Optical measurement of depth and duty cycle forbinary diffraction gratings with subwavelengthfeatures

John R. Marciante, Nestor O. Farmiga, Jeffrey I. Hirsh, Michelle S. Evans, andHieu T. Ta

We describe a new and unique method for simultaneous determination of the groove depth and duty cycleof binary diffraction gratings. For a near-normal angle of incidence, the �1 and �1 diffracted orders willbehave nearly the same as the duty cycle is varied for a fixed grating depth. The difference in theirbehavior, quantified as the ratio of their respective diffraction efficiencies, is compared to a look-up tablegenerated by rigorous coupled-wave theory, and the duty cycle of the grating is thus obtained as afunction of grating depth. Performing the same analysis for the orthogonal probe-light polarizationresults in a different functional dependence of the duty cycle on the grating depth. By use of both TEand TM polarizations, the depth and duty cycle for the grating are obtained by the intersection of thefunctions generated by the individual polarizations. These measurements can also be used to assessqualitatively both the uniformity of the grating and the symmetry of the grating profile. Comparisonwith scanning electron microscope images shows excellent agreement. This method is advantageoussince it can be carried out rapidly, is accurate and repeatable, does not damage the sample, and useslow-cost, commonly available equipment. Since this method consists of only four fixed simple measure-ments, it is highly suitable for quality control in a manufacturing environment. © 2003 Optical Societyof America

OCIS codes: 050.1950, 050.2770, 120.0120, 120.3940, 220.4000.

1. Introduction

Diffraction gratings play an important role in manytechnologies, including spectroscopy, display systems,and wavelength-division multiplexing architectures intelecommunications. For telecommunications, theirability to map wavelength into a propagation anglemakes them suitable for use in add–drop filters, mul-tiplexers and demultiplexers, and other wavelengthconfigurable devices.

Generally, a diffraction grating consists of a peri-odically varying boundary between two dissimilarmaterials. The spectral properties of the grating aredetermined solely by the grating period �, and therefractive indices of bounding materials n1 and n2.

The authors are with Corning Rochester Photonics Corporation,115 Canal Landing Boulevard, Rochester, New York 14626. Thee-mail address for J. R. Marciante is johnm@lle.rochester.edu.

Received 14 October 2002; revised manuscript received 20 Feb-ruary 2003.

0003-6935�03�163234-07$15.00�0© 2003 Optical Society of America

3234 APPLIED OPTICS � Vol. 42, No. 16 � 1 June 2003

This can be calculated simply with the grating equa-tion1

n1 sin��1� � no sin��m� �m�

�� 0, (1)

where � is the wavelength of the incident light invacuum and m is the order of the diffracted light.The angles �1 and �m are the respective incident andexit angles of the light with respect to the gratingnormal, and no is either n1 or n2, depending onwhether the grating is reflective or transmissive.Although the spectral properties of the grating aredetermined by Eq. �1�, the efficiency of the grating isdetermined by the profile of the periodic structureand requires rigorous vector wave modeling to pre-dict accurately.

Figure 1 depicts a cross-sectional view of a binarygrating, for which the periodic structure is rectangu-lar in nature. The parameters that control such agrating are depth d and the duty cycle of the grating,here defined as a��, the duty cycle of material 3.Using these parameters, one can accurately designgratings with the desired spectral characteristics and

efficiency. However, in the fabrication of such dif-fractive elements, deviations can occur from the orig-inal design. Some are embodied in deviations of theparameters defined in Fig. 1, and others includesloped sidewalls, rounded corners, or other defectsthat change the grating profile from the desired form.

Generally, once a fabrication process is sufficientlydeveloped for production volumes, only certain pa-rameters need to be inspected for quality control.Measuring the grating depth and duty cycle generallyrequires the use of sophisticated and expensiveequipment. For example, a scanning electron micro-scope �SEM� can be used, but operator experience andinspection time required per sample are prohibitiveto a manufacturing process, and the process itself isoften damaging to the sample under inspection. Foranother example, profilometry with a confocal scan-ning laser microscope is difficult owing to the sub-wavelength features of the gratings.2 These types ofmethod are also not suitable for in situ monitoring ofthe grating fabrication.

