In micromachining for micro-optics, a special tech-nique is required to create continuous analog-resistprofiles for analog micro-optical elements such as mi-crolenses and diffractive lenses. The binary square-type resist profile has been researched and optimizedextensively for the past twenty years to reach thehighest possible level of resolution with the opticalstepper of deep-ultraviolet (DUV) wavelength. How-ever, the development of a fabrication technique forlarge analog-resist profiles in a thick photoresist be-gan relatively recently.
To meet this challenge for analog-resist profiles,several new types of photomask have appeared, suchas gray-scale, halftone, and binary phase masks, forcertain applications other than high-resolution inte-grated circuit devices.1–8 Gray-scale masking exploitsthe continuously varying optical density in the spe-cial patented high-energy-beam-sensitive (HEBS)glass plate to form a continuous relief profile in thephotoresist.2 Gray-scale masks have two main draw-
backs. One is high cost, and the other is strict depen-dence on the optical density of the photoresist beingused. It is necessary to characterize the thickness ofthe resist in terms of the optical density for a specificexposure tool to design a proper optical-density mapon a gray-scale mask. Halftone masks create analogoptical transmittance by use of a square pixel arrayrepresenting continuous optical density.1,3 Halftonemasks can be made capable of creating analog opticaltransmittance for incident light exposure by varyingthe pixel density or size. However, this techniquesuffers from the pixel aperture diffraction effect andalso requires the adjustment of pixel density for aspecific exposure tool. Pure binary phase masks aresometimes used for fabricating high-frequency sinu-soidal gratings in the photoresist with half of theperiod of the mask. However, to date, the phase-grating-mask technique has not been used for creat-ing an analog-intensity profile as an enhancement togray-scale or halftone masks.
The idea of utilizing an alternating phase shift onthe photomask was first introduced in the early 80sas a technique for enhancing the resolution by over-coming the diffraction limit of the imaging system.9Now the phase-shift mask is a mature, standard tech-nique for resolution enhancement in the semiconduc-tor industry and is used mainly for dense periodicpatterns of submicrometer resolution. The phaseshift is usually implemented in the alternating open-ing region of a binary amplitude chrome mask by
The authors are with the School of Optics, University of CentralFlorida, 4000 Central Florida Boulevard, Orlando, Florida 32816.E. G. Johnson’s e-mail address is [email protected].
Received 13 June 2005; revised 11 August 2005; accepted 12August 2005.
etching the mask substrate for a phase shift of ahalf-wavelength.10
In this study a new phase-mask technique is pro-posed, which allows for the fabrication of an analogmicro-optic profile in a thick photoresist. This tech-nique is fundamentally different from the gray-scale and halftone mask techniques in that itutilizes a phase function on the mask plane to cre-ate an analog optical intensity on the wafer plane,while the other two techniques exploit only analog-amplitude functions on the mask plane. We inves-tigated the potential of controlling the opticaltransmittance of the phase mask in a continuousfashion by changing the parameters of the binaryphase grating. Using only the zeroth-order diffrac-tion from the phase grating with a � phase shift, theoptical transmittance can be controlled by simplyvarying the duty cycle of the phase grating. We findthis technique to be a promising alternative to gray-scale and halftone masking techniques in the fieldof analog photolithography. The feasibility of thistechnique was investigated and demonstrated in astandard photolithographic environment.
2. Background
A. Stepper Optics
The stepper being used in photolithography is basi-cally an imaging system for a photomask combinedwith an illuminating system called a condenser. Thephotomask receives collimated illumination fromthe condenser upon exposure and has a complex-amplitude transmittance go�x, y�. This complex-amplitude transmittance function can be consideredthe object function to be imaged by the stepper. If weconsider the stepper as a coherent imaging system,then the imaging amplitude gi�x, y� is given by thefollowing equations10:
G(f, g) � H(f, g)FT�go(x, y)�, (1a)
gi(x, y) � FT�1[G(f, g)]. (1b)
In the above equations H�f, g� is the transfer functionof the stepper, and G�f, g� is the Fourier spectrum ofthe object amplitude go�x, y� at the pupil plane of thestepper. The partial coherence of the source is takencare of by multiplying the object amplitude functiongo�x, y� with the plane-wave component exp�j2��fsx� gsy�� from the source point of the spatial frequency�fc, gc� and by taking the summation of the intensi-ties from all the point sources in the source area.
