profiles where the maximum �1 phase depth is 8�.he four plots show variation in (a) efficiency for �1, (b) efficiency
or �2, (c) MSE for �1, and (d) MSE for �2.
ig. 6. Effect of applying different bulk shifts to the two desiredhase profiles where the maximum �1 phase depth is 6.8�. Theour plots show variation in (a) efficiency for �1, (b) efficiency for
, (c) MSE for � , and (d) MSE for � .
2 1 2
33 nm pattern, the 442 nm pattern shows little change.his is because the particular choice of maximum phaseepth and number of levels tends to give better values ofSE for 442 nm than for 633 nm, so we are effectively op-
imizing the MSE for the 633 nm pattern.
. EFFECTS OF WAVELENGTH CHOICEhen designing and fabricating multiple-wavelengthOEs it is important to consider how the choice of wave-
ength influences the quality of the resulting images. Tonvestigate this we have used the same, two squares, tar-et output intensity profile as in Section 3. Using the 8�6-level case, we left the first wavelength constant at 300m and varied the second from 300 to 1500 nm in steps ofnm (i.e., 300 steps). It is worth highlighting at this stage
hat, in terms of the design process, it is not the absolutealue of the two wavelengths that is important but the ra-io given by
= �n��2� − 1��1/�n��1� − 1��2. �7�
hus when we design a DOE for wavelengths of 300 and00 nm, it will be equally valid for 700 and 1937 nm (forused silica), the only difference being that all of the etchepths will be increased in the latter case. Since the prop-rty we are interested in is , we can justifiably leave �1onstant and vary only �2 in our investigation.
The algorithm was run for three different settings; therst was with no enhancement applied, then with depth
Fig. 7. Modeled results for optimizing the MSE.
ig. 8. Modeled results where the worst values of MSE areelected.
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A. J. Caley and M. R. Taghizadeh Vol. 23, No. 1 /January 2006 /J. Opt. Soc. Am. A 197
ias, and finally with bias phase. The effect of varying n �1, �2, �R1, and �R2 is shown in Fig. 9.
We can draw a number of conclusions from theseraphs. First, and least surprising, as approaches 1, i.e.,1=�2, the efficiencies become very low and the nonuni-ormity very high in all three cases. This is expected, ashe difference in phase for the two wavelengths is verymall (approaching zero as approaches 1) so there isery little scope for generating different phase profiles forhe two wavelengths. The graphs in Fig. 9 indicate thathis is an important effect down to a value of of �0.8. As
continues to decrease below 0.8, the general trend is ofmproving quality, but this becomes much more gradual.
Also of note is that when the second wavelength is ap-roximately a harmonic of the first, giving values of qual to 0.5, 0.33, 0.25, etc., there is a corresponding dropn quality, the efficiencies fall, and the nonuniformitiesise. This effect appears more pronounced for lower-orderarmonics; i.e., =0.5 is the worst. This is a little moreifficult to explain. If we consider our desired phase, ��1or �1, we will get good results when the actual phase isround ��1
+2n�. This will give a phase delay for the sec-nd wavelength of ���1
+2n��. This expression showshat for =0.5, no matter how many values of n are avail-ble to us, we effectively get only two choices, 0.5��1
or.5��1
+�. All the others are simply 2� additions. For 0.33 we get three choices, for =0.25 we get four, and son. Thus the effect becomes less pronounced as the num-er of choices approaches our maximum value of n-four inhis case.
The final point that we can make from analysis of theseraphs is that bias phase consistently gives the best-uality measurements. Depth bias gives better nonunifor-ity than with no enhancement, but there is little differ-
nce in the efficiency. This is because nonuniformity is theuality measure used in the selection of depth bias. Theepth bias results are not as good as bias phase, but thisay be because the original choice, 8�, is close to opti-um. We have not run depth bias and bias phase to-
ig. 9. Results of applying the algorithm with no enhancement,ias phase, and depth bias on (a) nonuniformity for �1, (b) non-niformity for �2, (c) efficiency for �1, and (d) efficiency for �2, forifferent values of .
ether in this work. It is likely that doing so would giveetter results, although independently varying all the pa-ameters involved would be computer intensive. It may behat the optimum choice of maximum phase depth is in-ependent of the phase profiles, since it is the value thatives the best distribution of available phase levels. Thisossibility has not been investigated here, but if it is thease, the choice of maximum phase could be calculatedrior to the application of bias phase with little increasen computer run time. Although the graphs in Fig. 9 comerom just a single target image, the explanations put for-ard to explain the observed features are generally appli-
able.