Several methods have been derived by use of thediffraction characteristics of the grating to determinethe grating structure. By scanning the diffractionpattern at a fixed wavelength3,4 or measuring dif-fracted power while scanning the incident wave-length,5 one can extract various physical parametersby fitting experimental and theoretical curves gener-ated by rigorous coupled-wave analysis.6 Eventhough these methods can be quite powerful in theirpredictive capability, a large amount of data needs tobe collected.

A technique has been developed for which a limitedset of parameters can be predicted, such as the grat-ing depth, duty cycle, and sidewall angle.7 Themethod utilizes scatterometry used in previouswork,3 and likewise involves the collection of a largeamount of data.

Another method has been developed in which thediffraction efficiencies of the first and second dif-fracted orders are measured, and their ratio yieldsthe duty cycle from an analytic expression.8 Thismethod cannot account for variations in the gratingprofile, and since it requires that the second dif-fracted order be used, the minimum size of the grat-ing period that can be measured is limited because ofthe shorter optical wavelengths required.

In this paper we develop a new metrology methodfor rapidly and accurately determining the depth andduty cycle of one-dimensional binary diffraction grat-ings based on the ratio of two optical power measure-ments. This new metrology method meets thefollowing requirements:

�a� can be carried out quickly�b� is accurate and repeatable�c� requires very little data collection�d� uses low-cost, commonly available equipment�e� does not damage the sample�f � is adaptable to a manufacturing environment.

The paper is organized as follows. Thesymmetric-order efficiency ratio is defined and ex-plored in Section 2. Experimental measurementsvalidating the process defined in Section 2 are pre-sented in Section 3. The methodology for the simul-taneous prediction of depth and duty cycle from foursimple optical power measurements is described inSection 4. A discussion of the caveats of this metrol-ogy method, as well as concluding remarks, are givenin Section 5.

2. Symmetric-Order Efficiency Ratio

The basis of our measurement method hinges onwhat we define as the symmetric-order efficiency ra-tio �SOER�, which is simply the ratio of the measuredpowers in the m � �1 and m � �1 diffracted orders.For our measurement technique, depicted in Fig. 2,the grating is illuminated at an angle that is near-normal incidence with linearly polarized light. Thepower in the m � �1 and m � �1 diffracted orders ismeasured, and the ratio is compared to rigorouscoupled-wave theory by use of the known �i.e., previ-ously measured� grating depth. To briefly describethe rigorous coupled-wave theory, the diffractiongrating profile is broken into a large number of thinplanar sheets. In each of these sheets, the electricfield is expanded in terms of diffracted orders. Thepartial waves in any given sheet solve a set ofcoupled-wave equations and are coupled to the othersheets through electromagnetic boundary conditions.The net solution of this approach yields the reflectionand transmission phase and amplitudes for each dif-fracted order. Complete details of this method can

Fig. 1. Depiction of a binary grating: n1 and n2 are the refractiveindices of the media that share the periodic boundary, which gen-erally consists of materials of indices n3 and n4; d is the gratingdepth; � is the grating period; and a�� is the duty cycle of thematerial with refractive index n3.

Fig. 2. Experimental configuration of the SOER method shown inthe transmission mode. The TE and TM polarizations requiredfor the measurements are indicated in gray.

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be found in Ref. 6. The material refractive indicesused for the calculations throughout the paper aresummarized in Table 1.

To gain some physical insight into the SOER mea-surement process, consider illumination of the grat-ing at normal incidence. Because of symmetry, the�1 and �1 orders must contain the same fraction ofincident power. However, as the incident angle de-viates from normal, the behavior of these two orderswill change. For small deviations, the change in be-havior should not be drastic and can thus be usedwith confidence. Using larger angles can increasethe sensitivity of the measurement. However, suf-ficiently large angles of incidence can result in manydiffracted orders besides the two of interest �m � �1and �1�. The exact angles at which these higherorders are excited are strongly dependent on thephysical structure, and can thus be a source of errorfor minor tooth-shape deviations if the interrogationangle is near such a threshold. As such, the incidentangle is chosen to be as far away from normal as pos-sible while still remaining a few degrees away from thepossible excitation of higher-order diffraction.