The stepper is acting as an imaging system witha spatial-frequency filter, cutting off the high-frequency components from the mask object by thepupil edge of the stepper. The size of the pupilradius is proportional to the numerical aperture(NA) of the stepper, and this sets the radius of thecircular transfer function H�f, g�.
B. Binary Phase-Grating Mask for an Analog-IntensityProfile
For periodic patterns like binary gratings, the mini-mum period for cutoff is determined by the wave-length, NA, and partial coherence factor of thestepper as follows10:
pc ��
(1 � �)NA. (2)
Our GCA G-line stepper has 436 nm of source wave-length, 0.35 of NA, and 0.6 of partial coherence factor,giving 0.78 �m of minimum resolvable period on thewafer. The actual cutoff period of the binary gratingon the mask for a 5� stepper is five times bigger to be3.89 �m. So, if the period of the binary grating on themask is smaller than the cutoff period, the 1 andhigher-order diffracted light from the mask would becut off by the pupil aperture at the stepper, and onlythe zeroth order would contribute to form a uniformflat intensity on the wafer. This is the resolutionlimitation for a binary amplitude grating mask.
To create analog-resist profiles with a photomask,we should make the intensity profile coming throughthe mask continuous. This can be achieved by creat-ing an analog-light amplitude with a halftone mask,3as illustrated in Fig. 1(a). This type of mask uses a setof subresolution opaque pixels and a fixed center-to-center subresolution period (pitch). Since the periodis subresolution, smaller than the value of Eq. (2),only the zeroth diffraction order will pass through thestepper system to form the image on the wafer. Theimaging intensity on the wafer plane is proportionalto the relative amount of area not blocked by theopaque pixels and can be calculated with the follow-ing equation:
I � 1 � Apixel�Apitch. (3)
In this equation Apixel and Apitch represent the areas ofpixel and pitch, respectively. Thus, by varying thearea of pixels as a function of position on the mask, itis possible to create analog-intensity profiles comingout of the mask and to fabricate a desired analog-resist profile with the proper resist exposure and de-velopment times.
A drawback of this approach for an analog-resistprofile is that it is plagued by light scattering at theedge of the pixels. This is due to the fact that there isa sharp transition in light amplitude from 0 to 1 atthe boundary of the chrome pixel. This scattered lightcan be stray light in the optical stepper system andcan contribute to degradation in the smoothness ofthe imaging intensity. Avoiding this stray-light prob-lem entails creating the analog-intensity profile with-out using the binary chrome pattern such as a half-tone mask. To address this issue, we came up withthe idea of using a phase-only binary grating maskwith � phase depth. The layout of this mask is similarto the halftone mask shown in Fig. 1(b), but eachpixel square has a � phase shift relative to the back-
ground area instead of opaque chrome. The phase-shifting pixels can be made by patterning thephotoresist, which is transparent for the stepperwavelength. If the period of this two-dimensional(2D) phase grating is smaller than the value of Eq. (2)and the phase shift of the pixels is �, then only thezeroth order will go through the stepper system. Fur-thermore, the intensity of the zeroth-order diffractedlight will depend on the fill factor of the pixel. So thesize of this square pixel can be varied to control thezeroth-order efficiency from this grating. The fill fac-tor is defined as the ratio of the area of pixel to thearea of pitch and is also related to the duty cycle asfollows:
F � w2 � a2�2. (4)
The amplitude of this 2D binary phase grating with a� phase shift can be expressed in the following form:
t(x, y) � �2 rect�x � a�2a ,
y � a�2a � 1
�1
2 comb� x
,y. (5)
By taking the 2D Fourier transform of this expressionand putting �0, 0� in the spatial frequency, we arrived
at the zeroth-order diffraction (DE) efficiency equa-tion below:
DE(w) � 1 � 4w2 � 4w4. (6)
The zeroth-order diffraction (DE) efficiency versusthe duty-cycle plot for Eq. (6) is nonsymmetric withzero at w � 1��2 � 0.707, as shown in Fig. 2. This is
Fig. 1. Two different types of photomask for analog-resist-profile formation: (a) halftone mask and (b) binary phase-grating mask. a, sizeof the pixel; �, pixel pitch.