. CONCLUSIONn conclusion, we have discussed an algorithm for the de-ign of dual-wavelength DOEs and have further investi-ated the benefits of extending the bias phase techniqueo dual-wavelength DOE design and the effect of wave-ength choice on the result. In the examples shown, thehoice of bias phase can produce a variation of up to 2% inhe efficiency and can improve the MSE by up to a factorf 2. We have further established that bias phase shoulde used primarily to optimize MSE and that a spread ofifferent phase shifts needs to be used in the first run.urthermore, we have illustrated the effect the choice ofias phase can have on the quality of the output pattern.e have shown that the choice of the two wavelengths
an have an important effect on the quality of the far-fieldmages. The ratio of the two wavelengths should be lesshan 0.8 (assuming that �1 is the shorter wavelength).econdary wavelengths in the region of the primaryavelength’s harmonics should also be avoided. It is also
uggested that further work should be carried out to de-ermine whether depth bias can be optimized indepen-ently of the desired phase profiles.
CKNOWLEDGMENTShis work was supported by an Engineering and Physicalciences Research Council (EPSRC) studentship and Ba-ic Technology grant GR/S85764. Partial support of theetwork of Excellence on Micro-Optics is also gratefullycknowledged. Corresponding author Adam J. Caley’s-mail address is [email protected].
EFERENCES1. M. J. Thomson and M. R. Taghizadeh, “Diffractive
elements for high-power fibre coupling applications,” J.Mod. Opt. 50, 1691–1699 (2003).
2. K. Balluder and M. R. Taghizadeh, “Regenerative ring-laser design by use of an intracavity diffractive mode-selecting element,” Appl. Opt. 38, 5768–5774 (1999).
3. M. R. Taghizadeh and A. J. Waddie, “Micro-optical andoptoelectronic components for optical interconnectionapplications” in Proceedings of the From Quantum Optics toPhotonics Conference, Acta Phys. Pol. 101, 175–187 (2002).
4. M. W. Farn, M. B. Stern, W. B. Veldkamp, and S. S.Medeiros, “Color separation by use of binary optics,” Opt.Lett. 18, 1214–1216 (1993).
5. H. Hua, Y. Ha, and J. P. Rolland, “Design of an ultralightand compact projection lens,” Appl. Opt. 42, 97–107 (2003).
6. J. R. Sze and M. H. Lu, “Design and fabrication of the
1
1
1
1
1
1
1
1
198 J. Opt. Soc. Am. A/Vol. 23, No. 1 /January 2006 A. J. Caley and M. R. Taghizadeh
diffractive phase element that synthesizes three-colorpseudo-nondiffracting beams,” Opt. Eng. (Bellingham) 41,3127–3135 (2002).
7. Y. Ogura, N. Shirai, J. Tanida, and Y. Ichioka,“Wavelength-multiplexing diffractive phase elements:design, fabrication, and performance evaluation,” J. Opt.Soc. Am. A 18, 1082–1092 (2001).
8. M. Lo, B. Z. Dong, B. Y. Gu, and P. Meyrueis, “Non-periodicdiffractive phase element for wavelength-division(de)multiplexing,” Opt. Commun. 173, 217–221 (2000).
9. J. Bengtsson, “Kinoforms designed to produce differentfan-out patterns for two wavelengths,” Appl. Opt. 37,2011–2020 (1998).
0. H. Dammann, “Color separation gratings,” Appl. Opt. 17,2273–2279 (1978).
1. D. W. Sweeney and Gary E. Sommargren, “Harmonicdiffractive lenses,” Appl. Opt. 34, 2469–2475 (1995).
2. I. M. Barton, P. Blair, and M. R. Taghizadeh, “Dual-
3. A. J. Caley, A. J. Waddie, and M. R. Taghizadeh, “A novelalgorithm for designing diffractive optical elements for 2colour far-field pattern formation,” J. Opt. A, Pure Appl.Opt. 7, 276–279 (2005).
4. K. Balluder and M. R. Taghizadeh, “Optimized phasequantization for diffractive elements by use of bias phase,”Opt. Lett. 24, 1756–1758 (1999).
5. R. W. Gerchberg and W. O. Saxton, “A practical algorithmfor the determination of phase from image and diffractiveplane pictures,” Optik (Stuttgart) 35, 237–246 (1972).
6. J. S. Liu and M. R. Taghizadeh, “Iterative algorithm for thedesign of diffractive phase elements for laser beamshaping,” Opt. Lett. 27, 1463–1465 (2002).
7. M. R. Taghizadeh, P. Blair, B. Layet, I. M. Barton,A. J. Waddie, and N. Ross, “Design and fabrication ofdiffractive optical elements,” Microelectron. Eng. 34,219–242 (1997).
