Subwavelength transmission grating retarders for use at 106 μm

Subwavelength transmissiongrating retarders for use at 10.6 mm

D. L. Brundrett, E. N. Glytsis, and T. K. Gaylord

Designs are given for gallium–arsenide subwavelength grating retarders operating at 10.6 mm. Adesign procedure is detailed that takes into account the reflections at all surfaces and that uses numericaloptimization to improve the transmittance of the retarders to nearly 100%. It is shown that thehomogeneous uniaxial layer model for subwavelength gratings can be used to provide starting points forthe Nelder–Mead simplex optimization, obviating the need for stochastic optimization techniques such assimulated annealing. An analysis of the designs with respect to wavelength, angle of incidence, andfabrication tolerances indicates that such grating retarders will perform favorably compared with com-mercial alternatives. © 1996 Optical Society of America

Key words: Subwavelength gratings, infrared retarders.

1. Introduction

Retarders are needed at mid-IR and far-IR wave-lengths for applications such as laser cutting, wherecircularly polarized 10.6-mm laser radiation providesa greatly improved cut over that of linearly polarizedlight.1 Further, the growing interest in wirelesscommunications in the mid-IR will lead to the needfor retarders at a number of wavelengths in the 3-mmto 12-mm range. Currently there are several typesof devices commercially available to accomplish re-tardation for mid-IR applications. Transmissive re-tardation plates in CdS can be antireflection ~AR!coated for greater than 98% transmission, but theyare expensive relative to visible-wavelength platesand require careful mounting and alignment of twoplates if synthesized zero-order performance is to beachieved. Fresnel rhombs can also be fashionedfrom lossless isotropic materials such as ZnSe andGaAs, but again they are expensive when comparedwith visible wavelength devices and are bulky andextremely sensitive to alignment errors. Alterna-tively, reflective retarders based on thin-film designs2are also available, but these require extra routingoptics to direct the beam where desired.

The authors are with the School of Electrical and ComputerEngineering and the Microelectronics Research Center, GeorgiaInstitute of Technology, Atlanta, Georgia 30332.Received 21 December 1995; revised manuscript received 25

March 1996.0003-6935y96y316195-08$10.00y0© 1996 Optical Society of America

In recent years, a number of authors have beguninvestigating the possibility of using subwavelengthgratings ~for which all nonzeroth diffracted orders arecut off ! as retarders. Such investigations can be di-vided into two categories, i.e., those involving reflec-tion gratings3–5 and those involving transmissiongratings6–9; here we concentrate on the latter type.Transmission grating retarders exploit the form bi-refringence of one-dimensional subwavelength di-electric surface-relief gratings. This is an effect thathas long been known and understood in the micro-wave community,10–12 and that has been extensivelyinvestigated by the optics community in recentyears.13–19 One-dimensional subwavelength grat-ings behave as layers of negative uniaxial material,with optic axes perpendicular to the grating grooves,and with effective principal indices of refraction thatare dependent on the wavelength-to-grating-periodratio, the grating ridge and groove materials, and thegrating filling factor, F ~the fraction of one periodfilled with the ridge material!. Thus a plane wavenormally incident upon the grating and linearly po-larized along the grating grooves behaves as an ordi-nary wave, whereas the wave polarizedperpendicular to the grating grooves behaves as anextraordinary wave.Most transmission grating retarder research to

date has concentrated on binary gratings, withgroove depths d related to the retardation phase bythe simple amplitude transmittance expression,

d 5 dyk0uDnu, (1)

1 November 1996 y Vol. 35, No. 31 y APPLIED OPTICS 6195

where k0 is the free-space wave number and d is theretardation phase in radians. The effective birefrin-gence Dn 5 nE 2 nO has been assumed to depend onthe grating parameters according to the relationships

nO 5 @Fnr2 1 ~1 2 F!ng

2#1y2, (2)

nE 5 @Fynr2 1 ~1 2 F!yng2#21y2, (3)