Fig. 2. Zeroth-order diffraction efficiency versus duty-cycle curvefor a 2D binary phase grating.
due to the fact that the 2D binary phase gratingmakes the zeroth-order efficiency at the half-fill fac-tor �F � 1�2�. The cutoff period for a 2D binary phasegrating is given by Eq. (2), but with the factor of 1��2owing to the presence of diagonal diffraction orders.
This 2D efficiency equation can be solved for theduty cycle as well. So the 2D grating duty cycle w�x, y�as a function of position �x, y� on the mask can beobtained from the desired zeroth-order intensity-transmittance profile I�x, y�. Thus it is possible todesign the 2D binary phase-grating mask for arbi-trary 2D analog-resist profiles.
3. Design and Fabrication of the Phase Mask
A. Exposure Characteristic of the SPR 220-7 Photoresist
To properly design the duty cycle of a binary phase-grating mask for the desired analog-resist profile,we must characterize the exposure response of thethick photoresist in terms of the duty cycle of thebinary phase-grating mask. The SPR 220-7 pho-toresist can be spun to form 7–12 �m of thickness ona fused-silica wafer with good uniformity, and italso works well for both I- �365 nm� and G-line�436 nm� exposure tools. It was spun on a fused-silica wafer at a thickness of 12 �m and was ex-posed with our GCA G-line stepper for exposuretimes of 0.3–3.6 s in a dose-matrix form on thewafer. The exposure intensity at the wafer in ourstepper is 150 mW�cm2, corresponding to a45–540 mJ�cm2 exposure dose range. This waferwas developed in an MF CD-26 developer for4 min,and the developed depth (in micrometers)versus the exposure dose (in mJ�cm2) is plotted inFig. 3. The SPR 220-7 resist responds slowly to thedelivered exposure energy until the exposure timereaches 0.5 s �75 mJ�cm2� and begins to rise rapidlypast that point. After 0.8 s of exposure time, theslope of the exposure curve is slightly reduced and
maintains a linear form until it reaches saturationat �2.8 s (420 mJ�cm2 of dose). Because of this non-linear response of the resist to the exposure dose, aproper amount (0.6–0.8 s) of flat bias exposure isnecessary before exposure with the phase mask.After undergoing bias exposure without the mask,the SPR 220-7 will respond to the delivered analogdose in a smooth, almost linear fashion.
It is possible to numerically convolve the zeroth-order efficiency versus the duty-cycle curve (see Fig.2) of the phase grating with the exposure curve (seeFig. 3) to make a curve of the remaining resistthickness d (in micrometers) versus the duty cycleresulting from the sequential bias exposure and theexposure through the phase mask. This convolutioncurve is useful for estimating the response of theresist to the duty cycle of the phase-grating mask. Italso helps to determine the appropriate range ofduty cycle for a linear response in the resist profile.
To perform this numerical convolution, we need tomatch the scales of the efficiency and exposurecurves. The zeroth-order efficiency DE�w� as a func-tion of the grating duty cycle w can be regarded as theintensity transmittance of exposure light, which isthe ratio of the zeroth-order intensity I�w� to the flatexposure intensity I0. To obtain the absorbed dose,one should multiply DE(w) by the flat exposure in-tensity I0 and exposure time t (in seconds). Thereshould also be an additional term for the flat biasexposure, which is performed before exposure withthe phase-grating mask. This is simply the product ofthe flat exposure intensity I0 �150 mW�cm2� and thebias exposure time tb (in seconds). Thus the totalamount of absorbed dose D�w� (in mJ�cm2) as a func-tion of duty cycle w is given by
D(w) � I0�tb � DE(w) � t�. (7)
Fig. 3. Developed thickness of the SPR 220-7 resist versus expo-sure time in the stepper.