Consider a grating that is etched into a fused-silicasubstrate. Using rigorous coupled-wave analysis,we calculate the diffraction efficiency of TM-polarizedincident light �light that is polarized in the plane ofFig. 1� of the m � �1 and m � �1 transmitted orders,

as shown in Fig. 3, using the parameters listed in thefigure caption. Note that the dependences on theduty cycle for each order are quite similar yet distin-guishable.

If we calculate the ratio of the transmitted diffrac-tion efficiencies for these two orders, taking into ac-count the Fresnel reflections at the bottom of thesubstrate, the behavior becomes quite linear in theregion of 50% duty cycle, as shown in Fig. 4. Welabel this ratio T�1�T�1 to signify that we are takingthe ratio of the fraction of power contained in the m ��1 transmitted order to that in the m � �1 trans-mitted order. Also shown in the plot is a similarratio for the m � �1 to m � 0 orders, labeled T�1�T0.The power in the first diffracted order, normalized tothat in the zeroth order, is often used to assess thegrating. However, this ratio is a slowly varying andmultivalued function around 50% duty cycle, makingit impossible to use in determining the grating dutycycle in this range. On the contrary, using the T�1�T�1 SOER, the measurement becomes almost twiceas sensitive and is single valued. Also worth notingis the small fraction of light in the diffracted orderscompared with the zeroth-order transmitted light.This can have an effect on the measurement accuracywhen using the T�1�T0 comparison.

Another advantage to using the SOER method isreduced sensitivity to other fabrication errors. Anyirregularities in the grating will typically result in alower diffraction efficiency of the nonzero orders, withthe excess energy ending up in the zeroth order.This will result in a T�1�T0 ratio that is too small,resulting in an inaccurate measurement of the dutycycle. On the contrary, near-normal incidence fab-rication errors will affect the m � �1 and m � �1orders rather similarly. Thus, we expect the SOERmethod to be less sensitive to fabrication errors. Asan example, we investigate the effect of sidewallslope, the deviation from the tooth wall angle fromthe surface normal. Figure 5 shows the deviation in

Table 1. Refractive-Index Values Used in Numerical Calculations forFused Silica and Photoresist at Various Optical Wavelengths

Wavelength �nm� Fused Silica Photoresist

632.8 1.4570 1.6406611.9 1.4577 1.6439594.1 1.4582 1.6470543.5 1.4602 1.6574442 1.4662 1.6901

Fig. 3. Diffraction efficiencies of the m � �1 and m � �1 trans-mitted orders as a function of duty cycle. The grating underinvestigation is a fused-silica binary grating, such that n1 � n4 �1 �air� and n2 � n3 � 1.457 �fused silica at 632.8 nm�. The probelight is TM polarized with a wavelength of 632.8 nm and is incidenton the grating at 8 degs from normal. The grating pitch is 750nm, and the grating depth is 220 nm.

Fig. 4. Ratio of diffraction efficiencies of the m � �1 and m � �1transmitted orders �black curve� and the m � �1 and m � 0 orders�gray curve� as a function of duty cycle. The grating under inves-tigation and the probe technique used are identical to that used inFig. 3.

3236 APPLIED OPTICS � Vol. 42, No. 16 � 1 June 2003

the T�1�T�1 and T�1�T0 ratios for a range of sidewalltilt angles. The SOER method shows a factor of 2less sensitivity to these deviations than the T�1�T0method, as expected.

For the SOER method to give accurate duty cyclevalues, the depth of grating d must be known tosufficient accuracy. This is the case for gratingsthat are fabricated by etching into a film that hasbeen deposited onto a substrate with an etch-stoplayer in between, or at least a high differential etchrate between the film and the substrate. However,other types of grating might be fabricated by simplyetching into the substrate itself. For such gratings,the etch depth is uncertain to some degree. Figure 6shows the T�1�T�1 ratio as a function of duty cyclefor various depths of the fused-silica grating. Whilethe different curves exhibit similar behavior, the ra-tio values are distinct enough that the correct dutycycle cannot be predicted without accurate knowl-

edge of the grating depth. Using calculations simi-lar to those shown in Fig. 6, uncertainty in depthmeasurements can be translated to uncertainty inthe duty cycle measured with the SOER method.