provided by the first-order homogeneous uniaxiallayer model ~HULM!,19 where nr and ng are the re-fractive indices of the ridge and groove materials,respectively, and F is the grating filling factor. Werefer to this as the amplitude transmittance ap-proach. This is a one-pass approach, i.e., it neglectsall reflections at the cover–grating and grating–substrate interfaces and assumes equal transmittedamplitudes for the ordinary and extraordinarywaves. This is equivalent to assuming that the in-dex difference between the ridge and groove materi-als is small, which makes Dn small and in turn fromEq. ~1! makes the groove depth large. If, for in-stance, the groove material is air and the ridge ma-terial is glass with F 5 0.5, then the effectivebirefringence is roughly 0.1, which implies a groovedepth d 5 2.5l0 for d 5 py2. Also, Eqs. ~2! and ~3!assume a quasistatic wavelength-to-period ratio,which combined with a large groove depth can pro-duce impractically extreme grating ridge aspect ra-tios.Subwavelength transmission grating retarders de-

signed by the amplitude transmittance approachhave been used for many years in microwaveapplications.20–22 Optics research in the area beganin 1983, when Flanders measured the retardationphase produced by silicon nitride binary gratingswith periods of 240 nm.6 The effective birefringenceand index discontinuities were not too large in thiscase, and good agreement was obtained between themeasured retardation phase and that predicted byamplitude transmittance theory. In the same year,Enger and Case fabricated gratings with 300-nm pe-riods in photoresist and fused quartz with resultssimilar to those obtained by Flanders.7 To avoid theaspect ratio problem caused by their low-index grat-ings, the authors suggested cascading two gratings inseries to achieve the desired retardation. Later,Cescato et al.8 used the amplitude transmittance ap-proach together with the film stack method of Ono etal.23 to explore the phase shifts achievable in sinu-soidal surface-relief photoresist gratings on glasssubstrates, with results similar to those of Enger andCase. Davidson et al.9 attempted to use the ampli-tude transmittance approach to design a space-variant half-wave retarder in GaAs for use at 10.6mm; cascaded gratings were again used to achieve thedesired retardation. The large effective birefrin-gence of the GaAs gratings reduced the requiredgroove depths considerably but also produced un-equal transmitted amplitudes for the ordinary andextraordinary waves, causing the amplitude trans-mittance design to fail. This was demonstrated in

6196 APPLIED OPTICS y Vol. 35, No. 31 y 1 November 1996

the authors’ experimental results, wherein a trans-mittance of 40% was measured at a point where thedevice should have produced a transmittance of 0%.In this paper the design of grating retarders in

GaAs is re-examined; one possible application is de-picted in Fig. 1, where a grating is used to replace theCdS wave plate in an electro-optic amplitude modu-lator. Noting from the results of Ref. 9 that theamplitude transmittance approach can fail for thiscase, we seek a more complete treatment of the de-sign problem. In Section 2 we detail an efficient nu-merical design procedure for subwavelengthtransmission grating retarders, which takes into ac-count large-form birefringences, multiple reflectionsfrom all surfaces involved, and nonquasistaticwavelength-to-period ratios in determining the effec-tive refractive indices for the grating; we use thisprocedure to design quarter-wave and half-waveGaAs subwavelength grating retarders for use at 10.6mm. In Section 3 we investigate the effects of fabri-cation errors on the performance of these designs;Section 4 presents a discussion and summary.

2. Design Procedure

The basic structure under consideration is shown inFig. 2~a!, where a rectangular-groove surface-relief~binary! grating of period L, groove depth d, and fill-ing factor F ~the ridge width is FL! is confined be-tween semi-infinite cover and substrate regions ofrefractive indices nc and ns, respectively. It is usefulto generalize this structure to that shown in Fig. 2~b!.The gratings treated here will be standard binary-optic structures for which ng 5 nc and nr 5 ns. Also,in the analysis that follows, coherent reflections fromthe back of the substrate are assumed to be sup-

Fig. 1. ~a! Electro-optic amplitude modulator, normally config-ured, with a quarter-wave transmissive retarder preceding theelectro-optic material. ~b! The volume, weight, and expense of theretarder can be eliminated by etching an appropriate subwave-length grating into the entrance surface of the electro-optic mate-rial.

pressed, either by a standard thin-film AR coating orby a two-dimensional subwavelength AR grating.17,24With illumination by a normally incident unit am-

plitude plane wave of free-space wavelength l0, lin-early polarized with equal x and y components, thecomplex amplitude of the transmitted field at thegrating–substrate interface is

E 5 xTx exp~ jFx! 1 yTy exp~ jFy!, (4)

where Ti and Fi ~i 5 x, y! are real numbers, andretardation phase d is defined as

d ; Fy 2 Fx. (5)

To specify the polarization state conveniently, we alsouse auxillary angle a and ellipticity ε defined inRef. 25:

a ; tan21~TxyTy!, (6)

ε ; tan@sin21~sin 2a sin d!y2#. (7)

For transmissive retarders, d should be mpy2, m 51,2, and angle a should be py4; it is also of interest tomaximize the total transmitted power ~or transmit-tance!, T 5 ns~Tx

2 1 Ty2!ync.