Fig. 4. Remaining thickness versus duty cycle obtained by nu-merical convolution of the resist characterization curve with azeroth-order efficiency curve of a binary 2D phase grating for 0.6 sof bias and 2.6 s of exposure time.
The next step is the polynomial curve fitting of theexposure characteristic curve. The developed depthh (in micrometers) versus the exposure dose D (inmJ�cm2) can be fitted to a sixth-order polynomialh�D� of D. Once we have the coefficients for thissixth-order polynomial, we can simply substitutethe dose parameter with D�w� of Eq. (7), and then itbecomes the curve h�w� of the developed depth h (inmicrometers) versus the duty cycle w. The remain-ing resist curve d�w� (in micrometers) is simply theinitial thickness d0 minus the developed depthcurve h�w�. Figure 4 shows the remaining resistthickness d�w� resulting from this convolution for0.6 s of bias and 2.6 s of exposure with a 2D binaryphase-grating mask. As shown in this curve, the 2Dsquare phase-grating mask undergoes rapid varia-tion in the remaining thickness with duty cycle ofthe phase-grating mask. This property makes the2D phase-grating mask ideal for high-sag micro-optics.
B. Designing the Phase-Grating Mask for anAnalog-Resist Profile
To demonstrate the feasibility of this binary phase-grating-mask approach for the fabrication of analogmicro-optics, we designed a simple square micro-prism of 100 �m width. The height of the microprismwas chosen to be 7 �m, with a base thickness of0.5 �m underneath it. From an optical viewpoint,this base thickness is redundant and is unnecessary,but if the resist height goes to zero at the edge of theelement, the corresponding grating duty cycle on themask would be almost zero, which is practically im-possible. We chose 0.5 �m as the minimum linewidthfor a phase-grating mask, which corresponds to 0.24of duty cycle and 0.6 �m of actual pixel size for the 2Dphase-grating mask with a 2.5 �m period. Accordingto the convoluted remaining resist curve, this wouldresult in 0.5 �m of remaining resist, giving the mi-croprism the desired resist pedestal. Following thesame procedure, we also designed a V-groove elementwith a 7 �m depth and a total width of 100 �m.
There are two approaches to the design of the duty-cycle map of the binary phase grating for the fabri-cation of a desired resist profile. The first is to simplystart from the zeroth-order grating-transmittanceprofile, whose form is the reverse of that of the de-sired resist profile. This is the simplest way of design-ing the duty cycle and it assumes that the response ofresist to the exposure intensity is linear. For a mi-croprism profile such as this, the transmittance pro-file should be of inverted linear-ramp form with thewidth. From the transmittance profile I�x, y�, theduty-cycle map W�x, y� can be computed by solving
Fig. 5. Intensity-transmittance and duty-cycle profiles of thephase-grating mask for (a) the 100 �m microprism and (b) theV-groove.
Fig. 6. One-dimensional target resist profile (solid curve) anddeveloped resist profile (dotted curve) of the microprism resultingfrom the numerical convolution of the exposure curve of the SPR220 and the designed duty-cycle map: (a) from the intensity-transmittance-based phase-grating mask and (b) from the resist-profile-based phase-grating mask.
The above equation for the duty-cycle function is forlower-duty-cycle solutions of the grating-efficiencyequation. For the higher-duty-cycle solution, the mi-nus sign in the equation should be changed to a plus.When designing the desired grating-transmittanceprofile, one should ensure that it is normalized prop-erly to prevent the minimum duty cycle from becom-ing too small to fabricate. The grating-transmittanceprofile was normalized to 0.78 of the maximum trans-mittance, which corresponds to 0.24 of minimumduty cycle for the 2D grating mask.