The SOER method is rather general and can beextended to other gratings. For example, considerthe binary fused-silica grating used in the previouscalculations. To make such a grating, one needs tocreate a photoresist grating, which is then used as anetch mask to create the silica grating. In this case,it is extremely important to predict the duty cycleaccurately. If an incorrect photoresist duty cyclecan be detected, then the sample can be discarded,saving an etch run that would have resulted in afused-silica grating with the incorrect duty cycle.

Figure 7 shows the SOER method as applied to abinary photoresist grating on a fused-silica substrate.Note that at the standard �red� He–Ne wavelength�632.8 nm�, the useful duty cycle range is between0.50 and 0.75. When testing for duty cycles around50%, this configuration will not work. However, wecan compensate for the modified behavior of thethicker grating �with respect to the fused-silica grat-ing� by changing the probe wavelength. Note that,as we use successively shorter wavelengths, the use-ful predicable duty cycle range moves toward shorterduty cycles. At the green He–Ne line �543.5 nm�, theusable duty cycle range is 0.40–0.70, making thiscase appropriate for accurate measurement of dutycycles around 50%.

Another control that can be used to optimize theprocess is the polarization state of the probe light.Figure 8 shows T�1�T�1 ratios as a function of dutycycle for both the fused-silica and the photoresistgratings by use of TE-polarized light, which is lightthat is polarized perpendicular to the plane of inci-dence �or polarized parallel to the grating teeth�.Using the probe wavelength as an optimization pa-rameter, one can find a usable duty cycle range as

Fig. 5. Precent deviation of duty cycle prediction as a function ofsidewall tilt for the T�1�T�1 SOER method �black line� and T�1�T0

method �gray curve�. The grating under investigation and theprobe technique used are identical to that used in Figs. 3 and 4.

Fig. 6. Ratio of diffraction efficiencies of the m � �1 and m � �1transmitted orders as a function of duty cycle for various gratingdepths, as labeled in the figure. Otherwise, the grating underinvestigation and the probe technique used are identical to thatused in Figs. 3–5.

Fig. 7. T�1�T�1 ratio versus duty cycle for various He–Ne laserwavelengths, as labeled in the figure. The grating under inves-tigation is a binary photoresist–air layer 550 nm deep with a750-nm pitch on a fused-silica substrate. The probe techniqueused is identical to that used in Figs. 3–6 except for the incidentwavelength.

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was noted previously. The TE case tends to be moresensitive to parameter changes, making the properselection of wavelength more difficult compared withthe TM case. Of course the polarization of the inputlight need not be strictly TM or TE. The relative mixof these two polarization states can be used as an-other parameter that can be varied to optimize theT�1�T�1 ratio for investigation of a given duty cyclerange.

So far, the SOER method that we have describedhas focused on measuring the power contained in thetransmitted diffracted orders. However, it is not arequirement that one must use transmitted orders.For the gratings we have used as examples, thepower in the transmitted orders is significantlyhigher, so use of the SOER method in the transmittedmode reduces experimental errors that would other-wise have been induced by measurement of lowpower levels. For reflective gratings, the same prin-ciples described above for the SOER method can be

used to define an accurate metrology by use of probewavelength and polarization state to optimize theR�1�R�1 ratio curve. Figure 9 shows the results ofuse of the SOER method in the reflective mode for thesame gratings investigated previously. The duty cy-cle of the fused-silica grating can be determined alonga wide range, from 0.35 to 0.90, whereas the photore-sist grating is only usable provided the grating has aduty cycle in the 0.40–0.70 range. In both cases, theSOER method in the reflective mode proves effectivein being able to determine an accurate duty cycle in awide range around 50%.

One additional use of the SOER method is quali-tative assessment of the grating. By changing theincident angle from � to ��, the symmetry of thegrating profile can be qualitatively assessed. Uni-formity of the grating can likewise be inspected bymeasurement of the SOER at various locations acrossthe grating.