For this example the center wavelength is l0 510.6 mm, the cover material is air, and the substrate

Fig. 2. ~a! Gratings consist of binary structures of period L,groove depth d, and filling factor F, with cover and groove refrac-tive indices nc, and substrate and ridge refractive indices ns. ~b!Extension of the structure in ~a! to three layers; all layers have thesame period, each layer with its own filling factor Fi and groovedepth di, i 5 1, 2, 3.

material is GaAs ~ns 5 3.269!. To cut off all non-specular orders in the substrate while remaining in areasonable fabrication regime, we set the period at3.0 mm. This wavelength-to-period ratio is too smallfor the quasistatic approximation to hold, which pre-cludes the use of Eqs. ~2! and ~3!. Instead thehigher-order HULM as outlined in Ref. 19 is used, inwhich effective refractive indices nO and nE for thegrating are determined by solving the following equa-tions:

~nr2 2 nO

2!1y2 tanFpFL

l0~nr

2 2 nO2!1y2G5

2~ng2 2 nO

2!1y2 tanFp~12 F!L

l0~ng

2 2 nO2!1y2G , (8)

~nr2 2 nE

2!1y2

nr2 tanFpFL

l0~nr

2 2 nE2!1y2G5

2~ng

2 2 nE2!1y2

ng2 tanFp~12 F!L

l0~ng

2 2 nE2!1y2G . (9)

For F 5 0.5, the solutions to these equations are nO 52.6911 and nE 5 1.7073, so that Eq. ~1! gives aquarter-wave groove depth of dpy2 5 2.6937 mm anda half-wave groove depth of dp 5 5.3874 mm. How-ever, an analysis of uniaxial layers with these indicesand depths produces d 5 0.942 py2 and T 5 79.13%for the quarter-wave case, and d 5 0.988p and T 581.72% for the half-wave case. These deviationsfrom ideal performance are due to the index discon-tinuities at the cover–grating and grating–substrateinterfaces, as well as to the large effective birefrin-gence of the grating. To improve the performance ofthese retarders, we use the higher-order HULM to-gether with the Berreman layer analysis26 to searchthe F–d plane around dpy2 and dp for the desiredbehavior of the transmitted fields.

A. Homogenous Uniaxial Layer Model

In Fig. 3~a!, contours of constant ellipticity ε andtransmittance T are plotted in the dpy2 region of theF–d plane; for the quarter-wave case we seek combi-nations of F and d for which ε 5 1, which ensures d 5py2 and a 5 py4. The desired ellipticity occurs atF ' 0.45 and d ' 2.8 mm, but the transmitted powerat that point is less than 85%, with the remaining15% lost to reflection. This power loss may be un-derstood directly from the HULM and Eqs. ~8! and~9!, which yield effective index values of nO 5 2.6063and nE 5 1.5418 at F 5 0.45. Layer theory predicts100% transmission for either wave if the effectiveindex for that wave takes the value neff 5 ~ncns!

1y2 51.8080 and if the groove depth is d 5 l0y~4neff! 51.4657 mm. Although neither of these conditions issatisfied for either wave in this case, the transmit-tance can be improved by using the three-layer grat-ing of Fig. 2~b! to provide a more tapered indextransition from cover to substrate. Analysis resultsfor this structure with each of the layer depths con-strained to be equal and the filling factors con-


strained to be F2 5 2F1, F3 5 3F1 are shown in Fig.3~b!, where ε 3 1 at F3 ' 0.48 and d1 5 d2 5 d3 '0.970 mm. Also, for these values the transmittedpower is seen to be approximately 92%. To improvefurther on this performance one must remove theconstraints on groove depths and filling factors andemploy numerical optimization, as discussed below.In Fig. 4~a!, contours of constant ellipticity ε, trans-

mittance T, and angle a are plotted in the dp regionof the F–d plane; for the half-wave case we seekcombinations of F and d for which ε 5 0, and forwhich simultaneously a 5 py4. There is a contin-uum ofF–d combinations for which ε 5 0, but a 5 py4only at F ' 0.38 and d ' 4.83 mm, where againtransmitted power T is less than 85%. The perfor-mance of the constrained three-layer grating is given