The second approach is to use the numerical de-convolution method to determine the required doseprofile D�x, y� (in mJ�cm2) for the desired 2D resistheight profile d�x, y� (in micrometers) with certainbias and exposure times and to compute the intensitytransmittance I�x, y� and the duty-cycle profileW�x, y� from that dose profile. The desired 2D resistheight profile d�x, y� (in micrometers) can be used todetermine the required exposure dose profile to pro-duce it by using the polynomial-fitting expression forthe exposure curve h�D� (in micrometers) of the SPR220 resist. The exposure characteristic curve in Fig. 3can be put on the exposure dose D (in mJ�cm2) versus
the developed resist depth h (in micrometers) scaleand be represented by the sixth-order polynomialfunction D�h�:
D(h) � a0 � a1h � a2h2 � a3h
3 � a4h4 � a5h
5 � a6h6.(9)
This function gives the required dose value D (inmJ�cm2) for a certain developed resist depth h (inmicrometers). Since the relation between the devel-oped resist depth h and the desired remaining resistheight d is d0 � h�x, y� � d�x, y� for the total resistthickness of d0, it is possible to substitute the desiredresist depth profile h�x, y� � d0 � d�x, y� in the poly-nomial expression D�h� to obtain the desired doseprofile D�x, y� � D(h�x, y�). Equation (7), relating theexposure dose and intensity, is still valid with thespatial-coordinate parameters �x, y� in place of dutycycle w. Thus, with proper bias and exposure times toclear the SPR-220 resist, the desired dose profile canbe used to compute the required transmittance pro-file by solving Eq. (7) for I�x, y�:
I(x, y) � �D(x, y)�I0 � tb��te. (10)
From the exposure curve of the SPR 220-7, we deter-mined the bias and exposure times to be 0.6 and 2.6 s,respectively. Finally the duty-cycle distribution
Fig. 7. Microscope image of the phase-grating mask fabricated on the PMMA e-beam resist by using the e-beam direct-writing tech-nique.
W�x, y� can be obtained from the required transmit-tance profile I�x, y� by using Eq. (8).
To verify this principle and design method for thephase-grating mask for the analog-resist elements,we first designed simple prism profiles, which basi-cally consist of a linear ramp in a resist profile. Theprism is 100 �m square and 7.5 �m high. The 2Dphase-grating-mask design based on both the trans-mittance and the resist profiles was used to fabri-cate this structure in the SPR 220-7 resist. Thetarget transmittance profile is a linear ramp oftransmittance ranging from 1.66% to 78%. For theresist-profile-based phase-mask design, the linearlyramping resist profile going from 0.5 to 7.5 �m ofresist height was used. Next we designed aV-groove structure by using a 2D binary phase-grating mask designed from both transmittanceand resist profiles. For the transmittance-profile-
based phase-mask design, an inverted V-type pro-file with 0.78 of transmittance at the center top wasused to generate a duty-cycle map with Eq. (8). Theresist-profile-based phase-mask design was ob-tained from the V-like trough profile with 7 �m ofdepth and 100 �m of width, using the previouslyexplained numerical deconvolution procedure forextracting the duty-cycle map. The range of inten-sity variation and duty cycle are similar to those inthe prism cases. The one-dimensional (1D) trans-mittance profiles and duty-cycle maps for thephase-grating mask designed from the resist pro-files of the microprism and the V-groove are shownin Fig. 5.
Once we have designed the duty-cycle map W�x, y�for the 2D phase-grating mask by using the resist-profile-based approach, it is possible to analyticallypredict the developed analog-resist profile by utiliz-
Fig. 8. Two-dimensional Zygo profiles of fabricated analog elements on the SPR 220 resist: (a) microprism and (b) V-groove.