3. Experimental Results

To validate the SOER method, several fused-silicaand photoresist gratings were tested in the transmis-sion mode with TM-polarized light at 632.8 nm undersimilar conditions as given in Figs. 3 and 7. Depthdata were obtained by use of SEM and scanningprobe microscopy techniques. As an example, Fig.10 shows the SEM image for a binary photoresistgrating. From this picture, the grating duty cycle ismeasured to be 0.47 0.02, the uncertainty originat-ing primarily from the extraction of exact dimensionsfrom the image. By use of the measured depth of0.500 m for this grating, the SOER measurementspredict a duty cycle of 0.459 0.006, which is inexcellent agreement with the directly measuredvalue.

In all the measurements that were performed, thecomparison between the SOER method and the SEMdata is extremely favorable, with the values obtainedby the SOER method falling within the uncertainty ofthe SEM measurements. This experimental valida-

Fig. 8. T�1�T�1 ratio versus duty cycle by use of TE-polarizedlight for the fused-silica and photoresist gratings used in Figs. 3–7.The probe technique used is identical to that used in Figs. 3–7except for the incident wavelength and polarization.

Fig. 9. R�1�R�1 ratio versus duty cycle by use of TM-polarizedlight for the fused-silica and photoresist gratings used in Figs. 3–7.The probe technique used is identical to that in Figs. 3–7 except forthe incident wavelength and use of reflected rather than transmit-ted orders.

Fig. 10. SEM image of a photoresist grating on a fused-silicasubstrate with similar parameters as those used in Fig. 7.

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tion confirms that the SOER metrology is an accuratemethod for quickly measuring the duty cycle of one-dimensional diffraction gratings.

4. Dual-Polarization Symmetric-Order Efficiency RatioMethod

As outlined above, measuring the SOER only givesinformation about the duty cycle provided the depthof the grating is known. To gather informationabout the depth, another measurement needs to beobtained. As discussed previously, light that is in-cident from TE versus TM polarizations will yielddifferent results of the SOER as a function of dutycycle. Consider rotating the linear polarizer be-tween the probe laser and the test grating, as shownin Fig. 2. The added complexity of the experimentalprocedure over the SOER method is minimal, yet anadditional and critical piece of information can beextracted from this process. This polarizer can berotated by 90 deg to allow the light incident on thegrating to be either TE or TM polarized, and theSOER is measured for each of these polarizations.An additional experimental requirement is that, ifthe laser is polarized, the laser should be rotated byapproximately 45 deg with respect to the TE and TMaxes as defined by the polarizer and the grating.This allows for nearly equal power in each of theSOER measurements. The exact value for this an-gle is not critical since a ratio is being measured, butit should allow for appreciable power levels in bothTE and TM cases such that the signal-to-noise ratiocan be roughly equivalent for both cases.

Consider use of the SOER method without knowl-edge of the grating depth. A given SOER valuewould lead to a value for the duty cycle that is linkedto the grating depth, or more formally, a duty cyclethat is a function of the grating depth. If we use theSOER method for a different polarization, we willmeasure a different SOER and generate a different

function describing the dependence of the duty cycleon the depth. Figure 11 shows an example of this fora binary grating made of fused silica. The intersec-tion of the TM and TE curves yields the correct valueof the depth and duty cycle of the grating. For thiscase, the resultant duty cycle is 45.4%, and the re-sultant grating depth is 232 nm. It is important tonote that each curve need not be single valued inde-pendently, only their intersection needs to be uniqueto have a unique solution of depth and duty cycle.

In Fig. 12 we plot TM and TE curves for severaldifferent SOER values. The configuration was cho-sen to optimize the response in the range of 40–60%duty cycle and 200–250-nm depth. Since the mea-surements will necessarily come with some uncer-tainty, the solutions that can be used with mostconfidence are those for which the curves intersectnearly perpendicularly. Note that, for the case plot-ted in Fig. 12, as the grating depth is reduced, the TEand TM curves become nearly parallel. In thisrange, the depth and duty cycle cannot be determinedexcept in terms of broad estimates of the actual value.However, the useful range of this particular calcula-tion is for grating depths greater than �200 nm.