Fig. 3. ~a! Response of a L 5 3.0 mm period single layer GaAsgrating to a normally incident plane wave of wavelength l0 5 10.6mm as a function of filling factor F and groove depth d around F 50.5, d 5 dpy2, as given by the HULM. ~b! Response of a GaAsthree-layer grating as a function of filling factor F3 and total groovedepth D 5 d1 1 d2 1 d3 for the same conditions. For this plot thedesign parameters are constrained such that F2 5 2F1, F3 5 3F1,and d1 5 d2 5 d3.


in Fig. 4~b!where ε3 0 and a3 py4 at F3 ' 0.55 andd1 5 d2 5 d3 ' 1.84 mm; the transmittance is im-proved to approximately 95%.

B. Rigorous Coupled Wave Analysis Optimization

It is possible to optimize these HULM results beforeproceeding, but it is more advantageous to go directlyto optimization by using an exact electromagneticanalysis of the grating, such as the rigorous coupledwave analysis27 ~RCWA! or the rigorous modal anal-ysis,28 because it has been demonstrated for nonqua-sistatic cases that the diffracted phases predicted bythe HULM may deviate somewhat from those pro-duced by rigorous analysis.18,29 For this study thethree-layer grating retarders were optimized by us-ing the RCWA, with 39 diffracted orders retained foreach grating, and objective functions of the form

fpy2 5 W1u1 2 εu 1 W2~1 2 Tx 2 Ty!, (10)

fp 5 W1uεu 1 W2~1 2 Tx 2 Ty! 1 W3upy4 2 au, (11)

whereWi are the weights initially set equal to unity.The inputs to the optimization procedure are the fill-

Fig. 4. Same as Fig. 3, but now in the region F 5 0.5, d 5 dp.

Table 1. Optimized Quarter- and Half-Wave Retarder Designs and Their Performance

Optimizationa F1 F2 F3 d1 ~mm! d2 ~mm! d3 ~mm! εb T ~%!b

RCWApy2NM 0.1642 0.3839 0.5984 1.5497 0.8050 1.0802 1.0000 98.53

RCWApy2SA 0.1452 0.4513 0.7776 1.4166 1.4669 0.8606 0.9918 99.18

HULMpy2NM 0.1468 0.3740 0.5650 1.4988 0.6475 1.1751 1.0000 99.90

RCWApNM 0.1879 0.2481 0.4749 1.9623 2.4330 1.6468 17.14 3 1026 95.67

RCWApSA 0.2261 0.4486 0.5543 1.3889 2.3405 1.9280 21.38 3 1022 96.09

HULMpNM 0.1625 0.3153 0.5034 2.1530 2.0598 1.6102 23.35 3 10210 96.50

aDetermined by the HULM, and RCWA with 39 orders, using NM and SA optimization.bPerformance of the designs is given in terms of ε and T.

ing factor and groove depth vectors F 5 @F1F2F3# andd 5 @d1d2d3#, and Eqs. ~10! and ~11! are minimizedsubject to the constraint that F1 , F2 , F3.Because of the accuracy of the HULM, the filling

factors and groove depths arrived at above can beused as starting points for the Nelder-Mead ~NM!simplex optimization30 with reasonable assurance ofconvergence to a suitable minimum. This results inthe design and performance data shown in Table 1.In the half-wave case it was necessary to setW2 5 20to achieve the maximum transmittance. The mini-mum feature widths are 493 nm for the quarter-wavedesign and 564 nm for the half-wave design.A second commonly employed optimization tech-

nique is the simulated annealing ~SA! algorithm.31Because of its stochastic nature, the SA algorithmgenerally requires many more function calls than theNMalgorithm, but it is much less susceptible to beingtrapped in local minima of the objective functions ofEqs. ~10! and ~11!. Use of the SA algorithm in thiscase results in two additional designs included inTable 1, for which the ellipticity is slightly degradedbut for which the transmittance is slightly improved.For comparison purposes, simplex optimization re-sults for the layer-model retarders using the HULMare also included in Table 1. Apparently, the pres-ence of evanescent diffracted orders, which are notaccounted for by effective medium theories such asthe HULM, causes the transmittances of the opti-mized grating retarders to differ from those of theoptimized layer-model retarders. In the quasi-staticlimit, the transmittances of the grating retarders andthe layer-model retarders approach each other.