ing a similar numerical convolution method used forproducing the curve of Fig. 4. We can simply put theduty-cycle function W�x, y� into Eq. (6) to obtain theintensity-transmittance profile I�x, y� coming out ofthis mask. Then we can use Eq. (10) to obtain theexposure dose profile D�x, y� (in mJ�cm2) from theintensity profile I�x, y� with a certain bias time tb andexposure time t. It is then straightforward to performthe numerical convolution with the exposure curve ofthe developed resist depth h (in micrometers) versusthe dose D (in mJ�cm2) to finally obtain the developedresist profile d�x, y� (in micrometers). Regarding thephase-grating mask designed from the transmittanceprofile, the procedure for analytically predicting thedeveloped resist profile is much simpler, as theintensity-transmittance profile I�x, y� does not needto be computed. The analytically predicted micro-prism and V-groove profiles were compared with thetarget profile as shown in Fig. 6. As is evident fromFigs. 5 and 6, the transmittance-based phase-gratingmask produces a larger mismatch on the lower sideedge of the element than does the phase-gratingmask based on the resist profile. This is due to thefact that the zeroth-order intensity transmittance ofEq. (6) rapidly becomes nonlinear near the minimumduty-cycle region, where the intensity is high and theremaining resist is low. However, the predicted resistprofile of the resist-profile-based phase-grating maskis in excellent agreement with the target resist pro-file.
C. Fabricating the Phase-Grating Mask withElectron-Beam Direct Writing
The phase-grating masks for the 100 �m microprismand V-groove elements were designed with the 2Dsquare-pixel phase grating, as already discussed. The
grating period p was chosen to be 2.5 �m according tocutoff Eq. (2). First, the desired duty-cycle map wascreated in a MATLAB code according to the above-mentioned numerical procedure, and the resultingnumerical matrix of the duty cycle was saved as anASCII file. We developed a GDS2 file-writing programwith C language and ran this program with the ASCII
text file to create the GDS2 mask file of a binaryphase-grating mask. It is possible to fabricate thephase-grating mask with a conventional photolitho-graphic technique by using a chrome mask and UVexposure tools such as a stepper and a mask aligner.However, electron-beam (e-beam) direct writing witha PMMA resist has proved to be the best way in termsof low absorption and cleanness of the fabricatedgrating surface. We used 495 PMMA A6 from Micro-Chem for the e-beam resist and coated a 127 mmfused-silica mask plate with it for a thickness of420 nm. This thickness makes a � phase shift at theG-line �436 nm�. The e-beam writing was performedwith a Leica 5000� EBPG machine in the nanopho-tonics clean room of the Center for Research andEducation in Optics and Lasers (CREOL) at the Uni-versity of Central Florida. The optimum e-beam dosefor clearing the PMMA resist was found to be450 �C�cm2. After the e-beam writing was finished,the mask plate was developed in the MIBK�IPA �1:3�developer for 70 s and then rinsed with isopropylalcohol. The microscope picture of the fabricatedphase-grating mask for the microprism is shown inFig. 7.
4. Fabrication and Analysis of Analog-Resist Profiles
A. Fabrication Process of Micro-Optics Profiles on theSPR 220-7 Photoresist
For the fabrication of these analog elements on theSPR 220-7 resist with our fabricated phase-gratingmask, we coated a 101.6 mm fused-silica wafer withan SPR 220-7 resist for an initial thickness of 12 �m.This is the maximum thickness possible with an SPR220-7 resist when it is spun at 1000 rpm with a spincoater. Next it was soft baked on a hot plate at 115 °Cfor 90 s. We used our GCA G-line stepper to exposethis wafer with the phase-grating mask. The bias andexposure times were 0.6 and 2.6 s, respectively, asthe duty-cycle map on this mask was designed forthat exposure condition. The exposed SPR 220-7 re-sist should sit for at least 45 min before the postbakeis applied, since it takes this length of time for thephotochemical reaction to finish and stabilize. Thepostexposure bake was performed in the same way asthe soft bake. After the postexposure bake was fin-ished and the wafer was cooled down, the wafer wasdeveloped by immersing it in MF CD-26 developer for4 min. Finally it was rinsed with de-ionized waterand dried with nitrogen.