This method can be practically implemented formanufacturing quality control by generating a tableof TE and TM SOER ratios for a very fine grid ofgrating depths �steps of 1 nm� and duty cycles �stepsof 0.5%�. The two SOER values �comprised of fourpower measurements� can then be compared againsta look-up table for evaluation of the grating duty cycleand depth.

So far, the analysis presented has been for gratingsmeasured in the transmission mode. Like its rootfunction the SOER, the dual-polarization SOER canbe used in the reflection mode as well, taking the ratioof the �1 and �1 reflected orders to calculate theSOER ratios. The only caveats are to ensure suffi-cient diffraction efficiency such that the measuredoptical signals are significantly above the noise levelof the detectors.

Fig. 11. Calculated duty cycle versus depth curves for orthogonalpolarizations by use of a TM SOER value of 0.95 and a TE SOERvalue of 1.00. The probe light has a wavelength of 632.8 nm, andis incident on the grating at 7.5 deg from normal. The binarygrating has a grating pitch of 760 nm and is made of fused silica�n1 � n4 � 1 �air� and n2 � n3 � 1.457 �fused silica at 632.8 nm� .

Fig. 12. Calculated duty cycle versus depth curves for orthogonalpolarizations and various values of the TE and TM SOER. Theprobe configuration and grating are identical to those used in Fig.11.

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There are several alternative measurement config-urations that can be used for this method that aredifferent from that which is depicted in Fig. 2. It ispossible to use a polarizer in front of each detectorinstead of a single polarizer after the laser. Thissolution leads to additional equipment cost because ofthe additional polarizer, as well as increased usererror since the polarizers must be changed equally ateach detector. It is also conceivable to measure theSOER for two different wavelengths instead of twodifferent polarizations. Similar to the dual-polarization method, the two functions of duty cycleversus depth obtained by use of two wavelengthswould share a common point from which the dutycycle and grating depth could be extracted. How-ever, this dual-wavelength method not only requiresan additional laser source, but also that either fourfixed detectors be used, or that two detectors be mov-able. This alternative method leads to additionalequipment costs, and possibly an additional source ofuser error because of the moving detectors.

5. Discussion and Conclusion

There are a few caveats for use of the SOER method.First, total internal reflection of the diffracted ordersin the grating substrate or superstrate must be con-sidered when selecting the probe wavelength. Sec-ond, the usable prediction range given by the SOERis only that, a prediction range. Measurements ongratings fabricated with duty cycles outside thisrange will yield incorrect answers that are due toeither the small slope or multivalued nature thatcould be present in certain duty cycle ranges of theSOER curves. Generally, the physical gratings pro-duced should be bounded within a limited range, andthe SOER method designed appropriately for thatrange. However, if the depth and duty cycle areboth being solved for, it is unlikely that the TE andTM cases will degenerately produce several identicalpairs of depth and duty cycle for a unique pair of TEand TM SOER values, since it is in fact the differenceof the behaviors between the cases that allows useof the dual-polarization SOER method to beginwith. Again, since the gratings under test should

be bounded within a limited range, the dual-polarization SOER method can be designed appro-priately for that range.

In conclusion, we have invented an accurate andrepeatable method of rapidly measuring the duty cy-cle and grating depth of one-dimensional diffractiongratings. The dual-polarization symmetric-order ef-ficiency ratio method is a low-cost nondestructive in-spection metrology suitable for manufacturingenvironments for quality control purposes. Thetechnique uses an optical probe wavelength as a con-trol for optimization of the prediction ranges and canbe used in the reflective or the transmission mode.The measurement process requires four simple powerreadings, involving a binary change in only one opti-cal element.

The authors thank Tasso Sales for the use of hisnumerical vector diffraction model.

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4. B. K. Minhas, S. A. Coulombe, S. S. H. Naqvi, and J. R. McNeil,“Ellipsometric scatterometry for the metrology of sub-0.1-m-linewidth structures,” Appl. Opt. 37, 5112–5115 �1998�.

5. E. W. Conrad and D. P. Paul, “Method and apparatus formeasuring the profile of small repeating lines,” U.S. Patent5,963,329 �5 October 1999�.

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