3. Design Performance

Because the retarders designed here differ funda-mentally from traditional devices, it is interestingand important to determine the performance degra-dation that can be expected from variations in fillingfactors and groove depths caused by fabrication er-rors. With six design parameters to consider, wehave chosen to perturb the original designs by intro-ducing random additive errors to all of the designparameters and then to examine the wavelength andfield-of-view sensitivities of an ensemble of 1000 suchperturbed gratings. For every grating in the ensem-ble, vectors F0 and d0 of design filling factors andgroove depths are altered by means of F 5 F0 1 F9and d 5 d0 1 d9, where F9 is a vector of uniformly

distributed random filling factor errors in the range20.02 , Fi9 , 10.02 ~equivalent to changes in thefabricated ridge width of no more than 660 nm!, andd9 is a vector of uniformly distributed random groovedepth errors in the range 250 nm , di9 , 150 nm.To reduce the amount of computation time re-

quired for this simulation, we used the HULM ratherthan the RCWA; this is reasonable, because the be-havior of the optimized layer-model retarders withrespect to perturbation of any particular design pa-rameter is representative of the behavior of the grat-ing retarders with respect to the same perturbation.Support for this claim is given in Fig. 5, where retar-dation phase d for the quarter-wave layer-model de-sign is evaluated with layer analysis, and thequarter-wave NM design is evaluated with theRCWA, both as a function of wavelength. Curvesare given for both unperturbed and maximally per-turbed designs ~F9 5 Fmax9, d9 5 dmax9!, and it isevident that both designs are affected in a similarmanner by the perturbations. Plots of a and ε showsimilar behavior. The simulation was therefore car-ried out with layer analysis and the HULM, keepinginmind that perturbed grating designs analyzedwith

Fig. 5. Comparison of the retardation phase sensitivity to themaximum allowed fabrication errors for both the RCWA andHULM designs as a function of wavelength. The kinks in theRCWA curves are due to the turn on of the i 5 61 diffracted ordersin the substrate.


the RCWA would produce similar results. An im-portant exception to this statement is seen at theshort-wavelength end of Fig. 5, where d begins tochange rapidly because of the turn on of the i 5 61forward diffracted orders at the extreme left of theplot. This behavior was not accounted for in thelayer analysis simulations.The distributions of d values produced by the

quarter-wave ensemble at the center wavelength andat the wavelength extremes are indicated in Fig. 6.The distributions are quasi-Gaussian, albeit nar-rower at the long-wavelength end of plot; this is asexpected, because the perturbations are smaller frac-tions of the wavelength at that extreme. The solidline in the d–l0 plane is the curve for the unperturbeddesign, which in this case coincides with the ensem-ble mean, and the dashed lines in the plane representthe ensemble mean 6 1 standard deviation ~STD!.Similar distributions were found for all quantities inboth the quarter-wave and half-wave cases.The curves for the unperturbed designs as well as

ensemble means and standard deviations for thequarter-wave and half-wave cases are given as afunction of wavelength in Figs. 7 and 8. For thesake of clarity, the definition of the ellipticity hasbeen extended for the quarter-wave case to includevalues of ε . 1. Also, some caution must be used ininterpreting the statistical curves for transmittancein both the quarter-wave and half-wave cases; be-cause the maximum transmittance is 100%, the en-semble STD’s narrow by necessity as the meantransmittance rises. The lowering of the meantransmittance from that of the unperturbed designsis also reasonable, because optimization has maxi-mized the transmittance and almost any perturba-tion reduces it. All quantities were calculated for arange of incidence angles out to 5° in the x–z and y–zplanes, and none exhibited noticeable changes fromtheir normal incidence values.It is also appropriate to compare the performance

of the grating retarders designed here to the perfor-mance of commercially available alternatives.These alternatives include wave plates, thin-film-