B. Analog-Resist Profiles
To measure the surface profile of the fabricated ana-log elements on a thick photoresist, we used opticalprofilometry. The contact profilometry machine was
Fig. 9. Comparison of the numerically predicted 1D microprismprofile (solid curve) and the fabricated 1D microprism profile (dot-ted curve): (a) microprism and (b) V-groove.
not used because it could damage the photoresistsurface and it is also more limited in the height mea-surement range. The developed resist profile wasmeasured with a Zygo white-light interferometer andwas viewed in our scanning electron microscopy ma-chine. The three-dimensional Zygo pictures of themicroprism and V-groove are shown in Fig. 8. The 1Dsurface profiles through the center of the elementswere taken from the measured surface data in theZygo interferometer. These experimental surface pro-files were compared with the initial target resist pro-file from which the phase-grating mask wasdesigned.
The result of the profile measurement shows goodagreement with the predicted resist profile obtainedfrom the numerical convolution method. We com-pared the 1D numerically predicted resist profile
with the fabricated resist profiles of the prism andV-groove as shown in Fig. 9. Both elements wereobtained by using the resist-profile-based phase-grating mask. The slightly higher experimentalprofile on the low side of the prism is due to thesomewhat oversized small phase-grating pixels onthose areas. Using the numerical convolution com-putation with the introduction of pixel size error,we estimated this oversizing error of the gratingpixels to be 3%–4%. This oversizing is most likelycaused by slight overdeveloping of the PMMA resistafter e-beam writing. Both the transmittance-basedphase-grating design and the resist-profile-basedphase-grating design produce final resist profilesthat are in good agreement with the predicted resistprofile. However, in the case of the transmittance-based phase-grating mask, the profile near the bot-
Fig. 10. Two-dimensional Zygo profiles of fabricated analog-vortex elements on the SPR 220 resist: (a) vortex with charge number equalto one and (b) vortex with charge number equal to three.
tom edge of the prism creates a larger mismatch tothe designed prism profile than the profile from theresist-based phase-grating mask. Because the lin-ear ramp in the intensity profile does not producethe linear-resist profile as a result of the nonlinearexposure response of the SPR 220 resist, there ismore deviation from the designed prism profile thanfrom the prism profile made with the resist-profile-based phase-grating mask. Thus it is evident thatthe resist-based phase-grating mask should be usedto fabricate the most exact analog-resist element.
We also designed and fabricated analog-vortex el-ements with resist-profile-based phase-gratingmasks. The vortex is the analog surface profile ramp-ing up in angular fashion. The analytic expression forthe vortex of height d0 is given by
d(x, y) � d0[tan�1(my�x) � �]�(2�). (11)
In the above equation m is the integer charge num-ber for the vortex. For m � 1, the vortex profilemakes one circular ramping from 0 to 360 deg. Form � 2 or higher integers, there will be two or moreangular ramping profiles in 360 deg. We used Eq.(11) with 7 �m of the total height value d0 andcharge numbers of m � 1 and 3 to design and fab-ricate the resist-based phase-grating mask as ex-plained in Subsection 3.B. The resulting analog-vortex profiles after exposure and development areshown in Fig. 10. We checked the angular linearityof the vortex profile by sampling the points lying onthe circular ring within the vortex profile by usingthe Zygo interferometer. This is plotted in Fig. 11and shows a good linearly ramping height on theangular scale. Thus the vortex profile provides an-other example that demonstrates the accuracy ofthe resist-profile-based phase-grating-mask methodfor analog elements.
5. Discussion and Conclusion
A. Discussion
In the fabrication of the phase-grating mask for an-alog elements, the control of linewidth and pixel sizewas critical for making the analog surface exactly asdesigned. There are a few parameters affecting theprofile, such as e-beam dose and current, developingtime for the PMMA e-beam resist, and initial thick-ness of the PMMA resist. We performed experimentaltuning of these parameters to make a phase maskwith the exact duty-cycle distribution as designed. Ifthere is significant error in the duty cycle of the ac-tual phase-grating mask, it will result in an incorrectresist height profile since the transmittance of thephase-grating mask will deviate from the desired pat-tern.