Fig. 6. Distributions of retardation phases d at the center wave-length and at the wavelength extremes. The solid line in the d–l0plane is the curve for the unperturbed design; the ensemble meanlies on the same curve. Dashed lines in the plane are the ensem-ble mean 6 1 STD. Histogram bins are 0.5° wide.


coated reflective retarders, and Fresnel rhombs.The quarter-wave case is shown in Fig. 9~a!, wherethe retardation phase produced by the RCWA NMoptimized grating retarder is plotted along with thatproduced by ideal l0y4 and 5l0y4 CdS plates, a 20-layer ThF4yZnS copper substrate reflective retarderoptimized in themanner of Ref. 2, and a GaAs rhomb,all as a function of normalized wavelength. Thesame quantites are plotted in Fig. 9~b! for a 5° angleof incidence in the x–z plane. It should be noted inthis plot that the slight kink in the grating retardercurve at l0 5 10.07 mm is due to the turn on of the i 511 forward diffracted order; at this point the trans-mitted amplitudes of both waves also drop sharply,and this effect can be used experimentally to ensureproper alignment of the grating.

4. Discussion and Summary

From Figs. 9~a! and 9~b! it is evident that gratingretarders behave as nearly ideal zero-order waveplates, thus making them relatively insensitive tochanges in temperature and wavelength. Further,the performance of the grating retarders given herechanges minimally for small deviations from normalincidence, compared with wideband reflective retard-ers and Fresnel rhombs, provided that all diffractedorders remain cut off. Figures 7 and 8 indicate that

Fig. 7. Quarter-wave design and ensemble statistical curves forthe retardation phase, ellipticity, and transmittance as a functionof wavelength.

such retarders can perform reasonably well whileallowing for realistic fabrication tolerances.The procedure used to arrive at these designs is a

two-step process: First, using the higher-orderHULM and layer analysis, one locates the approxi-mate solution within the parameter space; second,using the RCWA, one can optimize the design effi-ciently, with the approximate solution from the firststep serving as a starting point. It is important tounderstand that this process exploits the accuracy ofthe HULM in step 1 to reduce the number of functioncalls used in step 2. Without a good starting pointthe NM simplex algorithm is easily trapped in anylocal minima of Eqs. ~10! and ~11!, forcing one to usethe SA strategy or a similar stochastic minimizationstrategy. It is interesting to note that the SA de-signs differ substantially from those obtained by NMoptimization, indicating the probable existence of anumber of approximate solutions to the optimizationproblem. However, although the retarder transmit-tance wasmarginally improved in the SA designs, thenumber of function evaluations used by the SA algo-rithm was approximately 7500 compared with 320used by the NM algorithm. Because each functionevaluation took roughly 70 s in this case ~running onan IBM RS6000 Model 930!, use of the HULM to

Fig. 8. Half-wave design and ensemble statistical curves for theretardation phase, ellipticity, and transmittance as a function ofwavelength.

allow NM optimization provided a drastic reductionin total computation time needed.We have presented a numerical design procedure

for determining the parameters necessary to producesubwavelength transmission grating retarders insemiconductor materials for use in the mid-IR. Theprocedure takes into account large-form birefrin-gences and reflections from all surfaces; these areeffects that must be included in the high-index semi-conductor case. Using this procedure, we have de-signed quarter-wave and half-wave retarders inGaAs for use at 10.6 mm. The grating retarders de-signed here have minimum feature sizes of approxi-mately 500 nm, making their fabrication possiblewith e-beam writing or dry-etching techniques, andour analysis indicates that these retarders comparefavorably with commercial alternatives.

Fig. 9. ~a! Retardation phase at normal incidence as a function ofwavelength for the quarter-wave grating retarder and commercialalternatives. ~b! The same curves, but now at a 5° angle of inci-dence in the x–z plane. The kink in the GR curve near l0 5 10.6mm is due to the cut on of the i 5 11 forward diffracted order in thesubstrate. GR, grating retarder; ZP andMP, l0y4 and 5l0y4 idealCdS wave plates, respectively; RR, reflective retarder design ~seetext for details!; and FR, GaAs Fresnel rhomb.


This research was supported in part by grantDAAH-04-93-G-0027 from the Joint Services Elec-tronics Program.

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Subwavelength transmission grating retarders for use at 106 μm