The smallest square pixel was 0.5 �m. This is hardto achieve with the chrome mask and contact maskaligner but is easily attainable with e-beam directwrite. Another advantage of e-beam direct writing isthat it allows for a very small increment of pixel size.To make an extremely smooth analog profile on thethick resist, the duty-cycle increment should be 0.01or smaller. This corresponds to 25 nm for a 2.5 �mperiod on the mask. We used a minimum beam res-olution of 10 nm to make this fine variation of pixelsize on the phase-grating mask. It will be hard orimpossible to achieve this level of pixel size variationby using other photolithographic techniques. So, inaddition to the low absorption and cleanness of thePMMA resist, this fine level of pixel variation is an-other good reason for using the e-beam direct-writingtechnique for fabricating phase-grating masks.
Maintaining good repeatability of analog-resistprofiles from wafer to wafer requires that the processbe tightly controlled. The resist and developer shouldalways be in good condition and be used well beforetheir expiration dates. Appropriate resist processingshould be followed, such as the proper baking condi-tions and sitting time prior to each process step. Also,the stepper system should be maintained in goodcondition to ensure consistent exposure results. Asfor mask fabrication, the PMMA resist process shouldbe well controlled to avoid any incorrect gratingshape on the phase mask. Once the proper e-beamdose and PMMA resist developing time are deter-mined, the resist and developer should be kept underthe proper conditions and the resist process should beperformed in a consistent manner. This requirementon process control is quite standard for lithographicprocessing, making it simple to maintain. Thus thisanalog-resist-profile fabrication process using thephase-grating mask is easily transferable to a prod-uct manufacturing environment.
B. Conclusion
We have demonstrated a phase-grating-mask tech-nique based on e-beam direct writing, which is apromising alternative to gray-scale and halftonemask techniques for the fabrication of analog-resistprofiles. The principle of this technique is that the
Fig. 11. Measured angular-vortex height profile: (a) upper curveis from the analog vortex with 0.6 s of bias time and (b) lower curveis from the analog vortex with 0.8 s of bias time.
zeroth-order efficiency of the binary phase grating of� phase depth depends on the duty cycle of the grat-ing. Using a small enough grating period makes itpossible to allow only the zeroth-order light to passthrough the stepper system to form an aerial imageon the wafer. The numerical convolution of thezeroth-order diffraction efficiency and exposure char-acteristic curve of the thick resist was used to predictthe response of the resist height to the duty-cyclevariation of the phase-grating mask under a partic-ular exposure condition. This numerical computationseemed to agree well with the experimental resultson the SPR 220 resist. We also developed a numericaldeconvolution technique to compute the requiredduty-cycle variation for a certain analog-resist profileto be made on a thick resist. This approach is simple,and only the zeroth-order diffraction efficiency curveand the exposure characteristic curve of the particu-lar resist are needed. The phase-grating-mask designthat is based on this numerical deconvolution of thetarget resist profile proved to yield a more accurateanalog-resist profile than the simple transmittance-profile-based phase-grating-mask design.
Thanks to the high e-beam resolution of the Leica5000� EBPG system, we were able to represent thefine variation of the linewidth and pixel sizes in thebinary phase-grating mask required to fabricateanalog-resist profiles. The PMMA e-beam resist wasused to form the phase grating with varying dutycycle on the mask because of its high transmittanceat the G-line and its chemical durability. We fabri-cated an analog microprism, a V-groove, and a vortexelement on the SPR 220-7 resist to verify the feasi-bility of this technique. But this technique can beapplied to other types of positive thick photoresistand is not limited to the thickness range of the SPR220-7 resist. The fabricated analog-resist elementscan be followed by a dry-etching process to transferthe analog-resist profile to the substrate.
While this paper has demonstrated the feasibilityof analog-resist-profile fabrication by using thephase-grating mask, further development and char-acterization of the dry-etching process to transfer the
analog-resist pattern to a fused-silica substrate willmake this technique a viable option for the fabrica-tion of a variety of refractive–diffractive optical ele-ments.
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